Hedging Variable Annuity Guarantees

p. 1/4 Hedging Variable Annuity Guarantees Actuarial Society of Hong Kong Hong Kong, July 30 Phelim P Boyle Wilfrid Laurier University Thanks to Yan Liu and Adam Kolkiewicz for useful discussions.

p. 2/4 Outline Introduction Dealing with embedded options Hedging: Some practical spects The Guaranteed Minimum Withdrawal Benefit(GMWB) Basic features Valuation of basic GMWB contract Semi static hedging Concluding remarks

p. 3/4 Background: Variable Annuities Investors like to have possibility of upside appreciation Investors also concerned about downside risk Many insurance and annuity products that combine both features Structured products, Equity Indexed Annuities and Variable Annuities(VA) These products contain different types of exotic options

p. 4/4 New product: Sequence of Steps Aim: Profitable return on capital Design the contract Manage the risks Risk management Reserving hedging the embedded options Pricing

p. 5/4 Ways to Hedge Buy options from another institution Can be expensive, Lapses Farm out hedging to a third party Set up managed account. Hedge in house Keeps control. Need resources. Hedging program can become politicized

p. 6/4 Hedging Program: Issues Need very clear objectives P and L should break even. Not a profit centre Educate senior management B of Directors may not fully understand hedging Communicating hedging performance Important Attribution of performance Which metrics to use.

p. 7/4 Dynamic Hedging (DH) Calibrate model to market options Value embedded options using calibrated model Construct portfolio of traded assets with same Greeks as liability portfolio Repeat this procedure periodically Rebalance the asset portfolio each time Update for lapses and new business

p. 8/4 Features of DH Does not protect against model risk Suppose market jumps. Gives rise to gap risk DH does not work if markets are closed for some reason eg post nine eleven. DH can be a challenge if gamma changes sign frequently (cliquet options) Dynamic hedging assumes you know the model, continuous prices, that markets are open and there is enough liquidity.

p. 9/4 Semi Static Hedging (SSH) SSH overcomes many of these problems. Use example Hedging a ten year European option with a portfolio of one year standard options Construct a portfolio of the short term options that replicates the long term option There is an exact expression for this. Assume t < τ < T. S t stock price at t

p. 10/4 SSH Formula Let V t (S t, T) be market price at time t, of long option to be hedged. We have (Carr Wu) (1) V t (S t, T) = k=0 g(k)p t (S t, t,τ, k)dk where P t (S t, t, τ, k) is the European put price at time t with maturity τ and strike k.

p. 11/4 SSH Formula The coefficients g are given by g(k) = 2 V (S, τ, T) S 2 S=k These correspond to the gammas of the long term derivative at the horizon time τ with stock price equal to k. Can approximate this integral with a finite sum.

p. 12/4 GMWB GMWB adds a guaranteed floor of withdrawal benefits to a VA Provides a guaranteed level of income to the policyholder. Suppose investor puts $100,000 in a VA Policyholder can withdraw a certain fixed percentage every year until the initial premium is withdrawn

p. 13/4 GMWB Assume the withdrawal rate is 7% per annum. Our policyholder could withdraw $7,000 each year until the total withdrawals reach $100,000. This takes 14.28 years. Note the policyholder can withdraw the funds irrespective of how the investment account performs Here is an example. Market does well at first and then collapses.

p. 14/4 Example Year Rate Fund before Fund after Amount Balance on fund withdrawal withdrawal withdrawn remaining 1 10% 110,000 103,000 7,000 93,000 2 10% 113,300 106,300 7,000 86,000 3-60% 42,520 35,520 7,000 79,000 4-60% 14,208 7, 208 7,000 72,000 5-2.8857% 7,000 Zero 7,000 65,000 6 r% 0 0 7,000 58,000...... 14 r% 0 0 7,000 2000

p. 15/4 The GMWB The GMWB guarantees a fixed level of income no matter what happens to the market Fee for the GMWB is expressed as a percentage (say fifty basis points) of either The investment account or The outstanding guaranteed withdrawal benefit.

p. 16/4 Assumptions Perfect frictionless market Ignore lapses, partial withdrawals mortality etc. Assume max amount taken each year Fixed term contract over [0, T]. Index investment fund dynamics di t = µi t dt + σi t db t where B t is a Brownian motion under P µ is the drift σ is the volatility.

p. 17/4 Investor s account Let A t be the value of the investors account at time t. A has an absorbing barrier at zero. Suppose first time it hits zero is τ. Dynamics of A for 0 < t < τ are da t = [(µ q)a t g]dt + σa t db t where q is the fee and g is the withdrawal rate. If the initial investment amount is A 0 then g = A 0 T.

p. 18/4 Pricing the contract Put Option decomposition Investment often in mutual fund. Insurer guarantees to pay remaining withdrawal benefits if A(t) reaches zero in [0, T]. Guarantee provided by the insurance is a put option with random exercise time. If policyholder s investment account stays positive there is no payment under the put option. The option is exercised automatically when the account balance first becomes zero.

p. 19/4 Put Option decomposition When this happens the insurer agrees to pay the remaining stream of withdrawal benefits of g per annum. T τ g e ru du Put option has a random exercise time τ. Put is funded by the fee payable until time τ. Guarantees backed solely by the claim paying ability of XYZ insurance Co.

p. 20/4 A reaches zero in year 11 100 90 80 70 Account value 60 50 40 30 20 10 Exercise 0 0 5 10 15 Time

p. 21/4 Valuation of GMWB Use numerical methods to value the contract (Monte Carlo or pde) Here we just value a very simple contract assuming full utilization. We ignore lapses and mortality and utilization choice. (Could include them.) Assume simple lognormal model for convenience and deterministic interest rates.

p. 22/4 Numerical Example Benchmark contract, fifteen year term. No lapses no deaths. All policyholders start to withdraw funds at max rate from the outset. Input parameters are Parameter Symbol Benchmark value Initial investment A 0 100 Contract term T 15 years Withdrawal rate g 6.6667 Volatility σ 0.20 Riskfree rate r 0.05 Now compute pv contributions and put prices for different values of q.

p. 23/4 Put values for basic GMWB Value of q Present value of Put option basis points Contributions 0 0 3.98 10 0.96 4.07 25 2.36 4.20 48 4.40 4.40 75 6.79 4.67 100 8.87 4.91 200 16.33 5.98 300 22.63 7.17

p. 24/4 Contributions and put option 25 20 15 10 5 0 0 0.005 0.01 0.015 0.02 0.025 0.03

p. 25/4 No arbitrage Values For this example the no arbitrage Value of q is q = 0.004751 Contributions and Put values when q = 0.004751 Entity Value (sd) Value of contributions 4.4012(0.0003) Value of put option 4.4014(0.0008)

p. 26/4 Sensitivity to interest rate Suppose we have benchmark contract. Fix q = 0.004751. Explore sensitivity of put value and present value of contributions to inputs. First vary the interest rate Interest Assumption Present Value of Put Option Contributions 0.04 4.29 6.14 0.05 4.40 4.40 0.06 4.50 3.09

p. 27/4 Sensitivity to interest rate 7 6.5 6 Contribution & put values 5.5 5 4.5 4 3.5 3 2.5 2 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 Interest rate

p. 28/4 Sensitivity to volatility Now vary the volatility Volatility Present Value of Put Option Assumption Contributions 0.15 4.37 2.08 0.20 4.40 4.40 0.25 4.44 7.07

p. 29/4 Sensitivity to volatility 8 7 Contribution & put values 6 5 4 3 2 15 16 17 18 19 20 21 22 23 24 25 Volatility

p. 30/4 Sensitivity to Account Value Vary A 0 and keep g fixed at 100 15 = 6.6667 A 0 Present Value of Put Option Contributions 90 3.73 6.10 100 4.40 4.40 110 5.07 3.20 Conclusion: Moneyness matters

p. 31/4 Semi static Hedging SSH Hedge the risk with a portfolio of one year options Re hedge after one year This overcomes some of the disadvantages of delta hedging SSH hedging is more robust than delta hedging For GMWB there is mild path dependence

p. 32/4 Distribution of Index in one year 200 180 160 140 120 100 80 60 40 20 0 40 60 80 100 120 140 160 180 200 220 240

p. 33/4 Evolution of A over year Pick one sample path. Withdraw 1.667 per quarter. Time t I t A t 0 100 100.00 0.25 110 108.33 0.50 120 116.52 0.75 110 105.14 1.00 100 93.91 A 1 = 93.91 for up down path.

p. 34/4 Evolution of A over year Pick another sample path. Withdraw 1.667 per quarter. Time t I t A t 0 100 100 0.25 90 88.33 0.5 80 76.85 0.75 90 84.79 1 100 92.55 A 1 = 92.55 for down up path.

p. 35/4 Distribution of Liability in one year For each value of the index at time one I 1 there is a range of A 1. Conditional on I 1 the future GMWB liability V 1 (I 1 ) takes a range of values. There is a different value of V 1 (I 1 ) for each A 1. So we can write V 1 = V 1 (I 1, A 1 ) Very messy simulation to get them all Two ways. First way use Brownian bridge construction

p. 36/4 Ten paths of a Brownian bridge 2 1.5 1 0.5 0 0.5 1 1.5 2 0 0.5 1 1.5 2 2.5 3 3.5 4 Time in quarters

p. 37/4 Liability Calculation Fix I 1 Use Brownian bridge to get distribution of A 1 I 1 For each pair (A 1, I 1 ) find the GMWB liability at time one Find average value of liability given I 1 (2) V (I 1 ) = E A1 [V 1 (I 1, A 1 )] Use put options with different strikes to construct SS Hedge based on V (I 1 ).

p. 38/4 Distribution of A 1 Fix a value of the index at time one I 1. Conditional on I 1 find the distribution of A 1 I 1. We have seen that A 1 is path dependent. Next graph assumes I 1 = 100 and gives distribution of A 1

p. 39/4 Distribution of A 1 I 1 = 100 1200 1000 800 600 400 200 0 91 91.5 92 92.5 93 93.5 94 94.5

p. 40/4 Distribution of A 1 We can get all the moments of A 1 I 1 in closed form It turns out that A 1 I 1 is almost normal. We fix I 1 and write A 1 = Ā1 + σ A z where Ā1 = E[A 1 ] and where z has mean zero and variance one.

p. 41/4 V 1 as a function of A 1 Fix I 1. We have V (I 1, A 1 ) = V (I 1, Ā1 + σ A z) Using Taylor series and taking expectations we have (3) E A1 [V (I 1, A 1 )] = V (I 1,Ā1) + (σ2 A ) 2 V (I 1,Ā1)

p. 42/4 SS Hedging Two expressions for E A1 [V (I 1, A 1 )] The last one equation (3) is much more convenient. Find portfolio of put options to replicate average liability E A1 [V (I 1, A 1 )]. Optimization procedure or apply equation (1). Repeat after one year

p. 43/4 Profile of average net liability 40 30 Green line net liability 20 10 0 10 20 40 60 80 100 120 140 160 180 200 220 240 Index value after one year

p. 44/4 Summary and future work We discussed Semi static hedging Application to GWMB Future work deal with path dependent structure Stochastic volatility Other products Natural hedges