Compute DV01, duration and convexity at the level of a fixed-income portfolio.
The idea
A portfolio's interest rate risk is aggregated differently for each measure. DV01 is additive: portfolio DV01 is the sum of the position DV01s, with short positions entering as negatives. Effective duration and convexity are market-value-weighted averages of the position figures, so portfolio duration = Σ(position value × duration) ÷ portfolio value. Portfolio convexity is weighted in the same way, by market value. Weights come from market values, not from face amounts or a simple average. For a long-short book, the dollar duration is long value × duration minus short value × duration, and dividing by net value can give very large figures, so many managers quote DV01 instead. Once you have portfolio duration D and convexity C, the estimate for a yield change Δy is the same as for a single bond: percent change = −D × Δy + 0.5 × C × (Δy)². The exam error is averaging durations without weights or adding durations together.
Formulas
Portfolio DV01
DV01p = Σ DV01i
- DV01i
- the DV01 of position i, negative for a short position
DV01 is additive in currency. Durations are not; they must be value-weighted.
Portfolio effective duration
Dp = Σ (Vi × Di) ÷ Vp
- Vi
- the market value of position i
- Di
- the effective duration of position i
- Vp
- the portfolio's total market value
Weights are market values, not face amounts. Convexity is aggregated the same way. For a long-short book a small net value can make this very large.
Percentage price change from duration and convexity
%ΔP = −D × Δy + 0.5 × C × Δy²
- D
- the effective duration
- C
- the convexity
- Δy
- the yield change, as a decimal
For a portfolio use the market-value-weighted D and C. Forgetting the 0.5 is the usual error.
A practice question
A portfolio has three positions with DV01s of USD 1,200, USD 2,500 and USD 800. The portfolio DV01 is:
- USD 4,500 - correct
- USD 1,500Why not: Takes the simple average of the position DV01s rather than their sum.
- USD 2,886Why not: Combines the DV01s as if they were independent, as a root-sum-of-squares, which is not valid for aggregating.
- USD 1,851Why not: Weights the DV01s by themselves instead of summing them.
DV01 is additive across positions: 1,200 + 2,500 + 800 = USD 4,500. Short positions enter with a negative sign.
A harder one
A portfolio is long USD 80 million market value of bonds with an effective duration of 5.5 and short USD 30 million market value of bonds with an effective duration of 8.0. The portfolio DV01 is closest to:
- USD 68,000Why not: Adds the absolute values of the long and short legs rather than netting.
- USD 33,750Why not: Uses net market value of USD 50 million times the simple average duration of 6.75.
- USD 20,000 - correct
- USD 200,000Why not: Uses 0.001 rather than 0.0001 as the basis point factor, which is a 10bp figure.
Net dollar duration = 80 x 5.5 - 30 x 8.0 = 440 - 240 = 200 million. DV01 = 200,000,000 x 0.0001 = USD 20,000.
The hardest tier
A USD 100 million portfolio has USD 60 million in bonds with duration 4.0 and convexity 25, and USD 40 million in bonds with duration 9.0 and convexity 110. If yields rise 75bp, the estimated change in portfolio value using duration and convexity is closest to:
- A loss of USD 4.50 millionWhy not: Uses portfolio duration of 6.0 alone and ignores the convexity term.
- A loss of USD 4.67 millionWhy not: Subtracts the convexity term instead of adding it.
- A loss of USD 4.17 millionWhy not: Omits the factor of 0.5 on the convexity term.
- A loss of USD 4.33 million - correct
Portfolio duration = 6.0 and convexity = 0.6 x 25 + 0.4 x 110 = 59. Change = 100 million x (-6.0 x 0.0075 + 0.5 x 59 x 0.0075²) = 100 million x (-0.045 + 0.00166) = about -4.33 million.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Applying Duration, Convexity, and DV01
- Outline a one-factor interest rate model and give examples of factors that are commonly used in it.
- Given the move in rates and the price change that results, define a fixed-income security's DV01 and compute it.
- Given the DV01 of each bond, calculate how much face amount is required to hedge a position that is sensitive to interest rates.
- Explain what effective duration is, interpret it, and calculate it for a fixed-income security from a given change in rates and the price change that results.
- Compare DV01 and effective duration as gauges of price sensitivity, noting how they are alike and how they differ.
- Explain what convexity is, interpret it, and calculate it for a fixed-income security from a given change in rates and the price change that results.
- Explain how effective duration and convexity are used to hedge a position.
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