Chapter 11 · Private Credit: Default, Recovery, and Covenants

Chapter at a glance

Private credit is priced by decomposing a quoted spread into what each basis point compensates: expected loss, a default-risk premium, illiquidity, and covenant or information rents. The chapter builds a hybrid default intensity riding the borrower’s leverage path, state-dependent recovery drawn at the worst time, a covenant three-state engine, and prepayment — then insists each block be labeled by its measure and its address. Its organizing display is the spread waterfall.

Learning Outcome Statements

LOS 11.1 — Specify the cash-flow anatomy of a bilateral loan—coupons, fees, default, recovery, prepayment, amendment—as a marked point process with state-dependent.

LOS 11.2 — Derive, with proof, survival probabilities and defaultable claim prices in the doubly stochastic (Cox) framework, including the recovery-of-market-value reduction.

LOS 11.3 — Connect reduced-form intensities to the structural first-passage view of the leveraged borrower, and exploit the private lender’s information advantage in the hybrid.

LOS 11.4 — Quantify wrong-way recovery risk through the covariance identity for expected loss, and model state-dependent loss-given-default. The answer to teach: wrong-way risk is a covariance with a sign you can prove.

LOS 11.5 — Represent covenant life as a three-state process, value the covenant as a lender option, and price the removal of that option (cov-lite).

LOS 11.6 — Analyze prepayment as the borrower’s call, derive its negative-convexity consequences, and adjust effective duration accordingly.

LOS 11.7 — Decompose a private-credit spread into expected loss, default risk premium, illiquidity, and residual—each located at one address—and reconcile physical and.

Laboratory (book §11.9)

Module: Private Credit Engine — open in the Laboratory

Price bilateral loans with all four layers live; the spread waterfall is the organizing display. E1 decompose the 575 (the waterfall with bands, the sentence each block licenses in committee English); E2 price cov-lite (covenant value by regime, verifying the three monotonicities); E3 wrong-way stress (sweep the recovery–regime link, separating the P-effect from the priced effect); E4 reconcile the committee (back out the P/Q wedge and convert it to a default-risk Sharpe contribution — the Chapter 7 audit applied to credit).

Downloads: Python notebook · Excel workbook · Slides

Exercises

Solutions are distributed to instructors with the Instructor’s Solutions Manual; they are not posted here.

Conceptual Problems

11.1 Classify each component of the loan (11.1) in the taxonomy of Table 2.1 (rate, event intensity, mark), and state which components the lender’s information covenant makes better-observed than a public bondholder’s filtration would.

11.2 Explain, for a non-quantitative committee, why the agent’s 1.9% and the desk’s 4.5% can both be right, using only the ideas “measured frequency” and “price of bearing”—then state what would constitute an inconsistency between them.

11.3 A sponsor argues that quarterly certified reporting makes covenants redundant: “you’ll see trouble coming either way.” Separate the information content of the covenant from its control-rights content, and say which one Section 11.6’s option value prices.

11.4 Why does the wrong-way covariance of Proposition 11.5 bite harder for a portfolio than for a single loan? Connect to the common stress factor and to the variance floor of Section 11.8.

Mathematical Problems

11.5 Derive the recovery-of-face pricing formula: with recovery 𝑅f of par paid at 𝜏d , show ∫ 𝑇 h∫ 𝑇 i ∫𝑠 ∫𝑇 ∫𝑠 − 0 (𝑟+𝜆) − 0 (𝑟+𝜆) 𝑉0 = e 𝑐𝑒 𝑠+𝑒 + 𝑅f 𝜆 𝑠 𝑒 − 0 (𝑟+𝜆) 𝑠 , 0 0 and quantify the wedge against Proposition 11.3 for constant parameters.

11.6 In the hybrid model (11.2) with lognormal EV, derive the dynamics of 𝜆 𝑡 = Λ(𝑑𝑡 ) by Itô for smooth Λ, and show the intensity’s volatility is proportional to |Λ′ (𝑑𝑡 )|: default risk is most volatile exactly near the covenant region.

11.7 Prove the association inequality used in Proposition 11.5 for nondecreasing 𝑓 , 𝑔 of a real variable, and extend the wrong-way identity to the pricing measure: show the -expected loss adds a term plus a measure-wedge cross term, and identify each in the waterfall (11.3).

11.8 Model the covenant as a barrier option: with two barriers 𝐵 < 𝐵c on lognormal EV, express the value of the covenant as a difference of first-passage functionals, and prove monotonicity in 𝐵c and in 𝜎EV via pathwise coupling.

11.9 Carry out the reconciliation of E4 analytically: given (𝜆 ¶ , 𝜆, 𝑅), define the defaultrisk Sharpe contribution over horizon Δ as excess spread over expected loss divided by loss volatility; compute it for (1.9%, 4.5%, 40%) at Δ = 1 and compare with a good-deal ceiling of 0.5.

Computational Problems

11.10 Reproduce Figure 11.1 from the printed seed; then vary the recovery–regime link and regenerate the right panel, reporting the covariance correction in spread terms across the sweep.

11.11 Implement the covenant three-state engine on Chapter 9 borrower paths; reproduce E2’s cov-lite repricing by regime and verify the three monotonicities numerically.

11.12 Build the prepayment panel: with intensity 𝜆p increasing in the loan’s above-par value, plot loan value against borrower quality with and without call protection; report effective and spread durations and locate the negative-convexity region.

11.13 Waiver terms are negotiated, not drawn from exogenous intensities: model the breach state as a bargaining game between a lender with acceleration rights and a sponsor with reinvestment options, characterize equilibrium waiver fees and repricings as functions of distance to insolvency and of the sponsor’s fund-level incentives 178 11 Private Credit: Default, Recovery, and Covenants (Chapter 9’s carry convexity), and design an empirical test using amendment-andwaiver pricing across the cycle.