Answer to Question #211481 in Macroeconomics for danial

Question

Suppose that an individual invests $2,500 at the end of each of the next 6 years and

earns an annual interest rate of 8%. Calculate the present value of this series of annuity payments

B)Suppose that an individual invests $2,500 at the beginning of each of the next 6 years and

earns an annual interest rate of 8%. Calculate the value of this series of annuity payments. How

does this result compare with the solution to Part A?

Answer 1

In the first question we will find the present value of ordinary annuity and in the second question, it is annuity due.

(a)To find the present value of ordinary annuity, we will use the following formula:

$PV = P \times \frac{[ (1- (1+r)^{-n} )}{ r}]$

Where:

PV = present value

P = periodic payments

r = discount or interest rate

n = number of periods

$PV \space ordinary = 2500 \times[\frac{ (1- (1+0.08^{)-6} )} { 0.08}]\\ = 2500 \times [\frac{ (1- 0.063169627) }{ 0.08}]\\ = 2500 \times{(0.369830373 }{ 0.08})\\ = 2500 \times 4.622879664\\ = 11557.19916 = \$11,557.2 (rounded)$

Present value of ordinary annuity is $11,557.2.

b)

To find the present value of annuity due, we will use the following formula:

$PV \space due = P + P \times [ \frac{(1- (1+r)^{-(n-1) }}) { r} ]$

where:

PV = present value

P = periodic payments

r = discount or interest rate

n = number of periods

$2500 + 2500 \times [ \frac{(1- (1+0.08)^{-(6-1)} ) }{ 0.08 }]\\ = 2500 + 2500 \times [\frac{ (1- 0.680583197 ) }{ 0.08} ]\\ = 2500 + 2500 \times (\frac{ 0.319416803 }{ 0.08} )\\ = 2500 + 2500\times 3.992710037\\ = 2500 + 9981.775093\\ = 12481.77509\\ = \$12,481.78 (rounded)$

Present value of ordinary due is $12,481.78.

 If we compare both, present value of annuity due has higher value than present value of ordinary annuity. This happens due to time value money principle, as annuity due payments are received earlier and when calculating the present value, payment received earlier is worth more than payment received later.