--- title: "Capital-Structure Arbitrage" book: "Strategies II: Volatility, Relative Value, Macro and the Bank Desks" subject: quant language: en chapter: 9 exercises: 8 source: https://one-course.com/books/quant/9/en/chapter/9-capital-structure-arbitrage --- # Chapter 9 — Capital-Structure Arbitrage A company’s shares and its credit default swaps both price its default risk. When they disagree, a trader buys one, sells the other, and waits for the disagreement to close, or for the model that measured it to prove wrong. Yu tested the trade on 135 759 daily CDS spreads of 261 North American companies, using the CreditGrades model. Single trades could lose heavily, because spreads and share prices were only loosely correlated. An equally weighted portfolio of all trades earned Sharpe ratios like those of other fixed-income arbitrage strategies. On this chapter’s sixty synthetic firms the trade earns a Sharpe ratio of 1.40, and 48% of its trades lose. When leveraged buyouts, which double a firm’s debt while its shares jump, come four times as often, the Sharpe ratio falls to 0.39. The build is `firm.capstruct`. ## 9.1 From equity to credit spread A structural model (Book 6, chapter 14) treats equity as a claim on a firm’s assets and default as the assets falling below what is owed. The version used here is in the spirit of CreditGrades. Assets per share are the share price plus the part of the debt that would be recovered, $V = E + LD$ for debt per share $D$ and recovered share $L$. Asset vol is the equity vol scaled down, $\sigma_E E/V$, and default is the first time $V$ touches $LD$. The five-year survival probability $q$ from the first-passage formula gives a spread, $(1 - R)(-\ln q)/5$ for CDS recovery $R$. **Definition 9.1 (Equity-implied spread).** The *equity-implied spread* of a company is the credit default swap spread that a structural model gives from its share price, equity volatility and debt per share; comparing it with the market spread measures whether the equity and credit markets disagree about the company’s default risk. With a share price of 50, debt of 40 a share, 35% equity vol, $L = 0.5$ and $R = 0.4$, the implied five-year spread is 55.8 basis points ([Figure 9.1](#fig-s2-capital-structure-arbitrage-curve)). At debt of 80 a share it is 115.4. Spreads rise as the share falls, but only to 189.8 basis points at a share price of 10 with debt of 40. The model’s asset vol falls with the equity, a feature of the approach, which keeps distressed spreads lower than a constant-vol model would. ![The chapter’s equity-implied five-year spread by share price, at 35% equity vol, for debt of 40 and 80 a share. The arrow is a leveraged buyout: the share rises from 50 to 60 on the premium while the debt doubles, and the spread goes from 55.8 to 98.8 basis points. Data: s2_capstruct.curve.](https://one-course.com/images/onecourse/chapters/quant-9/s2-capital-structure-arbitrage/fig-4efbbd923894.svg) ***Figure 9.1.** The chapter’s equity-implied five-year spread by share price, at 35% equity vol, for debt of 40 and 80 a share. The arrow is a leveraged buyout: the share rises from 50 to 60 on the premium while the debt doubles, and the spread goes from 55.8 to 98.8 basis points. Data: `s2_capstruct.curve`.* ## 9.2 The trade and its hedge ratio **Definition 9.2 (Cross-asset hedge ratio).** A *cross-asset hedge ratio* is the quantity of one asset that offsets the model sensitivity of a position in another asset of the same issuer; in capital-structure arbitrage, the shares per unit of CDS notional that offset the CDS’s value change when the share price moves, minus the risky annuity times the implied spread’s derivative in the share price. A trader who finds the market spread above the implied one sells protection: the CDS is paying too much for a risk the equity says is smaller. If the share then falls, the implied spread rises and the protection seller loses. So the trader also shorts shares, as many as the hedge ratio says. At a share price of 50 and debt of 40, that is 0.00067 shares per unit of notional: about 6 700 shares, or $0.34 million of stock, against $10 million of protection. A buyer of protection does the reverse and buys the shares. `firm.capstruct` runs the trade on sixty firms for ten years ([Listing 9.1](#lst-s2-capital-structure-arbitrage-trades)). Each firm’s market spread is the model’s spread at the firm’s true barrier and debt, times a mean-reverting mispricing (10% in logs, half-life 34 days). The trader does not know a firm’s true barrier and assumes $L = 0.5$ everywhere, while the truth varies by firm with a dispersion of 25%. The trader estimates the equity vol from the trailing year. It enters when the market spread is more than 40% (in logs) from the implied spread, and exits when the gap falls below 5% or after 180 days. ## 9.3 Evidence and losses | sixty firms, ten years | leveraged buyouts 0.05 a firm-year | none | 0.2 a firm-year | | --- | --- | --- | --- | | trades | 440 | 413 | 433 | | mean per trade (bp of notional) | 26.1 | 25.3 | 9.3 | | share losing | 48% | 44% | 52% | | worst trade (bp) | $-347$ | $-179$ | $-378$ | | share closed at 180 days | 56% | 55% | 57% | | Sharpe ratio of the book | 1.40 | 1.66 | 0.39 | The trades look like Yu’s. Almost half lose, the worst lose several percent of notional, and the book of all of them earns a steady Sharpe ratio ([Figure 9.2](#fig-s2-capital-structure-arbitrage-book)). Most of the return comes from the CDS leg, 20.4 of the 26.1 basis points, and the share hedge adds 5.6. More than half the trades never converge and are closed at 180 days. They were entered on a gap that was really the trader’s wrong barrier, not a mispricing, and those gaps do not close. Duarte, Longstaff and Yu found that the fixed-income arbitrage strategies needing more “intellectual capital” earned significant alphas after bond and equity factors and after fees, and that many had positively skewed returns, not the negative skew of picking up nickels in front of a steamroller. ![The synthetic capital-structure book’s cumulative P&L per average open trade (about 27 at a time), as a share of one trade’s notional, with leveraged buyouts at 0.05 and 0.2 a firm-year. Data: s2_capstruct.cumulative.](https://one-course.com/images/onecourse/chapters/quant-9/s2-capital-structure-arbitrage/fig-cc86d7a0cebe.svg) ***Figure 9.2.** The synthetic capital-structure book’s cumulative P&L per average open trade (about 27 at a time), as a share of one trade’s notional, with leveraged buyouts at 0.05 and 0.2 a firm-year. Data: `s2_capstruct.cumulative`.* ## 9.4 Model risk A leveraged buyout breaks the trade in both legs at once. The firm’s debt doubles and its share jumps 20% on the takeover premium, and the trader learns the new debt from filings three months later. The market spread jumps at once. A trader who had sold protection loses on the CDS, and loses again on the short shares, which rose. Of the trades open when their firm’s leverage shifted, the five that had sold protection lost 98 basis points on average (the worst 316). The seven that had bought protection earned 220. When buyouts come four times as often, 23 protection sellers were caught and lost 174 basis points each on average. The structural model is the trade’s only measure of mispricing, and every input it takes is uncertain: the barrier, the equity vol, the debt a share. A persistent gap is more often a wrong input than a market error. The trader needs rules for when to stop believing the model. Close gaps that do not converge. Leave out firms in play for a takeover. Update debt from filings as they arrive. ## 9.5 Strategy files **Strategy file 9.1 — Equity-implied spread convergence.** **Who pays you, and why.** Investors in one market who price a company’s default risk differently from investors in the other, until the two converge. **Instruments and venues.** Single-name CDS; the company’s shares for the hedge. **Signal.** Market spread against the [equity-implied spread](#def-s2-capital-structure-arbitrage-implied) from a structural model. **Sizing and execution.** Many small positions; hedge ratio from the model; exit on convergence or after a time limit. **Costs.** CDS bid-ask spreads; borrow on short shares. **How it dies.** Gaps that are model error; takeovers and recapitalisations that move equity and credit the same way. **Horizon, capacity, infrastructure.** Months; a structural model fed with filings, equity vol and CDS quotes. **Backtest honestly.** Debt as known at the time; include the trades closed at the time limit. **Sources.** Yu (2006); this chapter: a book Sharpe ratio of 1.40 and 48% of trades losing. **Strategy file 9.2 — Long CDS, short equity on weak equity.** **Who pays you, and why.** Credit investors slower than equity investors to price bad news. **Instruments and venues.** CDS protection bought; shares sold short. **Signal.** A falling share price with a CDS spread below its implied level. **Sizing and execution.** Sized to the hedge ratio; small in names close to distress. **Costs.** Protection premium; borrow on hard-to-borrow shares. **How it dies.** Recovery of the shares; borrow recall. **Horizon, capacity, infrastructure.** Weeks to months. **Backtest honestly.** Borrow costs on falling shares. **Sources.** No performance figure verified. **Strategy file 9.3 — Debt–equity dislocation after events.** **Who pays you, and why.** Holders forced to sell one side after an event (a downgrade, an index exclusion, a buyout announcement). **Instruments and venues.** CDS, bonds and shares of the same issuer. **Signal.** The event and the direction it moves each claim. **Sizing and execution.** Small; the event may change the capital structure itself. **Costs.** Wide spreads around events. **How it dies.** The event is a leverage shift: equity and credit are both repriced, correctly. **Horizon, capacity, infrastructure.** Days to weeks; event monitoring. **Backtest honestly.** Event dates and filings as known then. **Sources.** No performance figure verified; this chapter: protection sellers open through a buyout lost 98 basis points on average. ## 9.6 Tutorial: same default, two prices **Goal.** Simulate firms with CDS and equity driven by a structural model with planted noise and leverage shifts, trade convergence, and measure the losses when the model is wrong. **End state:** the table and the two figures. 1. **The trades**. `def trades (sim: dict , cfg: CapConfig | None = None ) -> dict : """Convergence trades from day YEAR on, one unit of protection notional each; side +1 sells protection (market spread rich). Returns the closed trades, the book's daily P&L and the number of open trades each day.""" cfg = cfg or CapConfig() E, s, T = sim[" E " ], sim[" spread " ], sim[" E " ].shape[0 ] implied = implied_spread(E, sim[" D_known " ], sim[" vol " ], cfg) gap = np.log(s / implied) shift_days = {} for d, j in sim[" shifts " ]: shift_days.setdefault(j, []).append(d) out, daily, n_open = [], np.zeros(T), np.zeros(T) for i in range (E.shape[1 ]): t = YEAR while t < T - 1 : if abs (gap[t, i]) <= cfg.enter: t += 1 continue side, t0 = (1.0 if gap[t, i] > 0 else -1.0 ), t h = side * float (hedge_ratio(E[t, i], sim[" D_known " ][t, i], sim[" vol " ][t, i], cfg)) cds = equity = 0.0 daily[t] -= cfg.cost * cfg.rpv01 while t < T - 1 : c = side * (s[t, i] / YEAR - cfg.rpv01 * (s[t + 1 , i] - s[t, i])) e = -h * (E[t + 1 , i] - E[t, i]) cds, equity = cds + c, equity + e daily[t + 1 ] += c + e n_open[t + 1 ] += 1 t += 1 if abs (gap[t, i]) < cfg.exit or t - t0 >= cfg.max_days: break daily[t] -= cfg.cost * cfg.rpv01 cds -= 2 * cfg.cost * cfg.rpv01 through = any (t0 < d <= t for d in shift_days.get(i, [])) out.append({" firm " : i, " entry " : t0, " exit " : t, " side " : side, " cds " : cds, " equity " : equity, " pnl " : cds + equity, " through_shift " : through}) t += 1 return {" trades " : out, " daily " : daily, " open " : n_open}` **Listing 9.1.** Entry on the gap, CDS and share legs day by day, exit on convergence or time. code/firm/capstruct/firm_capstruct.py 2. **Results**. `def stats (shift_rate: float = 0.05 ): """Per trade (bp of notional) and for the book (daily P&L per average open trade, annualised).""" _, sim, res = run(shift_rate) p = np.array([x[" pnl " ] for x in res[" trades " ]]) days = np.array([x[" exit " ] - x[" entry " ] for x in res[" trades " ]]) d = res[" daily " ][YEAR:] / res[" open " ][YEAR:].mean() return {" trades " : len (p), " mean_bp " : float (p.mean() * 1e4 ), " lose " : float ((p < 0 ).mean()), " worst_bp " : float (p.min() * 1e4 ), " days " : float (days.mean()), " timeouts " : float ((days >= 180 ).mean()), " sr " : float (d.mean() / d.std() * math.sqrt(YEAR)), " ann " : float (d.mean() * YEAR), " open " : float (res[" open " ][YEAR:].mean()), " shifts " : len (sim[" shifts " ]), " cds_bp " : float (np.mean([x[" cds " ] for x in res[" trades " ]]) * 1e4 ), " equity_bp " : float (np.mean([x[" equity " ] for x in res[" trades " ]]) * 1e4 )} def through_shifts (shift_rate: float = 0.05 ): """Trades open when their firm's leverage shifted, by side: count, mean and worst P&L in bp of notional.""" _, _, res = run(shift_rate) out = {} for side, name in ((1.0 , " sold protection " ), (-1.0 , " bought protection " )): p = np.array([x[" pnl " ] for x in res[" trades " ] if x[" through_shift " ] and x[" side " ] == side]) out[name] = {" n " : len (p), " mean_bp " : float (p.mean() * 1e4 ), " worst_bp " : float (p.min() * 1e4 )} return out` **Listing 9.2.** Per-trade and book statistics, and trades caught by a leverage shift. code/strategies-2/09-capital-structure-arbitrage/python/s2_capstruct.py 3. **Run** `stats()` , `stats(0.2)` , `through_shifts()` , `curve()` and `fig_capstruct.py` . **What to change next.** Give the trader the true barrier and measure how much of the loss was model error; skip firms whose share jumped more than 15% in a day; enter only after the gap has persisted for a month. ## 9.7 Build: capital-structure arbitrage **Purpose.** [Equity-implied spreads](#def-s2-capital-structure-arbitrage-implied) from a first-passage model, hedge ratios, convergence trades and a leverage-shift stress. **Interface.** `CapConfig(…)`, `implied_spread(E, D, sigma_E, cfg)`, `hedge_ratio(E, D, sigma_E, cfg)`, `simulate_firms(cfg)`, `trades(sim, cfg)`. **Rules.** The trader sees debt with a lag and assumes one barrier for all firms; one unit of notional per trade; costs at entry and exit. **Acceptance tests.** `code/firm/capstruct/tests/`: spreads rise with leverage and fall with equity; the survival formula by hand; no mispricing and no model error give no trades. **Stretch.** Barrier uncertainty as in CreditGrades; bond legs; recovery risk. Sources and further reading - F. Yu, “How profitable is capital structure arbitrage?”, *Financial Analysts Journal* 62(5), 2006. - J. Duarte, F. A. Longstaff and F. Yu, “Risk and return in fixed-income arbitrage: nickels in front of a steamroller?”, *Review of Financial Studies* 20(3), 2007. ## 9.8 Exercises **Exercise 9.1 ★.** A five-year survival probability is 0.954 and recovery is 40%. What spread does the chapter’s formula give? **Solution of Exercise 9.1.** $0.6 \times (-\ln 0.954)/5 = 0.00565$: 56.5 basis points. **Exercise 9.2 ★.** The hedge ratio is 0.00067 shares per unit of notional and the share price is 50. How much stock hedges $10 million of protection? **Solution of Exercise 9.2.** $0.00067 \times 10\,000\,000 = 6\,700$ shares, about $0.34 million at 50. **Exercise 9.3 ★.** A protection seller with a risky annuity of 4.5 sees the spread rise from 56 to 99 basis points. What does it lose per unit of notional? **Solution of Exercise 9.3.** $4.5 \times 0.0043 = 0.0194$: 193.5 basis points of notional, before the carry received. **Exercise 9.4 ★★.** Why do more than half the synthetic trades close at the time limit rather than on convergence? **Solution of Exercise 9.4.** Many gaps come from the trader’s wrong barrier, not from a mispricing: the firm’s true $L$ differs from the assumed 0.5, so the market spread sits persistently away from the implied one and the gap never closes. **Exercise 9.5 ★★.** Why does a leveraged buyout hurt a protection seller in both legs? **Solution of Exercise 9.5.** The debt doubles, so the true spread jumps and the protection seller loses on the CDS; the share jumps on the takeover premium, so the short hedge loses too. The model assumed equity and credit move in opposite directions; a buyout moves them the same way. **Exercise 9.6 ★★.** Yu found large losses on single trades and a steady portfolio. How can both be true? **Solution of Exercise 9.6.** Spreads and share prices are loosely correlated day to day, so any one trade’s P&L is noisy and can be large; across hundreds of trades the noise averages out and the mispricing’s reversion remains. **Exercise 9.7 ★★★.** *Coding.* Run `stats(0.0)`. What does removing leverage shifts do to the worst trade and the Sharpe ratio, and what risk remains? **Solution of Exercise 9.7.** The worst trade improves from $-347$ to $-179$ basis points and the Sharpe ratio rises from 1.40 to 1.66; model error in the barrier remains, and 44% of trades still lose. **Exercise 9.8 ★★★.** *Find the flaw.* “The CDS has been 50% above its equity-implied level for six months, so the convergence trade is more certain than ever.” **Solution of Exercise 9.8.** A gap that persists for six months is more likely a wrong input (the barrier, the debt, the vol) or a change in the firm than a mispricing awaiting correction; in the chapter’s simulation the non-converging gaps are the model’s errors. ## 9.9 Problem: Same Default, Two Prices **Problem 9.1.** Weekend problem — capital-structure arbitrage The chapter’s synthetic firms and the public record. **Part I — The model.** 1. Describe the first-passage model and its inputs. 2. Define the [equity-implied spread](#def-s2-capital-structure-arbitrage-implied) and give it at a share price of 50 with debt of 40 and 80. 3. Why do the model’s spreads stay moderate as the share falls? 4. What did Yu use and find? **Part II — The trade.** 5. Define the [cross-asset hedge ratio](#def-s2-capital-structure-arbitrage-hedge) and compute the hedge for $10 million. 6. Describe the entry and exit rules. 7. What does the trader not know about each firm? 8. Where does most of the return come from? **Part III — The results.** 9. Give the book’s statistics with buyouts at 0.05 a firm-year. 10. Why do so many trades close at the time limit? 11. What did Duarte, Longstaff and Yu find? 12. How does the book compare with no buyouts? **Part IV — The verdict.** 13. State the *named result* : the convergence trade’s Sharpe ratio and the losses when leverage shifts. 14. What happened to trades open through a buyout, by side? 15. Why is the barrier the model’s weakest input? 16. What rules would you add? 17. How would you backtest the trade honestly? 18. Which strategy file is most exposed to buyouts? 19. How does this chapter relate to Book 6’s structural model? 20. In one sentence: what does a capital-structure arbitrageur bet on? **Solution of Problem 9.1.** 1. Assets $E + LD$ , asset vol $\sigma_E E/V$ , default when assets touch $LD$ ; inputs: share price, equity vol, debt per share, $L$ and recovery. 2. The CDS spread the model gives from the equity: 55.8 basis points with debt of 40 and 115.4 with 80. 3. The model’s asset vol falls with the share price. 4. CreditGrades, 135 759 daily spreads on 261 companies; large single-trade losses, a portfolio Sharpe ratio like other fixed-income arbitrage. 5. Shares per unit of CDS notional offsetting the CDS’s move with the share price; about 6 700 shares, $0.34 million. 6. Enter at a 40% gap in logs; exit below 5% or after 180 days. 7. Its true barrier and, for three months after a buyout, its debt. 8. The CDS leg: 20.4 of 26.1 basis points a trade. 9. 440 trades, 26.1 basis points each, 48% losing, worst $-347$ , Sharpe ratio 1.40. 10. Their gaps were model error. 11. Strategies needing more intellectual capital earned significant alphas with positively skewed returns. 12. Sharpe ratio 1.66, worst trade $-179$ . 13. **Named result.** The convergence book earns a Sharpe ratio of 1.40 with 48% of trades losing; protection sellers caught by a leveraged buyout lost 98 basis points on average, and with buyouts four times as frequent the Sharpe ratio falls to 0.39. 14. Protection sellers lost 98 basis points on average, buyers earned 220. 15. It cannot be observed and differs by firm; the trade takes it as known. 16. Close non-converging gaps, skip firms in play, update debt from filings, cap position size. 17. Debt and vol as known at the time, costs, trades closed at the limit, buyouts in the sample. 18. Debt–equity dislocation after events. 19. It uses Book 6’s first-passage survival to turn equity into a spread. 20. That equity and credit will agree again about the company’s default risk. ## 9.10 Interview questions **Interview question 9.1 ★ researcher.** How would you get a credit spread from a share price? **Solution of Interview question 9.1.** With a structural model: treat equity as a claim on assets, back out asset value and vol from the share price and equity vol, set a default barrier from the debt, compute survival and convert it into a spread. **Interview question 9.2 ★★ trader.** A company’s CDS is far above its equity-implied level. Walk through the trade and what could go wrong. **Solution of Interview question 9.2.** Sell protection and short shares at the model’s hedge ratio; earn the carry and the convergence. Risks: the model is wrong (barrier, vol, debt), a buyout or recapitalisation moves equity and credit the same way, borrow is recalled, and the gap widens before it closes. **Interview question 9.3 ★★ risk.** How would you stress a capital-structure book? **Solution of Interview question 9.3.** A leverage shift on the largest positions, a market-wide widening of spreads with equity steady, a change in equity vol, and a borrow recall; with the book’s gross notional and the loss on trades that never converge. **Interview question 9.4 ★★ researcher.** How would you tell a mispricing from a model error? **Solution of Interview question 9.4.** A mispricing reverts; a model error persists or tracks an input. Check whether the gap moves with the firm’s own news or filings, test other barriers, and look at bonds and options of the same firm. **Interview question 9.5 ★★ developer.** What data does the book need each day, and which of it arrives late? **Solution of Interview question 9.5.** CDS quotes, share prices, equity vol and borrow daily; debt per share from quarterly filings (late); corporate events as announced. The debt is the input most likely to be stale. **Interview question 9.6 ★★★ researcher.** For assets following a driftless geometric Brownian motion (in log terms, drift $-\sigma^2/2$) and a constant barrier $B$, derive the probability of not touching the barrier by time $t$ from the reflection principle. **Solution of Interview question 9.6.** For $X_t = \ln V_t$ with drift $\nu = -\sigma^2/2$ and $x = \ln(V_0/B)$, the reflection principle for Brownian motion with drift gives $P(\min_{u \le t} X_u > \ln B) = N\big((x + \nu t)/(\sigma\sqrt t)\big) - e^{-2\nu x/\sigma^2}N\big((-x + \nu t)/(\sigma\sqrt t)\big)$, and with $\nu = -\sigma^2/2$ the factor $e^{-2\nu x/\sigma^2}$ is $e^{x} = V_0/B$.