*relative*to an index. Individual's utility functions are closer to

*relative*than absolute wealth metrics.

Falkenstein suggests a relative utility function based on the individual's wealth \(w_i\) and also the wealth of the only other agent, which we denote \(w_{-i}\). \begin{equation} \label{relative} U^r(w_i;a,w_{-i}) = U^a(w_i-w_{-i};a) \end{equation} Indexing the only two humans alive by by \(1\) and \(-1\) helps keep the symmetry. Here \(a\) is a coefficient of risk aversion, shared by both participants, and $$ U^a(x) = - e^{-a x} $$ is the familiar utility function, more usually applied with \(x=w_i\).

The author introduces two assets into this two agent economy. One is riskless with deterministic return \(\nu\) and one is risky with mean \(\mu\) and variance \(\sigma^2\). Normally we would imagine \(\mu = 1.2\) for example and \(\nu = 1.05\) perhaps. But as we'll see momentarily, Falkenstein observes that risky asset return will be *exactly the same* as the riskless asset return if both agents' utility is relative, as in (\ref{relative}) above.

**From Neumann-Morgenstern to Markowitz**

This is to be compared to the case of absolute utility where the risky asset experiences excess return proportional to the risk aversion parameter \(a\) and the variance of the risky asset \(\sigma^2\). Here is a quick refresher.

Recall that there is strong motivation for maximizing expected utility *of some sort* and weaker justification for the particular form above. Leaving that philosophy alone we certainly can say that so long as \(x\) is gaussian, there is a purely mathematical equivalence between the task of maximizing \(U^a(x)\) and the task of maximizing a rather famous expression where the naive mean is penalized by a variance term:
\begin{eqnarray}
\arg\max_x E[U(x)] & = & \arg\min_x \left( -E[U(x)] \right) \nonumber \\
& = & \arg\min_x \left(\log -E[U(x)] \right) \nonumber \\
& = & \arg\min_x \left(\log E[e^{-ax}] \right) \label{exp} \\
& = & \arg\min_x \left(-aE(x) + \frac{1}{2} a^2 Var[x]\right) \label{var} \\
& = & \arg\max_x \left(E(x) - \frac{1}{2} a Var[x] \label{blah} \right)
\end{eqnarray}
Unsurprisingly (\ref{blah}) with \(x=w\) representing wealth is sometimes taken as an ansatz for investing - subject to the gaussian assumption we need to get from (\ref{exp}) to (\ref{var}). In fact there is another, lesser appreciated caveat I have summarized in an earlier post, but it isn't germane to Falkenstein's point either. His objection relates more to what \(x\) should be: absolute utility with \(x=w_i\) or relative utility with \(x=w_i - w_{-i}\).

**Absolute utility. Trading mean for variance.**

We first review the former. There should be no first order change in utility for an agent transferring a small amount of his wealth from cash to stock.

For this to be true the increase in expected terminal wealth \(E(w)\) must balance the variance penalty term above (that is, the term \(\frac{1}{2} a Var[w]\)).

Without loss of generality the terminal wealth is \(w_1(\lambda_1) = (1-\lambda_1) y + \lambda_1 x\) where \(y=\nu\) represents cash. A small increase in portfolio holding from \(\lambda_1\) in the risky asset to \(\lambda_1 + \delta\) changes the expected terminal utility by \begin{equation} \label{equilibrium} \Delta U = \overbrace{ \delta ( \overbrace{\mu}^{risky} - \overbrace{\nu}^{riskless} ) }^{increased\ mean} - \overbrace{\frac{1}{2} a \sigma^2 \delta\lambda_1}^{increased\ variance\ penalty} = 0 \end{equation} where \(\mu = E[x]\) must exceed the riskless expectation \(\nu\) of cash. From this equation, expressing the fact that there is no marginal benefit in risk adjusted terms of increasing or decreasing exposure, the excess return of the risky asset in this two agent economy can be expressed: $$ \overbrace{\mu}^{risky} - \overbrace{\nu}^{riskless} = \frac{1}{2} a \sigma^2\lambda_1 $$ where importantly, the variance penalty is proportional to the risk aversion \(a\), the variance of the risky asset, but also how much the agent already holds in the risky asset (i.e. \(\lambda_1\).

**Relative utility. No variance tradeoff exists. **

But Falkenstein (pages 90 onwards) argues against the relevance of this traditional volatility tradeoff. We introduce a second participant in the economy so that our first agent can be jealous of them. The second participant holds \(\lambda_{-1}\) in the risky asset and the rest in cash. Under the posited relative utility assumption (\ref{equilibrium}) is now quite different, as it involves the second agent. $$ \overbrace{ \delta ( \overbrace{\mu}^{risky} - \overbrace{\nu}^{riskless} ) }^{increased\ mean} - \overbrace{\frac{1}{2} a Var[x]\delta \underbrace{(\lambda_1-\lambda_{-1})}_{= 0} }^{increased\ variance\ penalty} = 0 $$ The variance term vanishes since each agent is identical and therefore, in equilibrium, must hold the same amount of the risky asset. There is no excess return. We simply have: $$ \overbrace{\mu}^{risky} = \overbrace{\nu}^{riskless} $$ In other words, in the limit where keeping up with the Joneses is all that influences the decision making of participants, there is there is no reward for risk taking. That's intuitive, because there is no such thing as systematic risk. Participants might as well ride that rollercoaster because in the event of a stockmarket crash, say, they will still happy. Their neighbours will lose too.

**References**

The reader is of course referred to Erik's book and blog articles.

## No comments:

## Post a Comment