A Model for Fixed-Income Returns

When you lend money, you get interest—simple enough.

The holding period return of a bond is a little more interesting and complicated, though. There's the coupon, sure. And if you buy a bond at 99 of 100 par, with a coupon of 4, then is your return based on the 99 or the 100?

Of course.

No. You didn't pay the 100. It's based on the 99.

You probably remember this simple ratio of coupon divided by the price as the __current yield__, which it still is. For decomposing bond returns, it's going to be referred to as the __coupon income__. Same thing, really. Now if you buy this bond and hold it, and things like the yield curve remain constant, what do you expect to happen to the price of the bond?

No. It will rise to par over time.

Right!

It's going to end up at 100; they always do (again, ignoring things like default for now). So this question of, "If the yield curve is unchanged, what is the price difference over my holding period?" gets at the __rolldown return__ as the bond "rolls down" (or really, "along") the yield curve to maturity. Discount prices rise to par. What would premium prices do?

Exactly.

No. Premium prices would have to fall to get to par.

So the rolldown return is negative for a premium priced bond. Adding these first two components together, the coupon income and the rolldown return sum to make the __rolling yield__. Think about the relationship between coupons and bond prices. What would you expect the rolling yield to be?

No. It really should always be on the same side of zero; consider investor motivations.

No. That wouldn't leave the investor with much reason to invest.

Yes! Coupons that are "too high" lead to premium prices, so investors get high coupon income and a negative rolldown return. Coupons that are "too low" lead to discount prices, so, a small coupon income and positive rolldown return. But even for zero-coupon bonds, this sum will be positive. The investor has to get some sort of reasonable expected return, so a positive rolling yield is quite reasonable.

Then there might be an expectation of a currency change if the bond isn't in the investor's domestic currency. Finally, there are expected credit losses. If there's a 99.9% chance of full payment, would you want to include that?

Actually, it would be a good idea.

Absolutely.

Past that, there are a few other considerations, and these are all in expectation since they are just forecasts. The investor might expect the benchmark yield to rise or fall, leading to this typical price adjustment. $$\displaystyle E[\Delta P_{\mbox{Bond}}] = -ModDur \times \Delta \mbox{Yield} + \frac{1}{2} \times \mbox{Conv} \times (\Delta \mbox{Yield})^2 $$ You've seen that before. Nothing new. Although, if the bond has an embedded option, you will want to use effective convexity rather than modified convexity. Also, if you're working with spreads instead of yields, you can use the related calculation: $$\displaystyle E[\Delta P_{\mbox{Bond}}] = -ModSpreadDur \times \Delta \mbox{Spread} + \frac{1}{2} \times \mbox{Conv} \times (\Delta \mbox{Spread})^2 $$

In fact, that's about right. For US investment-grade bonds, the average default rate has been about 0.1% in recent decades. Multiply that estimated default rate by the expected loss severity, and you have the expected loss to consider for defaults. But it's so tiny, that the CFAI removed this piece from the decomposition formula in recent years. This all adds up to this decomposition: $$\displaystyle E(R) \approx \text{Coupon income} + \text{Rolldown return} $$ $$\displaystyle +/- E [\Delta \text{Price due to investor's view of benchmark yield}] $$ $$\displaystyle +/- E[\Delta \text{Price due to investor's view of yield spreads}] $$ $$\displaystyle +/- E[\Delta \text{Price due to investor's view of currency value changes}] $$.

To summarize: [[summary]]

However, there are some natural assumptions which can limit the modeling of fixed-income returns. These include a YTM reinvestment rate for coupons and a local richness/cheapness effect. The local richness/cheapness effect are deviations fitted from the yield curve for individual maturity segments that result from certain financing advantages. In addition, some bonds can offer a very small return advantage due to the repo market. However, the local richness/cheapness factor and repo yield are usually too small to include in the return model.

99

100

It should remain the same

It should increase, getting closer to par

Fall

Rise as well

Always positive

Always negative

It depends on the bond price

No

Yes

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