Prospect Theory, Investor Utility, and CAPM Assumptions

# Validity of CAPM


# Validity of CAPM












I came across some literature regarding "Framing Theory" or "Prospect Theory", and the validity of CAPM. I was wondering if you could shed some light on a few questions I have in this regard:

According to this theory, people place much more weight on the outcomes that are perceived more certain than that are considered more probable. Investors also prefer dollar gain over dollar loss, and are also affected by “framing effect”: the way a problem is posed to the decision maker.

So I am assuming that under this theory, the utility curve would no longer be concave, but rather convex-concave (convex in losses and concave in gains). Therefore, is the CAPM still valid under this assumption? If not, how could we rewrite CAPM with these assumptions?

Any help or brainstorming would be appreciated! Thanks!

## Answer by guihp (score 2)

https://quant.stackexchange.com/a/14996

That is true. Utility would not be concave anymore under prospect theory (only for gains), but convex for losses, which is evidence against CAPM.

CAPM is valid either :

-if the utility function is quadratic (which is nonsense in terms of economic interpretation, and in general, Von Neumann- Morgenstern utility describes poorly reality and should be rejected as a descriptive Theory)

-if the distribution of the returns is elliptic, that is a general class that includes the normal distribution (which is nonsense also because there is skewness in real life).

In fact, CAPM is never verified, and fails miserably empirically. But even Prospect Theory cannot fix the problem at an equilibrium/aggregate level, since it doesn't solve the Pricing Kernel Puzzle. That is because Prospect Theory doesn't aim to solve this problem, but rather describes individual behavior in certain circumstances (simple lotteries).

If you want to really describe what happens, the utility has to alternate concavity and convexity, both in the domain of gains and losses (double alternation, with a more pronounced concavity for losses and convexity for gains, the exact opposite of Prospect Theory), while being globally concave and keeping the framing effect which is still very important.

In fact, we can derive what I call Real Utility functions (that does not depend of preferences nor risk aversion but rather imply both of them), that arises from overconfidence, underestimation of extrema and non bayesian learning. But the approximation of real utility functions is what is called the CAPM^2 (for continuous asymetric polynomial models).That was my master thesis actually,so I know it well.

How to integrate that utility into a Beta model requires 3 pages of calculus, but it is doable and has been done. The beauty is that it works smoothly and beats off Fama French factors plus momentum.

## Answer by emcor (score 1)

https://quant.stackexchange.com/a/14673

The general problem of the investor is:

$$ \max_{w\in[0,1]^n} U(\mu_p(w),\sigma_p(w))\quad s.t. \sum_{i=1}^n w_i=1$$

where $w$ being the portfolio weights, and $U$ utility function of portfolio risk $\sigma_p$ and return $\mu_p$.

CAPM assumes investors with concave utility function $U=\mu_p-\frac{1}{2}\sigma_p^2$, from which then follows that all investors mix the market portfolio with the riskfree asset according to their desired minimum risk/maximum return level.

I believe using a different nonconcave utility function would indeed change the model.

There is a paper on the necessary conditions for the CAPM utility function here which states:

## Answer by markowitz (score 0)

https://quant.stackexchange.com/a/27665

In a few words. The CAPM assume the concave utility function because its, implicitly, assume the validity of mean-variance approach. In utility function way the concavity is related with the concept of risk aversion and risk=variance of return. If utility function is convex the investor is prone to risk and CAPM is not valid and mean-variance as well. If as in Prospect Theory there are different risk-preference choice in gain and loss field ... CAPM is valid only in gain field and mean-variance as well.

## Answer by T123 (score 0)

https://quant.stackexchange.com/a/77228

Not so fast!

Levy and others have shown that even under Prospect Theory, CPT-efficent Portfolios ((Tversky, Kahneman (1992)) are identical with mean-variance portfolios

https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1468-036X.2009.00530.x

Here is an alternative version of a slightly modified value function of Prospect Theory, that guarantees that there is an equilibrium in any case:

https://openaccess.nhh.no/nhh-xmlui/handle/11250/163727

I checked this and can confirm that these portfolios are identical for the relevant section of the efficient frontier:

Note that Prospect Theory, although convex in the domain of losses, still preserves an overall concavity if loss aversion is considered. Huang, Santos and Barberis have shown that there is still a preference for higher returns and a lower variance under Prospect Theory, thus the basic assumptions of the CAPM seem to be intact:

https://nicholasbarberis.github.io/bhs_nonac.pdf