Key Rate Duration

Key rate duration (partial duration) is a financial metric that measures the sensitivity of a bond’s price to changes in interest rates at specific points along the yield curve. On the other hand, effective duration gauges sensitivity to overall parallel shifts in the benchmark curve. Key rate durations sum up to the effective duration. To compute key rate durations, only specific points on the yield curve (e.g., 2-year, 5-year rates) are adjusted rather than the entire curve.

Key rate duration helps identify shaping risk–a bond’s reaction to changes in the shape of the yield curve. For bonds with embedded options (e.g., callable bonds), the shape of the curve matters. A downward shift can impact the bond’s price due to its negative convexity.

Key rate duration can be calculated using the following formula:

\[KeyRateDur_{k} = – \frac{1}{PV} \times \frac{\Delta PV}{\Delta r_{k}}\]

Where:

\(\Delta r_{k}\) represents the change in the kth key rate.

\(\bigtriangleup PV\) is the change in the bond’s price.

\(PV\) is the bond’s initial price.

The sum of key rate durations results in effective duration as per the following formula.

\[\sum_{k = 1}^{n}\mspace{2mu}\text{KeyRateDur}_{k} = \text{EffDur}\]

The percentage change in bond price is expressed mathematically as:

\[\frac{\Delta PV}{PV}(\%\Delta PV) = – \text{KeyRateDur}_{k} \times \Delta r_{k}\]

Example: Analyzing Bond Price Sensitivity to Non-parallel Shifts in the Benchmark Government Par Curve

Consider a scenario with non-parallel shifts in the benchmark government par curve, characterized by a pronounced steepening at longer maturities. We will determine the expected price change for each of the two bonds, Bond C and Bond D, as a result of these shifts.

Shifts in Benchmark Government Par Curve:

$$\begin{array}{|c|c|}
\hline
\textbf{Maturity} & \textbf{Expected Change} \\
\hline
1 \text{ year} & +80 \text{ bps} \\
\hline
5 \text{ years} & +120 \text{ bps} \\
\hline
10 \text{ years} & +180 \text{ bps} \\
\hline
20 \text{ years} & +230 \text{ bps} \\
\hline
30 \text{ years} & +280 \text{ bps} \\
\hline
\end{array}$$

Bond Details

$$\begin{array}{|c|c|c|}
\hline
\text{Bond} & \text{Tenor} & \text{Key Rate Duration} \\
\hline
\text{Bond C} & 5 \text{ years} & 2.30 \\
\hline
\text{Bond D} & 10 \text{ years} & 3.60 \\
\hline
\end{array}$$

We can compute the expected percentage price change for each bond using the following formula:

\[\frac{\Delta PV}{PV}(\%\Delta PV) = – \text{KeyRateDur}_{k} \times \Delta r_{k}\]

For Bond C:

\[\%\Delta PV_{\text{Bond }C} = – 2.30 \times 1.20\% = – 2.76\%\ \]

For Bond D:

\[\%\Delta PV_{\text{Bond }D} = – 3.60 \times 1.80\% = \ – 6.48\%\]

Bond D’s price is more sensitive to the shifts, decreasing by roughly \(6.48\%\), compared to Bond C’s decrease of \(2.76\%\). This is attributed to Bond D’s higher key rate duration.

While understanding the portfolio duration and the general shift of the benchmark yield curve offers a rapid assessment of potential profits or losses, employing key rate durations enables a portfolio manager to adjust weights in specific tenors to optimize the risk-adjusted return.

Question #1

Which of the following best describes the key rate duration of a bond?

1. The bond’s sensitivity to a uniform change in all yields of the benchmark yield curve.
2. The bond’s sensitivity to a change in the benchmark yield at a specific maturity.
3. The bond’s sensitivity to changes only in the short-term rates of the benchmark yield curve.

Solution:

The correct answer is B.

Key rate duration (or partial duration) measures a bond’s sensitivity to a change in the benchmark yield at a specific maturity.

A is incorrect: This description aligns with effective duration, which measures a bond’s sensitivity when all yields of the benchmark change uniformly.

C is incorrect: Key rate duration refers to sensitivity at a specific maturity, not just short-term rates.

Question #2

In the context of key rate durations, “shaping risk”for a bond most likely refers to:

1. The risk that all yields on the benchmark curve will change by the same amount.
2. The risk associated with changes in the shape of the benchmark yield curve, such as steepening, flattening, or twisting.
3. The risk that only short-term rates on the benchmark curve will change.

Solution:

The correct answer is B.

“Shaping risk” refers to a bond’s sensitivity to changes in the shape of the benchmark yield curve, such as it becoming steeper, flatter, or undergoing a twist.

A is incorrect: This is a description of a parallel shift, not shaping risk.

C is incorrect: Shaping risk refers to changes in the entire shape of the curve, not just short-term rates.