CR 9 – Estimating Default Probabilities

Moody’s, S&P and Fitch publish ratings that summarise how creditworthy a corporate bond is. Strictly speaking a rating belongs to a particular bond issue rather than to the company behind it, but a company’s bonds very often all carry the same grade, so people talk loosely about a company being rated A or BB. A rating is meant to convey credit quality, and you might expect it to move every time good or bad news about the issuer reaches the market. In practice ratings change rather rarely, and that is a design choice rather than an accident.

The agencies want their ratings to be stable. They try hard to avoid a reversal, in which a bond is downgraded and then restored to its old grade a few weeks later, so they adjust a rating only when they believe the issuer’s creditworthiness has shifted for the long term. The reason lies with the people who use ratings most heavily. Bond traders and fund managers often operate under rules that restrict what grades they may hold; many bond funds, for instance, may own only investment-grade paper. If ratings jumped around, these holders would be forced into a great deal of buying and selling, with the transaction costs that come with it, merely to stay inside their own mandates.

Rating through the cycle

A closely related idea is that agencies try to rate through the cycle. Picture an economic downturn that pushes up the chance of a company failing within the next six months but leaves its prospects over a three to five year horizon almost unchanged. An agency would normally leave the rating where it is, because the rating is meant to describe the long-horizon picture rather than the short-term weather.

Not every provider works this way. Firms such as Moody’s KMV and Kamakura sell default probability estimates built from a company’s equity price and other market variables. Stability is not one of their goals, so these estimates react much faster to fresh market information than an agency grade does. The models behind them are covered later in this lesson, in the section on the Merton model.

Internal credit ratings at banks

Agency ratings are normally available only where a company has bonds trading in the public market. Many small and medium-sized businesses have never done so, which means a bank lending to them has no agency grade to lean on. For that reason, almost every bank runs its own procedure for producing an internal credit rating for its corporate and retail clients.

Internal-ratings-based approaches usually start from familiar accounting ratios. Profitability measures such as return on assets sit alongside balance sheet measures such as the current ratio (current assets divided by current liabilities) and the debt to equity ratio. Banks also know that a loan is repaid with cash, not with accounting profit. So a typical credit analyst takes the financial statements a borrower supplies and rebuilds them into a cash flow statement, which shows far more directly how comfortably the company can meet its interest and principal payments.

So the two kinds of rating differ in reach and in purpose. Agency ratings cover issuers of public debt, are published, and are deliberately slow to move. Internal ratings cover a bank’s whole book, including unrated private firms, and lean heavily on cash flow analysis of the borrower’s own figures.

Edward Altman was among the first to show that accounting ratios carry real predictive power for default. In 1968 he applied discriminant analysis, a statistical method that finds the weighted combination of variables that best separates two groups, to a set of failed and surviving firms. The result was the Z-score, built from five ratios.

For publicly traded manufacturing companies, the original weights were as follows.

A higher score points to a safer company. Altman read the score against four zones, set out below.

The weights came from a sample of 66 publicly traded manufacturers, 33 of which failed within a year and 33 of which did not. When the equation was tried on companies outside that sample, it held up well. Both kinds of misclassification were small: a Type I error is a firm the model says will survive but which goes bankrupt, and a Type II error is a firm flagged for bankruptcy that in fact survives. For a commercial bank’s lending desk a Type I error usually costs more, because it ends in a loan loss, whereas a Type II error only costs a forgone lending opportunity. Later versions adapt the weights for private manufacturers and for firms outside manufacturing, and the method has been revised many times in the more than 50 years since the original study.

The same arithmetic lets a lender sort a whole group of borrowers into credit-quality buckets, which is often how the score is used in practice.

Rating agencies publish tables that follow groups of issuers forward from the date they held a given rating and record what share of them had defaulted by each later horizon. The table below holds S&P average cumulative issuer-weighted default rates for 1981 to 2020. Every entry is cumulative, so it gives the percentage of issuers in a rating class that had defaulted at some point up to the end of that year.

Source: S&P Global. Shown with horizons as rows.

The table answers the most basic question about the link between a borrower’s rating and its probability of default: the lower the grade, the higher the default rate at every horizon. Reading down a column tells you the cumulative probability; for a bond first rated BBB, there is a 0.16% chance of default by the end of year one and a 0.43% chance by the end of year two.

Default probability in a single year

Subtracting adjacent cumulative entries gives the probability that default happens during one particular year, as judged from today. For the BBB bond, the chance of defaulting in its second year is 0.43 − 0.16 = 0.27%. Doing the same for the first four years of two very different grades reveals a striking contrast.

For the A-rated bond the yearly figure creeps upward: 0.05%, 0.08%, 0.09%, 0.11%. That pattern is typical of investment-grade issuers (BBB and above) over the first few years. Such a company starts out sound, and the longer the horizon, the more room there is for its finances to deteriorate.

The CCC/C bond behaves in the opposite way, with yearly probabilities of 28.3%, 10.03%, 5.09% and 2.94%. For a weak issuer the next year or two are the danger zone. A company that gets through that stretch has usually repaired its finances to some degree, so its chance of failing in any later year is lower.

Go back to the CCC/C row. The chance of defaulting during year three, as seen from time zero, is 43.42 − 38.33 = 5.09%. This is called an unconditional default probability, because it makes no assumption about what happened in years one and two. The probability that the bond is still alive at the end of year two is 100 − 38.33 = 61.67%, and so the chance of defaulting in year three given that it has survived to the start of that year is 0.0509 / 0.6167 = 8.25%. This second figure is a conditional default probability, and it is always larger than the unconditional one because it is measured only over the survivors.

From a one-year conditional probability to an instantaneous rate

The 8.25% above is a conditional probability measured over a full year. Shrink that interval to a tiny length Δt and you obtain the hazard rate, also called the default intensity. The hazard rate λ(t) is defined so that λ(t)Δt is the chance of default in the short interval after time t for a company still alive at t.

Let V(t) denote the cumulative probability that the company survives to time t, which is its survival probability. Seen from today, the probability of default inside that small interval is the fall in survival across it, V(t) − V(t + Δt). Dividing by V(t) turns it into a conditional probability, and matching that to the definition of the hazard rate gives the first line below. Rearranging and letting Δt shrink to zero gives a simple differential equation.

Integrating with V(0) = 1 gives the survival function, and the cumulative default probability Q(t) is simply one minus survival.

Writing the integral as t times the average hazard rate between zero and t gives the form used most often in practice.

When a company becomes bankrupt, everyone it owes money to lodges a claim. In the United States a bondholder claims the face value of the bond plus accrued interest. Sometimes the creditors agree to a reorganisation in which they accept part payment; in other cases a liquidator sells the assets and shares out the proceeds as far as they go. Some claims rank ahead of others and are paid more completely.

For a bond, the recovery rate is normally measured as the market price of the bond roughly one month after default, expressed as a percentage of principal. Moody’s data on recoveries for United States corporate bonds between 1983 and 2021 show how strongly seniority matters.

Source: Moody’s. The final column is 100 minus the recovery rate.

A first lien bond has the first claim on pledged collateral and a second lien bond the next claim. Among bonds with no collateral, the ranking runs senior unsecured, then senior subordinated, then subordinated, then junior subordinated. Average recoveries fall from 54.6% at the top to 22.3% at the bottom. The order is not perfectly monotonic in the data, since subordinated bonds (32.1%) recovered slightly more on average than senior subordinated bonds (31.0%), which is a reminder that these are historical averages over different sets of defaults rather than a mechanical rule.

Recovery rates and default rates move in opposite directions

The housing market in the 2007 to 2008 crisis showed the mechanism clearly. As more homeowners defaulted, more foreclosed houses came onto the market, house prices fell, and the amount lenders recovered on each defaulted mortgage shrank. A high default rate went hand in hand with a low recovery rate.

Corporate bonds show the same negative dependence. Altman, Brady, Resti and Sironi documented it, and Moody’s reports a high correlation between each year’s average recovery and that year’s default rate on non-investment-grade bonds. In a year with few defaults the economy is usually healthy, and recoveries on the bonds that do default might reach 60%. In a year with many defaults the economy is usually weak, and recoveries might be as low as 30%. So a lender is hit twice in a bad year: more borrowers default, and each default costs more.

A credit default swap (CDS) is a contract through which credit risk itself can be bought and sold. In its plain form it works like insurance against the default of one named company, the reference entity. A default by that company is a credit event. The protection buyer gains the right to sell bonds issued by the reference entity at their face value if a credit event occurs, and the protection seller promises to buy them at face value in that case. The total face value of bonds that may be sold this way is the swap’s notional principal.

In exchange, the protection buyer pays the seller a regular premium until the contract matures or a credit event occurs, whichever comes first. These payments are normally made quarterly in arrears. The total paid each year, as a percentage of notional principal, is the CDS spread.

Settlement, quotes and maturities

With physical settlement the protection buyer hands over bonds and receives their face value. With cash settlement, which is now the norm, a two-stage auction held several days after the credit event establishes a mid-market price for the cheapest bond that could be delivered, and the seller pays the difference between face value and that auction price.

Several large banks make markets in CDS. A dealer’s quote on five-year protection might read 250 bid and 260 offered, in basis points. That means the dealer will buy protection by paying 250 basis points a year (2.5% of principal) and will sell protection for 260 basis points a year (2.6% of principal).

Many companies and countries serve as reference entities. Five-year contracts are the most popular, but one, two, three, seven and ten year tenors are traded too. Contracts usually end on the 20th of March, June, September or December, so the true life of a new contract is close to, but not exactly, its stated tenor. Suppose five-year protection is bought by phoning a dealer on November 15, 2024; the contract would probably run to December 20, 2029. The first payment, due December 20, 2024, would cover the stub from November 15 to December 20, and quarterly payments would follow. (When the first standard date is less than a month away, the first payment is normally made on the second standard date instead.)

What counts as a credit event

The contract must define default precisely. A credit event usually includes failure to pay an amount when due, a restructuring of debt, or bankruptcy. Restructuring is sometimes left out of North American contracts, especially where the reference entity’s debt already carries a high yield.

Growth, stress and criticism of the market

The International Swaps and Derivatives Association produced a standard over-the-counter CDS contract in 1998 and 1999, and the market then expanded quickly. When the Bank for International Settlements began collecting statistics in December 2004, outstanding notional stood near $6 trillion. That figure reached a peak of $58 trillion in December 2007, then shrank to less than $10 trillion by December 2021. Financial institutions sit on both sides of the market; banks tend to be net buyers and insurance companies net sellers. A bank that hedges a loan with a CDS ends up in an odd position, since the institution bearing the credit risk is no longer the one that did the original credit checks.

From August 2007, regulators worried that CDS were a channel of systemic risk. AIG had sold large amounts of protection on AAA-rated tranches built from mortgages, the losses were severe, and its failure would have spread losses widely, so the United States government rescued it in September 2008. Even so, plain vanilla single-name CDS kept trading through 2007 and 2008, at much wider spreads, while many more complex credit derivatives stopped trading, largely because the simple contract was easy to understand. When Lehman Brothers failed in September 2008, the auction set recovery near eight cents per dollar of face value, leaving protection sellers to pay out roughly 92% of notional; despite fears of further failures, settlement on October 21, 2008 went smoothly. Later, in the euro area debt crisis, some lawmakers argued that speculation through CDS had deepened the troubles of Greece and other governments. Europe prohibited naked sovereign CDS positions, where protection on a country is held without any exposure to it, from 2013.

A deeper concern is asymmetric information. Most other over-the-counter contracts are driven by market variables such as rates, currencies, stock indices and commodities, about which nobody has a natural information advantage. A CDS spread depends on one company’s chance of default, and a bank that lends to that company, advises it and arranges its securities issues is likely to know more about it than a counterparty with no such relationship. Contracts can also be gamed. In a case the Financial Times described in January 2018 (January 11), the hedge fund GSO Capital Partners offered the housebuilder Hovnanian attractive financing on condition that it skip an interest payment, manufacturing a technical default that triggered a large payout on protection GSO had bought. Such engineered credit events have to be ruled out if the market is to survive, and flaws like these may help explain its declining popularity.

A CDS normally allows several different bonds of the reference entity to be delivered after default. They usually share the same seniority, yet they need not trade at the same percentage of face value once default has happened. One reason is that the bankruptcy claim equals face value plus accrued interest, so a bond with a lot of accrued interest tends to be worth more just after default; another is that the market may expect some bondholders to fare better than others in a reorganisation. The protection buyer will naturally deliver whichever eligible bond is cheapest. That right is the cheapest-to-deliver bond option, and in cash-settled contracts the auction is designed to find the value of that bond, which in turn sets the payoff.

Credit indices

Market participants built indices to track CDS spreads across many names, and in 2004 agreements between rival index producers consolidated the field. Two standard portfolios dominate.

Each portfolio is rolled every March and September: names that have lost investment-grade status are removed and new investment-grade names are added. The series number counts the rolls: when Series 37 of iTraxx Europe and Series 38 of CDX NA IG were defined in March 2022, it meant that by the end of that month the European portfolio had been updated 36 times and the North American one 37 times.

There is a liquid market in index protection at three, five, seven and ten year tenors, and these contracts usually mature on December 20 or June 20. So a “five-year” index trade actually lasts somewhere between 4¾ and 5¾ years. Roughly, the index level equals the average CDS spread of its members, but more precisely it sits a little below that average. Imagine an index of just two names, one quoted at 1,000 basis points and the other at 10. Protection on both costs slightly less than 505 basis points per name, because the risky name is likely to default sooner, so its high premium is expected to be paid for a shorter time and deserves less weight. A further wrinkle is that CDX NA IG counts restructuring as a default event, while single-name contracts on its members may not; iTraxx Europe does not have this mismatch.

Trading with fixed coupons

In practice CDS and CDS indices trade much like bonds. Each standard contract specifies a fixed coupon and a recovery rate, and the protection buyer pays that coupon regularly, while the market quotes a spread. At the trade date, the present value of the difference between paying the quoted spread and paying the fixed coupon over the remaining life is worked out by an agreed procedure, and one side pays it upfront. If the quoted spread is below the coupon, the protection seller pays the buyer, since the buyer will be overpaying through the coupon. If the quoted spread is above the coupon, the protection buyer pays the seller. From then on the buyer pays the coupon on the principal of whichever index members have not yet defaulted.

A credit spread is the extra annual return investors demand for bearing a particular credit risk. The CDS spread is one measure of it. A second is the bond yield spread, the amount by which a corporate bond’s yield exceeds that of a comparable risk-free bond. An arbitrage argument shows that the two should be close.

Picture a five-year corporate bond bought at par with a 7% yield, hedged on the same day with five-year CDS protection on its issuer costing 200 basis points a year. Netting the premium against the coupon, the investor earns 5% a year as long as the issuer survives. If the issuer defaults, the investor has earned 5% up to that point, delivers the bond under the CDS for its face value, and can reinvest that cash at the risk-free rate for the rest of the five years. The combined position is approximately a risk-free bond, so its 5% return should sit close to the risk-free rate.

So, the n-year CDS spread ought to be approximately equal to the n-year corporate bond’s par yield minus the n-year risk-free par yield, where the par yield is the coupon that would make a bond sell exactly at face value. If the CDS spread were well below that excess, holding the bond plus protection would return more than the risk-free rate. If it were well above, a short sale of the bond combined with selling protection would amount to borrowing below the risk-free rate. Neither trade is a perfect arbitrage, and liquidity constraints and other frictions limit how far either can be pushed.

Choosing the risk-free rate

A CDS spread is a direct reading of credit risk. A bond yield spread depends on which risk-free rate is subtracted. Bond traders usually measure it against a Treasury of similar maturity, quoting a corporate bond as, say, 250 basis points over Treasuries.

Researchers have turned the arbitrage argument around to infer the risk-free rate the market actually uses, matching CDS and bonds of the same maturity. With a five-year bond yield of 4.7% and five-year protection at 80 basis points, the risk-free rate implied for five years is 4.7% − 0.8% = 3.9%. Rates implied this way have come out above Treasury yields; one study put them near the short-term borrowing rate of a AA-rated company.

Asset swaps

An asset swap gives traders a direct estimate of how far a bond’s yield exceeds a floating reference rate. Suppose the asset swap spread on a bond is quoted as 150 basis points. Company A pays the bond’s promised coupons, while Company B pays floating plus 150 basis points. The coupons are exchanged whether or not the bond defaults. How the swap starts depends on where the bond is priced.

Built this way, the present value of the 150 basis point spread equals the amount by which a similar risk-free bond (discounted at the floating reference rate) is priced above the corporate bond. In symbols, if B is the corporate bond’s price per dollar of principal and B* its default-free value, the spread’s present value, V, satisfies V = B* − B, which is the present value of expected default losses under that reference rate.

The CDS-bond basis measures how far a company’s CDS spread exceeds its bond yield spread.

The arbitrage reasoning of the previous section says the basis should be close to zero. Several features of real contracts and real bonds push it away from zero, some in each direction.

The sign of the basis has changed with market conditions. Before the turmoil that began in 2007 it was usually positive; De Wit estimated the average basis in 2004 and 2005 at 16 basis points. During the Global Financial Crisis it was at times strongly negative, and as Bai and Collin-Dufresne (2011) explain, institutions could not trade it away because liquidity was scarce and other constraints bound. Since the crisis the basis has usually been small and negative.

Take a company whose five-year credit spread, whether read from a CDS, a bond yield spread or an asset swap, is 240 basis points, and suppose the expected recovery rate on default is 40%. A bondholder is compensated with 2.4% a year for expected default losses, so the spread can be thought of, roughly, as an average annual loss rate. Each default costs 1 − 0.4 = 60% of the exposure, which means defaults must be occurring at an average rate of 0.024 / 0.6 = 4% a year among surviving companies over the five years. In general:

When spreads are available at several maturities, the same idea lets you bootstrap a term structure of hazard rates, at least approximately. Because the average hazard from zero to T multiplied by T is the accumulated hazard, the average over a later interval follows by subtraction.

A more exact calculation from bond prices

The quick formula works well for CDS spreads, and for bond yield spreads or asset swap spreads when the bond trades near par. When a bond is far from par, a more careful method values the expected default loss explicitly.

Consider a five-year corporate bond with principal 100 and a 6% coupon paid semiannually, yielding 7% with continuous compounding. A comparable risk-free bond yields 5%, also continuously compounded. At those yields the corporate bond is worth 95.34 and the risk-free bond 104.09, so the present value of expected default losses over the five years is 104.09 − 95.34 = $8.75. To keep things simple, assume that the unconditional probability of default is the same value Q in every year, that default can occur only at 0.5, 1.5, 2.5, 3.5 and 4.5 years (just before a coupon date), that the risk-free rate is 5% continuously compounded at every maturity, and that recovery is 40% of principal.

At each possible default time the loss equals the bond’s risk-free value at that moment (computed from forward risk-free rates) less the 40 recovered. At 3.5 years, for instance, the default-free value is as follows.

The loss given default is 104.34 − 40 = $64.34, its present value is 64.34 × e^{−0.05 × 3.5} = 54.01, and the expected loss contribution is 54.01Q. Repeating this for every date gives the table.

Setting 288.48Q equal to the 8.75 expected loss gives Q = 8.75 / 288.48 = 3.03% per year.

The method extends naturally. Defaults can be allowed more often than once a year; a constant hazard rate, or any chosen time pattern, can replace a constant unconditional probability; and with several bonds you can estimate several parameters of the default term structure. Given four bonds, maturing at years 3, 5, 7 and 10, and a default probability shaped as a step function, the 3-year bond fixes the yearly default probability for years 1 to 3, the 5-year bond the fourth and fifth years, the 7-year bond years 6 and 7, and the 10-year bond years 8, 9 and 10. This mirrors the bootstrap used to build a zero-coupon interest rate curve.

Default probabilities estimated from history and default probabilities implied by credit spreads turn out to be very different. The comparison uses two data sets. The first is the seven-year cumulative default rate for each rating, taken from the 7-year column of the S&P table. The second is the average credit spread on bonds of each rating between December 1996 and June 2007, computed from Merrill Lynch yield data on bonds with an average maturity of about seven years, with the risk-free rate taken as 10 basis points below the seven-year swap rate. The spread data deliberately stop before the 2008 crisis; including that period, when spreads soared, would make the gap described below far more extreme.

Default rates from S&P Global; spreads from Merrill Lynch bond yields. The first row is printed as AA in the source; the 0.51 figure matches the AAA entry of the S&P table.

Turning each column into a hazard rate

From equation 9.2, if Q is the seven-year cumulative default probability then Q = 1 − e^−7λ̄, so the historical average hazard rate is found by taking logs.

The spread-based hazard rate comes from equation 9.3 with recovery set at 40%.

The final column is derived. For CCC/C the printed difference of 8.207 does not reconcile with its own columns: 18.395 − 10.118 = 8.277.

The spread-implied hazard is higher at every rating, and the absolute gap widens steadily as credit quality falls, from about 0.5 percentage points for AAA to more than 8 for CCC/C. Bond investors are being paid well for bearing credit risk, more than an actuarial estimate of expected default losses would justify, and that excess compensation grows with the amount of credit risk taken on. It is not constant over time either; when spreads are high, as in the 2008 crisis, the excess return tends to rise.

Why spreads imply more default than history

Asking why the two hazard rates differ is the same as asking why corporate bondholders earn more than the expected cost of default. Illiquidity is part of the answer, since corporate bonds are less liquid than Treasuries and investors want paying for that, but in ordinary conditions it accounts for only around 25 basis points of the extra return. Another possibility is that bond traders’ own subjective default probabilities are much higher than the historical record, perhaps because they allow for depression scenarios worse than anything in the data, yet it is hard to see this accounting for much of the gap.

The main reason is that defaults are not independent. Default rates swing widely from one year to the next; for all rated companies they ranged from 0.15% in 1981 to 4.19% in 2009. That variation is systematic risk, which cannot be diversified away, and bondholders earn an excess expected return for bearing it in the same way that equity investors do. Equity investors are often assumed to earn an extra 5% or 6% a year for systematic risk. Bondholders earn much less than this on high-quality bonds, but as credit quality falls a bond behaves more like equity and the excess return rises.

Default correlation has two broad sources. One is the economy: good times reduce default probabilities for all companies together and bad times raise them together. In Vasicek’s one-factor model the common factor F can be read as the overall health of the economy. The other is contagion, in which trouble in one sector or country spills into unrelated ones. Russia’s 1998 default triggered a rush into safe assets that widened spreads across the bond market; the same happened from 2007, and the recession that followed brought a record number of corporate defaults in 2009. In 2011 Greece’s difficulties made investors wary of Spanish, Irish, Portuguese and Italian debt, and spreads on those countries’ bonds rose sharply. Researchers still debate the cause; investors may grow more risk averse after losing money in one area, or bad news in one sector may darken their view of others. Either way, issuers in unrelated sectors can find funding harder to obtain and become more likely to default.

Nonsystematic risk adds to the premium. A stock investor can diversify away issuer-specific risk with a portfolio of around 30 stocks and so should not be paid for bearing it. Bond returns are far more skewed, with little upside. Over one year a single bond might return 7% with probability 99.75% (no default) and −60% with probability 0.25% (default). Diversifying that profile properly takes tens of thousands of bonds, and real portfolios are usually far from fully diversified, so bond investors may also earn a premium for idiosyncratic risk.

Which estimate for which job

Spread-based estimates are risk-neutral default probabilities; historical estimates are real-world default probabilities, also called physical probabilities. For valuation it is standard to work in a risk-neutral world, one in which investors require no extra return for risk, and so the spread-implied probabilities are the right input: a bond can be valued by applying them and discounting at the risk-free rate. For scenario analysis, including credit value at risk and other forecasts of what losses might actually occur, real-world probabilities from historical data are the appropriate choice.

Ratings are revised only now and then, so a default probability read from a rating table can lag events. Equity prices update continuously, and some analysts argue they offer a fresher source of default information. The foundation for this idea is the model Robert Merton proposed in 1974, in which a company’s equity is an option on its assets.

Keep the balance sheet simple: the firm has a single zero-coupon bond that matures at time T. The notation is listed below.

At time T, if V_T < D it is rational (in theory) for the company to default, and the shareholders are left with nothing. If V_T > D the debt is repaid and the shareholders keep V_T − D. Equity at maturity is therefore the payoff of a call option on the firm’s assets with a strike price equal to the debt repayment.

Applying the Black-Scholes-Merton formula, with no dividends, gives today’s equity value.

Default corresponds to the option finishing unexercised, and the probability of that is N(−d₂). Computing it needs V₀ and σ_V, neither of which can be observed directly. For a listed company, though, E₀ is observable, so equation 9.4 supplies one condition the two unknowns must satisfy. The equity volatility σ_E can also be estimated, and Ito’s lemma links it to asset volatility through the equity’s delta, ∂E/∂V, which for a call is N(d₁).

Equations 9.4 and 9.5 form a pair of simultaneous non-linear equations in V₀ and σ_V. In practice they are solved numerically: express both as functions equal to zero and let an optimiser such as Excel’s Solver choose V₀ and σ_V to minimise the sum of the two squared errors.

Distance to default

The output of Merton-type models is often summarised as the distance to default: the number of standard deviations by which the asset value would have to fall for default to be triggered T years ahead. One definition is simply d₂.

The smaller the distance to default, the more likely default becomes. For the company in Example 9, default over one year lies 1.14 standard deviations away.

Extensions and real-world vs risk-neutral output

The basic model has been extended in several ways. In one version default occurs as soon as asset value hits a barrier, rather than only at maturity. Another allows debt payments to fall due at several dates. Hull, Nelken and White (2004) show how implied volatilities can be used with the model.

The model also explains why risk-neutral default probabilities exceed real-world ones. In a risk-neutral world the firm’s assets are expected to grow at the risk-free rate. Real-world expected asset growth is normally higher, since investors require a risk premium. With a lower expected growth rate, the asset value is more likely to end below the face value of the debt, so the risk-neutral default probability is higher. Strictly, N(−d₂) is a risk-neutral probability, since it comes out of an option pricing model.

How good are the model’s default probabilities

Tested against actual default experience, Merton’s model and its extensions turn out to rank companies well by default probability, whether real-world or risk-neutral. Because the ranking is good, a monotonic transformation can be estimated that maps the model’s raw output to a reliable probability of either kind. Moody’s KMV and Kamakura provide a service of exactly this type, converting a Merton default probability into a real-world one, while other firms use the model to estimate credit spreads, which are closely linked to risk-neutral probabilities. Using a risk-neutral model output to produce real-world estimates may look strange, but it works because the calibration assumes only that the rankings produced by the model, by real-world probabilities and by risk-neutral probabilities are all the same. So the absolute level of N(−d₂) should not be taken at face value, while its ordering of firms is the part that carries information.