Shortfall probability and the corresponding utility function

am wondering what form the utility function might take before and after discontinuity at the minimum required return?

in the example in chapter 2 where the discontinuity is at the solvency level, the investor is risk-seeking before the discontinuity point and risk-averse after.

what i have deduced from this example is that the parameters of the investor's utility function can change according to the investor's circumstances (unfortunately the core reading does not elaborate much on this topic).

So, in case of using shortfall as a risk measure, can the investor's utility function be any form? for example, the investor can be risk-averse both below and above the required return level. or, the investor is normally risk-loving, but when he receives a return below the required return, he becomes risk-averse? or, the utility function below the required return simply cannot be constructed?


what are your thoughts on this.

 

Actually, it can be shown that the utility function corresponding to the shortfall probability consists of two straight line segments either side of the discontinuity at the benchmark level of return, each segment having the same gradient. This corresponds to the investor being risk-neutral both above and below the benchmark level of return.

This seems to make sense, as use of the shortfall probability suggests the investor cares only about whether the return is actually above or below the benchmark, and not about its variability above or below the benchmark.

As you're not told this in the Core Reading you shouldn't need to worry about it. In any case, exam questions on this section of Core Reading have been very few and far between.

 

oh, thanks so much. it does make sense.