Chapter 4- Doubt in quadratic utility function

[nluashok]

Hi,

I am putting my 3 doubts in one page. All are related to quadratic utility function. Request you for helping in understanding of same.

1.Page 18 i.e. Summary page in Chapter 4says "if expected return and variance are used as the basis of investment decisions, it can be shown that this is equivalent to a quadratic utility function."

I am not able to understand these lines. It would be very helpful if these are explained?

2. quadratic utility function has increasing absolute aversion and increasing relative risk aversion. I understand that these means investor will keep lesser amount in risky asset with increase in his asset. HOwever, generally though investor is risk averse, I think he will has decreasing absolute aversion. THen why we use quadratic utility function?

3. Also not able to understand page 11 lines about quadratic utility function "if an investor has a quadratic utility function, the function to be
maximised in applying the expected utility theorem will involve a linear
combination of the first two moments of the distribution of return. Thus variance
of return is an appropriate measure of risk in this case."
Why only two moments needed to explain QUF?

-Ashok

 

[Whippet1]

Hi,

We always assume that the investor prefers more to less (ie a higher mean investment return) and dislikes risk (variability of investment return). However, there are several different possible measures of investment risk we could use.

Consequently, there is a theorem (beyond the course) shows which type of utility function is consistent with the use of a particular measure of risk. As the quadratic utility function can be described in terms of its two moments, so it can be shown that its use corresponds to using variance (second moment) as the measure of investment risk.

With a quadratic utility function \(U(w) = w + d{w^2}\)

Hence the investor's expected utility is given by:

\(\begin{array}{ccccc}
E\left[ {U(w)} \right] & = \sum {p(w) \times (} w + d{w^2})\\
\; = \sum {p(w) \times } \;w\; + \;\sum {p(w) \times } \;d{w^2}\\
\;\; = \sum {p(w) \times } \;w\; + \;d\sum {p(w) \times } \;{w^2}\\
\;\; = E\left[ w \right] + d \times E\left[ {{w^2}} \right]
\end{array}\)

So, with a quadratic utility function, the expected utility depends only on the first two moments.

Finally, yes, the Core Reading suggests that the quadratic utility function is inconsistent with the assumption that the absolute risk aversion and relative risk aversion decrease and are constant respectively as wealth increases. Consequently, it is not very realistic.

However, its use is consistent with using mean-variance portfolio theory (MVPT) to find the investor's optimal portfolio, which maximises his expected utility. This is because MVPT is consistent with the more general principle of maximising expected utility (EU) only if the EU depends only on the first two moments (ie mean and variance), which it does if we assume the investor has a quadratic utility function (and/or conversely if the distribution of investment returns is defined only in terms of its first two moments, as is the case for a normal or lognormal distribution.)

I hope this makes things clearer.

Graham

 

[nluashok]

Hi Graham,

Many thanks for your comprehensive explanation. It has given me very good insight.
I appreciate your efforts and valuable time for making my doubts clear.
Great help.