Aggregate Loss VaR for a Homeowners Book
A homeowners writer has 50,000 policies in force. Claim frequency is modeled as Poisson with lambda = 0.08 per policy-year, so 4,000 claims are expected. Severity is modeled as gamma with mean $20,000 and standard deviation $30,000. Frequency and severity are assumed independent, and claims are assumed independent across policies. The chief actuary proposes holding economic capital at the 99.5% VaR of the aggregate loss distribution implied by this model, computed with a normal approximation to the aggregate. The company's catastrophe model, which is excluded from the aggregate calculation above, puts the net 1-in-200 hurricane loss at $40 million.
1. Compute the mean, standard deviation, and 99.5% normal-approximation VaR of aggregate losses, and assess whether the normal approximation is defensible on distributional grounds.
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Moments (in $000s). Expected claims = 50,000 x 0.08 = 4,000.
E[S] = 4,000 x 20 = 80,000, i.e. $80.0M.
E[X^2] = 30^2 + 20^2 = 1,300. Var(S) = 4,000 x 1,300 = 5,200,000, so sd(S) = $2.280M.
99.5% VaR (normal). 80.0 + 2.5758 x 2.280 = $85.87M, i.e. capital of about $5.9M.
Is the normal defensible? Check the skewness of S. For the gamma, alpha = (20/30)^2 = 0.4444 and theta = 900/20 = 45, so E[X^3] = 45^3 x alpha(alpha+1)(alpha+2) = 91,125 x 1.569 = 1.430e5. Then
skew(S) = lambda E[X^3] / (lambda E[X^2])^1.5 = 5.72e8 / 1.186e10 = 0.048.
The aggregate is almost symmetric even though individual severities are strongly skewed (skew 3): with 4,000 expected claims the central limit theorem has largely done its work. So the shape objection to the normal is weak; the error from skewness at the 99.5% point is a fraction of one standard deviation. The serious problems lie elsewhere.
2. The catastrophe model implies a net 1-in-200 hurricane loss of $40 million. Reconcile this with the $85.87M VaR and state what capital figure you would actually recommend.
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The two numbers are incompatible, and the cat number governs.
The aggregate model assumes independent claims across policies. Hurricane losses are the opposite: a single event drives thousands of claims simultaneously, and the resulting loss is not a draw from the compound-Poisson distribution at all. If a 1-in-200 event produces $40M of net loss, then by definition the 99.5% quantile of the true aggregate distribution is at least $40M of event loss on top of ordinary non-cat activity, not $85.87M total including an $80M expected loss.
Framing the recommendation. Compare on the same basis - loss, not loss-plus-expected:
- Model-implied 99.5% exceedance of non-cat losses above the mean = $5.87M.
- Cat model 1-in-200 = $40M.
A defensible approach is to hold capital against the convolution: simulate or analytically combine the non-cat aggregate with the cat loss distribution and read the 99.5% point off the combined distribution. As an approximation, capital must be at least the cat figure plus a contribution from non-cat variability, i.e. of the order of $42-46M rather than $5.9M. The $85.87M VaR level is not the issue; the $5.9M of capital above the mean is, and it is understated by close to an order of magnitude.
Also missing: parameter and trend uncertainty (the 20/30 severity parameters are themselves estimates), and any dependence between frequency and severity, which is usually positive in property lines.
3. The regulator is choosing between a 99.5% VaR standard and a 99% TVaR standard. Which fits this book, and how does the answer change the treatment of the catastrophe?
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VaR vs TVaR for this risk. VaR asks only where the distribution crosses a threshold; it says nothing about how bad things are beyond it, and it is not subadditive, so two legal entities each passing a VaR test can fail when combined. For a book whose tail is a small number of severe, correlated events, those are exactly the wrong properties: the 99.5% VaR can be met while the average shortfall in the worst 0.5% of years is several times the capital held.
How the choice interacts with the cat. Under VaR, a single 1-in-200 event sits right at the threshold, so the capital requirement is knife-edge: small changes in the cat model move the answer discontinuously, and the model can be gamed by adjusting the event set. Under 99% TVaR, the requirement is driven by the average of the worst 1% of outcomes, which weights the severe hurricanes that actually threaten solvency and is far more stable to reparameterization. TVaR is also coherent, which matters if the group has multiple legal entities.
Practical recommendation. If the standard must be VaR, hold it on the combined non-cat-plus-cat distribution and add an explicit cat margin on top. If TVaR is available, use 99% TVaR on the combined distribution and disclose the dependence assumption between cat events and non-cat frequency. Either way, do not report the $5.9M figure to a board as economic capital.