Unexpected Credit Loss and Loss Volatility

Why Credit Loss Volatility Matters

Expected credit loss describes the average loss across many comparable outcomes or repeated horizons. It does not describe how much an individual outcome may differ from that average. For a single exposure, the loss is often zero if the borrower does not default and positive if the borrower defaults. That two-outcome structure creates volatility even when the loss amount conditional on default is known exactly.

This tutorial derives the variance and standard deviation of that loss under a Bernoulli default assumption. It also considers the case where loss given default can vary. Here, unexpected loss means the standard deviation of loss around its expected value. Some capital frameworks define unexpected loss differently, for example using a specified tail measure; those measures require additional distributional analysis.

Definition: For a specified horizon, let $B$ be a default indicator: $B=1$ if the exposure defaults during the horizon and $B=0$ otherwise. Under the Bernoulli default assumption, $P(B=1)=p$ and $P(B=0)=1-p$. If the loss conditional on default is a fixed amount $L$, the realized loss is $X=BL$. The expected loss is $E[X]$, the loss variance is $\operatorname{Var}(X)$, and the standard deviation $\sqrt{\operatorname{Var}(X)}$ is the volatility measure used here for unexpected loss.

In a simple exposure model, $L$ can be calculated as exposure at default multiplied by loss given default. For example, an exposure at default of $400,000 and a loss-given-default rate of 25% imply a fixed conditional loss of $100,000. This tutorial treats the exposure and loss-given-default rate as fixed when using the fixed-$L$ formula. It assumes no loss if default does not occur and does not include other losses, such as collection costs, unless they are included in $L$.

Fixed Loss Given Default

With a fixed loss amount $L$, $X$ has two possible values: $L$ with probability $p$, and zero with probability $1-p$. Its expected value is $pL$. To calculate variance, use the definition $\operatorname{Var}(X)=E[X^2]-(E[X])^2$. Since the squared loss is $L^2$ on default and zero otherwise, $E[X^2]=pL^2$.

E[X]=pL,\qquad \operatorname{Var}(X)=pL^2-(pL)^2=p(1-p)L^2,\qquad \operatorname{SD}(X)=L\sqrt{p(1-p)}.

The factor $p(1-p)$ reflects the uncertainty in whether default occurs. If $p=0$ or $p=1$, the outcome is certain and the variance is zero. For a fixed $L$, the variance is largest at $p=0.5$, because $p(1-p)$ reaches its maximum there. The standard deviation has the same units as the loss, while variance has squared currency units.

Worked Example: Fixed Loss With a Low Default Probability

A hypothetical lender has a $500,000 exposure at default. For the one-year horizon, assume a default probability of 0.0400 and a fixed loss-given-default rate of 50%. Calculate expected loss, variance, and unexpected loss as standard deviation. Report money to the cent.

Method and assumptions. The exposure and loss-given-default rate are fixed, and loss is zero if there is no default. The conditional loss is $L=\$500{,}000(0.50)=\$250{,}000$. The loss is therefore $X=BL$, with $p=0.0400$.

Expected loss. $E[X]=pL=0.0400(\$250{,}000)=\$10{,}000.00$.

Variance. $p(1-p)=0.0400(0.9600)=0.0384$, and $L^2=(\$250{,}000)^2=62{,}500{,}000{,}000$ dollars squared. Thus $\operatorname{Var}(X)=0.0384(62{,}500{,}000{,}000)=2{,}400{,}000{,}000$ dollars squared.

Unexpected loss as standard deviation. $\operatorname{SD}(X)=\sqrt{2{,}400{,}000{,}000}=\$48{,}989.79$, rounded to the nearest cent.

Check and common mistake. The expected loss is 4% of the conditional loss, as it should be. The standard deviation is not the same as the expected loss: here it is $48,989.79 rather than $10,000.00. Calling the conditional loss itself “unexpected loss” would also be incorrect under the definition used here; the volatility measure is calculated around the expected loss.

When Loss Given Default Is Variable

A fixed conditional loss may be too restrictive. Recovery amounts can vary, for example, so the loss conditional on default can take several values. Let $Z$ denote the loss amount conditional on default, with conditional mean $m=E[Z\mid B=1]$ and conditional variance $s^2=\operatorname{Var}(Z\mid B=1)$. The realized loss is zero when $B=0$ and is $Z$ when $B=1$.

The total variance can be derived by conditioning on whether default occurs. If there is no default, the loss is zero and its conditional variance is zero. If default occurs, the conditional loss variance is $s^2$. The conditional means are zero without default and $m$ with default. Thus the expected conditional variance is $ps^2$, and the variance of the conditional means is $p(1-p)m^2$.

\begin{aligned} \operatorname{Var}(X) &=E[\operatorname{Var}(X\mid B)] +\operatorname{Var}(E[X\mid B])\\ &=ps^2+p(1-p)m^2. \end{aligned}

The same result follows directly from moments. Since loss is zero without default, $E[X]=pm$. Conditional on default, $E[Z^2\mid B=1]=s^2+m^2$, so $E[X^2]=p(s^2+m^2)$. Subtracting the square of the mean gives $p(s^2+m^2)-(pm)^2=ps^2+p(1-p)m^2$. This formula separates two sources of loss variance: variation in severity conditional on default, $ps^2$, and uncertainty about default, $p(1-p)m^2$. When severity is fixed, $s^2=0$, and the formula reduces to the fixed-loss result.

Worked Example: Variable Loss Given Default

A hypothetical $500,000 exposure has a one-year default probability of 0.0200. Conditional on default, loss given default is 20% of exposure with probability 0.5000 and 60% with probability 0.5000. Calculate the conditional mean severity, conditional severity variance, expected loss, total loss variance, and standard deviation. Report money to the cent and variance in dollars squared.

Method and assumptions. Loss is zero without default. Conditional on default, the loss $Z$ is either $\$500{,}000(0.20)=\$100{,}000$ or $\$500{,}000(0.60)=\$300{,}000$, each with probability 0.5000.

Conditional mean severity. $m=0.5000(\$100{,}000)+0.5000(\$300{,}000)=\$200{,}000.00$.

Conditional severity variance. The two deviations from the conditional mean are $-\$100{,}000$ and $+\$100{,}000$. Therefore $s^2=0.5000(100{,}000^2)+0.5000(100{,}000^2)=10{,}000{,}000{,}000$ dollars squared.

Expected loss. $E[X]=pm=0.0200(\$200{,}000)=\$4{,}000.00$.

Total variance. The conditional-severity component is $ps^2=0.0200(10{,}000{,}000{,}000)=200{,}000{,}000$ dollars squared. The default-uncertainty component is $p(1-p)m^2=0.0200(0.9800)(200{,}000^2)=784{,}000{,}000$ dollars squared. Hence $\operatorname{Var}(X)=200{,}000{,}000+784{,}000{,}000=984{,}000{,}000$ dollars squared.

Standard deviation and check. $\operatorname{SD}(X)=\sqrt{984{,}000{,}000}=\$31{,}368.77$, rounded to the nearest cent. As a moment check, the conditional second moment is $s^2+m^2=50{,}000{,}000{,}000$; therefore $E[X^2]=0.0200(50{,}000{,}000{,}000)=1{,}000{,}000{,}000$. Subtracting $E[X]^2=4{,}000^2=16{,}000{,}000$ gives the same variance, $984{,}000{,}000$.

Common mistake avoided. One might calculate $p s^2$ and then add $p(1-p)m^2$. Both terms are needed. Omitting the second term ignores uncertainty about whether default occurs; multiplying the second term by $p$ again would also be wrong.

Interpreting Unexpected Loss

Expected loss and standard deviation answer different questions. Expected loss is the probability-weighted average loss. Standard deviation measures the spread of loss outcomes around that average, under the specified model. A positive standard deviation does not mean that this amount is the loss expected in addition to expected loss in every period. Nor does it specify a particular percentile or guarantee a maximum loss.

The distinction is especially useful when comparing exposures. Two exposures can have the same expected loss but different default probabilities and loss amounts. Since a large conditional loss can make outcomes more variable, equal expected losses do not imply equal standard deviations. The Bernoulli model captures that single-exposure volatility, but it does not by itself describe dependence among exposures or a portfolio’s aggregate loss distribution.

Worked Example: Equal Expected Loss, Different Volatility

Two hypothetical, fixed-severity exposures are assessed over the same one-year horizon. Exposure A has a conditional loss of $300,000 and default probability 0.0200. Exposure B has a conditional loss of $150,000 and default probability 0.0400. For each, calculate expected loss and standard deviation, then compare. Report money to the cent.

Method and assumptions. For each exposure, use $X=BL$ with fixed conditional loss and Bernoulli default. No loss occurs without default.

Exposure A. $E[X_A]=0.0200(\$300{,}000)=\$6{,}000.00$. Its variance is $0.0200(0.9800)(300{,}000^2)=0.0196(90{,}000{,}000{,}000)=1{,}764{,}000{,}000$ dollars squared. Therefore $\operatorname{SD}(X_A)=\sqrt{1{,}764{,}000{,}000}=\$42{,}000.00$.

Exposure B. $E[X_B]=0.0400(\$150{,}000)=\$6{,}000.00$. Its variance is $0.0400(0.9600)(150{,}000^2)=0.0384(22{,}500{,}000{,}000)=864{,}000{,}000$ dollars squared. Therefore $\operatorname{SD}(X_B)=\sqrt{864{,}000{,}000}=\$29{,}393.88$.

Comparison and check. Both expected losses are $6,000.00, but Exposure A has the larger standard deviation by $\$42{,}000.00-\$29{,}393.88=\$12{,}606.12$. Exposure A has a lower probability of default but a larger loss if default occurs. The calculation shows why expected loss alone does not determine loss volatility. The method avoids treating equal expected losses as equal risk.

Common Exam Traps

Reporting expected loss as unexpected loss. The expected loss is $pL$; under the fixed-severity Bernoulli model, standard deviation is $L\sqrt{p(1-p)}$. They are generally different quantities.
Dropping the nondefault probability. The variance is $p(1-p)L^2$, not $pL^2$. The latter is the second moment $E[X^2]$, before subtracting the squared mean.
Using only one variance component for variable severity. With variable conditional severity, both $ps^2$ and $p(1-p)m^2$ are required.
Confusing variance units with standard-deviation units. Variance is in dollars squared; its square root is in dollars.
Interpreting standard deviation as a tail threshold. Standard deviation is a measure of spread, not a specified percentile or a maximum possible loss.

Key takeaway: For fixed loss $L$ conditional on default, expected loss is $pL$ and the standard deviation is $L\sqrt{p(1-p)}$. When conditional severity varies, total variance is $ps^2+p(1-p)m^2$. State which model applies before calculating, and keep expected loss distinct from volatility.

Check Your Understanding

Use the stated single-exposure assumptions. Unless a question says otherwise, loss is zero without default and money answers are rounded to the nearest cent.

A hypothetical exposure has EAD $700,000, fixed LGD 30%, and default probability 0.0200 over the horizon. Find the conditional loss, expected loss, variance, and standard deviation. (Answer: conditional loss $210,000.00; expected loss $4,200.00; variance 864,360,000 dollars squared; standard deviation $29,400.00)
A $500,000 exposure has default probability 0.0500. Conditional on default, its loss is $80,000 or $200,000 with equal probability. Find expected loss and standard deviation. (Answer: expected loss $7,000.00; standard deviation $33,331.67)
Exposure A has fixed conditional loss $300,000 and default probability 0.0200. Exposure B has fixed conditional loss $150,000 and default probability 0.0400. Compare expected loss and standard deviation for the two exposures. (Answer: each expected loss is $6,000.00; standard deviations are $42,000.00 for A and $29,393.88 for B)
A hypothetical exposure has fixed conditional loss $180,000 and default probability 0.0500. Calculate its expected loss, variance, and standard deviation. (Answer: expected loss $9,000.00; variance 1,539,000,000 dollars squared; standard deviation $39,230.09)
A fixed-severity exposure has conditional loss $300,000 and expected loss $9,000 over the horizon. Infer the default probability, then calculate the standard deviation. (Answer: default probability 0.0300; standard deviation $51,176.17)

What You'll Learn