Key Rate Duration

One Sided Durations

The computation of effective duration discussed earlier relied on computing the value of the bond for equal parallel shifts of the yield curve up and down (by the same amount). This metric captures interest rate risk reasonably well for small changes in the yield curve and for option free bonds.

For a callable bond, when the call option is at or near the money, the change in price for a decrease in yield will be less than the change in price for an equal amount of increase in yield. The value of a callable bond is capped by its call price: the bond’s value will not increase beyond the call price regardless of how low interest rates fall. Similarly, the value of a putable bond is more sensitive to downward movements in yield curve versus upward movements.

For bonds with embedded options, one sided durations, durations that apply only when interest rates rise (or, alternatively, only when rates fall), are better at capturing interest rate sensitivity than simple effective duration. When the underlying option is at the money (or near the money), callable bonds will have lower one sided down duration than one sided up duration: the price change of a callable when rates fall is smaller than the price change for an equal increase in rates. Conversely, a near the money putable bond will have larger one sided down duration than one sided up duration.

Key Rate Duration

Key rate durations or partial durations capture the interest rate sensitivity of a bond to changes in yields (par rates) of specific benchmark maturities. Key rate duration is used to identify the interest rate risk from changes in the shape of the yield curve (shaping risk).

The process of computing key rate duration is similar to the process of computing effective duration described earlier, except that instead of shifting the entire benchmark yield curve, only one specific par rate (key rate) is shifted before the price impact is measured.

Figure 25.2 shows the key rate durations for several 15 year option free bonds, each with a different coupon rate and YTM = 3%.

Figure 25.2: Key Rate Durations of Various 15 Year Option Free Bonds With Different Coupon Rates

Figure 25.3 shows the key rate durations for several 15 year European style callable bonds, each with a different coupon rate (callable in 10 years at par).

Figure 25.3: Key Rate Durations of Various 15 Year Callable Bonds With Different

Coupon Rates

Figure 25.4 shows the key rate durations for several 15 year European style putable bonds, each with a different coupon rate (putable in 10 years at par).

Figure 25.4: Key Rate Durations of Various 15 Year Putable Bonds With Different

Coupon Rates

The following generalizations can be made about key rates:

1. If an option free bond is trading at par, the bond’s maturity matched rate is the only rate that affects the bond’s value. Its maturity key rate duration is the same as its effective duration, and all other rate durations are zero.

In Figure 25.2, the 3% bond’s (i.e., the par bond) 15 year key rate duration is same as the bond’s effective duration, and all other rate durations are zero.

1. For an option free bond not trading at par, the maturity matched rate is still the most important rate. In Figure 25.2, the 15 year key rate duration is highest.
2. A bond with a low (or zero) coupon rate may have negative key rate durations for horizons other than the bond’s maturity. This is evidenced by some negative key rate durations for 1% and 2% coupon bonds in all three figures.
3. A callable bond with a low coupon rate is unlikely to be called; hence, the bond’s maturity matched rate is the most critical rate (i.e., the highest key rate duration corresponds to the bond’s maturity). For the 1% bond in Figure 25.3, the 15 year key rate duration exceeds all other key rate durations.
4. All else equal, higher coupon bonds are more likely to be called, and therefore the time to exercise rate will tend to dominate the time to maturity rate. For the 8% coupon bond in Figure 25.3, the 10 year key rate duration is highest.
5. Putable bonds with high coupon rates are unlikely to be put, and thus are most sensitive to their maturity matched rates. For the 8% bond in Figure 25.4, the 15 year key rate duration is the highest.
6. All else equal, lower coupon bonds are more likely to be put, and therefore the time to exercise rate will tend to dominate the time to maturity rate. For the 1% coupon bond in Figure 25.4, the 10 year key rate duration is highest.

Straight bonds have positive effective convexity: the increase in the value of an option free bond is higher when rates fall than the decrease in value when rates increase by an equal amount. When rates are high, callable bonds are unlikely to be called and will exhibit positive convexity. When the underlying call option is near the money, its effective convexity turns negative; the upside potential of the bond’s price is limited due to the call (while the downside is not protected).

Putable bonds exhibit positive convexity throughout.