Hi

@mrporters
The daily volatility is 2.0% and the VaR is concerned with the worst expected loss at some confidence level. So σ(1-day)*CF = 2.0%*2.0 = 4.0% means that we expect

*better than *(i.e., greater than) a one-day return of -4.0% on 97.7% of days or 97.70/100 days; or that we expect

*worse than* (i.e., less than) a one-day return of -4.0% on 2.3% of days or 2.3/100 days. By multiply 2.0% by ~2.0 we are scaling the volatility according to confidence level (while assuming a distribution), it's one confidence level among many:

- 95.0% one-day %VaR = 2.0%σ * NORM.S.INV(95%) = 2.0% * 1.645 = 3.290%,
**97.7% one-day %VaR = 2.0%σ * NORM.S.INV(97.7%) = 2.0% * 1.995 = 3.991%**
- 99.0% one-day %VaR = 2.0%σ * NORM.S.INV(99%) = 2.0% * 2.326 = 4.653%
- 99.9% one-day %VaR = 2.0%σ * NORM.S.INV(99.9%) = 2.0% * 3.090 = 6.180%,

We further scale the volatility/VaR by multiplying by the square root of (target days/current days). If you wanted a 10-day VaR, you would multiply by sqrt(10/1) per 2.0% * 1.995 * sqrt(10/1) = 12.62% because you'd be scaling over time from a 1-day VaR ("1" in the denominator) to a 10-day VaR ("10" in the numerator). The square root rule says volatility scales by the square root of time (because variance is linear. Why is this if daily returns are independent?). So now that we've scaled volatility by confidence (to get VaR) and over time, we have:

97.7% T-day %VaR = 2.0%*1.995*sqrt(T/1); in the case of a one-day var:

97.7% 1-day %VaR = 2.0%*1.995*sqrt(1/1) = 2.0%*1.995 = 3.99%.

That's the valid % version of VaR, which you just multiply by the initial wealth to get the dollar-based version, per the formula you are showing:

The 97.7% 1-day $VaR = 2.0%*1.995*sqrt(1/1) * (500 * $800) = 2.0% * 1.995 * 1.0 * $400,000 = $15,963.

So the formula is actually pretty cool: aside from multiplying by the wealth to translate % to $, the two other multipliers just scale a one-day volatility over time and by confidence. Importantly, we've

*assumed *a normal distribution for the returns. We used CF = 2 because we assumed a normal distribution, but a different distribution will have a different one-sided quantile (CF) at the 97.7% confidence level. I hope that helps,

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