FI 4 – Credit Analysis Models
A corporate bond yields more than a government bond of the same maturity. That difference is the credit spread, known in practice as the G-spread, and it is the single most widely quoted measure of credit risk. It pays the investor for two distinct things: the chance that the issuer fails to make a scheduled payment in full and on time, and the size of the loss that would follow such a failure.
Those two things are not the same, and this reading keeps them apart deliberately. Default risk is the narrow idea: how likely is an event of default. Credit risk is the wide idea: how likely is default and how much is lost when it happens. A heavily collateralised loan can carry high default risk and low credit risk at the same time, because the collateral makes the loss small even though the borrower is shaky.
Throughout the analysis the corporate bond and the government bond are assumed to be identical in taxation and liquidity. That is plainly a simplification. Government bonds are issued in far larger size, attract a broader institutional buyer base, are easier to pledge in repo and for centrally cleared derivatives, and in some markets receive favourable tax treatment: interest on United States corporate bonds is taxed federally and at state level, while federal government debt escapes state tax. Setting those differences aside is what allows default risk and expected loss to be isolated as the drivers of the spread.
The first factor: expected exposure to default loss
Expected exposure is the amount of money at risk on a given date before any recovery is taken into account. Start with the simplest possible case: a one-year, 4% annual payment corporate bond priced at par. Only one payment date exists, and on it the investor is due the final coupon plus the principal. The exposure is therefore 104 per 100 of par value.
The idea becomes harder once several periods and volatile interest rates enter, because the price of the bond on each future date is itself uncertain. Section six of this lesson handles exactly that case using a binomial interest rate tree.
The second factor: the recovery rate and the loss given default
The recovery rate is the percentage of the exposure that a defaulted bondholder eventually gets back. It depends on the industry, on where the issue sits in the capital structure, on total leverage, and on whether the issue is secured or otherwise collateralised. A 40% recovery rate is the common baseline assumption and is used here.
Loss given default follows directly. With exposure of 104 and 40% recovery, the loss given default is 62.4 per 100 of par value, because 104 × (1 − 0.40) = 62.4. The investor who suffers a default therefore collects 41.6, since 104 − 62.4 = 41.6. The complementary phrase is loss severity: a 40% recovery rate is a 60% loss severity.
Two simplifications are embedded here and are worth naming. Recovery is assumed to be instantaneous, whereas real bankruptcies drag on for years before cash appears. And the recovery rate is assumed to apply to accrued interest as well as to principal.
The third factor: risk-neutral against actual default probability
The probability of default is where most of the subtlety lives. Credit modelling insists on a distinction borrowed from option pricing. In risk-neutral valuation the expected payoff is discounted at the risk-free rate, not at the security own yield, and for that to give the right answer the probabilities used to form the expectation must be risk-neutral probabilities rather than observed historical frequencies.
Work the one-year bond through both routes. Suppose a rating agency has assembled a long history of one-year bonds issued by companies with the same business profile. In that history 99% of the bonds pay in full and 1% default with an average recovery of 40%. The actual default probability is therefore 1%.
Using the actual probability, the expected value at maturity is (104 × 0.99) + (41.6 × 0.01) = 103.376. Discounting at the risk-free rate of 3% gives 103.376 ÷ 1.03 = 100.365. But the bond is observed trading at 100. The historical probability produces a value that is too high by 0.365 per 100 of par.
So ask the reverse question: what default probability would reproduce the observed price of 100? Call it P*, so that the survival probability is 1 − P*.
Why is the risk-neutral probability the larger of the two? Two reasons. First, the historical frequency contains no default risk premium, and investors demand one because they are uncertain not only about whether a loss occurs but about when it occurs. Second, the spread observed in the market is not pure credit compensation: liquidity and tax differences are baked into it as well, and the model attributes all of that to default.
An analyst is ranking one-year credit risk across three corporate bonds already held in a portfolio. Her estimates of exposure, default probability and recovery are below. The three bonds have very similar yields to maturity; the exposures differ because the coupon rates differ.
| Bond | Exposure (per 100 of par) | Probability of default | Recovery (per 100 of par) |
|---|---|---|---|
| A | 104 | 0.75% | 40 |
| B | 98 | 0.90% | 35 |
| C | 92 | 0.80% | 30 |
Bond A: LGD = 104 − 40 = 64; expected loss = 64 × 0.75% = 0.480.
Bond B: LGD = 98 − 35 = 63; expected loss = 63 × 0.90% = 0.567.
Bond C: LGD = 92 − 30 = 62; expected loss = 62 × 0.80% = 0.496.
The ranking from highest to lowest credit risk is B, C, A. Bond B has both the largest default probability and, once recovery is netted off, a loss given default close to the largest, so it leads on both counts.
Note what the question does not support. Without the current market prices of the three bonds there is no basis for a trading recommendation. A high expected loss is not a reason to sell if the price already reflects it.
Where ESG considerations enter
Environmental, social and governance factors feed into credit analysis through exactly the same two parameters. A polluting company faces fines and operating restrictions; a company with poor labour practices faces reputational damage, boycotts and litigation; a company with weak governance is more likely to produce aggressive or fraudulent accounts. Each of those affects the estimated probability of default, the estimated loss given default, or both, and should be incorporated there rather than treated as a separate overlay.
Some instruments carry an explicit ESG link. Climate bonds, also called green bonds, earmark proceeds for environmentally beneficial projects and sometimes carry tax incentives to widen their appeal. Catastrophe and pandemic bonds sit further from conventional debt and closer to insurance: the World Bank issued pandemic bonds in 2017 that paid high coupons in exchange for the investor accepting a loss of capital if a pandemic occurred, with the proceeds directed as aid to poor nations facing a serious outbreak. As of July 2020, almost all the principal on those bonds had been written off because COVID-19 caseloads and deaths passed the trigger thresholds.
The credit valuation adjustment, or CVA, is the value of a bond credit risk expressed in present value terms today. It is built date by date from the chain in Figure 1: exposure, recovery, loss given default, probability of default, expected loss, then discounting. Add the discounted expected losses across every date and the total is the CVA.
Once the CVA is known, valuation is a subtraction:
A five-year, zero-coupon corporate bond
Take a five-year zero-coupon corporate bond against a government yield curve that is flat at 3.00%. Default is assumed to occur only at a year end, on Dates 1 through 5, and never on Date 0. The recovery rate is 40% of exposure and the annual default probability is 1.25%.
The exposure on each date is the value the bond would have on that date if no default had happened, discounted from maturity at the risk-free rate. On Date 5 that is simply the par value of 100. On Date 1 it is 100 ÷ (1.0300)4 = 88.8487, and the intervening dates follow the same pattern.
The default probabilities are conditional: each year figure assumes the bond has survived every earlier year. Statisticians call these hazard rates. The Date 1 probability of default is the raw hazard rate of 1.25%, so the probability of arriving at Date 2 alive is 98.75%. The Date 2 probability of default is therefore 1.25% × 98.75% = 1.2344%, leaving survival of 98.75% − 1.2344% = 97.5156%. Continue in the same way and the Date 3 probability of default is 1.25% × 97.5156% = 1.2189%, with survival of 96.2967%.
There is a shortcut for the survival probability that is used repeatedly later: raise one minus the hazard rate to the number of years. Surviving all five years has probability (100% − 1.25%)5 = 93.9043%. The five annual default probabilities sum to 6.0957%, and 6.0957% + 93.9043% = 100%, as they must.
| Date | Exposure | Recovery amount | LGD | POD | POS | Expected loss | Discount factor | PV of loss |
|---|---|---|---|---|---|---|---|---|
| 1 | 88.8487 | 35.5395 | 53.3092 | 1.2500% | 98.7500% | 0.6664 | 0.970874 | 0.6470 |
| 2 | 91.5142 | 36.6057 | 54.9085 | 1.2344% | 97.5156% | 0.6778 | 0.942596 | 0.6389 |
| 3 | 94.2596 | 37.7038 | 56.5558 | 1.2189% | 96.2967% | 0.6894 | 0.915142 | 0.6309 |
| 4 | 97.0874 | 38.8350 | 58.2524 | 1.2037% | 95.0930% | 0.7012 | 0.888487 | 0.6230 |
| 5 | 100.0000 | 40.0000 | 60.0000 | 1.1887% | 93.9043% | 0.7132 | 0.862609 | 0.6152 |
| Total | 6.0957% | CVA = 3.1549 |
POD is the probability of default on that date given no prior default, POS the probability of survival, DF the default-risk-free discount factor. Figures are per 100 of par value. Calculations are carried out at full spreadsheet precision and reported rounded, so intermediate products may differ marginally from the rounded inputs shown.
Read one row across to see the chain working. On Date 3 the exposure is 94.2596; at 40% recovery the loss given default is 56.5558; the conditional default probability is 1.2189%; multiplying gives an expected loss of 0.6894; the discount factor 0.915142 turns that into a present value of 0.6309.
From CVA to fair value to credit spread
The discounted expected losses total 3.1549. A default-free five-year zero would be worth 100 × 0.862609 = 86.2609, so the fair value of the risky bond is 86.2609 − 3.1549 = 83.1060.
Now convert that price into a yield. Solving 100 ÷ (1 + Yield)5 = 83.1060 gives 3.77%. The five-year government zero yields 3.00%, so the credit spread is 77 bps.
A rule of thumb worth carrying into the exam room approximates the credit spread as the annual default probability multiplied by one minus the recovery rate. Here that gives 1.25% × (1 − 0.40) = 0.75%, against the exact 0.77%. Close, and quick.
The key idea is that the compensation for credit risk can be stated two ways that mean the same thing: as a CVA of 3.1549 per 100 of par in present value on Date 0, or as a credit spread of 77 bps per year for five years.
Why the timing of default matters so much
An investor who buys at 83.1060 faces a wide fan of outcomes, and the internal rate of return depends heavily on when default strikes rather than merely on whether it does.
| Outcome | Cash received | Annual rate of return |
|---|---|---|
| Default on Date 1 | 35.5395 | −57.24% |
| Default on Date 2 | 36.6057 | −33.63% |
| Default on Date 3 | 37.7038 | −23.16% |
| Default on Date 4 | 38.8350 | −17.32% |
| Default on Date 5 | 40.0000 | −13.61% |
| No default | 100.0000 | 3.77% |
An early default is punishing because the recoverable amount is small and there is almost no time over which to spread the loss: 83.1060 = 35.5395 ÷ (1 + IRR) gives IRR = −57.24%. A default at maturity is the mildest case at −13.61%, since the investor at least receives 40.0000 after five years. The most likely single outcome, with probability 93.9043%, is no default at all and a return of 3.77%, which is exactly the yield to maturity. That is the useful reminder buried in this table: a yield to maturity on a risky bond is the return conditional on no default, not an expected return.
This spread of outcomes is itself a source of the default risk premium. The probability of default embedded in a credit model is carrying information about the likely timing of default as well as about its likelihood.
A hedge fund trader sees a three-year, 5% annual payment corporate bond quoted at 104 per 100 of par value. The research team puts the risk-neutral annual default probability used to build the conditional PODs at 1.50%, with a recovery rate of 40%. The government bond yield curve is flat at 2.50%.
5 + 5 ÷ (1.0250)1 + 105 ÷ (1.0250)2 = 109.8186.
| Date | Exposure | Recovery amount | LGD | POD | POS | Expected loss | Discount factor | PV of loss |
|---|---|---|---|---|---|---|---|---|
| 1 | 109.8186 | 43.9274 | 65.8911 | 1.5000% | 98.5000% | 0.9884 | 0.975610 | 0.9643 |
| 2 | 107.4390 | 42.9756 | 64.4634 | 1.4775% | 97.0225% | 0.9524 | 0.951814 | 0.9066 |
| 3 | 105.0000 | 42.0000 | 63.0000 | 1.4553% | 95.5672% | 0.9169 | 0.928599 | 0.8514 |
| Total | 4.4328% | CVA = 2.7222 |
The CVA is the column total, 2.7222. Were the bond default free it would be worth
5 ÷ (1.0250)1 + 5 ÷ (1.0250)2 + 105 ÷ (1.0250)3 = 107.1401.
Fair value is therefore 107.1401 − 2.7222 = 104.4178. The bond trades at 104, so it is undervalued by 0.4178 per 100 of par value.
Default on Date 1: 104 = 43.9274 ÷ (1 + IRR)1, giving −57.76%.
Default on Date 2: 104 = 5 ÷ (1 + IRR)1 + 42.9756 ÷ (1 + IRR)2, giving −33.27%.
Default on Date 3: 104 = 5 ÷ (1 + IRR)1 + 5 ÷ (1 + IRR)2 + 42.0000 ÷ (1 + IRR)3, giving −22.23%.
No default: 104 = 5 ÷ (1 + IRR)1 + 5 ÷ (1 + IRR)2 + 105 ÷ (1 + IRR)3, giving 3.57%, which is the yield to maturity at the purchase price.
One caveat on interpretation: an internal rate of return implicitly assumes coupons are reinvested at the same rate, so these figures set aside reinvestment risk.
Lenders rarely build a CVA table before extending credit. They lean on third-party assessments, and those come in two forms serving two different markets. Credit scores dominate retail lending to individuals and small businesses. Credit ratings dominate the wholesale market for corporate bonds, sovereign bonds and asset-backed securities. Both are ordinal measures focused mainly on the likelihood of default.
Credit scoring and the FICO score
Scoring methodologies differ by country. Some jurisdictions record only adverse information such as delinquent payments or outright default, so in effect every borrower starts with a clean score until something goes wrong. Others draw on a wider information set. A score reflects observed behaviour rather than opinion. Credit reporting agencies tend to operate nationally, because legal systems and privacy rules do not travel.
The FICO score, a registered trademark of the Fair Isaac Corporation, is used by roughly 90% of retail lenders in the United States. It is computed from consumer credit files held by the three national bureaus, Experian, Equifax and TransUnion. Five factors carry the following weights in the proprietary algorithm:
- 35%, payment history. Delinquency, bankruptcy, court judgments, repossessions and foreclosures, or the absence of them.
- 30%, debt burden. Credit card debt-to-limit ratios, the number of accounts carrying a positive balance, and the total amount owed.
- 15%, length of credit history. The average age of the accounts on file and the age of the oldest account.
- 10%, types of credit used. Installment payments, consumer finance and mortgages.
- 10%, recent searches for credit. Hard inquiries generated by applications for new borrowing, but not soft inquiries such as employment verification or a consumer checking a personal score.
Fair Isaac also publishes what the score deliberately excludes: age, sex, marital status, race, colour and national origin, together with salary, occupation, employment history, home address, and any child or family support obligations. Scores range from a floor of 300 to a maximum of 850.
| Score band | October 2005 | April 2009 | April 2017 |
|---|---|---|---|
| 300–499 | 6.6% | 7.3% | 4.7% |
| 500–549 | 8.0% | 8.7% | 6.8% |
| 550–599 | 9.0% | 9.1% | 8.5% |
| 600–649 | 10.2% | 9.5% | 10.0% |
| 650–699 | 12.8% | 12.0% | 13.2% |
| 700–749 | 16.4% | 15.9% | 17.1% |
| 750–799 | 20.1% | 19.3% | 19.0% |
| 800–850 | 16.9% | 18.2% | 20.7% |
Source: Fair Isaac Corporation. The three dates are chosen to straddle the global financial crisis: before it, at its depth, and well after it.
The pattern is what the economics would predict. The weakest three bands together account for 23.6% of borrowers in October 2005, rise to 25.1% in April 2009 as conditions deteriorate, then fall to 20.0% by April 2017. The strongest band moves the other way, from 16.9% to 18.2% to 20.7%. The average score across the three months ran 688, then 687, then 700.
Tess Waresmith checks her FICO score every year. She plans eventually to buy a two-family house, live in one unit and let the other to help meet the mortgage payments. This year her score has risen from 760 to 775.
A. Age itself is explicitly excluded from the algorithm. What is included is the average age of the accounts on file and the age of the oldest account, both of which rise automatically with the passage of time. Combined with a clean payment record, that lifts the score. This is why age and credit score correlate strongly even though age is not an input.
B. The credit card debt-to-limit ratio sits inside the 30% debt burden component. Raising the limit from $1,000 to $2,500 while holding the balance constant cuts the ratio, which helps.
C. A new car loan adds a fresh type of credit in use, part of the 10% types-of-credit component, and it carries no late payments. The effect is positive.
D. Checking a personal score is a soft inquiry. Soft inquiries are not fed into the calibration at all, so refraining from them changes nothing.
Credit ratings, notching and rating history
In corporate and sovereign bond markets the three major global agencies are Moody’s Investors Service, Standard & Poor’s, and Fitch Ratings. Each rates issuers and specific issues.
The mechanism that carries loss given default into the rating is notching. An issuer rating normally applies to senior unsecured debt. A subordinated issue of the same issuer is then marked down, or notched, by one or two levels to reflect its weaker priority of claim. Because the rating embeds expected loss and not merely the chance of default, the agencies call these credit ratings rather than default ratings.
Beyond the letter grade, agencies attach an outlook, positive, stable or negative, and can place an issuer on watch. The value of that machinery shows in a long rating history. Standard & Poor’s carried RadioShack Corporation from BBB− in May 1969 up to BB+ in 1978, then all the way to AAA in January 1983, back down through the investment-grade range across the 1990s and 2000s, to BB in October 2006, and finally to D in February 2015.
Two lessons come out of that path. Ratings can sit unchanged for long stretches: RadioShack was A+ from 1984 to 1991 and A− from 1993 to 2005. And a default is usually preceded by a visible sequence of downgrades rather than arriving without warning, a point that matters when the two families of credit models are compared in section five.
A bond does not have to default to hurt. It only has to be downgraded. Agencies publish transition matrixes summarising, from historical experience, the probability that an issuer at one rating is at each possible rating a year later. Combine such a matrix with a set of representative credit spreads and a modified duration, and a yield to maturity can be converted into an expected return that allows for rating migration.
| Rating today | AAA | AA | A | BBB | BB | B | CCC to C | Default |
|---|---|---|---|---|---|---|---|---|
| From AAA | 90.00 | 9.00 | 0.60 | 0.15 | 0.10 | 0.10 | 0.05 | 0.00 |
| From AA | 1.50 | 88.00 | 9.50 | 0.75 | 0.15 | 0.05 | 0.03 | 0.02 |
| From A | 0.05 | 2.50 | 87.50 | 8.40 | 0.75 | 0.60 | 0.12 | 0.08 |
| From BBB | 0.02 | 0.30 | 4.80 | 85.50 | 6.95 | 1.75 | 0.45 | 0.23 |
| From BB | 0.01 | 0.06 | 0.30 | 7.75 | 79.50 | 8.75 | 2.38 | 1.25 |
| From B | 0.00 | 0.05 | 0.15 | 1.40 | 9.15 | 76.60 | 8.45 | 4.20 |
| From CCC to C | 0.00 | 0.01 | 0.12 | 0.87 | 1.65 | 18.50 | 49.25 | 29.60 |
| Spread at that rating | 0.60% | 0.90% | 1.10% | 1.50% | 3.40% | 6.50% | 9.50% |
D denotes default. The bottom row gives representative credit spreads on a 10-year corporate bond at each rating. Every row of transition probabilities sums to 100.
Read the A row across. An A rated issuer has an 87.50% chance of still being A in a year. It has a 0.05% chance of climbing to AAA, which is what RadioShack managed in 1983, and a 2.50% chance of reaching AA. On the downside there is an 8.40% chance of BBB, 0.75% of BB, 0.60% of B, 0.12% of the CCC to C band, and 0.08% of outright default.
Turning migration into a price change
For each possible destination rating, the expected percentage price change is the negative of modified duration multiplied by the change in the credit spread. Take an A rated 10-year corporate bond with a modified duration of 7.2 at the end of the year, holding benchmark yields and spread levels otherwise stable. The current A spread is 1.10%.
Applying that to each destination gives: to AAA, −7.2 × (0.60% − 1.10%) = +3.60%; to AA, −7.2 × (0.90% − 1.10%) = +1.44%; to BBB, −7.2 × (1.50% − 1.10%) = −2.88%; to BB, −7.2 × (3.40% − 1.10%) = −16.56%; to B, −7.2 × (6.50% − 1.10%) = −38.88%; and to the CCC to C band, −7.2 × (9.50% − 1.10%) = −60.48%. Staying at A produces no spread change and therefore no price change.
Weight each of those by its transition probability from the A row and add:
(0.0005 × 3.60%) + (0.0250 × 1.44%) + (0.8750 × 0%) + (0.0840 × −2.88%) + (0.0075 × −16.56%) + (0.0060 × −38.88%) + (0.0012 × −60.48%) = −0.6342%.
So the expected return over the coming year is the yield to maturity minus 63.42 bps, assuming no default. For an investment-grade issuer the default column can be left out of the calculation without much damage; for a bond below investment grade the transition to D would have to be included, and the loss on that branch is far larger than any spread-widening loss.
Why migration almost always subtracts from return
The drag is not an accident of these particular numbers. Two forces push it the same way. First, the transition probabilities are not symmetric around the current rating: downgrades are more likely than upgrades for most ratings. Second, the spread penalties for downgrades are much larger in size than the spread benefits from upgrades, because credit spreads rise convexly as quality falls. For the A rated bond above, one notch up to AA tightens the spread by 20 bps and adds 1.44% to the price, while one notch down to BBB widens it by 40 bps and takes 2.88% off.
Manuel Perello manages wealth for several Latin American families who hold part of their assets in very high-quality corporate bonds. He tells them that a yield to maturity should be adjusted for possible spread widening before it is used as an expected return. His illustration uses a 10-year, AAA rated corporate bond with a modified duration of 7.3 at the end of the year, and the transition matrix above. He concludes that the expected return over the coming year is roughly the yield to maturity less 32.5 bps, even with no default.
AAA to AA: −7.3 × (0.90% − 0.60%) = −2.19%.
AAA to A: −7.3 × (1.10% − 0.60%) = −3.65%.
AAA to BBB: −7.3 × (1.50% − 0.60%) = −6.57%.
AAA to BB: −7.3 × (3.40% − 0.60%) = −20.44%.
AAA to B: −7.3 × (6.50% − 0.60%) = −43.07%.
AAA to CCC, CC or C: −7.3 × (9.50% − 0.60%) = −64.97%.
Notice that every branch is negative. AAA is the top rating, so there is no upgrade available and the implied floor on spreads means migration can only hurt.
Second, weight by the AAA row of the matrix:
(0.9000 × 0%) + (0.0900 × −2.19%) + (0.0060 × −3.65%) + (0.0015 × −6.57%) + (0.0010 × −20.44%) + (0.0010 × −43.07%) + (0.0005 × −64.97%) = −0.3249%.
That is the 32.5 bps deduction, to the nearest half basis point. The AAA to D probability in the matrix is 0.00, so no default branch is needed here.
Credit analysis models divide into two families that answer two different questions. Structural models ask why a company defaults. Reduced-form models ask when it defaults. Neither is fully general, and the choice between them turns on what data the modeller can actually see.
Structural models
Structural models trace back to the 1970s and to Fischer Black, Myron Scholes and Robert Merton. Their insight was that a company defaults when the value of its assets falls below the value of its liabilities, and that the probability of that happening behaves exactly like an option.
The name comes from the balance sheet: the model depends on the structure of assets, liabilities and equity. It is equivalently called a company-value model, because the driving variable is the asset value of the company. Picture the asset value wandering over time above a horizontal default barrier set by the liabilities. Default is the event of the asset value dropping through that barrier.
The probability of default is endogenous to the model: it is simply the part of the probability distribution of the future asset value that lies below the barrier. That immediately delivers a set of comparative statics. Default probability rises with the variance of the future asset value, rises with the length of the horizon, and rises with financial leverage. Cutting debt lowers the barrier and reduces the default probability.
Debt and equity as options on the assets
The most valuable feature of the structural approach is the option interpretation of the capital structure. Let A(T) be the uncertain asset value at time T, and suppose for simplicity that all liabilities are zero-coupon bonds maturing at T with face value K, which is the default barrier. Write D(T) and E(T) for the values of debt and equity at T.
Equity is a purchased call because its value rises with the asset value and, like an option, it can never go negative. Debtholders receive a premium for writing the call, and that premium is the value of holding priority of claim: if the assets fall below K, equity is wiped out and the debtholders take what is left.
Check the two branches. If A(T) > K the call is in the money, so E(T) = A(T) − K and D(T) = A(T) − [A(T) − K] = K: the debt is repaid in full and shareholders keep the surplus. If A(T) < K the call expires worthless, so E(T) = 0 and D(T) = A(T) − 0 = A(T): the debtholders take the whole company and still fall short. The identity holds in both cases, and at A(T) = K as well. Limited liability is built into the model rather than assumed separately.
Carol Feely is a junior credit analyst at a major rating agency. She accepts that equity behaves like a call option on the asset value but dislikes the idea that debtholders implicitly own the assets and write that call. She proposes instead that shareholders own the net value of the company, A(T) − K, and that their limited liability is a long put option struck at K. Debtholders, on her account, own a risk-free bond worth K at time T plus a short position in that put.
E(T) equals A(T) − K plus max[K − A(T), 0].
D(T) equals K less max[K − A(T), 0].
Case 1, A(T) > K. The put is out of the money, so the max term is zero.
Equity becomes A(T) − K + 0, which is A(T) − K.
Debt becomes K − 0, which is K.
These are exactly the call-model values for the solvent branch.
Case 2, A(T) < K. The put is in the money and pays K − A(T).
E(T) = A(T) − K + [K − A(T)] = 0.
D(T) = K − [K − A(T)] = A(T).
Again identical to the call model on the default branch.
Ms Feely is correct, and her framing adds something useful. It puts a price on limited liability: it is worth precisely the value of the put option that shareholders buy from the debtholders.
Reduced-form models
Reduced-form models appeared in the 1990s and were built to sidestep a real weakness in the structural approach. Option pricing in the Black–Scholes–Merton tradition assumes the underlying asset is actively traded. That is unobjectionable for a stock option, but the assets of a company do not trade.
Reduced-form models therefore stop treating default as endogenous. Default becomes an exogenous event that arrives at random. Rather than explaining why default occurs, the model describes statistically when it occurs. The default time can be modelled with a Poisson stochastic process whose key parameter is the default intensity, the probability of default over the next small time increment. That is why these models are also called intensity-based models or stochastic default rate models.
Strengths, weaknesses and the choice between them
Structural models give genuine economic insight but are demanding to implement. The modeller must pin down the value of the company, its volatility, and a default barrier derived from the liabilities. Determining that barrier is straightforward on paper and awkward in practice, because the necessary data are often unavailable or unreliable. The list of companies that concealed debt, including Enron Corporation, Tyco International, WorldCom, Parmalat and Lehman Brothers, shows how the barrier can be badly mismeasured precisely when an accurate reading would matter most.
Reduced-form models run on observable inputs. Default intensity is estimated by regression on company-specific variables such as the leverage ratio, net income to assets and cash to assets, together with macroeconomic variables such as the unemployment rate, GDP growth and measures of stock market volatility. Including macro variables lets the credit risk measure track the business cycle directly, which structural models do not do naturally.
The weaknesses of the reduced-form family are the mirror image. They do not explain the economics of default, and they treat default as a surprise that can strike at any moment. Reality is usually less abrupt: as the RadioShack history in the previous section illustrates, an issuer is normally downgraded repeatedly before the final event.
The practical resolution is that the choice depends on who is asking and why. Structural models need information best known to company management, and perhaps to their commercial bankers and rating agencies, which suits them to internal risk management, banks internal credit measures and published credit ratings. Reduced-form models need only what markets already publish, which suits them to valuing risky debt securities and credit derivatives.
One final point stops the structural family from looking merely theoretical. Its practical value has come through implementation. Rating agencies and consultancies, most prominently Moody’s KMV Corporation, use option pricing methods to estimate default probabilities and losses given default, using the history of the company equity price to estimate the volatility input that the option model requires.
Everything so far assumed a flat government curve and no interest rate volatility, which made the exposure on each date a single certain number. Relax both assumptions and the exposure becomes an expectation across the nodes of a binomial interest rate tree. The credit machinery does not change: only the first column of the table becomes harder to compute.
The plan has two steps and one subtraction. First compute the value assuming no default, written VND, which is what the corporate bond would be worth if it were default free. Then compute the CVA using expected exposures taken from the same tree. Fair value is VND minus CVA.
Step one: build the benchmark curve
The starting point is the par curve for annual payment benchmark government bonds. Because every bond is priced at par and every maturity is a whole number of years, there is no accrued interest and each coupon rate equals its yield to maturity.
| Maturity | Coupon rate | Price | Discount factor | Spot rate | Forward rate |
|---|---|---|---|---|---|
| 1 | −0.25% | 100 | 1.002506 | −0.2500% | |
| 2 | 0.75% | 100 | 0.985093 | 0.7538% | 1.7677% |
| 3 | 1.50% | 100 | 0.955848 | 1.5166% | 3.0596% |
| 4 | 2.25% | 100 | 0.913225 | 2.2953% | 4.6674% |
| 5 | 2.75% | 100 | 0.870016 | 2.8240% | 4.9664% |
The forward rate column gives the one-year rate beginning at the end of the previous year. All figures were produced on a spreadsheet and are reported rounded.
The negative one-year yield is not a misprint. It reflects conditions seen in several markets in recent years. On a par curve every bond is shown at 100, so the one-year security carries a negative coupon rate; the actual instrument would be a zero-coupon bond priced at a premium of 100.2506, since (100 ÷ 100.2506) − 1 = −0.0025.
Discount factors are bootstrapped sequentially from the par bonds. The first equation is 100 = (100 − 0.25) × DF1, giving DF1 = 1.002506. The second uses the first: 100 = (0.75 × 1.002506) + (100.75 × DF2), giving DF2 = 0.985093. Continue in the same manner through DF5 = 0.870016.
Spot rates then come out of the discount factors. The two-year spot rate solves (1 ÷ 0.985093)1/2 − 1 = 0.007538, and the four-year spot rate solves (1 ÷ 0.913225)1/4 − 1 = 0.022953. Forward rates are simply ratios of adjacent discount factors: the one-year rate two years forward is 0.985093 ÷ 0.955848 − 1 = 3.0596%, and the one-year rate four years forward is 0.913225 ÷ 0.870016 − 1 = 4.9665%.
Step two: calibrate the binomial tree
A no-arbitrage binomial tree of one-year forward rates is built to be consistent with those benchmark prices and with an assumed level of future interest rate volatility, here 10%. Rates at each date are spread around the corresponding implied forward rate, with the spacing widening as the assumed volatility rises.
| Date | Rate (probability of reaching the node) | ||||
|---|---|---|---|---|---|
| 0 | −0.2500% (1.0000) | ||||
| 1 | 1.9442% (0.5000) | 1.5918% (0.5000) | |||
| 2 | 3.7026% (0.2500) | 3.0315% (0.5000) | 2.4820% (0.2500) | ||
| 3 | 6.2197% (0.1250) | 5.0922% (0.3750) | 4.1692% (0.3750) | 3.4134% (0.1250) | |
| 4 | 7.2918% (0.0625) | 5.9700% (0.2500) | 4.8878% (0.3750) | 4.0018% (0.2500) | 3.2764% (0.0625) |
Rates run from the highest node to the lowest across each row. On Date 4 the range runs from 7.2918% down to 3.2764%.
The tree is arbitrage free only if it reprices the benchmark bonds. Test it on the 2.75% annual payment government bond, which the par curve says is worth 100. Working backwards from maturity, the five Date 4 values are 102.75 divided by one plus each Date 4 rate: 102.75 ÷ 1.072918 = 95.7669, then 96.9614, 97.9618, 98.7964 and 99.4903 at the successively lower rates. The four Date 3 values average the two Date 4 values that follow, add the coupon and discount:
At the top Date 3 node, [(0.5 × 95.7669) + (0.5 × 96.9614)] + 2.75, all divided by 1.062197, equals 93.3105. The remaining Date 3 values are 95.3559, 97.0816 and 98.5301. Continuing back to Date 0 produces exactly 100.0000, which confirms the calibration.
Step three: value the corporate bond assuming no default
Now take a five-year, 3.50% annual payment corporate bond. The analyst assigns an annual default probability of 1.25%, a recovery rate of 40%, and 10% volatility in benchmark rates.
Running the same backward induction with a 3.50% coupon gives a VND of 103.5450 per 100 of par value. The Date 4 node values are 96.4659, 97.6692, 98.6769, 99.5175 and 100.2165 from the highest rate down. The same answer comes far more quickly from the benchmark discount factors:
(3.50 × 1.002506) + (3.50 × 0.985093) + (3.50 × 0.955848) + (3.50 × 0.913225) + (103.50 × 0.870016) = 103.5450.
The reason for bothering with the tree is not the VND. It is that the same tree delivers the expected exposures, which the discount factors cannot.
Step four: expected exposure, CVA and the credit spread
The expected exposure on a date is the probability-weighted average of the bond values across that date nodes, plus the coupon paid on that date. For Date 4:
(0.0625 × 96.4659) + (0.25 × 97.6692) + (0.375 × 98.6769) + (0.25 × 99.5175) + (0.0625 × 100.2165) + 3.50 = 102.0931.
| Date | Expected exposure | LGD | POD | Discount factor | CVA per year |
|---|---|---|---|---|---|
| 1 | 103.2862 | 61.9717 | 1.2500% | 1.002506 | 0.7766 |
| 2 | 101.5481 | 60.9289 | 1.2344% | 0.985093 | 0.7409 |
| 3 | 101.0433 | 60.6260 | 1.2189% | 0.955848 | 0.7064 |
| 4 | 102.0931 | 61.2559 | 1.2037% | 0.913225 | 0.6734 |
| 5 | 103.5000 | 62.1000 | 1.1887% | 0.870016 | 0.6422 |
| Total | 6.0957% | CVA = 3.5394 |
The Date 4 loss given default is 102.0931 × (1 − 0.40) = 61.2559. The Date 4 conditional default probability is 1.25% × (100% − 1.25%)3 = 1.2037%, that is, the hazard rate multiplied by the probability of having survived the first three years. Multiplying LGD by POD and discounting at 0.913225 gives that year contribution of 0.6734.
The CVA totals 3.5394, so the fair value is 103.5450 − 3.5394 = 100.0056 per 100 of par value. Solving the price for a yield gives 3.4988%. The five-year benchmark par yield is 2.75%, so the credit spread is 3.4988% − 2.75% = 0.7488%, or 74.88 bps.
One caveat travels with that 74.88 bps. It is a credit spread only if the whole observed gap between corporate and government yields is credit. In practice liquidity and tax differences are also in there, and this analysis quietly attributes them to credit risk. In practice too, the spread is usually measured against the actual yield on a comparable-maturity government bond, which may itself be trading away from par.
Lori Boller runs a long-only high-yield mandate and hunts for bonds whose credit spread implies too high a default probability or too low a recovery rate. She is looking at a three-year, 4.00% annual payment bond priced at 104 per 100 of par value. In her view it should be priced on an annual default probability of 2.25% with a recovery rate of 40%, and she is comfortable assuming 10% volatility in government bond yields. Use the benchmark par curve and the 10% volatility tree above.
Step 1. VND from the tree. The bond matures on Date 3, so the Date 2 values are 104 divided by one plus each Date 2 rate:
104 ÷ 1.037026 = 100.2868; 104 ÷ 1.030315 = 100.9400; 104 ÷ 1.024820 = 101.4812.
Roll back to Date 1:
[(0.5 × 100.2868) + (0.5 × 100.9400) + 4] ÷ 1.019442 = 102.6183.
[(0.5 × 100.9400) + (0.5 × 101.4812) + 4] ÷ 1.015918 = 103.5621.
And to Date 0:
[(0.5 × 102.6183) + (0.5 × 103.5621) + 4] ÷ 0.997500 = 107.3586.
Step 2. Expected exposures.
Date 1: (0.50 × 102.6183) + (0.50 × 103.5621) + 4 = 107.0902.
Date 2: (0.25 × 100.2868) + (0.50 × 100.9400) + (0.25 × 101.4812) + 4 = 104.9120.
Date 3: the bond simply pays 104.
Step 3. LGD and POD. At 40% recovery, 107.0902 × (1 − 0.40) = 64.2541, 104.9120 × (1 − 0.40) = 62.9472, and 104 × (1 − 0.40) = 62.4000. The conditional default probabilities are 2.25% on Date 1, then 2.25% × (100% − 2.25%) = 2.1994% on Date 2, and 2.25% × (100% − 2.25%)2 = 2.1499% on Date 3.
| Date | Expected exposure | LGD | POD | Discount factor | CVA per year |
|---|---|---|---|---|---|
| 1 | 107.0902 | 64.2541 | 2.2500% | 1.002506 | 1.4493 |
| 2 | 104.9120 | 62.9472 | 2.1994% | 0.985093 | 1.3638 |
| 3 | 104.0000 | 62.4000 | 2.1499% | 0.955848 | 1.2823 |
| Total | 6.5993% | CVA = 4.0954 |
Does the assumed volatility matter?
Repeat the whole exercise on a no-arbitrage tree calibrated to 20% volatility instead of 10%. The rate ranges widen considerably: the Date 4 rates now run from 2.0948% up to 10.3757%, against 3.2764% to 7.2918% at 10% volatility. Every node value in the tree changes.
The VND does not. It is still 103.5450. That is the expected result and a further check on the calibration: assumed future rate volatility has no effect on the value of a default-risk-free bond without embedded options.
The CVA does change, very slightly. The expected exposures on Dates 2, 3 and 4 come out marginally lower at 101.5423, 101.0233 and 102.0636, against 101.5481, 101.0433 and 102.0931 at 10% volatility. Those small differences run through the loss given default and the yearly contributions, and the total CVA becomes 3.5390 rather than 3.5394. Fair value is therefore 100.0060 rather than 100.0056.
The direction of that tiny effect is worth understanding, because it is not obvious. Interest rate trees built on a lognormal assumption spread rates around the implied forward rate asymmetrically. The one-year forward rate four years ahead is 4.9665%. At 20% volatility the top Date 4 node sits 5.4092% above it, at 10.3757%, while the bottom node sits only 2.8717% below it, at 2.0948%. The upside in rates is larger than the downside. Higher rates mean lower bond values, so the top of the tree carries less exposure to loss, and that reduction more than offsets the greater exposure created at the bottom of the tree. Net expected exposure falls, so the CVA falls and fair value rises.
The general point matters more than the arithmetic. A change in assumed volatility usually moves a bond value only when there is an embedded option. Here it moves the value of an option-free bond, and the channel is credit risk.
The same arbitrage-free framework values a risky floating-rate note. Consider a five-year floater paying, annually, the one-year benchmark rate plus 0.50%. That 50 bp add-on is the quoted margin and is normally fixed for the life of the security.
Two mechanical points govern the cash flows. Interest is paid in arrears: the rate is set at the start of a period and paid at the end of it. So the payment landing on a given date is determined by the rate at the previous date, which is why the payments differ across nodes. The Date 1 payment is fixed at inception, because the Date 0 rate is known: (−0.25% + 0.50%) × 100 = 0.25. The final payment on Date 5 depends on the Date 4 rate; at the middle Date 4 node, where the rate is 4.8878%, it is (4.8878% + 0.50%) × 100 + 100 = 105.3878.
The value assuming no default
Backward induction through the 10% volatility tree gives a VND of 102.3633 per 100 of par value. The Date 4 values are close together and only just above par, which is exactly what a floater is designed to do: 107.7918 ÷ 1.072918 = 100.4660, and at the successively lower rates 100.4718, 100.4767, 100.4808 and 100.4841. Had the quoted margin been zero, every one of those values would have been exactly 100.0000.
The Date 3 values follow the usual rule, but the coupon added at each node is the one set by that node rate one period earlier. At the top Date 3 node the payment is 6.7197 and the value is [(0.5 × 100.4660) + (0.5 × 100.4718) + 6.7197] ÷ 1.062197 = 100.9122. Continuing gives Date 2 values of 101.3689, 101.3911 and 101.4098, Date 1 values of 101.8442 and 101.8707, and finally
[(0.5 × 101.8442) + (0.5 × 101.8707) + 0.2500] ÷ 0.997500 = 102.3633.
Expected exposure when payments and values use different probabilities
Floaters introduce one wrinkle in the exposure calculation. The bond value on a date follows the node probabilities for that date, but the interest payment arriving on that date was fixed one period earlier and therefore follows the node probabilities for the previous date. Both pieces must be weighted correctly. For Date 4:
Bond values: (0.0625 × 100.4660) + (0.25 × 100.4718) + (0.375 × 100.4767) + (0.25 × 100.4808) + (0.0625 × 100.4841).
Interest: (0.125 × 6.7197) + (0.375 × 5.5922) + (0.375 × 4.6692) + (0.125 × 3.9134).
The two together give an expected exposure of 105.6535.
Credit parameters that change over the life of the bond
This example also illustrates a deterioration in credit quality. Over Years 1 to 3 the annual default probability runs at 0.50% against a recovery rate of 20%. The issuer is then assumed to deteriorate, so Years 4 and 5 carry a 0.75% annual default probability and only a 10% recovery rate.
| Date | Expected exposure | LGD | POD | Discount factor | CVA per year |
|---|---|---|---|---|---|
| 1 | 102.1074 | 81.6859 | 0.5000% | 1.002506 | 0.4095 |
| 2 | 103.6583 | 82.9266 | 0.4975% | 0.985093 | 0.4064 |
| 3 | 104.4947 | 83.5957 | 0.4950% | 0.955848 | 0.3955 |
| 4 | 105.6535 | 95.0881 | 0.7388% | 0.913225 | 0.6416 |
| 5 | 105.4864 | 94.9377 | 0.7333% | 0.870016 | 0.6057 |
| Total | 2.9646% | CVA = 2.4586 |
Credit assumptions: Dates 1 to 3, annual default probability 0.50% and recovery rate 20%; Dates 4 and 5, annual default probability 0.75% and recovery rate 10%.
Follow the probabilities through the switch. The Date 2 default probability is 0.50% × (100% − 0.50%) = 0.4975%, and the Date 3 figure is 0.50% × (100% − 0.50%)2 = 0.4950%. Survival into the fourth year is (100% − 0.50%)3 = 98.5075%, so the Date 4 default probability jumps to 0.75% × 98.5075% = 0.7388%. Survival into the fifth year is 98.5075% − 0.7388% = 97.7687%, giving a Date 5 default probability of 0.75% × 97.7687% = 0.7333%. Cumulative default probability over the five years is 2.9646%.
The recovery change bites hard on the loss given default. On Date 2 the LGD is 103.6583 × (1 − 0.20) = 82.9266, whereas on Date 4 it is 105.6535 × (1 − 0.10) = 95.0881. The last two years contribute more than half the total CVA even though they carry less than half the cumulative default probability.
The CVA is 2.4586, so the fair value of the floater is 102.3633 − 2.4586 = 99.9047.
The discount margin
For a floating-rate note the yield measure corresponding to the credit spread on a fixed-rate bond is the discount margin. Because this floater is priced below par, its discount margin must exceed its quoted margin of 0.50%.
The discount margin is found by trial and error. Add a trial margin to the benchmark rate at every node, value the note by backward induction, and adjust the trial margin until the Date 0 value equals the fair value of 99.9047. The answer is 0.52046%. At that margin the Date 2 values are:
[(0.5 × 99.9629) + (0.5 × 99.9623) + 4.2026] ÷ (1 + 0.037026 + 0.0052046) = 99.9445, and similarly 99.9436 and 99.9429 at the lower nodes.
The two routes agree, which is the point of the exercise. Valuing the note as VND minus CVA and valuing it by adding a discount margin to every benchmark rate give the same Date 0 price.
Omar Yassin assesses distressed high-yield bonds. One candidate is a floater with three years left, paying annually at the one-year benchmark rate plus a quoted margin of 2.50%. It is rated CCC and is priced at 84 per 100 of par value. From research reports and the issuer credit default swap prices, he puts the probability of default in the coming year at about 30%. If the issuer files for bankruptcy at any point, he expects recovery of at least 50%, possibly as much as 60% because of valuable property holdings. If the issuer survives the coming year, he believes the default probability falls to about 10% for each of the remaining two years. He assumes 10% interest rate volatility.
Step 1. Cash flows. Each payment is the benchmark rate at the start of the year plus 2.50%, times 100. The Date 0 rate is −0.25%, so the Date 1 payment is (−0.25% + 2.50%) × 100 = 2.2500. If the Date 2 rate is 2.4820%, the maturity payment on Date 3 is (2.4820% + 2.50%) × 100 + 100 = 104.9820.
Step 2. VND. Date 2 values: 106.2026 ÷ 1.037026 = 102.4107; 105.5315 ÷ 1.030315 = 102.4264; 104.9820 ÷ 1.024820 = 102.4395.
Date 1: [(0.5 × 102.4107) + (0.5 × 102.4264) + 4.4442] ÷ 1.019442 = 104.8248, and [(0.5 × 102.4264) + (0.5 × 102.4395) + 4.0918] ÷ 1.015918 = 104.8557.
Date 0: [(0.5 × 104.8248) + (0.5 × 104.8557) + 2.2500] ÷ 0.997500 = 107.3586.
Step 3. Expected exposures.
Date 1: (0.5 × 104.8248) + (0.5 × 104.8557) + 2.2500 = 107.0902.
Date 2: (0.25 × 102.4107) + (0.5 × 102.4264) + (0.25 × 102.4395) + (0.5 × 4.4442) + (0.5 × 4.0918) = 106.6938.
Date 3: (0.25 × 106.2026) + (0.5 × 105.5315) + (0.25 × 104.9820) = 105.5619.
Step 4. Default probabilities. Date 1 is the raw 30%. Survival into the second year is 70%, so the Date 2 default probability is 70% × 10% = 7.00%. Survival into the third year is 70% − 7% = 63%, so the Date 3 default probability is 10% × 63% = 6.30%. Cumulative default probability is 43.30%.
| Date | Expected exposure | LGD at 50% | CVA at 50% | LGD at 60% | CVA at 60% | POD | Discount factor |
|---|---|---|---|---|---|---|---|
| 1 | 107.0902 | 53.5451 | 16.1038 | 42.8361 | 12.8830 | 30.0000% | 1.002506 |
| 2 | 106.6938 | 53.3469 | 3.6786 | 42.6775 | 2.9429 | 7.0000% | 0.985093 |
| 3 | 105.5619 | 52.7810 | 3.1784 | 42.2248 | 2.5427 | 6.3000% | 0.955848 |
| Total | 22.9608 | 18.3686 | 43.3000% |
Step 5. Recommend. He should buy. At 50% recovery the note is close to fairly priced at 84, and at 60% recovery it is materially undervalued. The entire decision turns on the recovery assumption, not on the default probability, which is common in distressed credit.
Step 6. The return if it survives. Cumulative default probability is 43.30%, so the probability of no default over the three years is the complement, 56.70%, which is also 0.70 × 0.90 × 0.90. In that case the discount margin measures the return, in the same way a yield to maturity does for a fixed-rate bond. A trial-and-error search gives a discount margin of 8.9148%, far above the 2.50% quoted margin because the note trades at a deep discount. At that margin the Date 1 values are
[(0.5 × 94.3039) + (0.5 × 94.2698) + 4.4442] ÷ (1 + 0.019442 + 0.089148) = 89.0600, and 88.9969 at the lower node, and Date 0 comes back to 84.0000.
Corporate yields, benchmark yields and the gap between them move every day. The analyst task is not to record the move but to say what caused it. Doing that requires knowing what is inside a yield in the first place.
The benchmark yield captures the macroeconomic factors that touch every debt security: the expected inflation rate and the expected real rate of return, plus whatever risk premium investors demand for being uncertain about those two. The spread over the benchmark captures the microeconomic factors specific to the issuer and to the particular issue. The dominant one is expected loss from default, but liquidity and taxation sit in there too, together with a premium for uncertainty about all three.
These components are hard to separate in practice, and they interact. A security whose default probability and recovery rate are difficult to assess tends to trade less often, so credit uncertainty produces illiquidity. An uncertain tax treatment of gains and losses raises the time and cost of valuing the bond, which also makes it less liquid.
The XVA family
Banks and consultancies have extended this thinking into derivatives valuation. The method starts with a value computed on benchmark discount factors, in practice derived from overnight indexed swap rates, which reference an average daily interest rate such as the effective federal funds rate in the United States. That OIS value plays the same role as the VND in the previous sections. It is then adjusted for the other factors, and those adjustments are known collectively as the XVA.
The credit valuation adjustment is the most developed and the most widely used member of the family. Alongside it sit a funding valuation adjustment, a liquidity valuation adjustment and a taxation valuation adjustment. In principle the same decomposition applies to debt securities: the observed spread between corporate and benchmark yields is the sum of these adjustments. This reading isolates the credit component.
It is worth being honest about scale. The models used in practice, sometimes called XVA engines, run Monte Carlo simulations over thousands of interest rate paths. The five-year binomial tree used here contains just sixteen paths. It is a small working replica of the real machinery, useful because every number in it can be checked by hand.
How default probability drives the spread
Return to the five-year, 3.50% corporate bond with a VND of 103.5450. Instead of assuming a default probability and deriving a spread, run the model in reverse: for each credit spread, find the annual default probability consistent with it at a 40% recovery rate.
Start with the AAA spread of 0.60% from the transition matrix. Trial and error gives an annual default probability of 1.01%. The expected exposures and losses given default are unchanged from the earlier table, because those depend on the interest rate tree and the recovery rate rather than on the default probability. Only the POD column and the yearly CVA contributions move.
| Date | Expected exposure | LGD | POD | Discount factor | CVA per year |
|---|---|---|---|---|---|
| 1 | 103.2862 | 61.9717 | 1.0100% | 1.002506 | 0.6275 |
| 2 | 101.5481 | 60.9289 | 0.9998% | 0.985093 | 0.6001 |
| 3 | 101.0433 | 60.6260 | 0.9897% | 0.955848 | 0.5735 |
| 4 | 102.0931 | 61.2559 | 0.9797% | 0.913225 | 0.5481 |
| 5 | 103.5000 | 62.1000 | 0.9698% | 0.870016 | 0.5240 |
| Total | 4.9490% | CVA = 2.8731 |
The CVA of 2.8731 gives a fair value of 103.5450 − 2.8731 = 100.6719, a yield to maturity of 3.35%, and a credit spread of 3.35% − 2.75% = 0.60%, which is the target. Repeating the search across every rating in the transition matrix produces the following.
| Credit rating | Credit spread | Annual default probability | Cumulative default probability |
|---|---|---|---|
| AAA | 0.60% | 1.01% | 4.95% |
| AA | 0.90% | 1.49% | 7.23% |
| A | 1.10% | 1.83% | 8.82% |
| BBB | 1.50% | 2.48% | 11.80% |
| BB | 3.40% | 5.64% | 25.19% |
| B | 6.50% | 10.97% | 44.07% |
| CCC, CC, C | 9.50% | 16.50% | 59.41% |
Cumulative figures are over the five-year life of the bond.
How the recovery rate drives the spread
Now hold the default probability fixed and move the recovery rate instead. Take the same bond with an annual default probability of 1.83%, which corresponds to an A rating and a 1.10% spread at 40% recovery. Suppose analysts conclude the recovery rate will be only 30%.
| Date | Expected exposure | LGD | POD | Discount factor | CVA per year |
|---|---|---|---|---|---|
| 1 | 103.2862 | 72.3003 | 1.8300% | 1.002506 | 1.3264 |
| 2 | 101.5481 | 71.0837 | 1.7965% | 0.985093 | 1.2580 |
| 3 | 101.0433 | 70.7303 | 1.7636% | 0.955848 | 1.1923 |
| 4 | 102.0931 | 71.4652 | 1.7314% | 0.913225 | 1.1300 |
| 5 | 103.5000 | 72.4500 | 1.6997% | 0.870016 | 1.0714 |
| Total | 8.8212% | CVA = 5.9781 |
Cutting the recovery rate from 40% to 30% lifts every loss given default by one sixth and pushes the CVA up to 5.9781. Fair value falls to 103.5450 − 5.9781 = 97.5670, the yield to maturity rises to 4.05%, and the credit spread becomes 4.05% − 2.75% = 1.30%, twenty basis points wider.
That twenty basis points is notching made quantitative. If the issuer is rated A on the strength of a 1.83% default probability and 40% recovery on its senior unsecured debt, a subordinated issue of the same issuer with a 30% expected recovery would carry the wider 1.30% spread, and the agency could justify rating it A− or BBB+ rather than A. Same issuer, same default probability, different priority of claim, different rating.
Edward Kapili is a summer intern on a fixed-income trading desk. He is asked to value a three-year, 3% annual payment corporate bond using a binomial tree calibrated to 20% volatility and the benchmark par curve, with an assumed annual default probability of 1.50% and a recovery rate of 40%. His supervisor then asks whether the credit spread over the three-year benchmark yield of 1.50% would rise more if the default probability doubled to 3.00% or if the recovery rate halved to 20%. Mr Kapili expects the doubling of the default probability to matter more.
Base case. On the 20% volatility tree the Date 2 values are 103 ÷ 1.043999 = 98.6591, 103 ÷ 1.029493 = 100.0492, and 103 ÷ 1.019770 = 101.0032. Rolling back gives Date 1 values of 100.2313 and 102.0770 and a VND of 104.4152. The expected exposures are 104.1541 on Date 1, 102.9402 on Date 2 and 103.0000 on Date 3.
| Assumption | LGD, Dates 1 to 3 | POD, Dates 1 to 3 | CVA | Fair value | YTM | Credit spread |
|---|---|---|---|---|---|---|
| POD 1.50%, recovery 40% | 62.4925 / 61.7641 / 61.8000 | 1.5000% / 1.4775% / 1.4553% | 2.6984 | 101.7168 | 2.40% | 0.90% |
| POD 3.00%, recovery 40% | 62.4925 / 61.7641 / 61.8000 | 3.0000% / 2.9100% / 2.8227% | 5.3174 | 99.0978 | 3.32% | 1.82% |
| POD 1.50%, recovery 20% | 83.3233 / 82.3522 / 82.4000 | 1.5000% / 1.4775% / 1.4553% | 3.5978 | 100.8173 | 2.71% | 1.21% |
Doubling the default probability. The VND is unchanged at 104.4152, because nothing about the benchmark tree or the coupon has moved. The CVA rises to 5.3174, fair value falls to 99.0978, the yield to maturity rises to 3.32%, and the spread widens to 1.82%, an increase of 92 bps.
Halving the recovery rate. Again the VND is 104.4152. The CVA rises to 3.5978, fair value falls to 100.8173, the yield to maturity rises to 2.71%, and the spread widens to 1.21%, an increase of 31 bps.
Conclusion. Mr Kapili is right. Doubling the default probability widens the spread by roughly three times as much as halving the recovery rate. The reason is structural: doubling the hazard rate doubles the expected loss on every date, whereas halving recovery from 40% to 20% raises the loss given default only from 60% of exposure to 80%, an increase of one third.
A yield curve traces one government issuer borrowing costs across maturities. A credit curve does the same thing for the spread: it plots, for one issuer or one rating category, the spread over a benchmark security across the range of outstanding maturities.
The uses are practical and widespread. An issuer working with an underwriter looks at relative spreads across maturities before choosing the tenor of a new issue or tendering for existing debt. An investment-grade portfolio manager uses an issuer existing credit curve to set a bid on a new primary issue and to guide secondary trading. Where an issuer has no curve of its own, the curve for a rating category or a corporate sector can supply prospective pricing for a new issue or fair value spreads for outstanding paper, which is an extension of matrix pricing. High-yield investors use the curve to weigh risk against reward across maturities. Because monetary and fiscal policy transmit through risky debt markets, policymakers now watch the credit spread term structure alongside the default-risk-free yield curve.
What shapes the curve
Four groups of drivers matter.
Credit quality. For the highest rated securities, spreads are already close to the implied lower bound of zero, so migration can realistically go one way only. The credit term structure for such issuers therefore tends to be flat or gently upward sloping. Lower quality issuers are far more sensitive to the credit cycle, and their curves are steeper, whether steeply upward sloping because a weakening economy points to spread widening or steeply inverted because the market expects tighter spreads at longer maturities. Push far enough down the quality spectrum and the contractual cash flows themselves stop being credible: the price of distressed debt converges on the recovery rate regardless of the remaining time to maturity, which produces a steeply inverted curve.
Financial conditions. The credit risk of a bond depends on expectations for growth and inflation. A stronger economy usually brings higher benchmark yields but lower credit spreads, because improving cash flows and profitability cut default probabilities. That countercyclical relationship between spreads and benchmark rates is one of the most consistently observed patterns across the business cycle.
Supply and demand. Unlike developed-market government debt, corporate bond liquidity varies enormously, and the great majority of issues do not trade on a given day. New and recently issued securities account for most of the volume and most of the spread volatility, so the curve is set largely by the handful of most actively traded bonds. The effects can cut against intuition. A borrower refinancing near-term maturities with long-term debt might be expected to steepen its curve, yet a tighter bid–offer spread at the long end can partly offset that. Anticipated heavy supply in a particular tenor can flatten the curve as well. Infrequently traded bonds quoted with wide bid–offer spreads distort the shape, so the size and frequency of trading across the maturity spectrum has to be checked before the curve is trusted.
Company-value model results. Traditional analysis of the issuer industry and of financial ratios such as cash flow, leverage and profitability against sector and rating peers is now complemented by forward-looking structural models. These take equity market valuation, equity volatility and balance sheet data and derive an implied default probability. Anything at the company level that raises that implied probability, greater equity volatility being the clearest case, tends to steepen the credit spread curve, and a fall in equity volatility does the reverse.
Two practical cautions
The first concerns the benchmark. A frequently traded government security with the nearest maturity is the natural choice in a developed market, but the duration and maturity of the most liquid on-the-run government bonds rarely line up with those of the corporate bond being analysed, so interpolation between the two closest government maturities is often necessary. Because that interpolation can distort the analysis at less liquid maturities, the benchmark swap curve built on interbank rates is frequently substituted, since the swap market is more liquid at off-the-run maturities.
The second concerns comparability. A term structure should contain only bonds with similar credit characteristics, which in practice means senior unsecured general obligations. Issues with embedded options, first or second lien provisions or other unusual features should be excluded, because their spreads reflect those features rather than the issuer credit. One feature does apply across the whole curve: cross-default provisions mean that every security of an issuer becomes subject to recovery in the same bankruptcy.
Changing default expectations move the curve
The models already built show directly why the curve slopes as it does. Return to the five-year zero-coupon corporate bond from section two, with a flat 3.00% benchmark curve and a 40% recovery rate, and raise the annual default probability from 1.25% to 1.50%.
| Date | Exposure | LGD | POD | POS | Expected loss | Discount factor | PV of loss |
|---|---|---|---|---|---|---|---|
| 1 | 88.8487 | 53.3092 | 1.5000% | 98.5000% | 0.7996 | 0.970874 | 0.7763 |
| 2 | 91.5142 | 54.9085 | 1.4775% | 97.0225% | 0.8113 | 0.942596 | 0.7647 |
| 3 | 94.2596 | 56.5558 | 1.4553% | 95.5672% | 0.8231 | 0.915142 | 0.7532 |
| 4 | 97.0874 | 58.2524 | 1.4335% | 94.1337% | 0.8351 | 0.888487 | 0.7419 |
| 5 | 100.0000 | 60.0000 | 1.4120% | 92.7217% | 0.8472 | 0.862609 | 0.7308 |
| Total | 7.2783% | CVA = 3.7670 |
The CVA rises from 3.1549 to 3.7670, fair value falls from 83.1060 to 86.2609 − 3.7670 = 82.4939, the yield to maturity rises to 3.9240%, and the credit spread widens from 77 bps to 3.9240% − 3.00% = 92 bps. A flat credit curve therefore implies a stable expectation of default through time; an upward-sloping curve implies that investors want progressively more compensation for bearing default risk over longer horizons.
Make that explicit by holding the benchmark rate at 3.00% for 3-year, 5-year and 10-year zero-coupon bonds while letting the default probability rise with time: 1.00% in Years 1 to 3, 2.00% in Years 4 and 5, and 3.00% in Years 6 to 10, with recovery constant at 40%.
| Maturity | Cumulative POD | VND | CVA | Fair value | Yield to maturity | Credit spread |
|---|---|---|---|---|---|---|
| 3 years | 2.9701% | 91.5142 | 1.6308 | 89.8833 | 3.6192% | 0.6192% |
| 5 years | 6.8125% | 86.2609 | 3.5259 | 82.7350 | 3.8633% | 0.8633% |
| 10 years | 19.9767% | 74.4094 | 8.9187 | 65.4907 | 4.3235% | 1.3235% |
The 10-year table has ten rows of exposures, PODs and discounted expected losses; only the totals are shown here. In it the Date 6 default probability is 2.7956% and the Date 10 default probability is 2.4749%, reflecting the 3.00% hazard rate applied to a shrinking surviving population.
Positively sloped credit curves are typical of a high-quality issuer with a strong competitive position in a stable industry, low leverage, strong cash flow and a healthy profit margin. Such an issuer has very tight short-term spreads which then rise with maturity, because the further out one looks the more room there is for macroeconomic deterioration, competitive disruption, technological change or anything else that raises the implied default probability. Empirical work supports an upward-sloping credit term structure for investment-grade portfolios.
Downward-sloping curves show up among high-yield issuers in cyclical industries and usually carry a story. A leveraged buyout or private equity acquisition typically loads the balance sheet with debt, and an inverted curve afterwards can mean that investors expect the new owners to extract efficiencies, improving future cash flow and profitability to the benefit of debt investors. The other common story is cyclical position: an issuer in a business such as retail, or oil and gas exploration, may be sitting at the trough of its cycle while the market prices in a recovery and progressively tighter spreads.
The distressed case, where the curve stops meaning much
A different situation arises when investors no longer expect the contractual cash flows at all and simply expect the recovery rate in a bankruptcy. Suppose the 5-year and 10-year zero-coupon bonds of the same issuer both trade at 40, the assumed recovery rate, and cross-default provisions mean both are treated alike in insolvency. The credit valuation adjustment is then just the difference between the VND and the recovery value.
For the five-year bond the VND is 86.2609 and the CVA is 86.2609 − 40 = 46.2609, leaving a fair value of 40. That price implies a yield of 20.1124% and a credit spread of 17.1124%. For the 10-year bond the VND is 74.4094 and the CVA is 34.4094, again leaving 40, for a yield of 9.5958% and a spread of 6.5958%.
This is why such a term structure is better described as an optical phenomenon than as a genuine statement about the relative risk and reward of long-dated against short-dated debt from a single issuer. Once a bond is priced on recovery rather than on spread, the spread is a derived number with little content.
The interpretation still matters for anyone whose view differs from the market. Suppose a portfolio manager disagrees with a curve implying a high near-term default probability that then declines. She can sell short-term protection and buy longer-term protection in the credit default swap market. If the issuer does not default, she keeps the premium on the protection sold and can either hold the longer-dated contract or sell it back to realise a gain.
Most corporate and sovereign bonds are general obligations: the investor has a claim on the issuer as a whole. Securitized debt is different. It finances a specific pool of assets or receivables, such as mortgages, automobile loans or credit card balances, rather than a whole balance sheet. That difference changes the analysis from top to bottom.
Issuers choose this route because it raises debt capacity, reduces the regulatory capital the originator must hold, and limits the residual risk retained. Isolating a pool of assets usually lowers the financing cost of those assets on a stand-alone basis compared with general obligation borrowing by the originator, and freeing up capital lets the originator keep writing new business. Investors accept the extra complexity in return for diversification, cash flows that are more stable and predictable, and a yield above that of similarly rated conventional securities.
Four things then have to be analysed that do not arise for a corporate bond: the underlying collateral, the parties who originate and service the pool over the life of the deal, the issuing entity itself, and the structural and credit enhancement features of the transaction.
Granularity and homogeneity determine the method
Two characteristics of the asset pool decide how the credit work should be done.
Homogeneity is the degree to which the underlying obligations resemble one another. Credit card balances and auto loans that each had to pass strict eligibility criteria to enter the pool are homogeneous, and general conclusions can be drawn about them collectively. Leveraged loans, project finance debt and commercial real estate exposures differ so much from one another that each needs its own scrutiny.
Granularity is the number of separate obligations in the pool. A highly granular portfolio may contain hundreds of debtors, in which case portfolio summary statistics support a sound view of creditworthiness. A pool of a few discrete exposures does not, and each obligation warrants individual analysis.
Combining these with asset type and tenor gives the approach. Short-term vehicles holding granular, homogeneous assets are assessed statistically against the existing book of loans. Medium-term granular and homogeneous pools call for a portfolio approach, because the pool is not static and turns over during the life of the deal. Discrete or non-granular heterogeneous pools require loan-by-loan analysis.
| Deal type | Underlying collateral | Risk horizon | Granularity | Homogeneity | Credit analysis approach |
|---|---|---|---|---|---|
| Asset-backed CP | Discount credits or advances to commercial borrowers | Short | Granular | Homogeneous | Book basis |
| Auto ABS | Auto loans or leases | Medium | Granular | Homogeneous | Portfolio basis |
| CMBS | Commercial mortgages | Long, typically | Not granular | Heterogeneous | Loan by loan review |
| Consumer ABS | Consumer loans | Medium | Granular | Homogeneous | Portfolio basis |
| CRE loans | Commercial real estate loans | Long | Not granular | Heterogeneous | Loan by loan review |
| Credit cards | Credit card balances | Short | Granular | Homogeneous | Book basis |
| Credit-linked notes and repackaging | Any financial assets | Medium, typically | Single asset, typically | Not applicable | Pass-through rating, asset by asset |
| LL CLOs | Leveraged corporate loans | Medium | Not granular | Heterogeneous | Loan by loan review |
| PF CLOs | Project finance debt | Long | Not granular | Heterogeneous | Loan by loan review |
| RMBS | Residential mortgages | Long | Granular | Homogeneous | Loan by loan, or portfolio basis |
| SME ABS | Loans to small and medium-sized businesses | Medium, typically | Granular | Mixed | Loan by loan, or portfolio basis |
| Trade receivables | Commercial credit | Short | Granular, typically | Homogeneous | Book basis |
Adapted from the general structured finance rating methodology published by Scope Ratings AG in 2016.
A granular pool in practice
Take the March 2016 issue of $750,000,000 Series 2016-1 Asset Backed Notes out of the Synchrony Credit Card Master Note Trust. Its prospectus sets out the credit score composition of the receivables standing behind the transaction.
| FICO credit score range | Receivables outstanding | Percentage of outstanding |
|---|---|---|
| Less than or equal to 599 | $995,522,016 | 6.6% |
| 600 to 659 | $2,825,520,245 | 18.7% |
| 660 to 719 | $6,037,695,923 | 39.9% |
| 720 and above | $5,193,614,599 | 34.4% |
| No score | $64,390,707 | 0.4% |
| Total | $15,116,743,490 | 100% |
Source: the Series 2016-1 prospectus. Note that the $15.1 billion pool of receivables backs a $750 million note issue.
An investor in this security does not analyse individual cardholders. The probability of default, the recovery rate and the variance of losses are estimated for a portfolio of borrowers whose FICO distribution is given above. The prospectus supports that portfolio work with further detail: the age of the receivables, average outstanding balances and delinquency rates. A heterogeneous pool of a few large loans would demand the opposite: an individual assessment of whether each commercial property or leveraged company can meet its obligations, with default probability and recovery estimated asset by asset.
Origination, servicing and counterparty exposure
The second major difference from corporate credit analysis is that an ABS investor depends on other parties for the whole life of the deal. At inception the originator or servicer sets and enforces the loan eligibility criteria, secures and maintains documentation and records, and maximises timely repayment and enforceability where borrowers fall behind. After inception the investor is exposed to operational and counterparty risk: the servicer has to keep managing and servicing the pool competently for years. In an auto ABS that means repossessing and selling a vehicle near its residual value without undue delay; in a commercial real estate transaction it means identifying and replacing a tenant that stops paying. Where the composition of the pool changes over time, the investor is additionally exposed to the quality of the replacement obligors.
So the creditworthiness of the servicer matters, and so does its track record. That record is usually assessed by looking at how more seasoned transactions handled by the same servicer performed through a full credit cycle. In the Synchrony transaction the trust is serviced by Synchrony Financial, while collections on the receivables are received and processed mainly by Synchrony Bank in a sub-servicing role. A prospective investor would therefore examine not only how other credit card ABS have performed but how notes serviced by Synchrony have performed against those of competing servicers.
Structure: the issuing entity and credit enhancement
The obligor in a securitization is usually a special purpose entity whose only function is to acquire a defined pool of assets and issue notes to finance them. The critical question about that entity is its relationship to the originator, specifically how far the bankruptcy of the issuer is insulated from the bankruptcy of the originator. That bankruptcy remoteness normally turns on whether the transfer of assets from originator to SPE qualifies as a true sale, which is what allows the two credits to be separated later.
Beyond bankruptcy remoteness, transactions carry explicit credit enhancements. Payout or performance triggers protect investors against adverse credit events: a failure by the servicer or seller to make required deposits or payments, or other defined events, can trigger early amortisation of the security. For consumer transactions such as credit card and auto ABS, the primary protection against deteriorating asset quality is the excess spread, the additional return built into the structure over and above the expected or historical loss rate on the pool. Issuers also create subordinated tranches, tiering the priority of claims so that senior tranches are protected and enjoy a larger excess spread cushion over the life of the financing.
Covered bonds
Covered bonds originated in Germany in the eighteenth century and are now issued across Europe, Asia and Australia. They resemble structured finance instruments in some respects but differ in ways that matter for credit analysis.
The instrument is senior debt of a financial institution, and what marks it out is that the holder has recourse to two things at once: the issuing institution itself and a defined pool of collateral. Each jurisdiction specifies the eligible collateral types and the permissible structures; the collateral is most often commercial or residential mortgages meeting set criteria, or public sector debt. That dual recourse has been the defining feature since the instrument was invented, and in the European Union it was reinforced by the Bank Recovery and Resolution Directive, under which covered bonds enjoy protection unavailable to other bank liabilities in a restructuring or regulatory intervention. The issuing institution must also keep enough assets in the cover pool to satisfy covered bondholders at all times, and its obligations regarding the pool are supervised by public or other independent bodies.
The cover pool is dynamic, which is the second analytical difference. Where a static pool of mortgages exposes the investor to prepayment risk, as with United States mortgage-backed securities, a covered bond sponsor must replace any prepaid or non-performing asset so that cash flows remain sufficient through to maturity.
Third, several redemption regimes exist to keep cash flows close to the original schedule if the sponsoring institution fails:
- Hard-bullet covered bonds. If payments do not arrive on the original schedule, a bond default is triggered and payments are accelerated.
- Soft-bullet covered bonds. Default and acceleration are postponed to a new final maturity date, usually up to a year after the original one.
- Conditional pass-through covered bonds. If payments have not all been made by the original maturity date, the bond converts into a pass-through security.
Credit analysis for covered bonds then follows conventional lines, evaluating both the issuer and the cover pool. Because of the extra enhancements, recovery rates tend to be high and default probabilities low, which makes covered bonds a comparatively safe credit asset and explains why rating agencies frequently rate them several notches above the issuing financial institution itself.
Pulling the reading together
The thread running through every section is a single decomposition. Credit risk is exposure, multiplied by one minus recovery, multiplied by a default probability, discounted. Everything else is a way of estimating those inputs or of restating the answer. Credit scores and ratings are third-party estimates of default probability, with notching bringing loss given default into the letter grade. Structural and reduced-form models are two competing ways to generate the default probability. The binomial tree is a way to compute exposure when rates are volatile. The credit spread, the discount margin and the CVA are three presentations of the same compensation. The term structure of credit spreads is what happens when the default probability is allowed to vary with the horizon. And for securitized debt the same decomposition applies, but to a pool rather than to a company, which is why granularity and homogeneity decide the method.