# 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.