Measuring Diversification Benefit

What Diversification Benefit Measures

When risks are considered together, their aggregate risk measure may be less than the sum of the measures calculated for each risk separately. Diversification benefit quantifies that difference. It is a comparison of two amounts, not a risk measure on its own: one amount is the sum of standalone measures, and the other is the measure for the aggregate loss.

The comparison is meaningful only when both sides use the same definition of risk, confidence level, time horizon, currency, and loss basis. For example, combining one-year net-of-reinsurance capital for one risk with gross multi-year VaR for another does not produce a meaningful measure of diversification benefit.

Definition: Let $X_1,\ldots,X_n$ be losses measured on a consistent basis, and let $\rho$ be the same risk measure applied to each standalone loss and their aggregate $S=\sum_{j=1}^n X_j$. The absolute diversification benefit is

B_{\mathrm{abs}}=\sum_{j=1}^n \rho(X_j)-\rho(S).

If $\sum_{j=1}^n\rho(X_j)>0$, the relative diversification benefit is

$$ B_{\mathrm{rel}}= \frac{\sum_{j=1}^n \rho(X_j)-\rho(S)} {\sum_{j=1}^n \rho(X_j)}. $$ It is often reported as a percentage by multiplying this ratio by 100.

The risk measure $\rho$ might be VaR at a specified confidence level, TVaR at that level, or a defined capital measure. The formula does not select the measure; it tells us how to compare its standalone and aggregate applications. If the denominator is zero, the relative benefit is undefined. If it is very small, the percentage can be unstable and should be interpreted cautiously.

A positive $B_{\mathrm{abs}}$ means the aggregate measure is below the sum of standalone measures. A zero value means they are equal, and a negative value means the aggregate measure exceeds that sum. The definition itself does not guarantee a positive benefit. Also, the relative figure is scaled by the sum of standalone measures: it is not the percentage reduction from the aggregate measure.

Why the Comparison Works

The formula is a direct comparison of two ways of measuring the same set of risks. First measure each risk separately and add the results; then aggregate the losses and measure the total. Subtracting the second amount from the first gives the absolute difference. Dividing that difference by the standalone sum expresses it relative to the starting amount.

For two risks, the calculation is

B_{\mathrm{abs}}=\rho(X_1)+\rho(X_2)-\rho(X_1+X_2), \qquad B_{\mathrm{rel}}= \frac{B_{\mathrm{abs}}}{\rho(X_1)+\rho(X_2)}.

If the measure is VaR, specify the convention. Here, $\operatorname{VaR}_\alpha(X)=\inf\{x:F_X(x)\geq\alpha\}$, where $F_X$ is the cumulative distribution function and $\alpha$ is the confidence level. For a normal loss $X\sim N(\mu,\sigma^2)$, this convention gives

\operatorname{VaR}_{\alpha}(X)=\mu+z_\alpha\sigma,

where $z_\alpha$ is the $\alpha$-quantile of the standard normal distribution. The reason is that $X=\mu+\sigma Z$, with $Z$ standard normal, so the $\alpha$-quantile of $X$ is $\mu+\sigma z_\alpha$. For jointly normal losses, their sum is also normal; its mean is the sum of the means, and its variance includes the covariance term. Thus the aggregate VaR can be calculated from the aggregate distribution, rather than by adding standalone VaRs.

Worked VaR and Capital Comparisons

Worked Example: VaR Benefit for Correlated Normal Losses

A hypothetical insurer models two one-year losses, in millions of dollars, as jointly normal. Their means are $\mu_1=2$ and $\mu_2=3$, their standard deviations are $\sigma_1=3$ and $\sigma_2=4$, and their correlation is $0.50$. Calculate the absolute and relative diversification benefits using 90% VaR, with $z_{0.90}=1.2816$. There is no discounting or mortality component in this one-year loss model.

Method and assumptions. Use the normal VaR formula for each loss and for their sum. The covariance is $\operatorname{Cov}(X_1,X_2)=0.50(3)(4)=6$. This comparison uses the same one-year 90% VaR basis for all three calculations.

The standalone VaRs are

\operatorname{VaR}_{0.90}(X_1)=2+1.2816(3)=2+3.8448=5.8448,

\operatorname{VaR}_{0.90}(X_2)=3+1.2816(4)=3+5.1264=8.1264.

Their sum is $5.8448+8.1264=13.9712$ million dollars. For the aggregate loss $S=X_1+X_2$, the mean is $2+3=5$, and the variance is

\operatorname{Var}(S)=3^2+4^2+2(6)=9+16+12=37.

Therefore the aggregate standard deviation is $\sqrt{37}=6.0827625$ million dollars. The aggregate VaR is

\operatorname{VaR}_{0.90}(S) =5+1.2816\sqrt{37} =5+7.7956685 =12.7956685.

The absolute benefit is $13.9712-12.7956685=1.1755315$ million dollars, or $1.1755 million to four decimal places. The relative benefit is

\frac{1.1755315}{13.9712}=0.0841396\ldots,

or 8.414% to three decimal places. Check: the aggregate VaR, $12.7957 million, is below the $13.9712 million sum of standalone VaRs, and their difference agrees with the reported absolute benefit. The common mistake avoided here is adding standard deviations, or using an incorrect covariance term, instead of calculating the variance of the sum.

Worked Example: Relative Benefit From Square-Root Aggregation

A hypothetical capital model assigns two risks standalone capital amounts of $12 million and $9 million. Its specified square-root aggregation model uses a correlation parameter of $0.25$. Calculate the aggregate capital, absolute benefit, and relative benefit. The amounts are already on the same horizon and capital basis; the model has no interest-rate or mortality input.

Method and assumptions. Apply the stated square-root model to these standalone capital amounts. This is a model-based aggregation calculation; the inputs are not being treated as means or standard deviations unless the model defines them that way.

C_{\mathrm{agg}} =\sqrt{12^2+9^2+2(0.25)(12)(9)} =\sqrt{144+81+54} =\sqrt{279} =16.7033

The standalone sum is $12+9=21$ million dollars. The absolute benefit is $21-16.7033=4.2967$ million dollars, and the relative benefit is

\frac{4.2967}{21}=0.204605\ldots=20.46\%.

Thus the model produces aggregate capital of $16.7033 million, absolute benefit of $4.2967 million, and relative benefit of 20.46%. The calculation checks because the aggregate capital is below the standalone sum by the stated absolute amount. The common mistake avoided is reporting $4.2967/16.7033$ as the relative benefit: the denominator in the defined measure is the sum of standalone capital, $21 million.

Discrete VaR and the Importance of the Quantile Convention

For discrete losses, VaR is a quantile of the loss distribution, not an average of possible losses. In particular, the chosen confidence level may fall at a jump in the cumulative distribution. The definition $\inf\{x:F_X(x)\geq\alpha\}$ must therefore be applied consistently to each standalone loss and to the aggregate.

Worked Example: Exact Discrete VaR Benefit

Two hypothetical one-year losses are independent. Loss $X_1$ is $0 with probability $0.80$ and $10 with probability $0.20$, in thousands of dollars. Loss $X_2$ is $0 with probability $0.70$ and $8 with probability $0.30$, also in thousands. There is no discounting or mortality component. Calculate the absolute and relative diversification benefits using 90% VaR.

Method and assumptions. Use the stated independence to form the exact distribution of $S=X_1+X_2$, then apply the quantile definition. For $X_1$, $F_{X_1}(0)=0.80<0.90$ and $F_{X_1}(10)=1$, so its 90% VaR is $10 thousand. For $X_2$, $F_{X_2}(0)=0.70<0.90$ and $F_{X_2}(8)=1$, so its 90% VaR is $8 thousand. The standalone sum is $18 thousand.

Independence gives the aggregate outcomes and probabilities:

\begin{aligned} P(S=0)&=0.80(0.70)=0.56,\\ P(S=8)&=0.80(0.30)=0.24,\\ P(S=10)&=0.20(0.70)=0.14,\\ P(S=18)&=0.20(0.30)=0.06. \end{aligned}

The cumulative probabilities are $F_S(0)=0.56$, $F_S(8)=0.80$, and $F_S(10)=0.94$. Since $0.80<0.90\leq0.94$, the aggregate 90% VaR is $10 thousand. The absolute benefit is $18-10=$8$ thousand, and the relative benefit is $8/18=0.4444\ldots$, or 44.44% to two decimal places.

Check: the cumulative probability first reaches at least 0.90 at a loss of $10 thousand, confirming the aggregate VaR under the stated convention. The common mistake avoided is interpolating between discrete loss amounts or choosing $8 thousand because it is the second outcome: at $8 thousand, the cumulative probability is only 0.80.

Interpretation and Common Traps

Diversification benefit is a result for a specified set of risks, dependence assumptions, and risk measure. A change in any of these can change the result. In particular, calculating the standalone measures alone is not enough: the aggregate measure must reflect the joint distribution or the stated aggregation model.

Mixing bases: Confirm that currency, horizon, gross or net treatment, and confidence level match on both sides.
Using the wrong denominator: Relative benefit is divided by the sum of standalone measures, not by aggregate capital.
Assuming the benefit must be positive: The formula permits zero or negative results. Interpret the sign of the computed difference rather than presuming a saving.
Confusing model inputs with risk measures: A square-root formula applies only under its stated model and inputs; its output should not be relabelled as VaR without justification.
Rounding too early: Keep unrounded intermediate values where possible, then round the reported money or percentage at the final step.

In practice, report the absolute amount and relative percentage together. The absolute amount indicates the size of the difference in the chosen currency; the percentage makes comparisons across portfolios of different sizes easier, provided the standalone denominator is positive and meaningful. This distinction is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Check Your Understanding

For each problem, use the stated risk measure and basis, and calculate the requested benefit rather than assuming its sign.

A model gives standalone risk measures of $4 million, $7 million, and $5 million, and an aggregate measure of $13 million. Calculate the absolute and relative benefit. (Answer: $3 million; 18.75%)
A hypothetical capital model has standalone amounts of $6 million and $8 million and uses square-root aggregation with parameter $0.40$. Calculate aggregate capital and relative benefit. (Answer: $11.7644 million; 15.97%)
Independent one-year losses, in thousands of dollars, have distributions $X_1=0$ with probability $0.75$ and $12$ with probability $0.25$; $X_2=0$ with probability $0.80$ and $5$ with probability $0.20$. Using $\operatorname{VaR}_{0.90}(X)=\inf\{x:F_X(x)\geq0.90\}$, find the absolute benefit. (Answer: $5 thousand)
Two jointly normal one-year losses, in millions of dollars, have means 1 and 2, standard deviations 2 and 3, and correlation $0.20$. With $z_{0.95}=1.645$, calculate aggregate standard deviation, absolute 95% VaR benefit, and relative benefit. (Answer: $3.9243 million; $1.7696 million; 15.76%)

What You'll Learn