Author Topic: Using Merton share formula to optimally allocate to stocks  (Read 5272 times)

SeattleCPA

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Using Merton share formula to optimally allocate to stocks
« on: August 05, 2024, 06:38:10 AM »
This is not a post to argue you shouldn't be 100% in stocks if that's what you believe.

But if you're not in that camp and you're wondering about how to rationally allocate money to stocks and bonds, you might want to look at using Nobel Laureate Robert Merton's "Merton share" formula. (This is the basic subject of that popular book from last year, the Missing Billionaires.)

The formula, as I've mentioned before here, calculates the optimal allocation to equities like this:

(expected equity return - riskless return)/(standard deviation on equities^2*relative risk aversion "factor")

If you set the relative risk aversion factor to 1, which means you are very not risk averse, the simplified formula might look like this given current environment:

(6% expected return on equities - 5% expected return on Treasuries)/(.16^2)... which equals roughly 40%

And then what's maybe my main point here is, after a bit of experimenting, I think you can probably get something like ChatGPT 4o to do the math for you.

What you do is point to the expected returns source you want to use and then ask ChatGPT to make the calculations. Here's an example prompt using Vanguard's most recent market outlook:

Quote
Please calculate merton share assuming returns shown in this webpage https://advisors.vanguard.com/insights/article/series/market-perspectives#projected-returns with 60% in us stocks and 40% in international stocks. Please use the intermediate treasuries return as the riskless rate. Assume risk aversion coefficient equal to 1.

ChpBstrd

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Re: Using Merton share formula to optimally allocate to stocks
« Reply #1 on: August 05, 2024, 01:12:02 PM »
Quote
(expected equity return - riskless return)/(standard deviation on equities^2*relative risk aversion "factor")
One issue is the equation is extremely sensitive. Using 0.16 as our SD of equities and 1 as our risk aversion factor, we obtain:

Expected Stock Return Over Risk-Free       Stock Allocation
1%                                                         39%
2%                                                         78%
3%                                                         >100%

So if you think stocks will have a total return of "6 or 7 percent" that could be either a 39% or 78% allocation, depending on which of those very close numbers you guesstimate. This might force a person into false precision guesswork.

Then while you're working on figuring out a good guess for expected stock returns, the risk-free rate is constantly changing. The 10-year treasury, commonly cited as the risk free alternative to stocks, has ranged from 4.99% to 3.78% in the past year. As we can see, that 1.21% difference will have a 47% impact on the result.

So a wild guess plus a moving target are creating a result that is the temporary combination of the two.

SeattleCPA

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Re: Using Merton share formula to optimally allocate to stocks
« Reply #2 on: August 05, 2024, 04:47:53 PM »

One issue is the equation is extremely sensitive. Using 0.16 as our SD of equities and 1 as our risk aversion factor, we obtain:


Yeah, I agree with the sensitivity. And it's not just the equity premium going from 1% to 2% (which actually is a pretty big change). But stuff like whether you're using all US stocks or a blend of US and non-US.

What I like about the Merton share after learning about it, playing with it, and trying it, is it's quantitative and intuitive. Thus, you can work with a not-a-black-box formula to get out and then get back in.

I should say that I am not all in on Merton. E.g., I'm not at 40% equities. Or at least I wasn't at the start of the day.