Actuarial Risk Managament Flashcards
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.
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 S(T) 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)
What is a guaranteed minimum death benefit with a return of premium guarantee?
And is it a call or a put?
A return of premium guarantee is a guarantee which returns the greater of the account value and the original amount invested.
Max ( Account value; 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 S(T) 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(S(T) ,K )=S(T) +max( K−S(T) , 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 op
What is an embedded death benefit guarantee for a guaranteed minimum death benefit ?
An embedded death benefit guarantee for a guaranteed minimum death benefit 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.
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
What is an earnings-enhanced death benefit?
It 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.
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=S(0) be the initial amount invested, and S(T) 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)
call option
What is the value of the earnings-enhanced death benefit ?
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.
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)]=∫C(t) f Tx (t)dt
What is the value of the embedded accumulation benefit guarantee for a guaranteed minimum accumulation benefit (GMAB) with a return of premium guarantee?
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)
Determine which one of the following statements regarding mortgage guaranty insurance is FALSE:
A)
Mortgage guaranty insurance is designed to limit losses to the lender, and by extension the depositors and creditors of the lending institution, after a mortgage has gone into default and the foreclosure process has begun.
B)
Mortgage guaranty insurance is most often purchased for loans that are considered higher risk.
C)
Mortgage guaranty insurance is different from mortgage life insurance and mortgage disability insurance.
D)
A mortgage guaranty insurance contract is a put option with payment contingent on borrower default.
E)
The mortgage loan itself is a put option, one which the lender holds.
Statements A, B, C, and D are true.
Mortgage guaranty insurance is purchased by mortgage lenders as protection from borrower defaults. It is secured by physical property such as a home.
Mortgage guaranty insurance is purchased by the lender to limit the lender’s losses when the mortgage goes into foreclosure. Therefore, mortgage guaranty insurance is commonly used for higher risk loans, such as loans with little equity or having credit risk factors. By purchasing this insurance, lenders can reduce the credit risk they are exposed to.
Do not confuse mortgage guaranty insurance with mortgage life insurance or mortgage disability insurance, both of which are purchased by the borrower.
Let B denote the outstanding loan balance at default, C the total settlement costs, and R the amount recovered from the sale of the property (less selling fees). Then, the payoff to the lender is:
max(B+C−R,0)
Observe this expression is the payoff of a put option with K=B+C and S=R. Thus, a mortgage guaranty insurance contract is a put option with payment contingent on borrower default.
Statement E is false.
For an uninsured position, the loss to the mortgage lender is:
max(B+C∗−R,0)
where B again denotes the outstanding loan balance at default, and R is the amount recovered on the sale of the property.
C∗, however, is the lender’s total settlement cost, which is lower than the mortgage insurer’s settlement cost, since the lender is not responsible for past mortgage payments. For the lender, missed interest payments are lost income, not something which must be paid to another party.
Because the loss to the lender is in the form of a put option, the lender has a short position in the put option, with the borrower, i.e., the mortgage holder, having the long position.
Thus, the mortgage loan itself can be viewed as a put option, one which the borrower holds.
Determine which one of the following statements regarding guarantees on variable annuity products is FALSE:
A guaranteed minimum death benefit (GMDB) with a return of premium guarantee is similar to a European put option with expiration contingent on the death of the policyholder or annuitant.
A guaranteed minimum accumulation benefit (GMAB) with a return of premium guarantee is similar to a European put option with payment contingent on the policyholder surviving to the guarantee expiration date and the policy still being in force at that time.
A guaranteed minimum withdrawal benefit (GMWB) provides a guarantee that the account value will not be less than the guaranteed withdrawal benefit base at any future time.
A guaranteed minimum income benefit (GMIB) provides a guarantee on the future purchase rate for a traditional annuity.
An earnings-enhanced death benefit is an optional benefit available with some variable annuity products that acts as a European call option with strike price equal to the original amount invested.
Statement C is false. The guaranteed minimum withdrawal benefit guarantees the size of withdrawals, not the size of the account value.
All other statements are true.
