Actuarial Risk Managament

Question 1

Determine which one of the following statements regarding guarantees on variable annuity products is FALSE:

A
A guaranteed minimum death benefit (GMDB) with a return of premium guarantee is a put option with expiration contingent on the death of the policyholder.

B
The value of the embedded death benefit guarantee for a guaranteed minimum death benefit (GMDB) with a return of premium guarantee is the probability-weighted average of the values of European put options of varying times to expiration with probability weights defined by the random variable that represents the future lifetime of a policyholder.

C
An earnings-enhanced death benefit is an optional benefit available with some variable annuity products that pays the beneficiary an additional amount when the policyholder/annuitant dies based on the increase in the account value over the original amount invested.

D
The value of the earnings-enhanced death benefit is equal to a multiplier times the probability-weighted average of the values of European call options of varying times to expiration with probability weights defined by the random variable that represents the future lifetime of a policyholder.

E
The value of the embedded accumulation benefit guarantee for a guaranteed minimum accumulation benefit (GMAB) with a return of premium guarantee is equal to the probability that the policyholder is alive and the policy is still in force times the value of a European call option.

Answer 1

Statements A, B, C, and D are true:

Guaranteed Minimum Death Benefit with a Return of Premium Guarantee
A return of premium guarantee is a guarantee which returns the greater of the account value and the original amount invested. Thus, upon a policyholder’s death, a guaranteed minimum death benefit with a return of premium guarantee pays the larger of the account value and the original amount invested when the policyholder dies.

Let K be the initial amount invested and ST be the account value at time T when the policyholder dies. The beneficiary receives the larger of the two values at time T, max(ST,K). This expression can be rewritten as:

max(ST,K)=ST+max(K−ST,0)
Without any guarantee, observe the beneficiary would simply receive ST; thus, the additional value of this guarantee is max(K−ST,0), which is the payoff of a European put option with strike price K=S0 and time to expiration T.

Because the exact time of expiry, the policyholder’s time of death, is not known, this is a life-contingent put option, where Tx is the future lifetime continuous random variable for a policyholder aged x.

We will denote Tx’s density function as fTx(t) and P(Tx) as the value of the European put option at the policyholder’s time of death. Then, the probability-weighted average of the values of the European put options is:

E[P(Tx)]=∫∞0P(t)fTx(t)dt
Earnings-Enhanced Death Benefit
An earnings-enhanced death benefit pays the beneficiary an amount based on the increase in the account value over the original amount invested.

This benefit is paid at the time of the policyholder’s death, and only if the account value at that time is greater than the original amount. The amount received from this benefit may be used to offset taxable gains from the variable annuity.

Consider an example where a policyholder purchases an earnings-enhanced death benefit of 40% of the variable annuity’s gains. Again, let K=S0 be the initial amount invested, and ST be the account value at time T when the policyholder dies. The beneficiary will receive 40% of any gain or zero, whichever is larger:

=40% x max(ST−K,0)
Observe the above is a multiple of a life-contingent call option payoff with a strike price of K.

We can let C(t) be the value of the European call option at the time of a policyholder’s death. Thus, the average price of the benefit is the probability-weighted average of the European call options:

E[C(Tx)]=∫∞0C(t)fTx(t)dt
Statement E is false:

The value of the embedded accumulation benefit guarantee for a guaranteed minimum accumulation benefit (GMAB) with a return of premium guarantee is equal to the probability that the policyholder is alive and the policy is still in force times the value of a European put option.

Similar to a guaranteed minimum death benefit with a return of premium guarantee, a guaranteed minimum accumulation benefit with a return of premium guarantee has an embedded put option. It differs, however, because the benefit is contingent on the policyholder surviving to the end of the guarantee period.

Let T∗X be the future lifetime of a policy with a guarantee period ending m years from now. Assuming lapses are possible, the lifetime of the policy can be considered less than the lifetime of the policyholder. Hence, the distribution of T∗X is usually different from TX.

We can let P(m) be the put price with a time to expiration of m. The payoff of P(m) is the maximum of 0 and the difference between the specified minimum accumulation amount and the account value at time m , i.e., max(0,K−Sm). Multiplying the price of the option by the probability of policy survivorship past time m, the expected cost of the guarantee is:

=Pr(T∗X≥m) x P(m)