Debt Capital Markets puzzles, solved step by step

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  1. 074One-year and two-year zero rates are 7% and 8%. A two-year bond with a 10% annual coupon trades at 104.00. Is it rich or cheap to the zero curve, and by how much per Rs 100?Bond pricing and yieldHardSyndicate desksFixed income asset management

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    Against the zero curve, the bond is:

    Show the worked solution

    The bond is rich by about 0.35 per Rs 100. Price each cash flow at its own zero rate: the year-one coupon of 10 at 7% is worth 9.35, and the final 110 at 8% for two years is worth 94.31. Fair value is 103.65, so a price of 104.00 is 0.35 too high. In yield terms the bond yields about 7.76% against a fair 7.95%, roughly 19 basis points too low.

    Why price each cash flow at its own rate?

    Money due in one year and money due in two years are different goods, the way a train ticket for next week and one for next month carry different prices. A zero rate is the price of money for one specific date, so each of a bond's cash flows is discounted at the zero rate for its own date, and the bond is worth the sum. The year-one coupon of 10 is discounted at 7%; the year-two payment of 110 is discounted at 8% for two years.

    Price each cash flow at its own zero rate, then compare with the marketYear 1: coupon 10at 7% for 1 year9.35Year 2: 100 + 10 = 110at 8% for 2 years94.31Fair value from the curve103.65Per Rs 100, axis starts at 102Fair value103.65Market price104.00rich by 0.35: yield 7.76% against a fair 7.95%
    The coupon of 10 at 7% is worth 9.35 and the final 110 at 8% for two years is worth 94.31, a fair value of 103.65, so the market price of 104.00 is 0.35 rich and the bond yields about 19 basis points less than the curve says it should.
    The relationship
    P∗=101.07+1101.082=9.346+94.307=103.65104.00−103.65=0.35P^{*} = \frac{10}{1.07} + \frac{110}{1.08^{2}} = 9.346 + 94.307 = 103.65 \qquad 104.00 - 103.65 = 0.35
    P*the fair price implied by the zero curve
    1.07one plus the one-year zero rate
    1.08^2two years of growth at the two-year zero rate
    What it says in wordsFair value is each cash flow discounted at its own date's zero rate; rich or cheap is the market price against that.

    What does rich mean, and how would a desk use it?

    Rich means the price is above what the curve says the cash flows are worth, so the bond yields less than it should: about 7.76% against a fair 7.95%. An investor would rather own the same cash flows through the curve, and a relative value desk might sell the bond and buy a matching pair of zero-coupon bonds, locking in about 0.35 per Rs 100 if the prices converge. On Rs 100 crore of face that is about Rs 35 lakh.

    Say what could make the gap real rather than a mistake. A gap of 0.35 can reflect liquidity, a different tax treatment of high-coupon bonds, or the cost of actually building the matching zero-coupon pair, not only a mispricing. The check is whether similar bonds show the same gap. The limit: this assumes the zero rates are for the same issuer's credit; a government zero curve would value a corporate bond too high.

    Where candidates lose it

    The instinctive answer is that a 10% coupon against 7% and 8% rates must be cheap. The high coupon is already in the price; the question is whether 104.00 is the right price for those cash flows, and it is not.

    The second slip is discounting both cash flows at 8%, which gives 103.57 and overstates how rich the bond is. Match each cash flow to its own date's rate.

    What the interviewer asks next

    • At what yield would the bond need to trade to be fair to the curve?
    • How would you lock in the gap if you believed it was a pure mispricing?
    • If the two-year zero rate rises to 8.5%, is the bond still rich at 104.00?