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I have the following problem that asks me to solve for the "maximal growth portfolio."

Suppose that the equilibrium stochastic discount factor evolves as $$ \log S_{t+1} - \log S_t = \kappa_s(X_t, W_{t+1}). $$Solve the following maximization problem: \begin{equation*} \begin{aligned} & \underset{x}{\text{maximize}} & & E[\log(R_{t+1})] \\ & \text{subject to} & & E\left[ \frac{S_{t+1}}{S_t} R_{t+1}\, \middle | X_t = x \right] = 1. \end{aligned} \end{equation*}

It seems clear how we would derive the following solution: $R^*_{t+1} = \exp(-\kappa(X_t, W_{t+1})) = \frac{S_t}{S_{t+1}}$. However, my question is related to understanding the economics behind this problem. The constraint in this problem is clearly saying that whatever portfolio we construct, it must be fairly priced (given the stochastic discount factor process $S_t$). However, I don't understand why there would be a portfolio that produces a "maximal expected return." So, there are my two questions:

  1. Why isn't the objective here unbounded? Can't we always choose a portfolio with a higher expected return (with correspondingly higher risk)?
  2. Also, why is there a log in the objective? Would this problem work just the same as having the objective be $E[R_{t+1}]$?
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  • $\begingroup$ jmbejara - Just curious. When economists write about logs without specifying a base, is it standard to assume they mean the natural logarithm? $\endgroup$ – 123 Jan 23 '15 at 17:17
  • $\begingroup$ Yeah, that's right. I've never seen another base ever used in economics. $\endgroup$ – jmbejara Jan 23 '15 at 19:52
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Say you have a portfolio with returns described by a random variable X. Call the lowest possible realization of X: xmin. If you take a levered position in that portfolio with leverage A and financing cost r your returns are r(A-1)+AX. There will exist a value of A no larger than 1/xmin where when you get the worst return of X and the levered portfolio returns -100%. That's game over for your wealth.

So if you invest for the long run and reinvest after each period it should be clear that you can't simply arbitrarily maximize leverage to maximize long run returns. That expected log return maximization objective function is exactly the objective function of long run return maximization. This objective treats -100% returns as -infinity utility and so avoids it at all cost.

If instead you maximized expected returns this wouldn't happen. Your preference would be for unboundedly large leverage, in line with your intuition.

Another way to think about this is risk aversion. Log utility in wealth is analogous to a CRRA utility parameter of 1, while the expected return utility function a CRRA parameter of 0 (risk neutral). Since the risk neutral guy doesn't care about risk, he sees an asset with positive expected returns and wants a gigantic levered bet on it. The log utility guy is relatively risk tolerant but still worries about extreme outcomes and so doesn't want too much leverage.

And if the range of values taken by X are mostly in the range of +/- 30%, it turns out that E[log(X)] is nicely approximated by E[X]+0.5Var(X) so can be well approximated with Markowitz mean-variance portfolio optimization where lambda is 1/2. Where as E[X] preferences is analogous to lambda of zero which won't have a bounded solution without an additional constraint.

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  • $\begingroup$ Thanks, that makes sense. I should have thought of that. The wikipedia article on the "Kelly Criterion" says that to an individual with logarithmic utility, the Kelly bet maximizes expected utility. $\endgroup$ – jmbejara Jan 23 '15 at 16:41
  • $\begingroup$ Why are the returns $r(A-1)+AX$ as you describe? $\endgroup$ – jmbejara Jan 24 '15 at 6:14
  • $\begingroup$ A=1 is the unlevered return of X. R is the borrowing rate (say -1 percent per year for a current margin account). So if you have 2 to 1 leverage your returns are 2X-0.01. $\endgroup$ – BKay Jan 24 '15 at 13:25

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