The solution to Merton's portfolio problem suggests that an investor invest $\frac{\mu - r}{\sigma^2 \gamma}$ percent of their wealth in the stock market, where $\mu$ is the rate of return of the stock mareket, $r$ is the risk free interest rate, $\sigma$ the annual volatility of the stock market, and $\gamma$ a measure of the risk-aversion of the investor. Taking $\gamma = 1$ corresponds to logarithmic utility (quite risk averse).

As of writing, the current LIBOR rate is 1.75%. The historical SPX rate of return is on the order of 10%, while an (overestimate) of its volatility is 0.20 (of course excluding the current bear market). With these estimates, we come to

$$ \text{Estimate of Merton's Fraction} = \frac{(.1 - .0175)}{(0.2)^2} = 2.0625 $$

Even using the current value of the vix $\sigma = .35$, Merton's solutions suggests investing two thirds of your wealth in the stock market in these turubulent times.

What causes these aggressive suggestions?

  • 2
    $\begingroup$ Logarithmic utility is quite risk tolerant. For risk averse I would suggest $\gamma=3$ or higher. $\endgroup$
    – nbbo2
    Apr 25, 2020 at 14:17

1 Answer 1


A few suggestions -

  1. Taking gamma to be 1 is actually quite risk tolerant, as suggested by noob2 in the comments.
  2. Historical annualised stock returns of 10% were achieved with much higher risk-free rates, so it’s not appropriate to use this expected return when interest rates are much lower.
  3. With valuations as high as they are right now, you need to be very bullish to assume excess returns of 7-8%. I think that 3-5% is more appropriate for the next 5-10 years.

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