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Bonds with embedded options have asymmetrical price sensitivity to up or down interest rate movements of the same magnitude. When the embedded option is in the money, the price of a callable bond has limited upside potential, while that of a putable bond has limited downside potential.

* One-sided durations* are effective durations when interest rates move upwards or downwards. One-sided durations capture the sensitivity to interest rates of bonds with embedded options better than two-sided durations, especially when the embedded option is near the money.

This measure suggests that callable bonds are more sensitive to interest rate rises than to interest rate falls. On the contrary, putable bonds are more sensitive to interest rate falls than to interest rate rises.

* Key rate duration* or partial durations measure bond price sensitivity to changes in the shape of the benchmark yield curve. Note that only key points rather than the entire benchmark yield curve are shifted, one at a time.

It is worth noting that the key rate durations of bonds with embedded options depend on both the time to exercise and the time to maturity.

$$ \textbf{Key Rate Durations of 20-Year Option-Free Bonds} $$ $$ \begin{array}{c|c|ccccc} \textbf{Bond} & \textbf{Annual} & \textbf{Key Rate} & \textbf{Duration} & & & \\ & \textbf{Coupon} & & & & & & \\ \hline & & \text{Price} & \text{Total} & 5- & 10- & 15- & 20 \\ & & \text{(% of par)} & & \text{year} & \text{year} & \text{year} & \text{year} \\ \hline A & 1\% & 68.51 & 12.07 & -0.20 & -0.41 & -0.58 & 12.78 \\ \hline B & 3\% & 100.00 & – & – & – & – & 10.75 \\ \hline C & 8\% & 143.72 & 0.08 & 0.29 & 0.79 & 1.22 & 7.87 \end{array} $$

From the above table, we can deduce the following:

- The maturity-matched rate is the only rate that affects the value of an option-free bond trading at par (The 3% coupon bond).
- The maturity matched rate is the most important rate for an option-free bond that is not trading at par. Notice that the 20-year key rate duration is the highest.
- Zero-coupon bonds or bonds with very low coupons can have negative key durations for some of the terms less than the maturity.

$$ {\textbf{Key Rate Durations of 20-Year Bonds Callable in 15 years with} } \\ {\textbf{varying coupon rates} } $$ $$ \begin{array}{c|c|ccccc} \textbf{Bond} & \textbf{Annual} & \textbf{Key Rate} & \textbf{Duration} & & & \\ & \textbf{Coupon} & & & & & & \\ \hline & & \text{Price} & \text{Total} & 5- & 10- & 15- & 20 \\ & & \text{(% of par)} & & \text{year} & \text{year} & \text{year} & \text{year} \\ \hline A & 1.00\% & 71.63 & 12.63 & -0.06 & -0.04 & -2.12 & 20.21 \\ \hline B & 3.00\% & 91.36 & 10.93 & – & – & 5.11 & 4.82 \\ \hline C & 8.00\% & 132.82 & 8.49 & 0.07 & 0.30 & 6.88 & 1.08 \end{array} $$

We can conclude the following from the above table:

- A low coupon callable bond is unlikely to be called.
- This implies that the maturity-matched rate will have the largest effect on the value of the callable bond.
- Notice that the 1% coupon bond has the highest key rate duration, which corresponds to the bond’s maturity of 20 years.

- The higher the coupon rate, the higher the possibility of the bond being called.
- This means that time to exercise has a greater influence on the key rate duration relative to the time to maturity as the coupon rates increase.
- Notice that for the 8% coupon bond, the 15-year key rate is the highest.

$$ {\textbf{Key Rate Durations of 20-Year Bonds Putable in 15 years with}} \\ \textbf{Varying Coupon rates} \\ \\ \begin{array}{c|c|ccccc} \textbf{Bond} & \textbf{Annual} & \textbf{Key Rate} & \textbf{Duration} & & & \\ & \textbf{Coupon} & & & & & & \\ \hline & & \text{Price} & \text{Total} & 5- & 10- & 15- & 20 \\ & & \text{(% of par)} & & \text{year} & \text{year} & \text{year} & \text{year} \\ \hline A & 1.00\% & 73.76 & 8.81 & -0.06 & -0.04 & 8.27 & 1.00 \\ \hline B & 3.00\% & 91.36 & 10.03 & – & – & 4.89 & 5.14 \\ \hline C & 8.00\% & 152.50 & 9.63 & 0.07 & 0.30 & 0.43 & 8.66 \end{array} $$

We can make the following conclusions from the above table:

- A high coupon rate putable bond is unlikely to be put. This makes it more sensitive to the maturity-matched rate (20-year rate).
- A low coupon putable bond is likely to be put, hence more sensitive to time to exercise (15-year rate).
- Notice that the 1% coupon bond has the highest 15-year key duration rate.

## Question

The following table shows the key rate durations of 20-year bonds putable in 10 years at par. The bonds have been valued using a 3% flat yield curve and assuming a 20% interest rate volatility.

$$ \begin{array}{c|c|c|c|c|c|c|c} \textbf{Bond} & \textbf{Coupon} & \textbf{Price} & \textbf{Total} & \bf{3-} & \bf{5-} & \bf{10-} & \bf{20-} \\ & & \textbf{(% of} & & \textbf{year} & \textbf{year} & \textbf{year} & \textbf{year} \\ & & \textbf{par)} & & & & \\ \hline A & 1\% & 78.39 & 7.96 & –0.12 & –0.32 & 7.71 & 0.69 \\ \hline B & 3\% & 109.01 & 15.27 & –0.02 & –0.06 & 5.56 & 9.79 \\ \hline C & 8\% & 209.41 & 13.00 & 0.06 & 0.18 & 2.09 & 10.71 \end{array} $$

Relative to Bond B, Bond A is

most likelysensitive to changes in the:

- 5-year rate.
- 10-year rate.
- 20-year rate.

Solution

The correct answer is B.Lower coupon puttable bonds are more likely to be put, hence more sensitive to time to exercise (10-year rate). We can see that Bond A, with a 1% coupon bond, has the highest 10-year key duration rate from the above table.

Reading 30: Valuation and Analysis of Bonds with Embedded Options

*LOS 30 (k) Describe the use of one-sided durations and key rate durations to evaluate the interest rate sensitivity of bonds with embedded options.*