21.2.1 Yield to worst

Unlike a vanilla bond, the date at which a callable bond will be redeemed is not known in advance. To calculate the YTM for such a bond, it is necessary to choose a redemption date in advance. The market convention is to take the most pessimistic assumption possible, which leads to the lowest return for the investor:

• If the bond is priced above par, the first possible redemption date is chosen. The reason is that interest rates have fallen, so the bond issuer will wish to exercise their option as soon as possible and issue a new bond at the prevailing lower rate.
• If the bond is priced below par, the last possible redemption date is chosen, because rates have risen and the issuer is paying less than the market rate for funds by leaving the bond uncalled.

The yield to maturity calculated under this assumption is called the yield to worst, as it is the lowest possible yield that the investor can receive from holding the bond. Naturally, if the actual redemption date differs from either of these assumptions, this calculated yield is of little use.

The yield to worst will always be lower than the security’s actual yield to maturity.

Yield to worst is seldom used in performance attribution. It is more useful when assessing the relative risks of different, comparable bonds.

21.2.2 Yield to call

Suppose a callable bond is trading above par, or that its market price is above its face value. In this case, its coupon will be higher than current interest rates. The bond’s issuer may then decide to call the bond at the next call date.

To take account of this when calculating the bond’s expected yield, the redemption date is assumed to be the next call date, rather than the maturity date. The resulting yield is called the yield to call.

In the opposite case where a bond trades below par, its coupon will be below current interest rates, so there will be little incentive for the issuer to redeem the bond early. The redemption date will then be the bond’s final maturity date, and the yield to call will be equal to the yield to maturity.

For callable bonds, yield to call is a more accurate guide to the actual yield available than yield to maturity. If available, it should be used in preference to yield to maturity; but note that option-adjusted spread (see below) is preferable to either.

A similar measure, called yield to put, exists for puttable bonds.

21.2.3 Option-adjusted spread and duration

Neither of the measures described above is ideal to measure the expected yield of a bond with embedded options, as they make no estimate of when the bond will actually be called. A more sophisticated approach uses the idea of option-adjusted spread.

While a full account of the technique is outside the scope of this book, the basic idea is to build a model of how interest rates evolve over the lifetime of the bond, and calculate the value of the embedded option at each point. This allows the calculation of a unique price for the bond at the current date.

This price will be lower than the price of an equivalent bond without call features, so its yield will be higher. Option-adjusted spread is a flat (equal across all maturities) spread that is added to the risk-free curve such that the sum of the security’s cash flows, priced using this new curve, equals the market price. In other words, it is a measure of the extra yield provided by the embedded option.

For bonds that have simple embedded options (i.e., without prepayment), the OAS is entirely due to the embedded option, so it is relatively straightforward to calculate the carry return generated by the option; it is just the OAS, minus any credit spread due to the bond, times the elapsed time. A more complex model is required for securities where prepayments can occur.

OAS is preferable to yield to call for attribution purposes, because it models the way in which interest rates may move in the future, rather than just using a snapshot of the bond’s current price. In other words, it makes some reasonable assumptions about when the bond will actually be called, and so provides a more accurate picture of the bond’s true yield, as well as explicitly splitting out the return from optionality. However, it is much more complex to calculate, and values of OAS may be difficult to acquire.

Just as spread duration measures the sensitivity of a security’s price to credit spread, option-adjusted duration (OAD) measures sensitivity to changes in option-adjusted spread. Some software systems describe all modified duration as OAD, even for securities with no optionality.

Your institution probably has its own model for calculation of OAS; there is no unique, globally accepted way to calculate it. If your attribution system also supplies a means of calculating OAS, consider whether the (probably different) values you will be using will cause problems.

The main reason I do not go into the construction of OAS in depth is that, most probably, your clients will already have values of this measure that were used to make the investment decisions in your portfolio. If you have a value available for OAS, use it. Otherwise, you may wish to try to integrate the pricing model used into your attribution system, if technically possible (and it may not be).