# Problem in calculating a simple VaR

In

Alexander, Gordon J. and Alexandre M. Baptista (2006). Does the Basle Capital Accord reduce bank fragility? An assessment of the value-at-risk approach. Journal of Monetary Economics 53(7), 1631–1660.

at page 1644 there is:

Consider the following example that is based on a 10-trading day investment horizon and 99% confidence level, as required by the Basle Capital Accord. Suppose that the expected rate of return and standard deviation of efficient portfolios $$S$$ and $$L$$ are given by: $$E[r_S] = 0.50\%$$, $$\sigma[r_S] = 0.40\%$$; $$E[r_L] = 1.00\%$$, and $$\sigma[r_L] = 0.60\%$$. It follows that $$V[0.99; r_S] = 0.43\%$$ and $$V[0.99; r_L] = 0.40\%$$, ...

and at page 1636 $$V$$ is defined as:

For any $$t\in(\frac{1}{2},1)$$, let $$z_t \equiv -\Phi^{-1}(1-t)$$, where $$\Phi(\cdot)$$ is the standard normal cdf. Using the assumption of normality, portfolio $$w$$’s VaR at $$100t\%$$ confidence level is: $$V[t,r_w]\equiv z_t\sigma[r_w]-E[r_w].$$

I tried to calculate the VaRs in the example but I don't get the same results, even if I scale the $$\sigma$$ by a $$\sqrt{10}$$ factor.

The calculation assumes that returns are normally distributed. VaR is a percentile of the returns distribution, which in turn can be expressed as a multiple (here labelled $$z$$) of the standard deviation of returns. (This works as along as the standard deviation exists for the assumed distribution.) For the $$99\,\%$$ confidence under a normal distribution, the multiple is $$2.33$$. So, in the first example, $$2.33 \times 0.4 - 0.5 = 0.43$$.