Optimal Allocation and Consumption with Guaranteed Minimum Death Benefits with Labor Income and Term Life Insurance

Transcription

1 Optimal Allocation and Consumption with Guaranteed Minimum Death Benefits with Labor Income and Term Life Insurance at the 2011 Conference of the American Risk and Insurance Association Jin Gao (*) Lingnan University Department of Finance and Insurance phone: , fax: Eric Ulm Georgia State University Department of Risk Management and Insurance phone: , fax: (*) Corresponding author

2 Optimal Allocation and Consumption with Guaranteed Minimum Death Benefits with Labor Income and Term Life Insurance February 17, 2011 Abstract Because human capital is often the largest asset an investor possesses when he is young, protecting human capital from potential risks should be considered as a part of overall investment advice. The risk of the loss of the policyholder s human capital - the mortality risk - to the household can be partially hedged by a term life insurance policy. Guaranteed Minimum Death Benefits (GMDB) in Variable Annuities (VA) can also help policyholders hedge the risk of the loss of human capital. Therefore, GMDB options and term life insurance can be considered as substitute goods. However, they are not perfect substitute as GMDB and term life have their own properties: Term life insurance has no correlation with equity markets, and it is purely regarded as a protection for human capital; the variable annuity products follow the performance of equity markets, and the GMDB is a protection against downside risks on equity markets as well as human capital. We find that fairly priced GMDB options fail to add value to a VA contract if a term life policy is available. 1

3 1 Introduction Guaranteed Minimum Death Benefits (GMDB) in Variable Annuities (VA) have been drawing a lot of interest recently. Variable Annuities have historically been used as an accumulation vehicle to provide for retirement. In recent times, various guarantees have been added to these accounts, including Guaranteed Minimum Death Benefits (GMDB), which promise more than just return of the account value on death. In this paper, we examine the suitability of Return of Premium GMDB options, which are similar to European put options with a random maturity date corresponding to the time of the individual s death. Financial professionals often argue that GMDBs are, at best, redundant and at worst, a poor investment. After all, term insurance is available at reasonable prices and it is therefore possible to buy a variable annuity without a rider and its associated fees, and a separate term life policy to protect the beneficiary. On the other hand, it isn t possible to fully replicate the GMDB payoff with term insurance, as the GMDB pays off more when assets are low, whereas the term insurance payoff is insensitive to asset levels. If the beneficiary is risk averse, he may prefer a GMDB to term life insurance in some instances. In this paper, we examine this question. In this paper, we assume that an individual owns a variable annuity contract (either with or without a GMDB rider and its associated fees) and makes decisions optimally in order to maximize the expected utility of lifetime consumption. The insured gets utility from consumption and has bequest motives. We include the effect on asset allocation from dissavings (consumption). We also reflect bequest motives by including the utility of the recipient of the policyholder s guaranteed death benefits. The policyholder maximizes the discounted expected utility of the policyholders and beneficiaries by investing dynamically in an underlying fixed account and variable fund and withdrawing optimally. We also assume the individual wishes to protect his beneficiary from loss of human capital. We assume labor income can be consumed in addition to the variable annuity withdrawals. Human capital is the present value of individual s remaining lifetime labor income, and 2

4 it will influence individual s asset allocation choices, consumption choices and insurance choices. The risks you can afford to take depend on your total financial situation, including the types and sources of your income exclusive of investment income (see Malkiel (2004) pg. 342). Hanna and Chen (1997) study optimal asset allocation by considering human capital. They conclude that the investors who have long investment horizons should apply an all equity portfolio strategy. Bodie, Merton and Samuelson (1992) study investment strategy given labor income. They find that younger investors should put more money in risky assets than should older investors. Chen, Ibbotson, Milevsky and Zhu (2006) take human capital into account, and argue that human capital affects asset allocation. There are roughly three stages of a person s life 1 : the first stage is the growing up and getting educated stage; the second stage is the accumulation stage, in which people work and accumulate wealth; the third stage is the retirement/payout stage. Human capital generates significant amount of earnings during the accumulation stage. As individuals save and invest, human capital is transferred to financial capital. Chen et al.(2006) provide an approach to making the individuals financial decisions in purchasing life insurance, purchasing annuity products and allocating assets between stocks and bonds. Finally, the policyholder optimizes the combined utility through payment of premiums for term life insurance to protect his income stream. We check if the guarantee options add value to the contract even if the term life policy is available. Many papers study life insurance demand. Campbell (1980) derives solutions to optimal life insurance demand on mortality risk. His model introduces an insurance market to hedge the mortality risk and allows for the possibility that future tastes may be state-dependent. In his work, mathematical demandfor-insurance equations were derived to explicitly describe household s optimal responses to human capital uncertainty. Grace and Lin (2005) examine the life cycle demand for different types of life insurance by using the Survey of Consumer Finances. They find a relationship between financial vulnerability and term life insurance demand, and that older people demand less term life insurance. There are also a number of papers studying the joint demand 1 This paper focuses on the accumulation stage. 3

5 of term life insurance and annuities. Hong and Rios-Rull (2007) construct an overlappinggenerations model to analyze social security, life insurance and annuities for households. It reveals that the existence of life insurance opportunities for people is important in welfare terms. Purcal and Piggott (2008) use an optimizing lifetime financial planning model to explore optimal life insurance purchase and annuity choices. Their model incorporates the consumption and bequests in an individual s utility function. Policyholders needs for life insurance and annuities varied across different levels of risk aversion and different bequest motives. We add to this literature by deriving the insured s optimal decisions in purchasing the term life policy, and allocating and withdrawing assets in his VA account. We price the GMDB from the insurer s perspective by incorporating the insured s choices in a risk neutral model. We also find that, when term insurance is available, policyholders have a lower utility when their VA contains a fairly priced GMDB than when it does not. 2 The Model We assume that the insured and his beneficiary are risk averse with the same utility function. We apply a constant relative risk aversion (CRRA) type utility which has a functional form c 1 γ, γ > 0, γ 1, u(c) = 1 γ ln(c), γ = 1. An individual gets utility from his consumption c. γ is the coefficient of relative risk aversion, and the reciprocal of γ measures the willingness to substitute consumption between different periods. 2.1 Without GMDB Case We first take a look at the without GMDB case: an individual purchases a variable 4

6 annuity contract without a GMDB option and makes a lump sum deposit to the variable annuity account. After the insured receives labor income at the beginning of period t, he will make his consumption decision. If his labor income is not enough to support his consumption, he will make a decision to withdraw from the VA account. At the same time, he will also decide whether he needs a term life policy to help his beneficiary to protect against his own premature death. After making his consumption, withdrawal and term life purchase decisions, all at time t, the policyholder will decide the optimal allocation between the fixed and variable subaccounts in the VA account. If the policyholder dies at time t, the amount in the VA account a t, will be inherited by his beneficiary. In addition to the bequest from the policyholder s VA account, the beneficiary also gets the term life policy payment F c t. If the insured survives until his retirement age, at the end of the policy period (T ), he will get the entire account value and will annuitize it for his retirement life. The insured s objective function in the without GMDB case can be written as (1) [ T max E β t ( t φ i)u(c t )+β T ( T T φ i)v T +1 (a T +1 )+ β t ( ] t 1 φ i)(1 φ t )ζv B (a t +Ft c ), ω t,d t,p t i=1 i=1 i=1 t=1 where ω t is the percentage of wealth held in the variable subaccount and 1 ω t is the proportion of wealth allocated in the fixed rate subaccount. To be more realistic, we assume ω t [0, 1], which means that there are no short sales. d t is the amount of the withdrawal from the policy. P is the premium of the term-life policy. F c is the face amount of the policy. β is the subjective discount rate. φ t is the survival rate at time t. v B is the beneficiary s value function, which depends on the policyholder s bequest motive ζ which measures the importance of the beneficiary s benefits to the policyholder and ranges from 0 to 1. If ζ = 0, the insured has no bequest motive and does not want to leave a bequest to his beneficiary; if ζ = 1, the insured has the strongest bequest motive and will treat his beneficiary like himself. Once the beneficiary receives the bequest, she will maximize her own utility by optimal t=1 5

7 allocations and withdrawals. The beneficiary s objective function can be written as (2) max ω B t,cb t [ TB E β tb t ( t B 1 φ i )u(c B t ) + β TB t ( ] T B φ i)v B (a B T i=t i=t B +1). t B =t In this case, if the insured dies at time t, the bequest amount received by the beneficiary is a B t = a t + F c t. When the insured deposits in the VA account, the beneficiary has T B years until her retirement age. When the insured dies, the beneficiary will receive the bequest with T B t years until retirement age and will transform the money into a lifetime payout annuity after she retires. In these T B t years, she will optimally withdraw to consume and will optimally allocate the amount between risky and risk-free investments. She will receive the terminal value a B T B +1 at the beginning of year T B + 1. We do not assume a bequest motive for the beneficiary. From the insured s objective function, we can derive his Bellman equation as follows (3) V t (a t ) = max ω t,d t,p t { ut (c t ) + (1 φ t )ζv B (a t + F c t ) + βφ t E[V t+1 (a t+1 ) a t, r t+1 ] } subject to c t y t + d t, 0 d t a t, a t+1 = (ω t (1 + r t+1 ) + (1 ω t )(1 + g t ))(a t d t ), 0 ω t 1, a T +1 = (ω T (1 + r T +1 ) + (1 ω T )(1 + g t ))(a T d T ), V T +1 (a T +1 ) = F c t = T max t=t +1 β t (T +1) ( t 1 i=t +1 φ i)u( c), P t (1 φ t )(1 + η), 0 P t < d t + y t, y t is the outside income at time t. The consumption amount for the insured c t is the sum of labor income y t and withdrawal amount d t from the VA account. The fixed account grows at a risk-free rate g t and the risky asset s expected rate of return is r t. η is the loading element, which refers to the amount that must be added to the pure premium to cover other 6

8 expenses, profit, and a margin for contingencies. T max is the longest time an individual can survive. After retirement, the insured will receive a life time pay-out annuity 2 from the VA account, and the level monthly payment is c. The retired insured consumes c and gets the utility. c can be derived from the terminal account value as follows, (4) a T +1 = (ω T (1 + r T +1 ) + (1 ω T )(1 + g t ))(a T d T ) = c T max t=t +1 t 1 i=t +1 φ i(1 + r f ) T +1 t, (5) c = Tmax t=t +1 a T +1 t 1 i=t +1 φ i(1 + r f ) T +1 t. The Bellman Equation is solved using a trinomial lattice with a t as the state variable. 2.2 With GMDB Case We next consider the with GMDB case: an individual purchases a variable annuity contract with GMDB options and makes a lump sum deposit to the variable annuity account. After the insured receives labor income at the beginning of period t, he will make his consumption decision. If his labor income is not enough to support his consumption, he will make a decision to withdraw from the VA account. Simultaneously, the GMDB strike level will be reduced proportionally with the withdrawal ratio. At the same time, he will decide if he needs a term life policy to help his beneficiary protect against his own premature death. After the consumption, withdrawal and term life purchase decision, all at time t, the policyholder will decide the optimal allocation choice between the fixed and variable subaccounts in the VA account. If the policyholder dies at time t, the amount in the VA account, which is protected by the GMDB (b t ), will be inherited by his beneficiary. In addition to the bequest from the policyholder s VA account, the beneficiary also gets the term life policy payment (F c t ). If the insured has labor income and is holding a term-life insurance policy, the objective 2 Life time payout annuity is an insurance product that converts an accumulated investment into income that the insurance company pays out over the life of the investor (Chen, et al. (2006)). 7

9 function is (6) [ T max E β t ( t φ i)u(c t )+β T ( T T φ i)v T +1 (a T +1 )+ β t ( ] t 1 φ i)(1 φ t )ζv B (b t +Ft c ), ω t,d t,p t i=1 i=1 i=1 t=1 where v B is the beneficiary s value function. The beneficiary gets the bequest (F c t + b t ) and maximizes her own utility by optimal allocations and withdrawals. The objective function for the beneficiary is as follows, [ TB (7) maxe β tb t ( t B 1 ωt B,cB i=t t t B =t t=1 φ i )u(c B t ) + β TB t ( ] T B φ i)v B (b B T i=t B +1). If the insured dies at time t, the bequest amount received by the beneficiary is b B t She will get the terminal value b B T B +1 at the beginning of year T B + 1. = b t +F c t. From the insured s objective function, we can derive the insured s Bellman equation (8) V t (a t, b t ) = max ω t,d t,p t { ut (c t ) + (1 φ t )ζv B (b t + F c t ) + βφ t E[V t+1 (a t+1, b t+1 ) a t, r t+1 ] } subject to a 1 = b 1, c t y t + d t, 0 d t a t, a t+1 = (ω t (1 + r t+1 ) + (1 ω t )(1 + g t ))(a t d t ), 0 ω t 1, k t+1 = k t (1 + r f ) a t+1 d t+1 a t+1, b t+1 = max(k t+1, a t+1 ), F c t = V T +1 (a T +1 ) = P t (1 φ t )(1 + η), 0 P t < d t + y t, T max t=t +1 β t (T +1) ( t 1 i=t +1 φ i)u( c). 8

10 k t is the GMDB strike level and will be reduced proportionally with the withdrawal ratio. c is the insured s periodic consumption after retirement, and we assume it is the level periodic payment from the variable annuity account. c can be derived from the terminal account value as in the without GMDB case. The Bellman Equation is solved using a trinomial lattice with the state variables a t and b t. We can see that if the policyholder buys a VA product with GMDB options and term life policy, the beneficiary is protected against the loss of the insured s human capital and investment risk simultaneously. Any policyholder with bequest motive will be better off purchasing both. Because these two products are partial substitutes for each other, the policyholder needs to consider the costs and benefits of having both products. 2.3 Two-Stage Bellman Equations Following Hardy (2003), we denote state variables as ( ) t,( ) t + to solve the maximization problem, i.e. the value immediately before and after the transactions at the discrete time t, respectively. We assume the policyholder is employed and receives labor income y t at t. Withdrawal, consumption and term life purchase also occur at t. Then the insured decides the amount to transfer between the fixed and the variable subaccounts at t +, which is still at time t but after the receipt of income, withdrawal, consumption and term life purchase. We also assume that the beneficiary gets the bequest immediately at t + just after the insured dies at t +. Therefore, the insured s Bellman equation can be expressed by two-stage Bellman equations. At the 1st stage from t to t +, the insured gets the utility from consumption which is equal to the sum of optimal withdrawal from the variable annuity account and labor income minus the term life premium. Since there is mortality risk and bequest motives, if the policyholder dies before retirement, the beneficiary will get the utility from the bequest at t +. At the 2nd stage from t + to t + 1, the insured maximizes his utility by allocating optimally the amount between two subaccounts. In the without GMDB case, the bequest amount is equal to the account value a t at 9

11 the moment of insured s death. The insured s two-stage bellman equations are as follows 1. From t to t + (9) V t (a t ) = max d t,p t { u(ct ) + ζ(1 φ t )v B (a t d t + F c t ) + V t +(a t +) } = max d t,p t { u(yt + d t P t ) + ζ(1 φ t )v B (a t d t + F c t ) + V t +(a t d t ) } = max d t,p t { u(yt + d t P t ) + ζ(1 φ t )ψ t (a t d t + F c t ) 1 γ + V t +(a t d t ) }. Under the CRRA assumption, we derive a constant factor 3 ψ t at period t to make v B (a t d t + F c t ) = ψ t (a t d t + F c t ) 1 γ, given ψ t < 0 if γ > 1,and ψ t > 0 if γ < From t + to (t + 1) (10) V t +(a t +) = max ω t { βφt EV t+1 (a t+1 ) }. By taking the first order condition of d t and P t on V t (a t ), we get (11) P t = 1 + A A 1 (1 φ t)(1+η) d t + y t A 1 a t 1 + A 1 (1 φ t)(1+η) [ ] 1/γ ζψt (1 γ) where A 1 =. 1 + η From the above equation, we can see that the term life premium P t is linearly related to withdrawal amount d t from the account, given 0 P t d t + y t. Since A 1 is positive, P t grows in pace with the withdrawal amount d t and income y t, and is negatively correlated with wealth level a t. If the insured withdraws from the VA account, the beneficiary will receive a lower bequest from the VA account, and the insured is willing to buy more term life insurance to transfer his wealth to his beneficiary. If the account value a t is high, term life is no longer important. If the variable annuity account contains a GMDB option, the insured s two-stage Bellman 3 It can be generated numerically. 10

12 equations are as follows, 1. From t to t + (12) (13) V t (a t, b t ) = max d t,p t { u(ct ) + ζ(1 φ t )v B (b t + + F c t ) + V t +(a t +) } { a t d t = max u(ct ) + ζ(1 φ t )v B (b t + Ft c ) + V t +(a t +) }, d t,p t a t { a t d t = V t (a t, b t ) = max u(yt + d t P t ) + ζ(1 φ)v B (b t + F t ) + V t +(a t d t ) } d t,p t a t = max d t,p t { u(yt + d t P t ) + ζ(1 φ)ψ t (b t a t d t a t + F t ) 1 γ + V t +(a t d t ) }. 2. From t + to (t + 1) (14) V t +(a t +, b t +) = max ω t { βφevt+1 (a t+1, b t+1 ) }. By taking the first order condition of d t and P t on V t, we get (15) P t = 1 + A 1b t a t 1 + A 1 (1 φ)(1+η) d t + y A 1b t 1 + A 1 (1 φ)(1+η) [ ] 1/γ ζψt (1 γ) where A 1 =. 1 + η With GMDB, the term life premium is also linearly related to the withdrawal amount d t from the account. As in the without GMDB case, P t grows linearly with the withdrawal amount d t and the income y t. If there is no withdrawal, i.e. d t = 0, P t is negatively correlated with the GMDB level b t, because the term life is partially substituted for by the GMDB protection. If d t > 0, we find that a higher asset level a t reduces the term life demand. The GMDB level b t is also negatively correlated with the term life demand. 3 Numerical Methodology We use two-stage Bellman equations to get the numerical results and apply a 2-Dimensional lattice. We solve the policyholder s utility optimization problem by backward induction 11

13 (month by month) from the retirement age t = T (at the beginning of age 65, T = 360) to t = 1 (at the beginning of age 35) by discretizing the beginning-of-period fund value A = [0, a max ], into 51 nodes, Â = { a 1, a 2,, a 51}, and Guaranteed Minimum Death Benefit { level B = [0, b max ], into 51 nodes, ˆB = b 1, b 2,, b 51}. Therefore, at any given period t, we will have a space (51 asset levels by 51 GMDB levels) and the total state space for the whole time period of a policyholder is (51 asset levels by 51 GMDB levels by 360 time periods, and 360 time periods correspond to 360 months from age 35 to 65). a 1 = b 1 = (1/u) 25, where u = e σ 3 t is the jump size, and a i = b i = a 1 u i 1, for i = 2, 3,, 51. All state variables are denoted as ( ) t, ( ) t +. We create the terminal value at time (T + 1) by using (16) c = Tmax t=t +1 a T +1 t 1 i=t +1 φ i(1 + r f ) T +1 t, (17) V T +1 (a T +1 ) = T max t=t +1 β t (T +1) ( t 1 i=t +1 φ i)u( c). After we get the terminal values, we can maximize the insured s utility backward from T + to 1. a. Transition from (t + 1) to t + As indicated before, the insured decides the allocation between the fixed and variable subaccounts in this time interval. We apply the trinomial tree to solve for the optimal allocation. Given GMDB level b i, let (18) a t+1 = a t + ( ) 1/u 1 u, for all nodes on a t +. Since a t+1 might not be always on those 51 nodes we defined in advance, we use cubic spline interpolation to get the values of V t+1 (a j+1, b i ), V t+1 (a j, b i ), and V t+1 (a j 1, b i ). 12

14 The probabilities p u, p d, and p m can be derived from the required mean and variance over the time periods: (19) (20) (21) p u = Aω2 + Bω + C (u 1)(u d), p d = ω(erh e gh ) + e gh 1 d 1 Aω2 + Bω + C (d 1)(u d), p m = 1 Aω2 + Bω + C (u 1)(u d) ω(erh e gh ) + e gh 1 + Aω2 + Bω + C d 1 (d 1)(u d), where A = e (2r+σ2 )h 2e (r+g)h + e 2gh, B = (e rh e gh )(2e gh d 1), C = (e gh 1)(e gh d). Then at any given GMDB level b i, we maximize (22) V t +(a j, b i ) = βφ(p u V t+1 (a j+1, b i ) + p m V t+1 (a j, b i ) + p d V t+1 (a j 1, b i )), over ω [0, 1] for i = 1, 2,, 51 and j = 1, 2,, 51, where V t +(, ) is a matrix at time t +. b. Transition from t + to t We then determine the insured s optimal withdrawal d t and term life premium P t. At any given GMDB level b i : b.1. Initialize the withdrawal amount d t,k = a j a j k for all k < j; the term life premium P t can be obtained from equation (15) corresponding to different d t,k, given 0 P t d t,k +y t ; b.2. For all k s, we derive (23) V k t (a j, b i ) = u t (d t,k + y t ) + (1 φ t )ζψ t (max(b i k, a i k ) + F c t ) 1 γ + V t +(a j k, b i k ) 13

15 where F c t = P t (1 φ t )(1 + η). From the above equation, one can see that as the policyholder withdraws money from the account, the GMDB level will also be reduced proportionally. Numerically, we need to reduce the V t along the diagonal of the lattice; b.3. Let V t (a j k, b i k ) = max(v t (b i 1, a i 1 ),, V t (b i k, a i k )), so we can locate the position of the maximum V t in the space. The maximum V t is not necessarily on the matrix nodes, but it must be between the diagonal points V t (a j k 1, b i k 1) and V t (a j k +1, b i k +1); b.4. Choose the optimal withdrawal d t to maximize V t by using quadratic interpolation, (24) a 2 j k 1 a j k 1 1 a 2 j k a j k 1 a 2 j k +1 a j k +1 1 α 1 α 2 α 3 V t (a j k 1) = V t (a j k ) V t (a j k +1) then we can derive the value of the parameters α 1, α 2 and α 3 which are used to estimate V t, (25) α 1 α 2 α 3 a 2 j k 1 a j k 1 1 = a 2 j k a j k 1 a 2 j k +1 a j k +1 1 V t (a j k 1) V t (a j k ) V t (a j k +1) 1 By using the calculated values of α 1, α 2 and α 3, we can solve for the optimal withdrawal amount d t and the optimal term life policy F c t by maximizing the following equation. { a j d t } (26) max ut (d t + y t ) + ζ(1 φ t )ψ t (b i + Ft c ) 1 γ + α 1 (a j d t ) 2 + α 2 (a j d t ) + α 3 d t a j 14

16 subject to 0 d t a j, A 1 = [ ζψ t(1 γ) ] 1/γ, 1 + η P t = 1 + A 1b t a t d t + y A 1b t A 1 (1 φ)(1+η) 0 P t d t,k + y t, F c t = P t (1 φ t )(1 + η). A 1 (1 φ)(1+η), We repeat the transition from (t + 1) to t + and the transition from t + to t until t = 1. We can get the optimal asset allocation ω t and optimal withdrawal amount d t from age 65 to age 35. In the remainder of this section, some numerical sensitivity tests have been done. We assume the base case parameter values are as follows, Table 1: Common Parameters in the Base case Loading element of Term Life η 0.1 Strength of Bequest Motive ζ 1 Subjective Discount Rate β 0.97 Risk Free Rate r f 0.03 Coefficient of Relative Risk Aversion γ 2 Growth Rate of Fixed Subaccount r g 0.04 Return of Risky Asset r 0.07 Volatility of Risky Return σ 0.15 GMDB roll-up rate r p 0 Annual Survival Rate φ 0.99 Annual Mortality Rate µ 0.01 Labor Income y 0.01 In the following analyses, all adjustments are made monthly. This is a little unrealistic, because the term life insurance demand F c t life. Results might change if F c t might not be able to be adjusted monthly in real is not so easily adjustable. In figure 1 and figure 2, the life 15

17 insurance demand at age 35 are checked, given different at-the-money asset levels 4, and we find that as the at-the-money wealth level decreases, the demand for the term life insurance increases at any risk aversion levels and any given insurance loadings. In the comparison of the term life insurance demand with γ = 2 and γ = 3 (Figure 3), one can see that when the asset level is low, people are willing to buy a term life insurance policy for their beneficiaries. From the figure, as η = 0, there is no difference in determining the term life insurance demands for poor people in the γ = 2 case and the γ = 3 case; as η = 0.5, the difference in the term life insurance demand between different risk aversion levels is also very small when the wealth level is low. The difference expands as the asset level rises, because rich insureds have alternative ways to give bequests to their beneficiaries. Therefore people s incentives to buy term life insurance will decrease as their wealth level increases. One can also find that risk aversion levels matter in determining the term life insurance demand: the higher the risk aversion is, the higher the term life demand is. Figure 1: Age 35 Term Life Insurance Demand with γ = 3 4 At any asset level, we use a 35 = b 35 16

18 Figure 2: Age 35 Term Life Insurance Demand with γ = 2 Figure 3: Term Life Insurance Demand comparison with different γ 17

19 The amounts in basis points which the insured is willing to pay for the GMDB at the beginning of the contract are derived in Table 2. A quadratic interpolation method is applied to estimate this value. We know that for an identical expected rate of return r 1, any policyholder with a bequest motive will be better off in a VA account with GMDB than in a VA account without GMDB protection. This implies that r 1 in an with GMDB account is equal to r 1 + ɛ in an without GMDB account for some ɛ > 0. Given three expected rates of return r 1 < r 2 < r 3, we get the policyholder s utility at age 35 in the no GMDB account; the utility the policyholder can get in a VA account with GMDB should be between the utilities of no GMDB accounts with r 1 and r 3 (given r 3 is large enough). We assume ṽ is the utility of the GMDB account with r 1 at age 35; v 1, v 2 and v 3 are the utilities of the no GMDB accounts with expected rate of return r 1, r 2 and r 5 3 respectively. v 1, v 2 and v 3 can be expressed by the quadratic forms of r 1, r 2 and r 3 as follows, (27) r1 2 r 1 1 r 2 2 r 2 1 r3 2 r 3 1 θ 1 θ 2 θ 3 = v 1 v 2 v 3 We can derive the parameter values of θ 1, θ 2 and θ 3 from (28) θ 1 θ 2 θ 3 r1 2 r 1 1 = r 2 2 r 2 1 r3 2 r v 1 v 2 v 3 Using these parameters, we get (29) ṽ = θ 1 (r 1 + ɛ) 2 + θ 2 (r 1 + ɛ) + θ 3. 5 We assume r 1 = 7%, r 2 = 7.2% and r 3 = 7.5%. 18

20 We solve the above equation (29) for ɛ, and then 10000ɛ is denoted as the basis points that the insured is willing to pay. From table 2, one can find that at any given risk aversion level, this amount decreases as the at-the-money asset level decreases. The less risk averse the policyholder is, the smaller the basis points he is willing to pay. This means less risk averse policyholders are reluctant to pay costs for the GMDB protection. This can be partially explained by the fact that our model does not allow policyholders to save their labor income, and the only way for poor people to leave bequests is to buy a term life policy. The poor insureds may regard the term life insurance as an efficient tool to transfer their wealth to their beneficiary. Therefore, a poor insured is willing to pay less for the GMDB and buy more term life insurance. Life insurance has some substitute effects on the GMDB. Also as η increases, the policyholder will have to pay more premium for the optimal term life benefit. Therefore, the attractiveness of term life will decrease, and the policyholders are willing to pay a little more for a GMDB and buy less term life coverage. It may also be explained by the fact that the term insurance is adjustable monthly in this model while the GMDB is locked-in at t = 0. By incorporating optimal choices of the insured, the insurance company can price the GMDB in a risk neutral way (Table 3). From that table, one can see that the gap between the price from the insurer s perspective and the price evaluated from the insured s perspective is very large, even when the term life loading factor is increased to η = 0.5. The reason there is little influence from the term life loading factor increase is that the increase of η does not change the insured s allocation and withdrawal choices significantly, but the insurer prices the GMDB by considering the insured s allocation and lapse choices. Therefore the GMDB price is still much higher than the amount the insured is willing to pay. 19

21 Table 2: GMDB at the money basis points at age 35 from insured s perspective At the money Account Value γ η = 0 η = 0.1 η =

22 Table 3: GMDB fair price VS. insured s expected price at age 35 γ η GMDB Insured s fair price willingness to pay At-the-money account value is a = b = 1 4 Conclusions In this paper, we apply a dynamic utility based model to derive optimal transfer and withdrawal choices for insureds who have variable annuity accounts with GMDBs, incorporating the insured s periodic labor income as well as term life insurance choices. We determine the number of basis points that a VA policyholder would be willing to pay to acquire GMDB protection for his beneficiary. We also use this model to price the GMDB option from the insurer s perspective using the derived optimal behavior. We find that term insurance is a quite satisfactory substitute for the GMDB and utility-maximizing policyholders are therefore willing to pay very little for this additional protection. Since GMDBs are put options, and therefore expensive, we find no overlap between the amount an insurance company should charge for these options and the amount a policyholder would be willing to pay. It would be useful to extend this model to include savings vehicles external to the VA and to include term-insurance policies that include transaction costs to adjust the face amount as the results may be sensitive to these assumptions. 21

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