 # Measuring Bond Exposure

Shaping risk is the sensitivity of a bond’s price to changes in the shape of the yield curve. An active bond investor trades based on the predicted shape of the yield curve.

Yield curve risk is the bond portfolio exposure to shifts in the yield curve. Yield curve movements can be explained by independent changes along any of its different dimensions, including level, steepness, and curvature.

## Level, Steepness, and Curvature

• Level: Relates to the parallel up or down movement of the yield curve. Empirical evidence shows that up and downshifts in the yield curve explain more than 75% of the yield curve’s total change.
• Steepness: Relates to the non-parallel shift in the yield curve, i.e., an increase in the long-term interest rates with a decrease in the short-term interest rates.
• Curvature: The rise in the long-term and short-term parts of the yield curve with the middle part falling or vice versa.

Let $$D_L$$, $$D_S$$, and $$D_C$$ be a given portfolio’s sensitivities to small changes in the level, slope, and curvature factors, respectively.

A small change in the level factor, slope factor, and curvature factor would result in a proportional change in the portfolio value as given below:

$$\cfrac {\Delta P}{P}\approx -D_L \Delta x_L-D_S \Delta x_S-D_C \Delta x_C$$

Where $$\Delta x$$ is the change in the respective factors.

#### Example: Level, Steepness, and Curvature

The yield curve risk of a certain bond portfolio is expressed as:

$$\cfrac {\Delta P}{P}\approx -2\Delta x_L-3\Delta x_S-\Delta x_C$$

Given that the following yield curve changes occurred: $$\Delta x_L=0.003, \Delta x_S=-0.002,\Delta x_C=-0.004$$, calculate the percentage change in the value of the portfolio.

#### Solution

$$\cfrac {\Delta P}{P}\approx-2×0.003-3×-0.002–0.004=0.004=0.4\%$$

Therefore, 0.4% is the predicted increase in the portfolio’s value resulting from shifts in the yield curve.

The yield curve risk can also be measured and managed using effective duration and key rate duration measures as discussed below:

## Effective Duration

The effective duration measures the price sensitivity to a small parallel shift in the benchmark yield curve, assuming that the bond’s credit spread remains constant.

The effective duration of a zero-coupon bond is equivalent to its maturity. This measure is inappropriate for identifying and managing the yield curve risk associated with non-parallel shifts.

## Key Rate Duration

The key rate duration measures a bond’s sensitivity to a small change in a benchmark yield curve at a specific spot rate, keeping all else constant.

Unlike effective duration, this measure allows for identifying and managing risk, i.e., interest rate sensitivity to non-parallel shifts in the yield curve.

The formula for key rate duration is the following:

$$\cfrac {\Delta P}{P}\approx -D_1 \Delta r_1-D_2 \Delta r_2-D_3\Delta r_3$$

Where:

$$D_i$$ is the key rate duration of the portfolio to the ith rate.

$$r_i$$ is the ith key rate.

## Question

Which yield curve risk measures is least appropriate for measuring shaping risk?

1. Effective duration.
2. Key rate duration.
3. A model that decomposes yield curve movements into changes in level, steepness, and curvature.

#### Solution

Shaping risk can be addressed by key rate durations and a measure based on sensitivities to level, slope, and curvature movements.

However, the effective duration is not an accurate measure of interest rate sensitivity to non-parallel shifts in the yield curve.

Reading 28: The Term Structure and Interest Rate Dynamics.

LOS 28 (i) Explain how a bond’s exposure to each of the factors driving the yield curve can be measured and how these exposures can be used to manage yield curve risks.

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