What this chapter is about

A key feature of modern finance is the ability to carve up the probability distribution of a risk factor into separate pieces and sell them individually. Doing so requires accurately pricing the risk in each piece.

The traditional tool employed by companies for pricing risk is the risk-adjusted discount rate. But this tool is meant to be applied just to a purchase or sale of the whole probability distribution. It is not the right risk price to charge for each of the separate pieces.

This chapter shows how to use separate prices of risk for the separate pieces of the risk distribution. Of course, averaged together they give back the traditional risk-adjusted discount rate, but having the separate elements will be useful for valuing and pricing many packages of risk.

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When the Whole is Worth Less Than The Sum of the Parts?

Financial engineering is supposed to enable the restructuring of risks in economically beneficial ways. Ideally, the repackaging of risk into a set of products with a total value that is greater than the total before the repackaging. But sometimes what happens is the creation of unfamiliar packages that are hard to value correctly. A package may appear attractive based on oversimplified risk and return calculations, when what it really contains are tiny doses of very expensive risks. This can happen if investors overlook how the price of risk varies across states.

This is exactly what researchers Joshua Coval, Jakeb Jurek and Erik Stafford found when they examined the risks and prices of senior tranches of collateralized debt obligations (CDOs) during the period September 2004 through September 2007.

A CDO is a claim to the cash flows from a portfolio of corporate debt securities. The CDOs are organized in tranches. The senior most tranche is paid first. The junior most tranche is paid last. If only a few of the debt securities in the pool default, then the junior most tranche will suffer a lower cash flow, but each of the more senior tranches will receive its promised payment. If more of the debt securities default, the junior most tranche may not receive anything, and the next most junior tranche will suffer a lower cash flow. The senior most tranche receives its promised payment, except in the extreme case that very many of the pool’s securities default. The system of tranching redistributes the risk of the underlying securities. The junior most tranche has a higher probability of default than the average security in the pool. This junior most tranche is sometimes known as toxic waste. The senior most tranche has a lower probability of default than the average security in the pool.

By setting up enough tranches, it is possible to create a senior tranche that is very, very secure, given a pool of underlying securities that is very risky. This can be very useful if there is a certain class of investors who are restricted to purchasing only very safe securities, meaning those with a triple-A rating.

How should the investor price the senior tranche? One approach would be to price the senior tranche by comparing it against other corporate debt securities with comparable default risk. Default risk is assessed by credit rating agencies. If triple-A rated corporate debt is priced with an X% discount, then the senior tranche of a CDO with a triple-A rating should be priced with an X% discount.

This would be a mistake. It isn’t just the probability of default that matters for pricing a debt instrument. What also matters is the appropriate price of risk. Default by the senior tranche of a CDO occurs in a unique set of circumstances, when many of the underlying securities all default at the same time. The state-of-the-world in which many underlying securities all default at the same time is an extreme realization of the underlying market risk factor. This extreme realization should have a higher risk price associated with it. An individual corporate debt security with the same probability of default will include some defaults that are exclusively due to the idiosyncratic problems at that firm and when the realization of the market risk factor was not bad. This realization should only have a moderate risk price associated with it. So if the triple-A rated senior CDO tranche security has the same default probability as the triple-A rated corporate debt security, then the senior CDO tranche security should have a lower price than the corporate debt security!

Were CDO senior tranche securities overpriced? And was this significant?

The authors answer yes to both questions. They studied the price of CDOs sold on the DJ CDX North American Investment Grade Index from September 2004 to September 2007. They find that the market price of the senior most tranches are much higher than properly risk-matched alternatives. According to their calculations, an investor in a AAA-rated CDO tranche on the CDX could have earned a yield spread four to five times larger by constructing the same payoff using the index options market.

What is the Price of Default Risk?

It has long been known that spreads on corporate bonds are too high. Too high, that is, under a simple set of assumptions. The difference between the yield on a corporate bond and a cash flow matched US treasury bond is the spread we are concerned with. This spread compensates the holder of the bond for the higher default risk on the corporate bond.

However, if one looks at historical experience to estimate the probability of default on corporate bonds, the probability is far below what would seem to be required in order to justify the spreads we see in the corporate bond market. This has been true for a long period of time, over many types of markets, and does not appear to reflect an underestimate of the probability of defaults—even of the probability of the financial crisis. The authors Almeida and Philippon give the following example: Suppose the rate on US treasuries is 5%, and that this is risk free. Moody’s estimates the historical probability of default on a BBB-rated corporate bond at 0.53%. Assuming a recovery in the event of default of 41%, these figures imply an investor would require a corporate bond yield of 5.33% or a spread of 0.33% to compensate for the risk of default. However, the typical spread on a BBB-rated corporate bond is 1.9% or 6 times the predicted spread.

This calculation assumes that the event of default is priced like all other cash flows. But it should’t be. Default occurs more often when the economy in general is doing poorly. Default risk contains some systematic risk, and should pay a premium risk price. How much? The answer is approximately 4 times as great!

See Heiter Almeida and Thomas Philippon, Estimating Risk-adjusted Costs of Financial Distress, Journal of Applied Corporate Finance 20(4):105-109.