# What is the expected return on the market portfolio?

### Three estimates and a confession

It is a common practice to proxy the stock market return with the S&P 500 index of blue-chip US equities. As a rough approximation, this is not too bad. But the stock market is actually considerably broader. The S&P 500 are neither 500 in number, nor the largest corporations by market capitalization; nor can the S&P 500 be held by investors in any strict sense (of course, there are cheap ETF proxies, but again, they also have tracking errors). Over the past thirty years, a total of roughly 57,000 stocks have been publicly listed at some point. Some 12,000 are currently listed. They’re worth $92tn. However, most of these stocks are rather small and quite illiquid. The universe of highly-liquid stocks is much smaller. The 1,000 largest firms by market cap are worth $69tn; the largest 500 are worth $56tn. We isolate 896 highly-liquid stocks as of today (based on standard liquidity filters) that are together worth $44tn. Over the past 30 years, there are 1,183 stocks that pass our liquidity filters and on which we have reliable data. This is essentially the universe of blue-chip US equities that are of interest to us. We proxy the market portfolio by a monthly-rebalanced portfolio of this filtered set of stocks, whose target weights are given by their relative market capitalization for the preceding month.

The graph of the cumulative returns on the market portfolio has the familiar shape. It looks very much like the S&P 500, doesn’t it? Indeed, the correlation between the two is 94.7%. However, plotting the two together, one can immediately see that the SPY is not a very good proxy of the market portfolio. In particular, the expected return on the SPY significantly overstates the expected return on the market portfolio. Whereas the unconditional return on the SPY is 11.1%; on the market portfolio, it is 9.8% per annum.

In what follows, we’ll test the idea that simply changing the estimator of expected returns and holding the rest of the portfolio construction process invariant can have major implications for portfolio performance. We construct three monthly-rebalanced, longonly portfolios, using only information that is available on the last trading day of the month. These are: (1) the market portfolio, which will serve as our benchmark; (2) the CAPM portfolio, where we optimize portfolios based on CAPM expected returns (ie, expected returns are assumed to be proportional to exposure to market risk); (3) the momentum portfolio, where we optimize weights by feeding expected returns using the estimator in Asness (1994). More precisely, we define momentum to be the 230-day cumulative return lagged by 22 days. We apply our proprietary desrisking technology to construct the latter two portfolios — everything is constant between the two except expected returns. Specifically, on each rebalance date, the same correlation matrix and volatility estimates are fed to the portfolio optimizer to obtain portfolios that maximize *ex ante* Sharpe ratios. The only difference between the CAPM portfolio and the momentum portfolio is the expected returns we feed to the optimizer.

We illustrate the performance of these three portfolios for all macro regimes over the past three decades, starting with the Clinton boom and ending with the Covid cycle.

During the Clinton boom but before the dot-com bubble, the market portfolio returned a scorching 18.3% per annum. By comparison, the mean return on the CAPM portfolios was 24.4% per annum, and that on the momentum portfolio, 25.9% per annum. The momentum portfolio sported a Sharpe ratio of 1.99, compared to 1.78 for the CAPM portfolio, and 1.53 for the market portfolio.

The Clinton boom set the stage for the dot-com bubble that quickly came to grief. The bubble lasted for a mere 90 trading days, but the dot-com winter lasted for years. In this period from late-1999 to the end of 2006, the market portfolio returned only 2.9% per annum; while the CAPM portfolio returned 15.5%, and the momentum portfolio returned 24.8% per annum. The Sharpe ratios tell an even starker story: while the market portfolio sported a Sharpe ratio of 0.18, the CAPM delivered 1.03 and the momentum portfolio delivered a Sharpe ratio of 1.66.

The high neoliberal era seems to have been the heyday of the momentum estimator. Even so, it continued to outperform. During the “long 2008”, 2007-2012, the market portfolio delivered 3.7% per annum. The CAPM portfolio delivered just 1.7%, while the momentum portfolio delivered a very respectable 9.5% per annum, all considered.

In the aftermath of the global financial catastrophe, the Fed became the only game in town. Extraordinary monetary accommodation turbocharged returns on the market portfolio, which returned 13.7% per annum in 2013-2019. The CAPM returned 11.8%, while the momentum portfolio returned 12.8% per annum — the first time the market portfolio beat the momentum portfolio for any length of time, and then too, by only 86 basis points.

But victory for the typical investor — by definition, the holder of the market portfolio — was short-lived. The market portfolio has averaged 13.0% per annum since Jan 2020; the CAPM has delivered 10.6%. Meanwhile, the momentum portfolio has returned 20.0% per annum over the same period.

Stock prices, of course, began falling around New Year’s Eve. Year-to-date, the market portfolio is down 22.3%, the CAPM portfolio is down 18.0%, but the momentum portfolio is down only 8.5%.

Over the whole sample, the market portfolio has an expected return of 9.7%, the CAPM portfolio has an expected return of 13.2%, and the momentum portfolio has an expected return of 18.5%. The Sharpe ratio of the momentum portfolio is twice as high as that of the market portfolio.

Unconditionally, the premium for pretending that Cliff’s momentum is a good estimator of expected returns is 8.8% per annum. There is considerable time-variation in this premium. And, it seems to be roughly counter-cyclical: high when expected returns on the market portfolio are low, and low when expected returns on the market portfolio are high. During the Clinton boom, for instance, the premium over the market portfolio was 7.6% per annum; during the dot-com winter, it averaged a mind-boggling 21.9% per annum. During the GFC winter, it was 5.8%; while it actually went to negative 0.9% in the post-GFC, pre-pandemic world when the Fed was the only game in town. In the Covid cycle, the premium revived to a healthy 7.0% per annum. The future is considerably more uncertain than the past. But I, for one, know which of these portfolios I want to own.

To sum up, the evidence is quite unambiguous that Cliff’s momentum is a good estimator of expected returns — in as much as the proof of the pudding is in the eating. Note that, for our purposes, the horizon for expected returns is the following month — we’re rebalancing monthly. Even if realized returns don’t match the momentum estimates well, or do so very noisily, we’re still dramatically better off pretending that they do. The point is that momentum *works*. It’s not entirely clear *why* it works; but it works. Not in the naive sense of “holding momentum stocks works,” rather it works when used rigorously within a larger strategy of systematic portfolio construction and disciplined rebalancing. This was basically what I was trying to convince Cliff on Twitter about. As with the Dirac equation, sometimes the creation is more intelligent than the creator.

What makes this particularly interesting at the moment is the divergence in the three estimates for expected returns on the market portfolio. Our unconditional estimate of expected returns on the market portfolio is 9.8% per annum (the annualized mean daily return on the market portfolio over the past 30 years); the rolling 252-day mean return is -5.4%; but Cliff’s momentum estimator gives an expected return estimate of 10.0% per annum. In other words, Cliff’s momentum estimator is suggesting that some of the recent sell-offs may reverse.

These are the numbers we obtain if we use the past performance of the market portfolio. What if instead of looking at the past performance of the market portfolio, we computed the weighted average of expected returns on individual stocks, with the weights given by relative market capitalization? In that case, the unconditional mean is 15.8% per annum (survival bias alert); the rolling 252-mean gives 0.3%; but Cliff’s momentum estimator yields 28.6%. So, the same pattern, except even more pronounced.

Those are the estimates. What is my confession? Well, I must confess that I have no idea which of these estimates is more reliable. What I do know is that I’ll need a very good reason to ditch Cliff’s momentum as an estimator of expected returns in my work.

Fama French market factor (Rm-Rf) is probably more representative (includes all of NYSE, AMEX and NASDAQ stocks). The risk premium Rm-Rf has a rolling 10y average of 8.6% p.a. over 1926-2021. It plots a near perfect sinusoidal curve :)

Tks! All very clear