Over the last few weeks, as High Yield indices’ yields have continued to fall, we have seen a lot of commentary from talking heads (which, presumably, once belonged to the now-headless chickens doing the actual trading) about a potential bubble in high yield corporate bonds. This talk is often accompanied with, at worst, a chart of the HYG share price and, at best, a chart of the yield of a high yield index. These are incomplete and inaccurate indicators of risk, prospective return, and future return attribution. In this post I hope to illustrate how someone whose only career objective is to maximize P and minimize L should approach and quantify expected risk and returns.

## The Laws of the Land

### Rational risk taking is done at the margin

For fixed income PMs, the world is a scary place right now. Low yields, low spreads and long durations make almost every purchase unappetizing. But our job is not to worry about that, it is to make money, and you make money by taking risks. That is why it is important to avoid getting caught-up in panic and to ignore every fund manager armed with Very Scary Macro Charts. Successful investors avoid getting caught-up in indecision and instead measure and compare risks, taking a diversified basket of the most attractive ones.

### Ex-ante cash-flows with a maturity date are exempt from asset price bubbles

Bubble assets are characterized by the promise of exponential returns. Outside of negative nominal rates, the maximum return of any fixed-income asset is, by definition, fixed. Therefore Fixed Income is not, has never been and will be a bubble. Bubbles are generally dependent on significant amounts of leverage to bid-up prices. Bond yields, if bid-up with borrowed money, would trend towards the cost of financing, at which point no rational party would finance the assets. That, however, doesn’t mean that buyers of bonds are not accepting uneconomic credit risk premiums leading to low, or negative, expected returns. Only once we accept this as truth, can we move past hubris propagated by talking-heads and make rational economic decisions.

### In credit, yield to maturity is not expected return

Bond issuers default, speculative issuers default more than others. Whenever one is analyzing a large basket of independent debtors, it is important to adjust yield to maturity for expected default rates, making sure to account for the loss of the next coupon. For speculative-grade issuers, 1982-2012 mean annual historic default, recovery and loss rates are reproduced (Moody’s Investors Service, Inc. , 2013).

As illustrated, high yield bonds default at a non-trivial rate, therefore it is important to adjust yields for expected credit losses; this is commonly referred to as loss-adjusted yield (LAY). Credit losses can be thought as a function of default rates and severities (1-recovery), a surface which can be visualized below where lighter color mean smaller credit losses and the x and y axes represent default rates and severities (in the second, you can think of the z axis as the remaining principal in a pool of bonds) (Wolfram Alpha)

A rough way to estimate expected returns for a static default rate and recovery rate for a pool paying annual coupons trading close to par would be:

`E[r] = (YTM * (1 - d)) - (d * (1-r))`

Where E[r] represents your expected return, YTM is Yield to Maturity, d is the annual default rate and r is the expected recovery rate. For example, assuming a 4.6% default rate, 42% recovery rates and 5% YTM the expected return would be about 2.1%.

`0.021 = .05 * (1 - 0.046) - 0.046 * (1 - 0.42)`

### Yields, spreads and break-even rates

For non-floating, non-callable bonds, the coupon rate is primarily composed of two rates, the risk-free interest rate, as measured by the US Treasury curve, and the credit spread. It is important to note here that credit spread must compensate buyers not only for credit losses, but also for the lost income on the base rate of the bond. Expanding the equation above,

`E[s] = s - (YTM * d) - (d * (1-r))`

Assuming a spread of 3% and keeping other variables constant this would mean that the additional return realized for taking the credit risk is only 0.1%. With a little bit of simple high-school algebra (or calculus if you are so inclined) we can set E[s] to zero, and solve for the default rate at which no risk premium would be realized, the break-even default rate, which would be 4.5% using the same assumptions. You can calculate the break-even default rate by rearranging the equation to:

`d = spread / (YTM + (1 – recovery))`

Because spread and YTM are known ex-ante, we can think about the break-even default rate as function of recovery rate and calculate default scenarios for various recovery scenarios. Below is an illustration of what the break-even default rate (y-axis) would be as a function of recovery rates (x-axis) using a 5% yield and 3% spread (Wolfram Alpha)

### The term premium and risk adjustments

Ex-ante cash-flows and zero credit risk make US Treasury strips the best benchmark to use when comparing risk premiums; we know with precision exactly what the final rate-of-return of an investment has to beat in order to be worth considering and because duration is equal to maturity we can estimate price volatility using past historical interest rate volatility. For example, a UST strip with a maturity of May 2018 (S 0 05/31/18 Govt 912834KJ6) traded at a mid-yield of 0.79% on May 8^{th}. This means that, assuming a 5y expected-life, in order for any isolated risk to be economic, it would need to have an expected return above 0.79%.

Louise Yamada never misses an opportunity to remind us that there are only two types of losses: losses of opportunity and losses of capital and that there will always be another opportunity if you protect capital. We can apply this philosophy to speculative grade bonds by considering negative loss-adjusted yields losses of capital and mark-to-market risk (adverse price volatility) a loss of opportunity. Limit the potential of loss of capital (negative loss-adjusted returns) and take measured opportunity risk (liquidity risk) when the compensation is adequate.

The foundation of measuring and comparing the aforementioned risk of loss of opportunity is that a risk-free 0.79% is superior to an uncertain 0.79%. There is different ways to adjust for this risk, the simplest of which is weigh the volatility of both assets and calculate how much additional expected return we are receiving for every unit of additional unit of price volatility. For the marginal risk taking to be rational, it must offer a volatility-adjusted premium in addition to being economic (unlikely to breach the break-even default rate for our recovery scenarios). Rational risk-taking should increase the ratio of expected-return (E[r]) to expected volatility (E[sd]). Because funding isn’t free and we can’t leverage efficient assets at a zero cost, it makes sense to subtract the cost-of-funding from our calculations of this ratio. If this seems familiar, it is because this measure is the Sharpe ratio.

### The dynamic nature of fair credit premiums

In the last section I differentiated between economic marginal risk-taking, where E[r] increases with marginal risk-taking, and rational marginal risk-taking, where E[r] marginally increases in a proportion greater than E[sd]. If marginal risk taking is performed on a relative basis, this means that—all else equal—lower short-term risk-free rates combined with lower term premiums (a flatter yield curve) would mean that lower credit spreads are justified as long as they remain economic. In other words, as credit-risk-free yields decline, the spread above expected losses at which a risk becomes rational to take declines. In simpler terms, lower term premiums justify lower credit spreads and vice-versa.

## Enforcing the Laws

### A simple example

On May 8^{th}, the BofA Merrill Lynch US High Yield CCC index had an effective yield (EY) of 8.31% and an option adjusted spread (OAS) of 7.53%. There is no average maturity published, but the 78bp gap between OAS and EY hints that it is probably close to 5 years (a May 31^{st} strip traded at a mid-yield of 0.79%).

According to the table above the average recovery rate for Senior Unsecured bonds rated Caa-C is approximately 36%, which would make the break-even default rate for this pool of bonds 10%, about 0.5sd below the long term mean. If we look at the historical mean annual credit loss rates, we can see that the spread only compensates the holders for approximately 50% of the mean credit loss rate. If past default trends are indicative of future default-trends, buying this index will likely be uneconomic and there is a significant risk of loss of capital.

We’ve identified what it takes for this risk to be uneconomic, but a more interesting question is: at what spread does it becomes rational to take this risk? Assuming the coupon rate is equal to the YTM and a 5-year life, the duration of the index would be 4.2y. Using historic data, the annualized standard deviation of the index price would be 14.5% and that of the aforementioned STRIP 5.34%. That means that, assuming a 15bp funding cost[i], the expected return of the CCC index would have to be higher than:

`min E[r] = funding-cost + (risky-price-volatility * (RF yield – funding cost) / RF-price-volatility`

`min E[r] = 2.77% = 0.15% + (14.5% * (0.79% - 0.15%)) / 3.54%`

Plugging in 2.77% and rearranging our E[r] equation for a function d(r):

`d = 0.0554 / (1.0831 - r) = (2.77% - 8.31%) / (r - 8.31% - 1) = (E[r] - YTM) / (r - YTM - 1)`

This means that, if assuming a recovery rate of 36%, the default-rate below which buying the index becomes rational is 7.6%. Alternatively, we could start with the median default rate of 17.7% and solve for the fair yield[ii]

`YTM = 16.5% = 2.77% + (17.7% * (1 - 36%)) / (1 - 17.7%) = (E[r] + (d * (1 - r))) / (1-d)`

### A look at the current environment

The following example applies the methods described above to the 5, 7, and 10-year STRIPS as well as the BofA Merrill Lynch US High Yield BB, B and CCC indices to examine, using present conditions, what the relative value of speculative grade credit is. You will notice that the following tables eschew default-rates and use historical credit-loss rates (default rate * severity) which we believe convey expected losses just-as, if not more, accurately and allow us to present the results in a more concise format.

The first table shows us the observed inputs for each asset class, the second table highlights our assumptions calculated from historical data for 3 different scenarios and the third table presents the results for each asset under each scenario.

The first thing you should notice is that expected returns for Ccc credit are negative in all three scenarios. There is significant risk of loss of capital, therefore comparing them to other assets based on price volatility does not make sense. Because the expected returns are positive or all remaining assets, we can now compare BB and B bonds vs a benchmark. To identify the bench mark, we attempt to find the most efficient risk-free investment with a similar holding period. In this case, the 7y STRIP has the highest expected ratio in both the adverse and baseline scenarios, and is close to being the highest in the optimistic scenario as well. We can consider the 7y STRIP our benchmark to test for economic viability of a risk. In the case of B bonds, we find that our expected value is inferior to the benchmark rate and, while the optimistic scenario is positive, the adverse scenario has a very low expected return. Taking credit risk in B-rated bonds is likely to be uneconomical. That leaves BB-rated bonds which have an expected yield that readily exceeds the benchmark in all scenarios and, additionally, are subject to less price volatility. Taking credit risk in these bonds would be both economical and rational according to the model.

### Uneconomic spreads and the inefficient market

Doe-eyed believers in efficient markets might be wondering how Ccc expected yields could be negative and why B-rated bonds are overpriced. We don’t pretend to know or care about why people willingly enter into investments likely to provide poor returns but, if you are interested, Eric Falkenstein wrote an excellent book about it, **The Missing Risk Premium: Why Low Volatility Investing Works**, in which—amongst other great ideas—he theorizes that risk premiums are negative for volatile assets with compelling data to support this thesis. Our go-to explanation is much simpler: people can’t resist a high-coupon and overestimate their ability to pick the bond that won’t default.

## Caveats

- No attempt has been made at quantifying returns from roll-down, bonds being called, tendered, or undergoing a change in rating other than default.
- The calculations presented ignore convexity for simplicity of the example
- Positive expected returns may result in losses due to spread widening
- As mentioned in, “The Dynamic Nature of Fair Credit Premiums” a steeper yield curve would, all-else equal, mean higher “fair” spreads. This means that interest rate risk is understated by duration alone.
- The speculative-grade market is relatively new and the data sample is limited
- Due to lack of time-series data, we cannot confirm the accuracy of recovery rates.
- We make no attempt to quantify or consider an additional liquidity premium
- The loss rates cited are for Senior Unsecured obligations. The bonds constituting the index may be junior or secured, leading to lower or higher expected rates of recovery respectively.

# Works Cited

Moody’s Investors Service, Inc. . (2013). *Annual Default Study: Corporate Default and Recovery Rates, 1920-2012.*

Standard & Poors. (2013, February 07). *Global Weakest Links And Default Rates: The Weakest Links Count Fell To 149 In February.* Retrieved from http://www.standardandpoors.com/ratings/articles/en/us/?articleType=HTML&assetID=1245348670006

Wolfram Alpha. (n.d.). *y=.03/(.05+(1-x)), x=0..1*. Retrieved from http://www.wolframalpha.com/input/?i=y%3D.03%2F%28.05%2B%281-x%29%29%2C+x%3D0..1&a=*MC.0!.!.1-_*NumberMath-

Wolfram Alpha. (n.d.). *z = 1-(y * x), y=0..1, x=0..1.* Retrieved from Wolfram Alpha: http://www.wolframalpha.com/input/?i=z+%3D+1-%28y+*+x%29%2C+y%3D0..1%2C+x%3D0..1&a=*MC.0!.!.1-_*NumberMath-

[i] Using a 0.015% cost of funds (the yield of a 1y STRIP) might seem low for investors borrowing at the call-money rate, but it isn’t too far from the rates available for repo of USTs or the embedded cost-of-funds in derivatives.

[ii] This example exaggerates the yield required due to the effect of convexity. If the pool’s yield was 16.5% the duration of the pool would diminish by ~15%.

[iii] In positively-sloped yield curve environment, bonds will “roll” and YTM understates their expected total return over shorter holding periods

[iv] Maturities were estimated using the difference between effective yield and Option-Adjusted Spreads.

[v] Assumed credit loss rates were derived using historical rolling 5-year loss data. The baseline, optimistic and adverse scenarios correspond to the annual loss rate of a 50, 25 and 75 percentile periods respectively

[vi] In positively-sloped yield curve environments, expected price volatility for STRIPS will be overstated due to the falling duration of yield not being internalized

drevertsVery interesting. Regarding using the treasury strips as proxies for duration, would it be correct in concluding that a bond with a duration of 10 year would have to yield 71 bps more than an equivalent bond with a 5 year duration (the 10 yr strip – the 5 yr strip from your example above)?

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