VAR and E(R)

Does a higher E® lead to a higher or a lower VaR ? Can someone explain it intuitively?

It does if we are thinking of basics where higher E® requires more risk. If you increase risk you will increase your variance. Assuming the same VaR settings for all your scenarios, VaR should be higher with more risk.

comparing portfolio with 50/50 stock/bond vs portfolio with 100% stocks.

It is ambiguous. If we consider Var=μ-zσ you can see there is a conflict as higher return is pulled down by the sigma component, but the magnitudes are not clear.

It is affected by all 3 variables, Est. R, sigma and zeta. And if higher return is achieved by option trading for example in one portfolio such outcomes might not be comparable to another option free portfolio by VaR.


Where did you find such question? There would not be such simple rules to remember. Maybe better solution would be to concentrate on advantages and drawbacks of each method.

VAR is a negative number because it is a loss #.

Higher E®, assuming other factors remain unchanged - would mean that VAR # comes down in absolute terms.

With an expected return of 0.135, we move 1.65 standard deviations along the x-axis in the direction of lower returns. Each standard deviation is 0.244. Thus we would obtain 0.135 – 1.65(0.244) = –0.268.36 At this point, VAR could be expressed as a loss of 26.8 percent.

Reading on

Some approaches to estimating VAR using the analytical method assume an expected return of zero. This assumption is generally thought to be acceptable for daily VAR calculations because expected daily return will indeed tend to be close to zero. Because expected returns are typically positive for longer time horizons, shifting the distribution by assuming a zero expected return will result in a larger projected loss, so the VAR estimate will be greater. Therefore, this small adjustment offers a slightly more conservative result and avoids the problem of having to estimate the expected return, a task typically much harder than that of estimating associated vol- atility. Another advantage of this adjustment is that it makes it easier to adjust the VAR for a different time period. For example, if the daily VAR is estimated at $100,000, the annual VAR will be $100,000 250 $1,581,139 . This simple conversion of a shorter-term VAR to a longer-term VAR (or vice versa) does not work, however, if the average return is not zero. In these cases, one would have to convert the average return and standard deviation to the different time period and compute the VAR from the adjusted average and standard deviation.

so making the return smaller - increases the VAR - so makes it more conservative.

this would imply - making the E® BIGGER - would make the VAR smaller.

Hi Cpk sir,

you’re only talking specific to changing time horizon right? not comparing 2 different portfolios of different risk and return. I guess my response isn’t correct because those factors wouldn’t be known?


I found similar vignette style question at Risk management EOC. The risk increase was not driven by asset return increase, assuming all else constant but with positive correlation between portfolios. Maybe, Jaffacake meant on this question.