Question

Fifteen years ago, you deposited $12,500 into an investment fund. Five years ago, you added an additional $20,000 to that account. You earned 8%, compounded semi-annually, for the first ten years, and 6.5%, compounded annually, for the last five years.

Required:
a)
What is the effective annual interest rate (EAR) you would get for your investment in the first 10
years?
b)
How much money do you have in your account today?
c)
If you wish to have $85000 now, how much should you have invested 15 years ago?

Answer 1

Part A: 8.16%

Part B: $56,577.5

Part C: $39,700

Step-by-step explanation:

Part A:

EAR = (1 + r / n )^ n - 1

Here

r is the nominal interest rate, which is 8%

n is the number of compounding periods in a year and here, for semiannual requirement it is 2.

So by putting values in the above equation, we have:

EAR = (1 + 8% /2) ^2 - 1

= 8.16 %

Part B:

Amount invested is $12500 which must be compounded for 10 years semi annually at the EAR. This means

The future value = Initial Investment + Interest Income for 10 year

Future Value after 10 years = $12500 + ( $12500 * 8.160%*10 years)

= $22,700

Similarly,

Amount invested for next 5 years is $42,700 ($22,700 and the additional $20,000 which was added to the account). This amount must be compounded for next 5 years at 6.5%. This means

Value Today = Initial Investment + Interest Income for 5 year

The Future Value = $42,700 + (42,700 * 6.5% * 5 years) = $56,577.5

Part C:

Let the initial amount that was deposited be "x".

As per the guidelines given in the question, for first 10 years the interest is EAR which is 8.16%.

Interest Earned = (x * 8.16% * 10 Years) = 0.816x

The same initial investment "x" would be investment for the next 5 years at 6.5% rate.

For next 5 years:

Interest Earned = (x * 6.5% * 5) = 0.325x

Future value of the investment at the end of the year 15 = $85,000

0.816x + 0.325x + x = $85,000

2.141x = $85,000

x = $85,000 / 2.141 = $39,700