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Different asset classes have different moments (mean, variance, covariance parameters) that have a more or less stable structure. We shall assume that the risk premia (expected returns in excess of the risk-free rate which we assume to be zero throughout) sported by different asset classes are stationary or time-invariant. This allows us to break the period for which we have data into a training period from 1994 to 2003 and a test period from 2004 to 2022. We shall optimize portfolio allocation weights for the major asset classes over the training period and test their performance over the test period.
In what follows, bonds refers to the AA-rated, investment-grade corporate bonds total return index, stocks or the market portfolio refers to the S&P 500 index, and longonly refers to our flagship longonly portfolio that harvests the overnight drift premium from the cross-section of blue-chip US equities. Before we model conditional strategic asset allocation, we should develop an unconditional target portfolio. In order to do that, we need to pay attention to the investor’s risk tolerance.
You can always generate very high returns in the short run at the price of greater risk (within the space spanned by investable assets). A mean-variance investor is one who measures gains by expected returns (here proxied by the annualized arithmetic mean over the training period) and costs by the variance of the returns of her portfolio. Investors can be ordered by their risk-tolerance hyperparameter (the coefficient of portfolio variance in the loss function). Equivalently, mean-variance investors can be ordered by their target volatility.
So, we compute the expected returns and the covariance matrix for bonds, stocks, our longonly overnight drift portfolio over the training period. We work throughout at the monthly frequency. The annualized return on bonds, stocks, and our longonly portfolio during the training period was 7%, 11% and 24% respectively. The volatilities were 5% for bonds and 14% for equities portfolios. The correlation between bonds on the one hand and the market and longonly portfolios of equities was 0.2 — that between the two stock portfolios came in at 0.9. All unconditional parameter estimates are important because they will get baked into the strategic asset allocation.
Our reference investor has a target volatility of 10% per annum. In a two-asset world, such an investor holds roughly 60% in stocks and 40% in bonds. Suppose our reference investor were to suddenly find herself in a three-asset world with our longonly offering, what would be her optimal asset allocation? We also assume that she is not leverage constrained.
The answer is easily obtained from a straightforward cvxpy optimization with the loss function determined by the expected returns and covariance matrix shown above. Our reference investor’s optimal (MVO) portfolio is 127% on our longonly portfolio, hedged by -77% exposure to the market portfolio and 50% exposure to investment grade corporate bonds. Ie, starting with a third of our portfolio in each of the three assets, we lever up our exposure to the longonly overnight drift portfolio by one and a quarter, hedge that risk by exposure to an inverse spy etf to the tune of three-quarters, and place two-quarters in investment grade bonds.
So, that’s how we construct our unconditional strategic asset allocation. How does it perform out-of-sample? We now restrict the data to 2004-2022 and look at the performance of the portfolios. The MVO portfolio is not only strongly preferable for a risk intolerant investor, with an expected return higher than or equal to the longonly portfolio (which has a lot of residual systematic risk despite derisking), it is also preferred by a risk neutral investor. But the performance is risk-adjusted terms is very attractive.
Let’s go back to where we began. We began with the assumption that the moments of the asset classes haven’t changed between the two periods. Our reference investor wants a target volatility of 10%. In 2004-2022 test sample, the MVO portfolio sported a realized volatility of 9.2%, so our risk target was more or less met over the past twenty years. Over the test period, the MVO portfolio had a max drawdown of 18.6%, compared to 11.3% for bonds, 39.8% for our longonly portfolio, and 50.8% for the market portfolio.
The message is really brought home when we look a simple bar graph of the Sharpe ratios. The market portfolio delivered a Sharpe ratio of 0.8 over the test period. Bonds yielded superior risk-adjusted returns: 0.9. The longonly overnight drift portfolio outperforms both handsomely: 1.8. While the MVO portfolio just hits it out of the park: 2.7.
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