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Sarah manages a hedge fund with a portfolio valued at \$2,000,000. The portfolio's daily returns have a standard deviation of \$3,000 and an average daily return of \$1,200. Calculate the five-day VAR at a 99% confidence level for Sarah's portfolio.

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    $\begingroup$ I tried ChatGPT, and it almost got it right, save for an arithmetic error. Perhaps it could give a head start for you, too. $\endgroup$ Nov 21, 2023 at 10:30
  • $\begingroup$ just tried it, the formula used is different. $\endgroup$ Nov 21, 2023 at 15:19

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It is really simple. The formula is just:

$VaR_{\alpha,T} = -\mu T + Z_{\alpha} \sigma \sqrt{T}$

Take note the time horizons should match between the drift and the vol terms. Also VaR is usually represented positively despite being a loss. Therefore, the PnL distribution is "reversed".

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  • $\begingroup$ The Z represents the one-sided normal deviate. $\endgroup$
    – KaiSqDist
    Nov 21, 2023 at 10:43
  • $\begingroup$ I'm actually not sure whether we're allowed to include the positive μ here. If Sarah has done better than risk-free returns hisorically, may we assume that her lucky streak will continue? Also, If the "daily standard deviation" σ is annualized, then do we need to divide by the number of trading days per year somewhere? $\endgroup$ Nov 21, 2023 at 12:35
  • $\begingroup$ Hi @DimitriVulis, I think OP is just asking a FRM question, so it depends if the question is asking for relative (in which case you are right) or absolute VaR. The lucky streak continuation part goes more into returns forecast and I think the question doesn't go that deep into what is best used as an expected return estimate. On the standard deviation the answer should be to just compute VaR first and then lengthen the horizon *5^0.5. But if the standard deviation is annualized then we should divide by the root of trading days, not by the number of days if I recall correctly. $\endgroup$
    – KaiSqDist
    Nov 21, 2023 at 12:49
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    $\begingroup$ Thanks, I believe you're right about 1/the number of trading days per year being inside the square root. Just thinking out loud further, in a historical VaR, if one has historically flipped a coin 250 times and got tails, say, 240 times, then there's no choice but to assume that this streak will continue. But in parametric VaR, forcefully assuming that all returns are normal or lognormal, and using their historical volatilities (and correlations), I'm genuinely not sure whether using also historical non-zero mean returns or excess returns complies with the spirit of Basel II. $\endgroup$ Nov 21, 2023 at 13:36
  • $\begingroup$ @Kai Thanks for the answer! I just had another doubt. The answer comes out be 10356.6 as per the equation. (-1200*5+3000*2.33*2.34). However, usually in excel we use the NORM.INV function to determine the min probability of the mean return and at 99% confidence, it's -0.002889522. Now the daily VaR is -5779.043622 and after multiplying it with the square root of 5 days, the answer is different than 10356 var. Any idea on this? $\endgroup$ Nov 21, 2023 at 14:47

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