Bond Risk and Return Using Duration and Convexity

The percentage price change of a bond, given a specified change in yield, can be more accurately estimated using both the bond’s duration and convexity compared to using duration alone. We will give an example to illustrate this.

Consider a 2-year, 4% semiannual coupon bond, settling on 10 June 2024, maturing on 10 June 2026, and yielding 4%—thus, priced at par. Suppose the investor has a position in the bond with a par value of USD50 million, and the yield-to-maturity increases by 100 bps.

Calculating the Actual Prices

\[PV_{0} = \frac{2}{1.02} + \frac{2}{{1.02}^{2}} + \frac{2}{{1.02}^{3}} + \frac{102}{{1.02}^{4}} = 100\]

100bp decrease in yield (3%)

\[PV_{-} = \frac{2}{1.015} + \frac{2}{(1.015)^{2}}\ + \frac{2}{(1.015)^{3}} + \frac{102}{(1.015)^{4}} = 101.9271923\]

\[\%\Delta PV^{Full} = \frac{101.9271923}{100} – 1 = 1.9272\%\]

100bp increase in yield (5%)

\[PV_{+} = \frac{2}{1.025} + \frac{2}{(1.025)^{2}}\ + \frac{2}{(1.025)^{3}}\ + \frac{102}{(1.025)^{4}}\ = 98.1190129\]

\[\%\Delta PV^{Full} = \frac{98.1190129}{100} – 1 = – 1.8810\%\]

Calculating the Modified Duration

\[ \begin{array}{c|c|c|c|c|c|c} \textbf{Period} & \textbf{Time to Receipt} & \textbf{Cashflow Amount} & \textbf{PV} & \textbf{Weights} & \textbf{Time to Receipt*Weight} & \textbf{Convexity of cashflows} \\ \hline 1 & 1.0000 & 2.0000 & 1.9608 & 0.0196 & 0.0196 & 0.0377 \\ \hline 2 & 2.0000 & 2.0000 & 1.9223 & 0.0192 & 0.0384 & 0.1109 \\ \hline 3 & 3.0000 & 2.0000 & 1.8846 & 0.0188 & 0.0565 & 0.2174 \\ \hline 4 & 4.0000 & 102.0000 & 94.2322 & 0.9423 & 3.7693 & 18.1146 \\ \hline \ & \ & \ & \textbf{100.0000} & \textbf{1.0000} & \textbf{3.8839} & \textbf{18.4805} \\  \end{array} \]

\[Annualized\ Macaulay\ duration\ \ = \frac{3.8839}{2} = 1.94195\]

\[Annualized\ convexity = \frac{18.4805}{2^{2}} = 4.620125\]

\[Modified\ duration = \frac{1.94195}{1 + \frac{0.04}{2}} = 1.9039\]

So, a 100bp increase (decrease) in yield-to-maturity results in \(\%\mathrm{\Delta}PVFull\ \)?–1.9039% (1.9039%)

Adding Convexity Adjustment to the Duration Estimate:

\[\%\Delta P_{VFull} \approx ( – \text{AnnModDur} \times \Delta\text{Yield}) + \left\lbrack \frac{1}{2} \times \text{AnnConvexity} \times (\Delta\text{Yield})^{2} \right\rbrack\]

\[\%\Delta P_{VFull} \approx ( – 1.9039 \times – 0.01) + \left\lbrack \frac{1}{2} \times 4.6201 \times ( – 0.01)^{2} \right\rbrack = 1.9270\%\]

\[\%\Delta P_{VFull} \approx ( – 1.9039 \times 0.01) + \left\lbrack \frac{1}{2} \times 4.6201 \times (0.01)^{2} \right\rbrack = – 1.8808\%\]

The results can be summarized in the following table:

\[ \begin{array}{c|c|c|c|c|c} \textbf{Change in yield} & \textbf{Actual} & \textbf{Estimated using} & \textbf{Difference from} & \textbf{Estimated using ModDur} & \textbf{Difference from} \\ & \%\Delta PV_{\text{Full}} & \textbf{ModDur} & \textbf{actual change} & \textbf{and Convexity} & \textbf{actual change} \\ \hline -100bps & 1.9272\% & 1.9039\% & -0.0233\% & 1.9270\% & -0.0002\% \\ \hline +100bps & -1.8810\% & -1.9039\% & -0.0229\% & -1.8808\% & 0.0002\% \\ \end{array} \]

Notice the enhanced precision after adding the convexity adjustment, shown by the decreased difference from the actual change.

Money duration and money convexity capture the first-order and second-order effects on the full price of a bond in currency units, respectively. The money convexity is calculated using the formula:

\[\text{MoneyCon} = \text{AnnConvexity} \times PV_{\text{Full}}\]

The change in the bond’s full price using Money Duration and Money Convexity is calculated using the formula:

\[\Delta PV_{\text{Full}} \approx – (\text{MoneyDur} \times \Delta\text{Yield}) + \left\lbrack \frac{1}{2} \times \text{MoneyCon} \times (\Delta\text{Yield})^{2} \right\rbrack\]

Question

An investor purchases a £5 million semi-annual 2.5% coupon bond with a yield-to-maturity of 1.75%, settling 01 July 2023 and maturing 01 July 2025. The bond’s ApproxCon using a 1 bp increase and decrease in yield-to-maturity is closest to:

1. 1.9464
2. 4.9277
3. 19.1756

Solution

The correct answer is B:

\[ApproxCon\ = \frac{\left( PV_{-} \right) + \left( PV_{+} \right) – \left\lbrack 2 \times \left( PV_{0} \right) \right\rbrack}{(\Delta\text{Yield})^{2} \times \left( PV_{0} \right)}\]

\[PV_{0} = \frac{1.25}{1.00875} + \frac{1.25}{{1.00875}^{2}} + \frac{1.25}{{1.00875}^{3}} + \frac{101.25}{{1.00875}^{4}} = 101.467753\]

\[PV_{-} = \frac{1.25}{1.00870} + \frac{1.25}{(1.00870)^{2}} + \frac{1.25}{(1.00870)^{3}} + \frac{101.25}{(1.00870)^{4}} = 101.4875066\]

\[PV_{+} = \frac{1.25}{1.00880} + \frac{1.25}{(1.00880)^{2}} + \frac{1.25}{(1.00880)^{3}} + \frac{101.25}{(1.00880)^{4}} = 101.4480044\]

\[ApproxCon = \frac{\left( (101.4875066 + 101.4480044) – (2 \times 101.467753) \right)}{(0.0001)^{2} \times 101.467753} = 4.9277\]