15Portfolio Credit

In early 2005 a popular trade sold protection on the equity tranche of the iTraxx Europe index, the first 3% of losses on 125 names, and hedged it against market-wide spread moves by buying protection on the index; the Bank for International Settlements put its carry at 300 to 400 basis points a year. In May, after the downgrades of General Motors and Ford (chapter 14), the correlation implied by equity tranche prices fell sharply, the equity tranche’s spread widened, and unwinds pushed the spreads of the mezzanine tranches down, at times in the opposite direction to the index. Positions hedged by every model their owners had lost on both legs in the same weeks. This chapter builds those models: the dependence of defaults through a copula, the loss distribution of a pool, the prices and risks of tranches, and why a correlation implied from a tranche is a quoting convention whose moves are a risk of their own. One Quant Book 2, chapter 24 introduced tranches, the large pool and base correlation; here the pool is finite, the tranches have a term structure, and the hedges are measured.

15.1 Default correlation and copulas

A copula (One Quant Book 4, chapter 15) joins the marginal distributions of the default times, each given by a name’s survival curve from chapter 13, into a joint distribution. The market’s choice is the simplest.

The asset correlation is not the default correlation. With a five-year default probability of 4.08% (a spread of 50 basis points, recovery 40%), an asset correlation of 25% gives a default correlation of 7.0%: rare events of two names coincide less often than their latent variables co-move.

15.2 The one-factor Gaussian copula and the large pool

Conditional on the common factor $Z = z$ the names are independent, each defaulting by $t$ with probability

For a finite pool, the distribution of the number of defaults given $z$ follows by adding names one at a time: with $P_k$ the probability of $k$ defaults among the first names, adding name $i$ gives $P_k \leftarrow P_k(1-p_i)+P_{k-1}p_i$, the recursion of Andersen, Sidenius and Basu (2003). Integrating over $z$ by quadrature gives the unconditional loss distribution (Figure 15.1).

def default_count_given_z(pz: np.ndarray) -> np.ndarray:
    """Distribution of the number of defaults for each z row of independent default probabilities."""
    nz, n = pz.shape
    dist = np.zeros((nz, n + 1))
    dist[:, 0] = 1.0
    for i in range(n):
        q = pz[:, i:i + 1]
        dist[:, 1:] = dist[:, 1:] * (1 - q) + dist[:, :-1] * q
        dist[:, 0] *= 1 - q[:, 0]
    return dist


def loss_distribution(p: np.ndarray, rho: float) -> np.ndarray:
    """Unconditional distribution of the number of defaults (Gaussian copula, quadrature over Z)."""
    return _W @ default_count_given_z(conditional_pd(p, rho, _Z))

The large-pool limit of One Quant Book 2 is fast and close for many names: the equity tranche’s five-year expected loss is 51.6% of its notional in the limit against 49.6% for 125 names at $\rho = 25\%$. The finite pool matters for the equity tranche and for pools whose names differ, which is what the recursion is for.

15.3 Pricing tranches: recursion, compound and base correlation

A tranche $[a,d]$ loses $\min(\max(L_t-a,0),d-a)/(d-a)$ of its notional when the pool has lost $L_t$. Its expected loss at each premium date, $\mathrm{EL}(t_k)$, gives the legs as for a default swap: the premium is paid on the expected outstanding notional, $\sum_k\delta_kP(0,t_k)\bigl(1-\tfrac12(\mathrm{EL}(t_{k-1})+\mathrm{EL}(t_k))\bigr)$, and the protection pays $\sum_kP(0,t_k)\bigl(\mathrm{EL}(t_k)-\mathrm{EL}(t_{k-1})\bigr)$. The equity tranche trades as an upfront plus 500 basis points running; the others trade at a running spread.

The compound correlations of the synthetic quotes are 15.0% for the equity, 12.9%, 18.4%, 25.2% and 50.0% for the tranches above 6%, and two values for the mezzanine, 2.1% and 89.9%. A mezzanine is long correlation at low correlation (more correlation spreads losses into it) and short at high correlation (they jump past it), so its spread is humped in $\rho$ (Figure 15.2). A quote above the hump’s 374 basis points, near $\rho = 29\%$, has no compound correlation at all. This is why the market moved to the base correlation of One Quant Book 2, defined on base tranches $[0,d]$ whose value is monotone in correlation; a mezzanine $[a,d]$ is then priced from two base correlations, $\mathrm{EL}_{a,d} = (d\,\mathrm{EL}_{0,d}(\rho_d)-a\,\mathrm{EL}_{0,a}(\rho_a))/(d-a)$.

15.4 Beyond the Gaussian: fat tails and tail dependence

15.5 May 2005 and 2008

The two episodes stress different parts of the model. In May 2005 the risk was idiosyncratic: one or two names blowing out hit the equity tranche, which absorbs the first losses, far more than an index-sized hedge, and the correlation implied by equity prices fell; the delta, computed for a parallel move with correlation held, hedged neither. In 2008 the risk was systematic: the factor moved, the senior tranches that the Gaussian copula had priced as nearly riskless lost value, and the correlations implied by senior tranches rose towards one.

15.6 Tutorial: tranches by recursion

15.7 Build: portfolio credit

15.8 Exercises

15.9 Problem: Long Equity, Short Mezzanine

15.10 Interview questions