Mathematics of Life Contingencies. Math 3281 3.00 W Instructor: Edward Furman Homework 1 Unless otherwise indicated, all lives in the following questions are subject to the same law of mortality and their times until death are independent random variables. 1. Assume that a decision maker s current wealth is 10,000. Assign u(0)=-1 and u(10,000)=0. a) When facing a loss of X with probability 0.5 and remaining at current wealth with probability 0.5, the decision maker would be willing to pay up to G for complete insurance. The values for X and G in three situations are given below. X G 10 000 6000 6 000 3 300 3 300 1 700 Determine three values on the decision maker s utility of wealth function u. b) Calculate the slopes of the four line segments joining the five points determined on the graph u(w). Determine the rates of change of the slopes from segment to segment. c) Put yourself in the role of a decision maker with wealth 10,000. In additional to the given values of u(0) and u(10,000), elicit three additional values on your utility of wealth function u. d) On the basis of the five values of your utility function, calculate the slopes and the rates of change of the slopes as done in part (b). 1
2. Consider a game of chance that consists of tossing a coin until a head appears. The probability of a head is 0.5 and the repeated trails are independent. Let the random variable N be the number of the trial on which the first head occurs. a) Show that the probability function of N is given by f(n) = ( 1 2 )n n = 1, 2, 3, b) Find E[N] and Var(N). c) If a reward of X = 2 N is paid, prove that the expectation of the reward does not exist. d) If this reward has utility u(w)=log w, find E[u(X)]. 3. A utility function is given by e (w 100)2 /200 w < 100 u(w) = 2 e (w 100)2 /200 w 100 a) Is u (w) 0? b) For what range of w is u (w) < 0? 4. If a utility function is such that u (w) > 0 and u (w) > 0, show that π[x] E[X]. A decision maker with preferences consistent with u (w) > 0 is a risk lover. 5. You are given: 1) The benefit of 1 on a ten-year endowment insurance is payable at the moment of death, or at the end of 10 years if (x) survives 10 years. 2) µ x (t) = 0.01 for t > 0. 3) δ = 0.06. 2
Determine the single benefit premium. Note that under constant force of mortality, the expectation of random variable for 1 unit of whole life insurance is µ µ+δ. A x:10 = µ µ + δ (1 e n(µ+δ) ) + e n(µ+δ) = 0.5685 6. A whole life insurance pays a benefit of 10e 0.05t at the moment of death if death occurs at time t. You are given: 1) µ = 0.02 2) δ = 0.04 Calculate the actuarial present value of the benefit. Netting the 0.05 rate of benefit increase against δ = 0.04 yields a net interest rate of -0.01. Letting A be the actuarial present value of the benefit, ( ) 0.02 A = 10 = 20 0.01 7. An n-year term insurance payable at the moment of death has actuarial present value of 0.0572. You are given: 1) µ x (t) = 0.007, t > 0. 2) δ = 0.05. Determine n. We have 0.05720 = µ µ + δ (1 e n(µ+δ) ) = 0.007 0.057 (1 e 0.057n ) n = 11 3
8. For a whole life insurance of 1000 on (x) with benefits payable at the moment of death: 0.04 0 < t 10 1) δ t = 0.05 10 < t 0.06 0 < t 10 2) µ x (t) = 0.07 10 < t Calculate the single benefit premium for this insurance. The single benefit premium for 1 is A x = A1 x:10 + 10 E x A x+10 = A x (1 10 E x ) + 10 E x A x+10 = 0.593869 Therefore, the single benefit premium for a benefit of 1000 is 593.868. 9. Dave wants to purchase a 5 year pure endowment with a single benefit premium. The amount of the endowment is $1,000. His insurance agent convinces him instead to use the same money to purchase a year endowment insurance policy which pays at the moment of death or at the end of five years, whichever comes first. You are given that µ = 0.04 and δ = 0.06. Calculate the benefit amount for this 5 year endowment. A x: 1 5 = e 5(0.04+0.06) = 0.606531, whereas A x:5 = 0.04 0.1 (1 e 0.5 ) + e 0.5 = 0.763918, so the benefit amount for endowment insurance is 100(0.606531/0.763918) = 793.97. 4
10. Bryon, a non-actuary, estimates the single benefit premium for a continuous whole life policy with a benefit of $100,000 on (30) by calculating the present value of $100,000 paid at the expected time of death. (30) is subject to a constant force of mortality µ x = 0.05, and the force of interest is δ = 0.08. Determine the absolute value of the error of Bryon s estimate. The expected value of an exponential distribution is the reciprocal of the force, or 1/0.05=20. The value of $100,000 paid in 20 years is 100, 000e 20(0.08) =100, 000e 1.6 = 0.05 20, 190. The true value of 100, 000A 30 is 100, 000( ) = 38, 462. The 0.05+0.08 error is $18,272. 11. Given: 1) i=5%. 2) The force of mortality is constant. 3) e x = 16 Calculate 20 A x. The mean survival time is 16, so the force of mortality is 1/16. Under the constant force of mortality, A x+20 = A x. Thus we have: ( ) 20 A x = 20 E x A x = e (ln1.05+1/16)(20) 1/16 1/16 + ln1.05 = 0.06065 12. For a continuous whole life insurance (Z = v T, T 0), E[Z]=0.25. Assume the forces of mortality and interest are each constant. Calculate Var(Z). 5
Since E[Z]=µ/µ + δ = 0.25, it follows that δ = 3µ. Then, E[Z 2 ] = µ µ + 2δ = 1/7 V ar(z) = 1 7 (1 4 )2 = 0.0804 13. Z is the present-value random variable for a whole life insurance of b payable at the moment of death of (x). You are given: 1) µ x+t = 0.01, t 0 2) δ = 0.05 3) The single benefit premium for this insurance is equal to Var(Z). Calculate b. For one unit, E[Z]= µ µ+δ = 1 6 and V ar(z) = µ µ + 2δ (E[Z])2 = 25 396 So we need b such that b 6 = 25b2 396 b = 396 6 25 = 2.64 GOOD LUCK! 6