Bond’s Percentage Price Change Using Curve-based Duration and Convexity

Effective duration and effective convexity are curve-based metrics that are crucial for assessing the interest rate risk of complex instruments, such as those with embedded contingency provisions. These metrics are typically determined from bond prices derived using an option valuation model, given specific changes in the underlying benchmark government yield curve. Effective duration and effective convexity can be used to estimate the percentage change in a bond’s full price for a given shift in the benchmark yield curve (∆Curve). It is calculated as:

\[\%\Delta PV^{Full} \approx ( – EffDur \times \Delta Curve) + \left\lbrack \frac{1}{2} \times EffCon \times (\Delta Curve)^{2} \right\rbrack\]

Example: Bond Price Impact from Par Curve Shifts

Consider the effective duration and effective convexity data for Bond X and Bond Y. We will examine the effects of a 200bps shift in the benchmark government par curve.

$$\begin{array}{|c|c|c|} \hline \text{Bond} & \text{EffDur} & \text{EffCon} \\ \hline \text{Bond } X & 7.425 & -295.0 \\ \hline \text{Bond } Y & 6.891 & -278.310 \\ \hline \end{array}$$

Upward Shift by 200 bps:

\[\%\Delta PV^{Full} \approx ( – EffDur \times \Delta Curve) + \left\lbrack \frac{1}{2} \times EffCon \times (\Delta Curve)^{2} \right\rbrack\]

Where \(\Delta Curve = 0.02\),

Bond X: \(\%\Delta PV^{\text{Full}} = ( – 7.425 \times 0.02) + \left( 0.5 \times – 295.0 \times (0.02)^{2} \right) = – 20.75\%\)

Bond Y: \(\%\Delta PV^{Full} = ( – 6.891 \times 0.02) + \left( 0.5 \times – 278.310 \times (0.02)^{2} \right) = – 19.35\%\)

The significant decline in bond prices in response to increasing interest rates can be attributed to the high effective durations of Bond X and Bond Y, indicating their increased vulnerability to changes in interest rates. The situation is compounded by the negative convexities, which amplify the declines in bond prices. Investors who hold these bonds are, therefore, at risk of incurring losses in the event of rising interest rates. The negative convexity also indicates that bond prices may not experience significant increases even if interest rates subsequently decrease, as we will see below.

Downward Shift by 200 bps:

\(\Delta Curve = – 0.02\).

Bond X: \(\%\Delta PV^{\text{Full}} = ( – 7.425 \times – 0.02) + \left( 0.5 \times – 295.0 \times ( – 0.02)^{2} \right) = 8.95\%\)

Bond Y: \(\%\Delta PV^{Full} = ( – 6.891 \times – 0.02) + \left( 0.5 \times – 278.310 \times ( – 0.02)^{2} \right) = 8.22\%\)

Despite declining interest rates, the price appreciation of both bonds is moderate. This seemingly counterintuitive behavior can be attributed back to the negative convexities of the bonds. In rate-declining scenarios, one would anticipate bond prices to rise more significantly. However, the negative convexity lowers this increase, causing the price appreciation to be less than would be predicted based solely on duration.

Question

The effective duration and effective convexity of a bond are 4.816 and 26.723, respectively. The percentage changes in the bond’s full price for ±100bp shifts in the benchmark government par curve are closest to?

1. -4.68% and 4.95%.
2. -4.47% and 4.71%.
3. -4.816% and 26.72%.

Solution

The correct answer is A.

\(\%\Delta PV^{Full} \approx ( – EffDur \times \Delta Curve) + \left\lbrack \frac{1}{2} \times EffCon \times (\Delta Curve)^{2} \right\rbrack\)

Upward shift:

\(\Delta Curve = 0.01\)

\(\%\Delta PV^{\text{Full}} = ( – 4.816 \times 0.01) + \left( 0.5 \times 26.723 \times (0.01)^{2} \right) = – 4.68\%\)

Downward shift:

\(\Delta Curve = – 0.01\)

\(\%\Delta PV^{\text{Full}} = ( – 4.816 \times – 0.01) + \left( 0.5 \times 26.723 \times ( – 0.01)^{2} \right) = 4.95\%\)