Duration and Convexity of a Bond Portfolio

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:

1. Using the weighted average of time to receipt of the aggregate cash flows.
2. Using the weighted averages of the durations and convexities of the individual bonds in the 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 a $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.

\[ \begin{array}{|c|c|c|c|c|c|c|} \hline \textbf{Bond} & \textbf{Maturity (Years)} & \textbf{Coupon (\%)} & \textbf{Price} & \textbf{YTM (\%)} & \textbf{Duration} & \textbf{Convexity} \\ \hline \text{Zero} & 4 & 0 & 87.1442228 & 3.5 & 3.8647 & 19.32367 \\ \hline \text{Semi-annual} & 5 & 4.5 & 101.115515 & 4.25 & 4.441605 & 23.12742 \\ \hline \end{array} \]

1. Calculate the weighted-average modified duration for the portfolio.
2. Calculate the weighted average convexity for the portfolio.
3. Calculate the estimated percentage price change of the portfolio given a 100 bp increase in yield-to-maturity on each of the bonds.

Calculating the weighted-average modified duration for the portfolio.

To compute the weighted-average modified duration:

Determine the market value for each bond.

Zero-coupon bond:

$$87.1442228 \times \frac{5,000,000}{100} = 4,357,211.14$$

Semi-Annual Bond:

$$101.115515 \times \frac{5,000,000}{100} = 5,055,775.75$$

Calculate the weight of each bond.

\[Total\ market\ value\ = \ 4,357,211.14 + 5,055,775.75\ = 9,412,986.89\]

\[Weight\ of\ Zero – coupon\ bond:\ 4,357,211.14/9,412,986.89\ = \ 0.46289357\]

\[Weight\ of\ Semi – Annual\ Bond:\ 5,055,775.75/9,412,986.89\ = \ 0.537106426\]

Multiply the weight of each bond by its duration and sum the results.

\[Weighted – average\ modified\ duration = (0.46289357 \times 3.8647 + (0.537106426 \times 4.441605) = 4.174559367\]

Calculating the weighted-average convexity for the portfolio.

Similar to the duration calculation above:

1. Determine the market value for each bond (which we have already done in step i).
2. Calculate the weight for each bond (which we have also done in step ii).
3. Multiply the weight of each bond by its convexity and sum the results.

\[Weighted – average\ convexity\ = (0.46289357 \times 19.32367 + (0.537106426 \times 23.12742)\ = 21.36668849\]

Calculating the estimated percentage price change of the portfolio given a 100 bp increase in yield-to-maturity on each of the bonds.

\[\%\Delta PV_{\text{Full}} \approx ( – \text{Duration} \times \Delta y) + \left\lbrack \frac{1}{2} \times \text{Convexity} \times (\Delta y)^{2} \right\rbrack\]

Where \(\Delta y\) is the change in yield (in decimal form).

For a 100bp change, Δy=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.

1. -2.438%.
2. -2.500%.
3. 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 Δy for 50 basis points is 0.005

\[\%\Delta PV_{\text{Full}} \approx \left( – \text{5} \times 0.005 \right) + \left\lbrack \frac{1}{2} \times \text{50} \times (0.005)^{2} \right\rbrack = – 2.438\%\]