Variable Annuities

and

Policyholder Behaviour

Prof. Dr. Michael Koller, ETH Zürich

Quantact Workshop 9.3.2018

Aim

• To understand what a Variable Annuity is,• To understand the different product features and how they interact,• To understand the risk management of Variable Annuities and their hedging,• To understand the consequences of policyholder behaviour on valuation and hedging.

1

What is a variable annuity?

A Variable Annuity is a Fund(s) plus additional Insurance Benefits.

• Insurance Benefits can be of different forms:– In case of Death ; Guaranteed Minimum Death Benefit (GMDB),– For saving ; Guaranteed Minimum Accumulation Benefit (GMAB), and– In case of regular income (annuity) ; Guaranteed Minimum Withdrawal Benefit (for

Life) (GMWB/GLWB)• The product has tax benefits.

2

GMDB

GMDB Example • GMDB: 160’000, initial fund value: 20’000, policyterm 25 years.

• We consider two possible outcomes (trajecto-ries) and assume that the person dies after 17years. Then we have the following:

“green” “red”Fund Value 130’000 35’000Death Benefit 160’000 160’000Loss Insurer 30’000 125’000

• Hence the insurer needs to be able to sell theunderlying fund at 160’000 if its value is belowthis amount, independently of its value. This iscalled a put option with strike 160’000.

• In the good outcome (“green”), this guaranteehas a value of 30’000. In the bad outcome(“red”), the guarantee has a value of 125’000.3

Insurance Protection within variable annuities

Variable Annuity Insurance ProtectionGMDB (Death Benefit) Protection in case of deathGMAB (Accumulation Benefit) Policyholder survives a certain timeGMIB (Income Benefit) Policyholder survives a certain time, regular

incomeGMWB (Withdrawal Benefit) Temporary Annuity (potentially deferred)GLWB (Life Benefit) (Deferred) Annuity, longevity

4

GMAB - Ratchet 5 years

GMAB Ratchet Example 5y • We assume an initial fund value of 1M.• This rider to the insurance policy ensures that

the policyholder receives the maximum value ofthe fund attained in the past.

• This maximum is evaluated at regular time inter-vals.

• In case of a 5 yr ratchet, we have the high watermarks as follows:

Time Fund Value Ratchet0 1.00 1.002 1.00 1.145 1.07 1.147 1.07 1.25

. . .20 1.79 1.7924 1.79 1.92 5

Example GMWB

GLWB Example

• The Policyholder has invested 100,000 at age60 and bought a GLWB (Guaranteed MinimumWithdrawal Benefit for Life), with the right to with-draw 5% pa.

• The benefit base GWB has increased if GWB =

c116′000 because he did not withdraw beforeage 65, so can now withdraw up to c5’800 pa.

• The fund is depleted at age 85 and the guaran-tee kicks in.

• The expected guarantee (“yellow”) reduces asmore and more policyholders die.

6

Balance Sheet

VA Funds

VA Guarantees

SH Capital

Assets Liabilities Risks

• Asset Performance• Basis Risk• Hedging Risk

• Market Risk• Interest Rate• Credit

• Asset Performance

7

Methods to value variable annuities

Valuation Tree

t=2

.............

..............

........................

..............

................................. . . . . . . . .................................

............... . . . . . . . .

............... . . . . . . . .

ω1

ω2

ω3

ω4

ω5

ω6

ω7

ω8

ω9

(10,10)

(11,9)

(11,10)

(8,11)

(14,9)

(10,13)

(10,8)

(14,9)

(10,13)

(10,9)

(12,10)

(7,15)

(7,10)

p

q

r

a

b

c

d

e

f

ghi

t=0 t=1

• Explicit formula or recursion (only insurance val-uation and very simple variable annuities),

• Solution of Black-Scholes-Merton differentialequation (different methods including treemethod),

• Monte Carlo Simulation (this is the approachmost often used for variable annuities).

Monte Carlo is most commonly used for VA since itis very versatile and can also cope with verycomplex option structures, such as ratchets.

8

Steps in Valuation of VA

Determine the number of people which benefit: Since only a tiny percentage of thewhole inforce dies within a given year, one only needs to provide the respective GMDBcover to them. Similarly the GMAB cover is paid only to the people surviving the entireterm of the policy. Hence we need to determine the respective percentages. This isdone by means of life decrement tables.

Calculate what these people receive: Then we need to know what the respective policy-holders are entitled to. Assume, for example, the people dying aged 40. They areentitled to get a GMDB at a certain level. Hence we need to determine the number ofthe corresponding units of guarantees. For our 40 year old person this would be putoptions at a strike price.

Calculate the value: At this point we know the valuation portfolio of guarantees represent-ing the VA guarantee (eg number of instruments and their characteristics). We nowneed to value them. For our example this is done via the Black-Scholes formula.

9

Value of Guarantee – Trading Grid

Equity Level π δ γ ρ ν

-50 % 65777.63 -33818.56 39673.78 -2341692.30 86714.87-40 % 59397.95 -36051.50 42961.58 -2241519.81 111455.01-30 % 53734.40 -37319.40 44743.99 -2136639.66 133559.18-20 % 48710.53 -37826.27 45431.79 -2030460.53 152596.75-10 % 44254.99 -37744.13 45168.28 -1925345.90 168470.76-5 % 42219.51 -37531.90 44705.65 -1873714.26 175247.490 % 40301.71 -37230.02 44052.73 -1822905.27 181284.49+5 % 38494.06 -36854.69 43243.46 -1773034.67 186616.49+10 % 36789.41 -36419.96 42315.00 -1724192.41 191283.08+20 % 33662.33 -35417.69 40235.70 -1629845.32 198789.30+30 % 30871.11 -34295.33 38026.66 -1540199.53 204141.09+40 % 28372.90 -33102.03 35808.25 -1455379.92 207655.90+50 % 26130.91 -31872.16 33640.82 -1375369.59 209618.73

10

VA risks

Risk Landscape

Financial Risks relate to capital markets, such asequity risks, but also the ability to trade at a cer-tain point in time.

Policyholder Behaviour Risks relate to the be-haviour of the policyholder at a given point intime.

Insurance Risks relate to the pure demographicrisks such as mortality.

Other Risks summarise the remainder of riskssuch as a rogue trader, etc.

11

Short Term vs Longer Term Risks

Short Term Longer Term

• Equity price,• Interest rates,• Operational risk / key man risk,• Lapses,• Liquidity,• Basis Risk.

• Longevity,• Long term volatility,• Interest rates,• Policyholder behaviour (lapses, . . . ).

Need to monitor short term risk closely and continuously. Regular MI and respective riskappetite statements are necessary.

12

Aim of Hedging

Approximation

The lower figure shows the P&L (in yellow) for a givenhedging strategy (upper figure) with the aim to ap-proximate the value of a VA, which depends amongother things on:• Equity prices, equity volatility,• Interest rates,• Mortality,• Lapses, utilisation . . .

Aim of hedging: immunising balance sheet ofinsurance company.

13

Hedging Strategy

A hedging strategy needs to consider and establish objectives in respect of:Economic Risks: Which economic risks are hedged and to what extent?Financial Statement Risks: How important are the risks regarding the publicly stated ac-

counts and to what extent is there a need to hedge them?Regulatory Capital Risks: What are the regulatory capital risks and to what extent need

they be hedged?

A hedging strategy needs to establish objectives with respect to the different dimensionsand define a corresponding risk appetite.

14

Hedge Strategies

∆ f (S, t,r,σ) =∂ f∂S︸︷︷︸

Delta

∆S+12

∂ 2 f∂S2︸︷︷︸

Gamma

(∆S)2+∂ f∂ t︸︷︷︸

T heta

∆t

+∂ f∂ r︸︷︷︸Rho

∆r+∂ f∂σ︸︷︷︸Vega

∆σ + . . .

Trivial Hedge: Nothing is hedged and the insurance company keeps entire risk.δ -hedge: Only the equity part is hedged, no interest rate hedge. A δ–γ–hedge is a variant

of this, where equities are hedged more accurately than with a pure δ–hedge.δ–ρ–hedge: Interest rates and equities are hedged.3 greeks hedge: δ , ρ and the equity volatility ν is hedged.Macro Hedge: The aim is to hedge the tail (or big movement) risks, potentially however

trading-off protection against the accuracy of the hedge for smaller magitude move-mens.

15

Balance Sheet

VA Funds

VA Guarantees

Capital

Assets LiabilitiesLiabilities

t = 0

Hedging Loss

∆ Hedge Lia.

Red. in Value

Red. in Capital

Assets Liabilities

t = 1

4

3

2

1

16

Risk Management for VA in general

• The best way to reduce VA risk is normally to have outperformance in the underlying in-vestment fund. Hence the choice and monitoring of fund performance is very important.

• The only way to really mitigate the VA risks completely is to sell them to a third party,otherwise there are always remaining residual risks.

Risk management for VA is vital.

17

Policyholder behaviour

Change Asset Allocation: The policyholder can change his asset allocation and invest indifferent assets, which are more or less risky.

Top up investment: The policyholder can invest an additional amount in the underlyingfund. This can change the guarantees.

Lapse: The policyholder can end the policy.Start withdrawing: The policyholder can start to withdraw money from the fund.Change amount of withdrawal: Within a given period, the policyholder can decide to with-

draw more or less.Partial Surrender: Withdrawing more than regularly allowed.Sell Policy: He can sell the policy to a third party to monetize the value of the policy.

Policyholder behaviour is a risk that needs to be considered, in particular for the productdesign. One needs to avoid product designs, that promote cristallisation of losses for

many policyholders at the same time.

18

Value of Guarantee at Inception

Instrument Strike Amount %age Value

0 Put Fund 100000 0.1 % 8.71 Put Fund 107177 0.2 % 17.72 Put Fund 114870 0.2 % 27.73 Put Fund 123114 0.2 % 39.54 Put Fund 131951 0.2 % 53.65 Put Fund 141421 0.2 % 70.6

. . .9 Put Fund 186607 0.3 % 146.010 Put Fund 200000 0.4 % 181.8. . .25 Put Fund 200000 86.6 % 34789.4

Total 40311.7

19

Impact of Lapse on Hedge Liability

To illustrate how the value of the hedge liability depends on lapse assumptions, assumethat the best estimate lapses (“BE”) indicated above (eg 4% for all years, except for year 10where lapses are 12%) were inaccurate and need to be revalued to 8% at year 10 and 2%thereafter (“New BE”). The following table shows the value of the valuation portfolio as attime 2. Note that maturity is now in 23 years.

Instrument Value Value Value ValueNormal BE New BE P&L

1 Put Fund 25.6 23.5 23.5 –2 Put Fund 39.7 35.1 35.1 –3 Put Fund 55.7 47.3 47.3 –

. . .7 Put Fund 156.7 112.6 112.6 –8 Put Fund 195.4 134.6 134.6 –9 Put Fund 241.2 149.3 149.3 0.0

. . .23 Put Fund 36986.0 10138.3 14802.2 -4663.9

Total 42844.5 12973.2 18006.3 -5033.1

20

Lapse depend on moneyness (1/3)

π(GMWB) = EQ

[∑

k∈Nvk max{0,(Rt−FVt)}× I?(k)

]= ∑

k∈Nvk EQ [max{0,(Rt−FVt)}× I?(k)]

= ∑k∈N

vk EQ[EQ [max{0,(Rt−FVt)}× I?(k)|Gk]

]= ∑

k∈Nvk EQ

[max{0,(Rt−FVt)}×EQ [I?(k)|Gk]

].

21

Lapse depend on moneyness (2/3)

t/x

sx

IT M = 0%

IT M = 100%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 1465 66 67 68 69 70 71 72 73 74 75 76 77 78 79

0%

2%

4%

6%

8%

10%

12%

14%

t:x:

22

Lapse depend on moneyness (3/3)

t/x

t p∗65

IT M = 100%

IT M = 0%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

65 66 67 68 69 70 71 72 73 74 75 76 77 78 79t:x: 23

... and so does the response function

Level

π

Using dynamical lapses

Lapse independent on ITM

50% 60% 70% 80% 90% 100% 110% 120% 130% 140% 150%

2.500

5.000

7.500

10.000

12.500

15.000

17.500

20.000

StS0

:

24

Price of a GMWB guarantee - Product definition (1/3)

FV (t) denotes fund value at time t (before withdrawal),

GV (t) denotes benefit base (guaranteed) value at time t,

R(t) annuity paid at time t

ψ(t, t +∆ t) denotes the fund performance from time t to t +∆t, and

T⊂ R+ denotes the set of times at which a ratchet takes place.

25

Price of a GMWB guarantee - Product definition (2/3)

In this example we have the following (assuming for sake of simplicity that T ⊂ {k×∆t|k ∈N0} and also that annuity payments only take place at direct times k ∆t for some k ∈ N0):

FV (0) = EE > 0

GV (0) = FV (0)

FV ((k+1)∆t) = (FV (k ∆t)−R(k ∆t))ψ(k ∆t,(k+1)∆t)

GV ((k+1)∆t) =

{max(GV (k∆t),FV ((k+1)∆t)−R((k+1)∆t)) if (k+1)∆t ∈ T,GV (k∆t) else.

The death benefit is defined as the maximum of the current fund’s value and the difference

between the current GV (t) and the annuities paid out before this point (egk ∆t≤t

∑k∈N0

R(k ∆t) until

age 85. Afterwards there is no death benefit.

26

Price of a GMWB guarantee - Product definition (3/3)

The annuity can be withdrawn at times S ⊂ {k×∆t|k ∈ N0} and it amounts at time t ∈ S

to ρ(ξ0)×GV (t)× I?(t)×ψ(t), ξ0 is the first time ξ0 ∈S where the person can withdraw.The person is allowed to withdraw less than this amount in line with the model as definedbeforehand.We assume a 65 year old policyholder who invests 100’000 $.

27

Utilisation - what is this?

t/x

ψt

ψ0 = 100%

ψ0 = 0%

0 1 2 3 4 5 6 7 8 9 10 11 12 13 1465 66 67 68 69 70 71 72 73 74 75 76 77 78 79

0%

20%

40%

60%

80%

100%

t:x:

28

Utilisation Model (1/2)

? † ‡

?0 ?1 ?k ?n

29

Utilisation Model (2/2)

Finally we look at an abstract example of the above concept. We assume for the sake ofsimplicity a time-homogeneous Markov chain and we consider a x = 65 year old man. Tomodel the transition matrix P(1) we assume the following:

100% 75 % 50% 25% 0%

100% 0.95 0.5 – – –75% 1−α 0.95α 0.05α – –50% 0.8(1−α) 0.2(1−α) 0.95α 0.05α –25% 0.1(1−β ) 0.1(1−β ) 0.8(1−β ) β –0% 0.1(1−β ) 0.1(1−β ) 0.8(1−β ) – β

with

α = 5√1−75%

β = 5√1−80%

30

Modified Model including Utilisation (1/2)

π(GMWB|ψo = i) = EQ

[∑

k∈Nvk max{0,(Rt×ψt−FVt)}× I?(k)|ψo = i

]= ∑

k∈Nvk EQ

[EQ [max{0,(Rt×ψt−FVt)}× I?(k)|Gk] |ψo = i

].

Since we have assumed stochastic independence of ψ from the capital market variables, weneed to first calculate

EQ [ψt|G0] = EQ [ψt|ψ0]

= ∑j∈S

j× pi j(0, t).

31

Modified Model including Utilisation (2/2)

Moreover if we assume that ψt and I?(t) are independent, we can calculate π(GMWB) asfollows:

π(GMWB|ψo = i) = EQ

[∑

k∈Nvk max{0,(Rt×ψt−FVt)}× I?(k)

∣∣ψo = i

]

= ∑k∈N

vk EQ[EQ [max{0,(Rt×ψt−FVt)}× I?(k)|Gk] |ψo = i

]= ∑

k∈Nvk EQ

[max{0,(Rt×E[ψt|ψo = i]−FVt)}×EQ [I?(k)|Gk]

]= ∑

k∈Nvk EQ[max{0,(Rt×{∑

j∈Sj× pi j(0, t)}−FVt)}×EQ [I?(k)|Gk]].

32

Utilisation example (1/2)

Level

π

Quarterly Ratchet

Yearly Ratchet

50% 60% 70% 80% 90% 100% 110% 120% 130% 140% 150%

2.500

5.000

7.500

10.000

12.500

15.000

17.500

StS0

:

33

Utilisation example (2/2)

Level

π

ψ0 = 100% (starts utilising)

ψ0 = 0% (waits withdrawing)

50% 60% 70% 80% 90% 100% 110% 120% 130% 140% 150%

2.500

5.000

7.500

10.000

12.500

15.000

17.500

StS0

:

Hedging depends on PH behaviour - Model risk!34

Hedging different effects

Easier to hedge More difficult to hedge

• Short dated options,• Equity prices,• Short term volatility,• Interest rates.

• Long dated options,• Long term volatility,• Long term interest rates,• Policyholder behaviour (lapses, . . . ),• Basis risk.

35

Modeling of Policyholder Behaviour Risk

In the following we want to draw our attention how policyholder behaviour risk can be mod-elled. We will look at both lapse and utilisation risk.Lapse Risk: The main risk is in regards to trend risk over time. A natural way to model this

is akin to longevity modelling:

st ; st×Zt,where(Zt)t≥0 is a compensated geometric BM, ie E[Zt] = 1.

Utilisation Risk: Modelled akin to credit default rating models based on Markov chains. Thestratey there is to replace the transition matrix P1 = P(t, t + 1) by a random transitionmatric P̃(t, t+1,ω). This can be done for example by means of Gaussian starves. Notethat the MC becomes time inhomogenous ∀ω .

In the following we want to have a look what this concretely means.

36

Lapse Risk

37

Utilisation Risk: base case (1/5)

t/x

ψx / 1− r(x)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 140

0.10.20.30.40.50.60.70.80.91

65 66 67 68 69 70 71 72 73 74 75 76 77 78 79t:x:

38

Utilisation Risk: stress (2/5)

t/x

ψx / 1− r(x)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 140

0.10.20.30.40.50.60.70.80.91

65 66 67 68 69 70 71 72 73 74 75 76 77 78 79t:x:

U+

π

U−

U+

π

U−

39

Utilisation Risk: capital (3/5)

40

Utilisation Risk: ψx0 = 0% (4/5)

41

Utilisation Risk: ψx0 = 100% (5/5)

42

Things to consider for VA risk management from a Boardperspective

The following dimensions need to be considered:Shortfall Risk: What is the intrinsic shortfall risk for the VA protfolio with respect to the

various metrics?Product Risk: What are the product risks within the portfolio and how are they managed?Hedging Risk: How does the hedging strategy address the risks and what are the risks

induced by the hedging strategy?

Clarity is also needed regarding risk appetite and the hedging strategy.

43

Good Practices

1. Define risk limits and take them seriously.2. Do not underestimate the benefits of diversification (in one’s business model).3. Carry out scenario analyses and stress tests.4. Monitor traders carefully.5. Do not blindly trust models.6. Do not sell clients inappropriate products.7. Do not ignore liquidity risk.8. Do not finance long-term assets with short-term liabilities.9. Make sure a hedger does not become a speculator.

44

Q&A

45