Portfolio Concentration and Credit Aggregation

Why Aggregation Changes Credit Risk

A portfolio can have the same expected credit loss under very different dependence assumptions, yet have a materially different range of possible aggregate outcomes. The key distinction is between how much loss is expected on average and how uncertain the total loss is. Adding expected losses requires no assumption about dependence. Calculating the variance or standard deviation of total loss does.

Concentration can arise because several positions are owed by the same obligor, so one default event affects all of them. It can also arise because different obligors are exposed to shared economic conditions, making their defaults dependent. The treatment here is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: Let $X_i$ be the credit loss on position $i$ over a specified horizon, and let $X_{\mathrm{tot}}=\sum_{i=1}^{n}X_i$ be total portfolio loss. The portfolio's expected loss is $E[X_{\mathrm{tot}}]$. Its aggregate loss variance is $\operatorname{Var}(X_{\mathrm{tot}})$, and its standard deviation is $\sqrt{\operatorname{Var}(X_{\mathrm{tot}})}$. Expected loss describes the mean; variance and standard deviation describe dispersion around that mean.

For the examples below, losses are measured over one period, at the end of that period, and are not discounted; equivalently, the effective annual interest rate used to discount them is $i=0$. There is no mortality component. Each loss is zero if default does not occur and is a stated fixed amount if it does. Thus each loss is a scaled Bernoulli random variable. These assumptions make dependence effects visible without introducing uncertainty in recovery or exposure at default.

Expected Loss Adds Even When Defaults Depend

For any finite set of losses with finite expectations, linearity of expectation gives

E[X_{\mathrm{tot}}]=E\left[\sum_{i=1}^{n}X_i\right]=\sum_{i=1}^{n}E[X_i].

This identity does not require independent defaults. Dependence matters when measuring the uncertainty of the sum. If $I_i$ is an indicator equal to 1 when obligor $i$ defaults and 0 otherwise, and the loss given default is a fixed amount $L_i$, then $X_i=L_iI_i$. With default probability $p_i$, the expected loss on that position is $E[X_i]=L_ip_i$.

Worked Example: Multiple Positions on One Borrower

A hypothetical lender has two unsecured positions on the same borrower. If the borrower defaults during the one-year horizon, the first position produces a loss of $240,000 and the second produces a loss of $160,000. The borrower's one-year default probability is 0.02. Assume fixed losses given default, no other loss events, and no discounting. Find the total expected loss, variance, and standard deviation. Then compare the variance with a model that incorrectly treats the two defaults as independent.

Method. Both positions are triggered by the same borrower's default indicator $I$, where $P(I=1)=0.02$ and $P(I=0)=0.98$. Therefore, $X_1=240{,}000I$, $X_2=160{,}000I$, and $X_{\mathrm{tot}}=400{,}000I$.

The expected loss is $E[X_{\mathrm{tot}}]=400{,}000(0.02)=\$8{,}000$. Equivalently, the two position-level expected losses are $240{,}000(0.02)=\$4{,}800$ and $160{,}000(0.02)=\$3{,}200$, which sum to $8,000.

For a Bernoulli indicator with probability $p$, the variance is $p(1-p)$. Thus the aggregate variance is $400{,}000^2(0.02)(0.98)=160{,}000{,}000{,}000(0.0196)=\$3{,}136{,}000{,}000$. The standard deviation is $\sqrt{3{,}136{,}000{,}000}=\$56{,}000$.

If the two default events were incorrectly treated as independent while retaining the same marginal probability, the variance would instead be $0.0196[(240{,}000)^2+(160{,}000)^2]$. Since $240{,}000^2=57{,}600{,}000{,}000$ and $160{,}000^2=25{,}600{,}000{,}000$, this is $0.0196(83{,}200{,}000{,}000)=\$1{,}630{,}720{,}000$. Its standard deviation is approximately $40,382, rounded to the nearest dollar.

Check and interpretation. The shared-obligor model has the same expected loss as the position-level sum, but a larger variance because both losses occur together on default. The independence calculation understates this concentration risk. Its standard deviation is approximately $40,382, not $40,381.

Common mistake avoided: Counting two positions as two independent default events simply because they are separate contracts. The borrower is the common event: when that borrower defaults, both stated losses occur.

Variance Requires Covariance Terms

For losses $X_1,\ldots,X_n$ with finite variances, expanding the squared deviation of their sum gives the general aggregation formula:

\operatorname{Var}\left(\sum_{i=1}^{n}X_i\right) = \sum_{i=1}^{n}\operatorname{Var}(X_i) + 2\sum_{1\leq i<j\leq n}\operatorname{Cov}(X_i,X_j).

The reason is that $X_{\mathrm{tot}}-E[X_{\mathrm{tot}}]=\sum_i(X_i-E[X_i])$. Squaring this expression produces each squared deviation and, for every pair $i,j$, two cross-products. Taking expectations turns the squared deviations into variances and each cross-product into a covariance. If the losses are independent, each covariance is zero and the variance is the sum of individual variances. Zero covariance is sufficient for that variance simplification, but it does not in general imply independence.

Specify the loss variables.
State the horizon and what each $X_i$ represents. If losses occur only on default, identify the default event and the loss amount conditional on default.

Calculate expected loss.
Find each $E[X_i]$ and add them. This step does not require independence.

Calculate the individual variances.
For a fixed loss $L_i$ on default with probability $p_i$, use $X_i=L_iI_i$ and $\operatorname{Var}(X_i)=L_i^2p_i(1-p_i)$.

Account for dependence.
Add twice the covariance for each distinct pair. Do not omit these terms unless independence or zero covariance is justified.

Worked Example: Two Different Obligors With Dependent Defaults

A hypothetical portfolio has one position on each of two different borrowers. The first produces a fixed loss of $100,000 on default and has default probability 0.10. The second produces a fixed loss of $60,000 on default and has default probability 0.20. The joint probability that both default during the one-year horizon is 0.05. Assume no other loss events, no severity uncertainty, and no discounting. Calculate expected aggregate loss, aggregate variance, and standard deviation.

Method. Let $I_A$ and $I_B$ be the respective default indicators. The joint distribution is $P(I_A=1,I_B=1)=0.05$, $P(I_A=1,I_B=0)=0.10-0.05=0.05$, $P(I_A=0,I_B=1)=0.20-0.05=0.15$, and $P(I_A=0,I_B=0)=1-0.05-0.05-0.15=0.75$. These probabilities are nonnegative and sum to 1.

Expected losses are $100{,}000(0.10)=\$10{,}000$ and $60{,}000(0.20)=\$12{,}000$, so expected aggregate loss is $\$22{,}000$.

The individual variances are $100{,}000^2(0.10)(0.90)=\$900{,}000{,}000$ and $60{,}000^2(0.20)(0.80)=\$576{,}000{,}000$. The covariance of the default indicators is $\operatorname{Cov}(I_A,I_B)=0.05-(0.10)(0.20)=0.03$. Therefore, the covariance of the dollar losses is $100{,}000(60{,}000)(0.03)=\$180{,}000{,}000$.

The aggregate variance is $900{,}000{,}000+576{,}000{,}000+2(180{,}000{,}000)=\$1{,}836{,}000{,}000$. The standard deviation is $\sqrt{1{,}836{,}000{,}000}\approx \$42{,}849$, rounded to the nearest dollar.

Check and interpretation. The correlation between the default indicators is $0.03/\sqrt{(0.10)(0.90)(0.20)(0.80)}=0.03/0.12=0.25$. It is positive, so the covariance increases aggregate variance. If the defaults were independent, the covariance term would be zero; the variance would be $1,476,000,000 and the standard deviation approximately $38,419.

Common mistake avoided: Using the joint default probability directly as a covariance. Covariance is $P(A\cap B)-P(A)P(B)$, not $P(A\cap B)$ alone.

Same Expected Loss, Different Concentration

A portfolio’s expected loss alone does not reveal whether losses are spread across many independent names or concentrated in a single name. For a simple comparison, assume fixed conditional losses and independent defaults across distinct obligors. In that setting, splitting an exposure among independent borrowers can reduce variance even if the total expected loss is unchanged. This conclusion depends on the independence assumption; common drivers or shared obligors can reduce or eliminate the diversification benefit.

Worked Example: Comparing One Large Exposure With Four Smaller Ones

Compare two hypothetical portfolios over one year. Portfolio A has one position that loses $100,000 if its borrower defaults, with default probability 0.10. Portfolio B has four positions on four different borrowers; each loses $25,000 on default, and each borrower has default probability 0.10. Assume the four defaults are mutually independent, losses given default are fixed, and there is no discounting. Compare expected loss, variance, and standard deviation.

Portfolio A. Its expected loss is $100{,}000(0.10)=\$10{,}000$. Its variance is $100{,}000^2(0.10)(0.90)=\$900{,}000{,}000$, so its standard deviation is $30,000.

Portfolio B. Each position has expected loss $25{,}000(0.10)=\$2{,}500$; across four positions the expected loss is $4(2{,}500)=\$10{,}000$. Each position’s variance is $25{,}000^2(0.10)(0.90)=625{,}000{,}000(0.09)=\$56{,}250{,}000$. Independence makes all pairwise covariances zero, so aggregate variance is $4(56{,}250{,}000)=\$225{,}000{,}000$. The standard deviation is $\sqrt{225{,}000{,}000}=\$15{,}000$.

Check and interpretation. Both portfolios have expected loss of $10,000, but Portfolio A’s standard deviation is twice Portfolio B’s. Under the stated independent-default assumptions, spreading the total exposure across four names reduces aggregate uncertainty. The comparison concerns variance and standard deviation, not a claim that all features of the two loss distributions are identical.

Common mistake avoided: Concluding that equal expected losses imply equal risk. Here the means match exactly while the variances differ by a factor of four.

Practical Interpretation and Exam Traps

The covariance term summarizes whether losses tend to occur together or offset one another. Positive covariance raises aggregate variance relative to the sum of individual variances; negative covariance lowers it. A zero covariance removes the cross-term in the variance calculation, but should not be mistaken for proof that the underlying events are independent.

When several positions share an obligor, the default event is shared even if the contracts have different exposures, collateral, or loss severities. With fixed losses on default, combine their conditional losses for that event before calculating the variance. For distinct obligors, do not assume independence merely because the names differ; use a specified joint distribution, covariance, or other justified dependence model. If loss given default is itself uncertain, default-indicator correlation alone is not enough to determine loss covariance: the dependence between severities and exposures also matters.

Expected loss is not a diversification measure. Its additivity holds whether losses are dependent or independent.
Variance is not generally additive. The covariance terms must be included unless they are known to be zero.
Use consistent units. Squaring dollar losses produces variance in dollars squared; the square root returns a dollar standard deviation.
Do not confuse a default probability with a covariance. For default indicators, covariance is joint default probability minus the product of the marginal probabilities.
Do not treat positions as obligors. Multiple contracts on one borrower can share one event and create concentrated losses.

Key takeaway: Sum position-level expected losses to obtain expected aggregate loss. To measure aggregate uncertainty, also account for dependence: shared obligors and positive covariance can make portfolio loss substantially more variable than an independence calculation suggests.

Check Your Understanding

For each problem, assume one-year, undiscounted losses, fixed loss amounts conditional on default, and no other loss events unless stated. Give dollar standard deviations to the nearest dollar.

A lender has two positions on the same borrower, with losses of $80,000 and $120,000 if that borrower defaults. The borrower’s default probability is 0.05. Find aggregate expected loss, variance, and standard deviation. (Answer: Expected loss $10,000; variance $1,900,000,000; standard deviation $43,589.)
Two independent borrowers have fixed losses of $50,000 and $100,000 on default, with default probabilities 0.04 and 0.03, respectively. Find aggregate expected loss and standard deviation. (Answer: Expected loss $5,000; variance $387,000,000; standard deviation $19,672.)
Two borrowers produce losses of $80,000 and $50,000 on default. Each has default probability 0.10, and their joint default probability is 0.04. Find the covariance of the default indicators, aggregate expected loss, and aggregate standard deviation. (Answer: Indicator covariance 0.03; expected loss $13,000; variance $1,041,000,000; standard deviation $32,265.)
Positions A and B are on the same borrower and produce losses of $70,000 and $30,000 if that borrower defaults; its default probability is 0.02. Position C is on a different borrower, produces a $40,000 loss on default, and has default probability 0.05. The two borrowers’ defaults are independent. Find total expected loss, variance, and standard deviation. (Answer: Expected loss $4,000; variance $272,000,000; standard deviation $16,492.)
Two borrowers have fixed losses of $200,000 and $100,000 on default, with default probabilities 0.20 and 0.10. Their joint default probability is 0.01. Find aggregate expected loss, the covariance of their dollar losses, and aggregate standard deviation. (Answer: Expected loss $50,000; dollar-loss covariance $-\$200,000,000$; variance $6,900,000,000; standard deviation $83,066.)

What You'll Learn