Effective Duration of Bonds with Embedded Options

Effective duration is the sensitivity of a bond’s price to a 1% parallel shift in the benchmark yield curve, assuming that the credit spread of the bond remains constant. Effective (option-adjusted) duration is the most appropriate measure for bonds with embedded options. It also works for straight bonds.

$$ \text{Effective duration} =\cfrac {P_{i-}-P_{i+}}{ 2×P_0(\Delta \text{Curve})} $$

Where:

\(P_{i-}\)estimated price if interest is decreased by \(\Delta \text{Curve}\)

\(P_{i+}\)= estimated value if interest is increased by \(\Delta \text{Curve}\)

\(P_0\) = Current price (per $100 of par value)

\(\Delta \text{Curve}\) is the change in the interest rates used to calculate the new prices.

The procedure for calculating the effective duration of an embedded bond is as follows:

Step 1: Determine the implied OAS for the benchmark yield curve using the current market price and a suitable interest rate volatility.

Step 2: Shift the benchmark yield curve down, and generate a new interest rate tree. Then calculate the value of \(p_{i-}\) using the OAS calculated in step 1.

Step 3: Shift the benchmark yield curve up by the same no. of basis points and generate a new interest rate tree. After that, calculate the value of \(p_{i+}\)  

Step 4: Calculate the effective duration of the bond.

Example: Effective Duration

Consider a three-year, 9% annual coupon bond callable at par in two years, currently priced at $105. Assume an interest rate volatility of 10%.

If the benchmark spot curve is shifted up by 30bps, the price falls to $103.48. If the benchmark spot curve is shifted down by 30bps, the price remains at par.

The effective duration for the bond is closest to:

Solution

$$ \begin{align*} \text{Effective duration} &=\cfrac {P_{i-}-P_{i+}}{ 2×P_0(\Delta \text{Curve})} \\ \\ & =\frac{$105-$103.48}{2\times$105\times0.003}=2.41 \end{align*} $$

This means that the value of the 9% annual callable bond will reduce by 2.41% as a result of a 100-bps increase in interest rates

Other yield duration measures, such as modified duration, can only be used for option-free bonds. This is because they assume that the bond’s cash flows do not change when the yield changes. However, recognize that the cashflows from embedded bonds change when the embedded option is exercised.

Also note that the effective duration of a bond does not exceed its maturity, except for tax-exempt bonds when analyzed on an after-tax basis. Moreover, the effective duration of cash is zero, while that of a zero-coupon bond is equivalent to its maturity.

Question

Consider a bond with the following characteristics

$$ \begin{array}{c|c} \textbf{Time to maturity} & \textbf{Four years from now} \\ \hline \text{Annual coupon} & 6.00\% \\ \hline \text{Type of bond} & \text{Callable at par 1,2 and} \\ & \text{ 3 years from today} \\ \hline \text{Current price} & $99.95 \\ \hline \text{Price after shifting down the} & $100.70 \\ \text{benchmark yield curve by 30 bps} & \\ \hline \text{Price after shifting up the} & $99.44 \\ \text{benchmark yield curve by 30 bps} & \end{array} $$

The effective duration of the bond is closest to:

1. 0.80.
2. 2.10.
3. 4.20.

Solution

The correct answer is B.

$$ \begin{align*} \text{Effective duration} &=\cfrac {P_{i-}-P_{i+}}{ 2×P_0(\Delta \text{Curve})} \\ \\ & =\frac{100.70-99.44}{2\times99.95\left(0.003\right)}=2.10 \end{align*} $$

An effective duration of 2.10 indicates that a 100-bps increase in interest rate would reduce the value of the four-year 6% callable bond by 2.10%.

C is incorrect. 4.20 exceeds the maturity of the bond (four years).

Reading 30: Valuation and Analysis of Bonds with Embedded Options

LOS 30 (i) Calculate and interpret the effective duration of a callable or putable bond.