Question

Consider John Smith, a new freshman who has just received a study loan and started college. He plans to obtain the maximum loan at the beginning of each year. Although John Smith does not have to make any payments while he is still in school, the 6.5 percent interest per year compounded monthly owed accrued and is added to the balance of the loan.

Study Loan Limits
Freshman	$26,250
Sophomore	$35,000
Junior	$55,000
Senior	$55,000

After graduation, John Smith gets a six-month grace period. This means that monthly payments are still not required, but interest is still accruing. After the grace period, the standard repayment plan is to amortize the debt using monthly payments for 10 years.

Required:

Using the standard repayment plan and a 6.8 percent APR interest rate, compute the monthly payments John Smith owes after the grace period.

Answer 1

The loan repayment starts only after the end of 4 years

Hence, the interest on the loan taken in first year accrues for 4 years, loan taken in second year accrues for 3 years, loan taken in third year accrues for 2 years and loan taken in fourth year accrues for 1 year

The future value of an amount Y after n periods and interest rate r = Y(1+r)ⁿ

Let us find the total accrued loan at end of 4 years

Value of accrued loan taken in 1st year = 26256(1+0.065/12)^4*12 = $34028.31

Value of accrued loan taken in 2nd year = 35000(1+0.065/12)^3*12 = $42513.51

Value of accrued loan taken in 3rd year = 55000(1+0.065/12)^2*12 = $62613.59

Value of accrued loan taken in 4th year = 55000(1+0.065/12)^1*12 = $58683.45

Total Value of loan pending loan amount at end of 4th year = 34028.31 + 42513.51 + 62613.59 + 58683.45 = $197838.86

Hence, Loan Amount P = $197838.86

Interest Rate = 6.8% or 0.068/12 monthly

Number of payment periods = n = 10*12 = 120 months

Let monthly payments made be X

Hence, the sum of present value of monthly payments must be equal to the value of the loan amount

=> X/(1+r) + X/(1+r)² +....+ X/(1+r)^N = P

=> X[1- (1+r)^-N]/r = P

=> X = rP(1+r)^N/[(1+r)^N-1]

Hence, Monthly Payments =  rP(1+r)^N/[(1+r)^N-1]

= 197838.86*( 0.068/12)*(1+ 0.068/12)¹²⁰/((1+ 0.068/12)¹²⁰-1) = $2276.74