Callable Bonds: A Quick Summary

1. Context

In this short video from FRM Part 1 curriculum, we summarize key aspects about callable bonds that we need to keep in mind. This topic of callable bonds is pretty testable both in respect of their key features, as well as with regards to their valuation and risk characteristics. The details of the readings in which this topic appears are given below:

Area	Financial Markets and Products
Reading	Corporate Bonds
Reference	Frank Fabozzi (editor), Chapter 12. Corporate Bonds In The Handbook of Fixed Income Securities, 8th Edition (New York: McGraw-Hill, 2012).

2. Video

3. Key Aspects to Remember

3.1 Definition

A callable bond is a bond with an embedded option, an option that grants the issuer to redeem / call back (all or part of) the bond prior to its stated maturity, at a price known as call price.

3.2 Issuer Motivation

The primary motivation to issue callable bonds is to achieve protect against against a decline in interest rates, which can come about if either the general market interest rate goes down or the credit quality of the issuer improves. In such a scenario, the issuer can call back the existing (now, relatively high coupon) bond and replace it with another bond that pays a lower coupon (and hence take advantage of the now lower borrowing rates).

3.3 Buyer Motivation

Buyers of callable bonds have in effect, sold a call option to the issuer, and hence need to be compensated for it. During times when this call option is worthy to be exercised i.e. times when interest rates have fallen, receiving the proceeds from the issuer’s call of the bond is not an attractive proposition to investors. Therefore, callable bonds carry reinvestment risk. To compensate for the above, callable bonds offer a higher yield (and hence a lower price) to buyers. This is in comparison to comparable straight coupon (non-callable) bonds.

3.4 The Mechanics of the Call

All details pertinent to the call are specified in the indenture. These details include the call schedule i.e. a table of dates / periods on which the bond can be called, and the associated call price (it can be fixed or make-whole). Since it calculates the call price by discounting remaining cash flows at a certain spread over Treasury yield of comparable non-callable bond, the make-whole provision is more attractive to bond holders than a fixed call price. To protect the interests of bondholders, there may be a lockout period, within which the issuer cannot call back the bond.

3.5 Value Decomposition

The value of a callable bond ($V_{callable}$ from the holder’s perspective) can be decomposed as: $$ V_{callable} = V_{non-callable} – V_{option} $$ where, $V_{non-callable}$ is the value of an equivalent non-callable bond with the same coupon and maturity as the callable bond and $V_{option}$ is the value of the embedded call option. Owing to the embedded option, pricing of callable bonds will require a numerical method (e.g. binomial trees) that incorporates the volatility of interest rates.

3.6 Price-vs-Yield Relationship

This relationship is shown below:

Few key observations:

• The difference between the two graphs (Non-callable and Callable) is the value of the call option.
• At high yields, it is the usual convex P-y relationship, since at high yields the embedded call option is out-of-the-money and hence a callable bond behaves like a non-callable bond.
• At low yields, the relationship turns concave i.e. shows negative convexity. This is because the embedded call option becomes valuable at these low yields and the bond suffers a price compression. Its price cannot rise above the call price, since the issuer will call back the bond before this happens.
• Owing to this transition from convex (positive convexity) to concave (negative convexity), there should be a yield at which the convexity of this P-y graph is a zero.

3.6 Interest Rate Risk: Duration

Since we have an embedded option to deal with, and hence there is no closed-form analytical expression to value the bond, it is the effective duration that is more relevant for callable bonds. It is also relevant because callable bonds do not have well-defined yield to maturity (ytm) that is used to define risk measures such as duration and DVO1. Effective duration can be calculated as: $$ D_{eff} = -\frac{V_+ – V_{-}}{V_0 \Delta y} $$ Effective duration of callable bond is very close to that of non-callable at high yields (interest rates), but the its duration is lower than non-callable bond when yields (interest rates) are low (owing to price compression).

3.7 Interest Rate Risk: Convexity

Again, we measure Effective Convexity. Formula is: $$ C_{eff} = \frac{V_+ + V_{-} – 2V_0}{V_0 (\Delta y)^2} $$ It is negative at low interest rates, and positive at high interest rates.

3.8 Option Adjusted Spread (OAS)

The OAS of any bond with embedded option is defined as: $$ \mbox{OAS} = \mbox{Z-spread} – \mbox{Option Value (bps / year)} $$ where $\mbox{Z-spread}$ is the spread the bond would have earned if interest rates were static i.e. with zero volatility (and hence embedded option will have no value and $\mbox{OAS} = \mbox{Z-spread}$. In callable bond the value of option (sold to the issuer) is positive and hence $\mbox{OAS} \lt \mbox{Z-spread}$.