Many people wrongly just look at yields. If you look at yields, the answer is simple: choose junk bonds. Junk bonds offer a higher yield.

The problem is that doesn't account for the fact that some junk bonds default and the losses that result from that.

The second issue is the uncertainty in modeling losses. I'm going to be using some models that require assumptions. These assumptions are not perfectly known. So we need to build in a margin of safety to make sure that we outperform treasuries.

I will be using a relatively simple model. I will assume that on each year, one of two events will occur:

- I will earn what the bonds yields (either yield to maturity or yield to worst).
- I will suffer losses due to default. This will result in my bond being worth the recovery rate. So my losses will be 1 minus the recovery rate.

So now we need some variables. I will call $p$ the probability of default (hence, the probability that the bond

*does not*default will be $1-p$.) I will call $r$ the recovery rate. Lastly I will call $y$ the yield on the bond. The returns on the bond will thus be:

$$\text{Returns} = (1-p) \times y - p \times (1-r)$$

Now, ultimately we want these to at least perform as well as treasury yields, which I'll call the variable $t$:

$$ (1-p) \times y - p \times (1-r) \ge t$$

So we can subtract $t$ from both sides and do some rearranging:

$$y - yp -t - p (1-r) \ge 0$$

Now I'm going to define the spread as $s = y - t$. This gives me:

$$ s - yp - p(1-r)\ge 0$$

Now ideally I'd like to get rid of that extra term $yp$. This term will be quite small. After all, if the probability of default is $p=2\%$ and the yield is $y=6\%$, this term is only $py=.12\%$. But instead of getting entirely rid of it, I'm going to simply replace it with $yp \approx sp$. This way we don't ignore the term but at the same time we can eliminate the $y$ from our equation. So that gets us to:

$$s(1-p) + p (r - 1) \ge 0 $$

So far so good. But I want to add in a term for uncertainty. To do that, I'm going to take the total differential of the left side (and look at their absolute values since we're not concerned with direction). I'm going to assume that the bond spreads are known with certainty ($\operatorname ds = 0$). I'll handwave the math and just show the result (hopefully I did it correctly):

$$\sqrt{p^2 \operatorname dr^2 + (r - s - 1)^2 \operatorname dp^2}$$

We're going to subtract this uncertainty term from the left side of our expression only (this is going to be our built in "margin of safety".) The final result then looks like this:

$$s(1-p) + p (r-1) - \sqrt{p^2 \operatorname dr^2 + (r - s - 1)^2 \operatorname dp^2} \ge 0$$

The idea here is that if this inequality is satisfied, then we can be reasonably convinced that we'll outperform treasuries for the given yield spread.

So what does this end up looking like? Well, we're going to need some inputs. We have 5 variables: $s, p, r, \operatorname dp, \operatorname dr$.

Let's start with the spreads: $s$. For this I will be using BofA Merill Lynch Index Spreads from Fred. For example, the Junk Bond Spread is around 4.5%. Investment grade spreads are about 1.5%. There are other types of break downs as well. You can look at 1-3 Year Investment grade spreads or BB rated spreads.

For simplicity, I'll assume we're dealing with investment grade bonds with a spread of $s=1.5\%$ and junk bonds with a spread of $4.5\%$.

Next, we'll need default rates. For this I will use some data presented in my blog on Downside Risk Investing. We'll be looking at Figure 1.

Now these default rates do not include the Great Recession period. So I would think we would need to consider default rates a little higher. I will assume $p=0.25\%$ for investment grade and $p=3.00\%$ for junk bonds. But we can vary those assumptions.

We can also use this figure for an estimate of $\operatorname dp$. We'll use the standard deviation. I'll round these off to $ \operatorname dp = 0.25\% $ for investment grade and $ \operatorname dp=3.00\% $ for speculative grade.

Lastly, we'll need the figures for recovery rates. I'll again be going back to my blog on Downside Risk Investing. This time we'll be looking at Figure 2:

I will assume that we're dealing with senior unsecured bonds. That would suggest a recovery rate of about $r=30\%$. We'll allow this to vary a bit as well.

Now what about the uncertainty in this estimate? Table 6 suggests that standard deviation for recovery rates is about 20-25%. So I will use what I think to be a conservative number of $ \operatorname dr = 25\%$. If someone has some insight into this you can let me know.

But instead of just plugging these numbers in, let's allow them to vary a bit. I will do this using a Google spreadsheet. Feel free to save a copy and play around with it a bit.

To use it, any time our metric is greater than 0%, it indicates that the spread provides an adequate margin of safety.

I'll go ahead and work through the formulas for a couple of examples.

__Investment Grade Example__

So there's really two parts to this formula: the returns and the uncertainty (to provide a margin of safety). I'll calculate both separately because I think it's good to see what the uncertainty estimates are.

Recall that we're going to assume $ s = 1.5 \% $, $ r = 30 \%$ and $ p = 0.25 \% $. Then our estimated returns (above treasuries) will be:

\[\begin{align*}

s(1-p) + p(r-1) &= 1.5\% (1-0.25\%)+0.25\%(30\% -1) \\

&= 1.32 \%

\end{align*}\]

Next, we'll calculate the uncertainty. We'll be using $ \operatorname dp = 0.25 \%$ and $ \operatorname dr = 25 \%$.

\[\begin{align*}

\sqrt{p^2 \operatorname dr^2 + (r - s - 1)^2 \operatorname dp^2} &= \sqrt{0.25\%^2 \times 25\%^2 + (30\% - 1.5 \% -1 )^2 \times 0.25\%^2 }\\

&= 0.19 \%

\end{align*}\]

The difference then will be $1.32 \% -0.19\% = 1.13\% $ which also corresponds to the table in the spreadsheet. Since this figure is greater than 0, it appears that investment grade bonds have an adequate spread.

__Speculative Grade Example__

So we'll break it down into two parts again. We'll be assuming that $ s=4.5 \% $, $ r = 30 \%$ and $ p = 3.00 \% $. Then our estimated returns (above treasuries) will be:

\[\begin{align*}

s(1-p) + p(r-1) &= 4.5\% (1-3.00\%)+3.00\%(30\% -1) \\

&= 2.27 \%

\end{align*}\]

Next, we'll calculate the uncertainty term. We'll be using $ \operatorname dp = 3.00 \%$ and $ \operatorname dr = 25 \%$.

\[\begin{align*}

\sqrt{p^2 \operatorname dr^2 + (r - s - 1)^2 \operatorname dp^2} &= \sqrt{3.00\%^2 \times 25\%^2 + (30\% - 4.50 \% -1 )^2 \times 3.00\%^2 }\\

&= 2.36 \%

\end{align*}\]

The difference then will be $2.27 \%-2.36 \% = -0.09\%$ which also corresponds to the spreadsheet table.

These results indicate that speculative grade bonds may not have enough spread to compensate for the risk... at least according this simple model of mine.

It's worth noting that there's a pretty large difference in the uncertainty terms for investment grade and speculative grade. The uncertainty term was pretty small for investment grade but quite substantial for speculative grade.

The evidence suggests that spreads on investment grade bonds are adequate. Speculative grade bonds spreads seem to be a bit on the tight side and therefore one should proceed with caution. At least that's what my simple model concludes.

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