Exercise 4.21 (a) Describe in words the insurance benefits with the present values given below. 20 v Tx if Tx 15. O if Tx 15. (b) Write down in integral form the formula for the expected value for (i) Z1 and (ii) Z2. (c) Derive expressions in terms of standard actuarial functions for the expected values of Z1 and Z2. (d) Derive expressions in terms of standard actuarial functions for the variance of both Z1 and Z2. (e) Derive an expression in terms of standard actuarial functions for the covariance of Z1 and Z2. (f) Assume now that x = 40. Calculate the standard deviation of both Z1 and Z2, using the Standard Ultimate Life Table, with UDD between integer ages, and interest at 5% per year