Question

Suppose your opportunity cost rate is 11 percent compounded annually. (a) How much must you deposit in an account today if you want to pay yourself $230 at the end of each of the next 15 years? (b) How m uch must you deposit if you want to pay yourself $230 at the beginning of each of the next 15 years?

Answer 1

Answer:

a. Amount = $1653.93

b. Amount = $1835.82

Explanation:

a.

The Present Value is the deposited amount of future payments.

The payments are annuity if they are made at the end of each year.

To compute the present value of an annuity with periodic payment, we'll make use of the following formula:

M(1 - (1 + r)^- T)/ r

Where

M = Periodic Payment = $230

T = Periods = 15

r = rate = 11% = 0.11

So, Amount of Deposit = 230(1 - (1 + 0.11)^-15)/0.11

Amount = 230(1 - (1.11)^-15)/0.11

Amount = 230 ( 1 - 0.209)/0.11

Amount = 230 * 0.791/0.11

Amount = 230 * 7.191

Amount = $1653.93

b.

In this case payments are made at the beginning of each period

This means that the payments are an annuity due.

To compute the present value of an annuity due with periodic payment, we'll make use of the following formula

M((1 + r) - ( 1 + r) ^ ( 1 - T))/r

Amount = 230(( 1 + 0.11) - (1 + 0.11) ^ (1 - 15))/0.11

Amount. = 230((1.11 - 1.11^-14))/0.11

Amount = 230(1.11 - 0.232)/0.11

Amount = 230 * 0.878/0.11

Amount = 201.94/0.11

Amount = $1835.82