# 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\times105\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

\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.

Shop CFA® Exam Prep

Offered by AnalystPrep

Featured Shop FRM® Exam Prep Learn with Us

Subscribe to our newsletter and keep up with the latest and greatest tips for success
Shop Actuarial Exams Prep Shop MBA Admission Exam Prep

Daniel Glyn
2021-03-24
I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
michael walshe
2021-03-18
Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.
Nyka Smith
2021-02-18
Every concept is very well explained by Nilay Arun. kudos to you man!