Diversification Effects in Business Decisions

Why Diversification Matters in a Business Decision

A business activity can have substantial risk on a standalone basis yet add less capital when combined with an existing portfolio. The difference arises because portfolio risk depends not only on the amount of risk in each activity but also on how their losses move together. A decision based only on standalone capital can therefore misstate the activity’s effect on the capital the organization must hold.

This tutorial uses a simple capital model to isolate that effect and connect it to an incremental business decision. The model is illustrative: an actual economic-capital framework may use a different risk measure, confidence level, horizon, or aggregation method. The calculations below are appropriate only under the stated assumptions.

Definition: Let $L_P$ be the centered loss of an existing portfolio and $L_A$ the centered loss of a proposed activity. In the two-component model used here, capital is $K=2\operatorname{SD}(L)$, where $\operatorname{SD}(L)$ is the standard deviation of loss. The activity’s marginal capital is $K_{P+A}-K_P$. The diversification benefit is the sum of the two standalone capital amounts minus combined capital: $K_P+K_A-K_{P+A}$.

The factor of 2 is a fixed scaling convention for these examples, not a universal definition of economic capital and not, by itself, a claim that capital equals a particular percentile of loss. All losses are measured over the same horizon, in the same currency, and on a consistent basis. The activity’s incremental profit is assumed to be measured over that same horizon.

Aggregation Under the Standard-Deviation Capital Model

Write $\sigma_P=\operatorname{SD}(L_P)$, $\sigma_A=\operatorname{SD}(L_A)$, and let $\rho$ be their correlation. For $\sigma_P>0$ and $\sigma_A>0$, correlation is covariance divided by the product of the standard deviations, so $\operatorname{Cov}(L_P,L_A)=\rho\sigma_P\sigma_A$. The combined loss is $L_P+L_A$.

\begin{aligned} \operatorname{Var}(L_P+L_A) &=\sigma_P^2+\sigma_A^2+2\operatorname{Cov}(L_P,L_A)\\ &=\sigma_P^2+\sigma_A^2+2\rho\sigma_P\sigma_A,\\ K_{P+A} &=2\sqrt{\sigma_P^2+\sigma_A^2+2\rho\sigma_P\sigma_A}. \end{aligned}

This formula follows from expanding the variance of a sum. Specifically, subtract the mean from each loss, square their sum, and take expectations. The squared terms contribute $\sigma_P^2$ and $\sigma_A^2$; the cross-product contributes $2E[(L_P-E[L_P])(L_A-E[L_A])]=2\operatorname{Cov}(L_P,L_A)$. Substituting covariance in terms of correlation gives the stated result. The square root converts variance to standard deviation, and multiplication by 2 applies the chosen capital convention.

The formula requires a valid joint model with the stated variances and correlation; in particular, $-1\leq\rho\leq1$. It describes the standard deviation of the sum. It does not automatically give an exact aggregation formula for a different capital measure, such as a tail risk measure, or for a model with nonlinear adjustments. In those cases, the capital measure’s own aggregation method must be used.

Worked Example: Measuring the Diversification Benefit

A hypothetical existing portfolio has loss standard deviation $$20{,}000$. A proposed activity has loss standard deviation $$15{,}000$, and its loss correlation with the portfolio is $0.40$. Using $K=2\sigma$, calculate combined capital, the activity’s marginal capital, and the diversification benefit relative to the sum of standalone capital amounts.

Method. Apply the variance-of-a-sum formula, take the square root, and multiply by 2. Keep amounts in dollars throughout.

1. Calculate the covariance contribution. $2\rho\sigma_P\sigma_A=2(0.40)(20{,}000)(15{,}000)=240{,}000{,}000$.

2. Calculate combined variance. $\sigma_{P+A}^2=20{,}000^2+15{,}000^2+240{,}000{,}000$. The squared terms are $400{,}000{,}000$ and $225{,}000{,}000$, so combined variance is $865{,}000{,}000$.

3. Calculate combined capital. $\sigma_{P+A}=\sqrt{865{,}000{,}000}=\$29{,}410.882\ldots$, so $K_{P+A}=2(\$29{,}410.882\ldots)=\$58{,}821.76$, rounded to the nearest cent.

4. Calculate marginal capital. Existing portfolio capital is $2(\$20{,}000)=\$40{,}000$. Therefore the activity’s marginal capital is $\$58{,}821.76-\$40{,}000=\$18{,}821.76$.

5. Calculate the diversification benefit. Standalone capital for the activity is $2(\$15{,}000)=\$30{,}000$. The standalone sum is $\$40{,}000+\$30{,}000=\$70{,}000$, so the benefit is $\$70{,}000-\$58{,}821.76=\$11{,}178.24$.

Check. Marginal capital plus the existing portfolio’s capital is $\$18{,}821.76+\$40{,}000=\$58{,}821.76$, which matches combined capital. Combined capital is also below the $$70{,}000$ standalone sum by exactly $$11{,}178.24$. The diversification benefit is not the same as the activity’s marginal capital.

From Marginal Capital to an Incremental Return Test

The relevant capital amount for deciding whether to add an activity is generally the change in portfolio capital, not the activity’s standalone capital. If the activity produces incremental profit $\Pi_A$, and marginal capital is positive, its marginal RAROC is incremental profit divided by marginal capital. Given a hurdle rate $h$, incremental economic profit is incremental profit less the hurdle charge on marginal capital.

\begin{aligned} \Delta K_A &= K_{P+A}-K_P,\\ \text{Marginal RAROC}_A &= \frac{\Pi_A}{\Delta K_A} \quad \text{when } \Delta K_A>0,\\ \text{Incremental economic profit}_A &= \Pi_A-h\Delta K_A. \end{aligned}

For positive marginal capital, the two tests are consistent: marginal RAROC exceeds the hurdle exactly when incremental economic profit is positive. Indeed, $\Pi_A/\Delta K_A>h$ is equivalent, after multiplying by the positive $\Delta K_A$, to $\Pi_A-h\Delta K_A>0$. If marginal capital is zero or negative, the ratio is not an ordinary return on positive capital; interpret the capital change and profit directly rather than applying this ratio mechanically.

Worked Example: Testing an Activity Against a Hurdle

A hypothetical portfolio has loss standard deviation $$30{,}000$. A proposed activity has standard deviation $$40{,}000$, correlation $0.50$ with the portfolio, and expected incremental profit of $$9{,}000$. Use $K=2\sigma$ and a hurdle rate of $14\%$. Find combined capital, marginal capital, marginal RAROC, and incremental economic profit, then assess the activity against the hurdle.

Method. First aggregate risk, then subtract existing capital to find the relevant incremental capital. Use that marginal amount—not standalone activity capital—in both return calculations.

1. Calculate combined variance. $30{,}000^2=900{,}000{,}000$, $40{,}000^2=1{,}600{,}000{,}000$, and $2(0.50)(30{,}000)(40{,}000)=1{,}200{,}000{,}000$. Thus combined variance is $3{,}700{,}000{,}000$.

2. Calculate combined capital. $\sqrt{3{,}700{,}000{,}000}=\$60{,}827.6253\ldots$, so combined capital is $2(\$60{,}827.6253\ldots)=\$121{,}655.25$, rounded to the nearest cent.

3. Calculate marginal capital. Existing capital is $2(\$30{,}000)=\$60{,}000$. Therefore $\Delta K_A=\$121{,}655.25-\$60{,}000=\$61{,}655.25$.

4. Calculate marginal RAROC. $\$9{,}000/\$61{,}655.25=0.14597\ldots$, or $14.60\%$ to two decimal places.

5. Calculate incremental economic profit. The hurdle charge is $0.14(\$61{,}655.25)=\$8{,}631.735$. Therefore economic profit is $\$9{,}000-\$8{,}631.735=\$368.265$, or $$368.26$ to the nearest cent.

Decision and check. The activity clears the $14\%$ hurdle: its marginal RAROC is about $14.60\%$, and its incremental economic profit is positive. The difference is small, so the calculation should retain precision until the final rounding. Using the unrounded capital amount gives the same result to the stated cent.

Correlation, Marginal Capital, and Interpretation

Lower correlation generally reduces combined standard deviation compared with higher correlation, holding the component standard deviations fixed. But “diversification benefit” and “negative marginal capital” are different conditions. For the standard-deviation model, the combined standard deviation is less than the existing portfolio’s standard deviation only if $\sigma_A^2+2\rho\sigma_P\sigma_A<0$. When both standard deviations are positive, this requires $\rho<-\sigma_A/(2\sigma_P)$. A negative correlation alone is not enough.

Worked Example: Negative Correlation Does Not Ensure Negative Marginal Capital

A hypothetical existing portfolio has loss standard deviation $$25{,}000$. A proposed activity has standard deviation $$20{,}000$, correlation $-0.30$, incremental profit $$500$, and a hurdle rate of $15\%$. Using $K=2\sigma$, calculate the activity’s marginal capital and incremental economic profit. Does its negative correlation mean it reduces portfolio capital?

Method. Calculate the combined variance with the signed covariance term. Then compare combined capital with existing capital; do not infer the sign of marginal capital from the correlation alone.

1. Calculate combined variance. The portfolio variance is $25{,}000^2=625{,}000{,}000$, the activity variance is $20{,}000^2=400{,}000{,}000$, and the covariance contribution is $2(-0.30)(25{,}000)(20{,}000)=-300{,}000{,}000$. Combined variance is therefore $625{,}000{,}000+400{,}000{,}000-300{,}000{,}000=725{,}000{,}000$.

2. Calculate combined and marginal capital. $\sqrt{725{,}000{,}000}=\$26{,}925.8240\ldots$, giving combined capital of $\$53{,}851.65$. Existing capital is $2(\$25{,}000)=\$50{,}000$, so marginal capital is $\$53{,}851.65-\$50{,}000=\$3{,}851.65$, rounded to the nearest cent.

3. Calculate incremental economic profit. The hurdle charge is $0.15(\$3{,}851.65)=\$577.7475$. Using unrounded marginal capital, the charge is $\$577.7472\ldots$, and economic profit is $\$500-\$577.7472\ldots=-\$77.75$ to the nearest cent.

Decision and check. Despite the negative correlation, combined capital exceeds existing capital, so marginal capital is positive. The activity does not clear the hurdle under this model because incremental economic profit is negative. The calculation avoids the common mistake of treating any negatively correlated activity as a capital reducer.

Common Mistake: Substituting Standalone Capital for Marginal Capital

A frequent error is to charge the proposed activity for its standalone capital even when the decision concerns the effect on the combined portfolio. In the previous example, standalone activity capital is $2(\$20{,}000)=\$40{,}000$. That is not its marginal capital. The correct marginal amount is combined capital minus existing capital: $\$53{,}851.65-\$50{,}000=\$3{,}851.65$.

The incorrect economic-profit calculation would be $\$500-0.15(\$40{,}000)=-\$5{,}500$. It uses the wrong capital denominator and overstates the hurdle charge for this portfolio decision. The correct calculation is $\$500-0.15(\$3{,}851.65)\approx-\$77.75$. In this case, the activity still fails the hurdle, but the size of the failure is very different. In another case, using standalone capital could reverse the decision.

A related arithmetic trap is to mishandle the covariance term. The correlation must be multiplied by both standard deviations, and the sign of the correlation must be preserved. In the previous example, the cross-term is $-300{,}000{,}000$, not a positive amount. However, the negative cross-term does not make the combined variance negative or imply that the combined standard deviation is below the existing portfolio’s standard deviation.

When to Use This Comparison

The calculation is useful when comparing a proposed activity’s contribution to portfolio risk with the profit it is expected to generate. It makes the diversification assumption visible: two activities with identical standalone volatility can have different marginal capital because their relationships with the existing portfolio differ. It also separates three quantities that should not be confused:

Standalone capital: the capital assigned to an activity considered on its own under the selected model.
Marginal capital: the change in portfolio capital after adding the activity.
Diversification benefit: standalone capital amounts in total minus combined capital.

The result depends on the portfolio definition, the risk measure, the dependence estimate, and the horizon. Correlation is a summary of linear co-movement and may not capture important tail dependence. If a more complete economic-capital model gives a different combined capital, use that model’s combined and existing capital consistently, then calculate the marginal amount as their difference. The return comparison should also use profit and capital measured on compatible bases and time horizons.

Key takeaway: For a portfolio decision, calculate how much total capital changes when the activity is added. A positive diversification benefit can coexist with positive marginal capital; negative correlation alone does not establish that an activity reduces capital. Compare incremental profit with the hurdle charge on marginal capital, subject to the assumptions and limitations of the chosen capital model.

Check Your Understanding

For each question, use $K=2\sigma$ and round dollar amounts to the nearest cent where needed.

A portfolio has loss standard deviation $$12{,}000$, and an activity has standard deviation $$9{,}000$, with correlation zero. Find combined capital, marginal capital, and the diversification benefit. (Answer: combined capital $$30{,}000$; marginal capital $$6{,}000$; diversification benefit $$12{,}000$)
A portfolio has standard deviation $$16{,}000$, and a proposed activity has standard deviation $$12{,}000$, with correlation $0.25$. Find combined capital, marginal capital, and the diversification benefit. (Answer: combined capital $$44{,}542.11$; marginal capital $$12{,}542.11$; diversification benefit $$11{,}457.89$)
An existing portfolio has standard deviation $$10{,}000$. An activity has standard deviation $$8{,}000$ and correlation $-0.50$. Find combined capital and marginal capital. Does the activity reduce capital under this model? (Answer: combined capital $$18{,}330.30$; marginal capital $-$1{,}669.70$; yes)
A portfolio has standard deviation $$18{,}000$. An activity has standard deviation $$15{,}000$, correlation $0.20$, and incremental profit of $$5{,}000$. At a $12\%$ hurdle, find marginal capital, marginal RAROC, and incremental economic profit. (Answer: marginal capital $$15{,}264.02$; marginal RAROC $32.76\%$; incremental economic profit $$3{,}168.32$)
An existing portfolio has standard deviation $$20{,}000$, and an activity has standard deviation $$10{,}000$. What correlation is required for combined standard deviation to be less than the existing portfolio’s standard deviation? If correlation is $-0.40$, calculate combined capital and marginal capital. (Answer: correlation must be less than $-0.25$; at $-0.40$, combined capital is $$36{,}878.18$ and marginal capital is $-$3{,}121.82$)

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