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FRM Part II · FRM Exam Part II · Arbitrage Pricing with Term Structure Models

A bank values a two-year bond with a face value of 100 that pays a 5 coupon at year 1 and 105 at year 2. The issuer may call the bond at year 1 for 100 (ex-coupon, immediately after the coupon is paid). The one-year rate is 5% today and 6% or 4% at year 1, with risk-neutral probabilities of 0.5 each. The issuer calls whenever calling is cheaper than the bond's continuation value. What is the value of the callable bond today?

The callable bond is worth about 99.55. In the down-rate state the ex-coupon value of 100.96 is capped at the 100 call price, giving an expected ex-coupon value of 99.53. Adding the 5 coupon and discounting one year at 5% gives 99.55, below the straight bond's 100.01.

  1. A94.79
  2. B99.53
  3. C99.55Correct
  4. D100.01

Explanation

Ex-coupon value at year 1 is 105/1.06 = 99.0566 (up) and 105/1.04 = 100.9615 (down). The issuer calls only in the down state, so the value there is capped at 100. The expectation is 0.5(99.0566 + 100) = 99.5283. Adding the 5 coupon and discounting at 5% gives 104.5283/1.05 = 99.55. The straight bond is 100.01, and 99.53 omits the coupon and the discounting.

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