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# Portfolio's Annual Value at Risk (VAR) at a 5 percent probability level

#### goodyhi11

##### New Member
Subscriber
Hi Turtles,

I have a question, when a question ask you about the Var at a 5% prob level, what is the z value that we use? (Var = Portfolio value * [(E(R) - z(sigma)])

I thought the z value at 95% prob level is 1.96, but apparently, the answer is 1.65. Can someone please shed some light on this? I am a bit confused with regards to what value to use, I bet it is related to the interpretation of single vs. double tails of the distribution).

Thanks much!

#### Nicole Seaman

Staff member
Subscriber
Hi Turtles,

I have a question, when a question ask you about the Var at a 5% prob level, what is the z value that we use? (Var = Portfolio value * [(E(R) - z(sigma)])

I thought the z value at 95% prob level is 1.96, but apparently, the answer is 1.65. Can someone please shed some light on this? I am a bit confused with regards to what value to use, I bet it is related to the interpretation of single vs. double tails of the distribution).

Thanks much!
Hello @goodyhi11

Do you have the actual practice question that you are referring to? It is helpful to see the full practice question and answer not only because it provides more detailed information, but also it is easier to search the forum to see if your question has already been answered.

Thank you,

Nicole

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @goodyhi11 I agree with @Nicole Seaman that a reference to the actual Q&A/example would be helpful, if only because it would be wrong. First, please do note that either Z = 1.65 or 1.96 refers to a normal distribution, as you may already know. But I like to remind folks that value at risk (VaR) does not require a normal distribution! (We tend to use the normal for things like testing regression coefficients because the central limit theorem often justifies a normal, but that's a totally different application ...).

Okay, then if we do mean to suggest that the appropriate or justifiable distribution for risk measurement purposes is the normal distribution, and we are relying on the analytical, parametric VaR approach (as opposed to simulation based approaches, HS or Monte Carlo), then we always want a one-side quantile, most often N(.950) = 1.645 or N(0.99) = 2.333. We would never retrieve a 95% normal VaR with Z = 1.96 or a 99% normal VaR with Z = 2.58 because these are two-sided; they are appropriate for two-side hypothesis testing. But VaR is only concerned with the one-side of the distributions where the worst losses might occur. I Hope that helps, we have much resource on this topic, in many detailed variations, in this forum!

#### Sixcarbs

##### Active Member
Subscriber
Hi @goodyhi11 I agree with @Nicole Seaman that a reference to the actual Q&A/example would be helpful, if only because it would be wrong. First, please do note that either Z = 1.65 or 1.96 refers to a normal distribution, as you may already know. But I like to remind folks that value at risk (VaR) does not require a normal distribution! (We tend to use the normal for things like testing regression coefficients because the central limit theorem often justifies a normal, but that's a totally different application ...).

Okay, then if we do mean to suggest that the appropriate or justifiable distribution for risk measurement purposes is the normal distribution, and we are relying on the analytical, parametric VaR approach (as opposed to simulation based approaches, HS or Monte Carlo), then we always want a one-side quantile, most often N(.950) = 1.645 or N(0.99) = 2.333. We would never retrieve a 95% normal VaR with Z = 1.96 or a 99% normal VaR with Z = 2.58 because these are two-sided; they are appropriate for two-side hypothesis testing. But VaR is only concerned with the one-side of the distributions where the worst losses might occur. I Hope that helps, we have much resource on this topic, in many detailed variations, in this forum!
Hi David,

I know one of the PQ's referred to the best way to measure the risk of a straddle, but your post above made me think. Could you use a 2-sided z number to measure the VaR of a short straddle? (Or maybe other negative gamma positions?)

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @Sixcarbs I'm always open to new possibilities and that's super clever, but I don't see how: it's true the short straddle experiences losses on both sides of the profit/payoff diagram, but such payoffs neverthess would be (should be) translated into a distribution of gains/losses where the losses are all on one side only. The profit/payoff plot is not the same thing as profit/payoff probability distribution, to my thinking .... stepping back, VaR is the quantile of a probability distribution, so we do need to estimate/approximate a probability distribution ...