# Are Shocks to Housing Priced in the Cross-Section of Stock Returns?

In the previous post I argued that the risk premium on property is due to the fact that the marginal investor in housing is your average homeowner who finds it extraordinarily hard to diversify away the risk posed by her single-family home to her balance sheet. If I am right, this means that housing wealth is a systematic risk factor that ought to be priced in the cross-section of expected stock excess returns (ie, returns in excess of the risk-free rate).

Assume that the marginal investor in the stock market is your average homeowner. Since it is so hard for her to diversify away the risk posed by fluctuations in property values, she should value stocks that do well when property markets tank. Conversely, stocks whose returns covary with returns on property should be less valuable to her. Given our assumption that your average homeowner is the marginal investor in equities, expected returns on stocks whose returns covary strongly with property returns should be higher than expected returns on stocks whose returns covary weakly (or better yet, negatively) with property returns. This is what it means for shocks to housing to be priced in the cross-section of expected stock returns.

We want our risk factor to capture broad-based fluctuations in housing wealth. Ideally, we would use a quarterly time-series for total returns (including both rent and capital gains) on housing wealth owned directly by US households. I am unaware of the existence of such a dataset—if you know where I can find the data, please get in touch.

We can also instrument fluctuations in housing wealth by using a property price index. Here we use the US property price index reported by the Bank of International Settlements. For return data we use 250 test assets from Kenneth French’s website. (The same dataset I used in my paper, “The Risk Premium on Balance Sheet Capacity.”)

We’re now going to jump straight into the results. For our econometric strategy please see the appendix at the bottom.

Figure 1 displays a scatterplot of the cross-section of expected stock returns. Along the X axis we have the betas (the sensitivity of the portfolio’s return to property returns) for our 250 test assets, and along the Y axis we have the mean excess returns of the portfolios over the period 1975-2016.

Guys, this is not bad at all. Our single factor explains 20 percent of the cross-sectional variation in expected excess returns. By comparison, the celebrated Capital Asset Pricing Model, for instance, is a complete washout.

It is very hard for single factor models to exhibit such performance. Table 1 displays the results from the second pass. We see that the mean absolute pricing error is large because the zero-beta rate does not vanish. Indeed, at 1.8 percent per quarter it is simply not credible. But the risk premium on property returns is non-trivial and significant at the 5 percent level.

Table 1. Property returns and the cross-section

 Estimate Std Error p-Value Zero-beta rate 0.018 0.006 0.002 Property return 0.007 0.004 0.049 R^2 0.195 Adj-R^2 0.192 MAPE 0.022

I have a lot of professional stake in the failure of this model actually. I have argued that stock returns are explained by fluctuations in the risk-bearing capacity of the market-based financial intermediary sector. In other words, the central thrust of my work is to say that we ought to pay less attention to the small-fry and considerably greater attention to the risk appetite of the big fish, for that is what drives market-wide risk appetite. Fortunately for my thesis, property shocks do well, but not nearly as well as balance sheet capacity.

Figure 3 displays yet another scatterplot for the cross-section. On the X axis we have the factor betas (the sensitivities of the portfolios to balance sheet capacity) and on the Y axis we have, as usual, mean excess returns over 1975-2016.

In Table 2 and Figure 3 we’re only looking at a single-factor model with balance sheet capacity as the sole systematic risk factor. That’s a parsimonious theory that says: exposure to fluctuations in the risk-bearing capacity of broker-dealers explains the cross-section of asset returns. The empirical evidence is pretty compelling that this is the right theory. We see that balance sheet capacity singlehandedly explains 44 percent of the cross-sectional variation in expected stock excess returns. What is also manifest is the vanishing of the zero-beta rate; and the attendant vanishing of the mean absolute pricing error. Other single factor models cannot even dream of competing with balance sheet capacity in terms of pricing error. Indeed, I have shown in my paper that even the pricing errors of standard multifactor benchmarks, Fama and French’s 3-factor model and Carhart’s 4-factor model, are significantly bigger than our single factor model’s 48 basis points. We can thus have good confidence that the evidence does not reject our parsimonious asset pricing model.

Table 2. The Primacy of Balance Sheet Capacity

 Estimate Std Error p-Value Zero-beta rate 0.002 0.010 0.440 Balance sheet capacity 0.095 0.038 0.007 R^2 0.442 Adj-R^2 0.440 MAPE 0.005

I know what you are thinking. If these things are priced in, there must be a way to make money off it. How do I get some of that juicy risk premium? Aren’t they non-traded factors? Yes, they are. But you can still harvest the risk premium on non-traded factors, eg by constructing factor mimicking portfolios. Briefly, you project your factor onto a bunch of traded portfolios and use the coefficients as weights to construct a portfolio that tracks your non-traded factor.

Figure 4 displays the risk-adjusted performance of portfolios that track benchmark risk factors and the two risk factors discussed in this essay. We report Sharpe ratios (the ratio of a portfolio’s mean excess return to the volatility of the portfolio return) rescaled by the volatility of the market portfolio for ease of interpretation.

The results are consistent with our previous findings. The stock portfolio that tracks property outperforms standard benchmarks convincingly. In turn, the portfolio that tracks balance sheet capacity outperforms the portfolio that tracks property. But let’s be very clear about what Figure 4 does not say. There is no free lunch. More precisely, there is no risk-free arbitrage.

The existence of these two risk premiums imply instead that there is risk arbitrage. That is, you can obtain superior risk-adjusted returns than the market portfolio by systematically harvesting these risk premiums. The existence of the two risk premiums is due to structural features. Specifically, the property premium exists because non-rich homeowners must be compensated for their exposure to housing; while the risk premium on balance sheet capacity exists because of structural features of the market-based financial intermediary sector—features that I explain in detail in the introduction of my paper. Since we can expect these structural features to persist, we should therefore not expect these risk premiums to vanish (or perhaps even attenuate much) upon discovery.

Appendix. Cross-Sectional Asset Pricing

We can check whether any given risk factor is priced in the cross-section of excess returns using standard 2-pass regressions where you first project excess returns $\left(R_{i,t}\right)$ onto the risk factor $\left(f_t\right)$ in the time series to obtain factor betas $\left(\beta_i\right)$ for assets $i=1,\dots,N,$

$R_{i,t}=\alpha+\beta_i f_{t}+\varepsilon_{i,t}, \qquad t=1,\dots,T,$

and then project mean excess returns $\left(\bar R_i\right)$ onto the betas in the cross-section to obtain the price of risk $\lambda$,

$\bar R_{i}=\gamma^{0}+\lambda\hat\beta_{i}+e_{i}, \qquad i=1,\dots,N.$

The scalar $\gamma^{0}$ is called the zero-beta rate. If there is no arbitrage, the zero-beta rate must vanish. If the zero-beta rate is statistically and economically different from zero, then that is a failure of the model. That’s why the mean absolute pricing error is a better metric for the failure of an asset pricing model than adjusted-R^2. It’s given by,

$\text{MAPE}:=|\gamma^{0}|+\sum_{i=1}^{N}\omega_{i}|\hat e_{i}|,$

where $\omega_{i}$ are weights that we will discuss presently.

If you try this at home, you need to know that (1) ordinary least squares (OLS) is inefficient in the sense that the estimator no longer has the lowest variance among all linear unbiased estimators; (2) OLS standard errors are an order of magnitude too low (and the estimated coefficients are attenuated, though still consistent) because their computation assumes that the betas are known, whereas we are in fact estimating them with considerable noise in the first pass.

The solution to (1) is well-known. Simply use weighted least squares (WLS) where the weights are inversely proportional to the mean squared errors of the time-series regressions,

$\omega_i \propto \left[\frac1{T}\sum_{t=1}^{T}\hat\varepsilon^2_{i,t}\right]^{-1},\qquad \sum_{i=1}^{N}\omega_{i}=1.$

The solution to (2) is to use errors-in-variable (EIV) corrected standard errors. In our work, we always use WLS for the second pass and report EIV-corrected standard errors wherever appropriate.