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Explain how spot rates and YTM are related.

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

The idea

A spot rate is the yield on a zero-coupon bond for a given maturity, the rate that discounts one cash flow at that date. A zero-coupon bond's YTM equals the spot rate for its maturity. A coupon bond is a bundle of zero-coupon payments. Its price is the sum of each cash flow discounted at its own spot rate, and its YTM is the single rate that reproduces that price. So YTM is approximately a weighted average of the spot rates at the cash flow dates, weighted by each flow's share of the price. When the spot curve is flat, every YTM equals the spot rate. On an upward-sloping curve a coupon bond's YTM is below the spot rate for its final maturity, because earlier, lower spot rates share the weight. On an inverted curve it is above. Spot rates are extracted from coupon bond prices by bootstrapping.

A practice question

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

Which statement best describes how a coupon bond's YTM relates to spot rates?

  1. The YTM equals the spot rate for the bond's final maturity, whatever the coupon.Why not: True only for a zero-coupon bond or a flat curve; earlier spot rates also influence a coupon bond's price.
  2. The YTM is the simple arithmetic average of the spot rates at the cash flow dates.Why not: Weights are shares of present value, not equal; the large final payment carries the most weight.
  3. The YTM is the single rate that reproduces the price found by discounting each cash flow at its own spot rate, so it acts as a weighted average of those spot rates. - correct
  4. Spot rates are the YTMs of coupon bonds, taken directly without adjustment.Why not: Coupon bond YTMs blend several spot rates, so spot rates must be bootstrapped from them.

A coupon bond is a bundle of zero-coupon payments. Its price uses each spot rate once; its YTM is the one rate that gives the same price, so it blends the spot rates in proportion to present values.

A harder one

The 1-year and 2-year spot rates are 1.00% and 3.00% (annual compounding). A 3-year bond pays a 6% annual coupon and trades at par (100). The 3-year spot rate implied by these prices is closest to:

  1. 6.00%Why not: Assumes the 3-year spot rate equals the bond's YTM, which is the par yield of 6%.
  2. 6.40%Why not: Subtracts the first two coupons (12) undiscounted instead of discounting them at their spot rates.
  3. 4.19%Why not: Treats the final cash flow as 100, leaving out the third coupon.
  4. 6.24% - correct

Value of the first two coupons = 6/1.01 + 6/1.03² = 11.60. The final payment of 106 must be worth 100 - 11.60 = 88.40, so (1 + s3)³ = 106/88.40 and s3 = 6.24%.

The hardest tier

Spot rates (annual compounding) are 8% for one year, 5% for two years and 3% for three years, an inverted curve. A 3-year bond pays a 10% annual coupon on face value 100. Its YTM is closest to:

  1. 3.00%Why not: Sets the YTM equal to the 3-year spot rate, ignoring the higher early spot rates that weigh on the price.
  2. 3.25% - correct
  3. 5.33%Why not: Takes the simple average of the three spot rates instead of solving for the single yield.
  4. 10.00%Why not: Reports the coupon rate, which would be the YTM only at a price of par.

Price = 10/1.08 + 10/1.05² + 110/1.03³ = 119.00. The YTM that gives this price is 3.25%, a little above the 3-year spot rate of 3.00% because the higher 8% and 5% spot rates apply to the early coupons.

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