Lognormal value at risk (VaR, FRM T5-01)
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- čas přidán 5. 06. 2024
- Welcome to the first video in this new playlist that is devoted to Topic 5 in the FRM. Topic 5, Market Risk, is the first topic in Part 2. We will start here by comparing normal to lognormal VaR and, specifically, we are going to generalize to absolute VaR. Absolute VaR generalizes the relative VaR so it's the complete version of VaR. The key thing that we are going to do here is look at four different use cases so we can compare normal VaR to lognormal VaR in the single-period case. Normal is when we assume that the arithmetic returns are normally distributed and lognormal is when we assume that the geometric returns are normally distributed.
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This is a great review... I am definitely recommending to my friends... Thanks!
Incredible explanation. Awesome!
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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.
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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)?
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,
@@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?
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?
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
So.
LogVar(%) = 1 - exp(-NorVar(%))