Question textYou own a bond with 15 years to maturity, a $1,000 par value and 7% p.a. interest paid annually. The bond's market price is $850, and your required rate of return is 10% p.a.Note: for calculation questions in the following parts, round the answers to 2 decimal places(a) What is the yield to maturity of this bond?The annual yield to maturity of this bond is Answer 1 Question 3[input]%(b)Determine the value of the bond to you, given your required rate of return.The value of this bond is $Answer 2 Question 3[input](c)You should Multiple choice 1 Question 3continue to hold the bond because the bond's yield to maturity is higher than your required rate of return and thus it is under-pricedcontinue to hold the bond because the bond's yield to maturity is lower than your required rate of return and thus it is over-priced.sell the bond because the bond's yield to maturity is lower than your required rate of return and thus it is over-priced.sell the bond because the bond's yield to maturity is higher than your required rate of return and thus it is under-pricedThe correct answer is: sell the bond because the bond's yield to maturity is lower than your required rate of return and thus it is over-priced.

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We start by restating the problem in our own words and then walk through the logical steps for each part.

Part (a) – What is the yield to maturity (YTM) for this bond?
- The bond pays a annual coupon of 7% on a 1,000 par value, so the annual coupon is 70.
- It has 15 years to maturity and a market price of 850. The YTM is the single discount rate that equates the present value of all future cash flows (the 70 annual coupons for 15 years plus 1,000 returned at the end) to the current price of 850. Mathematically, we solve for i in the equation: 70 * [1 - (1+i)^-15]/i + 1000/(1+i)^15 = 850.
- There is no closed-form algebraic solution for i; it must be found by a numerical method or a financial calculator. The value obtained by such calculation is approximately 8.84% per year.
- Why this makes sense: since the bond is priced below par (850 < 1,000) and its coupon rate (7%) is below your required yield, you must earn more than the coupon gives, so the YTM ends up higher than the coupon, here around 8.8%. If you compare to the price movement intuition, a deeper discount is needed to raise the yield to maturity to a level that makes the present value of cash flows equal to 850.
- Common pitfall notes: if you misinterpret the cash flows or forget the final 1,000 at maturity, you’ll get an incorrect YTM. Also, YTM assumes reinvestment of coupons at the YTM rate, which is an assumption worth noting.

Part (b) – Determine the value of the bond to you given your required rate of return (10%).
- The value to you is the present value of all expected cash flows discounted at your required rate of return, which is 10% in this case.
- Compute using: Value = 70 * [1 - (1.10)^-15]/0.10 + 1000/(1.10)^15.
- Step-by-step: First, compute the annuity factor a-angle-15 at 10%, which equals [1 - (1.10)^-15]/0.10 ≈ 7.605. Multiply by the annual coupon: 70 * 7.605 ≈ 532.35. Next, discount the par value: 1000/(1.10)^15 ≈ 239.47. Sum: 532.35 + 239.47 ≈ 771.82.
- Therefore, the value to you, given a 10% required return, is about 771.82.
- Why this result makes sense: since your required return (10%) is higher than the coupon yield (7%), you discount cash flows more aggressively, resulting in a price well below par, here around 771.82. If the required return were equal to or lower than the coupon rate, the value would be higher (closer to or above 1,000).

Part (c) – Which statement best describes whether you should hold or sell the bond?
- The provided multiple-choice options express various judgments about holding versus selling based on the relationship between YTM and your required return.
- The correct conceptual criterion is: if the bond’s YTM is lower than your required rate of return, the bond is overpriced relative to your return target, so you should consider selling. Conversely, if YTM exceeds your required return, the bond is underpriced relative to your target and you might buy or hold for upside.
- Evaluating the given statement choices:
  - Statement: "continue to hold the bond because the bond's yield to maturity is higher than your required rate of return and thus it is under-priced." This would hold if YTM > required return, indicating underpricing; however, in this problem the YTM is 8.84% and your required return is 10%, so this condition is not met. Therefore this statement is incorrect.
  - Statement: "continue to hold the bond because the bond's yield to maturity is lower than your required rate of return and thus it is over-priced." This matches the observed condition (YTM ≈ 8.84% < 10%), indicating the bond is overpriced relative to your return target, so not the best action if you’re aiming to maximize your return; this statement is inconsistent with the desired action if you want to adjust for a 10% target, so it is not the ideal recommendation.
  - Statement: "sell the bond because the bond's yield to maturity is lower than your required rate of return and thus it is over-priced." This correctly captures the logic: YTM < required return implies the bond is overpriced relative to your return objective, so selling is the sensible action. This aligns with standard fixed-income valuation logic.
  - Statement: "sell the bond because the bond's yield to maturity is higher than your required rate of return and thus it is under-priced." This misstates the relationship: if YTM were higher than your required return, the bond would be underpriced, and selling would not be the usual action; here the YTM is not higher than 10%, so this option is incorrect.
- In sum, the reasoning that aligns with the numbers is that you would sell because the YTM is lower than your required return, indicating the bond is overpriced relative to your return target.

MCD2170 - T2 - 2025 Week 6 Post class homework

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