Credit Risk

Study guides: Exam 9, Exam CP-341, Exam INV-101, Exam INV-201

Credit Risk is the risk of loss because a counterparty fails to pay what it owes (default) or because its credit quality deteriorates, lowering the value of the claim before any default occurs. An insurer's credit exposures are its bonds and loans, structured securities, reinsurance recoverables, premium receivables from agents and brokers, and derivative counterparties.

$E[\text{Credit loss}] = PD \times LGD \times EAD$

$PD$ is the probability of default over the horizon, $LGD = 1 - \text{recovery rate}$ the share of the exposure lost if default occurs, and $EAD$ the exposure at default, net of any collateral.
Spreads pay for more than expected loss. The annual expected loss on a bond is roughly $PD \times LGD$ , but observed credit spreads are several times larger: investors are also paid for defaults that cluster in recessions (systematic risk), for illiquidity, and for downgrade risk.
Prepayment is the opposite behaviour by a borrower — paying back early, usually when rates fall — and is grouped with default because it also changes the cash flows the insurer receives; see Prepayment Risk.
Loss before default. A downgrade widens the spread and cuts the price by about $-D \times \Delta s$ for spread duration $D$ . Credit Rating Migration matrices model movements between grades, so a portfolio's credit risk includes mark-to-market loss, not only default.
Correlation is the portfolio risk. Defaults cluster, so the loss distribution of a credit portfolio is skewed with a heavy tail whose size is set by default correlation. This is the central lesson of Structured Finance: senior tranches of pooled credit are safe only while defaults stay close to independent, and their losses fall in exactly the economic states where investors are worst off.
Managing it. Diversification and concentration limits by issuer, sector and reinsurer (Concentration Risk); rating floors; collateral; credit derivatives. Regulators charge capital for it — the U.S. Risk-Based Capital formula and, in Canada, the Credit Risk Margin of the MCT. Together with Interest Rate Risk it is the main financial risk of a bond portfolio.

Expected Loss versus Spread on a Corporate Bond An insurer holds $\$50$ M of BBB corporate bonds. The one-year probability of default is $0.25\%$ , the expected recovery $40\%$ , and the bonds yield $1.30\%$ over Treasuries. Compare the expected credit loss with the spread earned.
Answer
 $\begin{align*} E[\text{loss}] &= 0.0025 \times 0.60 \times \$50\text{M} \\ &= \$75{,}000 \\ \text{Spread income} &= 0.0130 \times \$50\text{M} \\ &= \$650{,}000 \end{align*}$ 
The expected loss is $0.15\%$ a year, so $\$575{,}000$ of the spread is compensation for bearing credit risk — systematic default, downgrade and illiquidity — rather than for expected default. Booking the full spread as profit overstates the portfolio's risk-adjusted return.

Default Correlation and the Joint Tail Two counterparties each have a one-year default probability of $5\%$ . Find the probability that both default if defaults are (a) independent, (b) correlated with default correlation $0.2$ . Also find the distribution of the number of defaults.
Answer For two default indicators with common probability $p$ and correlation $\rho$ , $P(\text{both}) = p^2 + \rho\,p(1-p)$ .
 $\begin{align*} P_{\text{indep}}(\text{both}) &= 0.05^2 \\ &= 0.25\% \\ P_{\rho}(\text{both}) &= 0.0025 + 0.2(0.05)(0.95) \\ &= 1.20\% \end{align*}$ 
Number of defaults:

Independent: $0$ w.p. $90.25\%$ , $1$ w.p. $9.50\%$ , $2$ w.p. $0.25\%$

Correlated: $0$ w.p. $91.2\%$ , $1$ w.p. $7.6\%$ , $2$ w.p. $1.2\%$

The expected number of defaults is $0.10$ in both cases, but the chance of losing both counterparties is almost five times higher. Expected loss is blind to correlation; the tail — what capital, a senior tranche, or a two-reinsurer panel is exposed to — is not.