Suppose that you are given the following information about two callable bonds that can be called immediately:

Estimated % price change if Benchmark Yield Curve Change by

BOND

-100 basis points

+100 basis points

ABC

+5.4%

-8%

XYZ

+22%

-17%

Calculate the effective duration of the bond that displays negative convexity.

(Round to two decimal places.)

Question

Suppose that you are given the following information about two callable bonds that can be called immediately:

Estimated % price change if Benchmark Yield Curve Change by

BOND

-100 basis points

+100 basis points

ABC

+5.4%

-8%

XYZ

+22%

-17%

Calculate the effective duration of the bond that displays negative convexity.

(Round to two decimal places.)

BlackTom AI · Accepted Answer

Negative convexity means the price increase for a given fall in yield is smaller in magnitude than the price decrease for the same rise in yield. Comparing the two bonds: ABC moves +5.4% for −100 bps and −8.0% for +100 bps (5.4% < 8.0%), while XYZ moves +22% for −100 bps and −17% for +100 bps (22% > 17%). Therefore ABC displays negative convexity.

Relevant knowledge points:
- Effective duration approximates the sensitivity of price to small parallel yield shifts: Duration ≈ (P↓ − P↑) / (2 × P0 × Δy), where P↓ is price if yields fall, P↑ is price if yields rise, P0 is initial price, and Δy is the yield change in decimal form.
- Percent price changes can be converted to dollar price changes by setting P0 to a convenient base (commonly 100).

Step-by-step calculation (using P0 = 100):
1. Convert percent changes to price levels:
   - For ABC: P↓ (yield −100 bps) = 100 × (1 + 0.054) = 105.4
                P↑ (yield +100 bps) = 100 × (1 − 0.08) = 92.0
2. Δy = 100 basis points = 1% = 0.01 (in decimal).
3. Plug into effective duration formula:
   Duration = (P↓ − P↑) / (2 × P0 × Δy)
            = (105.4 − 92.0) / (2 × 100 × 0.01)
            = 13.4 / 2
            = 6.7

Why this is correct:
- The formula uses symmetric yield shifts and the percent price changes provided; choosing P0 = 100 cancels units and yields the duration in years (or time units consistent with yield convention).
- ABC was identified as the negatively convex bond because its price rise (5.4%) is smaller than its price fall (8.0%) for equal magnitude yield moves, meeting the negative convexity criterion.

Why the other bond is incorrect:
- XYZ shows a larger price increase (22%) than price decrease (17%) for symmetric yield moves, indicating positive convexity; it is therefore not the bond with negative convexity.

Conclusion: The effective duration of the bond that displays negative convexity (ABC) is 6.70, matching the provided answer.

类似问题

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Consider the following statements. Choose the correct statement.

A 6% coupon bond paying interest annually has a modified duration of 10 years, sells for $800, and is priced at a yield to maturity of 8%. If the YTM increases to 9%, what is the predicted dollar change in price based on the bond’s duration? (Round to 2 decimal places.)

Question at position 18 Which bond is most sensitive to interest rate changes?All equal5-year, 8% coupon3-year, 10% coupon10-year, 3% coupon

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An investor buys a 6% annual payment bond with three years to maturity. The bond has a yield-to-maturity of 8% and is currently priced at 94.845806 per 100 of par. The bond’s Macaulay duration is closest to:

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