Lesson 8.3: Bonds with Embedded Options

Introduction

In financial markets, bonds often come embedded with options that give either the bondholder or the issuer certain rights regarding the bond. These embedded options can significantly influence the value, yield, and risk associated with the bond. In this lesson, we will focus on callable and putable bonds, two specific types of bonds with embedded options. Understanding how to value these bonds and compute metrics like effective duration, effective convexity, and option-adjusted spread is crucial for CFA Level II candidates. By the end of this lesson, you will be able to:

Value callable and putable bonds using binomial trees.
Compute and interpret the effective duration and effective convexity of bonds.
Understand and calculate the option-adjusted spread (OAS).
Explain the main concepts and terminology relevant to bonds with embedded options.

Callable Bonds

Definition and Features

Callable bonds are bonds that give the issuer the right to redeem the bond before its maturity date at predetermined prices. This embedded call option is beneficial to the issuer because they can choose to refinance the bond when interest rates decrease, which allows them to reduce their borrowing costs.

Valuing Callable Bonds with Binomial Trees

Valuing a callable bond involves determining the present value of its cash flows, taking into account the possibility that it may be called at various points in time. A binomial tree is a popular method to accomplish this.

Constructing the Binomial Tree

Set Up the Initial Parameters: Choose the number of periods (n), the interest rate (r), and the call price (C).
Create the Binomial Structure: In each period, assume the price can either go up (factor $u$) or down (factor $d$). We can express it mathematically as:

$$ P_t = P_{t-1} \cdot u \text{ or } P_t = P_{t-1} \cdot d $$

Calculate the Cash Flows: Determine the cash flows at each node, accounting for the possibility of calling the bond. If the bond is called, the cash flow will equal the call price.

Example: Valuing a Callable Bond

Consider a callable bond with:

Face value: $1,000
Coupon rate: 5%
Call price: $1,050
Maturity: 3 years
Interest rates (up factor $u = 1.10$, down factor $d = 0.90$)

Create the Binomial Tree: The tree for the interest rate over three periods will have 4 levels (0 to 3).
Calculate Bond Cash Flows: For each node, if the price exceeds the call price, calculate:

$$ C_t = \text{Min}(C_{\text{coupon}} + C_{\text{face value}}, C) $$

where $C_{\text{coupon}} = 50$ (interest payment). For the final cash flows, if the bond is called, then $C_t = C$.

Discounting: Calculate bond values at each node and discount them back using probabilities. Combine these prices to arrive at the present value of the callable bond.

Effective Duration and Convexity

Effective duration measures a bond’s sensitivity to interest rate changes, accounting for the embedded options that may change the expected cash flows.

The formula for effective duration $D_{eff}$ is given by:

$$ D_{eff} = \frac{C_{-} - C_{+}}{2 \cdot C_{0} \cdot \Delta y} $$

where:

$C_{-}$: Price of the bond if the yield decreases by $\Delta y$
$C_{+}$: Price of the bond if the yield increases by $\Delta y$
$C_{0}$: Initial price of the bond

Effective convexity is a measure that accounts for the curvature in the price-yield relationship when yields change. It is calculated as:

$$ C_{eff} = \frac{C_{-} + C_{+} - 2C_{0}}{C_{0} \cdot \Delta y^2} $$

Putable Bonds

Definition and Features

Putable bonds provide the bondholder with the right to sell the bond back to the issuer at a predetermined price before maturity. This option is advantageous for investors, especially when interest rates rise, allowing them to reinvest at higher rates.

Valuing Putable Bonds with Binomial Trees

Similar to callable bonds, putable bonds can be valued using a binomial tree approach, which considers the bondholder's right to sell the bond before maturity.

Example: Valuing a Putable Bond

Take a putable bond with:

Face value: $1,000
Coupon rate: 5%
Put price: $950
Maturity: 3 years
Interest rates (same factors as the callable bond)

Create the Binomial Tree: Construct the tree in a similar manner as callable bonds.
Calculate Cash Flows: For each node, determine cash flows based on whether the bondholder would choose to put the bond:

$$ C_t = \text{Max}(C_{\text{coupon}} + C_{\text{face value}}, P) $$

Discounting: As with callable bonds, discount the cash flows back to the present value to determine the price of the putable bond.

Option-Adjusted Spread (OAS)

Definition

The option-adjusted spread is a metric used to measure the yield spread of a bond, adjusted for the risk of embedded options. It helps investors compare bonds with options to similar bonds without options by taking into account the value of the embedded options.

Calculating OAS

The OAS can be computed using the following steps:

Determine the spread of the bond over the risk-free rate.
Adjust for the present value of the embedded option(s).
The formula for OAS is:

$$ OAS = \text{Z-spread} - \text{option cost} $$

Example Calculation of OAS

Assume a callable bond has:

Z-spread: 150 basis points (bps)
Option cost (value of the option): 50 bps

The option-adjusted spread would be calculated as:

$$ OAS = 150 \text{ bps} - 50 \text{ bps} = 100 \text{ bps} $$

Conclusion

In this lesson, we explored the important concepts related to callable and putable bonds with embedded options. We learned how to value these bonds using binomial trees, calculated effective duration and convexity, and computed the option-adjusted spread. Understanding these principles will aid you in making informed decisions when assessing the value and risk associated with fixed income securities with embedded options.

Study Notes

Callable bonds allow issuers to redeem the bond before maturity.
Putable bonds give bondholders the right to sell back to the issuer.
Valuation of callable and putable bonds utilizes a binomial tree approach.
Effective duration measures sensitivity while accounting for embedded options.
Effective convexity quantifies price curvature relative to yield changes.
The option-adjusted spread helps in comparing bonds with and without options.