Quantifying Risk Utility

The utility function for quantifying risk aversion shows up in two different places:

  1. Reading 21 - Asset Allocation: where U = E® - .005*(Ra)*Variance

where U is Utility from Risk adjusted expected return, E® is expected return, and Ra is the investor’s risk aversion


  1. Reading 27 - Equity Portfolio Mgmt: where U = E® - (Ra)*Variance

Here, U is Utility for active return, E® is the expected active return, and Ra is the active risk aversion

The first equation is for individual investors and the second is for managing a portfolio of managers

My question is why do we discount the individual’s risk aversion (by multplying by .005) and not the active risk aversion for a portfolio of managers? Is the formula for Ra different b/w the two?

i personaly jumped over that formula, i read the optimization stuff that came after which is the same old mean variance efficient frontier execept easier cause it assumes no correlation

as far as the utility funcation, can you please tell me what that function is usefull for? what can i do with it. i did not find it in the LOS, so i said f it

it seems to be like cfa just likes to pass formulas sometimes, many of which dont have much use or credibility, like the ones about how many stocks you have information about, and how strong is your information, seems like the inputs to the formula are so subjective it is does not mean much…

just my thinking, what do i know

I’ll have to look up the page #'s when I get home, all I know right now is the first utility function is in reading 21 and the second is in reang 27.

The functions basically help to put a quantity on risk aversion. they say if expected return is X then the investor will get Y equivalent return utilty from it.

yeh i see what your saying, so it helps you find the point where you want to be on the fronter, i just did not see it in the los as us being required to find that point, let me know

Thanks. The second formula is new to me…

There are a few pieces of difference

U = E® - .005*(Ra)*Variance

  • Variance = Variance of Asset Mix
  • Ra = Risk Aversion of investor
  • E(Rm) = Expected Return of Asset Mix

U = E® - (Ra)*Variance

  • Here E® = Expected Active Return of Manager Mix
  • Ra - is actually [λ] the investor’s trade-off between active risk and active return; measures
  • risk aversion in active risk terms
  • and Variance in this formula is the variance of active return

maybe the fact that the 2nd one is Active … removes the 0.005 stuff.

That was the only difference I could see as well, but I didn’t see any reasoning as to why the active return of a manager’s mix would cause you to not discount risk aversion.

The only thing I can think of is maybe because a manager’s bias toward risk aversion may not be as large as an individual’s and therefore you would need to discount the individual’s risk aversion to get an appropriate utility figure.

Broke - This is from Book 3 Reading 21 Pg 234, and Book 4 Reading 27 pg254.