Question1.18 Which of the following statements about the yield to maturity (YTM) on a 6-year, 5% annual coupon bond is true?Select one alternative: None of the options is correct. The YTM is equal to the realised yield if the bond is held until maturity and interest rates increase in the future. The YTM is greater than the realised compound yield if the bond is held until maturity and interest rates decrease in the future. The YTM is lower than the realised yield if the bond is held until maturity and interest rates decrease in the future. The YTM is equal to the realised yield if the bond is held until maturity. ResetMaximum marks: 2.5 Flag question undefined

Question

BlackTom AI · Accepted Answer

Yield to maturity (YTM) is the single discount rate y that makes the present value of all future coupon and principal payments equal to the bond’s current price P:
P = sum_{t=1}^{n} C/(1+y)^t + M/(1+y)^n,
where C is the annual coupon, M is the maturity (face) value and n = 6 here.

1. Key concepts
- YTM: the internal rate that prices the bond today.
- Realised (compound) yield when holding to maturity: the actual compound rate earned on the investment taking into account coupon receipts and their reinvestment.
- Equivalence condition: if coupons are reinvested at the same rate y, the realised compound yield equals y.

2. Step-by-step reasoning
Start from the pricing equation above. Multiply both sides by (1+y)^n to convert present values into accumulated (future) values at time n:
P(1+y)^n = C[(1+y)^{n-1} + (1+y)^{n-2} + ... + 1] + M.

3. Intermediate steps / algebra
- P = Σ_{t=1}^{n} C/(1+y)^t + M/(1+y)^n
- Multiply by (1+y)^n:
  P(1+y)^n = C[(1+y)^{n-1} + ... + (1+y)^0] + M.
- The right-hand side is exactly the future value at maturity of reinvesting each coupon C at rate y until time n, plus the principal M.
Thus if you pay price P and reinvest all coupons at rate y, the accumulated value at n grows at compound rate y from P to P(1+y)^n. The internal rate of return on the investment = y.

4. Why the correct answer is correct
Because YTM is defined to equate price and discounted cash flows; if you hold to maturity and coupons are reinvested at the YTM, the realised compounded return equals that same rate. The algebra above shows accumulation at y, proving equality.

5. Why the other options are incorrect
- “None of the options is correct.” — incorrect because the standard result (YTM equals realised yield when held to maturity with reinvestment at YTM) is valid.
- Statements that compare YTM to realised yield when rates increase/decrease are incomplete: realised yield depends on the actual reinvestment rates of coupons. If reinvestment rates differ from YTM, realised yield can be higher or lower accordingly, but no general unqualified statement (as given) is correct.

6. Conclusion
Therefore the correct choice is: “The YTM is equal to the realised yield if the bond is held until maturity.” This matches the supplied answer.

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