Variable Annuities and Policyholder Behaviour

Variable Annuities and Policyholder Behaviour Prof Dr Michael Koller, ETH Zürich Risk Day, 1192015

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, policy term 25 years We consider two possible outcomes (trajectories) and assume that the person dies after 17 years Then we have the following: green red Fund Value 130 000 35 000 Death Benefit 160 000 160 000 Loss Insurer 30 000 125 000 Hence the insurer needs to be able to sell the underlying fund at 160 000 if its value is below this amount, independently of its value This is called a put option with strike 160 000 In the good outcome ( green ), this guarantee has 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 GMDB (Death Benefit) GMAB (Accumulation Benefit) GMIB (Income Benefit) GMWB (Withdrawal Benefit) GLWB (Life Benefit) Insurance Protection Protection in case of death Policyholder survives a certain time Policyholder survives a certain time, regular income Temporary Annuity (potentially deferred) (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 of the fund attained in the past This maximum is evaluated at regular time intervals In case of a 5 yr ratchet, we have the high water marks as follows: Time Fund Value Ratchet 0 100 100 2 100 114 5 107 114 7 107 125 20 179 179 24 179 192 5

Example GMWB GLWB Example The Policyholder has invested 100,000 at age 60 and bought a GLWB (Guaranteed Minimum Withdrawal Benefit for Life), with the right to withdraw 5% pa The benefit base GWB has increased if GWB = c116 000 because he did not withdraw before age 65, so can now withdraw up to c5 800 pa The fund is depleted at age 85 and the guarantee kicks in The expected guarantee ( yellow ) reduces as more and more policyholders die 6

Balance Sheet Assets Liabilities Risks SH Capital VA Guarantees Basis Risk Hedging Risk Market Risk Interest Rate Credit VA Funds Asset Performance 7

Methods to value variable annuities Valuation Tree Explicit formula or recursion (only insurance valuation and very simple variable annuities), (10,10) p r q (11,9) (11,10) (8,11) t=0 t=1 a b c d e f g h i (14,9) (10,13) (10,8) (14,9) (10,13) (10,9) (12,10) (7,15) (7,10) t=2 ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 ω 7 ω 8 ω 9 Solution of Black-Scholes-Merton differential equation (different methods including tree method), Monte Carlo Simulation (this is the approach most often used for variable annuities) Monte Carlo is most commonly used for VA since it is very versatile and can also cope with very complex option structures, such as ratchets 8

Steps in Valuation of VA Determine the number of people which benefit: Since only a tiny percentage of the whole inforce dies within a given year, one only needs to provide the respective GMDB cover to them Similarly the GMAB cover is paid only to the people surviving the entire term of the policy Hence we need to determine the respective percentages This is done by means of life decrement tables Calculate what these people receive: Then we need to know what the respective policyholders are entitled to Assume, for example, the people dying aged 40 They are entitled to get a GMDB at a certain level Hence we need to determine the number of the corresponding units of guarantees For our 40 year old person this would be put options at a strike price Calculate the value: At this point we know the valuation portfolio of guarantees representing the VA guarantee (eg number of instruments and their characteristics) We now need 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 % 6577763-3381856 3967378-234169230 8671487-40 % 5939795-3605150 4296158-224151981 11145501-30 % 5373440-3731940 4474399-213663966 13355918-20 % 4871053-3782627 4543179-203046053 15259675-10 % 4425499-3774413 4516828-192534590 16847076-5 % 4221951-3753190 4470565-187371426 17524749 0 % 4030171-3723002 4405273-182290527 18128449 +5 % 3849406-3685469 4324346-177303467 18661649 +10 % 3678941-3641996 4231500-172419241 19128308 +20 % 3366233-3541769 4023570-162984532 19878930 +30 % 3087111-3429533 3802666-154019953 20414109 +40 % 2837290-3310203 3580825-145537992 20765590 +50 % 2613091-3187216 3364082-137536959 20961873 10

VA risks Risk Landscape Financial Risks relate to capital markets, such as equity risks, but also the ability to trade at a certain point in time Policyholder Behaviour Risks relate to the behaviour of the policyholder at a given point in time Insurance Risks relate to the pure demographic risks such as mortality Other Risks summarise the remainder of risks such 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 risk appetite statements are necessary 12

Aim of Hedging Approximation The lower figure shows the P&L (in yellow) for a given hedging strategy (upper figure) with the aim to approximate the value of a VA, which depends among other things on: Equity prices, equity volatility, Interest rates, Mortality, Lapses, utilisation Aim of hedging: immunising balance sheet of insurance 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 accounts 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 dimensions and define a corresponding risk appetite 14

Hedge Strategies f (S,t,r,σ) = f S + 1 }{{} S 2 Delta 2 f }{{} S 2 Gamma + f r + f σ + }{{} r }{{} σ Rho Vega ( S) 2 + f }{{} t T heta 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 movemens t 15

Balance Sheet Assets Lia Assets Lia Capital 4 Red in Capital VA Guar 3 2 Hedging Loss Hedge Lia VA Funds 1 Red in Value t = 0 t = 1 16

Risk Management for VA in general The best way to reduce VA risk is normally to have outperformance in the underlying investment 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 in different assets, which are more or less risky Top up investment: The policyholder can invest an additional amount in the underlying fund 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 withdraw 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 product design 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 01 % 87 1 Put Fund 107177 02 % 177 2 Put Fund 114870 02 % 277 3 Put Fund 123114 02 % 395 4 Put Fund 131951 02 % 536 5 Put Fund 141421 02 % 706 9 Put Fund 186607 03 % 1460 10 Put Fund 200000 04 % 1818 25 Put Fund 200000 866 % 347894 Total 403117 19

Impact of Lapse on Hedge Liability To illustrate how the value of the hedge liability depends on lapse assumptions, assume that the best estimate lapses ( BE ) indicated above (eg 4% for all years, except for year 10 where 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 at time 2 Note that maturity is now in 23 years Instrument Value Value Value Value Normal BE New BE P&L 1 Put Fund 256 235 235 2 Put Fund 397 351 351 3 Put Fund 557 473 473 7 Put Fund 1567 1126 1126 8 Put Fund 1954 1346 1346 9 Put Fund 2412 1493 1493 00 23 Put Fund 369860 101383 148022-46639 Total 428445 129732 180063-50331 20

Lapse depend on moneyness (1/3) [ ] π(gmwb) = E Q v k max{0,(r t FV t )} I (k) k N v k E Q [max{0,(r t FV t )} I (k)] = k N = [ v k E Q E Q [max{0,(r t FV t )} I (k) G k ] k N = [ ] v k E Q max{0,(r t FV t )} E Q [I (k) G k ] k N ] 21

Lapse depend on moneyness (2/3) s x 14% 12% 10% IT M = 0% IT M = 100% 8% 6% 4% 2% 0% t: x: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 t/x 22

Lapse depend on moneyness (3/3) 1 09 08 07 06 05 04 03 02 01 0 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 23 14 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 t/x

and so does the response function 20000 π 17500 15000 12500 Using dynamical lapses Lapse independent on ITM 10000 7500 5000 2500 S t Level S : 0 50% 60% 70% 80% 90% 100% 110% 120% 130% 140% 150% 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 N 0 } and also that annuity payments only take place at direct times k t for some k N 0 ): FV (0) = EE > 0 GV (0) = FV (0) FV ((k + 1) t) = { (FV (k t) R(k t))ψ(k t,(k + 1) t) max(gv (k t),fv ((k + 1) t) R((k + 1) t)) if (k + 1) t T, GV ((k + 1) 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 (eg k t t R(k t) until k N 0 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 N 0 } and it amounts at time t S to ρ(ξ 0 ) GV (t) I (t) ψ(t), ξ 0 is the first time ξ 0 S where the person can withdraw The person is allowed to withdraw less than this amount in line with the model as defined beforehand We assume a 65 year old policyholder who invests 100 000 $ 27

Utilisation - what is this? 100% E[ψ t ψ 0 ] 80% 60% 40% ψ 0 = 100% ψ 0 = 0% 20% 0% t: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 x: 28 t/x

Utilisation Model Finally we look at an abstract example of the above concept We assume for the sake of simplicity a time-homogeneous Markov chain and we consider a x = 65 year old man To model the transition matrix P(1) we assume the following: 100% 75 % 50% 25% 0% with 100% 095 05 75% 1 α 095α 005α 50% 08(1 α) 02(1 α) 095α 005α 25% 01(1 β) 01(1 β) 08(1 β) β 0% 01(1 β) 01(1 β) 08(1 β) β α = 5 1 75% β = 5 1 80% 29

Modified Model including Utilisation (1/2) [ ] π(gmwb ψ o = i) = E Q v k max{0,(r t ψ t FV t )} I (k) ψ o = i k N = [ ] v k E Q E Q [max{0,(r t ψ t FV t )} I (k) G k ] ψ o = i k N Since we have assumed stochastic independence of ψ from the capital market variables, we need to first calculate E Q [ψ t G 0 ] = E Q [ψ t ψ 0 ] = j p i j (0,t) j S 30

Modified Model including Utilisation (2/2) Moreover if we assume that ψ t and I (t) are independent, we can calculate π(gmwb) as follows: [ π(gmwb ψ o = i) = E Q v k max{0,(r t ψ t FV t )} I (k) ] ψ o = i k N = [ ] v k E Q E Q [max{0,(r t ψ t FV t )} I (k) G k ] ψ o = i k N = [ v k E Q max{0,(r t E[ψ t ψ o = i] FV t )} E Q [I (k) G k ] k N j p i j (0,t)} FV t )} E Q [I (k) G k ]] = k N v k E Q [max{0,(r t { j S ] 31

Utilisation example (1/2) π 17500 15000 12500 Quarterly Ratchet Yearly Ratchet 10000 7500 5000 2500 S t Level S : 0 50% 60% 70% 80% 90% 100% 110% 120% 130% 140% 150% 32

Utilisation example (2/2) π 17500 15000 12500 ψ 0 = 100% (starts utilising) ψ 0 = 0% (waits withdrawing) 10000 7500 5000 2500 S t Level S : 0 50% 60% 70% 80% 90% 100% 110% 120% 130% 140% 150% Hedging depends on PH behaviour - Model risk! 33

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 34

Things to consider for VA risk management from a Board perspective 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 35

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 36

Q&A 37