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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Term structure of interest rates

The spot yield curve is upward sloping. For an n-year coupon bond (n ≥ 2) with annual coupons, how does the n-year par yield compare with the n-year spot rate s_n?

On an upward-sloping spot curve the n-year par yield is below the n-year spot rate. It is a weighted average of the spot rates, and the early coupons are discounted at lower spot rates. The two are equal only when the curve is flat.

  1. AIt is below s_n, because early coupons are discounted at lower spot ratesCorrect
  2. BIt is equal to s_n for any shape of curve
  3. CIt is above s_n, because coupons are received before maturity
  4. DIt is equal to the average of s_1 and s_n
  5. It is equal to the one-year forward rate f(n-1, 1)

Explanation

The par yield is a weighted average of the spot rates, with weights that depend on the discount factors. On an upward-sloping curve the earlier coupons are discounted at spot rates below s_n, so the par yield is less than s_n. It would equal s_n only for a flat curve. The par yield is not a simple average of s_1 and s_n, and it is not a forward rate.

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