Question

Holly Krech is planning for her retirement, so she is setting up a payout annuity with her bank. She wishes to receive a payout of $1,700 per month for twenty years. (Round your answers to the nearest cent.)

(a) How large a monthly payment must Holly Krech make if she saves for her payout annuity with an ordinary annuity, which she sets up thirty years before her retirement? (The two annuities pay the same interest rate of 7.8% compounded monthly.) $

(b) How large a monthly payment must she make if she sets the ordinary annuity up twenty years before her retirement? $

Answer 1

The formula for annuity payments of $ P per month for n months is P = r(PV)/[1-(1+r)^-n] where r is the interest rate per period and PV is the present value of the deposit.

Here, P = $ 1700, r = 7.8/1200 = 0.0065 and n = 20*12 = 240.

(a). As per the above formula, the amount that Holly Krech must deposit is PV = (P/r) [1-(1+r)^-n] = (1700/0.0065)[1- 1/(1.0065)²⁴⁰] = (1700/0.0065)(1-0.211199548) = (1700/0.0065)(0.788800451) = $ 206301.66 ( on rounding off to the nearest cent). Thus, Holly Krech must deposit $ 206301.66 in order to receive a payout of $1,700 per month for twenty years.

If Holly Krech saves for her payout annuity with an ordinary annuity, which she sets up thirty years before her retirement, then the period and the maturity value of the other annuity must be 10 years and $ 206301.66 respectively.

The future value(F) of an ordinary annuity is given by F = P[(1+r)ⁿ -1]/r where P is the periodic payment, r is the interest rate per period and n is the number of periods. Here, F =$ 206301.66, r =7.8/1200 = 0.0065 and n = 10*12 = 120. Hence, 206301.66= P [(1.0065)¹²⁰ -1]/(0.0065) so that P = 206301.66*0.0065/[(1.0065)¹²⁰ -1] = 1340.96079/(2.175973024 -1) =1340.96079/(1.175973024) = $ 1140.30( on rounding off to the nearest cent).

Thus, Holly Krech must make a monthly payment of $ 1140.30 if she saves for her payout annuity with an ordinary annuity, which she sets up thirty years before her retirement.

b). The total amount that Holly will receive from her payout annuity is 240*$ 1700 = $ 408000