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Duration and convexity can be used to measure the interest rate risk of a portfolio of bonds, similar to a single bond. There are two methods to calculate the duration and convexity of a bond portfolio:
The first technique is theoretically more precise. However, its practical application proves challenging. Consequently, the emphasis tends to lean towards the second approach, largely due to its common usage among fixed-income portfolio managers. However, it operates under the assumption that yield changes are uniform across all maturities, leading to a parallel shift in the yield curve. Contrary to this assumption, yield curve shifts are typically observed as steepening, flattening, or even twisting yield curves, making pure parallel shifts rare.
Example: Calculating Weighted Average Duration and Convexity
An investor purchases $5 million par value of a 4-year, zero-coupon bond and a 5-year, fixed-rate semi-annual coupon bond. Details of the bonds are shown below.
\[ \begin{array}{c|c|c|c|c|c|c} \textbf{Bond} & \textbf{Maturity (Years)} & \textbf{Coupon (%)} & \textbf{Price} & \textbf{YTM (%)} & \textbf{Duration} & \textbf{Convexity} \\ \hline \text{Zero} & 4 & 0 & 87.1442228 & 3.5 & 3.8647 & 19.32367 \\ \hline \text{Semi-annual} & 5 & 4.5 & 101.115515 & 4.25 & 4.441605 & 23.12742 \\ \end{array} \]
To compute the weighted-average modified duration:
Zero-coupon bond: \(87.1442228 \times \$ 5,000,000\ = 435,721,114\)
Semi-Annual Bond: \(101.115515 \times 5,000,000\ = 505,577,575\)
\[Total\ market\ value\ = \ 435,721,114 + 505,577,575\ = \ 941,298,689\]
\[Weight\ of\ Zero – coupon\ bond:\ 435,721,114/941,298,689\ = \ 0.46289357\]
\[Weight\ of\ Semi – Annual\ Bond:\ 505,577,575/941,298,689\ = \ 0.537106426\]
\[Weighted – average\ modified\ duration = (0.46289357 \times 3.8647 + (0.537106426 \times 4.441605) = 4.174559367\]
Similar to the duration calculation above:
\[Weighted – average\ convexity\ = (0.46289357 \times 19.32367 + (0.537106426 \times 23.12742)\ = 21.36668849\]
\[\%\Delta PV_{\text{Full}} \approx ( – \text{Duration} \times \Delta y) + \left\lbrack \frac{1}{2} \times \text{Convexity} \times (\Delta y)^{2} \right\rbrack\]
Where \(\Delta y\) is the change in yield (in decimal form).
For a 100bp change, ?y=0.0100.
$$\%\Delta PV_{\text{Full}} \approx (-4.174559367 \times 0.01) + \left[ \frac{1}{2} \times 21.36668849 \times (0.01)^2 \right] = – 0.040677259 \approx – 4.0677\%$$
Question
Given that a bond portfolio has a duration of 5 years and a convexity of 50, estimate the percentage change in the portfolio’s value if there is an increase of 50 basis points in the yield-to-maturity.
- -2.438%
- -2.500%
- 2.563%
Solution
The correct answer is A.
Formula:
\[\%\Delta PV_{\text{Full}} \approx ( – \text{Duration} \times \Delta y) + \left\lbrack \frac{1}{2} \times \text{Convexity} \times (\Delta y)^{2} \right\rbrack\]
Where ?y for 50 basis points is 0.005
\[\%\Delta PV_{\text{Full}} \approx \left( – \text{5} \times 0.005 \right) + \left\lbrack \frac{1}{2} \times \text{50} \times (0.005)^{2} \right\rbrack = – 2.438\%\]