# Calculating the expected return of bonds given a default rate

In The investor’s manifesto book by W. Bernstein there is an example of calculating the expected return of corporate bonds. It goes like this:

Given that 10-year bonds yield an interest coupon of 7 percent and the estimated long-term bankruptcy rate of 2 percent the expected return would be 5 percent - the 7 percent interest coupon minus a 2 percent failure rate.

It doesn't feel right to subtract the failure rate for the coupon. What is a theoretical foundation for this formula?

## 1 Answer

Look at it this way.

Let's say you invest \$100,000 in bonds issued by 100 different companies. (Large, round numbers because those make it easier to do calculations.) The average bond returns 7% per year.

Without any credit events, after the first year, you will have collected \$7,000 (7%) in interest payments, thereby having a grand total of \$107,000.

Now apply a 2% credit event rate.

There are basically three ways we can do that; either 2% of companies go bankrupt and fail to honor any of their bond obligations, or companies fail to honor 2% of their bond obligations, or the interest rate on the bonds are reduced by 2%. These aren't the same thing!

In the first case, where 2% of companies go bankrupt, their bonds become worthless. Since you invested \$1,000 in each company's bonds, this means you lose \$2,000 due to bankruptcies. Assuming they paid the coupon (interest), you still got \$7,000, but lose \$2,000, so at the end of the year your bond portfolio is worth \$105,000 (\$2,000 less than in the no credit events case). You gained 5% during that year, instead of 7%.

In the second case, where companies fail to honor 2% of the face value of their bonds, your set of each company's bonds is now worth 98% * \$1,000 = \$980. The bonds are now worth \$98,000, plus the \$7,000 you were paid, or \$105,000. You again gained 5% during that year, instead of 7%.

In the third case, where the interest rate is reduced by 2% (not two percentage points; two percent), the interest rate on each bond is now 98% * 7% = 6.86%. This means that instead of getting \$7,000 in interest from \$100,000 in bonds, you would get \$6,860, or \$140 less. Thus you'd have \$106,860 instead of the expected \$107,000; you gained 6.86%, rather than 7%, that year. (If the interest rate was reduced by two percentage points, then it would be 5%, thereby netting you \$105,000 at the end of the year; again, \$2,000 less than you were expecting.) A reduction in interest rate is often accompanied by a fall in market price for the bond, but if you trust that the company can make good on its promise on the due date of the bond, you will still get the face value back on or near that day.

Therefore, assuming that credit events are total (in other words, that the company will no longer honor that bond at all), or that the coupon rate on the bonds is reduced by a given number of percentage points, you can simply subtract the number of percent of return. If the coupon rate is reduced by a given number of percent, then you have to figure out the new return and determine how that compares to the total return you were expecting. That's where you definitely can't rely on media reports; my experience is that journalists are notoriously bad at separating the concept of percent and percentage points, as well as average and median.

In the real world, of course, credit events for publicly traded bonds are a lot messier than this; for example, if a company writes off 2% of their debt, the market is unlikely to look at that as the end of it, and prices for that company's bonds are likely to fall further, driving their borrowing costs higher for when the debt is to be refinanced.

• Thanks for a detailed answer! I guess the author meant the first case but it would be reasonable to assume that if a company goes under it doesn't pay it's debt either so the result would be \$100,000 originally invested + 98 companies * \$1000 * 7% coupon - \$2000 = \$104,860 which is a 4.86% return. I believe the usual way to calculate the expected value of a random variable applies here. Suppose you bought a \$100 bond. There is 2% chance the company goes bankrupt and you loose \$100. There is a 98% chance you would be paid a \$7 coupon. The expected value is \$7 * 0.98 - \$100 * 0.02 = \$4.86. – epsylon Aug 25 '19 at 20:44
• In any case subtracting the bankruptcy rate from the interest rate looks to me as wrong as subtracting people from tables (even if the result of that operation aligns with the correct answer). This could probably be used as a shortcut rule given some assumptions but I wouldn't expect a respectable book to have statements like that without any explanations. A lot of investment bloggers recommend this book so I had high expectations but after reading part of it the doubt crept in. Is this a good book to read for a beginning investor? – epsylon Aug 25 '19 at 20:44