> For the complete documentation index, see [llms.txt](https://laurence-wilse-samson.gitbook.io/textbooks/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://laurence-wilse-samson.gitbook.io/textbooks/financial-economics-claims-prices-holders/part-iii-asset-markets/chapter_10_credit_risk.md).

# Chapter 10: Credit Risk — Corporate, Sovereign, and Municipal

*Part III: Asset Markets — Financial Economics: Claims, Prices, and Holders*

***

## Opening Episode: Two Downgrades in One Afternoon

On Wednesday, May 4, 2005, Kirk Kerkorian's investment vehicle Tracinda announced a tender offer for 28 million General Motors shares at $31 each, a bid to raise his stake to roughly nine percent of the company. GM's stock rose sharply. The next afternoon, Standard & Poor's cut General Motors' corporate credit rating two notches, from BBB− to BB, and Ford Motor Company's one notch, from BBB− to BB+, taking the finance subsidiaries — GMAC and Ford Motor Credit, which did most of the actual borrowing — down with them.

Two claims on the same two firms had moved in opposite directions in twenty-four hours. Section 10.2 explains why that is not a contradiction: a leveraged recapitalization that pleases an activist shareholder is, in the structural model of credit, a transfer from the bondholders to the equity. But the second event was the consequential one, and its consequence had almost nothing to do with the information in it. Nobody who followed the automakers learned anything on May 5. Both had been in visible trouble for years, losing market share and carrying pension and retiree health obligations that made the manufacturing business look like an appendage of an insurance company. The agencies were, as usual, late. What changed that afternoon was not the probability of default but the eligibility of the bonds.

Investment-grade indices admit only investment-grade bonds, and a very large share of the institutional money in corporate credit is managed against those indices under mandates that bind rather than advise. Contemporary estimates put the automakers' index-eligible debt at roughly 80 to 90 billion dollars, against a high-yield market then on the order of $700 billion. (Total obligations of the two groups, including non-index borrowings, ran to several times that.) The market was being told it would shortly have to absorb something like a tenth of itself, from holders not permitted to keep the bonds to holders who were, on a schedule anyone could read off the index rules.

Spreads gapped. The automakers' bonds fell hard, the broad investment-grade indices widened in sympathy, and the credit derivatives market convulsed in a way that surprised nearly everyone. Through 2004 and early 2005 a great deal of hedge fund capital had gone into the correlation trade: long the equity (first-loss) tranche of a standardized index of investment-grade default swaps, short the mezzanine tranche, hedged so a parallel move in the index would wash out. The trade was a bet that default correlation would stay high. A downgrade specific to two issuers is precisely an idiosyncratic shock, and idiosyncratic shocks hurt first-loss tranches while leaving mezzanine tranches comparatively unharmed. Implied correlation fell, the hedge failed, and losses forced further unwinding. Section 10.8 sets out the arithmetic; a *pricing convention* broke, not a borrower.

Then most of it came back. By the autumn the indices had retraced the great majority of the May widening, and neither company defaulted for another four years. The mechanical index exit, when it came, was staggered and later: index rules generally key on some combination of two or three agencies' ratings, and Moody's did not follow S\&P on GM until August or on Ford until the following year. The May move was the market pricing an exit that had not happened yet.

That is the chapter's thesis as an episode. A rating is a compressed opinion about default probability, and most of this chapter is about how to form such opinions properly. But a rating is also a *legal fact* — an entry in an investment policy, a trigger in a capital regulation, an index inclusion rule — and May 2005 is what happens when the legal fact moves and the economic fact does not. A corporate spread is not only compensation for expected loss. It is also, and in investment grade mostly, a price on risks the marginal holder finds expensive to bear and a record of who is permitted to hold the claim at all.

***

## 10.1 Default, Recovery, and the Credit Triangle

Start with the identity every credit desk uses and every credit desk knows to be incomplete. A defaultable zero-coupon bond promises one dollar at date $$T$$. With probability $$\mathrm{PD}$$ the issuer defaults and the holder recovers a fraction $$\mathrm{RR}$$ of face; the loss given default is $$\mathrm{LGD} = 1 - \mathrm{RR}$$. Price it with Chapter 3's machinery. Under the risk-neutral measure — in which every claim is priced by discounting its expected payoff at the riskless rate — the one-period bond is worth

$$
p = \frac{(1-\mathrm{PD}^{\ast}) + \mathrm{PD}^{\ast}\mathrm{RR}}{1+r\_f} = \frac{1 - \mathrm{PD}^{\ast}\mathrm{LGD}}{1+r\_f}
$$

where the star marks the risk-neutral default probability, per the notation convention. The bond's promised yield is $$y = 1/p - 1$$, and its **credit spread** is $$s = y - r\_f$$. Rearranging,

$$
s = \frac{(1+r\_f)\mathrm{PD}^{\ast}\mathrm{LGD}}{1 - \mathrm{PD}^{\ast}\mathrm{LGD}} \approx \mathrm{PD}^{\ast} \times \mathrm{LGD}
$$

This is the **credit triangle**: spread, default probability, and loss given default are three quantities of which any two determine the third. Quote a spread of 190 basis points and assume a 40 percent recovery, and you have asserted a risk-neutral default probability of about 3 percent. The identity is useful enough to be everywhere and simple enough to be dangerous. Three things it hides deserve naming.

**The star.** $$\mathrm{PD}^{\ast}$$ is not the probability that the firm defaults. It is that probability re-weighted by the state prices of Chapter 3 §3.4, inflated in proportion to how expensive the default state is. If a bond defaults only in a state of physical probability 2 percent in which the stochastic discount factor is 2.5, the state price on default is $$q = 0.05$$ and $$\mathrm{PD}^{\ast} = qR\_f = 0.052$$; at $$r\_f = 4$$ percent and $$\mathrm{RR} = 40$$ percent the spread is 335 basis points, of which only 120 is expected loss and 215 is the price of holding a claim that fails in bad times. Section 10.4 shows this wedge is the dominant term in investment-grade spreads.

**Recovery is not a constant.** Table 10.2 reports averages, and they are low exactly when default rates are high: in the worst credit years, average senior unsecured bond recoveries have run below 25 percent against a long-run average near 40 percent. Default and recovery are two draws from a common factor, so expected loss is worse than the product of the averages, and the losses arrive together.

**Neither leg is observed.** Spreads are. Default probabilities and recoveries are estimated, from history or from a model, and the rest of this chapter is about how.

### What history says

Table 10.1 gives the shape of the historical record. The numbers are approximate — rounded from the annual default studies the agencies publish free, and sensitive to sample period, universe, and weighting — but the shape has been stable for forty years.

**Table 10.1: Average Cumulative Default Rates by Rating (approximate, global corporates)**

| Rating        | 1 year | 5 years | 10 years | 10-yr expected loss at LGD = 60% |
| ------------- | ------ | ------- | -------- | -------------------------------- |
| Aaa / AAA     | 0.00%  | 0.10%   | 0.34%    | 0.20%                            |
| Aa / AA       | 0.02%  | 0.24%   | 0.66%    | 0.40%                            |
| A / A         | 0.06%  | 0.68%   | 1.93%    | 1.16%                            |
| Baa / BBB     | 0.16%  | 1.62%   | 3.70%    | 2.22%                            |
| Ba / BB       | 0.87%  | 6.70%   | 13.70%   | 8.22%                            |
| B / B         | 3.40%  | 18.90%  | 30.90%   | 18.54%                           |
| Caa-C / CCC-C | 10.00% | 34.60%  | 45.90%   | 27.54%                           |

*Source: Author's rounding of average cumulative issuer-weighted default rates from the Moody's Investors Service annual default study, with comparable figures in the S\&P Global Ratings annual default and transition study. The final column is the author's calculation, the ten-year default rate times a flat 60 percent loss given default. Both studies are free and are the source for the data exercise.*

![Figure 10.4: Cumulative default rates](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-b85c9f66dfe547b134bdf8c07f9160f5d7d6b4aa%2Ffig_10_04_cumulative_default_rates.png?alt=media)

**Figure 10.4: Cumulative default rates.** Table 10.1 on the logarithmic scale its numbers ask for, which is the only scale on which all seven ratings can be seen at once: the vertical distance from Aaa to Caa is more than two orders of magnitude, and that distance is the whole reason a letter is worth paying for. Convexity in the rating shows as the widening gaps down the ladder — the step from Baa to Ba multiplies the ten-year rate by nearly four, which is why the investment-grade boundary carries the weight Chapter 16 §16.2's capital schedule puts on it. Concavity in horizon shows as flattening: a good credit's hazard falls as it survives, so its curve bends over, while a distressed credit's is closer to linear and its spread curve can slope down. Aaa's one-year rate rounds to zero in the published tables, and rather than invent a floor for it the point is omitted. Two cautions on the levels. These are averages across decades in which annual speculative-grade rates ranged from under one percent to above twelve, so no year looks like this. And they are issuer-weighted rounded figures from the agencies' free studies, whose universes and weighting differ; the shape is stable across them, the third decimal is not. *Source: Table 10.1, the chapter's rounding of the Moody's and S\&P annual default studies.*

Three features carry through the chapter. Default rates are **convex in the rating**: Baa to Ba multiplies the ten-year rate by nearly four. They are **concave in horizon** for good credits and closer to linear for bad ones, which is why investment-grade spread curves slope up and distressed ones often slope down. And they are **enormously time-varying**: annual speculative-grade rates have ranged from under 1 percent to above 12 percent, clustered in recessions. Figure 10.1 shows the first and the third in market prices rather than in default counts.

![Figure 10.1: Credit spreads by rating](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-07a956b11da7a3a0b79fcceba9a6367d5d1f7599%2Ffig_10_01_credit_spreads_by_rating.png?alt=media)

**Figure 10.1: Credit spreads by rating.** **Interim free-data construction, pending the author's ruling on REVIEW\_FLAGS §1.** Panel (a) is the long history that is still free: Moody's Seasoned Baa less Aaa corporate bond yields, monthly from 1919, with each grade also measured against the ten-year Treasury constant maturity from 1953, NBER recessions shaded. Panel (b) is the rating ladder the specification asks for — ICE BofA option-adjusted spreads for AAA, BBB and high yield — over the only window FRED now serves, a rolling three years, with the truncation drawn at the left. The 1997-present window cannot be rebuilt from FRED, and no NBER recession falls inside the window that remains, which is why panel (a) carries the credit cycle here. The two measures are not interchangeable: the Moody's indices are long-maturity and not option-adjusted, and run roughly two and a half times the duration-matched OAS. *Source: Panel (a) Moody's Seasoned Aaa and Baa corporate bond yields and the 10-year Treasury (FRED AAA, BAA, GS10), monthly. Panel (b) ICE BofA option-adjusted spreads (FRED BAMLC0A1CAAA, BAMLC0A4CBBB, BAMLH0A0HYM2), which FRED now serves as a rolling three-year window only; the 1997-present history is no longer free. NBER business cycle dates.*

**Table 10.2: Average Recovery Rates by Instrument and Seniority (approximate)**

| Instrument             | Average recovery, % of face |
| ---------------------- | --------------------------- |
| First-lien bank loans  | \~67%                       |
| Senior secured bonds   | \~53%                       |
| Senior unsecured bonds | \~38%                       |
| Subordinated bonds     | \~28%                       |

*Source: Author's rounding of long-run average issuer-weighted recovery rates from the Moody's annual default study, measured by trading price roughly thirty days after default. Recoveries at ultimate resolution generally run higher and with far wider dispersion.*

The convention matters. The trading price a month after default is what a forced seller receives and what a mark-to-market model needs; the value ultimately distributed in reorganization is what a hold-to-maturity investor receives. The gap is itself a liquidity story, taken up in Chapter 11.

> **Box 10.1 — How a bond leaves an index**
>
> The opening episode turns on a mechanical exit, and the mechanism is a rulebook. It is worth reading, because the same structure recurs wherever a mandate is written against an index rather than against a portfolio.
>
> **Which ratings count.** A broad investment-grade corporate index does not use one agency's opinion. Index rules key on some combination of the major agencies — commonly the middle of three, or the lower of two — so a single downgrade may not move a bond at all. That is why GM's exit was staggered: S\&P cut in May 2005 and the index consequence waited on the others. A trader who knows the combination rule knows something about the timing that a reader of the press release does not.
>
> **When the exit happens.** Index membership is fixed at a rebalancing date, normally the end of the month, using ratings as of a stated cutoff. A bond downgraded on the fifth of the month remains in the index, and in every portfolio tracking it, until the month turns. The rebalance then does two things at once: it fixes the date on which index-constrained holders must be out, and, because the index's own return is computed at the rebalancing price, it fixes the price at which the exit is recorded. Everyone knows both in advance, which is what makes the trade in front of it possible.
>
> **The regulatory twin.** Section 939A of the Dodd-Frank Act ordered federal agencies to strike references to credit ratings from their rules and replace them with standards of creditworthiness of the agencies' own devising. The banking rules were rewritten accordingly. But a statute can remove a rating from a federal regulation without removing it from anything else, and §10.9's insurance capital schedule — the NAIC designations that Chapter 16's Table 16.3 prices — still runs on agency ratings in the ordinary case. The rating was demoted as a matter of federal law and retained as a matter of practice, which is a fair description of what happened to ratings generally after 2010.
>
> The general point is Chapter 1 §1.2's. An index rule is not a description of the market; for the holders bound by it, it is the market.

***

## 10.2 Structural Models: Credit as an Option on the Firm

Merton's (1974) insight is that a firm's capital structure is already an option position, so credit risk can be priced with the machinery of Chapter 8 rather than a new theory.

Let the firm's assets have market value $$V\_t$$, a geometric Brownian motion with volatility $$\sigma\_V$$. The firm has issued one zero-coupon bond of face value $$F$$ maturing at $$T$$, and equity that takes what is left. At $$T$$ the equity holders decide whether to pay: if $$V\_T > F$$ they pay and keep $$V\_T - F$$; if $$V\_T < F$$ limited liability lets them hand over the firm at no further cost. Equity's payoff is

$$
E\_T = \max(V\_T - F, 0)
$$

**Equity is a European call option on the firm's assets, struck at the face value of the debt.** Everything in Chapter 8 applies at once. And because the two claims exhaust the firm, $$V\_t = E\_t + B\_t$$, the debt follows:

$$
B\_T = \min(V\_T, F) = F - \max(F - V\_T, 0)
$$

**Risky debt is a riskless bond minus a put** — the bondholders have written the shareholders a put on the firm's assets struck at the face value of what they are owed, and the put premium *is* the credit risk. That sentence converts every question about credit into a question about an option.

### A worked example

Take $$V\_0 = 100$$, asset volatility $$\sigma\_V = 25$$ percent, a zero-coupon bond of face $$F = 80$$ maturing in $$T = 5$$ years, and a continuously compounded riskless rate of $$r = 4$$ percent. Black-Scholes gives

$$
d\_1 = \frac{\ln(V\_0/F) + (r + \tfrac{1}{2}\sigma\_V^2)T}{\sigma\_V\sqrt{T}} = 1.0365, \qquad d\_2 = d\_1 - \sigma\_V\sqrt{T} = 0.4774
$$

so $$N(d\_1) = 0.8500$$ and $$N(d\_2) = 0.6835$$. Then

$$
E\_0 = V\_0 N(d\_1) - Fe^{-rT}N(d\_2) = 85.00 - 65.498 \times 0.6835 = 40.23
$$

The debt is worth $$B\_0 = 100 - 40.23 = 59.77$$ — the same figure the put route gives, since the riskless bond is worth $$80e^{-0.04\times 5} = 65.50$$ and the put struck at 80 is $$Fe^{-rT}N(-d\_2) - V\_0 N(-d\_1) = 5.73$$. The promised yield is $$y = -\ln(59.77/80)/5 = 5.83$$ percent, so the credit spread is **183 basis points** and the risk-neutral default probability is $$N(-d\_2) = 31.7$$ percent.

Figure 10.2 is the model in two panels: what the two claims pay, and what the model believes about the thing they are claims on.

![Figure 10.2: Equity as a call on firm value](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-bc8e792f13d656e308c8fac8d2d2628758dd36f8%2Ffig_10_02_equity_as_a_call.png?alt=media)

**Figure 10.2: Equity as a call on firm value.** Panel (a) is the payoff of each claim against the firm's asset value at maturity, at the worked example's face value of 80. The equity line is a call struck at 80, kinked and flat to the left of it; the debt line is the firm's value capped at 80, which is the same thing as a riskless bond of face 80 minus a put struck there. The two sum to the dotted forty-five-degree line at every asset value, because the two claims exhaust the firm — that adding-up is what lets one option formula price both. The shaded wedge is the default region: the bondholders receive the firm and the shareholders walk away, which is not a penalty but the exercise decision limited liability gives them. Panel (b) is where the model thinks the asset value will land, under the risk-neutral measure that Chapter 8 §8.3.2 constructs. The shaded left tail is the 31.7 percent risk-neutral probability of default, and the gap between today's asset value and the default boundary, measured in standard deviations of log asset value, is d-two — 0.4774 here, and the quantity commercial implementations rename distance to default. Read the two panels together and the chapter's claim is visible: everything a credit analyst cares about is a statement about how far that boundary is and how wide the distribution around it is, which is to say about leverage and asset volatility, and nothing else.

From three inputs about the *firm* — what it is worth, what it owes, how uncertain its value is — the model produced a bond price, a spread, and a default probability jointly, with no free parameter for "credit risk." Table 10.3 varies one input at a time.

**Table 10.3: Merton Model Comparative Statics**

| Case                                                    | Credit spread | Risk-neutral $$\mathrm{PD}$$ |
| ------------------------------------------------------- | ------------- | ---------------------------- |
| Base: $$F = 80$$, $$\sigma\_V = 25$$ percent, $$T = 5$$ | 183 bp        | 31.7%                        |
| Lower leverage: $$F = 60$$                              | 77 bp         | 16.1%                        |
| Higher leverage: $$F = 90$$                             | 250 bp        | 39.5%                        |
| Lower asset volatility: $$\sigma\_V = 15$$ percent      | 41 bp         | 13.7%                        |
| Higher asset volatility: $$\sigma\_V = 35$$ percent     | 379 bp        | 44.1%                        |
| Shorter maturity: $$T = 1$$                             | 216 bp        | 17.7%                        |
| Longer maturity: $$T = 10$$                             | 135 bp        | 34.7%                        |

*Source: Author's calculation from the Merton (1974) model with an initial asset value of 100 and a riskless rate of 4 percent, varying one input at a time from the base case.*

**Leverage and volatility are the levers**, and they are the ones a credit analyst actually pulls. **Asset volatility raises the spread and equity value together** — the same $$\sigma\_V$$ that makes the put more valuable makes the call more valuable — the option-theoretic statement of the shareholder-bondholder conflict. Raising leverage does the same: moving $$F$$ from 80 to 90 transfers value from debt to equity though the firm's assets have not changed. That is May 4, 2005, in one line, and why bondholders write covenants (Chapter 24 §24.1).

**Maturity is non-monotonic.** Default probability rises with horizon, but the spread in the table *falls* between one and ten years. For a healthy firm it runs the other way: short-dated debt of a firm comfortably above its default point is nearly riskless, so its spread curve slopes up. The model generates both shapes and says which firms should show which.

### Distance to default

The quantity doing the work is $$d\_2$$, the number of standard deviations by which log asset value must fall to reach the default point. Commercial implementations — the KMV model, now part of Moody's Analytics — turn it into **distance to default**:

$$
\mathrm{DD} = \frac{\ln(V\_0/\text{default point}) + (\mu\_V - \tfrac{1}{2}\sigma\_V^2)T}{\sigma\_V\sqrt{T}}
$$

with three practitioner amendments: the drift is the *physical* expected return on assets rather than $$r$$; the default point is short-term debt plus roughly half of long-term debt, because firms rarely default the moment book liabilities exceed asset value; and $$\mathrm{DD}$$ maps to a probability through an empirical frequency table built from the default experience of firms at each distance, not through $$N(-\mathrm{DD})$$, because the normal has thinner tails than corporate reality.

$$V\_0$$ and $$\sigma\_V$$ are not observed. They are inferred by solving the Black-Scholes equation jointly with the identity $$\sigma\_E E\_0 = N(d\_1)\sigma\_V V\_0$$ from the market value and volatility of equity, as in Problem 6. The output is a firm-level, daily-updating default measure that typically moves well before the rating does — which is why the model is used, and the sharpest evidence that ratings carry little the market does not already have.

### What the model gets wrong

**Spreads are too low, and the failure is concentrated in investment grade.** Calibrate to a Baa-like firm: pick leverage and asset volatility so the model's *physical* ten-year default probability matches Table 10.1's 3.70 percent at an asset drift of 8 percent. At 50 percent leverage this requires $$\sigma\_V = 22$$ percent, and the model then produces a spread of **30 basis points** against an observed Baa level on the order of 150. Fed the right default probability, it misses the price by a factor of five.

That is the credit spread puzzle, and Section 10.4 is about it. Three further defects. Diffusions cannot jump, so short-horizon default probabilities and short-maturity spreads both go to zero, whereas actual one-year investment-grade spreads are small but decidedly positive. Default happens only at $$T$$, which first-passage extensions (Black-Cox and descendants) repair by letting default occur the first time $$V\_t$$ crosses a boundary. And one zero-coupon liability stands in for layered, covenanted, renegotiable capital structures whose seniority determines recovery. Each repair costs a parameter; none closes the investment-grade gap.

***

## 10.3 Reduced-Form Models

The structural approach explains default. The reduced-form approach declines to, and gains a great deal in exchange.

Treat default as the first jump of a point process. The default time $$\tau$$ arrives with **intensity** $$\lambda$$: over a short interval $$dt$$, conditional on survival so far, the probability of defaulting is $$\lambda\thinspace dt$$. In this section $$\lambda$$ is an intensity and never the price of risk. With a constant intensity, the survival probability is

$$
P(\tau > T) = e^{-\lambda T}
$$

and with a stochastic intensity it is $$E\[\exp(-\int\_0^T \lambda\_u\thinspace du)]$$. Under the risk-neutral measure the intensity is $$\lambda^{\ast}$$, and the two differ by exactly the compensation for default risk: the ratio $$\lambda^{\ast}/\lambda$$ is the credit market's version of the wedge between $$\mathrm{PD}^{\ast}$$ and $$\mathrm{PD}$$.

### ★ The Duffie-Singleton reduction

One result earned this formulation the market. Suppose that at default the bondholder recovers a fraction $$\mathrm{RR}$$ of the bond's *market value an instant before default* — "recovery of market value." Duffie and Singleton (1999) show that a defaultable zero-coupon bond is then worth

$$
B(0,T) = E^{\ast}\left\[\exp\left(-\int\_0^T \big(r\_u + \lambda^{\ast}\_u\mathrm{LGD}\_u\big)\thinspace du\right)\right]
$$

A defaultable bond is priced exactly like a riskless bond, with the short rate replaced by the **default-adjusted short rate** $$r\_t + \lambda^{\ast}\_t\mathrm{LGD}\_t$$. Every affine term-structure model of Chapter 9 therefore carries over unchanged: write $$\lambda^{\ast}\_t$$ as an affine function of state variables, add it to the short rate, and closed-form bond prices, factor decompositions, and estimation all apply to credit. With constant $$\lambda^{\ast}$$ and $$\mathrm{LGD}$$ the spread collapses to the credit triangle in continuous time,

$$
s = \lambda^{\ast}\mathrm{LGD}
$$

which is Section 10.1's identity with the horizon divided out.

### Which model, when

The choice is not about which is true; neither is. Use a **structural** model when the object is the link between a firm's economics and its credit — capital-structure decisions, covenant design, convertibles, an unlisted borrower, anything whose answer must respond to leverage and volatility. Use a **reduced-form** model to fit and interpolate observed prices — marking a derivatives book, stripping a default-swap curve, building a term structure of implied default probabilities. A structural model is a theory of the borrower; a reduced-form model is an interpolation of the market, and asking it why the intensity is what it is has no answer inside the model. That is the price of its tractability.

***

## 10.4 The Credit Spread Puzzle

Take Table 10.1's Baa row seriously. A ten-year cumulative default rate of 3.70 percent is an annualized hazard of 0.38 percent, so at a 60 percent loss given default the credit triangle puts the expected-loss component of a Baa spread at about **23 basis points** — roughly a seventh of the 150 or so basis points at which BBB option-adjusted spreads have averaged.

**Table 10.4: Expected Loss Against Observed Spread, by Rating**

| Rating    | Annualized default rate | Expected-loss spread at LGD = 60% | Order of magnitude of observed spread | Expected loss as share |
| --------- | ----------------------- | --------------------------------- | ------------------------------------- | ---------------------- |
| Aa / AA   | 0.07%                   | 4 bp                              | \~70 bp                               | \~6%                   |
| A / A     | 0.19%                   | 12 bp                             | \~95 bp                               | \~12%                  |
| Baa / BBB | 0.38%                   | 23 bp                             | \~150 bp                              | \~15%                  |
| Ba / BB   | 1.46%                   | 88 bp                             | \~250 bp                              | \~35%                  |
| B / B     | 3.63%                   | 218 bp                            | \~400 bp                              | \~55%                  |

*Source: Author's calculation. Default rates are annualized from the ten-year cumulative figures of Table 10.1; the observed-spread column states approximate long-run averages of ICE BofA US corporate index option-adjusted spreads by rating (the free FRED series used in the data exercise), rounded deliberately, since the average depends heavily on the sample window.*

![Figure 10.3: The credit spread puzzle](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-3b3d6309d95057ec838be6402601b60dffc5d0f5%2Ffig_10_03_the_credit_spread_puzzle.png?alt=media)

**Figure 10.3: The credit spread puzzle.** Table 10.4's observed spreads, split into the pieces that can be accounted for and the piece that cannot. Expected loss is the ten-year default rate at a flat sixty percent loss given default; the tax band is §10.4's wedge, about twenty-one basis points at a thirty-seven percent federal and five percent state rate, and it is drawn flat because it is a property of the instrument rather than of the rating. What is left is the residual the four explanations compete over. At Baa the arithmetic is stark: twenty-three basis points of expected loss and twenty-one of tax inside a spread of about a hundred and fifty, leaving more than a hundred that default cannot explain. The share expected loss accounts for rises monotonically down the scale, from about a sixteenth at Aa to more than half at B, which is the sense in which the puzzle is an investment-grade puzzle: precisely where credit is safest, the compensation is least about credit. The observed column is deliberately rounded to orders of magnitude, because the long-run average depends heavily on the window chosen — the decomposition is about proportions, and would survive a different window. *Source: Tables 10.1 and 10.4, and §10.4's tax calculation.*

Two facts organize what follows. The puzzle is **large in investment grade and modest in high yield** — the share of spread explained by expected loss rises monotonically down the rating scale. And it is a puzzle about *levels*, not *variation*: Collin-Dufresne, Goldstein and Martin (2001) found that the variables structural models say should drive spread changes explain only a small fraction of the observed variation, and that the residuals are dominated by a single common factor with no obvious counterpart in firm fundamentals.

Huang and Huang (2012) is the careful statement. They calibrate a family of structural models — Merton, first-passage, endogenous-default, stochastic-volatility, jump, countercyclical-risk-premium — imposing on each the discipline that it match the *historical* default frequency and recovery rate for its rating, and ask how much spread it then produces. For ten-year Baa bonds the answer is around 20 percent, less for higher ratings; for junk the models account for a majority. Four explanations compete for the remainder, and all four are partly right.

**Risk premia.** Defaults cluster in recessions, so credit fails in exactly the states Chapter 5 makes expensive, and must pay a premium beyond expected loss. That is the $$\mathrm{PD}^{\ast}/\mathrm{PD}$$ wedge of Section 10.1: matching a 150 basis point Baa spread at a 60 percent loss given default requires a risk-neutral intensity of 2.50 percent a year against a physical intensity of 0.38 percent, a ratio of 6.6. Large, but not obviously impossible — credit's payoff is left-skewed and its bad states are the bad states of Chapter 5's disaster and habit models. Whether the credit and equity premia are consistent under a single stochastic discount factor is a live question, and one of the cleanest tests of whether Chapter 5's resolutions resolve anything.

**Liquidity.** Corporate bonds trade rarely, a given issue may not print for weeks, and transaction costs run an order of magnitude above equity; Chapter 11 §11.5 shows that expected returns compensate for both average trading costs and exposure to systematic liquidity shocks. Longstaff, Mithal and Neis (2005) use default swaps to separate the components — a swap requires no funding of the bond and so carries far less illiquidity — and put the non-default component at a substantial share of investment-grade spreads.

**Taxes.** Treasury interest is exempt from state and local income tax; corporate interest is not. An investor facing 37 percent federal and 5 percent state rates keeps 63 cents of a Treasury dollar and 59.85 cents of a corporate dollar, so against a 4 percent Treasury yield a corporate bond must yield about 21 basis points more to break even before any credit risk at all — about 40 at a 9 percent state rate. Elton, Gruber, Agrawal and Mann (2001) find the tax term larger, in the investment-grade spread, than expected default loss. Measured from the other side of the same wedge, this is Chapter 9 §9.4's convenience yield: the Aaa-Treasury spread Krishnamurthy and Vissing-Jorgensen attribute to money-like services is the same gap, booked as Treasuries rich rather than as corporates cheap, and the two accounts must not be added together.

**The holder-based reading.** The first three take the marginal investor as given and ask what she is compensated for. The fourth asks who she is. If corporate credit is held predominantly by institutions whose demand is governed by capital charges, rating-based mandates, and accounting treatment, the spread is set where their constrained demand meets supply, and the compensation in it need not equal any unconstrained investor's required return. That is Chapter 20's demand-system logic applied to credit, and Section 10.9 makes the case. The four are not rivals; they are four names for the marginal holder's problem, and the last asks which holder is marginal. (*International Finance* Chapter 12 reaches the same decomposition from the practitioner's side; the models are here, the market description is there.)

***

## 10.5 Credit Default Swaps: What the Price Contains

In a credit default swap the protection buyer pays a periodic premium and, if a defined credit event occurs on a named reference entity, receives the difference between par and the post-default value of a deliverable obligation. The premium, quoted in basis points a year on notional, is the **CDS spread**.

Its price content is what makes it useful here. A cash corporate bond bundles three things: exposure to the issuer's default, exposure to the level of riskless rates, and a funding requirement — someone must put up money to own it, at whatever rate they can borrow. A default swap strips the last two away. It has no principal to fund and, being a swap, roughly zero value at inception, so its spread is close to a pure price of default risk: in the language of Section 10.3, a direct read on $$\lambda^{\ast}\mathrm{LGD}$$, and a term structure of quotes strips a term structure of risk-neutral intensities. That is why the empirical literature on the puzzle reaches for CDS data — the swap is the closest thing to the default leg on its own.

The residual is the **CDS-bond basis**: the swap spread minus the cash bond's spread over the swap curve. In principle it is near zero, because a static package — buy the bond, buy protection, fund the bond at the riskless rate — is riskless. It is riskless only for someone who can actually fund at that rate. In late 2008 and early 2009 the investment-grade basis reached roughly minus 250 basis points and high yield went further: cash bonds were extraordinarily cheap relative to protection, and the trade that would close the gap sat unexecuted for months.

That is not a mispricing in the ordinary sense but a measurement of the shadow price of balance sheet, in the sense of the limits-to-arbitrage argument of Chapter 15 §15.5 and the constrained-capital statement of Chapter 16 §16.5, with the dealer mechanics in Chapter 19. When the basis blows out, the reading is not that credit risk has been repriced but that the capital which normally enforces no-arbitrage has gone — and the basis measures it cleanly, because both legs reference the same default event, so everything that does not cancel is funding.

Market mechanics — standardized coupons and upfront conventions, credit-event determination, auction settlement, clearing and margin, and who is on each side — belong to *International Finance* Chapter 13.

***

## 10.6 Sovereign Credit as Theory

Corporate default is a legal process. Sovereign default is not.

No court can seize a sovereign's territory or compel its legislature to appropriate. Enforcement is limited to attaching whatever assets it holds abroad, and even that is bounded by immunity doctrines. What is left is not a bankruptcy procedure but a repeated game, so the question is not "can it pay?" but "will it choose to?" — **willingness to pay** rather than ability — and the answer comes from what the sovereign loses by defaulting.

Eaton and Gersovitz (1981) wrote the canonical version. A small open economy with volatile income borrows to smooth consumption and cannot commit to repay. Each period it compares repaying, which preserves market access, with defaulting, which delivers a one-time gain equal to the debt service but bars it from borrowing thereafter. Lenders, understanding this, lend only up to the amount that leaves repayment weakly preferable in every state they foresee: the model delivers an endogenous **credit ceiling** rather than a price for any quantity of debt. The mechanism is reputation and exclusion, not enforcement, and the ceiling is tighter for economies with more volatile income, because volatility makes market access valuable.

Bulow and Rogoff (1989) supplied the sharpest objection: if a defaulting sovereign can save abroad in the same instruments it was denied as a borrower, reputation alone cannot sustain any positive lending. Something else must be at stake — trade sanctions, disruption of trade credit, damage to the domestic banks holding the debt, political costs. Aguiar and Gopinath (2006) and Arellano (2008) embed the default choice in calibrated business-cycle models and reproduce the data: default is countercyclical, spreads rise as output falls, and emerging-market spread volatility appears without any exogenous credit shock. Sovereign spreads are, to a first approximation, a price on the *state of the borrower's economy* rather than on its balance sheet.

### Local currency, and why printing is not the answer

A recurring claim holds that a government borrowing in a currency it issues cannot default, because it can always create the money. It can always produce the nominal payment. Whether it will is a separate decision, and one it has often answered in the negative.

Inflating away a nominal debt *is* a default on the real value of the claim, and bondholders price it as one: the local-currency yield carries an inflation risk premium functionally identical to a credit spread, and a government that has once inflated pays it for a generation. The domestic costs of inflating are frequently larger than the costs of defaulting, because local-currency debt is typically held by domestic banks, pension funds, and insurers — the institutions this book keeps finding at the margin — so inflating destroys the domestic financial system's capital while a selective default on external holders does not. That asymmetry makes outright local-currency default rational, and it has been chosen: Russia's August 1998 default was on ruble-denominated GKOs, paper in a currency the government printed. Meanwhile the historical inability of emerging sovereigns to borrow abroad in their own currency — Eichengreen and Hausmann's "original sin" — has retreated, converting a share of what was pure credit risk into currency risk, which changes who bears the risk and which instrument prices it, not whether it exists.

The institutional machinery — collective action clauses, the pari passu litigation following Argentina's 2001 default, IMF programs, restructuring practice, and the crisis narratives — belongs to *International Finance*. What this chapter takes from sovereign credit is that a spread can price a *decision* rather than an *event*, and that the Merton apparatus, which assumes mechanical default at a boundary, has nothing to say about one.

***

## 10.7 Municipal Bonds and the Tax Story

The US municipal market is roughly $4 trillion of state and local government debt, and its defining feature is that interest is generally exempt from federal income tax and, for in-state holders, usually from state tax too. The exemption converts pricing into a tax problem.

The arithmetic is the **tax-equivalent yield**. A taxable bond yielding $$y$$ leaves an investor at marginal rate $$\theta$$ with $$y(1-\theta)$$; a muni yielding $$y\_m$$ leaves $$y\_m$$. The taxable yield matching a given muni is

$$
\mathrm{TEY} = \frac{y\_m}{1-\theta}
$$

At the top federal rate of 37 percent, a 3.00 percent muni is worth 4.76 percent of taxable yield; in the 24 percent bracket the same bond is worth 3.95 percent. The exemption is worth more to richer holders, which is the whole economics of the instrument's clientele and most of the economics of its price.

Inverting gives the **implied marginal tax rate**. If a muni yields $$y\_m$$ and a Treasury of the same maturity yields $$y\_t$$, an investor is indifferent when $$\theta = 1 - y\_m/y\_t$$. Applied across the curve, this produces the **muni puzzle**. At short maturities the ratio $$y\_m/y\_t$$ typically runs 65 to 70 percent, implying a marginal tax rate of 30 to 35 percent — close to the top statutory bracket, exactly as the theory says. At long maturities it typically runs 90 percent or higher, implying 10 percent or less. Nobody plausibly the marginal holder of a thirty-year municipal bond faces a 10 percent tax rate. Long-dated munis are too cheap for the tax story to explain.

![Figure 10.6: The muni puzzle](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-c7d13d22c926f896c99450d8562d2c95bb16e254%2Ffig_10_06_the_muni_puzzle.png?alt=media)

**Figure 10.6: The muni puzzle.** The marginal tax rate implied by the twenty-year muni-to-Treasury yield ratio, against the top statutory federal rate, from 1953 to 2016. An investor indifferent between the two bonds is revealing the rate at which the exemption is worth exactly its cost; across this sample that revealed rate averages 8 percent while the statutory top rate averages 57, and the shaded gap of 51 points is the puzzle in one image. It is at its most extreme in the high-tax decades: through the 1950s the top bracket was 91 percent and the ratio implies about 10. Two breaks are left visible rather than bridged. The Treasury issued no twenty-year bond between 1987 and 1993, so the comparison genuinely does not exist there; and the Bond Buyer 20-Bond index was discontinued in October 2016, which is where the window ends. The last stretch is the sharpest version of the puzzle and postdates most of the literature: from 2011 the ratio sits *above* one, meaning long munis yielded more than Treasuries outright, and the implied tax rate is not merely too low but negative — a number no tax story can produce at all, and which sends the explanation to the liquidity and tax-uncertainty channels above. *Source: Bond Buyer 20-Bond GO index and the 20-year Treasury constant maturity via the FRED mirror; top marginal federal individual income tax rates from IRS Statistics of Income. Author's calculations.*

The candidate resolutions are Section 10.4's four, reweighted. Default risk is real but small: municipal default rates have historically run far below corporate rates at the same letter rating, itself an artifact of the agencies having long used different scales for the two sectors. Munis carry embedded calls, worth more at long maturities. Liquidity is poor: roughly a million distinct CUSIPs, most trading a handful of times ever. Tax law is uncertain over thirty years in a way it is not over three, and the *de minimis* rule taxes discount bonds asymmetrically, biting hardest when rates rise. Longstaff (2011) argues from tax-exempt and taxable money-market rates that the exemption is worth *more* at the short end than the statutory rate can explain, sharpening the long-end puzzle rather than resolving it.

The holder-based reading is the one this book presses. Munis are held overwhelmingly by taxable US investors because they are worthless to anyone else: pension funds, IRAs, and foreign investors have nothing to gain from an exemption they do not need. Chapter 2's Table 2.5 shows the result — a claim class whose ownership is more concentrated in households than any other bond in the map, held directly and through muni-dedicated funds, with property and casualty insurers and banks taking the rest and their participation swinging with the corporate tax rate. Retail buyers take the short and intermediate maturities; the long end is left to a thin set of institutional buyers whose appetite depends on their own tax position. If that clientele is thin its required return is high, and the implied tax rate falling out of the ratio is not a tax rate at all — it is the residual after a segmented demand curve has cleared. The muni puzzle, on this reading, is the credit spread puzzle told where the segmentation is legible in the tax code.

***

## 10.8 Securitization: Pooling, Tranching, and Correlation

Securitization is claim design in its purest form. Assemble a pool of loans or bonds in a bankruptcy-remote vehicle and issue against it claims of different seniority. Cash flows are allocated top down — senior first, mezzanine next, equity or first-loss tranche last — and losses run the other way. Nothing is created: the pool's total expected loss is fixed by the assets. Tranching *redistributes* it across claims with different holders.

Pricing needs the **attachment** and **detachment** points, $$K\_A$$ and $$K\_D$$, as fractions of pool notional. With $$\mathcal{L}$$ the realized pool loss as a fraction of notional, the tranche absorbs

$$
\text{tranche loss} = \frac{\min(\mathcal{L},K\_D) - \min(\mathcal{L},K\_A)}{K\_D - K\_A}
$$

A tranche is a call spread on the pool's loss distribution, so its price depends on the whole distribution of $$\mathcal{L}$$, not merely its mean — and the shape of that distribution is governed by **default correlation**.

### A two-asset pool

Take a pool of two bonds, each of face value 100, each with a risk-neutral default probability of 10 percent and zero recovery, so the pool notional is 200 and each default costs 100. Split it into a junior tranche absorbing the first 100 of loss ($$K\_A = 0$$, $$K\_D = 50$$ percent) and a senior tranche taking the rest. The senior tranche loses only if *both* bonds default. Writing $$\rho\_D$$ for the correlation between the default indicators,

$$
P(\text{both default}) = \mathrm{PD}^2 + \rho\_D\mathrm{PD}(1-\mathrm{PD})
$$

**Table 10.5: Tranche Expected Loss as a Function of Default Correlation**

| $$\rho\_D$$ | $$P(\text{both})$$ | $$P(\text{at least one})$$ | Senior tranche expected loss | Junior tranche expected loss | Pool expected loss |
| ----------- | ------------------ | -------------------------- | ---------------------------- | ---------------------------- | ------------------ |
| 0.0         | 1.00%              | 19.00%                     | 1.00%                        | 19.00%                       | 10.00%             |
| 0.2         | 2.80%              | 17.20%                     | 2.80%                        | 17.20%                       | 10.00%             |
| 0.5         | 5.50%              | 14.50%                     | 5.50%                        | 14.50%                       | 10.00%             |
| 1.0         | 10.00%             | 10.00%                     | 10.00%                       | 10.00%                       | 10.00%             |

*Source: Author's calculation. Two bonds, each with risk-neutral default probability 10 percent and zero recovery; junior tranche 0-50 percent of pool notional, senior 50-100 percent. Expected losses are percentages of each tranche's own notional.*

Read the last column first: **the pool's expected loss is 10 percent regardless of correlation.** Correlation does not change how much is lost on average, only where the loss lands. As $$\rho\_D$$ rises the two bonds increasingly default together or survive together, which is bad for the senior tranche — the only state that hurts it becomes more likely — and good for the junior tranche, which a single default wipes out and which does not care whether there are two.

The magnitudes are the point. Against Table 10.1's ten-year expected-loss column, a senior tranche losing 1.00 percent sits inside the single-A row; the same tranche at $$\rho\_D = 0.5$$ loses 5.50 percent, between Baa and Ba. **Nothing about the underlying assets changed.** The whole difference between a top-rated senior claim and a speculative one is an assumption about a parameter that is not observable, cannot be estimated with any precision from a pool that has never seen a systemic downturn, and to which the senior tranche is exquisitely sensitive by construction. The standard device for imposing a correlation structure on a large pool became the one-factor Gaussian copula, in which each obligor defaults when a latent variable loaded on a common factor crosses a threshold; its convenience is that a whole loss distribution follows from one number.

That number is what moved in May 2005. The correlation trade of the opening episode was a position in $$\rho\_D$$ taken through the tranches of a standardized index, and the automakers' downgrade was a shock to the parameter rather than to the pool. Table 10.5 read from bottom to top is the trade losing money.

Figure 10.7 draws both halves of the argument: the mechanical one and the one about correlation.

![Figure 10.7: Tranching a loan pool](https://846781005-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2F3EupdX99vVBoNySDtmxb%2Fuploads%2Fgit-blob-c51c45a9b67965598b2e450f2547597867aea3d8%2Ffig_10_07_tranching_a_loan_pool.png?alt=media)

**Figure 10.7: Tranching a loan pool.** Panel (a) is the tranche-loss function of Section 10.8, drawn for the two tranches of the worked pool: a junior claim attaching at zero and detaching at half the pool, and a senior claim over the rest. Each is flat, then rises with slope one over the tranche's own thickness, then is flat again — a call spread on the pool's loss, which is why a tranche's value depends on the whole loss distribution and not on its mean. The markers above the panel are the only three losses this particular pool can realize, since it holds two bonds with zero recovery: nothing, half, or everything. Panel (b) is Table 10.5 as a picture. The pool's expected loss is the flat dashed line at 10 percent, and it does not move: correlation cannot change how much is lost on average, because that is fixed by the assets. What it changes is where the loss lands. As the default correlation rises the two bonds increasingly live or die together, which is precisely the state that hurts the senior tranche and is a matter of indifference to a junior tranche that one default already wipes out — so the senior line rises from 1.0 to 10.0 percent and the junior line falls from 19.0 to meet it. The vertical distance between the two lines is the whole of what tranching accomplishes, and it is a function of a parameter that is not observable, cannot be estimated from a pool that has never seen a systemic downturn, and to which the senior claim is exquisitely sensitive by construction.

Where this chapter stops is deliberate: pricing a tranche requires the loss distribution and correlation parameter above, and no more. The securitization pipeline — how collateral was originated and warehoused, how the ratings were produced and what the agencies' business model did to them, how the apparatus failed in 2007 and 2008 — is narrated in the companion volume on the 2008 crisis, which owns both the collateralized debt obligation story and the ratings-failure story. Chapter 13 §13.3 takes up mortgage-backed securities from the prepayment side, and Chapter 13 §13.2 sends the private-label structure back to this section.

***

## 10.9 Who Holds Credit Risk, and What Their Constraints Do to Its Price

Every section of this chapter has produced a residual. Expected loss explains about a seventh of an investment-grade spread (§10.4). The tax story explains the short end of the municipal curve and not the long end (§10.7). A CDS-bond basis of minus 250 basis points sat unarbitraged (§10.5). The residuals share a structure, and the structure is a fact about who is permitted to hold the claim.

Start with the size of the position. Chapter 2's Table 2.5 reports that insurers and pension funds together hold roughly $5 trillion of corporate and foreign bonds, more than any other domestic sector, and life insurers alone put on the order of 45 percent of their general accounts into corporate bonds (Chapter 16, Table 16.2). When a mechanism is claimed to work through insurers' demand for credit, the magnitude test of Chapter 2 §2.5 passes by a wide margin: this is the marginal holder.

What governs that holder's demand is not a view about expected returns. It is risk-based capital. Chapter 16 §16.2 sets out the machinery; what matters for pricing is the *shape* of the schedule in its Table 16.3. The capital charge on a life insurer's bond holding is roughly 0.4 percent at AAA, 1.3 percent at BBB, and 4.6 percent at BB — nearly flat within investment grade, and more than tripling at the boundary. Two consequences follow, and both are measurable.

**Within the buckets, insurers reach for yield.** If capital charges are constant across a rating category but yields are not, an insurer maximizing return on regulatory capital buys the highest-yielding bond in each bucket — the one the market thinks riskiest within it. Becker and Ivashina (2015) document exactly this: comparing insurers to other institutions holding bonds of the same rating at the same time, they find insurers systematically tilted toward the higher-yielding issues within each category, the tilt stronger for insurers closer to their capital constraint and when rates were low relative to their guaranteed crediting rates. This is reaching for yield *inside* the rules rather than around them, and it is what writing a regulation in terms of a coarse rating invites. Its pricing consequence is compression of within-bucket spread differentials: the marginal holder buys the risky end of each bucket, so the risky end is expensive relative to what an unconstrained investor would pay.

**At the boundaries, demand falls off a cliff.** The step from BBB to BB more than triples the capital charge on an asset whose cash flows did not change. Add the index mandates of the opening episode — an investment-grade index fund or index-benchmarked separate account may not hold a sub-investment-grade bond at all — and the boundary becomes a place where a large block of demand simply stops. That is a discontinuity in the demand curve at a point defined by a third party's opinion, the sharpest such line in the ecology this book describes. Nothing in Part II's asset pricing theory predicts that a claim's price should jump because its label changed.

But it does, and the jump has been measured. Ellul, Jotikasthira and Lundblad (2011) study downgrades across the boundary and show that the price decline is larger for bonds held more heavily by insurers under binding regulatory pressure, that those insurers are the sellers, and — the decisive finding — that the decline *reverses* over the following months. A permanent repricing would be information. A decline followed by a reversal is price pressure: a block of bonds transferred from constrained holders to unconstrained ones at a discount that pays the buyers for supplying balance sheet.

Which resolves the opening episode with this chapter's own tools. On May 5, 2005, S\&P delivered no news about General Motors' or Ford's cash flows. It delivered a change in eligibility affecting roughly 80 to 90 billion dollars of bonds held disproportionately by institutions about to become unable to hold them, into a high-yield market a fraction of the size. The widening was the price at which the market would arrange the transfer; the recovery was the Ellul reversal at the scale of a sector. The correlation convulsion was the same event refracted through Table 10.5 — a shock to a parameter a great deal of capital was positioned on, in instruments whose whole economic content is a loss distribution.

Three closing observations. This is the answer to Chapter 3 §3.7's "whose $$m$$?" in the credit market: the marginal holder of an investment-grade corporate bond is a capital-constrained insurer, and the stochastic discount factor implicit in the spread is hers, complete with a regulatory shadow price that appears in no household's utility function. The mechanism is symmetric — when regulators or index providers *loosen* a constraint the demand cliff becomes a demand surge, which is why an asset class's rating treatment is lobbied over as hard as a tax rate. And the reading is not that fundamentals are irrelevant. It is that an observed spread sums compensation for expected loss, a risk premium, an illiquidity premium, a tax term, and a term that exists because a rule says who may own the claim — and that in investment grade the last is not the small one. Credit is the market where that constraint is written into a public rulebook, so you can watch it work on a dated afternoon.

In Chapter 1 §1.2's terms, a spread that gaps at a downgrade is a constraint binding rather than news — a capital charge and an index rule, not a revision to the issuer's cash flows — and the reversal that follows is the flow completing, as the bonds reach holders for whom the rule does not bind.

> **Box 10.2 — The fallen angels of 2020**
>
> May 2005 is the mechanism without a policy response. Spring 2020 is the same mechanism with one, which is why the pair is worth reading together.
>
> On 25 March 2020 Ford Motor Company was cut to speculative grade, sending what was at the time the largest single block of debt on record across the investment-grade line. It was not alone. The pandemic quarter produced a wave of downgrades on the order of two hundred billion dollars of fallen-angel debt, arriving in a market whose dealers were at the limits of their balance sheets — the same fortnight Chapter 19's opening episode describes from the Treasury side.
>
> The arithmetic of §10.9 is what makes this dangerous rather than merely unpleasant. The investment-grade market is several times the size of the high-yield market, so a given fraction of the first arriving in the second is a large fraction of the second. The holders who must sell are index-constrained and the holders who may buy are a smaller set with their own mandates and their own drawdowns, and the price at which the transfer clears is set by the second group's capacity rather than by anybody's view of Ford's default probability.
>
> Then the Federal Reserve changed the buyer. On 9 April 2020 it broadened its corporate credit facilities to include debt of issuers that had been investment grade as of 22 March 2020 and were subsequently downgraded, subject to conditions. The cutoff date is the interesting part: the facility was defined not by what a bond was, but by what it had been on a particular Sunday. Fallen-angel spreads narrowed sharply on the announcement, well before any purchases under the facility were made, which is the cleanest available demonstration that what was being priced was the identity and capacity of the marginal holder rather than the credit.
>
> Section 10.9's mechanism, with a buyer of last resort attached. Chapter 19 §19.5 gives the pricing theory of what that buyer's arrival does.

***

## Elsewhere in the Series

* **Credit-spread components and sovereign credit described institutionally** — *International Finance*, Chapter 12. That chapter discusses the spread decomposition and the sovereign market from the practitioner's side; the models are here and the two are meant to be read together.
* **Credit default swap market mechanics** — *International Finance*, Chapter 13: standardized coupons and upfront conventions, credit-event determination, auction settlement, clearing and margin, and who is on each side of the market. Section 10.5 keeps only the price content.
* **Sovereign debt crises, restructuring practice, IMF programs, and the institutions of the international financial system** — *International Finance*. Section 10.6 keeps the theory of the repayment decision.
* **The securitization pipeline, collateralized debt obligations, and the industrial organization and failure of the rating agencies** — the companion volume on the 2008 crisis, which owns both stories. Section 10.8 stops at the depth needed to price a tranche.
* **This book**: state prices and the risk-neutral measure — Chapter 3 §3.4. Risk premia and why bad-state payoffs are expensive — Chapter 5. Option replication and Black-Scholes — Chapter 8. The riskless curve the spread is measured against — Chapter 9. Liquidity as a priced characteristic — Chapter 11 §11.5. Mortgage-backed securities and prepayment — Chapter 13 §13.3. Limits to arbitrage — Chapter 15 §15.5. Insurer balance sheets, risk-based capital, and the canonical constrained-capital statement — Chapter 16 §§16.2 and 16.5. Dealer balance sheets and the basis — Chapter 19. Demand-system asset pricing — Chapter 20. Covenants and the shareholder-bondholder conflict — Chapter 24 §24.1. The corporate issuer on the other side of §10.9's constrained holder — Chapter 23 §23.7 for credit-supply effects on leverage, and Chapter 25 §§25.1-25.2 for which firms reach the investment-grade door at all.

***

## Summary

1. **The credit triangle is the first-order identity and the first thing to distrust.** A spread is approximately the risk-neutral default probability times loss given default, so any two of the three determine the third. What the identity hides is that the probability in it is risk-neutral rather than physical, that recovery is low precisely when defaults are high, and that neither leg is observed.
2. **Historical default rates are convex in the rating and enormously time-varying.** Ten-year cumulative default rates run from a third of a percent at Aaa to nearly 4 percent at Baa and nearly 14 percent at Ba (Table 10.1), and annual speculative-grade rates have ranged from under 1 percent to above 12 percent, clustered in recessions.
3. **Merton's model makes equity a call on the firm's assets and risky debt a riskless bond minus a put.** Given asset value, asset volatility, leverage, and maturity, it produces a bond price, a spread, and a default probability jointly, with no free parameter for credit risk.
4. **The model's levers are the right ones and its levels are wrong.** Leverage and volatility drive spreads, volatility raises equity and debt risk together (the shareholder-bondholder conflict in one comparative static), and the maturity effect is non-monotonic. But calibrated to match the historical Baa default rate, the model produces a spread of about 30 basis points against an observed level near 150.
5. **Distance to default is the practitioner descendant.** Invert the Black-Scholes equations against observed equity value and equity volatility to recover asset value and asset volatility, measure the standard deviations to the default point, and map to a probability empirically. It typically moves before the rating does.
6. **Reduced-form models replace the default story with an intensity.** Under recovery of market value, a defaultable bond prices exactly like a riskless bond at the default-adjusted short rate $$r\_t + \lambda^{\ast}\_t\mathrm{LGD}\_t$$, so every affine model of Chapter 9 carries over. Structural models answer questions about the borrower; reduced-form models interpolate the market.
7. **The credit spread puzzle is an investment-grade phenomenon.** Expected loss accounts for roughly 6 percent of a AA spread, 15 percent of a BBB spread, and more than half of a single-B spread (Table 10.4). Risk premia, liquidity, the state-tax exemption of Treasuries, and segmented holder demand all contribute, and the four are complements rather than rivals.
8. **The CDS spread isolates default risk from funding and rate risk, and the basis measures what is left.** A basis of minus 250 basis points in 2008-09 was not a statement about credit; it was a price on balance sheet, and one of the cleanest available measurements of constrained arbitrage capital.
9. **Sovereign default is a decision, not an event.** With no bankruptcy court, repayment rests on exclusion and other costs of default (Eaton-Gersovitz, and Bulow-Rogoff's objection), spreads price the state of the borrower's economy, and a government that issues its own currency can still find local-currency default cheaper than inflation — as Russia did in 1998.
10. **Municipal pricing is a tax problem with a residual.** Tax-equivalent yields and implied marginal tax rates work at the short end and fail at the long end, where implied rates fall to 10 percent or less. Calls, liquidity, tax-law risk, and above all a thin, taxable, largely retail clientele are what fill the gap.
11. **Tranching redistributes a fixed pool loss, and correlation decides where it lands.** In the two-asset pool of Table 10.5, the pool's expected loss is 10 percent at every correlation, while the senior tranche's runs from 1.0 percent at $$\rho\_D = 0$$ to 5.5 percent at $$\rho\_D = 0.5$$ — on Table 10.1's scale, the distance from single-A to somewhere between Baa and Ba, produced entirely by an unobservable parameter.
12. **Insurers are the marginal holder of corporate credit, and the rating is a legal fact as well as an opinion.** A roughly $5 trillion position governed by a capital schedule that is flat within investment grade and triples at the boundary produces within-bucket reaching for yield (Becker-Ivashina) and demand cliffs at the boundary whose price footprint is a downgrade-day decline that reverses (Ellul-Jotikasthira-Lundblad). May 5, 2005, is that mechanism at the scale of a sector.

***

## Key Terms

* **Credit spread**: The yield on a defaultable claim minus the yield on an otherwise identical riskless claim, $$s = y - r\_f$$; the compensation for everything that distinguishes the two
* **Loss given default (LGD)**: One minus the recovery rate; the fraction of face value lost when default occurs. Recovery conventions differ — trading price shortly after default versus ultimate resolution value — and the two are not interchangeable
* **Credit triangle**: The identity $$s \approx \mathrm{PD}^{\ast} \times \mathrm{LGD}$$, in which any two of spread, risk-neutral default probability, and loss given default determine the third
* **Risk-neutral default probability**: The physical default probability re-weighted by state prices; larger than the physical probability whenever default occurs in high-marginal-utility states, and the source of the risk-premium component of the spread
* **Structural model**: A model in which default is derived from the firm's asset value crossing a boundary, so that credit risk is an option position on the firm (Merton 1974)
* **Distance to default**: The number of standard deviations of log asset value separating a firm from its default point; the KMV implementation's central statistic, mapped to a probability by an empirical frequency table
* **Reduced-form (intensity) model**: A model in which default arrives as the first jump of a point process with intensity $$\lambda$$, and the intensity is fitted to observed prices rather than derived from the firm
* **Default-adjusted short rate**: $$r\_t + \lambda^{\ast}\_t\mathrm{LGD}\_t$$, the rate at which a defaultable bond discounts under recovery of market value (Duffie-Singleton), which lets riskless term-structure machinery price credit
* **Credit spread puzzle**: The finding that observed investment-grade spreads greatly exceed both the expected default loss implied by historical default and recovery rates and the spreads that structural models calibrated to those rates can generate
* **CDS-bond basis**: The default swap spread minus the cash bond's spread; zero under a funding-free arbitrage, and therefore a direct measure of the shadow price of balance sheet when it is not
* **Willingness to pay**: The sovereign-credit concept that repayment is a choice weighed against the costs of default, rather than a constraint imposed by resources; the object Eaton-Gersovitz models and Merton's boundary cannot represent
* **Tax-equivalent yield**: $$y\_m/(1-\theta)$$, the taxable yield equivalent to a tax-exempt yield $$y\_m$$ for an investor at marginal rate $$\theta$$; its inverse gives the implied marginal tax rate embedded in a muni-Treasury ratio
* **Attachment and detachment points**: The fractions of pool loss at which a tranche begins and finishes absorbing losses; a tranche is a call spread on the pool's loss distribution
* **Default correlation**: The dependence between obligors' default events; it leaves a pool's expected loss unchanged while determining how that loss is distributed across tranches
* **Fallen angel**: A bond downgraded from investment grade to speculative grade, and therefore ineligible for the mandates and indices that governed a large share of its previous holders

***

## Readings

### Required

* Merton, R. C. (1974). "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates." *Journal of Finance* 29(2): 449-470. *The founding paper of structural credit modeling, and short. Read it for the two sentences that organize Section 10.2 — equity is a call on the firm, risky debt is a riskless bond minus a put — and then for how much follows from them.*
* Becker, B. and V. Ivashina (2015). "Reaching for Yield in the Bond Market." *Journal of Finance* 70(5): 1863-1902. *The cleanest identification of a holder constraint moving credit prices: insurers buy the highest-yielding bond inside each rating bucket, more so when their capital is tight. The empirical spine of Section 10.9.*

### Recommended

* Duffie, D. and K. Singleton (1999). "Modeling Term Structures of Defaultable Bonds." *Review of Financial Studies* 12(4): 687-720. *The reduction that made intensity models the market standard: a defaultable bond prices like a riskless one at a default-adjusted short rate.*
* Huang, J.-Z. and M. Huang (2012). "How Much of the Corporate-Treasury Yield Spread Is Due to Credit Risk?" *Review of Asset Pricing Studies* 2(2): 153-202. *The disciplined statement of the puzzle: several structural models, each forced to match historical default and recovery, and the spread each can then produce. Around 20 percent of a ten-year Baa spread.*
* Eaton, J. and M. Gersovitz (1981). "Debt with Potential Repudiation: Theoretical and Empirical Analysis." *Review of Economic Studies* 48(2): 289-309. *Sovereign lending sustained by exclusion rather than enforcement, and the endogenous credit ceiling that follows. Read alongside Bulow and Rogoff (1989), "Sovereign Debt: Is to Forgive to Forget?", American Economic Review 79(1): 43-50, which explains why reputation alone is not enough.*
* Longstaff, F. (2011). "Municipal Debt and Marginal Tax Rates: Is There a Tax Premium in Asset Prices?" *Journal of Finance* 66(3): 721-751. *Uses short-dated tax-exempt and taxable money-market rates to measure what the exemption is actually worth, and finds more than the statutory rate explains. The best entry point to Section 10.7's puzzle.*
* Ellul, A., C. Jotikasthira and C. Lundblad (2011). "Regulatory Pressure and Fire Sales in the Corporate Bond Market." *Journal of Financial Economics* 101(3): 596-620. *Downgrades across the investment-grade boundary, insurers as the forced sellers, and the price decline that reverses. The measured footprint of the opening episode.*
* Longstaff, F., S. Mithal and E. Neis (2005). "Corporate Yield Spreads: Default Risk or Liquidity? New Evidence from the Credit Default Swap Market." *Journal of Finance* 60(5): 2213-2253. *The decomposition that uses the swap to isolate the default leg, and the argument for a large non-default component.*
* Collin-Dufresne, P., R. Goldstein and J. S. Martin (2001). "The Determinants of Credit Spread Changes." *Journal of Finance* 56(6): 2177-2207. *Structural variables explain little of the variation in spread changes, and the residuals share a common factor. The puzzle stated in first differences.*
* Elton, E., M. Gruber, D. Agrawal and C. Mann (2001). "Explaining the Rate Spread on Corporate Bonds." *Journal of Finance* 56(1): 247-277. *The decomposition that put the state-tax term on the map, and found it larger than expected default loss for investment-grade bonds.*
* Arellano, C. (2008). "Default Risk and Income Fluctuations in Emerging Economies." *American Economic Review* 98(3): 690-712. *The modern quantitative Eaton-Gersovitz model: default is countercyclical, and calibrated spreads look like observed ones without an exogenous credit shock.*

***

## Discussion Questions

1. **Information or mandates?** A rating downgrade is followed by a large price decline. Two explanations compete: the agency revealed information the market did not have, and the downgrade changed who is permitted to hold the bond. State the observable predictions that distinguish them — think about timing relative to the announcement, the cross-section of which bonds move most, the behavior of the same issuer's equity and default swaps, and what happens over the following six months. Then apply your test to May 2005 and say which explanation the evidence supports, and how confidently. Finally: if mandates are doing most of the work, does it follow that ratings are uninformative, or could a rating be uninformative and still be the right thing to write into a regulation?
2. **The sovereign's choice.** A government owes the equivalent of 100 billion in debt denominated in its own currency and 40 billion denominated in foreign currency. A recession cuts tax revenue sharply. Set out the costs and benefits, to the government, of (a) repaying everything, (b) inflating the local-currency debt away, (c) restructuring the local-currency debt while paying the foreign-currency debt, and (d) defaulting on the foreign-currency debt while paying the local-currency debt. Who bears each cost, and which domestic constituency lobbies for which option? Russia in 1998 chose something close to (c). Under what conditions is that the rational choice, and what does its rationality imply for the claim that a government cannot default in a currency it prints?
3. **Where does the puzzle go?** Suppose a regulator replaced rating-based capital charges with a continuous, model-based risk measure applied to every bond individually. Using Section 10.9's mechanisms, predict what happens to (a) within-bucket spread differentials, (b) the size of the price drop at a downgrade across the investment-grade line, (c) the level of investment-grade spreads on average, and (d) insurers' portfolio composition. Which of the four predictions are you most confident about, and which of the four explanations of the credit spread puzzle would the reform let you test?
4. **A correlation you cannot observe.** Table 10.5 shows that a senior tranche's expected loss moves by a factor of five as $$\rho\_D$$ moves from 0 to 0.5, while the pool's expected loss does not move at all. You are asked to rate the senior tranche. What data would you use to estimate $$\rho\_D$$, and what is the fundamental problem with every source you can name? Then answer the harder question: given that the parameter cannot be estimated reliably, is issuing a rating on the tranche defensible at all, and if so, what would have to accompany the letter for it to be honest?
5. **The muni long end.** An advisor tells a client in the top federal bracket that thirty-year municipal bonds are "obviously cheap" because their implied marginal tax rate is 10 percent while the client's rate is 37 percent. Construct the strongest case that the advisor is right and the client should buy. Then construct the strongest case that the implied 10 percent is not an arbitrage at all but a correctly priced compensation for something. What single piece of data would move you most between the two positions?

***

## Problems

**Problem 1 — The credit triangle.** A one-year zero-coupon corporate bond has face value 100. The riskless rate is 4 percent. The market's risk-neutral default probability is 3 percent and the recovery rate is 40 percent of face.

(a) Compute the bond's price, its promised yield, and its credit spread. (b) Compare the exact spread to the approximation $$\mathrm{PD}^{\ast} \times \mathrm{LGD}$$. In which direction does the approximation err, and why? (c) The *physical* default probability is 1.2 percent. Compute the bond's expected return and its expected excess return over the riskless rate. (d) Decompose the spread into an expected-loss component and a risk-premium component, and verify that they sum to the spread. What is the ratio $$\mathrm{PD}^{\ast}/\mathrm{PD}$$, and what does it say about the state in which this bond defaults?

**Problem 2 — Merton.** A firm's assets are worth 120, with asset volatility 30 percent. It has one zero-coupon bond outstanding, face value 100, maturing in three years. The continuously compounded riskless rate is 3 percent.

(a) Compute $$d\_1$$, $$d\_2$$, the market value of equity, the market value of the debt, the value of the implicit put, the promised yield, the credit spread, and the risk-neutral default probability. (b) The firm pays a special dividend of 10, financed by cash, so that assets fall to 110. Recompute the debt value and the spread. How much value moved from bondholders to shareholders, and how does it compare to the dividend? (c) Return to the base case and raise asset volatility to 40 percent. What happens to equity value and to debt value? Explain the sign of each in one sentence, and name the covenant a lender would write to prevent it (Chapter 24 §24.1). (d) Return to the base case and raise the face value to 110. Compare the equity value to (a) and explain why the shareholders gain even though the firm's assets are unchanged.

**Problem 3 — Attachment and detachment.** A pool contains two bonds, each with face value 100, each with a risk-neutral default probability of 12 percent and a recovery rate of 40 percent. Three tranches are issued against the pool notional of 200: equity (0-15 percent), mezzanine (15-40 percent), and senior (40-100 percent).

(a) With zero default correlation, compute the probability of zero, one, and two defaults, and the loss on the pool in each case. (b) Compute each tranche's expected loss, in currency units and as a percentage of its own notional. Verify that the three sum to the pool's expected loss. (c) Repeat (a) and (b) with a default correlation of 0.4, using $$P(\text{both}) = \mathrm{PD}^2 + \rho\_D\mathrm{PD}(1-\mathrm{PD})$$. (d) State which tranche is helped and which is hurt by correlation, and explain the mezzanine's behavior — it is the tranche whose direction is not obvious in advance.

**Problem 4 — Tax-equivalent yield.** An investor faces a 37 percent federal marginal rate and a 9.3 percent state marginal rate, and holds bonds in a taxable account. Assume for simplicity that state tax is not deductible federally. She is choosing among an in-state municipal bond yielding 3.40 percent (exempt from both), a Treasury yielding 4.40 percent (exempt from state tax only), and a corporate bond yielding 5.20 percent (fully taxable). All have the same maturity.

(a) Compute the after-tax yield on each. (b) Compute the muni's tax-equivalent yield on a fully taxable basis and on a Treasury basis. Why are these two different numbers, and which one should she use? (c) What combined marginal rate would make her indifferent between the muni and the corporate bond? (d) Compute the marginal tax rate implied by the muni-Treasury ratio. Compare it to her actual federal rate and state, in two sentences, what Section 10.7 says the gap is measuring.

**Problem 5 — The puzzle in intensities.** Historical ten-year cumulative default rates for Baa issuers are about 3.70 percent, with a loss given default of 60 percent. Observed ten-year BBB spreads average roughly 150 basis points.

(a) Compute the implied constant physical default intensity $$\lambda$$ and the expected-loss spread $$\lambda\mathrm{LGD}$$. (b) Compute the risk-neutral intensity $$\lambda^{\ast}$$ implied by the observed spread, and the ratio $$\lambda^{\ast}/\lambda$$. (c) What ten-year default probability does the market appear to be pricing? Compare it to the historical 3.70 percent. (d) A colleague concludes that the market is irrationally pessimistic about Baa credit. Give three reasons the conclusion does not follow, one drawn from each of Sections 10.4, 10.5, and 10.9.

**Problem 6 ★ — Inverting the structural model.** A firm's equity has a market value of 40.23 and an equity volatility of 52.8 percent. Its debt is a single zero-coupon bond with face value 80 maturing in five years; the riskless rate is 4 percent.

(a) Set up the two-equation system in the unknowns $$V\_0$$ and $$\sigma\_V$$: the Black-Scholes equation for equity value, and the identity $$\sigma\_E E\_0 = N(d\_1)\sigma\_V V\_0$$. Derive the second equation from Itô's lemma applied to $$E\_0 = f(V\_0)$$. (b) Solve the system numerically and report $$V\_0$$ and $$\sigma\_V$$. (c) Compute the implied credit spread and the risk-neutral default probability. Then compute the distance to default and the physical default probability $$N(-\mathrm{DD})$$ under an expected asset return of 8 percent. (d) The physical default probability you obtain is far above anything in Table 10.1, while Section 10.2 reports that the same model produces spreads far *below* observed levels for investment-grade firms. Explain how a model can err in opposite directions for different firms, and what that implies for using $$N(-\mathrm{DD})$$ as a probability rather than as an ordinal ranking.

***

## Selected Solutions

*Solutions to Problems 1 and 3 follow. Solutions to the remainder are in the instructor materials.*

**Problem 1.**

(a) The expected risk-neutral payoff per dollar of face is $$(1 - 0.03) + 0.03 \times 0.40 = 0.9820$$. The price is $$0.9820/1.04 = 0.944231$$, or **94.4231** per 100 of face. The promised yield is $$1/0.944231 - 1 = 5.9063$$ percent, so the credit spread is $$5.9063 - 4.0000 = 1.9063$$ percent, or **190.6 basis points**.

(b) $$\mathrm{PD}^{\ast} \times \mathrm{LGD} = 0.03 \times 0.60 = 180$$ basis points. The approximation *understates* the exact spread by about 11 basis points, for two reasons visible in the exact formula: the numerator carries a factor $$(1+r\_f)$$, because the loss is discounted at the same rate as the promise, and the denominator $$1 - \mathrm{PD}^{\ast}\mathrm{LGD}$$ is less than one, because the yield is quoted on a price that has already been marked down. Both push the same way, and the gap grows with $$\mathrm{PD}^{\ast}\mathrm{LGD}$$.

(c) Under the physical measure the expected payoff is $$(1 - 0.012) + 0.012 \times 0.40 = 0.99280$$. The expected return is $$0.99280/0.944231 - 1 = 5.1438$$ percent, so the expected excess return over the riskless rate is **114.4 basis points**.

(d) The expected-loss component is the difference between the promised yield and the expected return, $$5.9063 - 5.1438 = 0.7625$$ percent, or **76.25 basis points** — the part of the quoted yield the investor does not expect to receive. The risk-premium component is the expected excess return, **114.38 basis points**. The two sum to 190.63, the spread, as they must by construction. (Note that 76.25 basis points exceeds the naive $$\mathrm{PD} \times \mathrm{LGD} = 72$$ basis points for the same discounting reason as in part (b).) The ratio $$\mathrm{PD}^{\ast}/\mathrm{PD} = 0.03/0.012 = 2.5$$: the state price attached to default is two and a half times its physical probability, which says that this bond defaults in states where a dollar is worth two and a half times what it is worth on average. Sixty percent of the spread is payment for that, not for the loss.

**Problem 3.**

(a) With $$\mathrm{PD} = 0.12$$ and $$\rho\_D = 0$$, the two defaults are independent: $$P(\text{both}) = 0.12^2 = 0.0144$$, $$P(\text{exactly one}) = 2(0.12)(0.88) = 0.2112$$, $$P(\text{none}) = 0.7744$$. Each default costs $$100 \times (1 - 0.40) = 60$$, so the pool loses 0, 60, or 120.

(b) The tranches have notionals of 30 (equity, 0-15 percent of 200), 50 (mezzanine, 15-40 percent) and 120 (senior, 40-100 percent). A loss of 60 exhausts the equity tranche and takes 30 of the mezzanine; a loss of 120 exhausts both and takes 40 of the senior.

**Table 10.6: Tranche expected losses in Problem 3, zero default correlation**

| Tranche          | Notional | Loss if 1 default | Loss if 2 | Expected loss | Expected loss (% of tranche) |
| ---------------- | -------- | ----------------- | --------- | ------------- | ---------------------------- |
| Equity 0-15%     | 30       | 30                | 30        | 6.768         | 22.56%                       |
| Mezzanine 15-40% | 50       | 30                | 50        | 7.056         | 14.11%                       |
| Senior 40-100%   | 120      | 0                 | 40        | 0.576         | 0.48%                        |

*Source: Author's calculation from the problem's assumptions.*

The three figures sum to 14.400, which is the pool's expected loss, $$2 \times 100 \times 0.12 \times 0.60 = 14.40$$.

(c) With $$\rho\_D = 0.4$$: $$P(\text{both}) = 0.0144 + 0.4(0.12)(0.88) = 0.05664$$, $$P(\text{exactly one}) = 2(0.12 - 0.05664) = 0.12672$$, $$P(\text{none}) = 0.81664$$.

**Table 10.7: The same tranches at a default correlation of 0.4**

| Tranche          | Expected loss | Expected loss (% of tranche) |
| ---------------- | ------------- | ---------------------------- |
| Equity 0-15%     | 5.501         | 18.34%                       |
| Mezzanine 15-40% | 6.634         | 13.27%                       |
| Senior 40-100%   | 2.266         | 1.89%                        |

*Source: Author's calculation from the problem's assumptions.*

The three figures again sum to 14.400 — correlation cannot change the pool's expected loss.

(d) The **senior tranche is hurt**, its expected loss rising from 0.48 percent to 1.89 percent, a factor of nearly four: it is damaged only in the joint-default state, and correlation is what makes that state likely. The **equity tranche is helped**, from 22.56 percent to 18.34 percent: it is wiped out by a single default and indifferent to the second, so anything that makes defaults arrive together rather than separately raises the probability that none arrives at all. The **mezzanine falls slightly**, from 14.11 to 13.27 percent, and its direction is genuinely ambiguous in general. A mezzanine tranche is hurt by the rise in $$P(\text{both})$$ and helped by the rise in $$P(\text{none})$$, and which effect wins depends on where its attachment point sits relative to the loss from a single default. That ambiguity is why mezzanine tranches were the instrument of choice for correlation trades, and why the trade of May 2005 was structured as a long-equity, short-mezzanine position rather than the other way around.

***

## Data Exercise: Spreads, Recessions, and a Downgrade

Everything in Parts A through C runs on free public data.

**Part A — The history of the spread (free data: FRED).** From the Federal Reserve Bank of St. Louis's FRED database, download the daily series `BAA10Y` (Moody's Seasoned Baa Corporate Bond Yield relative to the ten-year Treasury constant maturity, available from 1986), `BAMLH0A0HYM2` (ICE BofA US High Yield Index option-adjusted spread, from December 1996), and `BAMLC0A0CM` (ICE BofA US Corporate Index OAS).

1. Plot all three on one chart with recession bars, using FRED's `USREC` series to shade the NBER recession months. Report the full-sample mean, median, minimum, and maximum of each. Note where the mean sits relative to the median and explain the skew in one sentence.
2. Compute the correlation between daily changes in the high-yield OAS and daily changes in the ten-year Treasury yield. The sign is usually negative. Give the economic reason, and say what it implies for the difference between a bond's *spread* and its *yield* as a measure of credit risk.
3. Using Table 10.1, compute the expected-loss spread for Baa at a 60 percent loss given default. Draw it as a horizontal line on your Baa chart. On what fraction of days in the sample did the observed spread fall below twice the expected-loss line? What does the answer say about the credit spread puzzle as a statement about averages versus about episodes?

**Part B — The fallen-angel window (free data: FRED).** Download `BAMLC0A4CBBB` (BBB OAS) and `BAMLH0A1HYBB` (BB OAS), both daily.

1. Plot both from January 2005 through December 2005 and mark May 5. Measure the level of each on May 4, the maximum reached during May, and the level at the end of September. Report the widening and the retracement in basis points and as a percentage of the move.
2. Compute the BB-minus-BBB difference over the same window. Does the boundary spread widen or narrow around the downgrade, and what does the sign tell you about whether the event was priced as news about credit or as a transfer of inventory across the boundary?
3. Repeat the exercise for a second fallen-angel episode of your choosing — the March-April 2020 downgrade wave is the obvious candidate, with a much larger volume of debt crossing the line. Compare the size and the speed of the retracement, and offer an explanation for any difference that refers to who the buyers were.

**Part C — The default and recovery table (free data: agency default studies).** Obtain the most recent annual default study from Moody's Investors Service or S\&P Global Ratings. Both are published free, though registration is sometimes required.

1. Rebuild Table 10.1 with exact current figures for one-, five-, and ten-year cumulative default rates by rating. Note the sample period and the universe, and say how they differ from the table in this chapter.
2. Extract the annual speculative-grade default rate series and plot it with recession shading. Report the maximum and minimum annual rates and the years in which they occurred.
3. Extract the annual average recovery rate on senior unsecured bonds and plot it against the annual default rate on the same axes. Estimate the correlation. Then recompute the final column of Table 10.1 using a loss given default that varies with the default rate rather than a flat 60 percent, and report how much the expected-loss spread for Baa changes.
4. Locate the study's rating transition matrix. Report the one-year probability that a BBB issuer is downgraded to speculative grade. Multiply it by the size of the BBB market — available from the index provider's published fact sheets or from the Securities Industry and Financial Markets Association's free bond-market statistics — to produce an expected annual volume of fallen-angel supply. Compare it to the size of the high-yield market, and comment on the ratio in light of Section 10.9.

**Part D ★ (if you have WRDS).** Using TRACE, extract all transactions in the bonds of a single fallen-angel issuer over a window running from sixty trading days before its downgrade to one hundred and twenty days after.

1. Construct a daily volume-weighted average price and a daily trade count. Plot both, marking the downgrade date. Describe what happens to volume before, on, and after the event.
2. Split trades by size, using the customary threshold that separates institutional from retail-sized trades. Compute the net direction of institutional flow, using the dealer-side indicator, over the window. Who was selling, and when relative to the announcement?
3. Estimate the price decline from the pre-window average to the post-downgrade trough, and the fraction recovered by the end of the window. Compare your estimate to the magnitudes in Ellul, Jotikasthira and Lundblad (2011), and state two reasons a single-issuer estimate is not a clean measurement of the effect their paper identifies.