Ivo Welch, October 2023

Exploring a Low-Variance Portfolio

Treasury securities have terrible tax treatment. They realize tax obligations at a higher rate than equities — and they do so relentlessy all the time (every year end).

We can explore what it takes to buy a low-volatility portfolio and how low-vol we can get our portfolio doing this.

In the end, the approach to go about reaching this goal is similar to that of the BJSFF method — except that we want a low volatility portfolio rather than a high mean portfolio. On the very plus side, volatility is far more controllable by the investor than mean.

Note that it would also be easy to use this approach to try to form a high Sharpe-ratio portfolio. Again, this should be more controllable than the alpha fishing methods that most hedge funds are running.

Suggested Experimental Code

Take all CRSP stocks from 2000 to 2022.
In each year, obtain for each stock the intra-year daily standard deviation (call it $\sigma_{i,y}$).
Then, calculate within each year a rank based on this variable, call it $r\sigma_{i,y}$. You will use this as a signal.
In each year, obtain for each stock the end-of-year marketcap (call it $mcap_{i,y}$).
Then, calculate within each year a rank based on this variable, call it $r\sigma_{i,y}$. You will use this as another signal.
Think about the following: We usually do not simply use historical average rates of return to predict future average returns. Instead, we use some other variables (profitability, value/growth, firm/size, etc.) as indicators to help us improve the prediction of future average returns. In the same way, you can think of other variables as helping you predict the forward-looking behavior of the returns that you are really interested in.

Suggested Code 1 (Basics):

Take the lowest 20 $\sigma_{i,y}$ stocks (based on $r\sigma_{i,y}$). Forget all other stocks. Stocks that have the lowest signal (i,y) presumably should have lower future variance (sd), too. Invest equal amounts in these 20 stocks. This is your hypothetical investment strategy.
(Make sure you work with a zero-investment one-dollar strategy, although the results here will be almost the same. it’s more for the sake of understanding this and BJSFF regressions.)
Performance: calculate the daily rate of return on your portfolio for the following year. (Do not use monthlies!) Calculating for your portfolio: (a) the average rate of return, (b) the compound rate of return, and (c) the standard deviation of the rate of return. Hopefully, your strategy created a portfolio that gives you a low standard deviation the following year.
Compare your portfolio with your friends’ portfolios. Do you get the same 20 stocks each year? Do you get the same portfolio return moments the following year?

Suggested Code 2 (Marketcap weighting):

As in code 1, but you may want to value-weight your portfolio. Choose as your pfio-weight $w_{i,y} = (mcap_{i,y} / (\sum_i mcap_{i,y})$. The divisor assures that your signals add up to 1.0. This is your new hypothetical investment strategy.
Do the same performance evaluation as in Code 1.

Suggested Code 3 (Signal weighting):

As in codes 1 and 2, but stocks that have lower sigma seem more promising, so you want to invest more in them. Choose as your pfio-weight $w_{i,y} = 1.0 - (signal_{i,y} / (\sum_i signal_{i,y})$. The second term’s divisor assures that your signals add up to 1.0; the first term (1.0-) just makes sure your biggest investment is in the lowest ranks. This is your new hypothetical investment strategy.
Do the same performance evaluation as in Code 1.

Suggested Code 4 (Multi-Signal Investing):

As in code 2, but now you want to take two signals. You need to negate $rmcap_{i,y}$ in your sort (so that the AAPL has the lowest rank 1).
It is very important that when you select an investment strategy from two signals, you have similar scales on both variables. Using ranks assures this. (Otherwise, think of what you will get if you have two signals A and B, where A has no variation (or 1/100 the variation of B) and B has a lot of variation. How would your top 20 stocks look like in terms of your portfolio choice? How much would the A signal matter?)
Now take the lowest 20 $\sigma_{i,y}$ stocks and the largest 20 $mcap_{i,y}$ stocks. For these two signals and 20-40 stocks (there may be overlap), calculate a signal for each stock that is $signal_{i,y} = \lambda\times rmcap_{i,y} + (1-\lambda)\times rsigma_{i,y}$ where you choose different lambdas. Stocks that have the lowest $signal_{i,y}$ (lambda-weighted sum of both ranks) seem most promising. Thus, consider investing more in them, too (or not!). You can choose as weight $w_{i,y} = 1.0 - (signal_{i,y} / (\sum_i signal_{i,y})$.
Do the same performance evaluation as in Codes 1 and 2. What lambda gives you the lowest volatility?

Your Real Investment Strategy for the Year

Once you have Code 4, trying out different/more variables will be a breeze. It is more or less simply throwing in more variables into the same procedure.

Think Candidates

what lagged candidates would help you make a portfolio of lower variance?
own volatility?
marketcap?
industry?
what else?