Here is something I did for my own interest. It is what it is. It refers to a specific period of time. It's past performance. In this particular thread, I don't care to discuss the use of the arithmetic mean (appropriate here but I don't want to argue it), or the possibility of a rebalancing bonus (annual rebalancing is assumed), or the use of standard deviation as a measure of risk.

Because I personally am retired and very conservatively invested, I chose 40/60 as my starting point rather than 60/40 but will mention the 60/40 results afterwards.

It is based on real-world investments in actual mutual funds and thus could have been easily implemented by an ordinary retail investor using real dollars. The funds chosen for consideration were an initial portfolio of 40% Vanguard Total Stock, 60% Vanguard Total Bond. I explored the results of replacing some of Total Stock with the DFA US Small-cap Value Portfolio, DFSVX. I chose this fund because it is old, it is from a firm that has Fama and French among their directors, and because factor mavens in the forum have praised it. The time period covered is 1994-2019 inclusive, limited to whole calendar years and by the inception date of DFSVX.

1994-2019 includes a couple of business cycles. It includes about 26/28ths of the time since the "discovery" of the Fama-French three-factor novel. It includes one period (2000-2003) of stellar small-cap value performance.

The red dot is Total Stock, the green dot is Total Bond, the yellow dot is a 40/60 mix. The average (arithmetic mean of annual returns) is µ = 7.60%, and the standard deviation is σ = 7.73%.

Since 40/60 represents the amount of risk I wish to take, this is how I formulated the question. For a fixed "cost" of 7.73% in standard deviation, how much could I have increased return by including DFSVX in the portfolio

*without increasing the risk*, and what mixture of Total Stock and DVSVX would have given the greatest improvement?

In this chart, the aqua dot at right is DFSVX, and the maroon curve shows all of the possible µ and σ combinations obtained by mixtures of Total Stock and DFSVX. Interestingly, both Total Stock and DFSVX had almost identical Sharpe ratios, about 0.5, so DFSVX offered higher return at the cost of higher risk, and virtually identical risk-adjusted return.

However, because Total Stock and DFSVX had an imperfect correlation of ρ = 0.74. This means the efficient frontier of the two is not a straight line, but bulges upward.

Going back to the yellow line, then, we can choose to anchor the right end, not at the red Total Stock dot, but at some mixture of Total Stock and DFSVX. That will raise the right end of the yellow curve higher. Since DFSVX has higher risk than Total Stock, a mixture of 60% bonds with 40% combination-of-stocks will have higher risk; in order to bring hold risk constant, we need to reduce the total stock allocation, use a little less than 40% of a slightly-riskier combination of stocks.

It turns out that:

- The optimum mix of Total Stock and DFSVX, for this particular kind of optimum over this particular past time period, would have been 45% Total Stock, 55% DFSVX, i.e. more than half of the stock allocation is invested in small-cap value.
- To hold total portfolio risk constant, it is theoretically necessary to compensate by cutting the overall stock allocation, but in this particular case the reduction need turns out to be microscopic: 39.4% stocks, 60.6% bonds.
- The return of the portfolio was improved by 0.29%, 29 basis points, from 7.60% to 7.89%.

If we perform the same analysis but start with a 60/40 allocation, the return is improved by 0.43%, 43 basis points, from 8.84% to 9.27%. The optimum mix within the stock allocation was 56.5% to small-cap value. And in order to keep the risk constant at 7.73% it was necessary to raise the bond allocation to 41.1% and cut the stock allocation to 58.9%.

So, that's what I did and those are the numbers I got. They suggest to me that

*if* all the parameters were stable going forward--which I don't believe!--the theoretical optimum allocation to small-cap value would be quite high... but the

*amount* of improvement is not worth having unless it is all-but-guaranteed.

Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.