The Vasicek Single-Factor Credit Portfolio Model

Why a Single-Factor Model Is Useful

A portfolio can have a modest expected credit loss and still face a much larger loss in an adverse economic environment. The Vasicek single-factor model separates that shared economic influence from obligor-specific variation. It gives a tractable way to describe the loss distribution of a large, diversified portfolio and to calculate a high-confidence loss quantile.

The model is an idealized one-period credit model. It assumes obligors have the same unconditional default probability and asset correlation, and that their exposures and loss given default are sufficiently small relative to the portfolio for idiosyncratic variation to diversify. It is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: Let $p$ be the one-period unconditional default probability, $\rho$ the common asset correlation, and $g$ the loss given default as a proportion of exposure. Let $Z$ and $\epsilon_i$ be independent standard normal random variables, with the $\epsilon_i$ also mutually independent. Obligor $i$'s standardized latent asset variable is $A_i=\sqrt{\rho}Z+\sqrt{1-\rho}\epsilon_i$. Obligor $i$ defaults if $A_i<\Phi^{-1}(p)$. Here $Z$ is the common systematic factor and $\epsilon_i$ is the obligor-specific factor.

The parameter $\rho$ is the correlation between the latent asset variables $A_i$ and $A_j$ for distinct obligors. It is not, in general, the correlation between their binary default indicators. A low value of $Z$ is adverse in this convention: it lowers latent asset values and increases conditional default probabilities.

Conditional Default Probability and Asymptotic Loss

Given $Z=z$, the latent asset variable for an obligor is normal with conditional mean $\sqrt{\rho}z$ and conditional variance $1-\rho$. Consequently, its conditional default probability is

q(z)=P(\text{default}\mid Z=z) =\Phi\!\left(\frac{\Phi^{-1}(p)-\sqrt{\rho}\,z}{\sqrt{1-\rho}}\right).

Suppose a portfolio contains many obligors with equal exposure and the same $p$, $\rho$, and $g$. Conditional on $Z$, their default indicators are independent and have common mean $q(Z)$. As the number of obligors grows, the average default rate converges to $q(Z)$. Thus the portfolio loss as a proportion of total exposure approaches $gq(Z)$. The systematic factor remains in the portfolio loss; it is the idiosyncratic variation around the conditional rate that vanishes in the limit.

Theorem: In the asymptotic, equally weighted single-factor model, the portfolio loss rate is $L=gq(Z)$. Its unconditional expected loss rate is $E[L]=gp$. For $0<p<1$, $0<\rho<1$, and $g>0$, its loss quantile at confidence level $\alpha$ is $$ \operatorname{VaR}_{\alpha}(L) =g\Phi\!\left( \frac{\Phi^{-1}(p)+\sqrt{\rho}\,\Phi^{-1}(\alpha)} {\sqrt{1-\rho}} \right). $$ For total exposure $A$, the corresponding dollar loss quantile is $A\operatorname{VaR}_{\alpha}(L)$.

The formula describes the asymptotic portfolio loss rate, not the exact distribution of a finite concentrated portfolio. It also assumes a fixed loss given default and does not include migration losses, changing exposure, or uncertainty in the model parameters.

Why the Quantile Formula Has This Form

The conditional default probability $q(z)$ decreases as $z$ increases, so a high loss quantile corresponds to a low systematic-factor value. Since $Z$ is standard normal, the factor value at cumulative probability $1-\alpha$ is $\Phi^{-1}(1-\alpha)=-\Phi^{-1}(\alpha)$. Substituting that factor value into $L=gq(Z)$ gives the stated quantile.

\begin{aligned} \operatorname{VaR}_{\alpha}(L) &=gq\!\left(\Phi^{-1}(1-\alpha)\right)\\ &=g\Phi\!\left( \frac{\Phi^{-1}(p)-\sqrt{\rho}\,\Phi^{-1}(1-\alpha)} {\sqrt{1-\rho}} \right)\\ &=g\Phi\!\left( \frac{\Phi^{-1}(p)+\sqrt{\rho}\,\Phi^{-1}(\alpha)} {\sqrt{1-\rho}} \right). \end{aligned}

The last equality uses the symmetry of the standard normal distribution. The expected loss result follows from the law of total probability: averaging the conditional default probability over $Z$ recovers the unconditional default probability, so $E[q(Z)]=p$ and $E[L]=gE[q(Z)]=gp$.

Set the model inputs.
Identify the unconditional default probability $p$, asset correlation $\rho$, loss given default $g$, total exposure if a dollar result is required, and confidence level $\alpha$.

Find the normal quantiles.
Calculate $\Phi^{-1}(p)$ and $\Phi^{-1}(\alpha)$. For a high loss quantile, use the plus sign between these terms in the VaR formula.

Calculate conditional default probability at the tail.
Evaluate the normal CDF at the standardized argument in the formula.

Convert defaults to loss.
Multiply the conditional default probability by $g$ for a loss rate, and by total exposure as well for a dollar loss quantile.

Worked Example: A 99% Loss Quantile and the Factor at Zero

A hypothetical, highly granular portfolio has total exposure of $100 million, unconditional one-year default probability $p=0.05$, asset correlation $\rho=0.20$, and loss given default $g=0.60$. Find its conditional loss rate when $Z=0$, its expected dollar loss, and its 99% asymptotic dollar loss quantile.

Method. Use the conditional default formula for $Z=0$, the unconditional expected loss formula for the mean, and the Vasicek quantile formula for the 99% loss quantile. The normal quantiles are $\Phi^{-1}(0.05)=-1.644854$ and $\Phi^{-1}(0.99)=2.326348$.

Conditional loss rate at $Z=0$. The standardized conditional-default argument is $(-1.644854-0)/\sqrt{0.80}=-1.8380$, so $q(0)=\Phi(-1.8380)\approx0.0330$. Therefore the conditional loss rate is $0.60(0.0330)=0.0198$, or approximately 1.98%.

Expected dollar loss. The expected loss rate is $gp=0.60(0.05)=0.0300$. Multiplying by exposure gives $100(0.0300)=\$3.00$ million.

99% loss quantile. The numerator in the standardized argument is $-1.644854+\sqrt{0.20}(2.326348)=-1.644854+1.04037=-0.60448$. The denominator is $\sqrt{0.80}=0.894427$, so the argument is approximately $-0.67583$, and the conditional default rate at this tail factor is $\Phi(-0.67583)\approx0.2496$. The loss rate is $0.60(0.2496)\approx0.1498$, giving a dollar quantile of approximately $\$14.98$ million.

Check. The 99% loss quantile is greater than the $3.00 million expected loss, as expected for a high-confidence loss measure. The conditional loss at $Z=0$ is not the unconditional expected loss: $Z=0$ fixes the common factor rather than averaging over it.

Worked Quantile Calculations

Worked Example: High Confidence with a Low Default Probability

A hypothetical diversified portfolio has total exposure of $250 million, $p=0.01$, $\rho=0.25$, and $g=0.45$. Calculate its 99.5% asymptotic dollar loss quantile and compare it with expected loss.

Method. Apply the asymptotic VaR formula, using $\Phi^{-1}(0.01)=-2.326348$ and $\Phi^{-1}(0.995)=2.575829$.

The numerator is $-2.326348+\sqrt{0.25}(2.575829)=-2.326348+1.287915=-1.038433$. The denominator is $\sqrt{0.75}=0.866025$, giving a standardized argument of approximately $-1.1991$. Thus the conditional default rate at the tail factor is $\Phi(-1.1991)\approx0.1152$.

The loss rate quantile is $0.45(0.1152)=0.05184$, or approximately 5.18%. Multiplying by $250 million gives approximately $\$12.96$ million; using the unrounded normal probability gives approximately $\$12.97$ million. Expected loss is $250(0.01)(0.45)=\$1.125$ million.

Check. The tail quantile is substantially above expected loss, reflecting the possibility of a severe common-factor outcome despite the low unconditional default probability. The calculation is an asymptotic portfolio result, not an exact finite-portfolio probability.

Worked Example: Separating Systematic and Expected Loss

A hypothetical portfolio has total exposure of $800 million, $p=0.02$, $\rho=0.10$, and $g=0.40$. Find the 99% asymptotic loss quantile and expected dollar loss. Use $\Phi^{-1}(0.02)=-2.053749$ and $\Phi^{-1}(0.99)=2.326348$.

Method. First calculate the tail conditional default rate from the quantile formula. Then calculate expected loss separately from $gp$; the two measures use different quantities and should not be conflated.

The numerator is $-2.053749+\sqrt{0.10}(2.326348)$. Since $\sqrt{0.10}=0.316228$, the second term is approximately $0.73566$, giving a numerator of approximately $-1.31809$. The denominator is $\sqrt{0.90}=0.948683$, so the argument is approximately $-1.3894$. Therefore the tail conditional default rate is $\Phi(-1.3894)\approx0.0824$.

The 99% loss rate is approximately $0.40(0.0824)=0.03296$, or 3.30%. The dollar loss quantile is approximately $800(0.03296)=\$26.37$ million. Expected loss is $800(0.02)(0.40)=\$6.40$ million.

Check. The tail loss exceeds expected loss, while remaining below total exposure. The conditional default probability at the adverse 1% factor outcome is higher than the 2% unconditional probability, as the formula requires for a low factor value.

Common Exam Traps and Interpretation

A frequent mistake is to treat $p$ as the conditional default probability at $Z=0$. It is the probability averaged over the systematic factor. For example, with $p=0.05$ and $\rho=0.20$, the conditional rate at $Z=0$ is $\Phi(-1.644854/\sqrt{0.80})\approx0.0330$, not 0.05. The expected conditional probability over all values of $Z$ is still 0.05.

A second trap is using the wrong sign for the high-loss factor. In this convention, a bad outcome has low $Z$; the factor value for the $\alpha$-quantile of loss is $-\Phi^{-1}(\alpha)$. Substitution produces a plus sign before $\sqrt{\rho}\Phi^{-1}(\alpha)$ in the VaR formula. A minus sign there would incorrectly reduce the calculated high-confidence loss.

Finally, $\rho$ affects the spread of portfolio loss, not its expected loss under the model: $E[L]=gp$ for every admissible $\rho$. A higher asset correlation generally increases high-tail loss by making the common economic factor more influential. This does not mean that correlation alone determines risk: $p$, $g$, exposure, the confidence level, and the model’s assumptions also matter.

Key takeaway: The Vasicek model turns an unconditional default probability into a factor-dependent conditional rate, $q(Z)$. In a highly granular portfolio, the loss rate approaches $gq(Z)$; inserting the adverse factor quantile yields the asymptotic portfolio loss quantile. Keep the conditional rate, expected loss, and tail loss distinct.

Check Your Understanding

For each problem, use the asymptotic single-factor assumptions stated or implied. Round probabilities to four decimals and money to the nearest cent when a dollar amount is requested.

A hypothetical portfolio has $p=0.04$, $\rho=0.25$, and $g=0.50$. Find the conditional loss rate at $Z=0$. Use $\Phi^{-1}(0.04)=-1.750686$. (Answer: approximately 1.08%)
A diversified portfolio has $p=0.02$, $\rho=0.16$, and $g=0.40$. Find its 99% asymptotic loss rate. Use $\Phi^{-1}(0.02)=-2.053749$ and $\Phi^{-1}(0.99)=2.326348$. (Answer: approximately 4.41%)
A hypothetical portfolio has total exposure of $120 million, $p=0.03$, $\rho=0.20$, and $g=0.50$. Find its 95% asymptotic dollar loss quantile. Use $\Phi^{-1}(0.03)=-1.880794$ and $\Phi^{-1}(0.95)=1.644854$. (Answer: approximately $6.02 million)
A portfolio has $p=0.05$, $\rho=0.36$, and $g=0.70$. Find its conditional loss rate at $Z=-1$, using $\Phi^{-1}(0.05)=-1.644854$. (Answer: approximately 6.71%)
In the asymptotic model, a portfolio has total exposure $50 million, $p=0.03$, and $g=0.40$, but $\rho=0$. Find its 99% dollar loss quantile. (Answer: $0.60 million)

What You'll Learn