MRP Finance Internship Unit 07

Unit 07 · Practice solutions

Practice solutions

Bonds and the Price of Debt · Kenya · KSh

Worked answers to the optional practice questions. The required question is turned in to the TAs, and its answer is not published here.

Practice solutions

7.1

  1. Face value is a fixed contractual amount repaid at maturity, set when the bond is issued and never changing. Price is what the bond trades for today, set by the market's current required yield for a claim of this risk and maturity. They coincide only when the required yield happens to equal the coupon rate.
  2. Return = (100,000 − 88,500) / 88,500 = 11,500 / 88,500 = 12.99%.
  3. Infrastructure bond: 13.5% after tax. Corporate: 16% × 0.70 = 11.2% after tax. The infrastructure bond wins comfortably. A non-tax consideration favouring the corporate: many pension funds are in fact exempt from tax on investment income in some jurisdictions, which would make the corporate's full 16% comparable — so the first thing to check is whether the 30% actually applies to this investor. Alternatively, diversification away from concentrated sovereign exposure, or a shorter maturity better matched to the fund's obligations.

7.2

  1. The required yield is above 9%. The bond trades at a discount to par, which happens only when the coupon is less generous than what the market currently requires — buyers must be compensated by a capital gain from 88.50 up to 100 at maturity, on top of the coupon.
  2. Periodic coupon = 11/2 = 5.5. Periodic yield = 13/2 = 6.5%. Periods = 3 × 2 = 6. Price = 5.5 × [1 − 1.065⁻⁶]/0.065 + 100/1.065⁶ = 26.63 + 68.53 = 95.16.
  3. Because the bond's coupon is higher than what comparable investments currently pay, so the holder collects an above-market income stream for the remaining life of the bond. The premium paid is exactly the present value of that excess income. The investor loses 8 of capital at maturity but has collected more than 8 in extra coupons along the way, discounted to today. The market bids the price up until the total return on the price paid equals the required yield — so paying 108 is not generosity, it is the price at which the bond is fairly valued.

7.3

  1. Price = 11 × [1 − (1+y)⁻⁵]/y + 100/(1+y)⁵. At 12%: 39.65 + 56.74 = 96.40 (unrounded 96.395). At 12.5%: 39.17 + 55.49 = 94.66. The target of 95.20 lies between, so the yield to maturity is approximately 12.3% — to the nearest half percent, 12.5%.
  2. At 10%: price = 13 × [1 − 1.10⁻¹⁰]/0.10 + 100/1.10¹⁰ = 79.88 + 38.55 = 118.43, a gain of 18.43 per 100. The excess coupon is 13 − 10 = 3 a year for ten years; its present value at 10% is 3 × [1 − 1.10⁻¹⁰]/0.10 = 18.43 — exactly the gain. That is not a coincidence: a new par bond would pay 10, so the old bond is worth par plus the present value of the extra 3 a year it pays for the rest of its life. The price move *is* that present value, recognised today.
  3. Price each with C × [1 − (1+y)⁻⁵]/y + 100/(1+y)⁵. 10% bond: at 11.5% the price is 36.50 + 58.03 = 94.53; at 12% it is 36.05 + 56.74 = 92.79. 93.00 lies between, nearer 12%, so the YTM is about 11.9% — 12% to the nearest half percent. 16% bond: at 12% the price is 57.68 + 56.74 = 114.42; at 12.5% it is 56.97 + 55.49 = 112.46. 113.00 lies between, nearer 12.5%, so the YTM is about 12.4% — 12.5% to the nearest half percent. The 16% bond is the better buy: same issuer, same maturity, and it yields about half a point more. Its higher price only reflects its higher coupon; the yield is what puts two bonds with different coupons on the same footing.

7.4

  1. The secured bond has a claim on specific property, so if the issuer defaults the holder recovers substantially more — loss given default is lower. Since expected loss is probability of default times loss given default, and the probability of default is identical (same issuer), the entire 300-basis-point difference reflects the loss-given-default component.
  2. The market has become materially more pessimistic about this issuer specifically, since government yields are unchanged and therefore market-wide conditions did not move. I would look first at whether comparable corporates in the same sector also widened — if so it is sector news; if not, it is company-specific. Then at recent results, any covenant breach, refinancing due dates, rating actions, and whether the equity has moved correspondingly.
  3. Because an investor who may need to sell before maturity faces the risk of having to accept a poor price, or of being unable to sell at all, in a market with few buyers. They require extra yield to accept that risk. This is not compensation for default risk — it is a liquidity premium, and conflating the two leads analysts to overstate how risky an issuer is when the real problem is the thinness of the market it trades in.

7.5

  1. The 12-year holder is locked into a coupon three percentage points below the market for twelve years rather than three, so the total shortfall being suffered is four times as long. The present value of those extra nine years of shortfall is a substantial part of the price fall. Separately, the principal repayment is discounted over twelve years rather than three, so a change in the rate compounds through many more periods and reduces the value of the repayment far more sharply. Both effects push the same way, which is why price sensitivity rises steeply with maturity.
  2. The company has a serious duration mismatch: its assets are far longer than its liabilities. If rates fall, the value of its 4-year claim obligations rises modestly while its 15-year bonds rise a great deal, which is a windfall — but if rates rise, its assets fall sharply while its obligations barely change, potentially leaving it unable to meet claims without realising large losses. It should shorten the portfolio toward roughly 4-year duration, so that assets and liabilities move together and the firm is insulated from rate moves in either direction.
  3. She should lengthen the portfolio — buy long-dated bonds, which gain the most when yields fall. The risk is that she is wrong and rates rise instead, in which case those same long bonds fall three or four times as far as short ones would have, and the loss is realised immediately in mark-to-market terms. The position is a leveraged bet on the direction of rates, and it should be sized as one.