Hello all, I ran into these two formulas in both Asset Allocation and Equity Portfolio Management Chapters. They are the same concept but why would we have two different formulas? If so, how do we know which one to use? Thank you!
U=R-Lambda*Sigma^2
U=R-0.5*Lambda*Sigma^2 ?
I suspect that the readings were written by different authors.
It’s probably akin to the convexity controversy: some people divide the convexity by 2 and use it as is, while others don’t divide it by 2 but when they use it multiply it by ½.
If I were you, I’d write CFA Institute (info@cfainstitute.org), point out the discrepancy, and ask them how you’re supposed to know which formula you’re supposed to use in a given situation. Then let us know what they tell you.
You have the formula messed up for formula 1.
Asset Allocation:
U =E(Rm)−0.005Raσm^2
Ra is the level of risk aversion - which you have called lambda.
σm are expressed as percentages rather than as decimals
also read footnote at the bottom of the page…
See Bodie, Kane, and Marcus (2004, p. 168) for this expression. A standard expression for a mean– variance investor’s expected utility is U = E(R ) − 0.50R σ2 , where expected return and standard deviation are stated in decimal form and 0.5 is a scaling factor. Dividing 0.5 by 100 to get 0.005 in the text expression ensures that we can express expected return and standard deviation as percentages
so in 1 formula you use the direct percentage of the standard deviation (which is if they say it is 2% you use 2) (and use 0.005 as the scaling factor)
in the other when they say 2% is the standard deviation you would use 0.02 and use 0.5 as the scaling factor.
Thanks for your response! I just sent an email to CFAI. Once I recieve their answer, I will post it here. Thank you!
Thank you for your answer!
The first formula is on page 214 Equity Portfolio Management Chapter in Volume 4.
U=R-λ*σ^2
The second formula is the in the asset allocation chapter:
U =E(Rm)−0.05*Ra*σm^2
So, do you mean Ra*0.05=λ?
If I understand right - the two formulae are used in different contexts.
Formula in Asset Allocation:
U =E(Rm)−0.005Raσm^2
Um = the investor’s expected utility for asset mix m
E(Rm) = expected return for mix m RA = the investor’s risk aversion
σm^2 = variance of return for mix m
is used to determine an individual’s asset allocation mix given his level of risk aversion Ra.
Formula in Equity Management U=R-λ*σ^2 is to determine the MIX between different Equity Managers (Indexing, Active, Semi-Active) given the investor’s level of Risk Aversion
the three terms in this formula:
UA= expected utility of the active return of the manager mix rA= expected active return of the manager mix
λA = the investor’s trade-off between active risk and active return; measures risk aversion in active risk terms
The way I understand it - you are using one formula for a single individual, while the other is for an active manager mix. One measures regular risk and return (so uses 0.005 to quantify that) while the other is Active Risk and Active Return (no 0.005 because the usage of Active Risk and return already incorporate the regular risk and return terms). So watch out for the question saying “Active risk and return with an equity manager mix” vs. “individual’s asset allocation mix” and decide upon the formula accordingly.
That totally makes sense now! You are amazing!! Thanks a lot!