You can use rules of thumb to determine what percentage to allocate to stocks versus bonds. Like “60 percent to stocks and 40 percent to bonds.”
But Nobel Laureate Robert Merton developed a formula you can use to calculate a “Merton share” or optimal allocation to equities. The Merton share estimator below makes this calculation for you.
Note: Instructions and additional information appear beneath the calculator.
Merton Share Estimator
Merton Share Estimator Instructions
The Merton Share Estimator requires five inputs in order to calculate how much you or I should allocate to equities: equity return, equity standard deviation, risk-free return, risk-free standard deviation, and the constant relative risk aversion.
You should be able to get needed estimates of rates of return from the financial services company that holds your 401(k) or Individual Retirement Account. These firms also usually provide a volatility measure, too, which is the standard deviation.
Three tips here. First, if you’ve invested in several different equity classes with different expected returns? For example, if your equities allocation invests 50 percent in U.S. stocks expected to earn three percent and 50% in Non-US stocks expected to earn five percent? You calculate a weighted average return equal to four percent for the blended equities. For example:
50% * 3% + 50% * 5% = 4% weighted average return
A second tip: Adjust for inflation and work with real returns. That approach lets you use Treasury Inflation Protected Securities (TIPS) rates as the risk-free return. To adjust nominal equity returns for inflation, just subtract the inflation rate. For example, a six percent nominal equity return equates to a four percent real return if inflation equals two percent. For example:
6% nominal return – 2% inflation = 4% real return
A third tip: You can probably use historical standard deviations for your calculations. At least to start. Sometimes people say the standard deviation equals 15% roughly. Sometimes 20%. But an online tool like Portfolio Visualizer lets you calculate the actual standard deviation of blended portfolios of equities. Also, the CBOE VIX index shows the expected standard deviation on US stocks expected over the next month. (You can Google to get the most recent VIX value.)
Understanding Merton Share Estimator Calculations
A single formula calculates the Merton share. And that formula basically divides the equity premium by the squared standard deviation, or variance, of equities. So like this:
Equity Premium / Standard Deviation^2 = Equity allocation
For example, in a simple case where equities return two percent more than riskless investments and the standard deviation equals twenty percent? The formula might make this calculation and return .5, or 50%, thus signaling a 50 percent allocation to equities. Here’s the formula:
2% equity premium / (20% standard deviation ^2) = 50% equity allocation
But in practice, it’s a little more complicated. So let me drop down the rabbit hole for just a few sentences.
Nitty Gritty Details of Merton Share Estimator
To calculate the equity premium, the calculator assumes you’ve entered the expected, real, geometric mean return of equities and the expected, real, geometric mean return of risk-free bonds (like Treasury Inflation Protected Securities) along with the expected standard deviations of these two investment choices. (When a financial services company like Vanguard, Blackrock or Fidelity estimates the return you or I might earn from stocks or bonds over a decade? That’s a geometric mean, or average. It may also be nominal so including inflation. Or real, so adjusted for inflation.)
The calculator then estimates the real, arithmetic, mean return on equities and on risk-free bonds, and the difference between the two–which is the equity premium. (To make this estimate, the calculator uses a common but imprecise tweak: It adds half the variance, or the standard deviation squared divided by two, to the geometric return.)
To determine the appropriate allocation to equities, the calculator then does that simple division operation. But with another tweak, this one from Professor Merton. The formula actually divides by equity premium by the standard deviation squared, or the variance, times the constant relative risk aversion input. Thus the actual formula looks like this:
Equity Premium / (Standard Deviation^2*Constant Relative Risk Aversion)
The “constant” lets people assume different risk aversions. Research suggests many people have constant relative risk aversion equal to 2, a level which suggests some risk aversion. And the common range of constants runs from 1 (low risk aversion) to 5 (high risk aversion). For what it’s worth? I think my personal constant relative risk aversion equals 1 or 2. Most people’s relative risk aversion constant equals 2 or 3.
Note: Someone who is risk neutral or nearly so? Their relative risk aversion constant maybe equals 0. And in this case, they ignore risk and focus solely on the expected return.
Observations about Making Merton Share Estimator Calculations
Some quick observations about making Merton share calculations. And about using the calculation results to make better decisions.
First, the calculations suggest that we ought to often bear more risk than we do. Not always, no. And maybe not at the time I’m writing this in December of 2024, but usually individuals should bear more risk to earn higher expected returns.
A second point: The Merton Share Estimator’s calculations suggest that currently (late 2024) a smart way to dial down US investors’ risk is to invest more broadly than just in US stocks. If I model investing half in US stocks and half in international stocks, for example, the calculator suggests maybe a 70 percent allocation to equities for an average-ish risk aversion investor. If I model just investing in US stocks? It suggests less than a 40 allocation to equities for a low risk aversion investor and a 20 allocation to equities for an average-ish risk aversion investor.
Third, and this is personal and anecdotal… but I suspect the more you or I experiment with Merton share calculations? And the more you or I root around to get updated expected returns and standard deviations for stocks and bonds? Yeah. Okay. I think that may jack your or my constant relative risk aversion input. Thus, be careful.
Other Resources
The Super Safe Withdrawal Rate blog post provides a companion discussion and another calculator: Super Safe Withdrawal Rate Calculator.
Professor Merton’s research paper appears here: Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case
The authors of the book, “The Missing Billionaries,” use Merton’s share in their wealth advisory business. Lots of interesting insights appear at their website, including this one: Man Doth Not Invest by Earnings Alone. BTW, “The Missing Billionaires” is dense but a very interesting read.