 # Shortfall risk- does not assume normal dist?

Question 1.) in the fixed income book on page 148 (2011) – I thought shortall risk used standard deviation in the calc, so how is this not assuming a normal distribution?

is this the CFAI book or Schweser?

Do not have my books on me right now … will look it up after I get back home

Any random distribution has a standard deviation, not just normal distributions. For instance, let’s say you have these return outcomes [1%, 2%, 3%, 4%] with probabilities [25%, 30%, 15%, 30%]. Does it have a positive standard deviation? Of course. Is it normally distributed? Of course not.

I thought I remembered CFAI listeing a shortcoming of analycitcal VAR as the reliance on the assumption of a normal distrubition, due to it’s use of standard deviation?

An analytical VAR calculation might use standard deviations, and might assume a normal distribution. However, a normal distribution is not a prerequisite for standard deviation. If you believe this, you are mistaken.

Ya, here they define Shortfall risk refering to the probability of not achieving some minimum specified return. so no need to assume normal distribution.

Buy Hey, I have also seen using Shortfall risk in IPS section as the risk that portfolio will fall below some min acceptable level during a stated time horizon. Many time as (Rp - Multiple (say 1 or 2) of standard deviation) or (Rp - Min return) / std dev.

With Std Dev:

A normal curve will predict the % of outcomes that will be within (x) std devs of the mean, as in approx 68% will be within 1 std dev, 95% will be within 2 std devs, and 99% will be within 3 std devs.

For a non-normal curve (skewed), the % of returns falling within the same number of std devs will be higher than for a normal curve. for instance a non-normal curve may have 70% of returns within 1 std dev of its mean.

The point is that the normality assumption indicates a lowest probability of returns falling (x) std devs from the mean; or said a different way: a highest % of returns falling outside (x) std devs from the mean.

For shortfall risk, you are using the probablity of returns falling INSIDE (x) std devs from the mean. Meaning you are using the highest % of possible returns to perform the calculation.

For analytical VAR you are using the opposite, as in trying to find out how many returns will fall OUTSIDE (x) std devs. In this case you have to use the normality assumption because there may be a smaller portion of returns that fall outside (x) std devs if the distribution is skewed. If you use a skewed distribution and tell everyone that a specific loss has a probability of occuring 3% of the time and it happens 5% of the time, they will be upset.

Analytical VAR wants to know the probability of a specific loss in a given time frame, a normality assumption will give you the highest % that it will occur. Shortfall risk wants to know, given the return distribution, the likelihood of falling below a given return. Here you don’t need the normality assumption because you will be incorporating more returns within the (x) std devs for a non-normal distribution.

I hope that all makes sense.