Downside Risk

[hawayi_vgo]

Assume the spot and one-year forward price of gold is $1200 per ounce; i.e., neither contango nor
backwardation in gold forward curve. Assume the (initial) margin requirement for a gold futures
contract is 10% of the notional. Gold has an expected per annum return of 10% with continuous compounding and volatility of 10%. Assume risk is represented by the one-year 99% normally distributed value at risk (VaR). Under continuous compounding, if we have $12,000 to invest in gold, what is (i) the expected return and downside risk of a cash spot investment and (ii) the
expected return and downside risk of the same amount leveraged into gold futures?

The answer - (i) At 99%, the normal deviate is 2.33 such that the downside risk is 10%*2.33 = -23.3%

a) Why is the downside risk ---> volatility*2.33?
b) Is 2.33 the p value?
C) Isn't it that volatility of 10% alone signifies the downside risk...

I get a bit of the idea of the answer but can't 100% understand it. Please help. Thank you!

 

[ShaktiRathore]

Hello there,
The 2.33 is the z value its a standardized value for a normal distribution with mean 0 and standard deviation 1 which signifies that the value at 99% confidence level is 2.33 times standard deviations from the mean value 0. Higher the CL the higher the z value which means the value is further away from mean value. we assume normal distribution for any portfolio returns the z values are same for any normal distribution of a given mean and standard deviation because these z values are standardized values.
Now z value is given by x-hypothesized mean/std deviation =x%-0/10%=2.33 => x=10%*2.33 which mean that for the given normal distribution with mean 0 and standard deviation 10% the number of times of standard deviations that the actual value x is far from the mean value of 0% should be 2.33 at 99%CL. x represents the maximum loss level because it represents the worst case scenario of how returns can deviate from mean value at 99%CL assuming returns are normally distributed on the downside(1 tail). Thus the maximum loss in terms of returns that can occur at 99% CL is 10%*2.33=23.3%.
p value is a measure of significance p< significance level means that we reject null hypothesis and p>significance level we accept the null hypothesis. a p value represents area under normal distribution to the left of the given value x so if our z value for 99%CL is 2.33 significance level 0f 1-.99=1% and p value for the given loss of say x is .5% then it means that the value x lies to the left of z value i.e. x>z so that area to left of x(.5%) is less than to the left of z(1%) value that means x value is further from the mean than required by given hypothesis test that maximum that x value can deviate is 2.33 for x to be considered equal to hypothesized mean but as x deviates further from z critical value we reject the null hypothesis that x is equal to the mean. Thus as p<1% we reject the null hypothesis.
10 % alone would have signified the downside risk if we had not assumed a normal distribution, in case returns would have a distribution which is a uniform distribution of a straight line in that case we would simply say that returns can deviate maximum 10% on downside. i x be returns and s be std. deviation equation of line passing through mean 0 and s 1 is and slope as 1 is: x=1(s-1)=s-1, so that change in x is change in s and the maximum that x can deviate negatively is 10%. Here we do not consider the curvature of normal distribution that would result in a different value of downside risk.

hpe you understood
thanks

 

[hawayi_vgo]

Got you.

However, isn't based on the information presented, gold is normally distributed with mean of 10% and standard deviation (volatility) of 10% - Not standard normal distributed since it mean not equal to 0, and s.d. not equal to 1 and it hasn't been transformed into a Z score.

Since a random variable which in in this case VAR could be transformed into a Z score by

Z = (x - mean)/s.d. ; s.d. = volatility

and given Z = 2.33

Isn't it that

x = volatility*Z + mean
= 10%*2.33 + 10%

Basically, I'm saying that mean used in the calculation should be 10% and not 0. It is only X transformed into
a Z score that its distribution would be (mean = 0, s.d. = 1)

Am I missing something somewhere?

Thanks.

 

[ShaktiRathore]

Hi,
Basically we need to find the maximum loss that can occur for a portfolio or any position. VaR is defined as the maximum level of loss that a portfolio can suffer over a period at a given confidence level. As its a parametric approach we assume a normal distribution. We are interested in downside risk of the portfolio that is the negative side of normal distribution. So that the z value is negative,
Z = (x - mean)/s.d => -2.33=(x - mean)/s.d=>x=mean-s.d.*2.33 is the absolute VaR that accompanies the expected mean return with the loss level.
There is also the relative Var as is mentioned above where the loss level of portfolio does not consider the expected return and the loss is taken relative to 0% return. so that mean=0 and -2.33=(x - mean)/s.d=>x=0-s.d.*2.33 =-2.33*s.d. So you can see the difference. You have wrongly assume that loss is positive above but its negative that is we are interested only in the left side of the distribution that is how bad things can get that is what Var is and mean is on the positive side.

thanks

 

[Backwardation]

I would say that hawayi has posted the right formala to calculate the VaR, that is to include the mean. So recognizing the mean in the VaR calculation is not wrong and can also be found in the literature. However, in practice the mean is usually neglected because it is small - maybe its a bit confusing because the assumed mean of Gold is realtively high (expecially in relation to the STD) which is not very realistic.
But (obviously) for the exam (as in practice) you should not include the mean in your VaR-calculation.

 

[David Harper CFA FRM]

Hi Backwardation, I can't tell if i wrote the original question on this thread (it resembles a question I wrote based on Stulz, but it appears to have a subtle difference ... not sure ... ).
But, anyhow, can i just politely disagree with this assertion: But (obviously) for the exam (as in practice) you should not include the mean in your VaR-calculation

Historically GARP was a bit loose about the terms, but we can expect (in part due to much feedback) the "modern" FRM to be more precise. Specifically, as many practitioners justifiably assume inclusion of the mean (drift), modern questions will not (should not) leave it to chance:

• absolute var = -drift*∆T + sigma*deviate*SQRT(∆T); i.e., loss relative to initial value
• relative var = sigma*deviate*SQRT(∆T); i.e, loss relative to exp future value; aka special case of absolute var when drift can be assumed to be ~0, as in 10-day market risk VaR

As part of our feedback to GARP (based on associated customer feedback), we pointed out that the absolute VaR is given by the FRM-assigned Dowd (and supported by much literature, to agree wholeheartedly with your "I would say that hawayi has posted the right formala to calculate the VaR, that is to include the mean. So recognizing the mean in the VaR calculation is not wrong and can also be found in the literature."), so in a sense, it is the question's burden to specify if a relative VaR is wanted.

Beyond precision of the question, the realistic factor is timeframe: for a 10-day market risk VaR for equities, then a simplifying assumption rounds the drift down to zero, so it doesn't matter. But it does matter for credit and operation VaR and other cases where the horizon is longer (e.g, one-year). So, i just wanted to add some color b/c, question depending (and MVaR, CVaR, OpVar depending) it may be called for to "include" the mean (i.e., employ absolute VaR; I don't like "include/exclude" the mean because that gets confusing), thanks,