Mr. Robert borrows K26,000 and must repay K34,000 in 4 years, what is the yield to maturity (YTM) or annual interest rate? Mr. Peter borrows K50,000 and repays with equal annual payments of K5,224 for 15 years, what is the yield to maturity (YTM) or annual interest rate implied? Comment on these two types of credit instruments.

What you are solving

Both questions ask for the interest rate that makes the present value (PV) of the promised repayments equal to the amount borrowed. That rate is the loan’s yield to maturity (YTM), also called the implied annual interest rate.

1) Mr. Robert: single payment in 4 years

Here there is only one future cash flow.

• Amount borrowed (PV): $26{,}000$
• Amount repaid at maturity (FV): $34{,}000$
• Time: $n=4$ years

Use the compound interest relationship: $$PV(1+r)^n = FV$$ So, $$26{,}000(1+r)^4 = 34{,}000$$ $$1+r = \left(\frac{34{,}000}{26{,}000}\right)^{1/4}$$ $$r = \left(\frac{34{,}000}{26{,}000}\right)^{1/4}-1 \approx 0.0694$$ So the YTM is about $6.94\%$ per year.

2) Mr. Peter: level annual payments for 15 years (annuity)

This is a fixed-payment amortizing loan. The PV of an ordinary annuity is: $$PV = PMT\cdot\frac{1-(1+r)^{-n}}{r}$$ Substitute the values: $$50{,}000 = 5{,}224\cdot\frac{1-(1+r)^{-15}}{r}$$ Divide both sides by $5{,}224$ to see the annuity factor required: $$\frac{50{,}000}{5{,}224} \approx 9.571$$ Now find $r$ such that: $$\frac{1-(1+r)^{-15}}{r} \approx 9.571$$ Solving numerically (trial-and-improve or a financial calculator) gives: $$r \approx 0.0625$$ So the implied YTM is about $6.25\%$ per year.

Comment: what these credit instruments are

• Mr. Robert’s loan is a single-payment (discount / zero-coupon style) loan: no interim payments, and all repayment happens at maturity. This creates a larger “lump-sum” refinancing and liquidity burden for the borrower.
• Mr. Peter’s loan is a fixed-payment amortizing (installment) loan: equal annual payments, with interest paid early and principal gradually paid down over time. This is typically easier for cash-flow planning and reduces outstanding balance (credit risk) over the life of the loan.

Quick self-check

• If the repayment is bigger than the amount borrowed, $r$ must be positive.
• A longer amortizing loan with relatively small payments usually implies a moderate interest rate, consistent with the $\approx 6.25\%$ result.