# You're Probably Underweight Real-Estate

### Evidence from Doeswijk portfolios

Despite the catastrophe wrought by the dealers in the mid-2000s, real estate returns (as proxied by the etf, VNQ) have averaged 5.0% per annum at a compounded rate in the 15 years since April 2007. (All numbers are as of the closing prices on June 21, 2022. Expected returns are compounded or geometric rates of return.) Over the same period, stocks (VTI) averaged 8.6% per annum, while bonds (USIG) averaged only 3.6% per annum. In general, we proxy the asset classes in our investable universe by the following etf tickers.

"STOCKS": "VTI",

"REAL ESTATE": "VNQ",

"PRIVATE EQUITY": "PSP",

"BONDS": "USIG",

"TIPS": "TIPZ",

"HEDGE FUNDS": "HDG",

"CREDITS": "AGG",

"HIGH YIELD": "HYG",

"COMMODITIES": "DJP".

Our estimate of the equity risk premium (the expected return on stocks less the expected return on bonds) is thus 5.0% per annum. If forward-looking expected returns are close to these unconditional means, and these are the only three asset classes in the investable universe, your equity allocation ought to be 100%. Unfortunately, the future may not bear much resemblance to the past. For one, the 5.0% equity premium is not statistically significant. Recall that the standard error of the difference of means is the square root of the sum of the squared standard errors of the means. Formally, we obtain *t* = 1.18 if we use arithmetic means and *t* = 0.90 if we use geometric means. If it were a standard normal, the latter would be at the 82nd percentile. The weak result makes sense. Forward-looking expected returns on different asset classes can only be known with some uncertainty for otherwise risk arbitrage opportunities would be plentiful — but markets are efficient to zeroth order.

Doeswijk and coauthors gave us the best available estimate of the global market portfolio in 2020. In 2019, they published “Strategic Asset Allocation: Determining the Optimal Portfolio with Ten Asset Classes.” This is a fascinating paper on cross-asset allocation. I was struck by the clarity with which real estate emerges as a real alternative to equities. Here we replicate some of their work by documenting the results of an exercise where we only vary expected returns, leaving the correlation matrix and the standard deviations unchanged in the mean-variance optimization.

Before we do the controlled tests, we show some preliminary work. We obtain Doeswijk’s capital market assumptions and correlation matrix. These contain their forward-looking assumptions. You can think of these as ex ante meta-estimators: each of Doeswijk’s assumptions is informed by the literature, the data, and a healthy dose of common sense. Each of these numbers is defended in the detail.

Note the top-left 3x3 submatrix in the table above. The difference in expected returns between equities and real estate is 1.0% in geometric terms but 1.8% in arithmetic terms. However, that between private equity and US equities is 0.0% on a geometric basis and 2.5% on an arithmetic basis. The arithmetic assumptions are rosier than the geometric. And they are weighted more towards private equity at the expense of equities and real estate. This will show up clearly in the controlled experiment that follows. The premia look modest — the 4.0% unconditional equity risk premium seems correct, if a little bit on the conservative side. And the volatilities also look conservative (20% for equities and 7% for bonds; although real estate at 16% might be on the low side).

Here’s the correlation table. Again, each of these numbers is defended in the detail by Doeswijk and coauthors. The key column is the first. Correlation with stocks is a descaled version of CAPM beta. Commodities and inflation-protected bonds are uncorrelated (*r* = 0.0) with stocks; then come treasuries (*r* = 0.2), corporate bonds (*r* = 0.4), followed by high yield, hedge funds and real-estate (*r* = 0.6); and finally, private equity (*r* = 0.8). Asset duration for these assets is roughly proportional to these numbers.

We also examine the correlation matrix against three empirical estimators. The data in this case is daily returns at close from 2011-07-01 to 2022-06-21. Note that the empirical estimates were unavailable to hold throughout but *a priori* strategic allocations obviously were. So we shall not use the empirical estimates but use Doeswijk’s meta-estimates. The empirical estimates suggest a taxonomy whose first split is that between bonds and risk assets. Note that the assets are sorted on duration from top to down and left to right (stocks > PE > real estate > hedge funds > commodities > high yield > corporate bonds > treasuries > inflation-protected treasuries).

The low correlation of commodities and equities is picked up the Pearson estimator, and gets more pronounced for our shrinkage estimators. Note also the “core” of highly-correlated, high-duration risk assets: stocks, PE and real estate, best seen in the Ledoit Wolf shrinkage estimate at the top-left. We shall see that the optimal allocation *within this core* is the key to diversification for a patient asset-only investor. The objective of such an investor is to maximize wealth in period *t *+ 1. As the authors explain:

We take the perspective of an asset-only investor in search of the optimal portfolio. An asset-only investor does not take liabilities into account. The investment horizon is one year and the opportunity set consists of ten asset classes. The investor pursues wealth maximization and no other particular investment goals are considered.

Doeswijk and coauthors.

We create four monthly-rebalanced portfolios with time-invariant target weights. The risk premia and volatilities are assumed to be those reported in the capital markets assumptions table above. The correlation matrix used is also Doeswijk’s. We vary only the expected returns in a controlled experiment with the mean-variance machinery. “Doeswijk-GRP” is the MVO portfolio obtained by using the geometric risk premia as expected returns. “Doeswijk-ARP” likewise uses the arithmetic risk premia. “Broad-liquids” is just 50% on equities, and 10% each on real estate, commodities, high yield bonds, treasuries and investment grade corporate bonds. “MVO” is a test portfolio that uses the historical compounded expected returns. Note that the full-sample realized expected returns used in constructing the MVO portfolio were obviously not available in real time. We bake it only to test the Doeswijk assumptions against historical patterns.

We find that, as expected, using geometric premia results in a larger equity allocation (68%) than using arithmetic premia (42%). The MVO equity allocation is also large (63%). Now look at the allocation to real estate. Using arithmetic premia yields a fifth (20%) of capital be allocated to property, using geometric premia yields one-thirds (32%), and borrowing information from the future suggests an even larger allocation (37%). These are large numbers for what is treated as an “alternative asset class” and capped at 5-15% in the strategic asset allocations of institutional investors. This is why I say you’re probably underweight real-estate.

Note that we’re talking about an abstract financial portfolio of an asset-only investor. These are just ETFs. The question is what proportion should we hold them in? The different expected return assumptions (GRP vs ARP) yield allocations of 20-30%. These are based on conservative assumptions specified in the detail by the authors. Given the capital market assumptions (including the correlation matrix), the optimality of the two portfolios (Doeswijk-GRP and Doeswijk-ARP) follows mechanically from the mean-variance optimization.

The proof of the pudding, of course, is in the eating. So, how have these portfolios performed since 2007?

We find a premium of 3.0% using geometric premia instead of arithmetic premia. The longshort is a standard 130-30 longshort with $130 long on the geometric portfolio (with 32% on real estate) and $30 short on the arithmetic portfolio (with 20% on real estate). It has the highest expected return, a compounded rate of 8.5% per annum, and happens to have the Sharpe ratio of 0.36, same as our monthly-rebalanced portfolio tracking the market-weighted portfolio (“US-equities”). The quasi-equiweighted cross-asset portfolio (“Broad-liquids”) has the highest Sharpe ratio (0.42 vs 0.36 for equities). But the expected return on the geometric portfolio is 1.6% higher. That’s a lot of premium.

The 3% premium between geometric and arithmetic portfolios is so large that the most attractive portfolio is the longshort portfolio. The longshort offers an expected return premium of 1% over the market portfolio for the same Sharpe ratio. Because of our controlled experiment, we know that the only difference between the long and short of the longshort is the expected return assumption. We’re just telling the machine to modestly tune the relative expected returns between real estate, stocks, and private equity. This allows us to beat equities. Our estimates suggest that a real-estate allocation of as much as 20-30% may be warranted.

Because markets are efficient to zeroth order, this is as far as the machine itself will take you unconditionally. In the future, we want to do this exercise conditionally; at the monthly frequency. We’ll test whether dynamic optimization allows you do do even better.

So you want to vary asset allocation within bands, such as in the graph above (from here). What should the bands be? In our judgement, 40-70% for equities and 10-30% each for bonds, commodities, and real estate may be appropriate.

Equities are the bread and butter of the patient asset-only investor. A case can be even made that the default allocation to equities should be 100%. The case is perhaps stronger for the 60-40 benchmark in a two-asset world with only stocks and bonds. Thankfully, investors are not in a two-asset world. And especially when investing amid low expected returns and soon to be double-digits inflation, they need to look at other asset classes. Put another way, the conditional equity risk premium as of writing is at best at zero, if indeed not negative. Investors sorely need places other than equities to hide. We’ve also seen that housing, but not bonds or equities, can be expected to protect investors against inflation. So, the case is very strong for a large real-estate allocation in an inflationary market winter.

Conditionally, of course, the case is also very strong for commodities. While the SPY has lost 12% over the past year, our dynamically optimized portfolio of commodities etfs (“derisked_portfolio”) has returned 37%, whereas the equiweighted portfolio of Cliff Asness and coauthors has returned 24%. Both of these commodities portfolios have benefited from the rally in OIL, which has gained 70%. The difference between the two is that, while the target weights of the equiweighted portfolio are fixed, the target weights of the derisked portfolio are dynamically adjusted on each rebalance date using our proprietary derisking methodology.

Given the low expected returns on bonds and stocks in the near future, we think that an allocation of 40% to equities (our unconditional lower bound), and 30% each to commodities and real estate (our unconditional upper bounds) is warranted as of writing. Of course, one can do much better than holding etfs.

We have developed monthly-rebalanced portfolios of equities (our flagship portfolio) and commodities that beat their market and etf benchmarks. Now we want to construct a portfolio of real-estate assets that is as efficient. Then we can offer a cross-asset product whose within asset class portfolios harvest signal alpha and offer more attractive moments than the relevant market benchmarks — to be then combined in a conditionally-optimal way. Rotating both in and out of asset classes should be efficiently fast and prudently slow. We can optimize the learning rate out of sample, but an even more interesting second product would use a macrofinancial signal to control the strategic asset allocation — which, as we’ve seen, works beautifully in a two-asset world. Exciting times at the cutting-edge of quant research!

Anyway, you probably need to rethink your real-estate allocation.

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