Excess Return on risky bond

Can someone help me and tell me how this was derived?

I can’t get my head wrap around how we found it.

Why is it called excess return? My definition of excess return is the return over the risk-free rate

can someone pls explain me this in layman terms

XR ≈ (s * t) – (Δs * SD) – (t * p * L)

Excess return is often used for relative valuation purposes in the CFA curriculum. You can take different “credit risky” (i.e. corporate) bonds and evaluate their excess return above some benchmark given (1) their given yield spread, (2) how their price would change given some CHANGE in yield spread and/or (3) their risk of losses from potential default. You can compare corporate bonds against each other in this three-part manner. Long story short, the higher the excess return, the better a bond looks versus its peers that you’re comparing it to.

If there is no change in yield spread and no default risk, your excess return above whatever benchmark rate you’re using is simply the spread. Adjusted for the time period involved, this is simply equal to (s * t). That’s Part 1 of the equation.

But what if the spread changes over the period you hold it? How will one bond versus another be affected? This is evaluated by looking at the particular bond’s spread duration and multiplying it by the change in spread. Similar to how you would multiply a bond’s negative duration times a change in interest rates. But you’re doing this for spread change not interest rate changes. This is the second part of the equation above. Just like the formula for price changes in a bond due to interest rate changes, here the effect of spread duration is negative on a bond’s price too, due to increases in spread. So the second bracket is subtracted and not added in the formula. That’s Part 2 of the equation.

Third part is the credit losses effect that can make different bonds have greater or lesser expected returns versus each other, regardless of whether they have the same spreads or even if they react the same way to spread changes. The formula for predicting credit losses, as you know from #4 of the 5 standard components of total return for fixed income (covered at the beginning of your fixed income reading), is equal to the probability of default times the loss given default. This is the final bracket (t * p * L) in this formula. The “t” simply adjusts for the time period just like in the spread (s * t) first part of the equation. And that’s Part 3 of the equation.

WHY DO WE CARE ABOUT THIS?
This is a way to evaluate one corporate bond versus another. You will typically see 2 corporate bonds with similar spreads to the benchmark yield that you’re evaluating them against. But spreads don’t tell the whole story really. You want to know how changes in spread and also different levels of potential credit losses for the 2 bonds may make one more attractive than the other one. You can do this calculation to better see which one should to give you a better result at the end of the day. You’re comparing corporate bonds against each other to see which one has the best relative value versus the others.

So if you see a CFA question in Level 3 asking you to rank corporate bonds in order of preference, and you see they’ve given you “spread duration” in the question description as well as “recovery rate” and “probability of default,” chances are good that they’re actually asking you to do this equation for the bonds. They may not even mention the words “Excess Return” in the problem - you may need to spot this on your own, that this is what they’re asking for, and it silently explains why they gave you a spread duration somewhere in the problem. You will just need to key in on the odd appearance of spread duration in combination with the other usual suspects and do this equation as your answer method. The bond that you will prefer in your answer is the one with the highest numerical result (the greatest Excess Return).

And - in case it isn’t already abundantly clear due to the amount of specialized work you just did to evaluate individual bonds against each other using this formula - if you’re using this Excess Return approach to evaluate bonds, you are using a BOTTOM UP approach as your credit strategy. Just to try to tie things together with a nice bow in conclusion, as best I can here.

This is my layman’s understanding, if anyone has a correction or improvement please chime in. I tried my best!

Cheers - good luck on your exam - you got this :+1:

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You explained it so well. Thanks :slight_smile: