*Submitted by Salil Mehta of Stiatistical Ideas*

**The Value-At-Risk Fiasco**

If you were a looking at a simple portfolio that was a mix of 1 unit S&P 500 and 1 unit Shanghai Stock Exchange (SSE), then you are likely to consider value-at-risk (VAR) to feel cozy with your overall portfolio risk. This measure however is not considered a **coherent** risk measure that satisfies all of the properties of interest: monotonicity, translation invariance, homogeneity, and subadditivity. We'll explain the first three in a future article, but only focus here on how VAR violates the last of these four properties.

Subadditivity is where the risk associated with multiple holdings, in a portfolio, *should not be ** greater* than the sum of the individual holdings' risk. This construes the hallmark of diversification, and yet combined with the inappropriateness of VAR to measure market risk we see subadditivity levels violated. Risk events that should have only happened say one month every 1.5 years have occurred in each of the past three summer months.

**in the S&P, for each month starting more than 5 years ago in January 2010 (and through May 2015). A total of 65 months. We can set a probability tolerance of just over 6%, and state that the probability of seeing a**

*worst weekly losses***than this VAR should be**

*loss greater**~6% (or 1 in 16 months). We can vary this level about, but this is simply a foothold to initiate our analysis.*

**less than or equal to** VAR in this case, for the S&P, would come to a worst weekly loss of **6.0%**. Bear in mind that the * average *worst weekly loss over the 65 months for the S&P was a 1.9% loss. Now we do the same exercise for the SSE, and with the same probability tolerance of ~6% we get a VAR loss of

**5.3%**. Here the

*worst weekly loss over the 65 months for the SSE is a 2.6% loss. Note that the parametric mathematical relationship to estimate the overall VAR from blending two equally volatile stocks (or indexes) does relate to the correlation between those 2 indexes.*

**average**

*VAR*_{overall }

*= VAR*_{index}

*?[*

*(1/number of indexes) + (1-1/number of indexes)*

*r]*

*= VAR*_{index}

*?[½ + ½*

*r]*

**, versus that for the S&P, is a reflection of the nonparametric nature of equity returns. This turns out to be critical as we go through this article. Also keep in mind that these VAR express a fixed week period, and not any continuous range beyond that. We know the maximum-VAR was higher if we simply augment to the trading week, ending August 21, the following "Black Monday". And most also know by now, that these stock returns are not related to the normal distribution (or elliptical distributions for that matter), and now we also see these fat tails are not even related to one another. For more on the Student**

*loss level***distribution, see here, here.**

*t**, -6.8%, and the worst weekly loss before the summer of 2015 was*

**-6.6%***.*

**-7.5%** For the SSE, the changes were: **-5.3%**, * -6.6%*, and

*. The 3 months associated with the S&P above, and the 3 months associated here with the SSE, have one month in common (May 2010). The four*

**-6.9%****bolded**months of the six months noted (3 S&P and 3 SSE) are part of the worst 3

**joint**, "worst weekly losses". We show these 3 joint losses below, where again the portfolio constitutes 1 unit S&P and 1 unit SSE (for a portfolio that is 50% in both indexes you would take ½ of every loss and VAR for the purposes of comparison):

(-5.2%) +

**-7.5%**+ (-2.8%) = -10.3%(-5.2%) +

**-6.9%**= -12.1%**-6.6%**+**-6.6%**= -13.2% (this is May 2010)

*Note values in () are the paired worst weekly loss for the*

*bolded**S&P or the SSE worst weekly losses.*

**than the sum of the two holdings' VARs. Or 11.3% (**

*no greater***6.0%+5.3%**). And empirically we see above that the portfolio VAR comes to smaller than a 10.3% loss.

*June 2015: -0.7% + -14.3% = -15.0%*

*July 2015: -2.2% + -12.9% = -15.1%*

*August 2015: -5.9% + -12.3% = -11.9%*

In ** each **of these 3 months, the S&P always stayed within VAR yet the overall losses still were

*always***than the 10.3% VAR (and all were greater than the theoretical 11.3% VAR for that matter!) A 1 in 16 months event**

*greater***happening 3 months straight is**

*immediately***not**a

**quirky <0.02% (6%**

^{3}) probability situation. It was a case of incorrectly using VAR as

**nonparametric risk measure for the market we are modeling (e.g., "extreme" tail risk events). Despite how commonly it is endeared anyway by investors and middling stress testing regulators.**

*the preferred*