Am-II Course Notes Full

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Chapter 1 : Fudametals o! actuarial mathematics

"ectio 1 : #asic !iacial mathematics

1 !ASIC FI"A"CIAL MATHEMATICS

1#1 Compound interest

%.%.% Simplifying assumptions regarding interest rates

For most of the 0or in this course 0e 0ill assume interest rates are constant. hismeans t0o things

1. he term structure of interest rates is flat. he %ield curve is flat.

2. Interest rates do not change over time. 5olatilit% of interest rates is 6ero.

7ote that neither of these assumptions is realistic.

%.%.& efinitions of interest rate notation

$et i be an interest rate.

i 0ill appl% for a certain period a %ear9 half a %ear9 a quarter or a month t%picall%:

i 0ill be pa%able and this compounded: at some interval a %ear9 a month etc.:

Table %.%( Types of Interest rates

Type of interest rate Applies for Compounds Conversion to

Annual Effective

7ACA ; Annual /ffective7ominal Annual Compounded

Annuall%

%ear %earl% i

7AC+ ; i2

7ominal Annual Compounded+emi!annuall%

<ear 0ice a %ear

[1

i2

2

2

−1

]7AC= ; i4

7ominal Annual Compounded

=uarterl%

<ear =uarterl% [1 i4

4

4

−1]

Actuarial Mathematics II David Kirk

$a%e & ' 107





Type of interest rate Applies for Compounds Conversion to

Annual Effective

7AC> ; i12

7ominal Annual Compounded>onthl%

<ear >onthl%

[1 i12

1212

−1]7>C>7ominal >onthl% Compounded

>onthl%

>onth >onthl% [1i 12−1]

7=C>7ominal =uarterl% Compounded

>onthl%

=uarter >onthl%

[1

i

4

12

−1

]7ACC ;

7ominal Annual Compounded

Continuousl%

<ear Continuousl% e−1

his last ro0 is called the force of interest and is described in the follo0ingsections.

%.%.) *ontinuously compounded rates of interest

?delta@: is the s%mbol commonl% used for the continuousl% compounded rate ofinterest. It is also no0n as the force of interest. It is not generall% used to quote aninterest rate in the maret. It is used 0hen performing actuarial calculations 0henthe cashflo0s are paid continuousl% over time.

7ote that %ou ma% also have seen or 0ill see: the force of interest 0hen it comes tofinancial economics and the )lac!+choles option pricing formula.

%.%.+ erivation of

t =lim n∞ 1in

n nt

– 1

t =[ ln 1i ]t

t =t ⋅ln 1i i=e

−1


$a%e ( ' 107





t is the time!dependent force of interest applicable at duration t.

-ur usual simplification is that

t =∀ t

%.%., Application of force of interest in actuarial mathematics

/ample

(remiums are paid monthl% in advance. o calculate the interest applicable topremiums 0e could use a 7>C> interest rate such that the present value of a

premium in 1 monthBs time is

PV Premium1= Premium1⋅1i −1

-r9 0e could use an annual effective 7ACA: rate to perform the same calculation

PV Premium1= Premium1⋅1i −1

12

hese mae sense for premiums since premiums are paid as specified intervalsaccording to the polic% terms. +till9 0e could 0rite the same calculation using acontinuousl% compounded 7ACC in this case: force of interest lie this

PV Premium1= Premium1⋅e

−

12

o0ever9 0hen are claims paidD %picall%9 0e assume the% are paid at the end of theperiod 0e are considering. >ost of our eamples 0ill be annual eamples for

simplicit%: but 0e 0ill also 0or 0ith monthl% eamples. )ut is it correct to assumethat polic%holders 0ill 0ait until the end of the month for their death benefitD

A more accurate approach 0ould be to assume that claims are paid continuousl%over time. he (5 of benefits paid during a %ear could be 0ritten as follo0s

PV Benefits =SA⋅∫t =0

1

e−t ⋅e− x t ⋅ x t dt

Ehere x t is the instantaneous annualised probabilit% of death at age t. Ee

0onBt deal much 0ith continuous!time assurance functions. his is given as an

eample to better understand 0h% 0e ma% need a force of interest or continuousl%compounded rate of interest.


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%.%.- Interest# discounts and discount factors

If i is the annual effective rate of interest9 then the follo0ing also eist

,iscount factor vt =1i −t

he discount factor is used to tae present values of cashflo0s in a more concise 0a%.

,iscount d =1−v

he discount is the rate of interest 0hen the interest is ?paid in advance@. For

eample9 if interest rates are 1*G9 borro0ing +,1** for one %ear 0ill requirerepa%ment of the +,1** and +,1* of interest for a total pa%ment of

USD110=USD100⋅1i

o0ever9 some loans are structured so that the interest pa%able is deducted off the

amount borro0ed. If the loan is the discounted value of +,1**9 then the borro0erreceives

USD100⋅1 – d =USD90.91

and the total interest paid is

USD100 – USD100⋅1−d =USD100⋅[1 – 1−d ]=USD100⋅d

As an eercise9 prove that d = i

1i

d =1−v=1−1

1i=

1i−1

1i =

i

1i

1#$ Actuarial notation for financial mathematics

his section describes some of the basic actuarial notation used for non!contingent

financial maths. hat is9 financial maths that doesnBt not depend on contingentevent such as death.

%.&.% Annuity certain payable in arrears

Consider a regular stream of constant pa%ments of amount 19 pa%able annual inarrears for n %ears. Eith a constant annual effective interest rate of i9 the formula for

this can be 0ritten as


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PV =∑t =1

n

vt

his is a geometric sum 0ith geometric increases of1

1i or 1i −1 for n

periods. hus

PV =1 – 1i−n

i

If %ou need to9 sho0 the derivation of the above as an eercise.

he actuarial notation for this formula is

a n ∣=

1−1i −n

i

Chec that %ou can calculate the follo0ing

a10 ∣ = 6.14 for i = 10%

a10 ∣= 7.72 for i = 5%

a1 ∣ = 0.95 for i = 5%

%.&.& Annuity certain payable in advance

Consider a regular stream of constant pa%ments of amount 19 pa%able annual inarrears for 7 %ears. Eith a constant annual effective interest rate of i9 the formula for

this can be 0ritten as

PV =

∑t =1

n

vt −1

PV =1 – 1i−n

d

As an eercise9 prove that the previous t0o statements are equivalent

he actuarial notation for this formula is


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a n ∣=

1−1i −n

d

Chec that %ou can calculate the follo0ing

a 10 ∣ = 6.76 for i = 10%

a 10 ∣ = 8.11 for i = 5%

a 1 ∣ = 1.00 for i = 5%

7o0 sho0 that a n ∣=1i ⋅a n ∣=1a n−1 ∣

/plain in 0ords 0hat this relationship means.

%.&.) Accumulation factors

For each s%mbol ?a@ there is an equivalent s%mbol ?s@. a is the present value function for a

constant series of cashflo0s. s is the future value function for a constant series of cashflo0s.

(a%ments in arrears

FV =∑t =1

n

C ⋅v−t −1

sn ∣=1i n−1

i

(a%ments in advance

FV =∑t =1

n

C ⋅v−t

¨ s n ∣=1i n−1

d

(rove the follo0ing


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v

n⋅ sn ∣=a

n ∣

vn⋅¨ s n ∣

=a n ∣


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"ectio 2 : Itroductio to cocepts o! li!e coti%ecies

$ I"TR%&UCTI%" T% C%"CE'TS %F LIFE C%"TI"(E"CIES

$#1 Sur)i)al probabilities

he probabilit% of survival for a life aged for t %ears is sho0n as p xt .

If the t subscript is omitted9 it is assumed to be 1. hus9

p x= p x1

>ortalit% in each %ear is independent. hus9 the follo0ing equation holds

p x2 = p x1 ⋅ p x11

and further

p xn = p x1 ⋅ p x11 ⋅⋅ p xn−11

In terms of standard life table notation9 p xt =l xt

l x 0here l x is the number of lives

in the population aged x .

$#$ &eath probabilities

he probabilit% of death for a life aged 0ithin t %ears is sho0n as q xt .

q x=1− p

x

q xt =1− p

xt

(rove that

q x2 ≠q x⋅q x1


$a%e 10 ' 107





(roof

q x2 =1− p x2

q x2 =1− p x⋅ p x1

q x2 =1−1−q x⋅1−q x1

q x2 =1−[1−q x−q x1q x⋅q x1 ]q x2 =q xq x1−q x⋅q x1

Intuitivel%9 the second last line can be interpreted as The probability of dying within two years is the sum of the probability of dying in the first year and the

probability of dying in the second year, less the probability of dying in both years. ae a moment to ensure that %ou understand 0h% this mathematicall%: true

If q x=0.01 and q x1=0.02 0hat is q x2 D

q x2 =1− p x1 ⋅ p x11 =1−0.990.98=0.02980

&.&.% orce of mortality

In the same 0a% that for interest 0e have a force of interest that reflects the

continuous compounding of interest over time9 0e have a force of mortality 0hichcan be interpreted as the annualised instantaneous probabilit% of death at a

particular point.

p xt =e−∫

m= x

xt

mdm Ehere t is the force of mortalit% at time t .

If t = x∀ t ∈[ x , x1 then p x1 =e

− ∫m= x

x1

mdm simplifies to become p x1 =e− x

hus q x1 =1−e− x and so x=−ln 1− q x1 0hich is also x=−ln p x1

+o 0e have assumed a constant force of mortality 0ithin an age bracet in order to

calculate the force of mortalit% operating over that age bracet. his is a commonmethod of calculating monthl% mortalit% rates from annual life tables.

q x1

12

=1−e−

x/12

(rove the above equation.


$a%e 11 ' 107





If q45 ; *.*19 calculateq45

12

hen9 calculate q451

12assuming a constant force of mortalit% for lives aged 4". +ho0

%ou result to at least " decimal places.

Is this the same asq45

12D

If the% are different9 eplain 0h% this is.

Ee have used the assumption of a constant force of mortalit% in deriving theseresults. Another possible assumption is the uniform distribution of deaths or

,,.

nder ,,

q x1

12

=q x

12

or more generall%

q xt =t q x for an% integer and * H t H 1

$#* &eterministic modellin+

In this chapter9 0e consider deterministic models. A deterministic model is one

0here 0e consider onl% a single path of a random variable or time series. Ee do notconsider the potential variation around this sample path.

If 0e are going to use a single path9 the obvious choice is the Expected Outcome.For all our calculations9 0e 0ill be calculating epected future cashflo0s9 or equating

epected present values.

he epected value of a cashflo0 at time t for a poisson distribution is

E [CF t ]=∑ x=0

∞

pmf CF t

x⋅ x=∑ x=0

∞e−

x

x ! ⋅ x

Fortunatel%9 our modelling is a lot simpler. Ee are considering a single life 0hich is

either alive or dead in an% particular period. hus9 the random variable follo0s a)ernoulli distribution. If the cashf lo0 in question is a premium9 then a premium on 1is pa%able if the life is alive9 and * is paid if the life is dead.


$a%e 12 ' 107





E [CF t ]=∑

x=0

1

pmf CF t

x ⋅ x=1⋅ p xt 0⋅1− p

xt = p

xt

he Expected Present alue of that same premium pa%able as described above is

EPV [CF t ]=v

t ∑ x=0

1

pmf CF t

x⋅ x=vt [1⋅ p

xt 0⋅1− p

xt ]=v

n⋅ p xt

$#, %me+a a+e

$ife tables have a maimum age called the Omega Age. he probabilit% of death at

this age is 1. hus9 nobod% can live to be older than age .

Ee 0ill use this concept in later sections.


$a%e 1& ' 107




"ectio & : Itroductio to actuarial mathematics

* I"TR%&UCTI%" T% ACTUARIAL MATHEMATICS

*#1 Life-contin+ent annuit. factors

).%.% Immediate Annuity

An immediate annuity is an annuit% that begins pa%ing 1 immediatel% not deferreduntil some point in the future: but is paid annual in arrears for the lifetime of the

life. hus9 it is not reall% paid immediately but rather after 1 %ear. he first cashflo0taes place at the end of the period.

An immediate annuit% factor is the Epected Present alue of a stream of pa%ments

of 1 as long as the life aged is still alive.

EPV=v1⋅ p x1 v

2⋅ p x2 v

3⋅ p x3

In practice9 because our life table has a cut!off maimum age9 0e might epress this

as

EPV=v1⋅ p x1 v2⋅ p x2 v3⋅ p x3 v− x⋅ p x− x

he standard actuarial notation for this is

a x=v1⋅ p x1 v

2⋅ p x2 v

3⋅ p x3 v

− x⋅ p x− x

a x=∑

i=1

− x

vi⋅ p

xi

).%.& Immediate Annuity in advance

As the name suggests9 this is the epected present value of an annual pa%ment of 1 forever% %ear that the life currentl% aged is alive9 0ith pa%ments occurring at the startof the %ear.

a x=v0⋅ p x0 v

1⋅ p x1 v

2⋅ p x2 v

− x⋅ p x−

x

a x=∑

i=0

− x

vi⋅ p xi

+ho0 that a x=a x1


$a%e 1( ' 107





).%.) Temporary annuities

A temporar% annuit% is a contingent annuit% that is pa%able annuall% until the sooner

of death9 or a specified term.

In arrears

a x n ∣=v1⋅ p x1 v

2⋅ p x2 v

3⋅ p x3 v

n⋅ p xn

a x n ∣=∑

i=1

n

vi⋅ p xi

In advance

a x n ∣=v0⋅ p x0 v

1⋅ p x1 v

2⋅ p x2 v

n−1⋅ p xn−1

a x n ∣=∑

i=0

n−1

vi⋅ p xi

+ho0 that a x n ∣=1a x n−1 ∣

+ho0 that a x n ∣=a x−vn⋅ p xn ⋅a xn

*#$ 'remium pa.able for an annuit.

A polic%holder usuall% purchases an immediate annuit% b% pa%ing a singlepremium. he insurer invests this mone% to earn interest i: and pa%s regularpa%ments ever% %ear to the polic%holder as long as he or she is alive. The

policyholder is insured against the ris! of living too long. he insurer has taen

on the ris that the polic%holder lives longer than epected.he premium charged b% the insurer for the annuit% depends on the interest ratesavailable in the maret9 and the level of mortalit% epected.

/plain 0hether and 0h%: the insurer 0ill have to charge a higher or lo0erpremium if

interest rates rise


$a%e 1) ' 107





mortalit% rates decrease

If interest rates rise9 the discount rate 0ill increase. If the discount rate increases9the present value of future cashflo0s 0ill decrease. hus9 a lo0er premium can be

charged. Another 0a% of looing at this is to consider that 0ith higher interest rates9the premium received 0ill earn higher investment return interest: and 0ill thusgro0 faster to be able to pa% higher benefits. For the same benefit9 a lo0er premiumcan be charged.

If mortalit% rates decrease9 polic%holders are epected to survive for longer. Annuit%pa%ments are made as long as the polic%holder survives. hus9 lo0er mortalit%

implies more pa%ments. he present value of these higher pa%ments 0ill be higherand the premium charged 0ill have to increase.

o calculate the premium for an% t%pe of polic% in the traditional actuarial 0a%9 0eequate the epected present value of income to the epected present value of outgo.

In this case 0e are onl% considering premiums and benefits.

(remiums are paid b% the polic%holder to the insurer. )enefits are paid b% the

insurer to the polic%holder. nless told other0ise9 the premiums 0ill be constantand denoted as P

he first step should al0a%s be

EPV Premiums=EPV Benefits

7o09 0e fill in the $+ and + separatel% 0ith more detailed formulae.

!"S =EPV Premiums= P

+ince there is onl% one premium paid9 and it is paid at the start of the contract9 the/(5 is simpl% (.

7o0 if the annual benefit pa%ment in arrears: is )9

#"S =EPV Benefits = B⋅a x

Make sure you understand the above equation. Why is B⋅a x the expected present

value of benefits?

7et step is to complete the original equation


$a%e 1* ' 107





EPV Premiums =EPV Benefits P = B⋅a x

herefore the premium that the insurer should charge is simpl% B⋅a x .

Ehat is the premium pa%able for a temporar% annuit% paid annuall% in advance forup to 2" %earsD ,efine all s%mbols used. Eill this be smaller or greater than the

premium for an immediate annuit% paid in advanceD

*#* Assurance factors

).).% Assurance factor An assurance factor is the epected present value of a sum assured of 1 for a life aged

paid at the end of the %ear.

An assurance factor is the Epected Present alue of a stream of pa%ments of 1 in the

%ear of death of the polic%holder.

EPV=v1⋅ p x0 ⋅q xv

2⋅ p x1 ⋅q x1v

3⋅ p x2 ⋅q x2

In practice9 because our life table has a cut!off maimum age9 0e might epress thisas

EPV=v1⋅ p x0 ⋅q xv

2⋅ p x1 ⋅q x1v

3⋅ p x2 ⋅q x2v

− x1⋅ p x− x ⋅q

he standard actuarial notation for this is

A x=v1⋅ p x0 ⋅q xv

2⋅ p x1 ⋅q x1v

3⋅ p x2 ⋅q x2v

− x1⋅ p x− x ⋅q

A x=∑

i=0

− x

vi1⋅ p

xi ⋅q

xi

).).& Term Assurance factor

A x1

n ∣

=∑i=0

n−1

vi1⋅ p xi ⋅q xi=∑

i=1

n

vi⋅ p xi−1 ⋅q xi−1

/plain the above equation in 0ords. Ehat does the 1 over the signif%D

Confirm that %ou understand 0h% both definitions from i ; o and from i ; 1: are correct.


$a%e 17 ' 107





).).) /ure 0ndo!ment factor

A pure endo0ment is a benefit that pa%s onl% onl% survival to time n. he /(5 of the benefit

is given b% the follo0ing equation

A x n ∣

1 =vn⋅ p xn

Eho 0ould 0ant to bu% such a polic%D

Eh% might it be unpopularD

A polic%holder 0ith a specified financial need in future might purchase the polic%

as a savings vehicle9 provided the financial need eists onl% for that polic%holderand not for an% dependants. An eample might be

+aving for retirement 0ith no famil%:. If the polic%holder dies9 no savings areneeded.

+aving for universit% education. he education 0ill not be needed if thepolic%holder is dead.

It is often unpopular because no benefit is paid on death. -n death occurring

shortl% before maturit%9 a large number of premiums 0ill have been paid but no

benefit is received. (roblems arise through poor selling processes 0here the polic%ma% not be full% eplained. Further9 the famil% of the deceased polic%holder ma% beunhapp% that there is no benefit to be paid to them. As a result9 these policies are

usuall% not sold b% insurers an% more.

).).+ 0ndo!ment Assurance factor

An /ndo0ment Assurance often called Just an ?endo0ment@ ! for this course use the full

name to prevent confusion: is a polic% that pa%s a benefit at the end of %ear of death of thelife assured9 or at the end of %ear of maturit% given survival to that period.

he /(5 of benefits is given as

A x n ∣= A

x1

n ∣

A x n ∣

1 since it combines the benefit of the erm Assurance 0ith the

benefit of a pure endo0ment. -ne can also 0rite

A x n ∣

=∑i=1

n

vi⋅ p

xi−1 ⋅q

xi−1v

n⋅ p xn

+ho0 that this can further be 0ritten as


$a%e 1+ ' 107





A x n ∣

=∑i=1

n−1

vi⋅ p

xi−1 ⋅q

xi−1v

n⋅ p xn−1

An n!%ear endo0ment assurance pa%s a benefit at the end of %ear of death for n!1 %ears. -n

survival to n!1 %ears9 a benefit is paid on death or survival at the end of %ear n.

*#, (uaranteed Endo/ments

A guaranteed endo0ment is not reall% an insurance polic%. It pa%s a benefit at maturit% on

death or survival to maturit%. he ?guaranteed endo0ment factor@ is simpl% vn

+ho0 that vn is equal to v

n⋅ p xn ∑

i=0

n−1

p xi ⋅q

xi⋅v

n

A guaranteed endo0ment pa%s a benefit on death before time n at the end of %ear n9 and

also on survival to time n.

*#0 'remium calculations for assurance benefits

iven an epression for the regular annual premium paid in advance: for an endo0ment

assurance polic%.

emember9 start 0ith EPV Premiums=EPV Benefits .

ive similar epressions for

regular annual premium paid in advance: for a term assurance polic%

regular annual premium paid in advance: for a pure endo0ment

regular annual premium paid in advance: for an endo0ment assurance polic%

regular annual premium paid in advance: for a term assurance polic%

regular annual premium paid in advance: for a pure endo0ment

o0 does the single premium for a pure endo0ment polic% compare 0ith the amount

%ou 0ould need to invest in a ban account earning the same amount of interest9 over thesame periodD Eh% is thisD

*# Fractional .ear factors

All the above factors can be considered 0ith pa%ments more frequent than once per

%ear. his 0ill not be covered in this course to mae room for other concepts.


$a%e 1, ' 107





*#2 3oint-life factors

>an% of the above factors also eist in Joint!life versions. hese depend on the

survival or death of one or both of t0o lives t%picall% a married couple:. hese arenot frequentl% used in practice and have also been ecluded from this course to allo0other concepts to be covered in more detail.


$a%e 20 ' 107



Actuarial Mathematics II © 2007 David Kirk

Course Notes

Chapter $ Introduction to reser)es and liabilities

About

This chapter introduces the concepts of actuarial reserving and e"plores

the differences bet!een reserving principles for solvency# and reserving 1

liability principles for financial reporting. The Actuarial *ontrol *ycle !ill

be e"plored as a frame!or$ for solving actuarial problems 2including

reserve calculations3.

Chapter Contents1 he Actuarial Control C%cle.................................................................. .......................23

1.1 -vervie0......................................................................................................... ........23

1.2 Identif%ing the problem or ?defining the problem@:..........................................24

1.3 ,eveloping the solution.............................................................................. ..........24

1.4 >onitoring the eperience................................................................................. ...2"

1." /nvironment.......................................................................................... ................2"

1.# (rofessionalism.................................................................................. ...................2#

2 is management in contet of Actuarial Control C%cle...........................................2&

2.1 +teps in is >anagement.......................................................................... ..........2&

2.2 is Identification...................................................................... ..........................2&

2.3 is >easurement........................................................................ ........................2&

2.4 is >anagement.......................................................................................... .......28

2." is >onitoring........................................................................................ ............28

3 Cashflo0s arising from insurance products...............................................................2'

3.1 ,efinition of cashflo0.......................................................................................... .2'

3.2 -perating cashflo0s...................................................................................... .......2'

3.3 Cashflo0s of a capital nature.......................................................... .....................2'

3.4 >ismatch in timing of cashf lo0s in insurance products...................................3*

4 Actuarial reserves and liabilities.................................................. ...............................32

4.1 Actuarial reserves for solvenc% .............................................................................. 32



Chapter 2 : Itroductio to reserves ad lia-ilities

"ectio Chapter 2 : Itroductio to reserves ad lia-ilities

4.2 Actuarial liabilities for financial reporting............................................. .............32

4.3 ,ifference in purpose bet0een actuarial reserves and actuarial liabilities........33

4.4 Actuarial )alance +heet......................................................... ..............................34

4." %pical actuarial reserves................................................................................... ...3"

4.# +pecial reserves.................................................................................................... .3#


$a%e 22 ' 107




"ectio 1 : .he Actuarial Cotrol C/cle

1 THE ACTUARIAL C%"TR%L C4CLE

1#1 %)er)ie/

he Actuarial Control C%cle ?ACC@: is a problem solving tool used in actuarial 0or.here are control c%cles in man% professions and industries that follo0 a similar

form. he e% idea is that it is a cycle that controls ho0 to ensure a s%stematicapproach to ongoing problem solving.

It is a useful tool to solve actuarial problems in practice. It is also an etremel% usefultool for developing ans0ers for actuarial eams. It encourages thining around the

problem and follo0ing the problem through to its logical9 long!term result.


$a%e 2& ' 107

Illustration &.%( The Actuarial *ontrol *ycle

&e)elopin+

the Solution

Identif.in+

the problem

Monitorin+the

e5perience

'rofessionalism

The En)ironment





1#$ Identif.in+ the problem 6or 7definin+ the problem89

he first step is to identif% the problem. +ometimes this 0ill be apparent from the

real!life or eam: problem9 but sometimes significant 0or is required tounderstand 0hat the problem is. his is an active step rather than a passive step:that requires the actuar% to consider the drivers of the problems9 riss inherent in the

problem9 different staeholders 0ho ma% have an interest in the problem.

ip K thin about 0h% the problem isnBt trivial to solve. Ehat maes it difficult9 0h%

hasnBt this problem alread% been solvedD Ehat solutions have alread% been triedD Eho is happ% 0ith the current solution9 and 0ho is unhapp%D

he problem ma% be

-ur erm Assurance policies are not profitable

Ehat 0ill future mortalit% rates be for our polic%holdersD

-ur business is eposed to several ris

his last problem 0ill probabl% need to be devolved further b% identif%ing each of the

riss to identif% specific problems. his identification process is part of the ?Identif%the (roblem@ step.

his step is also called ?defining the problem@ since to accuratel% identif% the

problem9 it is necessar% to properl% define the e% aspects of the problem.

1#* &e)elopin+ the solution

-nce the problem has been properl% identified and defined 0e need to 0or to solve

it. his step is called ?,eveloping the +olution@. iven the information specified inthe previous step9 several solutions 0ill be proposed9 0ith fla0ed solutions being

discarded. he problem characteristics identified in the previous step should be eptin mind as an effective solution 0ill have to address these elements.

+olutions ma% involve

Ee 0ill increase term assurance premium rates

Ee 0ill estimate future mortalit% based on past mortalit% eperience9 allo0ing forapparent trends. Ee 0ill mae use of reinsurersB data to supplement our data

0here it is not credible.

Ee 0ill mae use of reinsurance and pooling to reduce riss to our business.


$a%e 2( ' 107





1#, Monitorin+ the e5perience

his step is 0hat mae the ACC a ?control c%cle@. -nce 0e have identified the

problem and developed a solution9 it is criticall% important to monitor theeperience. Ee monitor the results to ensure that an% assumptions 0e made indeveloping the solution are reasonable. Ee loo for unintended consequences and

gather ne0 information for the net step. he net step is to ?Identif% the (roblem@or redefine the problem based on updated information from our eperience9 and

given the solutions 0e have alread% implemented.

Ee must monitor sales of term assurances. If our increased premium rates lead to

significant lo0er volumes of business9 our unit epenses ma% increase resulting inlo0er profitabilit%.

Future mortalit% ma% differ from past mortalit%. -ur past eperience ma% not havebeen a reliable estimate of future mortalit%. >ortalit% improvements through

improved medical care: or 0orse mortalit% I5LAI,+ or pandemics such as birdflu: result in current and future mortalit% being ver% different from past mortalit%.

-ur reinsurance programme ma% not be sufficient and our business ma% still beeposed to too much ris. Alternativel%9 our reinsurance programme ma% be too

etensive9 0ith the result that our profits are too lo0.

1#0 En)ironment

his part of the ACC is not a step in the process. ather9 it is a factor that must beconsidered at each stage of the c%cle. he specific elements considered 0ill var%

depending on the problem at hand9 but this list gives an idea of the t%pical categoriesto consider

/conomic environment interest rates9 inflation9 economic gro0th9 currenc%stabilit%9 ,( per capita9 income inequalit%:

egularl% environment licensing requirements9 asset admissibilit% rules9polic%holder protection9 reporting requirements9 capital requirements9 taation

rules:

)usiness environment number9 si6e and strength of competitors9 distribution

channels9 emplo%ee considerations:

(olitical environment political stabilit%9 potential for political interference in

business9 impact on the economic environment:

+ocial environment polic%holder activism9 treating customers fairl%9 sociall%

responsible investing9 limits on under0riting9 limits on eclusions:


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1# 'rofessionalism

All aspects of actuarial 0or must be performed 0ith utmost professionalism and

ethics. >uch of our 0or relates to individuals long!term savings and other criticalevents in their life such as the loss of a bread!0inner for a famil%. rust in placed ininsurers companies to loo after individuals in their time of need and actuaries are

usuall% the custodians of this trust.

he follo0ing list represents Just a fe0 of the t%pical considerations underprofessionalism

Actuaries should consider the impact of their actions on all staeholders.

It ma% often be important to conve% this understanding to non!actuaries in a clear

manner.

Actuaries should not perform 0or for 0hich the% are not qualified or in 0hichthe% do not have sufficient eperience.

Actuaries should avoid conflicts of interest9 and if it is necessar% to 0or underconflicts of interest9 then this should be communicated to all relevant parties.

Actuaries must maintain the confidentialit% of their clients.

<ou 0ill receive a cop% of the MBs (rofessional Conduct +tandards 0hich governs the

professional conduct of members of the Institute and Facult% of Actuaries in the M.It also applies to Fello0s of the Actuarial +ociet% of +outh Africa. he content is not

eaminable for this course9 but %ou 0ill be epected to consider aspects ofprofessionalism 0here relevant in eam questions.


$a%e 2* ' 107




"ectio 2 : isk maa%emet i cotet o! Actuarial Cotrol C/cle

$ RIS: MA"A(EME"T I" C%"TE;T %F ACTUARIAL C%"TR%L C4CLE

$#1 Steps in Ris< Mana+ement

is Identification

is >easurement

is >anagement

is >onitoring

/ach of these steps is outlined in the sections belo0.

$#$ Ris< Identification

is Identification is commonl% performed through maintaining a ris! register. heris register is a structured record of all the important riss faced b% the organisation.It can be difficult to list all relevant riss in a useful 0a%9 as man% riss 0ill interact

0ith one another and these lins must be made clear in the ris register.

$#* Ris< Measurement

he riss should then be measured. here are three primar% methods used to

measure riss.

&.).% etailed modelled based on actual data

If reliable9 credible data is available from the compan%Bs o0n eperience9 or throughindustr% data9 then the riss can be modelled in detail through fitting probabilit%distributions to the magnitude and frequenc% of losses9 allo0ing for correlations and

dependencies bet0een riss.

his is t%picall% the case for insurance riss9 persistenc% riss and maret riss. Credit

riss is increasingl% being modelled accuratel% as bans collected relevant default

data for )asel II compliance.

&.).& Appro"imate modelling based on sub4ective estimates

+ome riss are difficult to model eactl%. Credit ris is still difficult to modelaccuratel% for some classes of business. -perational ris is another that is t%picall%

measured in an approimate manner. his can be based on subJective estimation ofdistribution functions9 or through an arbitrar% factor.


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"ectio 2 : isk maa%emet i cotet o! Actuarial Cotrol C/cle

&.).) 5ualitative assessment

+ome riss often operational or catastrophe riss: cannot be measured at all 0ith

current information and tools. For these9 a subJective assessment of the probabilit%or frequenc% or lielihood: and severit% or ?impact@: is made.

For eample

an earthquae might be ver% infrequent9 but have a severe impact

Fraud b% an emplo%ee might be fairl% frequent9 but lo0 in impact.

$#, Ris< Mana+ement

his section is called ?is >anagement@ 0hich is also the name of the overall

processN o0ever9 it is in this section that the riss alread% identified and measuredare acted upon. +ome of the possible actions include

Avoid the ris

educe the ris

>itigate the ris

ransfer the ris

(ool the ris

+hare the ris

edge the ris

Insure the ris

Accept the ris

$#0 Ris< Monitorin+

As 0ith the Actuarial Control C%cle9 it is criticall% important that riss are monitored

on a regular basis. iss ma% change in nature over time9 and unless the ris registersare regularl% updated and the measurements re!performed9 the ris management

actions ma% no longer be appropriate.

Additionall%9 if the riss are regularl% monitored9 it serves to focus managementBs

attention on ris management as a critical component of their Job9 rather than assomething peripheral to their main function.


$a%e 2+ ' 107




"ectio & : Cash!los arisi% !rom isurace products

* CASHFL%=S ARISI"( FR%M I"SURA"CE 'R%&UCTS

*#1 &efinition of cashflo/

Cashflo0 is the transfer of cash from one entit% to another. It onl% counts as acashflo0 0hen it has moved or ?flo0ed@. hus9 cash at a ban is not a cashflo0.

(remiums flo0 from the polic%holder to the insurance compan% and are thus acashflo0.

(rofit is not a cashflo0 since no mone% flo0s an%0here. It is an accounting result.

*#$ %peratin+ cashflo/s

-perating cashflo0s are those cashflo0s that occur during the normal operatingactivities of the compan%

).&.% *ash inflo!s

(remiums

Investment income interest and dividends9 proceeds from sale of financialinstruments:

einsurance claims

einsurance commission

).&.& *ash outflo!s

Claims

/penses

Commission

einsurance premiums

(rofit commission

a

*#* Cashflo/s of a capital nature

+ome cashflo0s arise due to the compan% increasing or decreasing the amount ofcapital it has. hese are not operational cashflo0s but are capital cashflo0s.


$a%e 2, ' 107





).).% *apital *ash inflo!s

Issue +hares

Issue (referred +hares or ?preference shares@:

Issue bonds L debentures borro0 mone% from investors:

).).& *apital *ash outflo!s

(a% dividends

(a% interest on bonds L debentures

epa% capital on bonds L debentures

)u%bac shares

>ae a capital distribution pa% out some share capital. 7ormal dividends are paidfrom retained earnings:

*#, Mismatch in timin+ of cashflo/s in insurance products

For most insurance products9 cash in!flo0s occur mostl% at the start of the polic%9 and

cash out!flo0s mostl% at the end of the polic%.

).+.% Immediate annuity e"ample

+ingle premium is received up front at t ; *

Claims and epenses are paid ever% month or %ear from t ; 1 for up to 3* or 4* %ears

).+.& %6 year 0ndo!ment Assurance e"ample

(remiums are paid regularl% from t ; *

+mall death claims are paid regularl% from t ; 1 to 1*. hese cashflo0s are muchsmaller than the premiums

/penses are paid regularl% from t ; * to 1*. hese cashflo0s are much smaller thanthe premiums

At t ; 1*9 a large maturit% pa%ment is made to the survivors. his is much largerthan the premium received in the last %ear.

).+.) Term Assurance e"ample

erm assurance has closer matching of in! and out!flo0s. he maJor mismatch isbecause the premium is level9 but the ris increases over time due to increasingmortalit% rates.


$a%e &0 ' 107





$evel premiums are paid each %ear

/penses are paid each %ear

Claims are small at the start of the polic%9 but increase over time

It is liel% that

(remiums O /penses )enefits for earl% durations

(remiums H /penses )enefits for later durations


$a%e &1 ' 107




"ectio ( : Actuarial reserves ad lia-ilities

, ACTUARIAL RESER>ES A"& LIA!ILITIES

his section covers the need for actuarial reserves and liabilities9 eplains the

difference bet0een them and the t%pes of reserves and liabilities calculated.

,#1 Actuarial reser)es for sol)enc.

As 0e sa0 in the previous section9 cashflo0s for insurance compan% are?mismatched@. hat is9 0e receive cash before having to pa% it out. +ometimes this

difference can be man% %ears. he case of the immediate annuit% is an eas% eample. Ee receive cash upfront and then have to mae pa%ments for man% %ears sometimes

2* or 3* or even 4* %ears:. It is important that 0e ensure 0e have enough mone% to beable to mae these pa%ments 0hen the% fall due.

An insurance company is solvent if it can meet is liabilities as they fall due.

hus9 actuaries calculated the amount of money that must be held in reserve to

ensure that liabilities can be met as the% fall due. It is common for this calculation tobe prudent since 0e have a responsibilit% to our polic%holders to loo after them.

In general9 the more prudence in the calculation of the solvenc% reserves the better.

,#$ Actuarial liabilities for financial reportin+

Financial reporting is the disclosure of

the compan%Bs financial position balance sheet:P and

financial performance income statement:

to allo0 current and potential future shareholders to

assess the value of their shareholdingP

estimate ho0 the compan% 0ill perform in futureP and

mae informed investment decisions about 0hether to bu%9 hold or sell thecompan%

and to assist management to

mae good operational and strategic decisions in the management of the compan%.


$a%e &2 ' 107





+.&.% 0"ternal reporting

+hareholders and investors need to no0 the current financial position and past

financial performance of the compan%. he% 0ill also be interested in the solvenc% ofthe compan% if the compan% is insolvent and cannot pa% polic%holders9 there 0ill beno mone% left for shareholders and other investors:. o0ever9 the% also need to no0accuratel% 0hat the balance sheet and income statement of the insurer loos lie.

+ince most insurance products have large cashflo0s upfront follo0ed b% negativecashflo0s at later durations9 0ithout calculating the liabilities due to polic%holders

and placing this on the balance sheet9 insurance companies 0ill sho0 large profits inthe earl% %ears of a polic%9 follo0ed b% large losses at the end of a polic%. his clearl%

doesnBt reflect the realistic scenario for the insurer.

+.&.& Internal reporting

Internal management also need to no0 the accurate financial position of the

compan% so that the% can manage it effectivel% and mae appropriate strategic andoperational decisions. he% need to no0 the assets and liabilities on the balance

sheet9 as 0ell as ho0 much profit the compan% maes as a 0hole9 and ho0 muchprofit is made b% each business unit or product line.

,#* &ifference in purpose bet/een actuarial reser)es and actuarial liabilities

Actuarial reserves for solvenc% are intended to protect polic%holders against the risthat the insurer becomes insolvent. (rudence is a good thing. >ore prudence isgenerall% better.

Actuarial liabilities for financial reporting are to provide shareholders and otherinvestors 0ith accurate information about the financial position of the compan%. Asmall element of prudence is good to ensure that profits are recognised prematurel%.o0ever9 too much prudence paints an unrealistic picture of the financial position of

the insurance compan% and can be misleading.

his is an important difference in the purpose of solvenc% calculations and financial

reporting.

A secondar% difference is that financial reporting generall% requires accurateinformation for individual business units and product lines to enable management tomae appropriate decisions on this level. -n the solvenc% side9 as long as the entire

compan% is solvent9 0e are less concerned although not unconcerned: 0ithindividual product lines.


$a%e && ' 107





,#, Actuarial !alance Sheet

he largest elements of an insurersB balance sheet are the actuarial reserves or

liabilities. he illustration belo0 gives an eample of a life insurance compan% 0ithtotal assets ; 1** and total liabilities ; &".

he t0o sides of the balance sheet are the assets and liabilities. he illustration above

sho0s the assets on the left hand side9 and t0o different breado0ns of the liabilitieson the right hand side.

+.+.% Insurance company assets

Insurance companiesB assets can be valued at maret value or some form ofhistorical cost or amortised cost.

+.+.& Liabilities

he middle column sho0s the most basic breado0n. &" of liabilities and 2" of?shareholdersB net asset value@9 0hich is also no0n as ?shareholdersB equit%@ or?shareholdersB capital@.


$a%e &( ' 107

Illustration &.&( Actuarial balance sheet

Actuarial Balance Sheet

Margins

A s s e t s ( M a r k e t V a

l u e o r

A m o r t i s e d C o s t )

Best estimate

liabilities

Risk Capital

Free assets

Liabilities

Shareholders Net

Asset Value

0

20

0

!0

"0

#00

Assets Liabilities Liabilities





he column on the right provides more detail. otal liabilities of &" consist of best

estimate liabilities of #* and margins for prudence of 1". he level of prudence 0illdiffer bet0een companies9 bet0een countries9 and ma% be different for solvenc%reserving and financial reporting liabilities. (rudence is usuall% introduced throughconservative assumptions.

+.+.) 7is$ capital and free assets

"is! capital is capital held as an additional buffer against fluctuations in eperienceand other riss to the insurer. egulators ma% stipulate the minimum level of ris

capital that must be held over and above the reserves. is capital belongs toshareholders9 but ma% be needed to pa% benefits to policies if eperience is much

0orse than epected.

,#0 T.pical actuarial reser)es

+.,.% /rospective reserves

7et (remium 5aluation

ross (remium 5aluation or ealistic ,iscounted Cashflo0 5aluation

+.,.& 7etrospective reserves

nit fund reserves

nearned (remium reserves

+.,.) Appro"imate reserves

+ome reserves or liabilities are difficult to calculate9 or there ma% not be sufficientinformation to calculate them accuratel%. In these cases9 approimate methods ma%

be used. his is often the case 0hen the overall si6e of these reserves L liabilities issmall.

Common approimations ma% include

multiple of premium multiple of sum assured

retrospective accumulation of premiums9 epenses and benefits

+.,.+ 8onlife reserves

7on!life insurance generall% does not used discounted cashflo0 reserves. hereserves commonl% used for non!life insurance are


$a%e &) ' 107





nearned (remium eserves (: or nearned (remium (rovision ((:

Additional nepired is eserves A: or Additional nepired is

(rovision A(: .

his is sometimes called the nepired is eserve or : but this is nottechnicall% correct

Incurred )ut 7ot eported eserve

-utstanding claims reserve.

,# Special reser)es

In addition to the above reserves9 special reserves are sometimes held to meet

specific needs.

+.-.% 9ad data reserves

If there are concerns about the data on 0hich the reserves are based number of

policies9 premium9 age etc.: then a bad data reserve ma% be held to reflect thepotential additional outgo in future arising from corrections to the data.

A bad data reserve should not be held forever. he data problems should be fied andthe reserves released. Ehen a reserve is no longer held it is said to be ?released@.

+.-.& Mismatch reserves A mismatch reserve ma% be held 0hen assets and liabilities are not matched.

+.-.) Investment :ption and ;uarantee ;uarantee 7eserves

Investment options and guarantees can present significant riss to an insurer. >an%

insurance compan% failures have been the result of problems 0ith investmentguarantees and options.

he reserves for this riss are usuall% calculated using stochastic models thateplicitl% model the possible variation in investment returns and interest rates.

+.-.+ 8onInvestment :ption < ;uarantee 7eserves

-ptions and guarantees that do not involve investment returns are also significantsources of ris. +ome eamples include

uaranteed Insurabilit% )enefit

-ptional premium increases

Implicit option to lapse a polic%


$a%e &* ' 107





+.-., 7egulatory special reserves

he insurance regulator ma% require additional special reserves to be held. hese

could be a percentage of premiums9 or a fied amount for each class of business.


$a%e &7 ' 107




Course Notes

Chapter * Actuarial reser)in+ techni?ues for life

insurance

About

This chapter covers the core techniques of actuarial reserving. =e !ill

cover the net premium valuation and gross premium valuation.

Chapter Contents

1 (ractical eserving (rinciples........................................................ .............................4*

1.1 (rospective valuation........................................................................................... ..4*

1.2 Assumptions required the ?basis@:........................................ .............................4*

2 ross (remium 5aluation...................................................................................... ......42

2.1 ,efinition of ross (remium 5aluation..............................................................42

2.2 Impact of basis changes on gross premium valuation........................................42

2.3 /amples of gross premium valuations......................................... ......................42

2.4 /ample of impact of basis changes on the gross premium valuation...............43

3 7et (remium 5aluation................................................................................. ..............44

3.1 ,efinition of 7et (remium 5aluation........................................................... .......44

3.2 Implicit allo0ance for epenses in net premium valuation................................44

3.3 Impact of basis changes on net premium valuation....................................... .....4"

3.4 /amples of net premium valuations........................................... .......................4"

3." +ingle premium policies under the net premium valuation...............................4#

3.# Alternative interpretations of net premium valuation.......................................4#

3.& /ample of impact of basis changes on the net premium valuation..................4&

4 etrospective versus prospective calculation of reserves.......................................... .48

4.1 Assumptions required for equalit% .................................................................... ...48

4.2 %pical form of relationship................................................................................ .48

4.3 Intuitive interpretation of relationship.................................................. .............48

4.4 ,erivation of equalit% for a Ehole of $ife Assurance.................................... .....4'



Chapter & : Actuarial reservi% techi3ues !or li!e isurace

"ectio Chapter & : Actuarial reservi% techi3ues !or li!e isurace

" Impact of reserves on profit and loss................................................. ........................."*

".1 7e0 business strain.......................................................................................... ....."*

".2 Qillmerisation............................................................................ ............................"1

".3 )asis changes................................................................................................. ........"2

".4 (attern of reserves over time.................................................................. .............."3


$a%e &, ' 107




"ectio 1 : $ractical eservi% $riciples

1 'RACTICAL RESER>I"( 'RI"CI'LES

1#1 'rospecti)e )aluation

In a prospective valuation9 0e tae the reserve to be the present value of futurecashflo0s.

If 0e invest the reserves in assets that earn the discount rate9 then the reserve 0ill besufficient to meet future cashflo0s.

1#$ Assumptions re?uired 6the 7basis84

In valuing the epected present value of future cashflo0s9 0e need to mae certainassumptions regarding 0hat these epected future cashflo0s 0ill be. heseassumptions are collectivel% called the basis.

%.&.% Mortality 2 q x 3

/pected mortalit% rates for each age are required. -ften9 this 0ill be different for

males and females. hese rates 0ill be based on

past mortalit% eperience of the insurance compan%

industr% data if this is available

data from reinsurers

mortalit% eperience from similar countries

no0n or epected trends9 such as annuitant mortalit% improvements anddeteriorating mortalit% from I5LAI,+

%.&.& iscount rate 2 i 3

he discount rate or valuation rate: is used to calculate the present value ofcashflo0s for the reserves. It can be set in t0o 0a%s

1. he epected future investment return or %ield: on the assets bacing thereserves

2. he maret!consistent %ield epected on similar instruments in the maret.

If the assets bacing the reserves closel% match the liabilit% cashflo0s9 then approach1 and 2 0ill produce the same result.


$a%e (0 ' 107




"ectio 1 : $ractical eservi% $riciples

7ote9 the discount rate 0e are considered is the epected future investment return

or %ield. It does not necessaril% reflect recent past investment returns and does notdirectl% affect the actual interest received on the investments.

%.&.) 0"penses

Future epenses commission9 salaries9 rent9 depreciation9 electricit% etc.: are

sometimes modelled eplicitl% t%picall% 0ith a ross (remium 5aluation9 but not fora 7et (remium 5aluation:. An assumption regarding the epected level of future

epenses per polic% 0ill be required.

%.&.+ :ther decrements

A decrement is a change from one state to another. For eample9 death is a

decrement from the state ?alive@ to the state ?dead@. -ther decrements that are ofinterest in insurance are

$apse the polic%holder stops pa%ing premiums so the contract is cancelled:

+urrender the polic%holder stops pa%ing premiums and receives a ?surrender

value@ and the contract is cancelled:

(aid!up the polic%holder stops pa%ing premiums9 b% the polic% continues under

modified terms9 usuall% 0ith a lo0er +um Assured:

7one9 some9 or all of these decrements ma% be modelled in a ross (remium 5aluation.

%.&., :ther assumptions

,epending on the product and valuation methodolog%9 a range of other assumptionsma% be required.


$a%e (1 ' 107




"ectio 2 : 5ross $remium 6aluatio

$ (R%SS 'REMIUM >ALUATI%"

$#1 &efinition of (ross 'remium >aluation

A #ross Premium aluation is a reserving technique that discounts the epected value of all future cashflo0s9 including the actual premium paid b% the polic%holder

the #ross Premium: and an% epenses epected to be paid in future.

Ee 0ill use the s%mbol $P for the ross (remium. In these eamples9 0e model

epenses as % pa%able annual in advance as a fied amount per polic%.

$#$ Impact of basis chan+es on +ross premium )aluation

If the basis is changed e.g. >ortalit%9 discount rate or epenses: then the calculation

of the gross premium reserve must tae these ne0 assumptions into account. hus9the basis change is directl% incorporated into the reserve.

$#* E5amples of +ross premium )aluations

he reserve for a polic% in force still active: at time t is calculated as V t as sho0n

for each regular premium product belo0.

&.).% =hole of Life Assurance

V t =SA⋅ A xt −$P ⋅a xt % ⋅a xt

&.).& Term Assurance

V t =SA⋅ A xt

1

n−t ∣−$P ⋅a xt n−t ∣ % ⋅a xt n−t ∣

&.).) 0ndo!ment Assurance

V t =SA⋅ A xt n−t ∣−$P ⋅a xt n−t ∣ % ⋅a xt n−t ∣

&.).+ Annuity For this annuit%9 0e assume pa%ments annuall% in arrears ; )enefit9 and epensesalso incurred annuall% in arrears.

V t = Benefit % ⋅a xt


$a%e (2 ' 107




"ectio 2 : 5ross $remium 6aluatio

$#, E5ample of impact of basis chan+es on the +ross premium )aluation

Table ).%.9asis change impact on gross premium valuation for Term Assurance

$asis change %mpact of basic change

on #P

%mpact of basis change

on "eserve

Increase mortalit% 7o change Increase

Increase discount rate 7o change ,ecrease

Increase epenses 7o change Increase


$a%e (& ' 107




"ectio & : Net $remium 6aluatio

* "ET 'REMIUM >ALUATI%"

*#1 &efinition of "et 'remium >aluation

A &et Premium aluation is a reserving technique that discounts the epected value of future benefit cashflo0s and future &et Premium cashflo0s. he &et

Premium is not the same as the ross (remium actuall% paid b% the insured. his isan important difference 0ith the 7et (remium 5aluation.

Ee 0ill use the s%mbol &P for the 7et (remium. he 7et (remium is calculated

as the premium necessar%9 at the outset of the polic% i.e. Ehere t; *: to meet futurebenefits.

7o epenses are modelled as is the nature of the 7et (remium 5aluation.

*#$ Implicit allo/ance for e5penses in net premium )aluation

he net premium valuation does not eplicitl% allo0 for epenses. o0ever9 the netpremium used is generall% lo0er than the gross premium actuall% paid b% the

polic%holder. he net premium is calculated 0ithout considering epenses9 so it islo0er b% approimatel% the amount required to cover epenses.

Ehen 0e use the net premium in the net premium valuation9 because it is lo0er than

the gross premium9 the reserve is higher b% approimatel% the same amount as theallo0ance for epenses in the gross premium.

Consider the eample 0here 7( ; ( K / for a Ehole of $ife Assurance

ross (remium 5aluation V t =SA⋅ A xt −$P ⋅a x t % ⋅a xt

7et (remium 5aluation V t =SA⋅ A

xt − &P ⋅a

xt

0hich is also V t =SA⋅ A xt −$P ⋅a xt $P − &P ⋅a x t

0hich is also V t =SA⋅ A xt −$P ⋅a x t % ⋅a xt

And so the net premium valuation and gross premium valuation are equal if 7( ;

( K /. his is not al0a%s the case9 particularl% 0here the basis has changed sincethe polic% 0as issued.


$a%e (( ' 107





*#* Impact of basis chan+es on net premium )aluation

If the basis is changed e.g. >ortalit%9 discount rate or epenses: then the calculation

of the net premium reserve must tae these ne0 assumptions into account. o0ever9there are t0o impacts for ever% basis change

1. he 7et (remium is calculated on the ne0 basis9 as at the start of the contract

using the standard formula. hus9 the 7et (remium changes 0hen 0e change the valuation basis or the ?reserving basis@ or the set of assumptions used to calculatethe reserve.

2. he 7et (remium 5aluation reserve is then calculated on the ne0 basis9 but alsoallo0ing for the recalculated 7et (remium.

he impact of a particular basis change on the 7et (remium 0ill be opposite of thedirect impact of the basis change on the reserve. his means that the overall impacton the net premium valuation 0ill be less than that for a gross premium valuation.

Ee 0ill consider some eamples of this later in this section.

*#, E5amples of net premium )aluations

he reserve for a polic% in force still active: at time t is calculated as V t as sho0n

for each regular premium product belo0.

).+.% =hole of Life AssuranceV t =SA⋅ A xt − &P ⋅a xt

where &P =SA⋅ A x

a x

).+.& Term Assurance

V t =SA⋅ A xt

1

n−t ∣− &P ⋅a xt n−t ∣

where &P =SA⋅ A x1

n ∣

a x n ∣

).+.) 0ndo!ment Assurance

V t =SA⋅ A xt n−t ∣− &P ⋅a xt n−t ∣

where &P =SA⋅ A x n ∣

a x n ∣


$a%e () ' 107





).+.+ Annuity

For this annuit%9 0e assume pa%ments annuall% in arrears. o0ever9 because there is

no premium to be valued9 0e cannot allo0 for epenses as the difference bet0eengross and net premiums. hus it is t%pical to allo0 eplicitl% for epenses even for thenet premium valuation. -nce method is given belo0.

V t = Benefit % ⋅a xt

*#0 Sin+le premium policies under the net premium )aluation

he net premium valuation allo0s implicitl% for epenses through the difference in

the net premium and the gross premium. o0ever9 for single premium policies9 thepremium is not a component of the reserve. nless 0e mae an adJustment to the net

premium valuation 0hen dealing 0ith single premium policies9 our reserves 0ill notbe sufficient.

).,.% Single /remium 0ndo!ment Assurance

V t =SA⋅ A xt n−t ∣expense reserve

he epense reserve could be calculated in different 0a%s. A common and sensible:approach 0ould be as follo0s

expense reserve= % ⋅a xt n−t ∣

In this case9 the net premium valuation and gross premium valuation are the same.

*# Alternati)e interpretations of net premium )aluation

+ome practitioners also use a net premium valuation to be a valuation using

net premium=gross premium−expenses

he difference from this and a traditional net premium valuation is that the netpremium is calculated eplicitl% as gross premium less epenses9 and that this

premium is not recalculated if the basis changes.

his is not strictl% a net premium valuation in the formal sense. here is also the

requirement to test the net premium to ensure there is a sufficient gap bet0een thenet and gross premium to cover future epenses


$a%e (* ' 107





*#2 E5ample of impact of basis chan+es on the net premium )aluation

Table ).&.9asis change impact on net premium valuation for Term Assurance

$asis change %mpact of

basic changeon &P

%mpact of

basis change'excluding &P

change( on"eserve

%mpact of &P

change on"eserve

Combined

%mpact ofbasis change

and &Pchange on

"eserve

Increasemortalit%

Increase Increase ,ecrease +mall Increase

Increase

discount rate

,ecrease ,ecrease Increase +mall ,ecrease

Increaseepenses

7o change 7o change 7o change 7o change


$a%e (7 ' 107




"ectio ( : etrospective versus prospective calculatio o! reserves

, RETR%S'ECTI>E >ERSUS 'R%S'ECTI>E CALCULATI%" %F RESER>ES

)oth the ross (remium 5aluation and the 7et (remium 5aluation are prospective

reserving techniques. (rospective means that 0e ?loo for0ard@ to future cashflo0sand tae the present values of these cashflo0s to calculate the current reserve.

A retrospective calculation of reserves considers past cashflo0s to estimate thecurrent reserve. nder certain assumptions9 0e can sho0 that the prospective and

retrospective calculation 0ill give the same result.

,#1 Assumptions re?uired for e?ualit.

Actual cashflo0s must be equal to epected cashflo0s under the basis used for theprospective valuation

(remiums must have been calculated on the same basis as the valuation

,#$ T.pical form of relationship

he equation belo0 gives the t%pical form of the relationship. his version is for a 0hole of life assurance.

V t 1 =

[ V t P ⋅1i −SA⋅q xt ] p xt

he equation belo0 gives the same equation for an annuit% 0ith annual benefits paid

in arrears.

V t 1 =V t ⋅1i −Benefit⋅ p xt

p xt

,#* Intuiti)e interpretation of relationship

he reserves at time t9 plus the premium received at the start of the %ear andinvestment return on the reserve and premium Just received9 is sufficient to pa%

claims during that %ear and set up the reserve required at the end of the %ear for allsurviving polic%holders.


$a%e (+ ' 107




"ectio ( : etrospective versus prospective calculatio o! reserves

,#, &eri)ation of e?ualit. for a =hole of Life Assurance

e!uire" to Prove#

V t 1 =[ V t P ⋅1i −SA⋅q xt ]

p xt

Proof#

V t 1 =SA⋅ A xt 1− P ⋅a xt 1

V t 1 =SA⋅[ A xt −q xt / 1i ]⋅1i/ p xt − P ⋅ a xt −1 ⋅1i / p xt

V t 1 =[SA⋅{ A xt −q xt /1i}⋅1i − P ⋅ a xt −1 ⋅1i ]

p xt

V t 1 =1i ⋅[SA⋅ A xt − P ⋅a xt P −SA⋅q xt ]

p xt

V t 1 =

[ V t P ⋅1i −SA⋅q xt ] p xt


$a%e (, ' 107




"ectio ) : Impact o! reserves o pro!it ad loss

0 IM'ACT %F RESER>ES %" 'R%FIT A"& L%SS

0#1 "e/ business strain

eserves are often calculated on more prudent or conservative assumptions. his isto ensure that

profits are not recognised prematurel%

the insurance compan% has sufficient funds to meet its obligations to polic%holdersas the% fall due

If the reserving basis is more prudent that the basis used to calculate the premium9

then the reserve at time * 0ill be greater than 6ero. hus leads to new businessstrain. 7e0 business strain is the loss incurred at the start of a polic% due to aprudent reserving basis and also initial epenses9 although this is not covered here:.his loss is temporar%9 and profits 0ill emerge over time as the prudent assumptions

turn out to be more conservative than actual eperience.

Consider the eample belo0

A polic% pa%s a benefit of 11* in one %earBs time9 regardless of death or survival. he valuation discount rate is assumed to be "G as this is a prudent assessment of the

epected future investment returns available in the maret.

o0ever9 the premium is calculated using a discount rate of 1*G because this is abest estimate of the epected future investment returns available in the maret. Ishould be eas% to see that the single premium pa%able at the start of the polic% is

$P =100

he reserve at time * is V 0 =110⋅1.05−1−100⋅1=4.$%0

hus9 before the first premium is received9 0e must create a reserve of 4.&#. his

0ill cause a loss of 4.&# at t ; *. his is the new business strain.

After the first premium is received9 the reserve 0ill simpl% be the present value of

the benefit of 11* at "G. his is 110⋅1.05−1=104.$% . his reserve 0ill earn

interest at 1*G not "G9 since 1*G is our best estimate of actual investment returns:.

At t ; 1 the assets have gro0n from 1*4.&# to 11".24 115.24=104.$%⋅1.10 :. Ee

must pa% the benefit of 11* at t; 19 0hich leaves us 0ith ".24 profit.


$a%e )0 ' 107





hus9 the ne0 business strain 0as a temporar% loss. It has been reversed b% the

profit of ".24 at t ; 1.

%mportantly, the present value of the profit at t ) * 5.24⋅1.10−1=4.$% which is the same as the new business strain or loss at t ) +.

he present value of profits arising from the contract does not depend on the level

of prudence 0ithin the reserve. It depends on the actual cashflo0s premiums9claims and epenses: incurred. he reserves affect onl% the timing of the profits and

losses.

0#$ @illmerisation

,.&.% Impact of initial e"penses on the ;ross /remiumQillmerisation is a technique used to allo0 for initial epenses incurred at the outset

of the polic%. his is onl% necessar% for the net premium valuation. For the grosspremium valuation9 the premium 0ould be calculated as follo0s

$P ⋅a x n ∣=SA⋅ A x n ∣

' % ⋅a x n ∣

$P =SA⋅ A

x n ∣ '

a x n ∣

%

Ehere / is the ongoing epenses incurred at the start of ever% %ear9 and I is the initial

epenses incurred once at the outset of the polic%.


$a%e )1 ' 107





,.&.& >illmerised 8et /remium ?aluation

he Qillmerised 7et (remium 5aluation is given as

V t =SA⋅ A xt n−t ∣− &P ⋅a xt n−t ∣− ' ⋅a xt n−t ∣

a x n ∣

where &P =SA⋅ A x n ∣

a x n ∣

' =initi&l expenses

At t ; *9 the third time in the above reserves equation ; !I. As t approaches n9 thethird term in the reserve equation above approaches 6ero. The reserve is decreased

by the amount of initial expenses initially to reduce new business strain. headJustment reduces over time as t approaches n and the polic% approaches maturit%.

his can be intuitivel% understood as increasing the net premium to reflect thecomponent of the ross (remium that 0ill be used to recover initial epenses is

recognised in the 7et (remium 5aluation. Eithout this Qillmer AdJustment9 the 7et(remium 0ill be too much lo0er than the gross premium and 0ill result in ver% high

ne0 business strain.

0#* !asis chan+es

,.).% ;ross /remium valuation

he full impact of the basis changes is reflected in the liabilit% valuation. his changeflo0s directl% through the income statement.

,.).& 8et /remium valuation

he impact of the basis changes is reduced because the net premium itself alsochanges 0hen the basis is changed. he basis change has a smaller impact on the

income statement than the gross premium valuation. ,.).) Impact on financial reporting

If basis changes result in a large change in reserves or liabilities9 then basis changes

0ill have a large impact on the profit of the compan% for a particular %ear.


$a%e )2 ' 107





For eample9 if the total liabilities of a life insurer are +,1*m and the average

annual profit is +,4**9***9 then a "G increase in liabilities resulting from a basischange ma%be an increase in mortalit%: 0ill produce a +,"**9*** increase in

liabilities9 0hich is a corresponding decrease in profits of +,"**9***. his 0illerase all the profits for the %ear and mae the insurance compan% mae a loss.

1. ,o %ou understand ho0 a basis change can lead to a change in liabilitiesD

2. ,o %ou understand ho0 a change in liabilities 0ill lead to a profit or loss in the

period in 0hich the basis change is madeD

3. ,o %ou understand 0h% one needs to be careful not to change the basisunnecessaril%D

4. ,o %ou understand ho0 basis changes might be used to manipulate profitsD

1. Changing assumptions changes the epected present value of income and outgo.+ince liabilities and reserves: are calculated as the epected present value of

outgo less the epected present value of income9 the liabilit% figure 0ill change.

2. he ?accounting equation@ is A ; / $ 0here A is assets9 / is equit% and $ isliabilities. A change in $ 0ithout a corresponding change in A must lead to achange in /. his change comes about through a profit or a loss.

3. nnecessar% basis changes 0ill lead to inappropriate profits and losses9distorting the financial performance of the compan%. his is particularl% relevant

given the subJectivit% involved in setting assumptions and issues of credibilit% ofdata and random fluctuations.

0#, 'attern of reser)es o)er time

Ee have alread% covered the graphs of V t over time in lectures.


$a%e )& ' 107




Course Notes

Chapter , Unit and non-unit reser)es

About

This chapter !ill not be covered in the course and as a result the notes

provide only the outline of the methodology..

Chapter Contents

1 nit!lined products........................................................................... .........................""

1.1 Charging structure............................................................................. ....................""1.2 )enefit fleibilit% ........................................................................................... ........"#

1.3 Investment fleibilit% ............................................................................ ................"#

1.4 Investment uarantees...................................................................... ..................."&

2 nit Fund or nit eserves................................................................ ........................."8

2.1 nit pricing........................................................................................................... ."8

2.2 nit reserve................................................................................................. .........."8

3 7on!unit reserves................................................................................................ ........."'

3.1 Introduction................................................................................ .........................."'

3.2 >aret practice around the use of non!unit reserves........................................."'

3.3 7ames for non!unit reserves............................................................. ..................."'

3.4 Calculation principles....................................................................... ...................#*



Chapter ( : it ad o8uit reserves

"ectio 1 : it8liked products

1 U"IT-LI":E& 'R%&UCTS

nit!lined products also no0n as 5ariable niversal $ife products: are

sophisticated life insurance products that combine ris cover and savings benefits ina fleible structure. he fleibilit% is t0o!fold

1. he mi of ris benefits and savings contributions can be specified b% thepolic%holder at inception of the polic%9 and often changed 0ithin limits during the

term of the polic%.

2. he investment vehicles: into 0hich funds are invested can be selected b% the

polic%holder at the outset9 and often changed 0ithin limits during the term of the

polic%.

1#1 Char+in+ structure

(remiums are paid b% the polic%holder. -f this premium9 a certain proportion is

allocated to the unit fund. he unit fund operates much lie a ban account and feesL charges are deducted from the unit fund. Investment return is credited to the unit

fund. he unit fund is also sometimes no0n as the unit account.

,eath benefits are paid first out of the unit fund9 and topped up b% the insurer if the

+um Assured is greater than the value of the unit funds. hus9 the ris premium

charged each month is calculated as '&x()−*nit +un" , 0 ⋅q xt .

An annual asset management fee is also charged on a monthl% basis. (olic% fees9administration charges9 advice levies9 bid!offer spreads and a range of other charges

ma% be levied.

-ne of the advantages of a unit!lined product is that the charging structure is

transparent in that the polic%holder can see eactl% 0hat charges are levied.o0ever9 in practice9 polic%holders often find that the charging structure is

etremel% complicated and difficult to understand. igh charges 0ill inevitabl% lead

to poor value for mone% for polic%holders.

he unallocated premium plus fees and charges plus ris premiums are income forthe insurer. /penses and death benefits in ecess of the unit fund are outgo for the

insurer.


$a%e )) ' 107





1#$ !enefit fle5ibilit.

%.&.% 7is$ cover

A 0ide range of benefits can be offered. /ach benefit is charged 0ith a separate rispremium. As long as the total premium paid b% the polic%holder is greater than theris premiums required for the ris benefits: and epenses9 the remaining premiumis added to the unit fund and 0ill gro0 the unit fund over time.

%.&.& Maturity benefits

At maturit%9 unless there is a guaranteed minimum maturit% value see investmentguarantees belo0: the unit fund 0ill be paid out as a maturit% value.

%.&.) Surrender benefits+urrender values are usuall% paid as the unit fund less some deduction for

unrecouped initial epenses incurred

some amount of the future profit foregone b% the insurer

(oor surrender values have been a maJor criticism of unit!lined policies around the

0orld. his is eacerbated b% complicated charging structures and poorcommunication of surrender terms.

1#* In)estment fle5ibilit.he investment return credited to the polic%holder is the actual investment return

earned on the underl%ing funds as selected b% the polic%holder. suall% severaldifferent funds are offered such as

)alanced portfolio equities9 bonds9 propert% and cash:

Aggressive portfolio mostl% equit%:

,efensive portfolio greater proportion of cash and bonds:

Cash!onl% portfolio investing in mone% maret instruments up to '* da%s:

International portfolios offering eposure to international currencies9multinational companies and the domestic companies of other countries:

+ector fund based on a particular sector of a maret9 e.g. financials or industrialsor commodities:

Absolute eturn Funds L edge Funds


$a%e )* ' 107





he funds can either be o0n!branded and managed: funds or funds offered b% other

financial service providers. +ome companies mae use of ?0hite labelled@ 0here the%repacage another compan%Bs fund under their o0n brand.

1#, In)estment (uarantees

Investment uarantees can be offered9 ensuring that at least a minimum return is

offered. hese 0ill generall% be charged for eplicitl% and require a great deal of careto ensure that the fee charged is appropriate given the riss involved. he riss 0ill

var% depending on the nature of the underl%ing fund selected b% the polic%holder sothe fee charged must also depend on the underl%ing fund or combination of funds

selected.


$a%e )7 ' 107




"ectio 2 : it Fud or it eserves

$ U"IT FU"& %R U"IT RESER>ES

$#1 Unit pricin+

he unit account or unit fund is calculated as a price per unit or unit price and anumber of units. he unit fund for a particular polic%holder is

UF t =UP t ⋅ & t

where UF t =*nit +un" &t time t

UP t =*nit Pri,e &t tme t

& = -umer of units &t time t

he nit (rice 0ill be the same for all polic%holders9 but the number of units 0illdiffer so the unit fund 0ill differ. Allocated premiums and charges are converted to

numbers of units before being added or subtracted from the unit fund. Investmentreturn increases the unit price of each unit.

his is the basic approach. he actual detail of unit pricing is comple and be%ondthe scope of this course.

$#$ Unit reser)e

he unit fund or unit reserve is taen as the maret value of the units.

UF t 1=UF t AP t −,h&rgest − #P t ⋅1investment return t

where AP =)llo,&te" Premium

#P =is/ Premiums

Allocated (remiums are the proportion of the total ross (remium paid b%polic%holders that is allocated to the unit fund. he unallocated premium is income

for the compan% as is effectivel% another form of charge. he is (remiums chargedare the actual ris premiums deducted from the unit fund to provide the ris

benefits.

his formula is a ver% simplistic version based on a ver% simple charging structure. In

practice9 the calculation can be more comple.


$a%e )+ ' 107




"ectio & : No8uit reserves

* "%"-U"IT RESER>ES

*#1 Introduction

7on!unit reserves are reserves held in respect of unit!lined business separate fromthe unit fund or unit reserve. It is calculated as the epected present value of non!

unit cashflo0s such as

unallocated premiums

fees and charges

epenses

non!unit riss benefits paid.

+ince some of the charges and benefits paid on unit!lined policies depend on the

si6e of the unit fund9 0e need to proJect the unit fund into the future in order tocalculate the epected future non!unit cashflo0s.

he total reserve for a unit!lined product is the sum of the unit reserve and the non!unit reserve.

7on!unit reserves can be positive or negative9 but are usuall% negative. A negative

non!unit reserve decreases the total reserve for a polic%.

*#$ Mar<et practice around the use of non-unit reser)es

+ince non!unit reserves are usuall% negative9 it is not uncommon for a compan% to setthe non!unit reserve to 6ero. here are t0o primar% reasons for this

1. (rudence. )% not holding a ?negative reserve@ the overall reserves are higher and

thus more prudent

2. +implicit%. he calculation of non!unit reserves is fairl% comple.

*#* "ames for non-unit reser)es

In man% countries9 non!unit reserve are named according the local currenc%. +o9 in

the nited Mingdom9 non!unit reserves are called terling "eserves and in +outh Africa the% are called "and "eserves. he term non-unit reserve is generic and can

be used in all marets.


$a%e ), ' 107




"ectio & : No8uit reserves

*#, Calculation principles

1. sing a set of assumptions9 proJect the unit fund for0ard allo0ing for allocated

premiums9 charges9 ris premiums and investment returns and taing decrementsinto account.

2. Calculate the epected future level of non!unit cashflo0s for all polic%holders.

3. Calculate the present value of these epected future cashflo0s

Eh% are non!unit reserves usuall% negativeD

Ehat does a positive non!unit reserve impl%D

7on!unit reserves are usuall% negative for t0o reasons

profitabilit% products 0ill have higher future income than outgo. If this differenceis greater than an% prudential margins included in the valuation basis9 the non!

unit reserves 0ill be positive. An advanced concept9 0hich is not eaminable and 0ill not be eplained in detail is that of distortions caused b% proJecting and

discounting the unit funds in a real 0orld rather than maret consistent:manner.

Initial epenses are usuall% large. he unit!lined product 0ill have beendesigned to recoup these costs through future charges. hese future charges arecapitalised to offset the cost of the high initial epenses.

A positive non!unit reserve is usuall% an indicator of an unprofitable or ver% lo0

profit product. his is unusual and the reasons behind positive non!unit reservesshould be investigated.


$a%e *0 ' 107




Course Notes

Chapter 0 Actuarial reser)in+ techni?ues for non-

life insurance

About

This chapter covers the core techniques of actuarial reserving for nonlife

insurance. =e !ill cover the material in less detail than for life insurance

Chapter Contents

1 Introduction to reserving for non!life companies.............................................. .........#3

1.1 ,ifferences bet0een life insurance non!life insurance........................................#3

1.2 7eed for reserves L liabilities for non!life insurance...........................................#4

1.3 /plicit versus implicit prudence................................................. ........................#4

2 nearned (remium eserves L (rovisions.................................................................#"

2.1 Introduction.................................................................................. ........................#"

2.2 ,istribution of ris over polic% term.......................................... .........................#"

2.3 ( under uniform distribution of ris................................................. .............##

2.4 ( under non!uniform distribution of ris............................................ ..........#&

2." ( for monthl% premiums.......................................................................... .......#8

3 Additional nepired is eserves L (rovisions......................................... .............#'

3.1 7eed for an A .............................................................................. ...................#'

3.2 Calculation of the nepired is eserve................................................ .........#'

3.3 Calculation of the A .................................................................... ..................#'

3.4 Alternative naming conventions........................................... ..............................#'

4 -utstanding Claims eserves L (rovisions............................................................. ....&*

4.1 Introduction........................................................................................ ..................&*

4.2 Case estimates........................................................................ ..............................&*

4.3 +tatistical methods.................................................................................. .............&*

4.4 Combined methods................................................................... ...........................&1

" Incurred )ut 7ot eported eserves L (rovisions........................................... ...........&2



Chapter ) : Actuarial reservi% techi3ues !or o8li!e isurace

"ectio Chapter ) : Actuarial reservi% techi3ues !or o8li!e isurace

".1 Introduction............................................................................ ..............................&2

".2 (ercentage of premium approach..................................................... ...................&2

".3 $oss ratio approach.................................................................................. .............&2

".4 )asic chain ladder methods.................................................... .............................&3

"." Average cost per claim method......................................................................... ....&4

".# )ornhuetter!Ferguson.................................................................................. ........&8


$a%e *2 ' 107




"ectio 1 : Itroductio to reservi% !or o8li!e compaies

1 I"TR%&UCTI%" T% RESER>I"( F%R "%"-LIFE C%M'A"IES

1#1 &ifferences bet/een life insurance non-life insurance

$ife insurance and non!life insurance are different in man% respects. he table belo0outlines some of these differences.

&on-life %nsurance ife %nsurance

(olicies are short!term often one %ear: (olicies are long!term9 often ten %earsor more

Claim frequencies are high Claim frequencies are ver% lo0

>ore than one claim can occur on a

polic%

suall% onl% one claim per polic%

Claim amounts are variable Claim amounts are usuall% fied

(olic%holders can easil% change insurer (olic%holders can change insurer9 butgiven the long!term nature of the

policies it doesnBt happen as often

he insured person is usuall% thepolic% beneficiar%

he insured person is usuall% not thebeneficiar%

Claims are usuall% for indemnit%against loss

Claims meet a general financial need

Claim amounts are affected b%inflation

Claim amounts are generall% fied

(olicies do not provide a savingsbenefit

(olicies frequentl% combine ris coverand a savings element

Table ,.%.*omparison of life and nonlife insurance

#roup ife Assurance '#A( policies have many of the same characteristics asnon-life insurance. Thus, many of the techni/ues used for non-life reserving

and also used for group life business.


$a%e *& ' 107




"ectio 1 : Itroductio to reservi% !or o8li!e compaies

1#$ "eed for reser)es liabilities for non-life insurance

%.&.% Solvency

7on!life insurance companies must hold reserves to ensure that the% can meetpolic%holder liabilities claim pa%ments and associated epenses: as the% fall due.

%.&.& inancial reporting

As 0ith life insurance companies9 accurate assessment of the compan%Bs financialposition and financial performance requires an assessment of the outstandingliabilities.

%.&.) Monitoring e"perience and repricing

Accurate pricing is criticall% important for non!life insurance. (remium rates areregularl% re!assessed based on claims eperience. o measure recent claimseperience accuratel%9 the insurer needs an assessment of claims that have been

incurred but have not %et been paid. If onl% paid claims are considered9 the totalclaims incurred 0ill be underestimated and premiums rates ma% be set too lo0.

1#* E5plicit )ersus implicit prudence

(rudence is usuall% introduced to non!life reserves in an implicit manner through

not discounting future cashflo0s

subJective increases in case estimates of claims

arbitrar% increases in the si6e of reserves held e.g. 1*G higher:


$a%e *( ' 107




"ectio 2 : eared $remium eserves ' $rovisios

$ U"EAR"E& 'REMIUM RESER>ES 'R%>ISI%"S

$#1 Introduction

(remiums are t%picall% received in advance of pa%ing claims. hese premiums 0henreceived should not be immediatel% counted as profit as future claims 0ould then

lead to losses.

At an% point in time9 a certain proportion of premiums received 0ill relate to ris

cover to be provided in future. Ee consider these premiums as unearned becausethe insurance compan% has not %et earned the premiums. -nce the ris cover is

provided to the polic%holders in future9 and claims have been incurred9 then thepremiums 0ill be considered to be earned.

In the simplest sense9 if premiums are paid annuall% in advance on 1 anuar% each %ear9 half the premiums 0ill have been earned b% 3* une. he other half of the

premium 0ill be unearned.

$#$ &istribution of ris< o)er polic. term

For most non!life insurance policies9 riss are uniform over the lifetime of the polic%.his means that the probabilit% of a claim9 and the epected si6e of claims9 is

constant over the term of the polic%.

+ome eceptions include

Crop insurance.Crop 0heat9 grapes9 fruit: insurance is highl% dependent on 0eather conditions

0hich are seasonal.

>otor vehicle accident insurance

In some countries9 hail9 sno0 and other adverse 0eather conditions lead to higherclaims in certain seasons t%picall% 0inter:.

+ome eamples 0here claims are uniform throughout the %ear

>otor vehicle theft insurance

he ris of theft of a motor vehicle 0ill usuall% be uniform over the term of a polic%.

(ublic liabilit% insurance


$a%e *) ' 107





Eill buildings and household contents insurance perils of fire9 flood9 0ater

damage9 subsidence etc.: depend on the time of %earD As an eercise9 give someeamples of claim t%pes that 0ill depend on time of %ear9 and some that are

uniform.

Flood and to some etent subsidence claims are usuall% season!dependent. +omefires ma% be created from heating appliances used in the house during 0inter. heftand most other fires should be uniform throughout the %ear.

If houses are frequentl% left unoccupied for etended periods summer houses9 orhouses during etended holida% periods 0hen families ma% vacate the house for

several 0ees at a time: ma% be correlated 0ith an increase in theft claims.

$#* U'R under uniform distribution of ris<

Ehen ris is distributed uniforml% over the term of the polic%9 the unearned

premium is taen as the proportion of the premium pa%ment interval that lies in thefuture for the most recentl% paid premium.

&.).% )-,ths method

he 3#"ths method consider the proportion of premium unearned measured in da%s

a %ear divided into 3#" da%s:.

*P =[ "&s until next premium

"&s from l&st premium to next premium ]⋅Premium

&.).& &+ths method

he 24ths method consider the proportion of premium unearned measured in half!months a %ear divided into 24 half!months:.

his method is an approimation to the 3#"ths method. (remiums are usuall%

assumed to be received half0a% through a month rather that on the eact date ofpremium receipt.


$a%e ** ' 107





$#, U'R under non-uniform distribution of ris<

If ris is not uniforml% distributed over the term of the polic%9 the ( needs to be

adJusted to reflect not onl% the proportion of time remaining for the most recentl%paid premium9 but the proportion of ris.

/ample

A polic% has annual premiums +,19***: paid in advance on the 1st of anuar% each %ear. is increases linearl% over the lifetime of the polic%. Ehat is the ( at 3*

une of each %earD

is/ t =(⋅t Ehere c is the gradient and t is the time in %ears

otal ris for the polic% %ear is

ot&l is/ =∫0

1

ris/ t dt

=∫0

1

(⋅t dt

=[(⋅t 2

2 ]0

1

=

(

2

is/ rem&ining t 0.5=∫0.5

1

ris/ t dt

=[(⋅t 2

2 ]0.5

1

=(

2−

(

8=

3

8⋅(

Per,ent&ge ris/ rem&ining t 0.5=(⋅3 /8

( /2=

%

8=$5

o the 0P" re/uired is 0123+


$a%e *7 ' 107





his can be epressed graphicall% as the area under the function of ris as a function

of time as sho0n belo0

As an eercise9 calculate the proportion of premiums unearned at 3* +eptember if

is/ t =e(⋅t

is/ t =−t 2t

is/ t =e(⋅t

therefore *P =e

0.$5,−1

e(−1 ⋅ premium

is/ t =−t 2t therefore *P =−2⋅0.$5

33⋅0.$52 ⋅ premium

$#0 U'R for monthl. premiums

For monthl% premium policies9 the ( at the end of the month 0ill al0a%s be 6ero.

Chec that %ou understand 0h% this is true.


$a%e *+ ' 107

Illustration ,.%( ;raphical representation of unearned ris$ under nonuniform distribution

of ris$

Earned Premiums under non-uniform distribution of risk

Risk $nearned

% &'(Risk )arned %

2'(0(

2'(

'0(

&'(

#00(

0( 2'( '0( &'( #00(

t

R i s k

Risk $nearned Risk )arned




"ectio & : Additioal epired isk eserves ' $rovisios

* A&&ITI%"AL U"E;'IRE& RIS: RESER>ES 'R%>ISI%"S

*#1 "eed for an AURR

If the future epected claims are higher than the ( calculated as described in theprevious section9 the insurance compan% 0ill need to hold additional reserves.

his usuall% occurs for one of t0o reasons

1. (remium rates are too lo0 and 0ill not be sufficient to meet normal levels offuture claims

2. +ignificant additional riss have arisen since the premiums 0ere set. For eample9

court rulings ma% create a ne0 precedent that 0ill epose the insurer to higherclaims in future. Alternativel%9 an unepected increase in vehicle replacementparts ma% mean that future motor accident claims 0ill be more costl% to pa%.

*#$ Calculation of the Une5pired Ris< Reser)e

he nepired is eserve or is the reserve for unepired riss. nepiredriss are riss arising from current policies. his is an estimate of future claimpa%ments for 0hich the insurance compan% has a legal obligation to pa%. his 0illreflect the remaining term on eisting policies as the insurer is not obligated to

rene0 policies and ma% increase the premium rates on polic% anniversar% in mostcases.

%picall%9 as is the case 0ith most non!life insurance reserves9 future claims are notdiscounted in this calculation as an implicit form of prudence. Although it might be

considered more accurate to discount claims and use an eplicit amount of prudence9this is not maret practice.

*#* Calculation of the AURR

he calculation of the A is straightfor0ard

)* ='&x * −*P , 0

If the is less than the (9 no A is held.

*#, Alternati)e namin+ con)entions

+ome insurance companies use the term to mean A.


$a%e *, ' 107




"ectio ( : 9utstadi% Claims eserves ' $rovisios

, %UTSTA"&I"( CLAIMS RESER>ES 'R%>ISI%"S

,#1 Introduction

he process to pa% a claim begins once a claim has been reported to the insurancecompan%. o0ever9 there ma% be a dela% of a fe0 da%s to several %ears from the date

the claim is reported to the date the claim is finall% settled.

>otor accident claims 0ill be paid 0ithin a short period of time. he dela%s here are

caused b% the need to eamine the vehicle for damage9 assess culpabilit% and liabilit%9repair the damage and for the final claim amount to be determined.

For product liabilit% claims9 it ma% tae several %ears for the courts to decide 0hatliabilit% is o0ed b% the insured.

-nce the claim has been reported to the insurance compan%9 the insurance compan%must create an outstanding claims reserve 'OC"( based on the epected claim

amount to be paid. here are t0o primar% methods of assessing outstanding claimsreserves.

,#$ Case estimates

+peciall% trained and eperienced: claims assessors or ?loss adJusters@ consider the

details of each claim on an individual basis and estimate the si6e of the claim. hisrelies on the epertise of the claims assessor.

his approach is useful 0hen

here are a small number of claims so the 0orload of assessing individual claims

is manageable.

he claims are large or unusual in nature so human Judgement is required.

he claims can be accuratel% assessed based on the details of the claim. For

eample9 for a specific model vehicle9 the cost of replacing a specific part orrepairing a specific bod% panel should be no0n 0ith high certaint%.

,#* Statistical methods

+tatistical methods are used 0hen there is a ver% large number of claims.Considerable time 0ould be required to estimate claims individuall%9 and the impactof errors in an% individual claim estimate should be small given the large number ofclaims.


$a%e 70 ' 107




"ectio ( : 9utstadi% Claims eserves ' $rovisios

he method might allo0 an average claim amount per claim t%pe9 polic% t%pe and

peril t%pe. >ore complicated approaches can be adopted if required.

-nl% ver% large insurers 0ith ver% large9 homogeneous boos use statistical methods.

/ample

Average claim amount per claim estimated over recent %ears has been +,2"*. Ee

have 34 reported claims that have not %et been paid. Ehat is the total -CD

6 =34⋅250=USD 87500

,#, Combined methods

In some cases9 both methods are used together. For eample9 claims assessors ma%mae case estimates for each individual claim. he insurance compan% ma% then use

statistical methods to adJust these estimates for inflation or assessor bias.


$a%e 71 ' 107




"ectio ) : Icurred #ut Not eported eserves ' $rovisios

0 I"CURRE& !UT "%T RE'%RTE& RESER>ES 'R%>ISI%"S

0#1 Introduction

Claims are not al0a%s reported to the insurance compan% immediatel%. ,ela%s ma%be

+everal da%s for a motor vehicle accident

+everal 0ees for a house burglar% 0hile the occupants 0ere a0a% on holida%

+everal %ears for product liabilit% 0here the product defect is not no0n for several %ears.

At an% given valuation date the date at 0hich 0e calculate reserves: there 0ill be anunno0n number of claims that have been incurred i.e. have happened: but have

not been reported. /ach claims 0ill have an unno0n si6e. his poses a seemingl%difficult question9 ?o0 do %ou reserve for claims that %ou donBt no0 eistD@

his is the subJect of this section. hese claims are called %ncurred $ut &ot"eported claims or simpl% I)7 claims. he reserves for these claims are no0n

strictl% as %$&" "eserves but often simpl% %$&"s.

0#$ 'ercenta+e of premium approachhe most basic approach is to calculate I)7s as a fied percentage of premium.iven that claims 0ill generall% increase 0ith increased premium volumes more

policies9 larger riss:9 this is a reasonable simple approach. he problem is that thisapproach alone does not help us to assess what percentage of premiums to use.

0#* Loss ratio approach

he simplest approach that does help to assess the absolute level of the I)7 reserveis called the loss ratio approach. A $oss atio $: is the ratio of total claims

incurred to total premiums earned over a certain period for a certain line of businessof other grouping.

=6l&ims n,urre"

Premiums E&rne"

he $ approach assumes that claims losses: can be epressed as a stable percentage

of premiums. his is t%picall% the 0a% policies are priced9 and for large portfolios ofbusiness the $ ma% be relativel% stable. he $ 0ill usuall% be estimated in

conJunction 0ith the pricing and under0riting departments:.


$a%e 72 ' 107





4ear Earned

Premiums

Assumed

oss "atio

Expected

Claims

"eported

Claims

0nreported

Claims

200x ! "#

$%

! & "# #$

'()# %

$ * #$

2**4 "*943* #*G 3*92"8 3*91** 1"8

2**" #&98** #*G 4*9#8* 3894** 2928*

2**# ##93** &*G 4#941* 41931* "91**

2**& &"9*&* ""G 41928' 239#"* 1&9#3'

Total %$&" 53,*22

Table ,.&( Loss 7atio approach to calculating I987 reserve

he table above sho0s the t%pical 0a% in 0hich the $ approach is applied.

0#, !asic chain ladder methods

he )asic Chain $adder )C$: method is a more sophisticated method of estimatingI)7 reserves.

,.+.% Assumptions required

he past development pattern of claims 0ill continue in future. Future reportingdela%s 0ill be similar to past reporting dela%s:

Future claims inflation 0ill be a 0eighted average of past claims inflation

he past claims eperience on 0hich the reporting dela%s are based includes at

least one %ear that is full% run!off. At least one past %ear of claims must be 1**Greported:


$a%e 7& ' 107





0#0 A)era+e cost per claim method

Another method often adopted is called the average cost per claim. It is similar to

the )C$ method in that it assumes past development patterns reporting dela%s: 0illcontinue in future. o0ever9 it assumes the past development pattern measured interms of number of claims 0ill be constant in future and similar to the past. he

)C$ assumes past development patterns measured in amounts of claims 0ill beconstant in future and similar to the past:.

he reserve is calculated in t0o steps

1. sing the past development pattern9 calculate ho0 man% unreported claims havebeen incurred

2. +eparatel%9 calculate an average cost per claim based on past claims information.his figure ma% be adJusted if necessar% to reflect no0n changes in claim si6esdue to inflation9 court precedents9 changes in polic% conditions etc.

>ore basic methods are sometimes used for step 1 such as subJective assessment ofthe number of unreported claims9 or the average number of unreported claims oversimilar periods in the past.


$a%e 7( ' 107





,.,.% :vervie! of approach

he approach can be separated into distinct steps. Ee 0ill consider the eample set

at 31 ,ecember 2**#. 7ote that in this eample 0e onl% use three %ears of claimsinformation. In practice more detail could be and generall% 0ould be: used.

1. abulate reported claims %ear in 0hich claim 0as incurred9 and the number of %ears dela% to the claim being reported.Cclaim %ear9 reporting dela% are sho0n in their correct positions in the table belo0

Table ,.).Tabulation of claims by year of incidence and reporting delay

"eporting 1elay + * 5

Claim 4ear

2**4 C2**49* C2**491 C2**49*

2**" C2**"9* C2**"91 7ot available

2**# C2**#9* 7ot available 7ot available

he ?not available@ cells are not %et available since at the end of %ear 2**#: 0e do not

no0 about claims that 0ill be reported in 2**& and 2**8.

2. hen 0e cumulate the claims in a similar table 0here each column includes thetotal claims reported up to that point as sho0n in the table belo0.

able ".4.abulation of cumulative claims b% %ear of incidence and reporting dela%


Claim 4ear

2**4 C2**49* C2**49*C2**491 C2**49*C2**491 C2**492

2**" C2**"9* C2**"9*C2**"91 7ot available

2**# C2**#9* 7ot available 7ot available


$a%e 7) ' 107





3. -nce 0e have the cumulative claims reported for each %ear and each reporting

dela%9 0e can calculate ?lin ratios@ that lin the claims reported b% reporting dela% to the claims reported b% reporting %ear 1.

lin) 071=C 200471C 200571

C 200470C 200570

lin) 172=

C 200472

C 200471

here are fe0er data points available for lin) 172 than for lin) 071 because of

the triangular nature of the information available.4. Ee can use the lin ratios to estimate the empt% cells of the cumulative claims

table

able ".".abulation of cumulative claims b% %ear of incidence and reporting dela%


Claim 4ear

2**4 C2**49* C2**49*C2**491 C2**49*C2**491C2**492

2**" C2**"9* C2**"9*C2**"91 C2**"9*C2**"91: link 192

2**# C2**#9* C2**#9* link *91 C2**#9* link *91 x link 192

". he right!most column no0 includes estimates of total claims incurred provided

the first ro0 is full% ?run!off@:. If 0e subtract total reported claims from the totalestimated claims incurred9 0e get an estimate of total I)7 claims.


$a%e 7* ' 107





#. A slight modification of the approach is to calculate cumulative development

factors cdf :based on the product lin ratios. he calculations are algebraicall%equivalent9 but the cumulative development factors are easier to 0or 0ith andhave another useful interpretation.

,"f 0=lin) 071⋅lin) 172

,"f 1=lin) 172

&.1

,"f t

) percentage of claims reported b% time t.

,.,.& More detailed !or$ed e"ample>ae sure %ou receive a cop% of the more detailed 0ored eample in an electronicspreadsheet.

,.,.) /roblems !ith 9*L method

he )C$ method 0ors 0ell if the assumptions underl%ing it are appropriate.

1. Ehen might the assumption of stable development patterns not holdD

2. Ehen might the assumption relating to inflation not holdD

1. he development pattern 0ill change if an% of the factors affecting an% of the

causes of dela%s change. For eample9 different broers ma% report claims 0ithdifferent dela%s9 so changes in broers or the amount of business arising from

certain broers could change the pattern. Administration s%stems and processesma% change resulting in shorter or longer dela%s usuall% longer in the short

term9 and hopefull% shorter in the long!term as the benefits of the s%stem changeare recognised:. -utsourcing of administration functions can have a dramatic

change in the reporting dela%s.

2. If the econom% has moved from a high inflation phase to a lo0 inflation phase or vice versa. his has happened to man% developing economies as tighter monetar%polic% and more careful fiscal polic% have led to greater stabilit% and lo0er

inflation.


$a%e 77 ' 107





0# !ornhuetter-Fer+uson#

>ore advanced methods eist for estimate I)7 reserves. 7one of these 0ill be

covered in this course. he% are generall% covered in a specialised non!life course.

he most commonl% used advanced method is no0n as the $ornhuetter-6erguson method or ?)F@ method. It uses )a%esian techniques to combine

estimates using the loss!ratio approach and the basic chain ladder method based onthe credibilit% of the available data.


$a%e 7+ ' 107




Course Notes

Chapter Assets and Liabilit. considerations

About

This chapter covers the some basic principles for considering assets and

liabilities together. The ma4or aim of this section is to introduce the

concepts of matching and immunisation.

Chapter Contents

1 <ield Curves................................................................................................ ..................8*

1.1 Factors affecting the shape of the %ield curve......................................................8*

1.2 Qero Coupon <ield Curve......................................................... .............................81

2 >atching of assets and liabilities............................................................ ....................82

2.1 Introduction to matching............................................................. ........................82

2.2 (erfect cashflo0 matching and the $a0 of -nce (rice......................................82

2.3 ,ifficulties of perfect cashflo0 matching.......................................................... ..83

3 >easures of the term of financial instruments....................................... ...................843.1 ,iscounted >ean erm......................................................................................... 84

3.2 5olatilit% ............................................................................................... .................8"

3.3 Comparison of ,> and volatilit% ........................................................ ..............8#

4 $inear approimations to change in prices....................................................... ..........8&

4.1 eneral form of first!order linear approimation...............................................8&

4.2 /ample for a 6ero coupon bond............................................... ..........................8&

4.3 /ample for 2 %ear coupon bond............................................... ..........................884.4 +econd!order approimation........................................................................... ....88

" Immunisation................................................................................. .............................'*

".1 (urpose of immunisation.................................................................. ...................'*

".2 Assumptions......................................................................................... ................'*

".3 equirement of immunisation...................................................... .......................'1

".4 /ample of immunisation............................................................................ ........'2

# >ismatch eample........................................................................ ..............................'4



Chapter * : Assets ad ia-ilit/ cosideratios

"ectio 1 : ;ield Curves

1 4IEL& CUR>ES

A %ield curve is a function or graph9 if presented graphicall%: of the 4ield To

7aturity <>: for fied interest securities 0ith term to maturit% .

An eample is given belo0

1#1 Factors affectin+ the shape of the .ield cur)e

his have been covered in lectures and are onl% listed belo0

/pectations of future short!term interest rates

(reference for liquid short!term: instruments

>aret +egmentation

A fourth factor9 called >aret (reference9 is sometimes included. It is a 0eaer formof the >aret +egmentation theor%9 0here different investors in the maret have apreference for instruments of a certain term. his contrasts 0ith >aret

+egmentation 0here the investors only purchase instruments 0ithin a particularrange of terms.


$a%e +0 ' 107

Illustration -.%( 0"ample of a @ield *urve

Yield Curve

0*0(

#*0(

2*0(

+*0(

*0(

'*0(

!*0(

&*0("*0(

,*0(

0 * ' # * ' 2 * ' + * ' * ' ' * ' ! * ' & * ' " * ' , * ' # 0 * '

# # * '

# 2 * '

# + * '

# * '

Term

Y T M




"ectio 1 : ;ield Curves

1#$ @ero Coupon 4ield Cur)e

A normal %ield curve sho0s the <> for a bond 0ith a given maturit% date. o0ever9

the <> is the single discount rate used to discount all the cashflo0s coupons andprincipal: for a given bond.

A 8ero coupon yield curve sometimes no0n simpl% as a 8ero curve: sho0s the

%ield to maturit% on 6ero coupon bonds for the given maturit%. his <> is appliesonl% to cashflo0s of a specific term. hus9 the rates from the 6ero coupon %ield curvecan be used to discount specific cashflo0s of the appropriate term.

7ote that the slope gradient: of the Qero Coupon <ield Curve is greater in

magnitude than the <ield Curve. he <ield Curve can be thought of as a form of 0eighted average of the Qero rates9 0here the 0eights are related to the si6e of thecashflo0s coupons and principal:.

Coupon bonds can also be priced using the 6ero coupon %ield curve b% discountingeach cashflo0 coupon and principal again: at the appropriate rate from the 6ero

coupon %ield curve.


$a%e +1 ' 107

Illustration -.&( ;raph comparing a ero coupon yield curve against a yield curve.

Yield Curve and Zero Couon Yield Curve

+*'0(

*'0(

'*'0(

!*'0(

&*'0(

"*'0(

,*'0(

#0*'0(

# 2 + '

term

Y i e l

-ield Cur.e /ero Coupon -C




"ectio 2 : Matchi% o! assets ad lia-ilities

$ MATCHI"( %F ASSETS A"& LIA!ILITIES

$#1 Introduction to matchin+

>atching assets and liabilities means to choose assets 0ith the same characteristicsas the liabilities to reduce ris. Assets and liabilities are both financial instruments

and ma% var% in value independentl%. If the assets and liabilities have similarcharacteristics9 0e 0ould epect them to var% in a similar fashion to changes in

eternal conditions.

he three primar% characteristics considered are

1. erm

2. Currenc%

3. 7ature fied or real:

o0ever9 there ma% be other factors such as sensitivit% to particular maret prices

that could be considered.

he concept of matching is similar to that of hedging.

$#$ 'erfect cashflo/ matchin+ and the La/ of %nce 'rice

If 0e have t0o instruments9 A and )9 0hich produce the eact same cashflo0s as eachother under all circumstances9 the% must have the same price. his is no0n as theaw of One Price.

his should be fairl% obvious9 but it can be eplained further b% considering 0hatmight happen if A and ) 0erenBt priced the same.

Assume P A≥P B

hen9 0e purchase instrument ) and sell instrument A. +ince P A≥P B 9 this %ields

profit =P A−P B . +ince all future cashflo0s from ) 0ill perfectl% match

cashflo0s required to pa% to the o0ner of A9 0e cannot mae a loss in future.

hus9 0e have made ris!-free profit. he profit 0e made9 9 is made 0ith no

ris. his breas the no arbitrage assumption.


$a%e +2 ' 107




"ectio 2 : Matchi% o! assets ad lia-ilities

Further9 0e could continue to bu% ) and sell A to mae profit over and over again.

o0ever9 bu%ing ) 0ould increase the price of )9 and selling A 0ould decrease theprice of A. /ventuall%9 the price of A and ) 0ould be equal. +ince there is no ris to

this activit%9 an%bod% 0ho found the price difference 0ould immediatel% bu% asmuch of ) as possible9 and sell as much of A as possible. hus9 the pressure toequalise prices A and ) 0ould act ver% quicl% and 0ith great force to mae theprices equal.

$#* &ifficulties of perfect cashflo/ matchin+

In practice9 it is often difficult to find t0o securities 0ith the eact same cashflo0sunder all circumstances. In this case9 0e mae use of more approimate matchingtechniques. Ee attempt to match the currenc%9 nature and average term of cashflo0s.-ne should also tr% to match the convexity of the instruments as closel% as possible.

Conveit% is covered in later sections in this Chapter.

1. Eh% is it important to match annuit% cashflo0sD

2. Eh% is it less important to match term assurance cashflo0sD

1. Annuities have a high ,> and large reserves. Changes in interest rates give rise

to a large percentage change in the reserves9 and a large absolute change since thereserves are large.

2. erm Assurances generall% have fairl% small ,> and small reserves. A change ininterest rates gives rise to a smaller percentage change in reserves than annuities9

and as the reserves themselves are usuall% ver% small compared 0ith annuities9the absolute financial impact is lo0.


$a%e +& ' 107




"ectio & : Measures o! the term o! !iacial istrumets

* MEASURES %F THE TERM %F FI"A"CIAL I"STRUME"TS

*#1 &iscounted Mean Term

he 1iscount 7ean Term '17T( is the average term of the financial instrument9 0here the term of each cashflo0 is 0eighted b% the (resent 5alue of that cashflo0.

)elo0 0e give the derivation of ,> for an 7!%ear bond 0ith annual coupons Cpa%able in arrears.

P i= pri,e of instrument

P i=

∑t =1

&

C ⋅vt

100⋅v

n

:' i=∑t =1

&

t ⋅* t

∑t =1

&

*t

*t

=C ⋅vt for t ∈[17 & −1]

* & =C ⋅v & 100⋅v

&

:' i=[∑t =1

&

t ⋅C ⋅vt n⋅100⋅v

n][∑t =1

&

C ⋅vt 100⋅v

n]From the equations above9 it should be clear than the ,> of an 7!%ear 8ero

coupon bond is 7.

A 6ero coupon bond is a bond that does not pa% coupons. It onl% repa%s the

principal on maturit%. hus9 the formula for the price of a 6ero coupon bond is

simpl% P=100⋅v &


$a%e +( ' 107





*#$ >olatilit.

5olatilit% in this sense means the sensitivit% of price to interest rates. It is related to9

but not the same as9 volatilit% in the sense of random variabilit% of maret prices offinancial instruments.

).&.% ;eneral form

P i= pri,e of instrument

∂Pi

∂ i/Pi=

P; i

Pi =vol&tilit

).&.& 0"ample for a bond

Consider an n!%ear bond 0ith coupons C pa%able annuall% in arrears.

P i=∑t =1

&

C ⋅vt 100⋅v

n

∂Pi

∂ i /Pi =

P;i

P i =

−[∑t =1

&

t ⋅C ⋅vt 1n⋅100⋅v

n1]

[∑

t =1

&

C ⋅vt 100⋅v

n

]=

−[∑t =1

&

t ⋅C ⋅vt n⋅100⋅v

n]1i ⋅[∑t =1

&

C ⋅vt 100⋅v

n] Ee obtain the derivative through use of the chain rule.

aing the derivative requires some care because v=1

1i


$a%e +) ' 107





*#* Comparison of &MT and )olatilit.

,> and volatilit% are similar measures. o0ever9 the% are defined slightl%

differentl% and differ in numerical value for the same instrument.

−:'i

1i =

P; i

Pi

hus9 the ,> for an 7!%ear 6ero coupon bond is 79 but the volatilit% is− &

1i .


$a%e +* ' 107




"ectio ( : iear approimatios to cha%e i prices

, LI"EAR A''R%;IMATI%"S T% CHA"(E I" 'RICES

,#1 (eneral form of first-order linear appro5imation

For a small change in interest rate from i to i+ 9 0e can approimate

P Ai+ as

P Ai+P A i +⋅∂ P A i

∂ i=P A i +⋅

∂ P Ai

∂ i⋅P Ai ⋅P A i

,#$ E5ample for a Bero coupon bond

ae an 7!%ear 6ero coupon bond 0ith price P Ai at interest rate i and par value ;

1**. he formula for the price is P Ai =100 v

& and has volatilit%

− &

1iand

derivative 0ith respect to i of− &

1i ⋅100 v

& .

o continue the eample9 0e 0ill assume & =10

Ee calculate the value of the bond at "G to be P A5 =100⋅v10=%1.39

7o09 if 0e need to calculate the value of the bond at ".1G9 0e can use the linear

approimation as

P A5 0.1=P A

50.1−0.001⋅ &

10.05⋅P A

5

=%1.39−0.01

1.05⋅%1.39=%1.39−

1

105=%1.39−0.58=%0.81

+o the approimation %ields #*.81. If 0e no0 calculate the value directl% using theprice formula9 0e get

P A5.1=100⋅1.051−10=%0.81

In this case9 for this instrument9 for a *.1G change in interest rates9 the result is the

same up to two decimals places.

Chec for %ourself that the results differ from the third decimal place.


$a%e +7 ' 107





,#* E5ample for $ .ear coupon bond

Consider a 2!%ear bond9 par value 1** that pa%s a "G annual coupon in arrears.

1. Calculate the price at i ; "G and at i ; #G directl% using the price formula.

2. Calculate the ,> at i ; #G

3. Calculate the volatilit% at i ; #G

4. Calculate the price at i ; #.2G using the first!order linear approimation

". Calculate the price at i ; #.2G directl% using the price formula.

,#, Second-order appro5imation

he first!order linear approimation is onl% useful for ver% small changes in i. Forlarger changes9 0e need to consider higher order derivatives to more closel% match

the shape of the price formula curve the graph of price against interest rates:.

he formula for the second!order approimation etends the approach for the first!

order linear approimation

P Ai+ P A i +⋅∂ P A i

∂ i +

2⋅∂ P A

2 i

∂ i2/2

he graph belo0 sho0s the blue curved line of the price of a 1*!%ear QC) for different

interest rates. he straight dotted orange line is the first!order linear approimation.he approimation is reasonabl% close close the point of tangent9 but gets

progressivel% 0orse as 0e move a0a% from that point.

he curve dotted green line is the second!order approimation. As can be seen from

the graph9 it is a much better fit at most interest rates.


$a%e ++ ' 107






$a%e +, ' 107

Illustration -.)( /rice sensitivity of a %6year >ero *oupon 9ond

!"-#ear Zero Couon Bond Price

#0*020*0

+0*00*0'0*0!0*0&0*0"0*0,0*0

#00*0

0 * 0 (

# * 0 (

2 * 0 (

+ * 0 (

* 0 (

' * 0 (

! * 0 (

& * 0 (

" * 0 (

, * 0 (

# 0 * 0 (

YTM

P r i c e

1rie First 3rder Appro4imation Seond 3rder Appro4imation




"ectio ) : Immuisatio

0 IMMU"ISATI%"

0#1 'urpose of immunisation

he purpose of immunisation is to ensure that no losses 0ill be made on changes ininterest rates. It is a method of matching that can be used 0hen eact cashflo0

matching is not possible or too epensive.

&ote that immunisation in practice is almost never possible. It requires the

assumption of a flat %ield curve 0here changes in interest rates occur onl% through aparallel shift in the %ield curve. his does not happen in practice9 since it implies an

arbitrage opportunity 0here ris!free profits can be made b% investors.

he discussion of arbitrage opportunities is be%ond the scope of this course.

0#$ Assumptions

Interest rates are equal for all terms. he %ield curve is flat.:

Changes in interest rates are equal at all durations.

7ote that this second point is a necessar% result of the first point. If the %ield curveis flat the same interest rate at all durations: then an% change in the %ield curve at

one point must result in the same change at ever% point on the %ield curve.

All changes in interest rates are small. (ortfolios can be rebalanced 0ithout cost afterall small changes in interest rates.

Again9 this assumption is not necessaril% true in practice. Immunisation onl%protects against small changes in interest rates. $arge changes occur in practice.

+econdl%9 there are costs to rebalancing portfolios to ensure an immunised position.


$a%e ,0 ' 107





0#* Re?uirement of immunisation

,.).% Some definitions

All definitions given for interest rate i

f A i =v&lue of &ssets

f ! i=v&lue of li&ilities

∂ f A i

∂ i =f A

;i=first "eriv&te of &sset v&lue

∂ f ! i

∂ i

=f !; i =first "eriv&te of li&ilit v&lue

∂2f A i

∂ i2 =f A

;; i=se,on" "eriv&te of &sset v&lue

∂2f !i

∂ i2 =f !

;;i =se,on" "eriv&te of li&ilit v&lue

,.).& 7equirements for immunisation

For immunisation9 the follo0ing three criteria must hold

f A i =f ! i he value of assets and liabilities must be equal

f A; i =f !

; i he volatility of assets and liabilities must be equal

f A;;i ≥f !

;;i he convexity of assets must be greater than conveit% of

liabilities

5olatilit% here is defined as the sensitivit% of the price of the instrument to changes in

interest rates. +imilarl%9 conveit% is defined as the sensitivit% of volatilit% to changesin interest rates.


$a%e ,1 ' 107





,.).) /ractical interpretation

Assets and liabilit% values must be the same i.e. +,1** of Assets ; +,1** of

$iabilities:

he volatilit% or ,> of assets and liabilities must be the same 0e call this being

duration matched(

he spread of assets b% term: must be greater than that of liabilities

0#, E5ample of immunisation

Ee have an obligation of +,1** in " %ears time. Current interest rates are "G perannum. Ee have t0o assets bacing this liabilit%.

Asset R pa%s "2." in %ear #

Asset < pa%s 4&.#2 in %ear 4.

Assets A ; R <

Chec that f ! 5=$8.35

f , 5 =39.18

f - 5 =39.18

f A 5 =f , - 5 =$8.35

f !i=100⋅1i−5

f !; i=−5⋅100⋅1i −%

f A; i =−%⋅52.5⋅1i −$−4⋅4$.%2⋅1i−5

f !;5 =f A

;5

7ote that volatilit%9 or sensitivit% to changes in interest rates is negative. his

confirms the principle that bond prices move inversel% 0ith changes in interestrates.

f !;;i=30⋅100⋅1i −$

f A; i =42⋅52.5⋅1i −820⋅4$.%2⋅1i −%

f A;;i ≥f !

;;i


$a%e ,2 ' 107





+o it appears that all the requirements have been met. $etBs consider 0hat happens

the the assets and liabilities if 0e increase or decrease interest rates. Ifimmunisation is wor!ing, we should ma!e no profit or a small profit 'A9( inboth cases.

6or i ) :;

Chec that

f ! 4 =82.19

f 4 =41.49

f - 4 =40.$0

f A 4 =f - 4 =82.20≥f ! 4 =82.19

6or i ) <;

Chec that f ! %=$4.$3

f %=3$.01

f - % =3$.$2

f A % =f - 4 =$4.$3≥f !% =$4.$3

For both an interest rate increase and decrease9 our assets are higher or not lo0er:than the liabilities. hus9 0e are immune to small changes in interest rates.


$a%e ,& ' 107




"ectio * : Mismatch eample

MISMATCH E;AM'LE

ae an immediate annuit% for a life aged "*. he ,> for this liabilit% is 2* at an

interest rate of 4G. he reserve or liabilit% is +,8*9***. Ee invest the assetsbacing the reserve into a Qero Coupon )ond 0ith term of ten %ears.

1. Ehat is the ,> of the QC)D

2. Ehat is the 5olatilit% of the QC)D

3. Ehat is the 5olatilit% of the annuit% liabilit%D

4. Ehat happens if interest rates increase to 4."GD

". Ehat happens if interest rates decrease to 3."GD

#. Ehat practical implications does this haveD

&. Ehat should %ou do to reduce this risD

8. Eh% might it be difficult to perfectl% match the liabilit% cashflo0s arising from

annuit% productsD

1. 1*

2. −10

1.04=−9.%2

3. −20

1.04=−19.23

4. Assets decrease to &#92"4. $iabilities decrease to &29#83. Ee mae a profit of

39"&1.

". Assets increase to 839'"*. $iabilities increase to 889*'". Ee mae a loss of 4914".

#. he compan% is eposed to interest rate ris since changes in interest rates havethe potential to cause fluctuations in financial performance. Further9 0e should

be particularl% concerned about the potential for interest rate decreases since this 0ill lead to a loss.


$a%e ,( ' 107




"ectio * : Mismatch eample

&. Ee should tr% to match our liabilities more closel% b% selling the 1*!%ear bond

and bu%ing longer term fied interest securities. he first step should be to havethe ,> of assets ; to that of liabilities. 7et steps 0ould be to consider perfect

cashflo0 matching9 or attempting immunisation. >ore complicated hedgingstrategies such as entering into derivative contracts such as s0aps could beconsidered.

8. Annuit% cashflo0s have a long average term. It might be difficult to find assets inthe maret 0ith sufficientl% long term. his 0ill prevent ,> matching9cashflo0 matching and immunisation.

/ven if available9 the assets ma% be ver% epensive due to large demand and

limited suppl% of these long!term instruments. /ven if the epected annuit%cashflo0s could be matched perfectl% matched9 the annuit% cashflo0s dependon mortalit% ris. If polic%holders eperience different mortalit% from assumedeither due to incorrect or basis or simpl% random fluctuations: then the actual

cashflo0s 0ill not be matched. his could lead to profits or losses9 but 0ouldalso require a rebalancing of the portfolio to match the updated cashflo0s. his

rebalancing can be epensive and the updated assets ma% not be available.

(erfect cashflo0 matching also requires the premiums received from ne0policies to be invested in eactl% matching assets as soon as the funds are

received. In practice9 ne0 business cashflo0s are matched monthl% in large9fairl% sophisticated companies. A fe0 companies match ne0 business on a 0eel%

basis. Again9 the costs of this frequent rebalancing can be large.


$a%e ,) ' 107




Course Notes

Chapter 2 'ension fund reser)in+

About

This chapter covers the basic actuarial techniques used for the calculation

of retirement fund reserves.

Chapter Contents

1 he 7ature of (ension Funds................................................................................ .......'&

1.1 ,efined )enefit ,): structures...................................................... .....................'&1.2 ,efined Contribution ,C: structure........................................................... .......'8

1.3 >ovement to0ards ,efined Contribution..........................................................'8

1.4 +pouse benefits.................................................................................... .................''

2 (re!funding versus cash!based approaches..............................................................1**

2.1 Cash or ?pa% as %ou go@ s%stem......................................................... ...................1**

2.2 Complete full funding from inception.................................................... ............1*1

2.3 (ractical pre!funding s%stems......................................................................... ....1*2

3 (roJected nit Credit >ethod............................................................................ ........1*3

3.1 eserving for benefits in pa%ment..................................................................... ..1*3

3.2 eserving for active emplo%ees....................................................... ....................1*3

3.3 Change in reserve.................................................................. ..............................1*"



Chapter 7 : $esio !ud reservi%

"ectio 1 : .he Nature o! $esio Fuds

1 THE "ATURE %F 'E"SI%" FU"&S

1#1 &efined !enefit 6&!9 structures

%.%.% Typical defined benefit structure

A defined benefit scheme is one 0here the benefit is defined and guaranteed b%the emplo%er. A t%pical benefit definition 0ould be

Pensin &#A=Salar/ &#A⋅Servi(e⋅ p

0here (ension7A is the annual pension pa%able at 7ormal etirement Age

+alar% 7A is the final salar% paid at normal retirement age

+ervice is the number of %ears service to the emplo%er

pG is the specified percentage9 usuall% some0here bet0een 1."G and 2."G.

his could also be defined as

Pensi.n0=Salar/ &#A⋅Servi(e⋅ p 0here (ension* is the pension pa%able *

%ears after 7ormal etirement Age.

he eact notation is not as important as the concept and the calculation. +ou mustconsider how the terms are defined and use them appropriatel% or define %our o0n.

7o equation is useful unless all terms have been appropriatel% defined. his is ageneral comment that applies across all sections and should appl% to %our other

courses too:.

Inflationar% increases in the pension pa%able during retirement are usuall% at thediscretion of the pension fund trustees and 0ill depend on available funds and

emplo%ee contributions. +ome level of increase either absolute or relative toinflation: ma% be implied or even guaranteed.

%.%.& Interpretation of typical defined benefit structure

For ever% %ear of service emplo%ment:9 the emplo%er promises to pa% pG of theemplo%ees final salar% as a pension once the emplo%ee retires at 7ormal etirement

Age.

If pG is 1."G and the emplo%ee 0ors for the compan% for 4* %ears9 the pension

pa%able is #*G of final salar%.


$a%e ,7 ' 107





If pG is 2."G and the emplo%ee 0ors for 3* %ears9 the pension pa%able 0ill be &"G

of final salar%.

he factor pG is part of each emplo%erBs remuneration pacage and is usuall% entirel%at the discretion of the compan%.

%.%.) Some variations on typical defined benefit structure

+alar% 7A ma% be an average of the last %ears of salar% often 3 %ears:. his reduces

the potential for manipulation and unequal treatment bet0een equal emplo%ees

7ormal etirement Age might be a band of ages rather than a single specific age9

0ith possibl% adJustments to the final benefit depending on actual retirement age

pG ma% ver% outside the usual range specified. pG ma% differ for different levels of

emplo%ees.

Additional %ears of service ma% be ?purchased@ through once!off contributions.

1#$ &efined Contribution 6&C9 structure

A defined contribution pension scheme is one 0here the contributions aredefined clearl%. he actual benefit paid depends entirel% on the investmentperformance of the underl%ing assets. his scheme operates as a savings vehicle 0ith

no guarantees made in terms of replacement of actual final salar%.

At retirement9 the value of accumulated funds is used to purchase a retirementincome product often from an insurance compan%:. ,epending on the eact legal9ta and regulator% environment9 some of the accumulated funds ma% be taen as

cash.

Ehen might it mae sense for a retiring emplo%ee to tae a portion in cash ratherthan in a lifetime annuit%D

Ehen 0ould it not mae senseD

1#* Mo)ement to/ards &efined Contribution,) schemes represent significant riss to the emplo%ers. >an% prominent companiesface severe difficulties in financing their ,) liabilities. he emplo%ers are eposed to

longevit% ris

investment ris

As a result9 man% companies are moving to0ards ,C schemes9 0here ris istransferred to the emplo%ees.


$a%e ,+ ' 107





,C schemes involve little actuarial valuation 0or. he liabilities are retrospectivel%

accumulated fund values and do not require discounted cashflo0 valuationtechniques.

he rest of this chapter deals 0ith the valuation of ,) schemes onl%.

1#, Spouse benefits

>an% pension schemes provide a spouse=s pension on death of the emplo%ee eitherduring emplo%ment9 during retirement or both. he spouseBs pension 0ill usuall% be

smaller than the emplo%ees pension.

1. Eh% might an emplo%er offer a spouseBs pensionD

2. Eh% might the spouseBs pension be lo0er than the emplo%eeBs o0n pensionD

3. Eould the spouseBs pension be different before and after the %ear in 0hich the

emplo%ee 0ould have reached normal retirement ageD

1. he emplo%er might offer the benefit as a 0a% to attract top emplo%ees to the

compan%. Also9 if the compan% did not offer the benefit9 and an emplo%ee 0ith afamil% died in emplo%ment especiall% shortl% before retirement: and not benefit

0as paid to the famil%9 the bad publicit% might force the compan% to pa% the

benefit in an% event. hus9 man% companies elect to offer the benefit in the firstplace.

2. he emplo%eeBs pension is intended to provide for hisLher income in retirement9

and a proportion for his famil%. -n the death of the emplo%ee9 he part intendedfor hisLher o0n income is no longer required9 and onl% the portion intended for

hisLher famil% is required. Also9 b% offering a lo0er spouseBs pension9 the cost of the benefit is reduced 0hich allo0s a higher salar% or emplo%eeBs pension to be paid at the same cost tothe compan%.

3. )efore 7A9 it is more liel% that the emplo%ee 0ill have a %oung famil% 0ithman% financial obligations. After retirement9 it is more liel% that maJorhousehold debts 0ill be paid off. his is Just an eample9 the eact nature of thebenefits 0ould depend on the rules of the pension fund as stipulated b% the

emplo%er or countr%!specific la0.


$a%e ,, ' 107




"ectio 2 : $re8!udi% versus cash8-ased approaches

$ 'RE-FU"&I"( >ERSUS CASH-!ASE& A''R%ACHES

(ension funds can be funded paid for: at different points in time. In this section9 0e

first address t0o etreme methods ?pa% as %ou go@ and complete full funding frominception: before describing the more moderate approaches more often adopted in

practice.

$#1 Cash or 7pa. as .ou +o8 s.stem

nder the cash s%stem9 pension benefit pa%ments to retired emplo%ees are paid 0hendue9 and no prior reserves are held. /ach pa%ment is an epense to the emplo%er

0hen it is paid.

he ris 0ith this approach to emplo%ees is that if the emplo%er can no longer meet

these obligations9 the% have no securit% and 0ill liel% lose their pension.

nder 0hat conditions might the pension be relativel% safeD

nder 0hat conditions might the pension be severel% at risD

A state government!baced: pension might be relativel% safe 0hile the population

is gro0ing rapidl%9 such that a large base of active 0orers are contributing to0ardsthe income of pensioners. o0ever9 as soon as this population gro0th slo0s9 the

?population p%ramid@ becomes a population column or even an inverted p%ramidand a small number of active 0orers must support the income of a large number of

pensioners. At this point9 the sustainabilit% of the s%stem is at ris.

he pension fund of a compan% depends similarl% on the ratio of active emplo%ees

to pensioners. As long as the compan% continues to gro0 in terms of number ofemplo%ees9 the funding requirements are lo0. o0ever9 inevitabl% this changes and

the pension 0ill become onerous to finance.

(a%!as!%ou!go s%stems are usuall% at ris in the long term.

A cash s%stem is used b% some companies in practice. It is more often used b% thegovernments of countries rather than individual countries.

Is a cash!based s%stem more safe or less safe if the sponsoring emplo%er is a

governmentD

Ehat is the current problem 0ith the nited +tates +ocial +ecurit% +%stemD


$a%e 100 ' 107





If the government is bacing the pension fund9 the% should be more able to pa% the

pensions as the% can raise additional funds through higher taes. o0ever9 since itis still the 0oring population that are pa%ing the taes9 the burden on the active

0ors can become high if the population gro0th rate slo0s do0n.

he + +ocial +ecurit% +%stem has man% problems. -ne of these is the high gro0thin the number of people claiming from the s%stem relative to the gro0th of those

contributing to the s%stem.

$#$ Complete full fundin+ from inception

,his approach is not adopted in practice.he emplo%er calculates the full present value of pension pa%ments due 0hen anemplo%ee Joins the firm. his amount is set aside as a reserve. he full cost of theentire pension is recognised at the date on 0hich the emplo%ee Joins the compan%.

Eh% is this liel% to be unpopular 0ith emplo%ersD

Are there an% riss left to the emplo%erD

Is this an accurate reflection of the cost of emplo%ee benefitsD

A ver% high9 upfront cost at the point of hiring an emplo%ee 0ill decrease profits 0hen emplo%ees are hired. his acts as a disincentive to hire emplo%ees 0ith full

benefits.

here are considerable riss left to the emplo%er. he assets bacing the pension funare eposed to maret fluctuations. A decline in asset values 0ill require additionalfunds from the emplo%er. Further9 if emplo%ees live longer than epected and

require an income after 7A from the compan% for a longer period than epected.

hese cashflo0 riss also give rise to assumption and valuation riss! at the point at 0hich future the compan% recognises changes in epected mortalit% or investmentreturns9 the reserves 0ill change 0ithout a change in asset values.

7o9 the cost of emplo%ee benefits should be matched 0ith the service provided b%the emplo%ees to the emplo%er. nder this method9 the entire epected cost isrecognised upfront.


$a%e 101 ' 107





$#* 'ractical pre-fundin+ s.stems

In practice9 man% pension funds are graduall% funded over the period of an

emplo%eeBs service such that at retirement9 the full pension has been reserved for.

#eserve0=0

#eserve &#A= Full PV .f pensi.n pa/ments

7A here means normal retirement age. he rate and pattern at 0hich

#eserve0 gro0s to #eserve &#A depends on the funding approach chosen.

In the net section 0e 0ill discuss the pro>ected unit credit method 0hich is one

of the more common approaches.


$a%e 102 ' 107




"ectio & : $ro<ected it Credit Method

* 'R%3ECTE& U"IT CRE&IT METH%&

he (roJected nit Credit >ethod is one popular method used to reserve for pre!

funded pension funds. It is covered here as an eample of an appropriatemethodolog%. -ther methods are not covered in this course but involve similar

concepts and techniques.

*#1 Reser)in+ for benefits in pa.ment

).%.% 7eserving methodology

he reserving methodolog% for benefits in pa%ment are the same as those used in

valuing an immediate annuit%. he epected present value of future cashflo0s iscalculated9 allo0ing for

probabilit% of survival

inflationar% increases in pension pa%ments

discount rates based on available returns in the maret and taing intoconsideration the assets bacing the pension liabilities.

).%.& 0conomic assumptions required

nominal discount rate and pension pa%ment inf lation rate

-

real discount rate

).%.) emographic assumptions required

base mortalit% b% gender9 and sometimes class of 0orer:

mortalit% improvements

*#$ Reser)in+ for acti)e emplo.ees ).&.% 7eserving methodology

he reserving methodolog% involves t0o e% steps

1. he first step is to calculate the epected present value of the pension pa%mentsbased on proJected final salar%.

2. he second step is to determine 0hat proportion of this liabilit% has been accruedbased on past service compared 0ith epected total service.


$a%e 10& ' 107





It is step 2 above that defines the (roJected nit Credit >ethod differentl% from

alternative pension reserving methodologies.

#eserve=EPV future pension p&ments ⋅ p&st servi,eexpe,te" tot&l servi,e

>ore rigorousl%9 0e might define this as

#eserve x=v

&#A− x⋅ salar/ x⋅1e &#A− x⋅ p⋅ &#A− %A ⋅a

&#A

2 ⋅ x− %A

&#A− %A

#eserve x= salar/ x⋅1e

1i &#A− x

⋅ p ⋅a &#A

2⋅ x− %A

where v=1i −1

i=nomin&l "is,ount r&tee=s&l&r infl&tion

salar/ x=,urrent s&l&r for emploee &ge" x

2=1i

1 3 −1=

1i−1 3

1 3 =

i− 3

1 3

3 = pension in,re&ses in retirement

x=,urrent &ge

E)=&ge &t st&rt of servi,e or emplomenta &#A

2=imme"i&te &nnuit in &"v&n,e &t "is,ount r&te 2

).&.& 0conomic assumptions required

nominal discount rate and salar% inf lation rate

A7,

nominal discount rate and pension pa%ment increase rate

-

real discount rate usuall% different rates for the period before and after retirement

).&.) emographic assumptions required

base mortalit%

mortalit% improvements


$a%e 10( ' 107





*#* Chan+e in reser)e

From t to t19 the reserve required 0ill change for both retired emplo%ees and active

emplo%ees.

).).% *hange in reserve for pensioners

he change in reserve here operates lie an immediate annuit%. he factors affectingthe change in reserve are

un0ind of the discount rate

benefit pa%ments made

+urvival * or 1 for each person: to net period actual versus epected mortalit%:

,ifference bet0een assumed pension increases and actual pension increases

declared actual versus epected pension increases:

Changes in mortalit% basis

Changes in the economic basis

For each of these factors9 mae sure %ou understand

1. Eh% it impacts the reserve

2. In 0hich direction it impacts the reserve

3. o0 large or frequent the changes are liel% to be

4. o0 the compan% might manage the riss of these factors

he ?un0ind of the discount rate@ is the increase in the reserve as a result of thepassage of time since future cashflo0s are discounted b% less time. his is not a ris

since it 0ill happen over time 0ith certaint%.

Ehen benefit pa%ments are made9 the% are in the past. +ince the reserve taes into

account future cashflo0s9 the% are no longer part of future cashflo0s and thusreduce the reserve. his also happens 0ith certaint% and so no ris management isrequired.


$a%e 10) ' 107





/ach pensioner has a survival probabilit%. o0ever9 each pension 0ith either die or

survive. For ever% pensioner that survives9 the reserve 0ill increase since mortalit% 0as lighter than epected: and for ever% pensioner that dies the reserve 0ill

decrease since mortalit% 0as heavier than epected:. If the total number ofpensioners that survive matches epectations9 the overall reserve 0ill not change asa result since actual deaths are the same as epected. )% having a large number ofpensioners9 the random fluctuations 0ill average out and the overall variabilit% of

reserves due to actual mortalit% being different from epected 0ill decrease. Accurate reserving assumptions 0ill help to ensure that overall mortalit%

epectations are accurate. rends such as annuitant mortalit% improvements:should be taen into consideration.

If actual pension increases are higher than assumed9 then the reserve 0ill increasesince the epected future cashflo0s 0ill no0 be higher than previousl% epectedsince the base current pension: is higher than previousl% assumed. his 0illgenerall% occur annuall% 0hen pension increases are decided. he approach

follo0ed to setting the pension increases should be taen into consideration 0hendetermining the pension increase assumption. Another form of ris management is

?automatic@ since a e% input into increases declared 0ill be the surplus available inthe pension fund assets less liabilities:. hus9 high pension increases 0ill result in

an increase in pension liabilities reserves: but this 0ill often correspond 0ith aprevious increase in asset values from good investment performance. hus9

although liabilities reserves: 0ill increase9 the surplus 0ill not be inappropriatel%decreased.

An% change in basis 0ill affect the reserves through a change in epected futurecashflo0s. he basis 0ill usuall% be considered annuall%. o0ever9 changes ma% be

made less frequentl%. he impact of economic basis changes can be managedthrough matching assets and liabilities such that assets and liabilities move in line

0ith each other. Changes to the mortalit% basis are more difficult to hedge. Careshould be eercised 0hen setting the basis in the first place9 and spurious changes

unnecessar% small changes that might be reversed in future: should not be made.

).).& *hange in reserve for active employees

he change in reserve for active emplo%ees is an important part of the actuarialcalculation for ,) schemes.

he change can be epressed as

! #eserve t = #eserve t 1 – #eserve t


$a%e 10* ' 107





he actual change depends on the approach adopted to move from #eserve0=0

to #eserve &#A=PV of pension enefits .

his ma% be considered along 0ith the Anal%sis of (rofit in A>III. his 0ill notform part of A>II.

Documents

Am-II Course Notes Full