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Financial Mathematics


Jonathan Ziveyi1

1University of New South Wales

Risk and Actuarial Studies, Australian School of Business

[email protected]

Module 1 Topic Notes

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Plan

Module 1: Time Value of Money and Valuation of Cash FlowsCash Flow ModelsA Mathematical Model of InterestSimple and Compound InterestDiscount InterestNominal InterestForce of InterestReal and Money InterestRelation between Cash Flow, Interest and Present ValueAnnuities: IntroductionTerm AnnuitiesNon-Level Annuities

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Module 1: Time Value of Money and Valuation of Cash Flows

Cash Flow Models

Plan


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Cash Flow Models

Cash flow models A cash flow is a series of payments (inflows oroutflows) over a period of time.A mathematical projection of the payments involved in a financialtransaction is referred to as a cash flow model.

Cash flows are characterised by their:

nature: inflow or outflow

amount

timing

probability (if contingent)

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Cash Flow Models

Comparing cash flows We want to compare different sets of cashflows:

why? compare 2 securities or investments compare scenarii for a given product (product development,

profit testing, solvency) compare potential new products (product development) etc. . .

how?

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Cash Flow Models

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Cash Flow Models

Procedure:

make the cash flow clear; draw a time diagram

choose any point in time now: present value, sometimes NPV (Net Present Value) in the future: accumulated value in the middle... should be convenient: all are equivalent!

"bring back or forth" all cash flows to the point of time youhave chosen

add them up

compare!

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Cash Flow Models

You want to buy a television from Bling Bees on 31/12/2009 thatis worth $3000. The super mega deal (yeahh) is that you can takethe television now and need only to pay $1000 on 31/12/2011 and$2000 on 31/12/2012. Their advertisement campaign is "Nointerest, no deposit until 2011!".But you are smart (of course, you are an actuary), and you knowthat if Bling Bee invests $1000 now, this investment will be worth$1100 in one year, $1210 in two years and $1331 in three years.Taking this information into account, what discount can youreasonably get from Bling Bee?

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A Mathematical Model of Interest

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Time value of money How much would you pay to buy a securitythat is guaranteed to give you $100 in 1 years time?

What if there was a chance of default?

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Time is money!

Interest is a mathematical tool to embody

the time preference of agents in the economyusually, agents are impatient (interest is positive)

risk (interest is raised to include a risk premium)

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Mathematical modelConsider an amount of money invested for a period of time.

A(0): principal = the amount of money initially invested

t: the length of time for which the amount has been invested

A(t): amount function or accumulated amount function this is the accumulated amount of money at time t

corresponding to A(0)

Assuming these are two equivalent cash flows at two different pointin time, how can we link them using interest?

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Accumulation function Let a(t) be the accumulation function:

a(t) the accumulated value at time t of an original investmentof 1 made at time 0

it is a scaled version of A(t) with a(0) = 1 and can thus bestudied independently of the amounts that are invested

it represents the way in which money accumulates with thepassage of time

We haveA(t + k)

A(t)=

a(t + k)

a(t),

which means

A(t + k) = A(t)a(t + k)

a(t).

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Effective Interest In mathematical terms, the effective interest It,kaccumulated between t and t + k (for a period k from t) is

It,k = A(t + k) A(t),

and then the effective rate of interest it,k for this same period is

it,k =A(t + k) A(t)

A(t)=

a(t + k) a(t)

a(t). (1.1)

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Homogeneity in time When the effective rate of interest is the samefor all t, then we have

a(t + k)

a(t)=

a(k)

a(0)= a(k)

A(t + k) = A(t)a(k)

it,k =A(t + k) A(t)

A(t)=

A(t + k)

A(t) 1

=a(t + k) a(t)

a(t)= a(k) 1.

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Forms of interest

a(t) is modeled with the help of interest

effective interest is always defined as in (1.1)

however, interest can be expressed in many different ways,depending on the situation (mainly conventions)

each way has a different set of assumption

each definition may lead to different forms for a(t)

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Assumptions about interest

1. how much interest is paid? usually expressed as a percentage per year (per annum, p.a.) amount can depends on the time period (inhomogeneity) term structure of interest, see module 4 non-constant force of interest

amount is sometimes stochastic deterministic vs stochastic interest, see module 6

2. how often is interest paid? as a rule, once per compounding period, whose number per

time unit needs to be determined (usually once a year) simple vs compound interest (time horizon) nominal vs effective interest (several times a year) force of interest (continuously)

3. when is interest paid? at the beginning or end of the compounding period discount interest (beginning)

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Simple and Compound Interest

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Simple and Compound Interest Example: Assume John deposits$1000 on his bank account on 01/01/2010 at an effective rate ofinterest of 5% p.a. At the following dates:

1. what is the balance of his account?

2. how much would he get if he closed his account?

3. how much interest has he earnt?

4. how much interest has been credited on the account?

30/06/2010

01/01/2011

30/06/2011

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Simple and Compound Interest Main difference:

with simple interest: no interest is ever earnt oninterestinterest is not compounded

with compound interest: interest is continuously earnt oninterestinterest is compounded

When to use one or the other?

What happens within a year (compounding period) is usuallysimple interest (short term securities, T-bills, . . . )

However, simple interest is not homogeneous in time

For cash flows spanning over periods of more thana year, compound interest is generally used (easier..!)

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Simple Interest Accumulation function: for simple interest i ,

a(t) = 1+ it,

and the accumulated amount function after a period t is given by

A(t) = A(0) a(t) = A(0) (1+ it).

Usually, t < 1 (days/360 or 365, or months/12).

Effective rate of interest is not constant in this case (decreasing):

a(t + k) = (1+ i(t + k)) 6= (1+ it)(1+ ik) = a(t)a(k)

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Numerical Example A Bank accepts deposits for terms up to 3years and pays interest on maturity. How much interest would itpay on a deposit of $20,000 for a term of 1 year and 33 days if theinterest rate is 5% p.a. simple?

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Compound Interest Accumulation function: for compound interesti ,

a(t) = (1+ i)t ,

and the accumulated amount function after a period t is given by

A(t) = A(0) a(t) = A(0) (1+ i)t .

In this case, effective interest is homogeneous:

a(t + k) = (1+ i)t+k = (1+ i)t(1+ i)k = a(t)a(k)

or alternatively

a(t + k)

a(t)= (1+ i)k = a(k), t, k 0.

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Numerical ExampleA Bank accepts deposits for terms up to 3 years and pays intereston maturity. How much interest would it pay on a deposit of$20,000 for a term of 1 year and 33 days if the interest rate is 5%p.a. effective?

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General questions

1. What happens to the accumulation if i ? i ?

2. What is the amount of interest earned during each unit periodunder compound interest? simple interest?

3. What is the effective rate of interest during each unit periodunder compound interest? simple interest?

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Discount Interest

Plan


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Discount Interest







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Discount Interest

Rate of Discount The rate of interest i applies to the principal now,for interest calculated at t = 1, whereas the rate of discount dapplies to the principal at the end of the period, for interestcalculated at t = 0.In other words, for i :

we focus on the principal now

we correct this figure by adding interest at the end of theperiod

and for d :

we focus on the principal at the end of the period

we correct this figure by subtracting interest now

Both methods are equivalent, and use of one or the other isdictated by the situation for convenience.

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Discount Interest

Remember: i =a(1) a(0)

a(0)

=A(1) A(0)

A(0)= a(1) = (1+ i)

Now: d =a(1) a(0)

a(1)

=A(1) A(0)

A(1)= a(1) =

1

1 d

Financial reasoning:

At rate of compound interest of i% p.a. the discounted valueof an instrument is known. Is the compound rate of discountthat produces an equivalent discounted value higher or lower?

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Discount Interest

Simple vs compound discount interest Since we want d and i to beequivalent (they are just formulated differently), we have

1+ i =1

1 d.

For simple interest:

a(t) = a(0)1

1 dt

and for compound interest:

a(t) = a(0)

(1

1 d

)t

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Discount Interest

Numerical ExampleIn the US Treasury Bills are quoted using simple discount on thebasis of a 360 day year.Consider a US T-Bill with a face value of 500,000 and maturity in180 days time. Suppose that this is sold to yield 6%p.a (simplediscount). What are the proceeds of the sale?

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Discount Interest

Relations between Interest and Discount i is the effective rate ofinterest, d is the effective rate of discount and v = 1/a(1) is thediscount factor.Show these are correct as an exercise and use financial reasoning.

i =d

1 d

d =i

1+ id = iv

d = 1 v

i d = id

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Discount Interest

Intuition behind d = 1 v?

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Discount Interest

Intuition behind i d = id?

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Nominal Interest

Plan


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Nominal Interest







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Nominal Interest

Nominal Interest Rate

usually the compounding period is one year

nominal interest rates are interest rates that are still expressed as % p.a. but that have several (m) compounding periods per year

Example of securities for which nominal rates are relevant: Some bonds pay interest yearly, some semi-annually and some

quarterly Home loans usually charge interest monthly Some bank accounts pay interest daily

Notation: i (m) nominal interest rate, payable mthly

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Nominal Interest

Nominal vs effective rates With nominal interest rates, the rate

i (m)

m

is an effective rate of interest for a period of 1/m years.Reminder:

the effective rate of interest for a period is the ratio between

1. the effective (actual amount of) interest earned and2. the principal at the beginning of the period (for interest) or at

the end of the period (for discount).

i (m), m > 1, is not an effective rate of interest

i is the effective rate of interest for a year, equivalent to i (m)

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Nominal Interest

Relationship to i , the effective interest rate In general, for rate i perannum

(1+ i) =

(1+

i (m)

m

)m

i (m) = m[(1+ i)1/m 1

]

i (m)i ??

Can you use your financial reasoning to convince yourself which iscorrect?

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Nominal Interest

Numerical Example A product offers interest at 8% p.a., payablequarterly. What is the effective annual rate of interest implied?

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Nominal Interest

Nominal Discount Rates

Interest is converted m times per year (period)

Notation: d (m) nominal discount rate converted mthly

Relationship to d the effective rate of discount

1

a(1)= (1 d) =

(1

d (m)

m

)m

Note that the nominal discount rate increases as the frequencyof conversion increases.

d < d (2) < d (4) . . .

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Nominal Interest

Exercise Show that d (m) = i (m)v1m

i

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Nominal Interest

Numerical Example Find the accumulated amount of $100 investedfor 15 years if i (4) = .08 for the first 5 years, d = .07 for the second5 years and d (2) = .06 for the last five years.

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Force of Interest

Plan


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Force of Interest







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Force of Interest

Force of Interest Consider

limm

(1+

i (m)

m

)m

= limm

1+m i (m)

m+

m (m 1)

2!

(i (m)

m

)2+ ...

= 1+ i () +

(i ()

)22!

+

(i ()

)33!

+ ...

= e i()

or e,

where is called the force of interest, or continuously compoundingrate of interest.

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Force of Interest

Force of Discount Similarly, consider

limm

(1

d (m)

m

)m

= limm

1m d (m)

m+

m (m 1)

2!

(d (m)

m

)2 ...

= 1 d () +

(d ()

)22!

(d ()

)33!

+ . . .

= ed()

,

where d () is the force of discount.

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Force of Interest

We have1 d = ed

()and 1+ i = e i

().

Now

1 d = v =1

1+ i.

Thus,i () = d ()

and, in general,

d < . . . < d (m) < . . . < d () = = i () < . . . < i (m) < . . . < i .

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Force of Interest

Force of interest that varies with time Let

A(0) be the principal invested at time 0

interest be paid continuously at a rate (t) at time t

We seek an expression for A(t).

Interest paid over a small interval t is

A(t +t) A(t) A(t)(t)t

and thus

(t) A(t +t) A(t)

A(t)t.

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Force of Interest

Taking the limit, as t 0,

(t) = limt0

A(t +t) A(t)

A(t) t

=1

A(t)d

dtA(t)

=A(t)

A(t)

=d

dtlnA(t).

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Force of Interest

Integrating both sides over [0, t],

ts=0

(s)ds = lnA(s)|t0

= lnA(t) lnA(0)

= lnA(t)

A(0).

Thus we have

A(t) = A(0) exp

[ t0

(s)ds

]

and

a(t) = exp

[ t0

(s)ds

].

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Force of Interest

Numerical Example Force of interest at time 0 is 0.04, andincreases uniformly to 0.06 after 5 years. Find the amount after 5years of an investment of $1.

For affine (t), the integral in exponential can be simplified:

a(t + k)

a(t)= e

k

2[(t)+(t+k)]

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Real and Money Interest

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Inflation When comparing cash flows, time has to be accounted

1. because of the time preference of agents in the economy (riskfree interest)

2. because there is a risk of default (risk premium)

3. because the value of money changes over time:inflation/deflation

Inflation:

Inflation (deflation) is characterized by rising (falling) prices, orby falling (rising) value of money.

A common way of measuring inflation is the change inConsumer Price Index (CPI) which itself measures the annualrate of change in a specified "basket" of consumer items.

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Notation Let

i% p.a. be the money interest rate

r% p.a. be the real interest rate

p(t) be the price index (with P(0) = 1)

pi% p.a. be the inflation rate

What relationships can be established among i , r , P(t) and pi?

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Main relations We have

a(0) = 1 and a(1) = 1+ i .

andp(0) = 1 and p(1) = 1+ pi

Thus, the value of accumulation at todays prices is given by

a(1)

p(1)=

1+ i

1+ pi.

Now, define

1+ r =1+ i

1+ pi r =

i pi

1+ pi.

Caution: this holds only for effective rates!

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How to deal with inflation Inflation is introduced in calculationseither by

1. considering the cash flow at its dates $ (nominal value) anduse money interest rates:

A(0) = A(t)

(1

1+ i

)t

2. or adapting the amounts of cash flows to todays dollars (realvalue) and use a (modified) real interest rate:

A(0) =A(t)

(1+ pi)t

(1

1+ r

)t

Both methods are equivalent.

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Example An investor will receive an asset in 10 years time with facevalue $100,000. Given a nominal (money) interest rate of 9% p.a.,quarterly compounding, and an expected inflation rate of 5% p.a.,(also quarterly compounding), what should you pay now:

Asset 1 if the payment on the asset will not change, failing to increasein line with inflation

Asset 2 if the asset maintains its real value (an inflation indexed bond)

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To determine the price, we must be consistent. Either we work with

Method 1: the nominal value, and discount with the moneyinterest rate, or

Method 2: the real value, and discount with the real interestrate.

The effective real quarterly rate is

.09/4 .05/4

1+ .05/4= 0.9876543%

Thus, r (4) = 3.9506% and r = 4.0095%.

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Asset 1 Method 1:

PV =100, 000(1+ .09

4

)40= 41, 064.58

Method 2:

Real value =100, 000(1+ .05

4

)40 = 60, 841.33PV =

60, 841.33

(1.00987654)40= 41, 064.57

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Asset 2 Method 1:

PV =100 000

(1+ .05

4

)40(1+ .09

4

)40= 67 494.53

Method 2:

PV =100, 000

(1.00987654)40

= 67, 494.54

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Relation between Cash Flow, Interest and Present Value

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Our fundamental problem Three inter-related (sets) of values:

a set of cash flows (inflows and outflows, timing, probability)

interest and its assumptions

a present value / accumulated value(for a security: the price / value at maturity)

Learning outcome A3:

Understand the relation between a present value, a set ofcash flows and interest, be able to determine one infunction of the others in a variety of situations, as well asunderstand the interest rate risk (duration, immunisation)

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Practical examples

Find the price of a security: determine an initial cash flow suchthat the NPV is 0, given interest and a set of cash flows

Find the yield of a security or a project: determine the rate ofinterest such that the NPV is 0 (IRR), given a set of cash flows

Find the minimum return on the reserves that is necessary toensure all current life annuities can be paid until the end, giventhe current level of mortality (pensions)

. . .

Note

If the NPV is 0, we have then an equation of value

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Examples of Common Financial Instruments

Cash on deposit - term deposits, cash management trusts

Notes: Treasury notes, promissory notes, bank bills

Equities - also known as shares, equity shares or common stock

Bonds: Coupon Bonds, Zero Coupon Bonds (ZCB),Government bonds

Annuities: annuities-certain, life annuities

Insurance applications: Term life insurance, Endowmentinsurance

See Broverman and Sherris for the main definitions and examples.

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Determine a PV in function of cash flows and interest The presentvalue PV of an amount A(t) accumulated at time t is given by

A(t) = PV a(t) PV =A(t)

a(t).

The present value or discount factor is then

v =1

a(1)=

1

1+ i

= 1 d

= d/i

Powers of the discount factor can be used to discount all cash flowsif interest is homogeneous with time(which is the usual assumption)

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Numerical Examples Example 1Consider a Coupon bond which pays $6 at times 1 and 2, and anadditional $100 at time 2. Find the Present Value of this bond at8% p.a. effective interest.

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Numerical Examples Example 2In Australia, Short term Government securities such as TreasuryNotes and Treasury Bonds (less than 6 months to maturity) arepriced using simple interest and a 365 day convention.Consider a Treasury-note with a face value of 500,000 and maturityin 180 days time. Suppose that this is sold at a yield (interest rate)of 6%p.a. What are the proceeds of the sale?

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Determine interest in function of a PV and cash flows If there aremore than 2 cash flows, it is generally impossible to solve forinterest analytically.

In that case, several approaches are possible:

use a financial calculator

use a computer (R, Goal Seek in Excel, etc. . . )

use a numerical method (e.g. Newton-Raphson)(the method to be used in quizzes and in the final exam)

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Newton-Raphson method (a recursive numerical method) (see, e.g.http://en.wikipedia.org/wiki/Newtons_method)

f (in) =f (in)

in in+1 in+1 = in

f (in)

f (in)

1. determine f (i) such that f (i) = 0

2. determine f (i)

3. choose initial value i0

4. perform recursions

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Numerical Example A Bond pays $100 in 1.5 years. Couponpayments of $5 are payable times 0.5, 1, and 1.5

1. Find the Price of the Bond if the yield is i (2) = 6%.

2. Suppose the Price of the Bond is 107.14. Find the Yieldimplied by this price.

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Annuities: Introduction

Plan


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Notation

m| a(p)x :n i

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Our main tool to value annuities-certain: the perpetuity

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Numerical example A foundation has $100,000,000. Assuming along term net return on investments of 5% p.a., how much moneycan it use every year without decreasing the capital?Determine the annual payment if it is made in arrears or inadvance, and in the two situations:

1. the capital should not decrease in nominal terms

2. the capital should not decrease in real terms

Assume a long term inflation rate of 3% p.a.

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Numerical example

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Term Annuities

Plan


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Term Annuities

Annuity-immediate (paid in arrears)

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Term Annuities

Numerical example A Bond pays $100 at time 3. Coupon paymentsof $5 are payable at times 1, 2, and 3

1. Find the Price of the Bond if the effective yield is i = 5%.

2. What is the Price if the effective yield is 6%?

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Term Annuities

Annuity-due (paid in advance)

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Term Annuities

Numerical example A Bond pays $100 at time 2. Coupon paymentsof $5 are payable times 0, 1, and 2. (i.e. the first payment occursimmediately after purchase).

1. Find the Price of the Bond if the effective yield is i = 6%.

2. What is the Price if the effective yield is 5%?

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Term Annuities

Deferred annuity

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Term Annuities

Numerical example Consider a Bond pays $100 at time 6. Couponpayments of $5 are payable times 4, 5, and 6. How much wouldyou be willing to pay to purchase the bond today? Assume i = 6%.

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Term Annuities

Payments more frequent than a year

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Term Annuities

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Term Annuities

Alternative method Alternatively, find the effective pthly rate ofinterest,

j = (1+ i)1/p 1 =i (p)

p.

Then

a(p)n i =

1

panp j

where j = (1+ i)1/p 1.

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Term Annuities

Numerical example Payments of 10 made at end of each month fornext 5 years. Calculate their present value at (i) 8% p.a. effective,and (ii) 8% p.a. convertible half-yearly.There are at least two ways to do these questions:

1. work according to cash flows and change i

2. work according to i (p) and change cash flows

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Term Annuities

Numerical example

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Non-Level Annuities

Plan


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Non-Level Annuities

Increasing annuity (arithmetic progression)

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Non-Level Annuities

Numerical example Value the following set of cashflows at 10%p.a.: A payment of$10 at time 1, $20 at time 2, $30 at time 3, $40at time 4. What is the present value at time t = 0?

What is the present value at time t = 1?

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Non-Level Annuities

Increasing annuity (arithmetic progression): general case

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Non-Level Annuities

Numerical example Value the following series of payments at 10%p.a.:

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Non-Level Annuities

Increasing annuity (geometric progression)

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Non-Level Annuities

Numerical example You can invest in a bond that pays couponsthat grow with inflation. The coupon received at the end of thefirst year is $25,000, and each annual payment will increase, withinflation, at rate 2.5% p.a. There are 10 annual payments and thebond matures in 10 years with a face value of $400,000 (notindexed to inflation). What is the price of the bond at a valuationinterest rate of 8%p.a.?

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Non-Level Annuities

Increasing annuity with p payments per annum

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Non-Level Annuities

Numerical example Determine the present value of a 10 yearannuity with half-yearly payments in arrears at rate 2 p.a. in thefirst year, 4 p.a. in the second year, . . . , 20 p.a. in the 10th year.Use a 10% p.a. convertible half-yearly compound interest rate.

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Non-Level Annuities

Decreasing annuity

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Non-Level Annuities

Numerical example Value the following set of payments at 10% p.a:$40 at time 1, $30 at time 2, $20 at time 3, $10 at time 4.

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