Jeremy loans $2,500 to his son on the condition that he repays with 5 annual payments consisting of interest on the outstanding balance and equal amounts of principal repayments. Payments are made at the end of the year. Jeremy deposits the total payment he receives into a fund that accumulates at the same interest rate. Given that his son repaid $517.50 at the end of the 5th year, how much has the fund accumulated to by the end of the 5 years?

$2,969.22

Explanation:

Equal principal repayment=$2,500/5

Equal principal repayment=$500

The fact that Jeremy's son repaid $517.50 at the end of the 5th year, means that the interest paid in year 5 is the difference between the amount repaid($517.50) and the equal principal repayment($500)

interest paid in year 5=$517.50-$500=$17.50

That also means that the balance outstanding at the beginning of year 5( at the end of year 4) is $500, which effectively means that the interest rate on the loan is the determined thus:

interest paid in year 5=balance at the end of year 4*interest rate

$17.50=$500*interest rate

interest rate=$17.50/$500

interest rate=3.50%

The schedule of repayment is attached

The first repayment would be invested for 4 years, since it is occurring at the end of year 1( in years 2-5), the year 2 repayment would be invested for only 3 years and so on.

FV value of reinvestment of repayment=$587.50*(1+3.50%)^4+$570.00*(1+3.5%)^3+$552.50*(1+3.5%)^2+$535.00*(1+3.5%)^1+$517.50

FV value of reinvestment of repayment=$2,969.22

The amount the fund accumulated to by the end of the 5 years is $2,648.23.

Explanation:

Step 1: Calculation of interest rate

The interest rate can be calculated using the following RATE function in Excel:

Interest rate = RATE(nper,pmt,-pv,fv,type) (1)

Where;

nper = number of periods = number of years = 5

pmt = Fixed annual payments = Amount repaid by his son at the end of the 5th year = $517.50 = 517.50

pv = present value = Loan amount = $2,500 = 2500

fv = future value = desired cash balance after last payment = 0

type = when payments are due (0 = end of period. 1 = beginning of period) = 0

Substituting the values into equation (1), we have:

Interest rate RATE(5,517.50,-2500,0,0)   (2)

Inputting =RATE(5,517.50,-2500,0,0) into a cell in an excel sheet (Note: as done in the attached excel file), we have:

Interest rate = 1.16%

Step 2: Calculation of the amount the fund accumulated to by the end of the 5 years

This can be calculated using the formula for calculating the Future Value (FV) of an Ordinary Annuity as follows:

FV = M * (((1 + r)^n - 1) / r) (3)

Where,

FV = Future value = The amount the fund accumulated to by the end of the 5 years =

M = Fixed annual payments = $517.50

r = Interest rate = 1.16%, or 0.0116

n = number of years = 5

Substituting the values into equation (3), we have:

FV = $517.50 * (((1 + 0.0116)^5 - 1) / 0.0116)

FV = $517.50 * 5.11735342258641

FV = $2,648.23

Therefore, the amount the fund accumulated to by the end of the 5 years is $2,648.23.

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