Question

A portfolio of stocks has a two-day 99% VaR of $1 million. Assuming the returns are identically and normally distributed, what would be an appropriate translation of this VaR to a 95% confidence level over a 10-day horizon?

Answer 1

To translate a two-day 99% VaR to a 95% 10-day VaR, adjust the original VaR by the ratio of the standard normal z-scores for 99% and 95% confidence levels and scale by the square root of the time factor.

Step-by-step explanation:

To translate a two-day 99% VaR of $1 million to a 10-day horizon at a 95% confidence level, we need to adjust for the difference in confidence levels and the time horizon. VaR scales with the square root of time due to the properties of variance under the assumption of normally distributed returns, and the confidence level adjustments are made by understanding the z-scores corresponding to the normal distribution.

First, identify the z-scores from standard normal tables: a 99% confidence corresponds to a z-score of approximately 2.33, and a 95% confidence corresponds to a z-score of approximately 1.65. Next, scale the VaR to a 10-day horizon and the desired confidence level by multiplying the original VaR by the ratio of the z-scores and by the square root of the scaling factor for time, which is the square root of (10/2) as we are going from 2 to 10 days. This results in a new VaR: $1,000,000 * (1.65/2.33) * sqrt(10/2).