Loan Repayment Comparison

Study guide: Exam FM-2

Loan repayment comparison problems compare two different repayment structures for a loan of the same Principal $L$ over $n$ periods at rate $i$ — most commonly level payments (standard Amortization) versus constant-principal payments (equal principal repaid each period, plus interest on the declining Outstanding Balance) — and equate or contrast the resulting payments or total interest.

$P_{\text{level}} = \frac{L}{a_{\overline{n}|i}}$

$\text{Total interest (level)} = n\,P_{\text{level}} - L$

$\text{Total interest (constant principal)} = i\,L\cdot\frac{n+1}{2}$

Under constant-principal repayment, the principal portion each period is $L/n$ , and interest is charged on the declining balance $L\cdot\frac{n-t+1}{n}$ at time $t$ ; summing the interest over all $n$ periods gives $iL\frac{n+1}{2}$ .
At the same rate, both methods charge $iL$ of interest in the first period, but level payments repay principal more slowly at first, so the balance stays higher and every later period carries more interest: total interest under level payments is larger. The two can be tied together by setting their total payments or total interest equal and solving for an unknown rate or payment.
The same comparison approach applies to two loans repaid by the same method at different Interest Rates or Term of Loans: write the equation of value for each and set the quantities of interest equal.

Solving for the Rate That Equates Two Repayment Methods A $ $3{,}000$ loan can be repaid over 5 years in one of two ways: (i) level annual payments at an annual effective rate of $10\%$ , or (ii) principal payments of $ $600$ per year plus interest on the unpaid balance at rate $j$ . The total payments under both options are equal. Find $j$ .

Answer Option (i): $P = 3{,}000 / a_{\overline{5}|0.10} = 3{,}000/3.79079 \approx \$791.39$ , so total paid $= 5(791.392) = \$3{,}956.96$ and total interest $= 3{,}956.96 - 3{,}000 = \$956.96$ .

Option (ii): Total interest $= j(3{,}000)\dfrac{6}{2} = 9{,}000j$ .

$9{,}000j = 956.96 \implies j \approx 10.63\%$

Total Interest Under Constant-Principal Repayment A borrower takes a $ $5{,}000$ loan at an annual effective interest rate of $6\%$ , to be repaid with principal payments of $ $1{,}000$ per year for 5 years plus interest on the outstanding balance. Find the total interest paid over the life of the loan.

Answer $\text{Total interest} = iL\cdot\frac{n+1}{2} = 0.06(5{,}000)\cdot\frac{6}{2} = 0.06(5{,}000)(3) = \$900$