Safety-first Ratio/Threshold Return Question

CFA Level I Quant
#1

Hello everyone, first post on the forum, sure there will be many more.

Q. Porfolio A has a safety-first ratio of 1.3 with a threshold return of 2%. What is the shortfall risk for a threshold return of 2%?

A. Using z-tables, the cdf for -1.3 is 9.68%, which is the probability of returns being less than 2%.

Can someone explain the logic/reasoning behind the answer, specifically why it is correct to use the safety-first ratio value as the value to refer to in the table i.e. how does it relate to standard deviations from the mean and also why it is correct to refer to negative 1.3 in the table?

Thanks in advance.

#2

A. “the SFR is the # of SD below the mean. Thus the larger SFR has the lower prob. of return below the threshold return.” Then it says “using the standard normal dist tables, we can find the prob in the left-hand tails as indicated.”

That’s it. So if SFR is 1.3, that means that we have to find F(-1.3), because 1.3 is the # of SD BELOW the mean. This is why we need to find F(-1.3), instead of F(1.3). And we use the cdf because shortfall risk is the risk of falling below the target. So F(-1.3) will give us the prob of falling below the 2% target, hence the shortfall risk.

which is 9.68%

#3

Just read my notes back and it makes sense, many thanks.

#4

I understand logically that for SFR > 0 the probability of shortfall risk should be less than 50%.

But could someone explain it mathematically? I know @cpk123 wrote about it, but I still don’t understand it…

For P(Z < x) = N(x) and for P(x < Z) = N(-x) = 1 - N(x)

So why different rules apply for SFR (I assume that SF is x)?

#5

I am just making the same mistake as you are, losterloc. I knew that Z values gave P(Z < x), and so I couldn’t get my head around why if I was looking to find the probability of getting below a target, i.e. the shortfall target, I couldn’t just do the same thing.

The trouble is, the Z values and SF Ratio’s are not the same and not interchangeable. By definition, SF Ratio is the ratio of the Standard Deviation below the mean. So immediately, if its below the mean of 0 for a normal distribution, we are working in minus numbers. It’s not the Z value where we standardise a random variable. Recall that the table for Z values only give’s positives, and when we have a negative number, F (-Z) = 1 minun (FZ), as you’ve said.

So it’s all about the definition for SF Ratio as “the ratio of SD below the mean”.