Question

(1) Find the present value (one period before the first payment) of an annuity- immediate that lasts five years and pays $3,000 at the end of each month, using a nominal interest rate of 3% convertible monthly. Then repeat the problem using an annual effective discount rate of 3%. Which is higher? Why?

Answer 1

The present value of an annuity immediate is calculated using the following equation

A[ (1+ r)* " – 1] PV r(1+ r)"

Nominal interest rate of 3% convertible monthly

The annual effective rate of a nominal interest rate of 3% convertible monthly is calculated as

12 0.03 1+ 12 Annual effective interest rate = –1 = 3.04%

12x5 0.0304 12") $3000 x -1 PV = 0.0304 0.0304 12x5 X(1+ 12 12

Present value = $82,319.94

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Annual effective discount rate of 3%

12x5 0.03 12) $3000 x -1 PV = +1) ×; x(1 0.03 0,03 12x5 12 12

Present value = $83,026.69

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Present value at an annual effective discount rate of 3% is higher than the present value calculated at a nominal interest rate of 3% convertible monthly because the effective rate of a nominal interest rate 3% convertible monthly is higher than 3% . Present value varies inversely with the discount rate and when the discount rate is higher the present value is lower.