An insurance company has a variable annuity linked to

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38) An insurance company has a variable annuity linked to the S&P 500 index. A guaranteed minimum death benefit (GMDB) specifies the beneficiary will receive the greater of the account value and the original amount invested, if the policyholder dies within the first three years of the annuity contract. If the policyholder dies after three years, the beneficiary will receive the account value. Out of every 1000 policies sold, the company expects 10 deaths in each of years one, two, and three. Thus they also expect that 970 will survive the first three years. Assume the deaths occur at the end of the year. You are given the following at-the-money European call and put option prices, expressed as a percentage of the current value of the S&P 500 index. Duration (years) Call Price Put Price 1 18.7% 15.8% 2 26.2% 20.6% 3 31.6% 23.4% Calculate the expected value of the guarantee when the annuity is sold, expressed as a percentage of the original amount invested. (A) 0.23% (B) 0.32% (C) 0.52% (D) 0.60% (E) 0.76%
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IFM-02-18 77 Key: D The guarantee value is the sum of the three put prices multiplied by the probability of dying each year: 0.01(15.8) + 0.01(20.6) + 0.01(23.4) = 0.598 ≈ 0.60%. Reference: SN IFM-22-18
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IFM-02-18 78 39) A variable annuity has the following guarantees: Guaranteed minimum death benefit with a return of premium guarantee. Guaranteed minimum accumulation benefit with a return of premium guarantee, effective 10 years from the date the policy is sold. Earnings-enhanced death benefit that pays the beneficiary an additional benefit equal to 20% of any increase in the account value. The following notation is used: P ( T ) denotes the value of a European put option on the annuity value, with the strike price equal to the original amount invested and time to expiration T . C ( T ) denotes the value of a European call option on the annuity value, with the strike price equal to the original amount invested and time to expiration T . x T denotes the future lifetime of the policyholder, and ( ) x T f t denotes the probability density function of x T . Assuming no lapses, which expression below represents the combined value of all guarantees? (A) 0 0 ( ) ( ) Pr( 10) (10) 0.2 ( ) ( ) x x T x T C t f t dt T P C t f t dt + × + × (B) 0 0 ( ) ( ) Pr( 10) (10) 0.2 ( ) ( ) x x T x T C t f t dt T P P t f t dt + × + × (C) 0 0 ( ) ( ) Pr( 10) (10) 0.2 ( ) ( ) x x T x T P t f t dt T P C t f t dt + × + × (D) 0 0 ( ) ( ) Pr( 10) (10) 0.2 ( ) ( ) x x T x T P t f t dt T P C t f t dt + × + × (E) 0 0 ( ) ( ) Pr( 10) (10) 0.2 ( ) ( ) x x T x T P t f t dt T P P t f t dt + × + ×
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IFM-02-18 79 Key: C GMDBs are similar to put options, contingent on dying. GMABs are similar to put options, contingent on surviving. Earnings-enhanced death benefits are similar to call options, contingent on dying. Reference: SN IFM-22-18
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IFM-02-18 80
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