Coherent risk measures and why VaR is not coherent (FRM T4-5)

[my xls is here trtl.bz/2ErWQl8] Coherence requires that a risk measure meets all four of the following conditions unconditionally: 1. Translation invariance (aka, adding cash reduces risk), 2. Positive homogeneity (aka, risk is proportional to size"), 3. Monotonicity (aka, If Y dominates X, then Y is less risky than X), and 4. Subadditivity (aka, the risk measure should not penalize diversification). Value at risk (VaR) is a popular risk measure but VaR is NOT coherent because it is not necessarily sub-additive (instead, VaR is only subadditive if the returns are normally distributed). We can illustrate VaR's lack of subadditivity by observing that the VaR of a single bond can easily be zero, yet when combined into a portfolio of identical bonds, the portfolio VaR is greater than zero. VaR is often not subadditive when such a property is most desired: when the tails are heavy. Lack of subadditivity is of practical significance.
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[@monkiedeinhau557]

This is the best explanation I have heard regarding these topics, thank you.

[@Narek173377]

Problem here is that the number of bonds can be only integer (whole number). That is to say you can not own 1.5 bonds or 0.7 of a bond. You ether have 1, 2, 0 etc. So 95% Var is perhaps the default of 1.5 bond and 96% Var is the default of 1.8 bonds. We can say loosing 1.5 bond or 1.8 bond is the same as loosing 1 bond in both cases as you simply can't loose .5 or .7 of a bond. thus 95% probability is equal to 96% probability. So may be the problem is not in VaR? May be the problem is in using VaR in Bernoulli distribution?

[@peiwang3223]

Thanks a lot fot this very clear and detailed explanation, really clarify my confusions,

[@rjmorpheus]

I love these videos...would it be too much to ask recommendations on viable textbooks for this subject in particular? VaR and ES?

[@ta55o5]

Kevin Dowd - Measuring Market Risk

[@rjmorpheus]

@@ta55o5 Thank you....

[@investwithvincent6329]

18:28 Why is that risk measures becomes more important of a concern when we have fat tails?

[@y5jeyfuyf]

skew and kurtosis, we cant go beyond 3 thus mean is not representative therefore, we use GARCH and EWMA

[@investwithvincent6329]

12:48 What does "0" represent?