Safety First / Threshold Problem

Problem 25, p. 273 Kaplan book

Portfolio 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%?

Answer 9.68%

After looking at the answer, I understand you go to the Z table look up 1.3 and get .9032 and then subtract that from 1 to get .0968 however I cant visualize and really understand what is going on. I’m lookin at the Z table and it seems that Z tables always show you what is less than or equal to Z. So if we are looking for what is below it, and thats what it gives you, why do you need to subtract from 1. I would think subtracting from 1 would give you whats above it.

Also if anyone can put in laymans terms what is really happening I would appreciate it. So the safety first ratio is the Z? That means its 1.3 standard deviations away from the mean? We are just ignoring the 2% here?

You are making a mistake in interpreting SFRatio, the link bwt SF ratio and z value is not what you are thinking.

The first question which arises when we try to compare SF ratio with z value is, Is SF ratio a Z value? and the answer is no. SF ratio is not z value because of difference in calculation. Formula of SF ratio is, Exp return of portfolio minus threshold return divided by std. dev.

whereas the formula of z value is, observation (X) minus population mean divided by std dev.

We know that (or should know) that Exp portfolio return is also called Mean and we can assume that observation (X) could be our threshold return. If i assign values to these two (Exp Return of Portfolio (Mean) = 10% ) and , X or threshold return = 2% ) then how numerators of SF ratio and Z would look like

SF ratio = 10% - 2 % = 8%

Z = 2% - 10% = - 8%

So the difference bwt SF ratio and Z value is -ve sign. If you want to convert SF ratio into z value just add minus sign with SF ratio and you get the z value. (so simple)

The question you mentioned above is exactly the same. Now add -ve sign with SF ratio and check the probability in Z table.

Hi there,

I feel that the above did not exactly answer your question as it looks more like you are having a hard time visualizing what z-values represent.

Let me try to do my best.

First, realize that the z-score is nothing but a standard deviation from zero (that is , the mean of the standard normal distribution).

Second, the number that the z-value points at when we look at the table is the Probability that everything else lies BELOW that point.

Now, thing of the bell shape. If you shade the whole shape all the way to the right you have a probability of 1 that everything lies behind (on the left ) of that point. You will see that the z-score that goes close to giving us the probability one is close to 3.49. All that means is that if you multiply the Standard deviation (which is one for the standard normal) by 3.49 times you are going far enough to the right to consider everything else lying below it / to the left.

Having that down let’s see what the problem said:

You find a value that shows 90.32% probability that everything is on the left. However, notice that the safety first talks about the OTHER side of the bell curve. You want to know what is beneath 1.3 standard deviations to the left, not to the right as you are currently doing.

Now the thing with the standard normal is that it is SYMMETRICAL. That means that whatever lies 1.3 times to the left, also mirrors , 1.3 times to the right.

Since the total bell covers an area of 1. And since you essentially “shaded” all the area to the positive 1.3 deviations, what you need to do to “see” on the other side of the curve subtract from unity the shaded area that you are currently looking at. What you will be left with is the probability that something lies either : OVER 1.3 deviations or UNDER 1.3 deviations.

I know this can be tricky the first time, but I hope I helped out.

EDIT : To aid your understanding , please look at the table here

The top table represents z values on the left side of the mean. The probabilities always show how much area lies to the left of that.

The bottom tables represents z values on the right side of the mean. Again the probabilities show how much area lies to the left.

That is why you can always arrive at the first table by doing 1 minus z-value.

Some people get confused thinking that the probabilities under the positive z-scores cover the area to the right. This is wrong. The probabilities ALWAYS show how much is to the left of that point.

EDIT2: Even more help here :

Good luck!

Hope this diagram helps you visualize how we always take NEGATIVE Z table to find out the CDF that represents the shortfall risk.

The key idea is:
Higher the safety-first ratio, lower the shortfall risk.
Lower the safety-first criterion, higher the shortfall risk.

The motive is to minimize the left-tail (the one below Threshold Level Return). When you transform the Normal Distribution to Standard Normal Distribution, the expected return becomes 0 and the threshold return becomes the -SFR point. As SFR value becomes higher, -SFR value becomes lower making the left-tail smaller. Considering NEGATIVE Z value for finding probability of the returns going below threshold level (shortfall risk) is the same as taking 1 minus POSITIVE Z value CDF.

I believe we are ignoring the 2% since on transformation of the graph from normal distribution to standard normal distribution it becomes 0.

Another relation that I’ve noticed (that justifies the diagram) is that the formula for standardizing a normal distribution is (minus) the formula for SFR. This could be the reason why we take -SFR.

With reference to the diagram, if ER is the mean that becomes 0 on standardizing (formula: X – μ / σ) the point TLR will be (TLR-ER / σ) which is the same as -SFR if you check the formula SFR = ER-TLR / σ .

Oopsie, I just realized that was 8 years ago😂
Originally looked it up cuz I had the same query.