The risk manager is conducting a backtest on a 1-day 99.5% Value-at-Risk (VaR) model over a 10-year period at a 95% confidence level. The model is considered correctly calibrated if the number of times the actual losses exceed the VaR is within an acceptable range. The daily returns are assumed to be independently and identically distributed, and there are 250 trading days in a year.

To determine the maximum number of acceptable exceedances, we use the following formula derived from the binomial distribution:

$Z = \frac{x - pT}{\sqrt{p(1-p)T}}$

Where:

• $Z$ is the Z-score for the confidence level.
• $x$ is the number of exceedances.
• $p$ is the probability of an exceedance, which is the left tail level of the VaR (1 - 0.995 = 0.005 or 0.5%).
• $T$ is the total number of observations (250 trading days/year * 10 years = 2500).

Given a 95% confidence level, the Z-score $z$ is 1.96. Plugging in the values:

$1.96 = \frac{x - 0.005 \times 2500}{\sqrt{0.005 \times (1 - 0.005) \times 2500}}$

Solving for $x$:

$x \approx 19.4$

Since we cannot have a fraction of an exceedance, we round down to the nearest whole number, which gives us 19. Therefore, the maximum number of daily losses exceeding the 1-day 99.5% VaR in 10 years that is acceptable to conclude that the model is calibrated correctly is 19.