Expected Loss Across Multiple Exposures

Why Expected Loss Is Added Across Exposures

A portfolio can contain many credit positions with different balances, default probabilities, and recoveries. A single portfolio-wide default probability or recovery rate can conceal these differences. A transparent calculation first estimates expected loss for each exposure using its own inputs, then adds those estimates.

The result is a mean loss over the specified horizon. It is useful for summarizing expected credit costs, but it does not say how much losses might vary or how severe a tail outcome could be. The calculation below focuses on expected loss, aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: For exposure $i$, let $EAD_i$ be its exposure at default over the chosen horizon, $p_i$ its probability of default over that same horizon, and $R_i$ its recovery rate if default occurs. Its loss given default is $LGD_i=1-R_i$. If $EAD_i$ and $LGD_i$ are fixed, its expected loss is $EL_i=EAD_i p_i LGD_i$. For $n$ exposures, portfolio expected loss is $EL_{\text{portfolio}}=\sum_{i=1}^{n} EL_i$.

All three inputs must refer to compatible definitions and time periods. For example, a one-year default probability should not be combined with an exposure estimate for a different horizon without adjusting the inputs. Recovery is expressed as a fraction of exposure at default, so an assumed recovery rate of 40% implies an LGD of 60%, not 40%.

The basic calculation assumes the stated exposure and LGD are fixed for the purpose of the estimate. The default event can be represented by an indicator $I_i$, equal to 1 if exposure $i$ defaults during the horizon and 0 otherwise. Its loss is $L_i=EAD_i LGD_i I_i$, with $P(I_i=1)=p_i$. Thus $E[L_i]=EAD_i LGD_i p_i$.

Why the Portfolio Formula Holds

Let total portfolio loss be the sum of the losses on its exposures. The expected value of a finite sum is the sum of the expected values. This property does not require the exposure losses, or default events, to be independent.

L_{\text{portfolio}}=\sum_{i=1}^{n}L_i \quad\Longrightarrow\quad E[L_{\text{portfolio}}] =E\left[\sum_{i=1}^{n}L_i\right] =\sum_{i=1}^{n}E[L_i] =\sum_{i=1}^{n}EAD_i p_i LGD_i.

The middle equality is linearity of expectation. Dependence can change the range and likelihood of possible combined outcomes, but it does not change this expected-loss sum when the exposure-level expected losses are correctly specified. If exposure at default or recovery is random, the calculation must instead respect their conditional values. In general, for a default indicator $I_i$,

E[L_i]=p_i E[EAD_i LGD_i\mid I_i=1].

The conditional expectation is a product of the two conditional means only under additional conditions, such as conditional independence of exposure and LGD. Otherwise, calculate the conditional expected product rather than multiplying separate averages.

Worked Example: Summing Different Exposure-Level Estimates

A hypothetical lender estimates one-year expected credit loss on four positions. The exposure at default is fixed for this estimate. Assume Bernoulli default indicators with the stated one-year probabilities; recoveries are fixed conditional on default. No independence assumption between defaults is made. There is no interest-rate or mortality assumption in this calculation.

Position A. $EAD_A=\$1{,}200{,}000$, $p_A=0.020$, and $R_A=0.40$. Therefore $LGD_A=1-0.40=0.60$, and

EL_A=\$1{,}200{,}000(0.020)(0.60)=\$14{,}400.

Position B. $EAD_B=\$800{,}000$, $p_B=0.015$, and $R_B=0.50$. Therefore $LGD_B=0.50$, and

EL_B=\$800{,}000(0.015)(0.50)=\$6{,}000.

Position C. $EAD_C=\$500{,}000$, $p_C=0.040$, and $R_C=0.20$. Therefore $LGD_C=0.80$, and

EL_C=\$500{,}000(0.040)(0.80)=\$16{,}000.

Position D. $EAD_D=\$2{,}000{,}000$, $p_D=0.005$, and $R_D=0.70$. Therefore $LGD_D=0.30$, and

EL_D=\$2{,}000{,}000(0.005)(0.30)=\$3{,}000.

Adding the four estimates gives the portfolio expected loss:

EL_{\text{portfolio}} =\$14{,}400+\$6{,}000+\$16{,}000+\$3{,}000 =\$39{,}400.

Check. Each position’s expected loss is smaller than its exposure at default, and the total, $\$39{,}400$, is smaller than the combined exposure of $\$4{,}500{,}000$. The method is to calculate each exposure’s own probability-weighted loss severity before summing. It avoids treating all four positions as if they had the same default probability or recovery.

Use Weighted Inputs, Not Simple Averages

An average default probability across names is not generally a suitable portfolio probability. Exposures with different balances contribute different amounts to expected loss. A simple average gives each name equal weight, even when one position is much larger than another. Likewise, a portfolio-wide recovery rate must be weighted consistently with the exposures and defaults it represents.

A valid shortcut can be formed when the portfolio inputs are constructed to preserve the exposure-level sum. For example, with a common LGD, an exposure-weighted average default probability can reproduce the result. But the individual exposure calculation is usually clearer and less prone to misweighting.

Worked Example: Why the Simple Average Default Probability Fails

A hypothetical investor holds two bonds for a one-year horizon. Bond A has $EAD_A=\$10{,}000{,}000$, $p_A=0.010$, and recovery $R_A=0.40$. Bond B has $EAD_B=\$1{,}000{,}000$, $p_B=0.100$, and recovery $R_B=0.40$. Assume fixed exposure and recovery, and Bernoulli default indicators. No interest or mortality assumption applies.

Both positions have $LGD=1-0.40=0.60$. Calculate each expected loss:

EL_A=\$10{,}000{,}000(0.010)(0.60)=\$60{,}000, \qquad EL_B=\$1{,}000{,}000(0.100)(0.60)=\$60{,}000.

The portfolio expected loss is therefore

EL_{\text{portfolio}}=\$60{,}000+\$60{,}000=\$120{,}000.

The unweighted average default probability is $(0.010+0.100)/2=0.055$. Applying it to the combined exposure would give

\$11{,}000{,}000(0.055)(0.60)=\$363{,}000,

which does not equal the exposure-level sum. The correct exposure-weighted default probability is

\frac{\$10{,}000{,}000(0.010)+\$1{,}000{,}000(0.100)} {\$11{,}000{,}000} =\frac{\$200{,}000}{\$11{,}000{,}000} \approx 0.01818.

Using this weighted probability gives $\$11{,}000{,}000(0.01818)(0.60)\approx \$120{,}000$. The exact weighted calculation is $(\$200{,}000/\$11{,}000{,}000)(\$11{,}000{,}000)(0.60)=\$120{,}000$. The method avoids the common mistake of giving a small, high-probability position the same weight as a large, low-probability one.

Exposure at Default Can Differ From Today’s Balance

For some credit products, the amount outstanding at default may differ from the current balance. A revolving facility, for instance, can have additional funds drawn before default. The expected-loss calculation should use an exposure-at-default estimate appropriate to the horizon, not automatically substitute the balance recorded today.

Worked Example: Using Expected Exposure at Default

A hypothetical lender estimates one-year loss on three facilities. For the revolving facility, the stated exposure at default is the expected amount outstanding conditional on default; it is not the current drawn balance. Treat each stated exposure at default and recovery rate as fixed inputs to this estimate. Defaults are Bernoulli events with the listed probabilities. No interest or mortality assumption applies.

Revolving facility. Current drawn balance is $\$400{,}000$, but estimated $EAD_1=\$700{,}000$. With $p_1=0.040$ and $R_1=0.50$, $LGD_1=0.50$, so

EL_1=\$700{,}000(0.040)(0.50)=\$14{,}000.

Term loan. $EAD_2=\$800{,}000$, $p_2=0.015$, and $R_2=0.60$, so $LGD_2=0.40$ and

EL_2=\$800{,}000(0.015)(0.40)=\$4{,}800.

Invoice-finance facility. $EAD_3=\$500{,}000$, $p_3=0.030$, and $R_3=0.20$, so $LGD_3=0.80$ and

EL_3=\$500{,}000(0.030)(0.80)=\$12{,}000.

The portfolio estimate is

EL_{\text{portfolio}} =\$14{,}000+\$4{,}800+\$12{,}000 =\$30{,}800.

Check. If the revolving facility’s current balance were incorrectly used, its estimate would be $\$400{,}000(0.040)(0.50)=\$8{,}000$, understating its expected loss by $\$6{,}000$. Using the specified $\$700{,}000$ exposure at default gives the appropriate portfolio total of $\$30{,}800$.

Common Mistakes and Practical Checks

Expected loss is a mean, not a forecast that the portfolio will lose exactly that amount in a particular period. It is also not a capital requirement or a measure of loss volatility. Those questions require more than an exposure-level expected-loss sum.

Using recovery as the loss rate. If recovery is 35%, LGD is $1-0.35=0.65$. Multiplying by 0.35 calculates an incorrect severity.
Mixing horizons. Match the default probability and exposure estimate to the same period and definition.
Using current balance instead of EAD. Use the exposure-at-default assumption provided, including expected future drawdown where applicable.
Taking an unweighted average. Calculate the exposure-level terms first, or use carefully constructed weights that reproduce their sum.
Assuming independence is needed. Independence is not required to add expected losses. It matters for the distribution of combined losses, not for this sum of means.

A useful final check is to verify every LGD, confirm the units and horizon of each input, and recalculate the portfolio total from the exposure-level results. Keep intermediate monetary values unrounded where necessary; report dollar results to the cent when the inputs warrant that precision.

Check Your Understanding

For each hypothetical portfolio, calculate expected loss over the stated horizon. Assume fixed exposure and recovery unless the question says otherwise.

A one-year exposure has $EAD=\$250{,}000$, default probability $0.020$, and recovery rate $0.40$. What is its expected loss? (Answer: $\$3{,}000$)
Two one-year exposures have $(EAD,p,R)=(\$600{,}000,0.030,0.25)$ and $(\$1{,}400{,}000,0.010,0.50)$. Calculate each expected loss and the portfolio total. (Answer: $\$13{,}500$ and $\$7{,}000$; total $\$20{,}500$)
A revolving facility has expected exposure at default of $\$900{,}000$, one-year default probability $0.040$, and recovery rate $0.70$. Its current drawn balance is only $\$500{,}000$. Calculate expected loss using the stated exposure at default. (Answer: $\$10{,}800$)
Three positions have $(EAD,p,R)=(\$1{,}000{,}000,0.020,0.50)$, $(\$2{,}000{,}000,0.010,0.40)$, and $(\$500{,}000,0.080,0.20)$. Calculate the expected loss for each and then sum them. (Answer: $\$10{,}000$, $\$12{,}000$, and $\$32{,}000$; total $\$54{,}000$)
Two positions have $EAD_1=\$3{,}000{,}000$, $p_1=0.010$, $R_1=0.60$, and $EAD_2=\$1{,}000{,}000$, $p_2=0.040$, $R_2=0.25$. Find the portfolio expected loss. Would dependence between their defaults change this mean? (Answer: $\$12{,}000+\$30{,}000=\$42{,}000$; no)

What You'll Learn