The prices of the zero-coupon bonds with a face value of $1000 and maturities of 1, 2, and 3 years are $980.58, $970.85, and $969.73, respectively. Which one of the following statements is true? (To avoid rounding issues, express spot rates as percentages with two decimals, e.g., 2.38%.)

Question

BlackTom AI · Accepted Answer

We start by extracting the implied one-year, two-year, and three-year spot rates from the given zero-coupon prices.

- For 1-year zero with face value 1000 and price 980.58:
  1000 / (1 + s1) = 980.58  =>  1 + s1 = 1000 / 980.58 ≈ 1.01982
  s1 ≈ 1.98%

- For 2-year zero with price 970.85:
  1000 / (1 + s2)^2 = 970.85  =>  (1 + s2)^2 = 1000 / 970.85 ≈ 1.03000
  1 + s2 ≈ sqrt(1.03000) ≈ 1.01493
  s2 ≈ 1.49%

- For 3-year zero with price 969.73:
  1000 / (1 + s3)^3 = 969.73  =>  (1 + s3)^3 = 1000 / 969.73 ≈ 1.03096
  1 + s3 ≈ (1.03096)^(1/3) ≈ 1.0102
  s3 ≈ 1.02%

These spot rates show a downward trend as maturity increases: the 1-year rate is about 1.98%, the 2-year rate about 1.49%, and the 3-year rate about 1.02%.

Now evaluate each answer option in light of these findings:

- The yield curve over the next 3 years is flat.
  A flat yield curve would imply that the spot rates are equal across maturities. Our calculations give noticeably different rates (1.98%, 1.49%, 1.02%), so this option is not supported.

- The yield curve over the next 3 years has a non-monotonic (i.e., “U” or inverted “U”) shape.
  A non-monotonic shape would require the rates to rise and fall in an alternating fashion across maturities. The computed sequence is steadily decreasing with maturity, not rising at any point, so this is incorrect.

- None of the answers are correct.
  Since we found a consistent interpretation from the prices that aligns with a downward-sloping curve, this option is not accurate.

- The yield curve over the next 3 years is downward sloping.
  With s1 ≈ 1.98%, s2 ≈ 1.49%, and s3 ≈ 1.02%, the longer maturities offer lower yields, which corresponds to a downward-sloping curve.

- The yield curve over the next 3 years is upward sloping.
  This would require longer maturities to have higher spot rates, which is contradicted by the calculated values above.

In summary, the computed spot rates from the given zero-coupon prices indicate a clear downward trend in rates with increasing maturity, leading to a downward-sloping yield curve for the next three years.

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