Why Merton’s Portfolio Fraction Can Recommend Aggressive Equity Weights

Summary

The document examines why the Merton portfolio formula can produce a large recommended allocation to equities. The fraction depends on expected excess return, volatility squared, and risk aversion. Using historical equity returns, a risk-free rate, and volatility estimates, the question illustrates how this formula can imply investing more than total wealth in stocks; substituting a higher volatility estimate still produces a substantial allocation.

The response highlights the sensitivity of the result to its inputs and interpretation. It argues that a risk-aversion parameter of one represents relatively high risk tolerance, historical equity returns may not be appropriate expectations when interest rates differ from the historical environment, and current valuations may support a lower expected excess return. The estimates are illustrative rather than a forecast or investment recommendation. The document does not develop parameter estimation, constraints on leverage, or a fuller treatment of the model’s assumptions, so its calculations should be read as a prompt to scrutinize inputs.

Key ideas

• Merton’s recommended equity share rises with expected excess return and falls with volatility squared and risk aversion.
• Using historical returns as expected returns can produce aggressive allocations.
• A risk-aversion parameter of one may represent more risk tolerance than the question assumes.
• Expected returns should reflect the interest-rate and valuation environment being analyzed.
• The formula’s output depends strongly on uncertain inputs and model assumptions.

Tags

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


# 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?

## Answer by Chris Taylor (score 2)

https://quant.stackexchange.com/a/53608

A few suggestions -

- Taking gamma to be 1 is actually quite risk tolerant, as suggested by noob2 in the comments.

- 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.

- 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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.