If the Treasury yield curve is downward sloping, how should the yield to maturity on a 20-year Treasury coupon bond compare to that on a 1-year Treasury bill (bond)?

Question

If the Treasury yield curve is downward sloping, how should the yield to maturity on a 20-year Treasury coupon bond compare to that on a 1-year Treasury bill (bond)?

BlackTom AI · Accepted Answer

A downward-sloping yield curve means yields on longer-maturity Treasury securities are lower than yields on shorter-maturity securities (short-term rates > long-term rates). Yield to maturity (YTM) is the single constant discount rate that equates a bond’s future cash flows (coupons and principal) to its current price. For coupon-bearing bonds the YTM effectively represents a weighted average of the relevant spot (term) rates for each cash flow.

1. Relevant concepts
- Yield curve: plot of yields by maturity; "downward-sloping" = long rates < short rates.
- Yield to maturity: the internal rate of return that discounts all coupon and principal payments to the present price.
- Relationship: for positive-coupon bonds the YTM lies between the minimum and maximum of the spot yields that apply to the bond’s cash flows.

2. Step-by-step reasoning
- If the 1-year spot (bill) rate is higher than all long-term spot rates, each of the 20 annual cash flows (19 coupons + final coupon+principal) is exposed mostly to the lower long-term rates rather than the high 1-year rate.
- The YTM for the 20-year coupon bond will therefore be a weighted average of long-term spot rates (which are lower) and possibly the short 1-year rate for the first coupon; overall this weighted average is below the high short-term 1-year yield when the curve is downward sloping.
- Numerical intuition: suppose 1-year rate = 4% and long-term rates ≈ 2%. A 20-year bond’s cash flows are largely discounted at rates near 2%, yielding a YTM close to 2–2.5% — clearly below 4%.

3. Calculational note
- No explicit algebra required, but formally YTM solves Price = Σ (C / (1+YTM)^t ) + (F / (1+YTM)^20). Since the spot rates used to price each term are lower for long maturities, the single YTM that equates price will be lower than the high 1-year spot.

4. Why the correct answer is correct
- Because a downward-sloping curve means long-term spot rates are lower than the short-term rate; the 20-year bond’s YTM (a weighted average of those lower rates) must be less than the 1-year bill yield.

5. Why other options are incorrect
- A: Wrong—this asserts the opposite of the yield-curve statement.  
- C: Wrong—yields would only be equal on a flat yield curve, not a downward-sloping one.  
- D: Wrong—the coupon rate does not change the ordering imposed by the term structure; YTM will still reflect the lower long-term rates.

Conclusion: The yield on a 20-year bond would be less than that on a 1-year bill (Answer B).

If the Treasury yield curve is downward sloping, how should the yield to maturity on a 20-year Treasury coupon bond compare to that on a 1-year Treasury bill (bond)?单项选择题

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