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Explain what the coupon effect is and how coupon rate, YTM and bond prices relate to one another.

FRM® Part I · Valuation and Risk Models · Bond Yields and Return Calculations · FRM1.VR.11.g

The idea

The coupon effect is the fact that bonds with the same maturity but different coupon rates have different YTMs whenever the spot curve is not flat. A high-coupon bond receives more of its value early, so its YTM leans toward the lower early spot rates. On an upward-sloping curve it has a lower YTM than a low-coupon or zero-coupon bond of the same maturity, and the order reverses on an inverted curve. Separately, coupon, YTM and price are linked. If the coupon equals the YTM the bond trades at par, if the coupon is above the YTM it trades at a premium, and if below at a discount. With the YTM unchanged, the price is pulled toward par as maturity shrinks. For the same yield move, low-coupon bonds change more in percentage price terms. Do not confuse the coupon effect on yields with this price-yield relation.

A practice question

From the question bank, with the reason each wrong option is wrong.

A bond has a coupon rate of 6% and a YTM of 5%. Relative to its par value, the bond trades at:

  1. A discount, because the YTM is below the coupon rate.Why not: Reverses the relation: a yield below the coupon means the price is above par.
  2. Par, because both rates are positive and close to each other.Why not: Par requires the coupon to equal the YTM, not merely be close to it.
  3. A premium, because the coupon exceeds the yield investors require. - correct
  4. An amount that cannot be determined without the bond's maturity.Why not: The premium or discount sign depends only on coupon versus YTM; maturity affects its size.

Price is above par when coupon is above YTM, equal to par when they are equal, and below par when the coupon is below the YTM. Here 6% is above 5%, so the bond sells at a premium.

A harder one

Two 10-year bonds with face value 100 pay annual coupons of 3% and 7%. Both have a YTM of 5% (annual compounding). Which pair of prices is correct, in the order (3% bond, 7% bond)?

  1. 84.56 and 115.44 - correct
  2. 115.44 and 84.56Why not: Swaps the two: the lower-coupon bond sells at a discount and the higher-coupon bond at a premium.
  3. 100.00 and 100.00Why not: Assumes equal YTMs force both bonds to par, which only holds when the coupon equals the YTM.
  4. 80.00 and 120.00Why not: Uses a straight-line price adjustment of 1 point per year, ignoring discounting.

3% bond: 3 x annuity(5%, 10) + 100/1.05^10 = 84.56. 7% bond: 115.44. The coupon is below the YTM for the first bond (discount) and above it for the second (premium).

The hardest tier

Spot rates (annual compounding) are 1% for one year, 2% for two years and 5% for three years. Three 3-year bonds are priced off this curve: a zero-coupon bond, a 3% coupon bond and a 9% coupon bond. Which set of YTMs, in the order (zero, 3% coupon, 9% coupon), is correct?

  1. 5.00%, 4.90%, 4.72% - correct
  2. 4.72%, 4.90%, 5.00%Why not: Reverses the order: on an upward-sloping curve the highest-coupon bond has the lowest YTM.
  3. 5.00%, 5.00%, 5.00%Why not: Sets every YTM equal to the 3-year spot rate, ignoring the coupon effect.
  4. 5.00%, 3.00%, 9.00%Why not: Equates each YTM with the coupon rate, which holds only for par bonds.

The zero-coupon YTM is the 3-year spot rate, 5.00%. Coupon bonds receive cash earlier, at lower spot rates, so their YTMs are lower: 4.90% for the 3% coupon and 4.72% for the 9% coupon. Higher coupon, lower YTM on an upward-sloping curve.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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