Expected and Unexpected Credit Loss Calculations

Expected Loss and Unexpected Loss

Credit loss depends on whether a borrower defaults, how much is exposed when default occurs, and what portion of that exposure is not recovered. Its average is useful for pricing, budgeting, and forecasting, but the average alone does not describe how widely actual losses may vary. Credit risk analysis therefore distinguishes expected loss from unexpected loss, with the latter defined relative to a chosen measure of variability or tail risk.

Unless stated otherwise, the examples use a one-year horizon and amounts measured at the end of that horizon. They treat exposure as fixed over the period and do not discount losses. A probability of default is the probability that the obligor defaults during the horizon; loss given default is the fraction of exposure lost if default occurs; and exposure at default is the amount exposed when default occurs. Recovery assumptions should be consistent with how loss given default is defined, including whether recovery costs are deducted.

Definition: Let $p$ be the probability of default, $g$ the loss given default as a fraction, and $e$ the exposure at default. For a single obligor with fixed $g$ and $e$, credit loss $X$ is $ge$ if default occurs and zero otherwise. Its expected loss is $E[X]=pge$. Here $0\leq p,g\leq1$ and $e\geq0$.

The product $ge$ is the amount lost conditional on default, not the expected loss. Multiplying by $p$ accounts for the possibility that default does not occur. When exposure or loss given default can vary, the expected loss is instead the probability-weighted average of the possible losses; one should not use $pge$ with an arbitrary average unless its assumptions are justified.

Fixed Exposure and Severity

For fixed exposure and fixed loss given default, the loss distribution has only two outcomes: zero with probability $1-p$, and $ge$ with probability $p$. Its variance and standard deviation follow directly from those two possible values.

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

The variance formula can be derived from the definition $\operatorname{Var}(X)=E[X^2]-(E[X])^2$. Since $X^2$ is $(ge)^2$ on default and zero otherwise, $E[X^2]=p(ge)^2$. Subtracting the square of the mean gives

\operatorname{Var}(X) =p(ge)^2-(pge)^2 =(ge)^2(p-p^2) =p(1-p)(ge)^2.

In this setting, standard deviation is one possible measure of unexpected loss. It is not the only one. Another common choice is a specified loss quantile less expected loss. These are different quantities, and a calculation must state which one it reports.

Worked Example: Fixed Loss Given Default

A hypothetical insurer has a one-year exposure at default of $1,500,000 to a commercial borrower. The probability of default is 2%, and loss given default is fixed at 40%. Calculate expected loss, the standard deviation of loss, and 99% value at risk. Also calculate value at risk less expected loss.

Method. Model $X$ as zero if there is no default and $ge$ if default occurs. The loss conditional on default is

$$ ge=0.40(1{,}500{,}000)=600{,}000. $$

Expected loss. With $p=0.02$,

$$ E[X]=0.02(600{,}000)=12{,}000. $$

Standard deviation. The formula gives

\operatorname{SD}(X) =600{,}000\sqrt{0.02(0.98)} =600{,}000(0.14) =84{,}000.

99% value at risk. Define $\operatorname{VaR}_{\alpha}(X)$ as the smallest $x$ for which $P(X\leq x)\geq\alpha$. Here $P(X=0)=0.98$, which is below 0.99, while $P(X\leq600{,}000)=1$. Therefore $\operatorname{VaR}_{0.99}(X)=600{,}000$, and value at risk less expected loss is $600{,}000-12{,}000=$588{,}000$.

The check is that the two loss probabilities, 0.98 and 0.02, sum to 1, and the probability-weighted loss is $12,000. The standard-deviation measure of unexpected loss is $84,000, whereas the stated 99% quantile measure less expected loss is $588,000. They differ because they measure different features of the loss distribution.

Uncertain Loss Given Default

Loss given default may itself be uncertain, for example because recoveries depend on the value of collateral or the cost and timing of recovery. When conditional severity has a discrete distribution, calculate the unconditional loss distribution by multiplying each conditional severity probability by the probability of default. The remaining probability is assigned to zero loss if there is no default.

Result: If default has probability $p$, exposure is fixed at $e$, and conditional loss given default $G$ has specified possible values, then $X=0$ on no default and $X=eG$ on default. Calculate $E[X]$ and $\operatorname{Var}(X)$ from the resulting unconditional distribution. In particular, $E[X]=p e E[G\mid\text{default}]$ when the conditional mean exists and exposure is fixed.

Worked Example: Two Possible Recoveries

A hypothetical lender has a one-year exposure at default of $800,000 and a 4% probability of default. Conditional on default, loss given default is 40% with probability 70% and 70% with probability 30%. Find expected loss and the standard deviation of loss. Then find the 99% value at risk and value at risk less expected loss.

Method. First convert each conditional loss-given-default outcome into a dollar loss. At 40% severity the loss is $0.40(800{,}000)=$320{,}000$; at 70% severity it is $0.70(800{,}000)=$560{,}000$. The unconditional probabilities are

P(X=0)=0.96,\quad P(X=320{,}000)=0.04(0.70)=0.028,\quad P(X=560{,}000)=0.04(0.30)=0.012.

These probabilities sum to $0.96+0.028+0.012=1$. The expected loss is

$$ E[X]=0.028(320{,}000)+0.012(560{,}000) =8{,}960+6{,}720 =$15{,}680. $$

The second moment is

E[X^2] =0.028(320{,}000^2)+0.012(560{,}000^2) =2{,}867{,}200{,}000+3{,}763{,}200{,}000 =6{,}630{,}400{,}000.

Thus

\operatorname{Var}(X) =6{,}630{,}400{,}000-15{,}680^2 =6{,}384{,}537{,}600, \qquad \operatorname{SD}(X)=79{,}903.30.

The cumulative probability through the $320,000 loss is $0.96+0.028=0.988$, which is less than 0.99. The cumulative probability through $560,000 is 1, so $\operatorname{VaR}_{0.99}(X)=$560{,}000$. Value at risk less expected loss is $560{,}000-15{,}680=$544{,}320$. The standard deviation is rounded to the nearest cent; the dollar loss quantile is exact.

Scenario-Based Loss Measures

A specified scenario distribution can be used even when the loss outcomes do not arise from a fixed-severity default model. Assign each scenario a probability and a loss, then calculate the mean, variance, and requested quantile directly. Such a calculation is only as representative as its stated scenarios and probabilities; it does not, by itself, establish that those probabilities are appropriate for every purpose.

Worked Example: Quantifying a Scenario Tail

A hypothetical credit exposure has the following one-year loss scenarios: no loss with probability 97%, a $400,000 loss with probability 2%, and a $700,000 loss with probability 1%. Calculate expected loss, standard deviation, 99.5% value at risk, and value at risk less expected loss.

Method. The scenario probabilities sum to $0.97+0.02+0.01=1$. Expected loss is

$$ E[X]=0.97(0)+0.02(400{,}000)+0.01(700{,}000) =8{,}000+7{,}000 =$15{,}000. $$

The second moment and variance are

E[X^2]=0.02(400{,}000^2)+0.01(700{,}000^2) =3{,}200{,}000{,}000+4{,}900{,}000{,}000 =8{,}100{,}000{,}000,

\operatorname{Var}(X)=8{,}100{,}000{,}000-15{,}000^2 =7{,}875{,}000{,}000, \qquad \operatorname{SD}(X)=88{,}741.20.

At a loss of $400,000, cumulative probability is $0.97+0.02=0.99$, which is less than 0.995. At $700,000 it is 1. Therefore $\operatorname{VaR}_{0.995}(X)=$700{,}000$, and value at risk less expected loss is $700{,}000-15{,}000=$685{,}000$. The standard deviation is rounded to the nearest cent.

Interpretation and Common Traps

Expected loss describes the probability-weighted average, not a guaranteed loss or a cap on loss. Unexpected loss requires an explicit convention. If it means standard deviation, report $\operatorname{SD}(X)$; if it means value at risk less expected loss at confidence level $\alpha$, report $\operatorname{VaR}_{\alpha}(X)-E[X]$. State the chosen definition and confidence level rather than using “unexpected loss” as though it had only one universal numerical formula.

Applying severity twice. If loss given default is already the fraction not recovered, do not multiply by the unrecovered fraction again. Calculate $ge$, not $g(1-g)e$.
Omitting default probability. The amount $ge$ is loss conditional on default. Expected loss is $pge$.
Confusing variance and standard deviation. Variance is measured in squared dollars; standard deviation is measured in dollars. Take the square root only when reporting standard deviation.
Misreading a quantile boundary. For a discrete distribution, check the cumulative probability at each loss outcome. A 99% quantile need not be the loss outcome at exactly 99% probability.
Combining incompatible inputs. Exposure, probability of default, and loss severity must refer to the same time horizon and consistent recovery assumptions.

For example, using the first worked example, the incorrect calculation $0.40(1-0.40)(1{,}500{,}000)(0.02)=$7{,}200$ applies a recovery adjustment twice. The stated 40% is already loss given default; the correct calculation is $0.40(1{,}500{,}000)(0.02)=$12{,}000$.

Key takeaway: Set out the horizon and loss model first. For fixed exposure and severity, expected loss is probability of default times loss given default times exposure at default. For unexpected loss, calculate the distributional measure requested and distinguish it from expected loss.

Check Your Understanding

For each problem, use the stated one-year assumptions and identify the loss measure being calculated.

A hypothetical insurer has exposure at default of $1,250,000, probability of default 1.2%, and fixed loss given default 55%. Find expected loss. (Answer: $8,250.)
A lender has exposure at default of $600,000, probability of default 3%, and fixed loss given default 40%. Find the standard deviation of loss. (Answer: $40,940.87.)
A single exposure produces a loss of $500,000 if default occurs and zero otherwise. Default probability is 2%. Find expected loss and standard deviation. (Answer: expected loss $10,000; standard deviation $70,000.)
A hypothetical exposure has losses of $0, $150,000, and $400,000 with probabilities 94%, 4%, and 2%, respectively. Find expected loss, standard deviation, and 97% value at risk less expected loss. (Answer: expected loss $14,000; standard deviation $62,482.00; 97% value at risk less expected loss $136,000.)
A $2,000,000 exposure has a 2.5% default probability. Conditional on default, loss given default is 30% with probability 60% and 60% with probability 40%. Find expected loss and standard deviation. (Answer: expected loss $21,000; standard deviation $139,136.62.)

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