# Why does Merton's fraction give unintuitive quantities using real world data?

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?

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