Portfolio Variance and Diversification in Treaty Pricing

Why Treaty Portfolios Need More Than Standalone Volatility

A treaty’s expected loss describes its average cost, but pricing risk also depends on how uncertain that loss is. When several treaties are considered together, their combined volatility is not determined by their individual volatilities alone. Losses driven by the same catastrophe, economic condition, or claims environment can move together; losses that tend to occur in different circumstances may offset one another.

The relevant quantity for a portfolio is therefore not just the variance of each treaty loss. It is also the covariance between treaty losses. Covariance measures whether two losses tend to be above or below their own means at the same time. It provides the link between dependence and portfolio volatility.

Definition: Let $X_1,\ldots,X_n$ be finite-variance random variables representing treaty payments over the same period and in the same currency. The portfolio loss is $S=\sum_{i=1}^n X_i$. Its variance is $$ \operatorname{Var}(S)=\sum_{i=1}^n \operatorname{Var}(X_i)+2\sum_{i<j}\operatorname{Cov}(X_i,X_j). $$ For two losses, $\operatorname{Cov}(X,Y)=E[(X-E[X])(Y-E[Y])]$. When both standard deviations are positive, their correlation is $\rho_{X,Y}=\operatorname{Cov}(X,Y)/(\sqrt{\operatorname{Var}(X)}\sqrt{\operatorname{Var}(Y)})$.

These results require finite variances and losses measured over a compatible period. They do not require a normal distribution. Independence implies zero covariance when the variances are finite, but zero covariance does not in general imply independence. Correlation is undefined if either loss has zero variance.

Deriving the Portfolio Variance Formula

The formula follows by expanding the squared deviation of total loss from its mean. For two treaty losses, linearity of expectation gives $E[X+Y]=E[X]+E[Y]$, so

\begin{aligned} \operatorname{Var}(X+Y) &=E\left[\bigl((X-E[X])+(Y-E[Y])\bigr)^2\right]\\ &=E[(X-E[X])^2]+E[(Y-E[Y])^2]\\ &\quad+2E[(X-E[X])(Y-E[Y])]\\ &=\operatorname{Var}(X)+\operatorname{Var}(Y)+2\operatorname{Cov}(X,Y). \end{aligned}

The same expansion for $n$ losses produces one variance term for each treaty and two copies of each distinct pair’s covariance. Equivalently, covariance can be written as $\operatorname{Cov}(X,Y)=E[XY]-E[X]E[Y]$. The covariance term is positive when the losses tend to be high together and negative when high losses in one are associated with low losses in the other.

For two losses with standard deviations $\sigma_X$ and $\sigma_Y$, the formula can also be written as

\operatorname{Var}(X+Y)=\sigma_X^2+\sigma_Y^2+2\rho_{X,Y}\sigma_X\sigma_Y.

This version is convenient when standard deviations and correlation are supplied. It also makes clear why assuming independence is a substantive modeling choice: it sets the correlation term to zero. A positive correlation increases portfolio variance relative to the independent case; a negative correlation decreases it.

Worked Examples: From Joint Losses to Portfolio Pricing

Worked Example: Shared Events and a Positive Risk Load

A hypothetical annual treaty portfolio has two loss payments, $X$ and $Y$, measured in thousands of dollars. Treaty A pays 100 if event A occurs and zero otherwise; treaty B pays 60 if event B occurs and zero otherwise. For the year, the joint event probabilities are: both events, 0.08; A only, 0.02; B only, 0.12; neither, 0.78. Use the standard-deviation pricing principle $P=E[S]+\lambda\sqrt{\operatorname{Var}(S)}$, with $\lambda=0.25$. Calculate the portfolio price and compare it with the sum of separately calculated treaty prices. Assume no discounting and use these probabilities as the full annual loss model.

Method. First calculate each treaty’s moments and their covariance from the joint event model. Then apply the same stated pricing principle to the portfolio and to each treaty separately.

For treaty A, $P(A)=0.08+0.02=0.10$. Thus $E[X]=100(0.10)=10$, and, because $X$ is a two-outcome loss, $\operatorname{Var}(X)=100^2(0.10)(0.90)=900$.

For treaty B, $P(B)=0.08+0.12=0.20$. Thus $E[Y]=60(0.20)=12$, and $\operatorname{Var}(Y)=60^2(0.20)(0.80)=576$.

Both losses are positive only when their respective events occur. Therefore $E[XY]=100(60)(0.08)=480$, while $E[X]E[Y]=10(12)=120$. The covariance is $480-120=360$.

The portfolio expected loss is $E[S]=10+12=22$. Its variance and standard deviation are

\operatorname{Var}(S)=900+576+2(360)=2{,}196,\qquad \sqrt{\operatorname{Var}(S)}=\sqrt{2{,}196}=46.8615.

The portfolio risk load is $0.25(46.8615)=11.7154$, so its price is $22+11.7154=33.7154$ thousand dollars. Rounded to the nearest cent, the price is $33,715.37.

Separately, treaty A has standard deviation $\sqrt{900}=30$ and price $10+0.25(30)=17.5$ thousand dollars. Treaty B has standard deviation $\sqrt{576}=24$ and price $12+0.25(24)=18$ thousand dollars. The sum of separate prices is $35.5$ thousand dollars, or $35,500.00. Under this pricing principle, the combined portfolio price is lower by $1,784.63 because the portfolio standard deviation is less than the sum of the standalone standard deviations. The calculation is checked by adding the portfolio’s expected loss, 22, and its risk load, 11.7154, to recover 33.7154 thousand dollars.

Common mistake avoided: Treating the events as independent would set the covariance to zero, even though the joint probability is 0.08 rather than the independent value $0.10(0.20)=0.02$. The stated joint probabilities must be used.

Worked Example: Mutually Exclusive Treaty Losses

Two hypothetical annual treaty payments, in thousands of dollars, have this complete joint model: $X=80,Y=0$ with probability 0.25; $X=0,Y=50$ with probability 0.20; and $X=0,Y=0$ with probability 0.55. Calculate the portfolio standard deviation and the combined price under $P=E[S]+0.20\sqrt{\operatorname{Var}(S)}$. Compare the portfolio risk load with the sum of standalone risk loads. Assume no discounting.

Method. Calculate the moments from the three possible outcomes. Since the two positive payments never occur together, the product $XY$ is zero in every outcome.

For $X$, $E[X]=80(0.25)=20$, and $E[X^2]=80^2(0.25)=1{,}600$. Hence $\operatorname{Var}(X)=1{,}600-20^2=1{,}200$.

For $Y$, $E[Y]=50(0.20)=10$, and $E[Y^2]=50^2(0.20)=500$. Hence $\operatorname{Var}(Y)=500-10^2=400$.

Because $XY=0$ in every outcome, $E[XY]=0$. Therefore $\operatorname{Cov}(X,Y)=0-(20)(10)=-200$. The portfolio expected loss is $20+10=30$, and

\operatorname{Var}(S)=1{,}200+400+2(-200)=1{,}200,\qquad \sqrt{\operatorname{Var}(S)}=\sqrt{1{,}200}=34.6410.

The combined risk load is $0.20(34.6410)=6.9282$ thousand dollars. The combined price is $30+6.9282=36.9282$ thousand dollars, or $36,928.20 to the nearest cent.

The standalone standard deviations are $\sqrt{1{,}200}=34.6410$ and $\sqrt{400}=20$. Their separate risk loads sum to $0.20(34.6410)+0.20(20)=10.9282$ thousand dollars. The portfolio risk load is lower by $4$ thousand dollars. As a check, direct calculation gives $E[S^2]=80^2(0.25)+50^2(0.20)=2{,}100$, so $\operatorname{Var}(S)=2{,}100-30^2=1{,}200$, matching the covariance calculation.

Common mistake avoided: Setting covariance to zero just because the losses cannot occur together is incorrect. Here $E[XY]=0$, but the covariance is $E[XY]-E[X]E[Y]=-200$, not zero.

Worked Example: Allocating a Portfolio Risk Load

Two hypothetical treaty losses have means $E[X]=100$ and $E[Y]=80$, in thousands of dollars, standard deviations $\sigma_X=40$ and $\sigma_Y=30$, and correlation $0.25$. For a single annual period with finite variances, calculate the portfolio standard deviation and allocate a portfolio risk load of $0.15$ times that standard deviation to each treaty using its marginal contribution to portfolio standard deviation. Assume no discounting.

Method. Use correlation to obtain covariance, then allocate the portfolio standard-deviation measure by each loss’s covariance with the total loss. This allocation adds exactly to the portfolio standard deviation.

The covariance is $\operatorname{Cov}(X,Y)=0.25(40)(30)=300$, in squared thousands of dollars. Thus

\operatorname{Var}(S)=40^2+30^2+2(300)=1{,}600+900+600=3{,}100, \qquad \sigma_S=\sqrt{3{,}100}=55.6776.

The portfolio risk load is $0.15(55.6776)=8.3516$ thousand dollars. For the allocation, $\operatorname{Cov}(X,S)=\operatorname{Var}(X)+\operatorname{Cov}(X,Y)=1{,}600+300=1{,}900$; similarly, $\operatorname{Cov}(Y,S)=900+300=1{,}200$. The standard-deviation contributions are therefore

\frac{\operatorname{Cov}(X,S)}{\sigma_S}=\frac{1{,}900}{55.6776}=34.1250,\qquad \frac{\operatorname{Cov}(Y,S)}{\sigma_S}=\frac{1{,}200}{55.6776}=21.5526.

They sum to $55.6776$, the portfolio standard deviation. Multiplying by 0.15 allocates $5.1188$ thousand dollars of risk load to treaty X and $3.2329$ thousand dollars to treaty Y. These sum to $8.3517$ thousand dollars, with the small difference from the unrounded portfolio load due only to display rounding. Including expected losses, the allocated prices are $100+5.1188=105.1188$ and $80+3.2329=83.2329$ thousand dollars, which sum to the portfolio price of $188.3516$ thousand dollars before rounding.

The allocation identity follows because $S=X+Y$:

\operatorname{Cov}(X,S)+\operatorname{Cov}(Y,S) =\operatorname{Cov}(S,S)=\operatorname{Var}(S).

Dividing by $\sigma_S$ makes the contributions sum to $\sigma_S$; multiplying each by the same loading factor makes the allocated loads sum to the total load. Common mistake avoided: Allocating the total load in proportion to expected losses would not, in general, reflect each treaty’s contribution to portfolio volatility.

Interpreting Diversification in Treaty Pricing

A common pricing illustration is the standard-deviation principle $P=E[S]+\lambda\sigma_S$, where $P$ is the indicated price, $\sigma_S=\sqrt{\operatorname{Var}(S)}$, and $\lambda$ is a nonnegative loading factor chosen for the pricing context. This is a stated pricing assumption, not a universal rule that all treaty prices must follow. Its advantage for this comparison is that it makes the portfolio effect explicit: under the same $\lambda$, the difference between separate and combined risk loads depends on the relationship between standalone and portfolio standard deviations.

For two losses, independence gives $\operatorname{Cov}(X,Y)=0$, so portfolio variance is the sum of the individual variances. Positive covariance makes it larger than under independence; negative covariance makes it smaller. Even with positive covariance, the combined standard deviation is no greater than the sum of the standalone standard deviations, since correlation cannot exceed 1. That does not mean every portfolio has substantial diversification: if losses are highly positively related, the reduction may be small.

In practical treaty analysis, a common peril or shared claims driver can create dependence. Historical losses may also give an unstable estimate of covariance, particularly when severe events are rare. A pricing analysis should therefore make its period, loss definitions, and dependence assumptions explicit, and test how reasonable changes in covariance affect the indicated volatility and risk load.

Define the portfolio.
Specify each treaty payment, the measurement period, currency, and whether the random variable represents gross or ceded loss.

Obtain the inputs.
Use variances and covariances, or standard deviations and correlations. Do not assume independence unless it is justified by the model.

Calculate combined volatility.
Add individual variances and twice each distinct pairwise covariance, then take the square root to obtain standard deviation.

Apply and reconcile the pricing rule.
Calculate the portfolio risk load using the chosen rule. If allocating that load, verify that component allocations sum to the portfolio amount.

Common Exam Traps

Omitting the factor of two. For two losses the cross term is $2\operatorname{Cov}(X,Y)$, not $\operatorname{Cov}(X,Y)$.
Confusing covariance and correlation. Covariance has squared currency units; correlation is unit-free. Convert using $\operatorname{Cov}(X,Y)=\rho_{X,Y}\sigma_X\sigma_Y$.
Assuming zero covariance means independence. The implication is not generally valid. Use the dependence information actually supplied by the loss model.
Comparing unlike quantities. Variances have squared units, while standard deviations and standard-deviation risk loads have the original currency units.
Adding standalone risk loads automatically. That is appropriate only if the pricing approach calls for separate standalone pricing. Under a portfolio-based load, use portfolio volatility and reconcile any allocation to that total.

Key takeaway: Treaty portfolio volatility depends on both individual loss variances and pairwise covariances. A defensible diversification adjustment follows from the stated dependence model and pricing rule; it should not be inserted merely because losses have been grouped together.

Check Your Understanding

For each problem, use the stated common-period loss model and keep monetary units consistent.

Two independent treaty losses, in thousands of dollars, have variances 400 and 900. Find the portfolio variance and standard deviation. (Answer: variance $1{,}300$ thousand-dollars squared; standard deviation $\$36{,}055.51$.)
Treaty A pays $40,000 with probability 0.10; treaty B pays $100,000 with probability 0.05. Both pay only in the joint event, which has probability 0.01. Otherwise each pays zero. Find the covariance and portfolio variance. (Answer: covariance $\$20{,}000{,}000$; portfolio variance $\$659{,}000{,}000$ squared.)
Two treaty losses have standard deviations $25,000 and $40,000 and correlation $-0.20$. Find the portfolio variance and standard deviation. (Answer: variance $\$1{,}825{,}000{,}000$ squared; standard deviation $\$42{,}720.02$.)
A portfolio has expected loss $80,000 and standard deviation $60,000. Under $P=E[S]+0.12\sigma_S$, calculate its price. (Answer: $\$87{,}200.00$.)
Treaty losses have standard deviations $20,000 and $35,000. What is the largest correlation that keeps the portfolio standard deviation at or below $45,000? (Answer: $2/7$, approximately $0.2857$.)

What You'll Learn