We first looked at how to value bonds way back in Chapter 3. We explained in that chapter what bond dealers mean when they refer to spot rates of interest and yields to maturity. We discussed why long-term and short-term bonds may offer different rates of interest and why prices of long-term bonds are affected more by a change in rates. We looked at the difference between nominal and real (inflation-adjusted) interest rates, and we saw how interest rates respond to changes in the prospects for inflation.
All the lessons of Chapter 3 hold good for both government and corporate bonds, but there is also a fundamental distinction between government and corporate issues. When a government borrows money, you can usually be confident that the debt will be repaid in full and on time. This is not true of corporate borrowing. Look, for example, at Figure 23.1. You can see that in 2009, following the financial crisis, companies defaulted on a record $330 billion of debt. Bondholders are aware of the danger that they will not get their money back and so demand a higher yield.
We begin our review of corporate bonds by looking at how yields vary with the likelihood of default. Then in Section 23-2, we look more carefully at the company’s decision to default. We show that default is an option; if the going becomes too tough, the company has the option to stop payments on its bonds and hand over the business to the debtholders. We know what determines the value of options; therefore, we know the basic variables that must enter into the valuation of corporate bonds.
Our next step is to look at bond ratings and some of the techniques that are used by banks and bond investors to estimate the probability that the borrower will not be able to repay its debts. We will look at statistical models that seek to identify common features of defaulting companies. And we will look at structural models that estimate the probability that a firm’s value will fall to the point at which it will choose to default.
23-1Yields on Corporate Debt
BEYOND THE PAGE
U.S. bond default rate, 1980–2017
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The year 2017 was not a happy one for telecom operator, Frontier Communications. Its share price fell by 85% over the year, and its 11% bonds maturing in 2025 were trading at 79% of face value, where they offered a yield to maturity of 15.8%. A naïve investor who compared this figure with the 2% yield on Treasury bonds might have concluded that Frontier’s bonds were a wonderful investment. But the owner would earn a 15.8% return only if the company repaid the debt in full. By 2017, that was looking increasingly doubtful. The company had recorded some hefty losses and was on life support. It had nearly $18 billion of debt in issue and just over $3 billion of book equity. Because there was a significant risk that the company would default on its bonds, the expected yield was much less than 15.8%.
FIGURE 23.1 Global face value of defaulting debt, 1990–2017, in billions of dollars
Source: Moody’s Investor Service, “Annual Default Study: Corporate Default and Recovery Rates, 1920–2017,” February 2018.
Corporate bonds, such as the Frontier Communications bond, offer a higher promised yield than government bonds, but do they necessarily offer a higher expected yield? We can answer this question with a simple numerical example. Suppose that the interest rate on one-year risk-free bonds is 5%. Backwoods Chemical Company has issued 5% notes with a face value of $1,000, maturing in one year. What will the Backwoods notes sell for?
If the notes are risk-free, the answer is easy—just discount principal ($1,000) and interest ($50) at 5%:
Suppose, however, that there is a 20% chance that Backwoods will default and that, if default does occur, holders of its notes receive half the face value of the notes, or $500. In this case, the possible payoffs to the noteholders are
|
Payoff |
Probability |
|
|
No default |
$1,050 |
0.8 |
|
Default |
500 |
0.2 |
The expected payment is .8($1,050) + .2($500) = $940.
We can value the Backwoods notes like any other risky asset, by discounting their expected payoff ($940) at the appropriate opportunity cost of capital. We might discount at the risk-free interest rate (5%) if Backwoods’s possible default is totally unrelated to other events in the economy. In this case, default risk is wholly diversifiable, and the beta of the notes is zero. The notes would sell for
An investor who purchased the notes for $895 would receive a promised yield of 17.3%:
That is, an investor who purchased the notes for $895 would earn a return of 17.3% if Backwoods does not default. Bond traders therefore might say that the Backwoods notes “yield 17.3%.” But the smart investor would realize that the notes’ expected yield is only 5%, the same as on risk-free bonds.
This, of course, assumes that the risk of default with these notes is wholly diversifiable so that they have no market risk. In general, risky bonds do have market risk (i.e., positive betas) because default is more likely to occur in recessions when all businesses are doing poorly. Suppose that investors demand a 3% risk premium and an 8% expected rate of return. Then the Backwoods notes will sell for 940/1.08 = $870 and offer a promised yield of (1,050/870) – 1 = .207, or 20.7%.
What Determines the Yield Spread?
Figure 23.2 shows how the yield spread on U.S. corporate bonds varies with the bond’s risk. Bonds rated Aaa by Moody’s are the highest-grade bonds and are issued only by a few blue-chip companies. The promised yield on these bonds has on average been about 1% higher than the yield on Treasuries. Baa bonds are rated three notches lower; the yield spread on these bonds has averaged about 2%. At the bottom of the heap are high-yield or “junk” bonds. There is considerable variation in the yield spreads on junk bonds; a typical spread might be about 6% over Treasuries, but spreads can rocket skyward as companies fall into distress.
FIGURE 23.2 Monthly yield spreads between corporate and 10-year Treasury bonds, 1980–2018
Source: The Federal Reserve Bank of St. Louis, https://fred.stlouisfed.org/
Remember these are promised yields and companies don’t always keep their promises. Many high-yielding bonds have defaulted, while some of the more successful issuers have called and paid off their debt, thus depriving their holders of the prospect of a continuing stream of high coupon payments.
Figure 23.2 also shows that yield spreads can vary quite sharply from one year to the next, particularly for low-rated bonds. For example, they were unusually high in 2000–2002, and 2008–2009. Why is this? The main reason is that, as Figure 23.1 shows, these were periods when defaults were more likely. However, the fluctuations in spreads appear to be too large to be due simply to changing probabilities of default. It seems that there are occasions when investors are particularly reluctant to bear the risk of low-grade bonds and so scurry to the safe haven of government debt.1
To understand more precisely what the yield spread measures, compare these two strategies:
Strategy 1: Invest $1,000 in a floating-rate default-free bond yielding 9%.2
Strategy 2: Invest $1,000 in a comparable floating-rate corporate bond yielding 10%. At the same time, take out an insurance policy to protect yourself against the possibility of default. You pay an insurance premium of 1% a year, but in the event of default, you are compensated for any loss in the bond’s value.
Both strategies provide exactly the same payoff. In the case of strategy 2, you gain a 1% higher yield, but this is exactly offset by the 1% annual premium on the insurance policy. Why does the insurance premium have to be equal to the spread? Because, if it weren’t, one strategy would dominate the other and there would be an arbitrage opportunity. The law of one price tells us that two equivalent risk-free investments must cost the same.
Our example tells us how to interpret the spread on corporate bonds. It is equal to the annual premium that would be needed to insure the bond against default.3
By the way, you can insure corporate bonds; you do so with an arrangement called a credit default swap (CDS). If you buy a default swap, you commit to pay a regular insurance premium (or spread).4 In return, if the company subsequently defaults on its debt, the seller of the swap pays you the difference between the face value of the debt and its market value. For example, when American Airlines defaulted in 2011, its unsecured bonds were auctioned for 23.5% of face value. Thus, sellers of default swaps had to pay out 76.5 cents on each dollar of American Airlines’s debt that they had insured.
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What exactly is a default?
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A most controversial trade
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Credit default swaps proved very popular, particularly with banks that need to reduce the risk of their loan books. From almost nothing in 2000, the notional value of default swaps and related products had mushroomed to $62 trillion by the start of the financial crisis.5 Many of these default swaps were sold by monoline insurers, which specialize in providing services to the capital markets. The monolines had traditionally concentrated on insuring relatively safe municipal debt but had been increasingly prepared to underwrite corporate debt, as well as many securities that were backed by subprime mortgages. By 2008, insurance companies had sold protection on $2.4 trillion of bonds. As the outlook for many of these bonds deteriorated, investors began to question whether the insurance companies had sufficient capital to make good on their guarantees.
One of the largest providers of credit protection was AIG Financial Products, part of the giant insurance group, AIG, with a portfolio of more than $440 billion of credit guarantees. AIG’s clients never dreamed that the company would be unable to pay up: Not only was AIG triple-A rated, but it had promised to post generous collateral if the value of the insured securities dropped or if its own credit rating fell. So confident was AIG of its strategy that the head of its financial products group claimed that it was hard “to even see a scenario within any kind of realm of reason that would see us losing one dollar in any of these transactions.” But in September 2008, this unthinkable scenario occurred when the credit rating agencies downgraded AIG’s debt, and the company found itself obliged to provide $32 billion of additional collateral within the next 15 days. Had AIG defaulted, everyone who had bought a CDS contract from the company would have suffered large losses on these contracts. To save AIG from imminent collapse, the Federal Reserve stepped in with an $85 billion rescue package.
23-2Valuing the Option to Default
The difference between a corporate bond and a comparable Treasury bond is that the company has the option to default whereas the government supposedly doesn’t.6 That default option is valuable. If you don’t believe us, think about whether (other things equal) you would prefer to be a shareholder in a company with limited liability or in a company with unlimited liability. Of course, you would prefer to have the option to walk away from your company’s debts. Unfortunately, every silver lining has its cloud, and the drawback to having a default option is that corporate bondholders expect to be compensated for giving it to you. That is why corporate bonds sell at lower prices and offer higher yields than government bonds.
We can illustrate the nature of the default option by returning to the plight of Circular File Company, which we discussed in Chapter 18. Circular File borrowed $50 per share, but then fell on hard times, and the market value of its assets fell to $30. Circular’s bond and stock prices fell to $27 and $3, respectively. Thus Circular’s market-value balance sheet is:
|
Circular File Company |
|||||
|
Asset value |
$30 |
$27 |
Bond |
||
|
3 |
Stock |
||||
|
$30 |
$30 |
Firm value |
|||
If Circular’s debt were due and payable now, the firm could not repay the $50 it originally borrowed. It would default, leaving bondholders with assets worth $30 and shareholders with nothing. The reason that Circular stock has a market value of $3 is that the debt is not due immediately, but one year from now. A stroke of good fortune could increase firm value enough to pay off the bondholders in full, with something left over for the stockholders.
Circular File is not compelled to repay the debt at maturity. If the value of its assets is less than the $50 that it owes, it will choose to default on the debt and the bondholders will get to keep the assets. Circular’s bondholders have, in effect, bought a safe bond but, at the same time, given the shareholders a put option to sell the firm’s assets to the bondholders for the amount of the debt. The exercise price of the put is $50, the face value of the bond. If the value of the company’s assets when the bond matures is greater than $50, Circular will not exercise its option to default. If the assets’ value is less than $50, it will pay Circular to exercise its option and to hand over the assets to settle the debt.
Now you can see why bond traders, investors, and financial managers refer to default puts. When a firm defaults, its stockholders are, in effect, exercising their default put. The put’s value is the value of limited liability—the value of the stockholders’ right to walk away from their firm’s debts in exchange for handing over the firm’s assets to its creditors. In the case of Circular File, this option to default is extremely valuable because default is likely to occur. At the other extreme, the value of IBM’s option to default is trivial compared with the value of IBM’s assets. Default on IBM bonds is possible but extremely unlikely. Option traders would say that for Circular File, the put option is “deep in the money” because today’s asset value ($30) is well below the exercise price ($50). For IBM, the put option is far “out of the money” because the value of IBM’s assets greatly exceeds the amount of IBM’s debt.
Valuing corporate bonds should be a two-step process:
The first step is easy: Calculate the bond’s value assuming no default risk. (Discount promised interest and principal payments at the rate offered by Treasuries.) The second step requires you to calculate the value of a put written on the firm’s assets, where the maturity of the put equals the maturity of the bond and the exercise price of the put equals the promised payment.
BEYOND THE PAGE
Try It! Valuing the default put
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In Chapter 18, we assumed that the value of Circular’s bond was $27. Now we can see where that figure could have come from. Suppose that the standard deviation of the returns on Circular’s assets is 60% a year and that the risk-free interest rate is 10%. The current value of Circular’s assets is $30, and the exercise price of the default option is $50. If you enter these data into the Black–Scholes model, you find that the value of Circular’s default put is $18.6. You can now value its bond:
The value of Circular’s equity is equal to the value of its assets less the value of its debt. So the equity is worth about $3 (30 – 26.9 = $3.1).
Before you get too gung-ho about valuing the default option, we should warn you that, in practice, you would encounter complications that make the valuation of corporate bonds considerably more difficult than it sounds. For example, we assumed that Circular File is committed to making a single payment of $50 at the end of the year. But suppose, instead, that it has issued a 10-year bond that pays interest annually. In this case, there are 10 payments rather than just one. When each payment comes due, Circular has the option to make the coupon payment or to default. If it makes the payment, Circular obtains a second option to default when the second interest payment becomes due. The reward to making this payment is that the stockholders get a third put option, and so on. (This is an example of a compound put option.)
Of course, if the firm does not make any of these payments when due, bondholders take over, and stockholders are left with nothing. In other words, if Circular decides to exercise its default option, it gives up all subsequent default options.
Valuing the 10-year bond when it is issued is equivalent to valuing the first of the 10 options. But you cannot value the first option without valuing the nine that follow. Even this example understates the practical difficulties because large firms may have dozens of outstanding debt issues with different interest rates and maturities, and before the current debt matures, they may make further issues. Consequently, when bond traders evaluate a corporate bond, they do not immediately reach for their option calculator. They are more likely to start by identifying bonds with similar maturity and risk of default and look at the yield spreads offered by these bonds.
Valuing the default put may be challenging, but, now we know that limited liability is an option, we also know what the value of that option depends on. The following table shows how the value of the option to default depends on the underlying variables:7
|
If there is an increase in: |
Value of default put: |
|
Value of company’s assets |
Declines |
|
Standard deviation of asset value |
Rises |
|
Amount of outstanding debt |
Rises |
|
Debt maturity |
Rises |
|
Default-free interest rate |
Declines |
|
Dividend payments |
Rises |
We have seen that corporate bonds sell for lower promised yields than comparable Treasury bonds. Can we explain the size of these yield spreads in terms of the default put that attaches to corporate bonds? Feldhütter and Schaefer believe that the answer is yes in the case of investment-grade bonds. For these bonds, they find that default risk can do a fairly good job of explaining the typical yield spread.8 However, yields on junk bonds appear to be higher than their default experience would justify. It seems that investors in junk bonds demand additional yield to compensate for their relative lack of liquidity.
The Value of Corporate Equity
We saw in Chapter 20 that the value of a put option is identical to the value of a call option with the same exercise price, plus the present value of the exercise price, and less the value of the underlying asset. Think what this means for the value of Circular File’s default put:
If you twist this formula around, you get:
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Try It! Leverage and debt betas
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The expression on the right-hand side is simply the value of Circular’s equity. This tells us that you can think of Circular’s bondholders as effectively owning the company now, but the shareholders have the option to buy it back from them at the end of the year by paying off the debt. Thus, the balance sheet of Circular File can be expressed as follows:
|
Circular File Company (Market Values) |
|||||
|
Asset value |
$30 |
$26.9 |
Bond value = asset value − value of call |
||
|
3.1 |
Stock value = value of call |
||||
|
$30 |
$ 30 |
Firm value = asset value |
|||
A Digression: Valuing Government Financial Guarantees
When American Airlines declared bankruptcy in 2011, its pension plan had liabilities of $18.5 billion and assets of just $8.3 billion. But the 130,000 workers and retirees did not face a destitute old age. Their pensions were largely guaranteed by the Pension Benefit Guaranty Corporation (PBGC).9
Pension promises don’t always appear on the company’s balance sheet, but they are a long-term liability just like the promises to bondholders. The guarantee by the PBGC changes the pension promises from a risky liability to a safe one. If the company goes belly-up and there are insufficient assets to cover the pensions, the PBGC makes up most of the difference.
The government recognizes that the guarantee provided by the PBGC is costly. Thus, shortly after assuming the liability for the American Airlines plan, the PBGC calculated that the discounted value of payments on defaulted plans and those close to default amounted to $98 billion.
Unfortunately, these calculations ignore the risk that other firms in the future may fail and hand over their pension liability to the PBGC. To calculate the cost of the guarantee, we need to think about what the value of company pension promises would be without any guarantee:
With the guarantee, the pensions are as safe as a promise by the U.S. government;10 without the guarantee, the pensions are like an ordinary debt obligation of the firm. We already know what the difference is between the value of safe government debt and risky corporate debt. It is the value of the firm’s right to hand over the assets of the firm and to walk away from its obligations. Thus, the value of the pension guarantee is the value of this put option.
In a paper prepared for the Congressional Budget Office, Wendy Kiska, Deborah Lucas, and Marvin Phaup show how option pricing models can help to give a better measure of the cost to the PBGC of pension guarantees.11 Their estimates suggest that the value of the PBGC’s guarantees was substantially higher than its published estimate.
The PBGC is not the only government body to provide financial guarantees. For example, the Federal Deposit Insurance Corporation (FDIC) guarantees bank deposit accounts, the Federal Family Education Loan (FFEL) program guarantees loans to students, the Small Business Administration (SBA) provides partial guarantees for loans to small businesses, and so on. The government’s liability under these programs is enormous. Fortunately, option pricing is leading to a better way to calculate their cost.
23-3Bond Ratings and the Probability of Default
Banks and other financial institutions not only want to know the value of the loans that they have made, but they also need to know the risk that they are incurring. Some rely on the judgments of specialized bond rating services. Others have developed their own models for measuring the probability that the borrower will default. We describe bond ratings first and then discuss two classes of model for predicting default.
The relative quality of most traded bonds can be judged by bond ratings. There are three principal rating services—Moody’s, Standard & Poor’s, and Fitch.12 Table 23.1 summarizes these ratings. For example, the highest-quality bonds are rated triple-A (Aaa) by Moody’s, then come double-A (Aa) bonds, and so on. Bonds rated Baa or above are known as investment-grade bonds.13 Commercial banks, many pension funds, and other financial institutions are not allowed to invest in bonds unless they are investment-grade.14
|
Moody’s |
Standard & Poor’s and Fitch |
|
Investment-Grade Bonds: |
|
|
Aaa |
AAA |
|
Aa |
AA |
|
A |
A |
|
Baa |
BBB |
|
Junk Bonds: |
|
|
Ba |
BB |
|
B |
B |
|
Caa |
CCC |
|
Ca |
CC |
|
C |
C |
TABLE 23.1 Key to bond ratings. The highest-quality bonds are rated triple-A. Investment-grade bonds have to be the equivalent of Baa or higher. Bonds that don’t make this cut are called “high-yield” or “junk” bonds.
Bonds rated below Baa are termed high-yield or junk bonds. Most junk bonds used to be fallen angels—that is, bonds of companies that had fallen on hard times. But during the 1980s, new issues of junk bonds multiplied 10-fold, as more and more companies issued large quantities of low-grade debt to finance takeovers. The result was that for the first time, corporate midgets were able to take control of corporate giants.
Issuers of these junk bonds often had debt ratios of 90% to 95%. Many worried that this threatened the health of corporate America and, as default rates on corporate debt rose to 10% in the early 1990s, the market for new issues of junk bonds dried up. Since then, the market for junk debt has had its ups and downs, but as interest rates on Treasuries dwindled in the years following the financial crisis, investors sought higher yields, and new issues of junk bonds boomed.
Bond ratings are judgments about firms’ financial and business prospects. There is no fixed formula by which ratings are calculated. Nevertheless, investment bankers, bond portfolio managers, and others who follow the bond market closely can get a fairly good idea of how a bond will be rated by looking at a few key numbers, such as the firm’s debt ratio, the ratio of earnings to interest, the operating margin, and the return on assets. Table 23.2 shows how these ratios vary with the firm’s bond rating.
TABLE 23.2 How financial ratios differ according to a firm’s bond rating. Median ratios for U.S. nonfinancial firms by bond rating.
Source: Moody’s Financial Metrics:, “Key Ratios by Rating and Industry for North American Non-Financial Corporations,” December 2013.
Figure 23.3 shows that bond ratings do reflect the probability of default. Since 1970, no U.S. bonds that were initially rated triple-A by Moody’s have defaulted in the year after issue and only 1 in 700 have defaulted within 10 years of issue. (The Aaa default rate is not plotted in Figure 23.3. It would be invisible.) At the other extreme, about half of Caa to C bonds have defaulted by year 10. Of course, bonds do not usually fall suddenly from grace. As time passes and the company becomes progressively more shaky, the agencies revise downward the bond’s rating to reflect the increasing probability of default.
FIGURE 23.3 Default rates of corporate bonds 1983–2017, by Moody’s rating at time of issue.
Source: Moody’s Investor Service, “Annual Default Study: Corporate Default and Recovery Rates: 1920–2017,” February 15, 2018.
Rating agencies don’t always get it right. When Enron went belly-up in 2001, investors protested that only two months earlier the company’s debt had an investment-grade rating. Rating agencies also did not win many friends during the financial crisis of 2007–2009, when many of the mortgage-backed debts that had been given triple-A ratings defaulted. And when agencies do downgrade a company’s debt, they are often accused of precipitate action that increases the cost of borrowing.
23-4Predicting the Probability of Default
Statistical Models of Default
If you apply for a credit card or a bank loan, you will probably be asked to complete a questionnaire that provides details about your job, home, and financial health. This information is then used to calculate an overall credit score.15 If you do not make the grade on the score, you are likely to be refused credit or subjected to a more detailed analysis. In a similar way, mechanical credit scoring systems are used by banks to assess the risk of their corporate loans and by firms when they extend credit to customers.
Suppose that you are given the task of developing a system that will help to decide which businesses are poor credits. You start by comparing the financial statements of companies that went bankrupt over a 40-year period with those of surviving firms. Figure 23.4 shows what you find. Panel (a) illustrates that, as early as four years before they went bankrupt, failing firms were earning a much lower return on assets (ROA) than firms that survived. Panel (b) shows that, on average, they also had a high ratio of liabilities to assets, and Panel (c) shows that EBITDA (earnings before interest, taxes, and depreciation) was low relative to the firms’ total liabilities. In each case, these indicators of the firms’ financial health steadily deteriorated as bankruptcy approached.
Rather than focusing on individual ratios, it makes sense to combine the ratios into a single score that can separate the creditworthy sheep from the impecunious goats. That means estimating an equation that relates the risk of bankruptcy to a set of financial variables. Most statistical bankruptcy models focus on a relatively small set of accounting ratios. There is general agreement that the probability of bankruptcy is higher for firms that have low and declining profitability, high debt ratios and low interest coverage, and decreasing cash reserves and working capital.16
FIGURE 23.4 Financial ratios of 544 failing and nonfailing firms
Source: W. H. Beaver, M. F. McNichols, and J. W. Rhie, “Have Financial Statements Become Less Informative? Evidence from the Ability of Financial Ratios to Predict Bankruptcy,” Review of Accounting Studies 10 (2005), pp. 93–122.
For small businesses, there may be little alternative to the use of accounting data, but for large, publicly traded firms, it is also possible to take advantage of the information in security prices. Low and volatile stock returns, a low market-to-book ratio, and a low stock price all seem to provide additional information on impending bankruptcy.
Before we leave the topic of these statistical models, we should issue a health warning. When you construct a risk index, it is tempting to experiment with many different combinations of variables until you find the equation that would have worked best in the past. Unfortunately, if you “mine” the data in this way, you are likely to find that the system works less well in the future than it did previously. If you are misled by the past successes into placing too much faith in your model, you could be worse off than if you had pretended that you could not tell one would-be borrower from another and extended credit to all of them. Does this mean that firms should not use credit scoring systems? Not a bit. It merely implies that it is not sufficient to have a good system; you also need to know how much to rely on it.
Structural Models of Default
Bankruptcy prediction models use a variety of techniques to estimate the relationship between the occurrence of bankruptcy and the set of financial variables. One of the earliest models that is still widely used is the Z-score model developed by Edward Altman. This used the technique of multiple discriminant analysis to come up with a credit score.17 Others have used hazard or probit models. In each case, the user picks a number of variables that he or she suspects might indicate approaching financial distress and then uses a statistical technique to find the combination of these variables that best predicts which firms will become bankrupt.
A different approach is to develop a structural model that builds on the insight that stockholders will exercise their option to default if the market value of the assets falls below the payments that must be made on the debt. The best known of these models is the Merton model, named after Robert Merton who first developed it,18 or Moody’s KMV model, named after the firm that produced a commercial version. We will illustrate with a simple example.
Imagine a company, call it Upsilon, whose assets have a current market value of $100. Its debt has a face value of $60, and the debt matures in one year. The return on the assets has a standard deviation of 30%, so the asset value when the debt matures could be more or less than $60. We assume that the risk-free rate of interest is 5%. Then, if the debt was risk-free, it would be worth 60/1.05 = $57.14. But Upsilon’s debt is risky: If the assets are worth less than $60, the shareholders will exercise their option to default and hand over the assets to the debtholders. The Black–Scholes model tells us that the value of this put option is $0.27. Therefore, the value of the debt is:
The value of the equity is:
Value of equity = Value of assets – value of debt − 100 – 56.87 = 43.13
To estimate the probability of default, we need to calculate the probability that the put option will be exercised. Figure 23.5 shows the distribution of possible asset values at the end of the year, assuming that investors are risk-neutral and happy to earn the risk-free interest rate on their holdings.19 The shaded area in the figure shows the probability in a risk-neutral world that the value of the assets at the end of the year will be less than $60 and Upsilon will default. If you look back at the Black–Scholes formula in Section 21-4, you will see the expression N(d2). The probability that the option will be exercised is equal to 1 – N(d2). In the case of Upsilon,
FIGURE 23.5 Upsilon has issued one-year debt with a face value of $60. The shaded area shows that there is a 4.3% risk-neutral probability that the value of the company’s assets at the end of the year will be less than $60, in which case, the company will choose to default.
Risk-neutral probability of default = 1 – N ( d2 ) = .043, or 4.3%
There is a 4.3% chance that Upsilon will default.
The Merton model of default has obvious attractions. It has a theoretical base. So the relevant variables are pretty well known, and you do not need to go prospecting among past data to find variables that may be indications of impending default. But when you apply the model in practice, you inevitably encounter complications. For example, unless you can observe the value of the company’s debts, you can’t observe the value of its assets or measure their volatility.20 Also, companies may have several debt issues, each with a different maturity. You could use an average time to maturity, but, if you are concerned with the probability of default in the short run, you may wish to place more weight on debt that will shortly need to be repaid.
SUMMARY
Corporations have limited liability. If companies are unable to pay their debts, they can file for bankruptcy. Lenders are aware that they may receive less than they are owed and that the expected yield on a corporate bond is less than the promised yield.
Because of the possibility of default, the promised yield on a corporate bond is higher than on a government bond. You can think of this extra yield as the amount that you would need to pay to insure the bond against default. There is an active market for insurance policies that protect the debtholder against default. These policies are called credit default swaps. There are no free lunches in financial markets. So the extra yield you get for buying a corporate bond is eaten up by the cost of insuring against default.
The company’s option to default is equivalent to a put option. If the value of the firm’s assets is less than the amount of the debt, it will pay for the company to default and to allow the lenders to take over the assets in settlement of the debt. This insight tells us what we need to think about when valuing corporate debt—the current value of the firm relative to the point at which it would default, the volatility of the assets, the maturity of the debt payments, and the risk-free interest rate. Unfortunately, most companies have several loans outstanding with payments due at different times. This considerably complicates the task of valuing the put option.
Because of these complications, bond investors do not regularly use option models to value the default option that is attached to a corporate bond. More commonly, they rely on their experience to judge whether the spread between the yield on a corporate bond and the yield on a comparable government issue compensates for the possibility of default. Spreads can change rapidly as investors reassess the chances of default or become more or less risk-averse.
When investors want a measure of the risk of a company’s bonds, they usually look at the rating that has been assigned by Moody’s, Standard & Poor’s, or Fitch. They know that bonds with investment-grade ratings (at least triple-B) are much less likely to default than bonds with a junk rating.
Banks, rating services, and consulting firms have also developed a number of models for estimating the likelihood of default. Statistical models take accounting ratios or other indicators of corporate health, and weight them to produce a single measure of default. Structural models, such as the Merton model, take a different tack and seek to measure the probability that the market value of the firm’s assets will fall to the point at which the firm will choose to default rather than try to keep up with its debt payments.
FURTHER READING
The websites of the main credit rating agencies contain a variety of useful reports on credit risk. (See in particularwww.moodys.com, www.standardandpoors.com, and www.fitch.com.)
Altman and Hotchkiss provide a review of credit scoring models in:
E. I. Altman and E. Hotchkiss, Corporate Financial Distress and Bankruptcy, 3rd ed. (New York: John Wiley, 2006).
Books that discuss corporate bonds and credit risk include:
A. Saunders and L. Allen, Credit Risk Measurement, 3rd ed. (New York: John Wiley, 2010).
J. B. Caouette, E. I. Altman, P. Narayanan, and R. Nimmo, Managing Credit Risk, 2nd ed. (New York: John Wiley, 2008).
D. Duffie, Measuring Corporate Default Risk (Oxford, U.K.: Oxford University Press, 2011).
D. Duffie and K. J. Singleton, Credit Risk: Pricing, Measurement and Management (Princeton, NJ: Princeton University Press, 2003).
PROBLEM SETS
Select problems are available in McGraw-Hill’s Connect. Please see the preface for more information.
1. Expected yield You own a 5% bond maturing in two years and priced at 87%. Suppose that there is a 10% chance that at maturity the bond will default and you will receive only 40% of the promised payment. What is the bond’s promised yield to maturity? What is its expected yield (i.e., the possible yields weighted by their probabilities)?
2. Bond ratings* In February 2018, Aaa bonds yielded 3.38%, Baa bonds yielded 4.51%, and comparable Treasuries yielded 2.86%.
a. What was the credit spread on Aaa bonds?
b. What was the spread on Baa bonds?
c. What do you think would be the difference in price (as a percent of face value) between a typical 5% 10-year Baa bond and a similar Treasury bond?
3. Bond ratings It is 2030 and the yields on corporate bonds are as follows:
|
Aaa |
A |
Ba |
|
8% |
10% |
12% |
4. Tau Corp wishes to raise $10 million by an issue of 9% 10-year bonds. What will be the likely issue price (as a percent of face value) if Tau is rated (a) Aaa, (b) A, or (c) Ba?
5. Default option The difference between the value of a government bond and a similar corporate bond is equal to the value of an option. What is this option, and what is its exercise price?
6. Default option* Other things equal, would you expect the difference between the price of a Treasury bond and a corporate bond to increase or decrease with
a. The company’s business risk?
b. The degree of leverage?
c. The time to maturity?
7. Default option Company A has issued a single zero-coupon bond maturing in 10 years. Company B has issued a coupon bond maturing in 10 years. Explain why it is more complicated to value B’s debt than A’s.
8. Default option How much would it cost you to insure the bonds of Backwoods Chemical against default? (See Section 23-1.)
9. Default option Digital Organics has 10 million outstanding shares trading at $25 per share. It also has a large amount of debt outstanding, all coming due in one year. The debt pays interest at 8%. It has a face value of $350 million but is trading at a market value of only $280 million. The one-year risk-free interest rate is 6%.
a. Write out the put–call parity formula for Digital Organics’s stock, debt, and assets.
b. What is the value of the company’s option to default on its debt?
10. Default option Square File’s assets are worth $100. It has $80 of zero-coupon debt outstanding that is due to be repaid at the end of two years. The risk-free interest rate is 5%, and the standard deviation of the returns on Square File’s assets is 40% per year. Calculate the present value of the company’s debt and equity.
11. Predicting default probability* A friend has mentioned that she has read somewhere that the following variables can be used to predict bankruptcy: (a) the company debt ratio; (b) the interest coverage; (c) the amount of cash relative to sales or assets; (d) the return on assets; (e) the market-to-book ratio; (f) the recent return on the stock; (g) the volatility of the stock returns. The problem is that she can’t remember whether a high value of each variable implies a high or a low probability of bankruptcy. Can you help her out?
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11. Predicting default probability* What variables are required to use the Merton model to calculate the risk-neutral probability that a company will default on its debt?
12. Predicting default probability Company X has borrowed $150 maturing this year and $50 maturing in 10 years. Company Y has borrowed $200 maturing in five years. In both cases, asset value is $140. Sketch a scenario in which X does not default but Y does.
13. Predicting default probability Discuss the problems with developing a numerical credit scoring system for evaluating personal loans. You can only test your system using data for applicants who have in the past been granted credit. Is this a potential problem?
14. Predicting default probability Look back at Section 23-4. Suppose that the standard deviation of the return on Upsilon’s assets is 50%. Recalculate the probability that the company will default.
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CHALLENGE
15. Default option Look back at the first Backwoods Chemical example at the start of Section 23-1. Suppose that the firm’s book balance sheet is
|
Backwoods Chemical Company (Book Values) |
|||||
|
Net working capital |
$ 400 |
$1,000 |
Debt |
||
|
Net fixed assets |
1,600 |
1,000 |
Equity (net worth) |
||
|
Total assets |
$2,000 |
$2,000 |
Total value |
||
16. The debt has a one-year maturity and a promised interest payment of 9%. Thus, the promised payment to Backwoods’s creditors is $1,090. The market value of the assets is $1,200, and the standard deviation of asset value is 45% per year. The risk-free interest rate is 9%. Calculate the value of Backwoods’s debt and equity.
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FINANCE ON THE WEB
1. Go to finance.yahoo.com and select three industrial companies that have been experiencing difficult times.
a. Are the companies’ troubles reflected in their financial ratios? (You may find it helpful to refer to Figure 23.4.)
b. Now look at the company’s bond rating. Do the two measures provide consistent messages?
1For evidence on the effect of changing risk aversion on bond spreads, see A. Berndt, R. Douglas, D. Duffie, and M. Ferguson, “Corporate Credit Risk Premia,” Review of Finance, 22 (March 2018), pp. 419–454.
2The interest payment on floating-rate bonds goes up and down as the general level of interest rates changes. Thus a floating-rate default-free bond will sell at close to face value on each coupon date. Many governments issue “floaters.” The U.S. Treasury does not do so, though some U.S. government agencies do.
3For illustration, we have used the example of a floating-rate bond to demonstrate the equivalence between the yield spread and the cost of default insurance. But the spread on a fixed-rate corporate bond should be effectively identical to that on a floater.
4In the case of low-grade bonds, when the regular spread does not sufficiently protect the seller against the possibility of an early default, the buyer of the default swap may also be asked to pay an up-front fee.
5Notional value refers to the total face value of bonds covered by CDS contacts. The present value of a CDS contract at its creation is usually zero. That is, the buyer of credit protection usually pays no money up front. Then the present value fluctuates, increasing as and if credit risk increases, but is always smaller than the notional value unless the bond turns out to be totally worthless. Data on credit derivatives are published by the International Swap Dealers Association (ISDA) at www.isda.org.
6But governments cannot print the currencies of other countries. Therefore, they may be forced into default on their foreign currency debt. For example, we saw in Chapter 3 how Argentina defaulted on $95 billion of foreign currency debt and, how Greece defaulted in 2012. Very occasionally, governments have even defaulted on their own currency’s debt. For example, in 1998, the Russian government defaulted on $36 billion of ruble debt.
7Notice that the effect of an interest rate rise on the value of a put option is the opposite of its effect on the value of a call. Circular’s option to extinguish the debt is more valuable when interest rates are high. The effect of changes in the interest rate is generally modest.
8P. Feldhütter and S. Schaefer, “The Myth of the Credit Spread Puzzle,” Review of Financial Studies 31 (August 2018), pp. 2897–2942.
9There are limits to pension payments made by the PBGC to retired employees. Employees with large pensions are not made whole.
10The pension guarantee is not ironclad. If the PBGC cannot meet its obligations, the government is not committed to providing the extra cash. But few doubt that it would do so.
11Congressional Budget Office, “The Risk Exposure of the Pension Benefit Guaranty Corporation,” Washington, DC, September 2005.
12The SEC has been concerned about the power wielded by the three bond-rating agencies. It has therefore approved seven new nationally recognized statistical rating organizations (NRSOs): DBRS, A.M. Best, Egan-Jones Ratings, Morningstar Credit Ratings (previously known as Realpoint), Kroll Bond Rating, HR Ratings de Mexico, and Japan Credit Rating.
13Rating services also provide a finer breakdown. Thus, a bond might be rated A-1, A-2, or A-3 (the lowest A rating). In addition, the rating service may announce that it has put an issue on its watch list for a possible upgrade or downgrade.
14Investment-grade bonds can usually be entered at face value on the books of banks and life insurance companies.
15The most commonly used consumer credit score is the FICO score, which is used by the three main credit agencies—Experian, TransUnion, and Equifax. The agencies also use their own proprietary scoring system, VantageScore.
16An early example of these models is the Z-score model proposed by Edward Altman in E. I. Altman, “Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy,” Journal of Finance, 23 (September 1968), pp. 589–609. For two more recent examples that make use of both accounting and market data, see T. Shumway, “Forecasting Bankruptcy More Accurately: A Simple Hazard Model,” Journal of Business 74 (2001), pp. 101–124; and J. Y. Campbell, J. Hilscher, and J. Szilagyi, “In Search of Distress Risk,” Journal of Finance 63 (December 2008), pp. 2899–2939.
17Altman’s Z-score model is described in E. I. Altman and E. Hotchkiss, Corporate Financial Distress and Bankruptcy, 3rd ed. (New York: John Wiley, 2006).
18See R.C. Merton, “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,” Journal of Finance 29 (1974), pp. 449–470.
19If you wish to estimate the actual, rather than risk-neutral, probability that Upsilon will default, you need to use the expected return on the assets rather than the risk-free interest rate.
20Merton proposed an ingenious way to back out asset value and volatility from the value and volatility of the equity.