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Scaling VaR to a longer horizon

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A position worth $5000$ has independent normal daily returns with mean $0$ and standard deviation $1.2%$ . Use $z = 2.326$ for the $99%$ quantile of $N (0, 1)$ . What is the $99%$ VaR over $9$ trading days, in currency units? Give an exact decimal.

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Hint

Variance adds across independent days, so the standard deviation grows like $T$ .

Solution

One day's standard deviation in currency is $1.2% \times 5000 = 60$ .
Over the horizon it scales by $9 = 3$ , giving $60 \times 3 = 180$ .
The VaR is $2.326 \times 180 = 418.68$ .

Answer: $418.68$

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