Lognormal value at risk (VaR, FRM T5-01)

[@rhercula]

This is a great review... I am definitely recommending to my friends... Thanks!

[@Blueshockful]

this is awesome. you're literally saving my life. Thanks a ton

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[@andreast2168]

Incredible explanation. Awesome!

[@MR-ys4ui]

Sorry, could you please explain how in Normal VaR calculations you obtained relative % drift and relative % standard deviation for arithmetic returns. Doesn't the assumption of arithmetic returns implies that the drift and standard deviation are in expressed in absolute (money) values rather than %? How do you calculate the drift and standard deviation for absolute returns such that the result is a % drift and % standard deviation? These two seem to be contradictory to me. Thank you

[@plotus1937]

This is a little tangential. But I tried comparing these parameterized versions of calculating not necessarily VaR, but the 5th percentile portfolio value of an asset in 30 years. So I scaled mean return and stdev to 30 years and just used norminv(0.05, scaled_mean, scaled_stddev). I noticed that this value seems very far from the 5th percentile portfolio value if I run a Monte-Carlo simulation. In fact the further away the percentiles are from median, the more extreme the difference in these two methods. Do you know why this would be the case? To set up my MC simulation I used the arithmetic mean instead of geometric to account for volatility drift, but still the same stddev of the normal returns.

[@nononnomonohjghdgdshrsrhsjgd]

In the case of log returns (after minute 14) the mean and variance are calculated from the ln(Pt/Pt-1), right? To be more exact in my question: in the VaR-term (1- exp(Mean-Sigma x Z)), the mean and sigma are derived from the ln(Pt-Pt-1)-returns, and in the VaR (Mean-Sigma x Z) the mean and variance are based on the arithmetic returns (Pt/Pt-1)?

[@bionicturtle]

Yes, that is *correct* ! An arithmetic (aka, simple) return = P_t/P_t-1 - 1 and "normal" aVaR = -μ + σ*Z assumes these arithmetic returns have a normal distribution. A geometric return = ln(P_t/P_t-1) and "lognormal" aVaR = 1 - exp(μ - σ*Z) assumes these geometric returns have a normal distribution which, in turn, implies the price, P_t, has a lognormal distribution. If r = ln(P_t/P_t-1), then it follows that the VaR quantile must be at P_t-1*[1 - exp(μ - σ*Z)]. Thanks,

[@nononnomonohjghdgdshrsrhsjgd]

@@bionicturtle I thank you for the answer. I have second question: GBM is the case of Brownian motion when we use the mean and variance from log(returns) in the Monte Carlo Simulation. Brownian Motion is called for the case of arithmetic returns?

[@elinab376]

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[@Fuad2016]

So. LogVar(%) = 1 - exp(-NorVar(%))

[@samirhantour8979]

Very nice and thank you very much! What if the mue is higher than sigma? Let's say mue = 2% and sigma = 1% at 95% confidence. How can the result be interpreted?