Expert Q&A; Done Around January 1, 1993, Barbra Streisand signed a con- tract wi

Question

Expert Q&A; Done Around January 1, 1993, Barbra Streisand signed a con- tract with Sony Corporation for S2 million a year for 10 years. Suppose the first payment was made on the day of signing and that all other payments were made on the first day of the year. Suppose also that all payments were made into a bank account earning 4% a year, com-pounded annually (a) How much money was in the account (i) On the night of December 31, 1999? (i) On the day the last payment was made? (b) What as the pet valuhetact on the

Explanation / Answer

(a)(i).The formula for the future value (A) of annuity, when the first installment is paid immediately, is A = [P(1+r)[(1+r)n -1]/r where P is the periodic payment, r is the rate of interest per period and n is the number of periods. Here, P = $ 2 million, r = 0.04, and n = 6 (as on 01.01.1999). Thus, as on 01.01.1999, the value of the accumulated oney is 2*(1+0.04)[((1+0.04)6 -1]/0.04 = (2.08/0.04)[(1.04)6 -1] = 52(0.265319018) = $ 13.79658896 million = $ 13796588.96 = $ 13796589 (on rounding off to the nearest dollar). Now, the interest on this amount at 4 % for an year is 0.04*$ 13796589 = $ 5518.66 = $ 5519( on rounding off to the nearest dollar). Thus, the amount of money in the account on the night of 31st December,1999 is $ 13796589 +$ 5519 = $ 13,802108.

   (ii). On the day, the last payment was made, we haven = 10 so that A = 2*(1+0.04)[((1+0.04)10 -1]/0.04 = (2.08/0.04)[(1.04)10 -1] =52*0.480244284 = $ 24.97270282 million = $ 24972702.82 = $ 24972703(on rounding off to the nearest dollar).

(b). The formula for the present value of annuity (PV), when the first installment is paid immediately, is PV = P(1+r)[(1-(1+r)-n]/r where P, r,n are as above. Here, P = $ 2 million, r = 0.04, n = 10 so that PV = 2(1+0.04)[ 1-(1+0.04)-10]/0.04 = (2.08/0.04)[1-(1.04)-10] = 52*0.324435831 = $ 16.87066322 million = $ 16870663.22 = $ 16870663 (on rounding off to the nearest dollar).   

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