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Sum of independent normal period losses

For n independent identically distributed normal period losses, the sum has mean nμ and dispersion √nσ.

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Define additive period losses

Let each period’s loss be normal with mean μ and variance σ², and assume the losses are independent and identically distributed. The aggregate is their arithmetic sum. This setup is not automatically a model of compounded returns, changing positions or dependent market observations.

Sum the means and variances

μsum=nμ,σsum=nσ,VaRc=nμ+zcnσ\mu_{\mathrm{sum}}=n\mu,\qquad\sigma_{\mathrm{sum}}=\sqrt{n}\sigma,\qquad\mathrm{VaR}_c=n\mu+z_c\sqrt{n}\sigma

Independence removes cross-period covariance; joint normality supplies the sum’s normal law.

Use the corresponding tail average

Under these assumptions, aggregate ES is nμ+√nσ φ(zc)/(1−c). The mean term scales with n, so multiplying the entire one-period VaR or ES by √n generally gives a different answer when μ is nonzero.

Check the required assumptions

Dependence adds covariance terms; different periods can have different moments; nonlinear aggregation changes the loss definition. The public calculator receives an already stated same-horizon mean and dispersion and performs no automatic horizon extension.

Further references