SSRN Paper: Expected Returns and the Expected Growth in Rents of Commercial Real Estate


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Investigations of whether the cap rate, that is, the rent-price ratio in commercial
real estate incorporates information about future expected real estate returns
and future growth in rents.

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SSRN Paper: Expected Returns and the Expected Growth in Rents of Commercial Real Estate

  1. 1. Expected Returns and the Expected Growth in Rents of Commercial Real Estate∗ † ‡ § Alberto Plazzi Walter Torous Rossen Valkanov ∗ We thank Andrea Berardi, Christopher Downing, Harrison Hong, Andrey Pavlov, Antonio Rubia, Stephen Schaefer and seminar participants at the following conferences for useful comments: the AREUEA, the XXXVI EWGFM, the FMA European meeting, the SAET meetings in Vigo, and the SAFE center at the University of Verona. We are especially grateful to an anonymous referee for many suggestions that have greatly improved the paper. All remaining errors are our own. † The Anderson School at UCLA and University of Verona, 110 Westwood Plaza, Los Angeles, CA 90095- 1481, phone: (310) 825-8160, fax: (310) 206-5455, e-mail: ‡ The Anderson School at UCLA, 110 Westwood Plaza, Los Angeles, CA 90095-1481, phone: (310) 825- 4059, fax: (310) 206-5455, e-mail: § Corresponding author. Rady School at UCSD, Pepper Canyon Hall, 9500 Gilman Drive, MC 0093, La Jolla, CA 92093, phone: (858) 534-0898, fax: (858) 534-0745, e-mail: Electronic copy of this paper is available at:
  2. 2. Abstract We investigate whether the cap rate, that is, the rent-price ratio in commercial real estate incorporates information about future expected real estate returns and future growth in rents. Relying on transactions data spanning several years across fifty-three metropolitan areas in the U.S., we find that the cap rate captures fluctuations in expected returns for apartments, retail, as well as industrial properties. For offices, by contrast, the cap rate does not forecast returns even though additional evidence reveals that expected returns on offices are also time-varying. We link these differences in the ability of the cap rate to forecast commercial property returns to differences in the stochastic properties of their rental growth rates with the growth in office rents having a higher correlation with expected returns and being more volatile than for other property types. Taken together, our evidence suggests that variation in commercial real estate prices is largely due to movements in discount rates as opposed to cash flows. JEL classification: G12, R31 Keywords: Cap rate, real estate, return predictability, rent growth. 2 Electronic copy of this paper is available at:
  3. 3. 1 Introduction U.S. commercial real estate prices fluctuate considerably, both cross-sectionally as well as over time. For example, the returns to apartments during the last quarter of 1994 ranged from 21.4 percent in Dallas, Texas to −8.5 percent in Portland, Oregon. Eight years later, during the last quarter of 2002, the returns to apartments in Dallas and Portland were 1.2 and 4.4 percent, respectively. Other types of commercial real estate, such as retail, industrial, and office properties, have experienced even larger return fluctuations. Understanding what drives these fluctuations is an important research question since commercial real estate represents a substantial fraction of total U.S. wealth. In particular, the value of U.S. commercial real estate as of the end of 1999 was estimated to be approximately six trillion dollars, which at the time represented almost half of the U.S. stock market’s value (Case (2000)). From an asset pricing perspective, the price of a commercial property, be it an office building, apartment, retail or industrial space, equals the present value of its future net rents, that is, rents minus any operating expenses adjusted for vacancies. This fundamental present value relation implies that the observed fluctuations in commercial real estate prices should reflect variations in future rents or in future discount rates, or both. In valuing commercial real estate, it is particularly important to consider the possibility that discount rates and rental growth rates are time-varying as it is often conjectured that both fluctuate with the prevailing state of the economy. For example, Case (2000) points out “the vulnerability of commercial real estate values to changes in economic conditions” by describing recent boom-and-bust cycles in that market. He provides a simple example of cyclical fluctuations in expected returns and rental growth rates that give rise to sizable variation in commercial real estate prices. Case, Goetzmann, and Rouwenhorst (2000) make a similar point using international data and conclude that commercial real estate is “a bet on fundamental economic variables.” Despite these and other studies, however, little is known about the dynamics of commercial real estate prices. In this paper, we investigate whether expected returns and the expected growth in rents of commercial real estate are time-varying by relying on a version of Campbell and Shiller’s (1988) “dynamic Gordon” model. A direct implication of this model is that the cap rate, defined as the ratio between a property’s net rent and its price, should reflect fluctuations in expected returns or in rental growth rates, or both. The cap rate is a standard measure of 1
  4. 4. commercial real estate valuation and corresponds to a common stock’s dividend-price ratio where the property’s net rent plays the role of the dividend. As an illustration, suppose that the cap rate for apartments in Portland is higher than the cap rate of similar apartments in Dallas. The dynamic Gordon model suggests that either future discount rates in Portland will be higher than those expected in Dallas or that future rents in Portland will grow at a slower rate than in Dallas, or both. Whether or not cap rates can forecast future returns or future rent growth is ultimately an empirical issue and depends on the variability of these processes, their persistence, and their mutual correlation. To investigate whether cap rates do capture future variation in returns or rental growth rates, we use a novel dataset of commercial real estate transactions across fifty-three U.S. metropolitan areas reported at a quarterly frequency over the sample period 1994 to 2003. For a subset of twenty-one of these regions, we also have bi-annual observations beginning in the last quarter of 1985. These data are available on a variety of property types including offices, apartments, as well as retail and industrial properties. The transactions nature of our commercial real estate data differentiates it from the appraisal data typically relied upon in other real estate studies. For example, unlike the serially correlated returns and rental growth rates found in appraisal data (Case and Shiller (1989, 1990)), we verify that returns and rental growth rates in our data are not serially correlated beyond a yearly horizon. Relying on these data, we document that higher cap rates do indeed predict higher future returns on apartment buildings as well as retail and industrial properties. Cap rates, however, do not predict future returns of office buildings. For apartments, retail, and industrial properties, the predictability of returns is robust to controlling for cross- sectional differences using fixed effects as well as variables that capture regional differences reflecting demographic, geographic, and various economic factors. In terms of the economic significance of this predictability, we find that a one percent increase in cap rates leads to an increase of up to four percent in the prices of these properties. This large effect is due to the persistence of the fluctuations in expected returns and is similar in magnitude to that documented for common stock (Cochrane (2001)). By contrast, we do not find reliable evidence that cap rates predict future movements in rental growth rates. Only for offices do we find limited evidence of lower cap rates predicting higher future rental growth rates and then only at long horizons. This, however, does not imply that the rental growth rates of commercial real estate do not vary over time. On the contrary, their variability is similar to, and, in the case of offices, even larger than that of the growth documented in dividends. 2
  5. 5. Our results point to a fundamental difference between apartments, retail and industrial properties, where cap rates do forecast returns, and office buildings, where they do not. This difference provides us with a unique opportunity to investigate under what conditions valuation ratios can or cannot predict future returns. In the context of the dynamic Gordon model, it is well known that the dividend-price ratio predicts time-varying expected returns under two conditions. First, expected returns and expected dividend growth rates must be uncorrelated. If they are correlated, then fluctuations in one series will offset, on average, fluctuations in the other and, as a result, the dividend-price ratio will remain unchanged. Secondly, the presence of extreme movements in dividend growth rates that are orthogonal to the time variation in expected returns will make it difficult to detect a statistically reliable forecasting relation. These conditions have been discussed by Campbell and Shiller (1988b), Campbell, Lo, and MacKinlay (1997), and more recently by Lettau and Ludvigson (2004) and Menzly, Santos, and Veronesi (2004) but only in the context of common stock. Following the logic in these papers, there are a number of possible interpretations of the documented lack of forecastability by office cap rates. For example, it may be the case that the expected returns of office buildings exhibit much less time-variation than the expected returns of other commercial property types. Alternatively, cap rates may fail to forecast the variation in future office returns because office rental growth rates are more correlated with expected returns or are more volatile than the rental growth rates of other properties. These alternatives have very different implications for asset pricing and portfolio allocation. We find that the future returns of all four commercial property types exhibit similar correlations with macroeconomic variables, for example, the term spread, the default spread, the rate of inflation, and the short interest rate. These variables are known to capture fluctuations in business cycle conditions and have been widely used in the finance literature to track the time-varying behavior of stock market returns (Campbell and Shiller (1988a), Campbell (1991), Fama and French (1989), Lettau and Ludvigson (2004), Torous, Valkanov, and Yan (2005) and, for a good review, Campbell, Lo, and MacKinlay (1997)). However, while we find that the rental growth rates of all four commercial property types are correlated with these macroeconomic variables, the correlations are significantly higher for offices. We also show that the volatility in rental growth that is orthogonal to the macroeconomic variables is much higher for offices than for the other property types and is even higher than the volatility of the stock market’s dividend growth rate over our sample period. Hence, we conclude that while the expected returns of offices are time-varying, the failure to forecast their future variation reflects the fact that when compared to the other property types the 3
  6. 6. growth in office rents is more correlated with the state of the economy and its orthogonal remainder is much more volatile. The view of commercial real estate that emerges from our research is that of an asset class characterized by fluctuations in expected returns not unlike that of common stock. All property types including offices exhibit time-varying returns that are forecastable using precisely the same business cycle proxies found to forecast common stock returns. This being the case, our results suggest that institutional investors and others attempting to hedge the cyclical variation of common stocks should carefully consider the inclusion of commercial real estate into their portfolios. In fact, among the commercial real estate alternatives, office properties would appear to provide the least effective hedge as their rental growth rates also vary with the state of the economy. Commercial real estate, however, differs from common stock in several important ways. For example, it is often argued that common stock dividends do not accurately reflect changing investment opportunities confronting a firm. Dividends are paid at the discretion of the firm’s management and there is ample evidence that they are either actively smoothed, the product of managers catering to investors’ demand for dividends, or the result of managers’ reaction to perceived mispricings (Shefrin and Statman (1984), Stein (1996), and Baker and Wurgler (2004)). Dividends are also subject to long term trends such as the recent decrease in the propensity of firms to pay dividends (Fama and French (2001)). By contrast, rents on commercial properties are not discretionary and are paid by tenants as opposed to property managers. Furthermore, rents, especially office rents, are more sensitive to prevailing business conditions. Because commercial real estate is not publicly traded and is characterized by higher transactions costs, our analysis focuses on long horizon predictability where these particular factors are expected to be less important. Finally, the prices of commercial properties are likely to be more sensitive than stocks to geographic, demographic, and local economic factors. To capture these sensitivities, following Abraham and Hendershott (1996), Capozza, Hendershott, Mack, and Mayer (2002), and Glaeser, Gyourko, and Saks (2004), we use population growth, per capita income growth, employment growth, the growth in construction costs, coastal region dummies, as well as several other urbanization proxies in an attempt to control for these cross-sectional differences. The plan of this paper is as follows. In Section 2, we present our valuation framework and discuss its application to commercial real estate taking special care to account for locational differences as well as differences across property types. In Section 3, we discuss 4
  7. 7. our commercial real estate data. The main predictive results are presented in Section 4 along with various robustness checks. The economic significance of the predictability and its implication for the volatility of commercial real estate prices is discussed in Section 5. In Section 6 we provide further evidence on the time-variability of expected returns to commercial real estate. We link the differences in predictability results across property types to differences in forecastability as well as to differences in variability of their rental growth rates. We offer concluding remarks in Section 7. 2 Real Estate Returns, Rents, and Growth in Rents 2.1 The Rent-Price Ratio Model We denote by Pt the price of, say, an apartment building at the end of period t and by Ht+1 its net rent, that is, rent minus any operating expenses adjusted for vacancies, from period t to t + 1. The gross return from holding the apartment building from t to t + 1 is: Pt+1 + Ht+1 1 + Rt+1 ≡ . (1) Pt The above definition of the return to commercial real estate is similar to that of common stock. The only difference is that a commercial property provides real estate services at a market value Ht+1 instead of paying dividends. If we define the log return as rt+1 ≡ log(1 + Rt+1 ) and the log net rent as ht+1 ≡ log(Ht+1 ), we can follow Campbell and Shiller (1988) and express rt+1 using a first-order Taylor approximation as rt+1 ≈ κ + ρpt+1 + (1 − ρ)ht+1 − pt , where κ and ρ are parameters derived from the linearization1 . Solving this relation forward, imposing the transversality condition limk→∞ ρk pt+k = 0 to avoid the presence of rational bubbles, and taking expectations at time t, gives the following present value relation for the log price 1 In particular, ρ ≡ 1/(1 + exp(h − p)), being h − p the average log rent - price ratio. Note that for the US commercial real estate market, the average log rent - price ratio in the period 1994 - 2003 has been about 8.5% annually, implying a value for ρ of 0.92 in annual terms, which is slightly lower than the 0.97 in the equity market for the same period. 5
  8. 8. pt ≡ log(Pt ) of a commercial real estate property: ∞ κ pt = + Et ρk [(1 − ρ)ht+1+k − rt+1+k ] . (2) 1−ρ k=0 Expression (2) states that a high property price today reflects the expectation of high future rents or of lower future expected returns or both. If commercial real estate markets are efficient, then information about future cash flows or future discount rates should be reflected in current property prices. Expression (2) has been previously used in the asset pricing literature to analyze the fluctuations of equity returns (see Campbell and Shiller (1988b) and Campbell (2003) for a review). If expected returns and rental growth rates are both stationary, expression (2) implies a log-linear approximation of the rent-price ratio which will facilitate our subsequent empirical analyses. In the commercial real estate literature, the rent-price ratio Ht /Pt is referred to as the cap rate (Geltner and Miller (2000)). Therefore, if we define capt ≡ ht − pt , from expression (2) we can write: ∞ ∞ κ k capt = − + Et ρ rt+1+k − Et ρk ∆ht+1+k . (3) 1−ρ k=0 k=0 The above equation is best understood as a consistency relation. It states that if a commercial property’s cap rate is high, then either the property’s expected return is high, or the growth of its rents is expected to be low, or both. This log-linearization framework was proposed by Campbell and Shiller (1988b) as a generalization of Gordon (1962)’s constant- growth model and explicitly allows both expected returns and dividend growth rates to be time-varying. Like any other financial asset, there are good reasons to believe that expected returns to commercial real estate and rental growth rates are both time-varying. We will use expression (3) as the starting point for our analysis of fluctuations in commercial real estate prices. To proceed, we must make additional assumptions about the dynamics of expected returns, Et rt+1 , as well as expected rental growth rates, Et ∆ht+1 . We will assume that expected returns follow a stationary autoregressive process of order one (AR(1)) with 6
  9. 9. autoregressive coefficient φ satisfying |φ| < 1: Et rt+1 = r + xt = r + φxt−1 + ξt (4) where r is the unconditional expected return and r + xt is the conditional expected return at time t. The vector xt contains conditioning information such as the term spread, the default spread, inflation, and the short interest rate that have been previously shown to capture time- varying economic conditions2 , while ξt is a white noise disturbance. This AR(1) specification provides a parsimonious representation of slowly evolving macroeconomic conditions. We further assume that the expected growth of rents is also time-varying, Et ∆ht+1 = g + τ xt + yt = g + τ xt + ψyt−1 + ζt (5) where g is the unconditional expected rental growth rate and ζt is a white noise disturbance. Here a non-zero value of the coefficient τ implies that rental growth and expected returns are correlated. For example, if τ = 1 then both rental growth and expected returns respond equivalently to changing economic conditions. The yt term represents the variation in rental growth that is orthogonal to the variation in expected returns. We allow yt to also be serially correlated.3 Using expressions (4) and (5) and ignoring the κ terms, the cap rate, expression (3), can be written as: r−g xt (1 − τ ) yt capt = + − . (6) 1−ρ 1 − ρφ 1 − ρψ The first term in expression (6) reflects the difference between the unconditional expected return and the unconditional rental growth rate. The second term in expression (6) captures the influence of time-varying fluctuations in expected returns and rental growth rates on cap rates. In particular, large deviations from unconditional expected returns (large xt ) or more persistent deviations (large φ) imply high cap rates. Also, fluctuations in expected rental growth that are orthogonal to expected returns (yt ) are negatively correlated with cap rates. 2 See Campbell (1991), Campbell and Shiller (1988a), Chen, Roll, and Ross (1986), Fama (1990), Fama and French (1988, 1989), Ferson and Harvey (1991), and Keim and Stambaugh (1986), among others. 3 Lettau and Ludvigson (2004) use a similar specification to model the correlation between expected returns and expected dividend growth in common stocks. 7
  10. 10. We can immediately see that if expected returns and expected rental growth rates are correlated, that is, τ lies between zero and one, then it will be difficult for the cap rate to forecast expected returns. In the extreme case where expected rental growth rates move one for one with expected returns, τ = 1, the cap rate will be unable to detect any fluctuations in expected returns because the variation in expected returns will be exactly offset by corresponding fluctuations in expected rental growth rates. Detecting a forecasting relation between cap rates and expected returns is also made difficult if the variation in the portion of rental growth rate orthogonal to the variation in expected returns, yt , is large. Expressions (3) and (6) impose testable restrictions on the time series behavior of asset prices including commercial real estate. The extant literature (see, e.g., Campbell and Shiller (1988b), Fama and French (1988), Campbell, Lo, and MacKinlay (1997), Lettau and Ludvigson (2004)) tests this relation using data from the US stock market and the main findings can be summarized as follows: (i) the dividend-price ratio is somewhat successful at capturing movements in future expected stock returns; (ii) the dividend-price ratio is a “smooth” variable and, as a result, is most successful at capturing movements in expected returns at longer horizons; and (iii) the dividend-price ratio does not seem to capture fluctuations in dividend growth rates which appear to be close to i.i.d.4 These stock market time series regressions require a lengthly series of data owing to well known small-sample biases induced by the persistence of the dividend-price ratio (Stambaugh (1999)). 2.2 Real Estate Expected Returns, Rents, and Cap Rates Across Metropolitan Areas in the US The dynamic Gordon model outlined in the previous section specifies the time series properties and resultant forecasting relations expected to prevail between cap rates and expected commercial real estate returns as well as expected rental growth rates. In principle, these relations are applicable to any category of commercial real estate located in any metropolitan area. In this paper, we consider four broad categories of commercial real estate: apartments, office buildings, industrial and retail properties, denoted by the superscripts A, O, I, and R, respectively.5 By their very nature, these property types differ in their rent 4 Although ongoing research is revisiting these results (Lettau and Ludvigson (2004)). 5 Hotel properties represent another major category of commercial real estate. Unfortunately, we do not have data on hotels and so they are excluded from our subsequent analysis. However, hotels represent less than four percent of the total value of U.S. commercial real estate. See Case (2000). 8
  11. 11. and risk characteristics while their sensitivities to economic conditions are also expected to vary. To fix matters, we concentrate on a particular type of commercial real estate property (denoted by a “l” superscript and where l = A, O, I, and R) located in two distinct metropolitan areas, say, Portland (denoted by an “i” subscript) and Dallas (denoted by a “j” subscript) whose cap rates are each given by expression (3). The difference in their cap rates at time t can be written as ∞ ∞ capl − capl = k l + Et i,t j,t ρk ri,t+1+k − rj,t+1+k l l + Et ρk ∆hl l j,t+1+k − ∆hi,t+1+k k=0 k=0 (7) where k l ≡ ki − kj , and capl denotes the cap rate for property type l in area i at time l l i,t 6 t. Expression (7) decomposes the difference in these cap rates into differences in expected returns and in expected rent growth rates. For example, suppose that the cap rate of the average apartment in Portland, area i, is higher than that of the average apartment in Dallas, area j. This implies that either expected returns of apartments in Portland are higher than those in Dallas, or that the future growth rate in apartment rents in Portland is lower than in Dallas, or both. Similar conclusions hold for the other commercial property types. Differences in cap rates for a particular commercial property type across metropolitan areas in expression (7) depend on the cross-sectional and time series variations in their expected returns and expected rental growth rates. To see this, suppose that expected returns to a particular property type l in each metropolitan area respond differentially to the same underlying fluctuations in xt l l l Et ri,t+1 = ri + δi xt (8) l l where ri is the unconditional return to property type l in area i while δi captures, in a reduced-form fashion, the effects of time variation in underlying economic conditions xt on expected returns to property type l in area i. In addition, we allow the growth in rents for a particular property type to differ across 6 The same convention applies to the remaining terms. This equation is obtained under the assumption that the linearization constant ρ is the same across both markets, which is a realistic assumption in our dataset where its values range from 0.911 to 0.925. 9
  12. 12. areas. For example, the rental growth rate of property type l in region i is Et ∆hl l l l i,t+1 = gi + τi xt + yi,t . (9) It follows that the cap rates of a particular commercial property type l in area i can be expressed as l l l ri − gi xt yi,t capl = i,t + δi (1 − τil ) l − . (10) 1−ρ 1 − ρφ 1 − ρψ In expression (10), the first term captures the difference in the unconditional moments which is simply a log-linearization of the standard Gordon constant-growth model. Clearly, the unconditional difference in cap rates for a particular property type across any two metropolitan areas must be due either to differences in their expected returns or in their expected rental growth rates. The second term in expression (10) captures time-varying expected returns and expected rental growth rates. l Metropolitan areas with more variable expected returns (higher δi ) will have higher cap rates even if the unconditional expected returns and growth rates in the two areas are equal. Similarly, the cap rate will be a better proxy for time-varying expected returns in areas where the growth in rents is less influenced by economic conditions (lower τil ). The expected returns of property types whose growth in rents are more sensitive to the time-varying economic conditions, τi close to one, will not covary with the cap rate. Also, the variability l of the growth in rents that is orthogonal to expected returns, yi,t , plays an important role in our ability to detect a relation between cap rates and future returns. Since this variability is orthogonal to the cap rate as well as to expected returns, it has the effect of noise in a predictive regression. To the extent that the variability in this component differs across property types, we should expect to see a stronger forecasting relation between cap rates l and expected returns for precisely those properties with lower variability in yi,t and a lower total volatility in their rental growth process. Notice that even small fluctuations in expected return can have a large effect on prices so long as these fluctuations are persistent, which as we will see below is an empirically reasonable assumption. This amplifying effect is captured by the denominator in expression (10). Hence, the introduction of time-varying expected returns is particularly important for commercial real estate where economic variations have sizeable effects on prices. See, for 10
  13. 13. example, Case (2000) who provides an example illustrating “the vulnerability of commercial real estate values to changes in economic conditions” along with a review of recent boom- and-bust cycles in that market.7 Similarly, using international data, Case, Goetzmann, and Rouwenhorst (2000) find that commercial real estate is “a bet on fundamental economic variables”. One of the empirical issues that we will subsequently investigate is how large this particular effect is likely to be. 3 The Commercial Real Estate Data Our data consists of prices and annualized cap rates of class A offices, apartments, retail and industrial properties for fifty-three U.S. metropolitan areas. The fifty-three sampled areas include more than 60% of the U.S. population (2000 data). A listing of these metropolitan areas is given in Table A1 of the Appendix. The data are provided by Global Real Analytics (GRA) and are available on a quarterly basis beginning with the second quarter of 1994 (1994:2) and ending with the first quarter of 2003 (2003:1). The prices and cap rates for each property category are averages of transactions data in a given quarter. Taken together, we have panel data with 1908 observations (36 quarters × 53 metropolitan areas). We also have a subset of these data for twenty-one of the areas going back to 1985:4 and ending at 2002:4 but only at a bi-annual frequency.8 Given annual cap rates, CAPt , and prices, Pt , of a particular property type in a given area, we construct quarter t’s net rents from expression (1) as Ht = (CAPt × Pt−1 )/4.9 The gross returns 1+Rt in quarter t are then obtained from expression (1) while Ht /Ht−1 gives one plus the growth in rents. For consistency with the previously derived expressions, we work with log cap rates, capt = ln(CAPt ), and log rental growth rates, ∆ht = ln(Ht /Ht−1 ). Also, we rely on log excess returns, rt = ln(1 + Rt ) − ln(1 + Rt bl ), where Rt bl is the three month T T Treasury bill yield. Table A1 in the Appendix also reports time-series averages of excess 7 In his example, Case (2000) assumes that expected returns increase and the expected rent growth decreases with economic conditions. 8 While not scientifically rigorous, perhaps a good indication of the data’s accuracy is the fact that it is used by many real estate, financial, and government institutions. A partial list of the subscribers includes Citigroup, GE Capital, J.P. Morgan/Chase, Merrill Lynch, Lehman Brothers, Morgan Stanley Dean Witter, NAREIT, Pricewaterhouse-Coopers, Standard & Poors, Trammell Crow, Prudential RREEF Funds Capital/Real Estate Investors, Washington Mutual, FDIC, CalPERS, and GMAC. 9 We obtain very similar results by modifying the timing convention and relying on the expression Ht = (CAPt × Pt )/4. 11
  14. 14. returns, rental growth rates, and cap rates for all property types across all metropolitan areas. We compute autocorrelations of the excess returns for each property type in each metropolitan area. In Table 1, Panel A, we summarize the autocorrelation structure of the excess returns at different quarterly lags (k). In particular, at each lag we display the 25th , 50th , and 75th percentiles of the autocorrelations of excess returns for a particular property type using data from across all areas. We also display at each lag the number of areas whose autocorrelations are significantly different from zero at the 5% level (denoted by N ) and specify the number that are significantly positive (denoted by +) and negative (denoted by -). For apartments, the median autocorrelation at a one-quarter lag is −0.007 and the corresponding inter-quartile range is from −0.101 to 0.170. The number of areas exhibiting significant autocorrelation in apartment excess returns at a one-quarter lag is five, with four of these being positive. In the case of industrial, retail, and offices buildings, the median first- order autocorrelations in the corresponding excess returns are higher than for apartments at 0.043, 0.168, and 0.287, respectively. For retail properties, all eight of the significant first- order autocorrelations are positive. Similarly, out of the twenty-five significant first-order autocorrelations for offices, twenty-four are positive. The autocorrelations of excess returns for all property types decrease rapidly with lag length, even for office buildings which display the highest degree of serial dependence.10 In general, after three quarters (k = 3) to one year (k = 4) the median autocorrelations as well as the number of significant autocorrelations are both small. These results suggest that while the excess returns of commercial real estate exhibit some degree of positive serial dependence at a one-quarter lag (k = 1), they are essentially uncorrelated at lags of one year or longer (k ≥ 4). Since these particular properties are likely to be held by large institutional investors, it is not surprising to see less serial dependence in their returns than in single-family home data (Case and Shiller (1989)) where market inefficiencies and frictions undoubtedly play a larger role. It is also possible that the lack of significant serial dependence in Panel A of Table 1 is due to our brief sample period as well as the greater volatility of commercial real estate returns. To illustrate this point, in Table A2 of the Appendix, we report estimates of the volatilities of quarterly commercial property returns, rental growth rates, and cap rates for all metropolitan areas as well as their mean, median, minimum and maximum across metropolitan areas. For example, the 10 For offices, almost half of the series exhibit significantly positive serial correlation at a one-quarter lag. 12
  15. 15. average volatility of apartment excess returns is 6.1% per year which, though lower than the stock market return volatility of 16.7%, is still an order of magnitude larger than the 0.5% time-series standard deviation of apartment cap rates. Because of this extreme variability, tests for serial dependence in returns will have low statistical power to reject the null of no predictability, especially in small samples. Since this issue is especially of concern for short- horizon predictive regressions, we follow the approach taken in the stock market predictability literature11 and focus primarily on long-horizon forecasting relations. We summarize the serial dependence of rental growth rates and cap rates in Panels B and C of Table 1, respectively. Using data across all sampled metropolitan areas, the median first-order autocorrelations for rental growth rates are 0.078 for apartments, 0.049 for industrial properties, 0.041 for retail properties, and 0.119 for offices. When areas exhibit significant first-order autocorrelations in rental growth rates, they tend to be positive more often than negative. Interestingly, retail properties and office buildings show less persistence at a one-quarter lag in rental growth rates than in excess returns. At lags of three quarters to one year (k = 3, 4), the rental growth series exhibit no serial correlation for any of the property types. By contrast, regardless of the property type, cap rates are extremely persistent with median one-quarter autocorrelations of 0.815 for apartments, 0.744 for industrial properties, 0.834 for retail properties, and 0.817 for offices. Almost all fifty- three individual cap rate series exhibit significant positive serial correlation at a one-quarter lag across all property types which tends to persist for the first two years (k ≤ 8). 4 Predictive Regressions 4.1 Methodology We are interested in whether the cap rate for a particular property type in a given metropolitan area reflects investors’ expectations of future returns or rental growth or both. The framework in Section 2 suggests the following two regressions ri,t+1→t+k = αk + βk (capi,t ) + εi,t+k (11) ∆hi,t+1→t+k = µk + λk (capi,t ) + υi,t+k (12) 11 Campbell and Shiller (1988a), Campbell (1991), Lettau and Ludvigson (2004), among others. 13
  16. 16. k where expected returns and rent growth rates are proxied by rt+1→t+k ≡ l=0 rt+1+l and k ∆ht+1→t+k ≡ ∆ht+1+l , respectively. We run these regressions for various quarterly l=0 horizons k using the pooled sample of fifty-three metropolitan areas over the 1994 to 2003 sample period for each of the four property types. The pooled data are first stacked for all areas in a given quarter and then for all quarters. It is important to emphasize that our pooled regressions differ from the time-series regressions used in the stock return predictability literature. Given the limited time period spanned by our data, the pooled approach has two main advantages. First, because we are primarily interested in long-horizon relations but unfortunately do not have a long enough dataset to run time-series regressions, the only reasonable way of exploring these relations is to rely on pooled data. Secondly, as shown in Tables A1 and A2 in the Appendix, there is considerable heterogeneity in returns, rental growth rates, and cap rates across metropolitan areas at a particular point in time. Therefore, tests based on the pooled regressions are likely to have higher power than tests based on time-series regressions in which the predictive variable has only a modest variance (Torous and Valkanov (2001)). Before presenting our results, a number of statistical issues surrounding our pooled predictive regression framework must be addressed.12 First, the overlap in long-horizon returns and rental growth rates must be explicitly taken into account. In our case, this overlap is particularly large relative to the sample size. In addition to inducing serial correlation in the residuals, this overlap also changes the stochastic properties of the regressors by inducing persistence. While this particular problem has been investigated by many authors in the context of time-series regressions, it is also likely to affect the small-sample properties of the estimates in our pooled regressions. Secondly, the predictors themselves are highly cross-sectionally correlated. For instance, the median cross-sectional correlation of apartment cap rates in our sample is 0.522 and is as high as 0.938 (between Washington, DC and Philadelphia, PA). As a result, because we effectively have fewer than fifty-three independent cap rate observations at any particular point in time, a failure to account for this cross-sectional dependence will lead to inflated t statistics. Finally, the cap rates are themselves persistent and their innovations are correlated with the return innovations. Under such conditions, it is well known that, at least in small-samples, the least squares estimate of the slope coefficient will be biased in time-series predictive regressions 12 We thank the referee for suggesting that we conduct inference in our pooled regression framework by using resampling methods that carefully take into account the small-sample features of the data. 14
  17. 17. (Stambaugh (1999)). In pooled predictive regressions, the slope estimates will also exhibit this bias because they are effectively weighted averages of the biased estimates of the time- series predictive regressions for each metropolitan area. Because of these issues, traditional asymptotic methods are unlikely to provide accurate inference in our pooled predictive regressions. That being the case, we rely on a two-step resampling approach. In the first step, as is customary in any predictive regression exercise, we run a pair of time-series regressions for each of the fifty-three metropolitan areas: one- period returns regressed on lagged cap rates and cap rates regressed on lagged cap rates. The coefficients and residuals from these regressions are subsequently stored. For each area we then resample the return residuals and cap rate residuals jointly across time (without replacement, as in Nelson and Kim (1993)). The randomized return residuals are used to create one-period returns under the null of no predictability. To generate cap rates, we use coefficient estimates from the original regression together with the resampled residuals from the cap rate autoregression. We then form overlapping multi-period returns from the resampled single-period returns and, as we do with the original data, we pool the newly generated data for all fifty-three areas. For each resampling i we then obtain a pooled ˆi estimate βk at each k-quarter horizon, where k ranges from 1 to 20 quarters. Because the data ˆ are generated under the null, the average βk across resamplings, denoted by βk = I I βk , 1 ˆi i=1 is an estimate of the bias in the pooled predictive regression at horizon k. The bias-adjusted ˆadj ˆ ˆ estimate of βk is obtained as βk = βk − β k , where βk is the biased estimate of βk from the pooled predictive regression. This procedure is in essence that suggested by Nelson and Kim (1993) but applied to a pooled regression. It corrects for the small-sample bias in the slope coefficient because the contemporaneous correlations between the return residuals and cap rate residuals in a given metropolitan area as well as across areas is preserved. However, while this procedure captures the overlap in the multi-period returns, it does not address the issue of cross-sectional dependence in cap rates. This then necessitates our second step. To account for the possibility that our predictability results are driven by cross-sectional correlation in cap rates, we resample the cap rates across metropolitan areas at each point in time for each return horizon k.13 Then, in a cross-sectional regression, we estimate the ˆ predictive regression at each point in time and so obtain T estimates βk , where T is the sample size. We repeat the entire resampling procedure 1,000 times which produces 1,000 × 13 The bootstrap is carried out with replacement. We also tried resampling without replacement and obtained very similar results. 15
  18. 18. ˆ T estimates βk . We use these 1,000 × T estimates to compute standard errors, denoted by se(βk ). This second step is very similar to a standard Fama and MacBeth (1973) regression with the exception that we are running the regressions with 1,000 replications of bootstrapped data rather than the original sample. The standard errors from the double resampling account for both time-series and cross-sectional dependence in the data because in the first resampling step the overlapping nature of the returns is explicitly taken into account while in the subsequent resampling step the cap rates are drawn at each point in time from the empirical distribution of cap rates. The bias-adjusted double resampling t statistic, denoted by tDR , is computed as ˆadj ˆ tDR = βk /se(βk ) = (βk − β k )/se(βk ). We use the tDR statistics in all predictive regressions in this paper involving the cap rate. The 95th and 99th percentiles of the bootstrapped distributions of the tDR statistic are the critical values in our tests. For the sake of clarity and conciseness, we report levels of significance next to the estimates (5% and 1%) rather than small-sample critical values because the latter are a function of the overlap as well as the cross-sectional cap rate correlation of a particular property type. We also display the ˆ ˆ bias-corrected β adj and, in some instances, the biased estimate βk for comparison. The same k methods are used in estimating and making inference regarding the slope coefficient λk in equation (12).14 4.2 Results Table 2 presents the results of forecasting commercial real estate returns using cap rates (expression (11)) for apartments, industrial properties, retail properties, and office buildings. For each property type, we report the least squares non-adjusted as well as the bias-adjusted ˆ ˆadj ˆadj estimates, βk and βk , the tDR statistic, and the adjusted R2 . Statistical significance of βk at the 5% and 1% level are denoted by superscripts “a” and “b”, respectively. 14 A few things are worth mentioning about the above sampling procedure. First, the second bootstrap is necessary in order to take into account the cross-sectional correlation in cap rates. Without it, the standard errors would not be corrected for the cross-sectional dependence in cap rates. We verified that if we use only the Nelson and Kim (1993) randomization, we obtain t statistics very similar to the Newey and West (1987) results. Second, it is interesting to note that the pooled regression does not produce unbiased estimates. The reason is that given the cross-sectional correlation in cap rates and the fact that cap rate fluctuations and return shocks are correlated, the pooled regression effectively yields a weighted average of the biased estimates that would have been obtained in time series regressions for each metropolitan area. Third, the bias correction is large at longer horizons, where the sample is smaller and the overlap is larger. 16
  19. 19. For all property types, we see that at short horizons of less than one year (k < 4), cap rates are positively correlated with future returns. However, the corresponding bias- corrected slope coefficients are never statistically significant at the 1% level while the R2 s are uniformly low. The bias-corrected slope coefficients do increase in magnitude as the horizon k increases and at k = 3 quarters are significant at the 5% level. The general lack of significance and low R2 s at horizons of less than one year are consistent with the large variability in quarterly commercial property returns documented in the Appendix.15 For longer horizons, k ≥ 4, in the case of apartments the bias-corrected slope estimates increase from 0.251 at a one-year horizon to 0.778 at a five-year horizon and both of these estimates are statistically significant at the 1% level. The R2 s of the regressions also increase to 16.6 percent at the five-year horizon. In the case of industrial and retail properties, cap rates best predict returns at horizons of between two and four years. The largest bias- corrected slope estimate for industrial properties is 0.758 at k = 16 quarters with a R2 of 12.9 percent. For retail properties, the largest bias-corrected slope estimate is 0.945 at k = 12 quarters with a R2 of 24.6 percent. In both of these cases, the estimates are statistically significant at the 1% level. Interestingly, in the case of offices there does not appear to be reliable evidence of predictability at any horizon. The largest bias-corrected slope coefficient in the case of offices is 0.290 at k = 8 quarters, which is statistically significant at the 5% level but not at the 1% level. Also, the corresponding R2 is only 3.5 percent. At most other horizons, the bias-corrected slope coefficients for offices are indistinguishable from zero.16 The results in Table 2 suggest that at least for apartments, industrial and retail properties, cap rates reliably forecast returns at horizons of between three and five years.17 For office buildings, by contrast, there is little or no evidence of predictability at any horizon. Recall from expression (3) that the presence of predictability and its magnitude in the context of the dynamic Gordon model depend critically on the persistence of cap rates and on the 15 Also the positive estimates partially reflect the serial correlation in returns at lags of one to three quarters observed in Table 1. 16 As the horizon increases, the number of observations in the predictive regressions decreases even though we compute rt+1→t+k and ∆ht+1→t+k with overlapping data. The last column of each Table shows that, at the one-year horizon, we have 1643 observations, but only 795 observations are available at the five-year horizon. The small number of observations reduces the power of our tests and should work against our detecting predictability. 17 It is interesting to note that we find somewhat similar β estimates to those reported in the stock predictability literature. For example, Campbell, Lo, and MacKinlay (1997) report coefficients estimates of 0.329, 0.601, 0.776, and 0.863 at the one, two, three, and four year horizons, when forecasting stock returns (these numbers refer to the period 1952-1994). These values are very similar to ours reported in Table 2 and similar patterns are displayed also for the R2 statistic. 17
  20. 20. forecastability of rental growth. Our results suggest that office buildings may differ from the other property types either in the persistence of their cap rates or in the properties of their rent growth. In Table 3, we present the results from estimating regression (12). It is immediately evident that for apartments, industrial and retail properties, there is no reliable evidence that cap rates forecast the future growth in their rents. At horizons up to two years, the bias- corrected slope coefficients are not significantly different from zero and the corresponding R2 s are small, between 0 and 3.2 percent. At longer horizons, the bias-corrected slope coefficients for apartments and industrial properties are significant at the 5% level but not at the 1% level. For retail, we observe significant, at the 1% level, bias-corrected slope coefficients at k=12 and k=16 quarters only. By contrast, in the case of offices, the bias-corrected slope coefficients are negative and insignificant up to k=8 quarters. The corresponding R2 s increase with horizon k, reaching a value of 7.7 percent at k = 20 quarters. At this particular horizon, the slope coefficient is significant at the 1% percent level. In other words, while for apartments, industrial and retail properties, there is little evidence that cap rates can forecast the future growth in rents, the evidence is stronger for office buildings and, unlike the other property types, the estimates have the theoretically correct sign.18 4.3 Robustness 4.3.1 Fixed Effects Cap rates capture not only time-variation in expected returns but also cross-sectional differences across metropolitan areas. These cross-sectional differences are related to various 18 Expressions (11) and (12) can be thought of as cross-sectional regressions estimated once a quarter whose coefficients are then averaged over time, similarly to the Fama and MacBeth (1973) procedure. The estimates from our pooled regressions should be identical to those obtained from the corresponding Fama and MacBeth (1973) regressions if the cap rates do not vary with time (Cochrane (2001)). In fact, the Fama-MacBeth approach produces very similar results when applied to our data. For example, in the case of apartments at the five-year horizon, we obtain a Fama-MacBeth estimate of 0.750 instead of 0.778. The corresponding Fama-MacBeth t statistic of 5.714, computed with Newey-West standard errors, is somewhat larger than the tDR statistic of 4.007 reported in Table 2. For retail properties at the same horizon, we obtain a Fama-MacBeth estimate of 0.660 and a Newey and West (1987) t statistic of 3.830, which are very close to the pooled estimate and tDR statistic of 0.699 and 3.309, respectively, reported in Table 2. While the Newey and West (1987) and the tDR statistics are not directly comparable, because one is asymptotic while the other one takes into account the small-sample features of the data, they nevertheless illustrate the robustness of our findings. We obtain similar results at all horizons and across property types, which suggests that the statistical significance of our small-sample results are quite conservatively stated. 18
  21. 21. demographic, geographic, and economic factors. Since it is quite plausible that rents and prices adjust quicker to time-series shocks in a given metropolitan area than to variations across metropolitan areas, we allow for the possibility of unobserved heterogeneity across areas by incorporating fixed effects into the previous regressions: ri,t+1→t+k = αk + βk (capi,t ) + ϕi + εi,t+k ˜ (13) ∆hi,t+1→t+k = µk + λk (capi,t ) + ςi + υi,t+k ˜ (14) where the fixed effects coefficients ϕi and ςi capture the heterogeneity across metropolitan areas. We estimate the regressions by including fifty-three area-specific dummy variables. Table 4 presents the results of estimating the fixed effects regression (13).19 The regressions are estimated by first regressing excess returns and rental growth rates on the cross-sectional dummies and then regressing the residuals on the cap rates. The bias-adjusted slope estimates and corresponding tDR statistics are computed as described previously. Notice that neither the slope estimates nor the tDR statistics change dramatically. More importantly, the predictive power of the cap rates at long-horizons is still evident in the case of apartments, industrial, and retail properties where we see the bias-adjusted slope estimates being significant at the 1% level for k ≥ 4 quarters. We conclude that unobserved heterogeneity across metropolitan areas is unlikely to account for our findings. 4.3.2 Longer Sample Period with Fewer Metropolitan Areas An obvious question to pose is whether our results will hold over a longer sample period or are they specific to the 1994 to 2003 period. Indeed, this particular sample period contains only one full business cycle and coincides with a general upward trend in real estate prices. To answer this question, we extend the sample back to 1985 by augmenting our data with the bi-annual observations on all of the property types available for a subset of twenty- one of the fifty-three metropolitan areas. These data are available from the second half of 1985 to the first half of 1994 and to this we add the post-1994 data for these particular twenty- one areas sampled at a bi-annual frequency. This gives a sample spanning the 1985 to 2002 time period containing thirty-five semi-annual observations for the twenty-one metropolitan 19 We do not report the results of estimating the fixed effects regression (14). Recall that in the absence of fixed effects, the coefficients are largely insignificant. Adding these fixed effects reduces their significance even further. 19
  22. 22. areas giving a total of 735 observations for each property type. We rely on this dataset to investigate the stability of our predictability findings but at a cost of fewer cross-sectional observations. We re-estimate regression (13) with the 1985 to 2002 dataset now using twenty-one area-specific dummy variables and present the results in Table 5. For all property types, cap rates can still be seen to forecast future returns and the predictability increases with the forecasting horizon. The bias-adjusted slope estimates are similar to those previously reported in Tables 2 and 4 and, in fact, are slightly larger. For example, for apartments, the slope estimate at the five-year horizon is 1.297 while the corresponding estimate in Table 2 is only 0.778. Similar comparisons hold for the other property types. More importantly, the bias-adjusted slope estimates remain statistically significant at the 1% level at long horizons. The predictability results obtain even though we have bi-annual rather than quarterly data over the 1985 to 2002 sample period and only on twenty-one rather than fifty-three metropolitan areas. It is also worth noting that future office returns, while still the least predictable of all four property types, are now more forecastable than in the 1994 to 2003 sample.20 We conclude that the ability of cap rates to capture future fluctuations in commercial real estate returns does not appear to be driven by the 1994 to 2003 sample. 4.3.3 Other Robustness Checks In this section, we describe several other robustness checks used to ensure the reliability of our empirical results. We will discuss these results without detailing them in Tables as they are in general agreement with our previous findings. First, we run the forecasting regressions using only non-overlapping returns. The predictive power of the cap rates remains. In particular, we obtain bias-adjusted slope estimates that are similar to those previously reported and, in fact, the estimates and corresponding tDR statistics at longer horizons are larger than those obtained when using overlapping returns. These results, however, should be interpreted with some caution as the number of observations at longer horizons is rather small. In the pooled regressions, all observations are weighted equally. This estimation approach is efficient under the assumption that the variances of the residuals are equal 20 The results from forecasting rental growth are statistically insignificant and are omitted. They are available upon request. 20
  23. 23. across metropolitan areas. The homoscedasticity assumption may not be appealing as more populous metropolitan areas are generally more diverse giving rise to more heterogeneity in the quality of a given property type. For example, it is unlikely that the variance of residuals for Los Angeles will be the same as that of, say, Norfolk, Virginia. To address this concern, we use weighted least squares under the assumption that the heteroscedasticity in the residuals is proportional to the population in a given area. In particular, the weight given to a particular metropolitan area is given by its population divided by the total population of all metropolitan areas in the previous year. We then divide the left- and right-hand side variables of our predictive regressions (11) and (12) by the square root of these computed weights. Interestingly, the bias-adjusted slope estimates now increase in magnitude but the corresponding tDR statistics are very similar. The results for rental growth are once again insignificant. As a final remark, note that there are no efficiency gains to be had from estimating regressions (11) and (12) jointly. In fact, given that the right-hand side variables are the same, the joint seemingly unrelated equations (SUR) estimator is identical to our equation by equation estimator. 5 Economic Significance of the Predictability From an economic perspective, it is easier to interpret the response of future commercial real estate returns to changes in cap rates rather than to log cap rates. To do so, we divide the estimated slope coefficients by the average cap rate where the cap rate is expressed in the same units as the returns (see, e.g., Cochrane (2001)). We compute these transformed coefficients for apartments, industrial properties, retail properties, and office buildings using ˆ the corresponding β adj estimates from Table 2 and their average cap rates of 8.7%, 9.1%, k 9.2%, and 8.7%, respectively.21 For apartments, a small increase in the cap rate implies that expected returns should increase between 1.789, if we take the five-year estimates (0.778/(5 × 0.087)), and 2.885, if we take the one-year estimates (0.251/0.087). Similarly, for industrial properties, retail properties, and office buildings, the sensitivities to a small increase in the respective cap rate are 3.736, 3.554, 1.793, respectively, if we rely on the one-year estimates, and 1.486, 1.520, −0.145, respectively, if we use the five-year estimates. 21 From the Table A1 in the Appendix. 21
  24. 24. Based on these calculations, as expected, there appears to be a difference between the predictability of apartments, industrial properties, and retail properties versus office buildings. For the first three property types, the sensitivities are between 1.5 and 3.7 whereas for offices they are in the range of between −0.1 and 1.7. Based on this, it is tempting to conclude that expected returns of office buildings are much less time-varying. However, such a comparison assumes that the rental growth of all property types are not only equally unforecastable but that they are also equally volatile. In the next section, we show that while the rental growth rate of offices is unforecastable, it is much more volatile than the rental growth rates of the other property types. To further appreciate the effect of time-varying expected returns on commercial real estate, it is useful to compare our results to those in the stock market literature. For the aggregate stock market, a one percentage point increase in expected returns results in about a 4 to 6 percent increase in prices (see, e.g., Cochrane (2001) for a summary of the evidence).22 Hence, the sensitivity of commercial real estate prices to changing expected returns is not very different than that of common stock. To the extent that the fluctuations in expected returns of the stock market and commercial real estate are both driven by changing economic conditions23 , our findings suggest that an investment in commercial real estate is not necessarily an effective hedge against stock market risk. 6 Understanding the Results Thus far we have documented that for apartments as well as industrial and retail properties, cap rates are significantly correlated, both statistically and economically, with their future returns but not with the future growth in rents. For offices, cap rates forecast neither future returns nor future growth in rents. Two immediate questions arise as a result. First, do cap rates proxy for demographic, geographic, economic or other cross-sectional differences in commercial real estate prices, or do they capture time variation in expected returns? Second, why are the results for office buildings so different from the other property types? 22 The difference in magnitudes is due mainly to the fact that the dividend yield of the market is about 4 percent, which is half of the cap rate of commercial properties. 23 Which is something we verify later in the paper. 22
  25. 25. 6.1 Is Predictability Due to Time-Varying Expected Returns? The predictability we have documented is due either to cross-sectional differences in unconditional returns or to time series variation in conditional returns. To understand this point, suppose that for a given property type, the cap rate at time t in area i, Portland, is higher than that in area j, Dallas. From expression (10), the relatively lower price in Portland can either be due to a lower unconditional expected return or to a higher unconditional expected growth in rents in Dallas. These cross-sectional pricing differences are not a function of time. Cross-Sectional Controls The pricing of commercial and residential real estate across metropolitan areas and its relation to demographic, geographic, and economic variables has been widely investigated. For example, Capozza, Hendershott, Mack, and Mayer (2002) find that house price dynamics vary with city size, income growth, population growth, and construction costs. Abraham and Hendershott (1996) document a significant difference in the time-series properties of house prices in coastal versus inland cities. Lamont and Stein (1999) show that house prices react more to city-specific shocks, such as shocks to per-capita income, in regions where homeowners are more leveraged.24 In light of this evidence, we now investigate whether cap rates are not merely proxying for these cross-sectional effects as opposed to capturing time variation in economic conditions. To test this “proxy” hypothesis, we use demographic, geographic, and economic variables to capture differences across metropolitan areas. More specifically, for each metropolitan area, we use the following variables: population growth (gpopt ), the growth of income per capita (ginct ), and the growth of employment (gempt ), all of which are provided by the Bureau of Economic Analysis at an annual frequency. We also use the annual growth in construction costs (gcct ) compiled by R.S. Means. The construction cost indices include material costs, installation costs, and a weighted average for total in place costs.25 In addition, after lagging by two years, we include log population (popt−2 ), log 24 While most of the cited papers focus strictly on the residential market, similar mechanisms are likely at play in commercial real estate. 25 There are missing data for some metropolitan areas in our construction costs database. For these series, we assigned the values of the closest area for which data is available. In detail, we assigned to Oakland and San Jose the value of San Francisco, to Nassau-Suffolk the values of New York City and to West Palm Beach the values of Miami. For the areas where merged data is present in the real estate database, a unique index is constructed as weighted average of the single areas’ construction costs, based on their population. 23
  26. 26. per capita income (inct−2 ), log employment (empt−2 ), and log construction costs (cct−2 ), to proxy for the level of urbanization (Glaeser, Gyourko, and Saks (2004)). We lag these level variables by two years to prevent a mechanical correlation with corresponding growth rates. We also include a dummy variable (coastt ) which equals one when the metropolitan area is in a coastal region.26 We account for these cross-sectional differences by augmenting the regressions (11) and (12) as follows: ri,t+1→t+k = αk + βk (capi,t ) + θk Zi,t + εi,t+k (15) ∆hi,t+1→t+k = µk + λk (capi,t ) + θk Zi,t + υi,t+k (16) where Zi,t is the set of pre-determined characteristics. If cap rates are proxying for differences across metropolitan areas and not capturing time variation in expected returns, then the inclusion of these cross-sectional proxies will lower the significance of the estimated cap rate coefficients while increasing the regression’s R2 . Similarly, under the proxy hypothesis, the exclusion of the cap rate from these regressions should not significantly alter the regression’s R2 .27 The results of estimating regressions (15) and (16) at a four-year horizon (k = 16 quarters) are presented in Tables 6 and 7, respectively. For each property type, we run three specifications. The first includes the cap rate as well as the growth rates of the economic variables (gpop, gemp, ginc, and gcc). In the second specification, we add the levels of these variables as well as the coastal dummy (pop, emp, inc, cc, and coast). The third specification includes the growth rates and the levels but excludes the cap rate. Table 6 present the results of forecasting expected returns. Several results emerge. First, the growth rate variables are not found to be significant for any of the property types 26 We also collected data on financing costs in various metropolitan areas, but there was very little variation across metropolitan areas. Time series variation in interest rates is already captured as we compute all returns in excess of the Tbill rate. We also tried including the rent-to-income variable, which can be motivated from the results in Menzly, Santos, and Veronesi (2004). However, this variable was highly correlated with some of the other controls and we decided against including it in the regressions. 27 The fixed effect regressions previously discussed can also be interpreted as tests of the proxy hypothesis. The fixed effects capture the cross-sectional differences of unconditional expected returns and unconditional growth in rents without specifying their origin. Recalling the results from Table 4, we observe that accounting for cross-sectional differences does not significantly decrease the forecasting power of the cap rate. The drawback of this approach is that the dummy variables are simply too coarse to capture variation that can be better explained if we specify the correct source of heterogeneity. 24
  27. 27. nor does their inclusion affect the R2 s. Secondly, the inclusion of the levels of the control variables increases the regression R2 s, but their coefficients are only significant in the case of office buildings. This result may be due to a strong correlation between the control variables. For offices, the control variables are significant indicating heterogeneity across metropolitan areas. Thirdly, the exclusion of the cap rate leads to a dramatic drop in the regression R2 s for apartments, as well as industrial and retail properties. In other words, after accounting for cross-sectional differences in these metropolitan areas, cap rates do appear able to capture additional time variation. In the case of offices, however, the exclusion of the cap rate does not result in a significant drop in the regression R2 . Fourthly, the bias-corrected cap rate estimates are larger than those in Table 2. This difference might be partially due to the fact that we use annual rather than quarterly data in the regressions, because the additional economic and demographic controls are only available at that frequency. The results are similar at other return horizons. Taken together, the evidence in Table 6 suggests that cap rates are proxying for more than simply differences in expected returns across metropolitan areas. Table 7 presents the results for forecasting rental growth rates at a four-year horizon. The addition of the cross-sectional controls does not markedly alter the cap rate’s significance. However, the regression R2 s can now be seen to increase from a range of 3 to 4 percent (Table 3) to between 16 and 27 percent, depending on the property type with retail having the highest R2 . Several control variable coefficients are significant, depending on the property type and the particular specification. The office building regressions have the highest number of significant control variable coefficients, suggesting that cross-sectional differences in economic conditions play a significant role in determining the future growth in office rents. Interestingly, the exclusion of the cap rate from these regressions does not result in a dramatic drop in R2 s as was the case for expected returns. Hence, it seems that the expected growth in rents for all commercial property types is primarily determined by area-specific characteristics. The regressions presented in Table 6 and 7 exhibit relatively high R2 s while the control variables have low t statistics and are rarely found to be significant, all of which are symptoms of multicollinearity in the regression specifications. The high correlation between the regressors in the vector Zi,t is not surprising as these variables all attempt to capture similar facets of the underlying economic and demographic conditions in a metropolitan area at a particular point in time. To reduce the number of correlated variables while at the same 25
  28. 28. time succinctly summarizing their information content, we perform a principal component analysis (PCA) of the control variables28 . Our PCA reveals that three out of eight principal components account for more than 70% of the overall volatility in Zi,t . The three extracted principal components (which, by construction, are orthogonal) are particularly correlated with the level and growth in population and income as well as with the level of construction costs. Tables 8 and 9 present the results of regressing future returns and future rental growth, respectively, on the cap rate, the three principal components and the coastal dummy variable. We can see that in both regressions the three principal components and the coastal dummy are statistically significant for most of the property types. In particular, either the first, the second or both principal components are significant at the 1% level while the cap rate is still significant when the principal components are added. Moreover, for all property types but offices, the R2 s of the future returns regressions decreases substantially when we omit the cap rate while for all property types the R2 s of the future rental growth regression are slightly lower. Hence, we conclude that the economic variables are indeed capturing significant cross-sectional variation in returns and rental growth, but this does not limit the predictive content of cap rates. Time-Variation in Expected Returns Our empirical evidence is consistent with cap rates for apartments, industrial, and retail properties being able to capture fluctuations in time-varying expected returns. Expected rental growth rates, by contrast, appear to be determined by cross-sectional determinants and do not seem to fluctuate over time. To more directly verify this conclusion, we regress future one-year returns and future yearly rental growth rates on regional dummies and variables that have previously been documented to capture time-varying economic conditions. The conditioning information here includes the term spread, the default spread, the CPI inflation rate, and the three month Treasury bill rate.29 These variables have been widely used in the stock predictability literature to capture time-varying behavior in aggregate stock market expected returns (Campbell and Shiller (1988a), Campbell (1991), Fama and French (1989), 28 Another possibility is to select the variables according to their ability to improve the R2 . However, the set of variables so chosen is likely to vary across property type and the method increases the risk of overfitting. 29 We also tried using the consumption-wealth variable “cay,” which Lettau and Ludvigson (2001) show forecasts future aggregate stock market returns. In the specification with the term spread, default spread, inflation, and the short rate, the cay variable was not significant. However, it was significant if any one of the other variables was dropped. 26
  29. 29. Torous, Valkanov, and Yan (2005) and, for a good review, Campbell, Lo, and MacKinlay (1997)). The term spread is calculated as the difference between the yield on 10-year and 1-year Treasuries. The default spread is calculated as the difference between the yield on BAA- and AAA-rated corporate bonds while the CPI inflation rate is the quarterly growth in the CPI index.30 Under the hypothesis that expected returns are time-varying, they should be forecasted by the macroeconomic variables. Similarly, we expect these variables to have only modest power in forecasting future rental growth. In estimating these regressions, we use the longer 1985 to 2003 sample period with fewer metropolitan areas in order to obtain more precise parameter estimates as the regressors vary across time but are the same across metropolitan areas. We present the results from these regressions in Table 10. Focusing on panel A, we see that the future returns of apartments, industrial and retail properties are explained by time variation in the term spread, default spread, inflation, and the short interest rate. For these property types, the macroeconomic variables explain between 10 and 24 percent of the time series fluctuations in cap rates. The coefficients of inflation (CPIRET), the short rate (TB3M), and the default spread (DSPR) are statistically significant for all property types at the 1% or 5% level. It is interesting to note that for office buildings approximately 23 percent of the time series fluctuations in future returns is explained by the macroeconomic variables and is comparable to that for apartments and retail properties. Industrial properties have the lowest R2 . These results suggest that expected returns of offices are time-varying.31 Notice that the coefficients on inflation and the short rate are significantly negative. This result is consistent with the evidence on predicting stock returns (Campbell (1987), Fama and Schwert (1977), Fama and French (1989), Fama (1981), and Keim and Stambaugh (1986), Fama and French (1993), and Lettau and Ludvigson (2001)). The coefficient on the default spread is also significantly negative. While earlier studies found that the default spread forecasted future stock returns with a positive coefficient (Fama and French (1989), Campbell (1991), and Fama and French (1993)), more recently Lettau and Ludvigson (2001), using a larger and more recent sample that includes most of our 1985 to2003 sample period, also find the coefficient to be negative. As such, our commercial real estate findings are in 30 All these data, except the three month Treasury bill rate, are from the FRED database. The three month Treasury bill rate is obtained from Ibbotson Associates. The statistical properties of these variables are well known and are not provided here.(see, e.g., Torous, Valkanov, and Yan (2005)). 31 The results from the shorter 1994-2003 sample are very similar, albeit less significant, with R2 s in the range of between ten and fifteen percent. 27
  30. 30. line with those in the stock market predictability literature.32 Our findings are not only consistent with cap rates capturing time variation in the expected returns of these property types, but also with the expected returns of stocks and commercial real estate responding in the same direction to changes in underlying economic conditions. Panel B of Table 10 presents the results of forecasting the yearly growth in rents with the macroeconomic variables. In contrast to the results of Panel A, the economic variables have much less ability to predict the future growth in rents of apartments, industrial properties, and retail properties. The goodness of fit in these regressions is in the range of between 4 and 7 percent. For office buildings, we observe a much stronger forecastability of rental growth. The corresponding R2 of 13.5 percent is about twice as large as that of other property types with most of the forecastability being driven by the default spread and inflation. As we discuss in the next section, this difference is important in understanding the lack of expected return forecastability observed for offices. 6.2 Why are Offices so Different? A recurring theme thus far has been the differences documented between office buildings versus the other commercial property types. Only for office buildings do we fail to detect an economically and statistically significant relation between cap rates and future returns. Based solely on this evidence, to conclude that the expected returns of offices are less susceptible to economic fluctuations than the other property types would be correct only if expected growth in rents for all property types are: (i) equally correlated with expected returns; and (ii) equally volatile. To see the necessity of assumption (i), consider expression (10) and suppose that expected returns of offices and, say, apartments are equally exposed to economic variation (δ A = δ O ). In addition, we assume that the growth in rents of offices is much more correlated with expected returns than is the growth of apartment rents (τ O = 1 while τ A ≈ 0), and that the variation in rental growth orthogonal to economic conditions O A is the same for offices and apartments (V (yt ) = V (yt )). Under these assumptions, the cap rate will better predict the expected returns of apartments despite the fact that the expected returns of both property types are time-varying. This result obtains because the variability in expected returns for offices is offset by the variability in rental growth and 32 We replicated the Lettau and Ludvigson (2001) results and found that, for the 1985-2003 sample, the default spread has a negative coefficient in predicting excess stock market returns. 28
  31. 31. the net effect, captured by the term xt (1 − τ O ), results in an absence of variability in their cap rates. This argument has been made for the aggregate stock market by Campbell and Shiller (1988b) and more explicitly by Lettau and Ludvigson (2004) and Menzly, Santos, and Veronesi (2004). Assumption (ii) must also hold if different property types are to comparably predict future returns. Using a similar logic, suppose that the variability in office rent growth that is O A orthogonal to economic conditions is greater than that of, say, apartments (V (yt ) > V (yt )), while their expected returns are equally exposed to economic variables (δ A = δ O ). From expression (10), it follows that the apartment cap rate will better predict expected returns. The additional variability in office rent growth that is orthogonal to the variation in expected returns only adds noise to the predictive regression for offices and so will decrease its power. Table 10 provides evidence against assumption (i). Panel B of the Table shows that office rental growth is more forecastable by macroeconomic variables than is the rental growth of the other three property types. Moreover, it is interesting to note that the same variables that forecast office rent growth also forecast future office returns. In particular, the default spread and inflation rate are both negatively correlated with future office rent growth as well as future office returns. This evidence suggests that office rent growth is time-varying and is also correlated with office expected returns. As noted earlier, this correlation would make it difficult to detect predictability using office cap rates even if the expected returns of offices are time-varying. The second assumption is also not supported by the data as the growth of office rents is more volatile than for the other property types. To document this fact, we estimate the volatility of rental growth that is orthogonal to economic fluctuations for each property type in each metropolitan area using the time series data from the 1985 to 2003 sample period. We do so by first regressing the one year rental growth rates on the macroeconomic variables as in Table 10. Using the residuals from these regressions, we estimate GARCH(1,1) models, which yield twenty-one time series estimates of volatilities for each property type. We then compute the median and the mean filtered volatility across metropolitan areas for each property type. The results are plotted in Figure 1. It is immediately evident that the median and mean standard deviations for office buildings (solid line) are almost invariably greater than that for the other property types. Moreover, office buildings exhibit more conditional autoregressive 29
  32. 32. heteroscedasticity than the other series. Notice that the volatilities of office rent growth are particularly high during the 1991 to 1993 and the 1997 to 1999 time periods. The first period coincided with a declining market and decreasing rents while the second period was one of increasing rents (Case (2000)). The mean values of these volatilities across time are 7.0 percent, 7.1 percent, and 5.6 percent annually for apartments, retail, and industrial properties, respectively. For offices, the volatility is significantly higher at 8.5 percent. As a comparison, we also computed the dividend growth rate of the CRSP value-weighted index, a proxy for the aggregate stock market. The volatility of the stock market’s dividend growth rate that is orthogonal to the macroeconomic variables over that period is only 8.1 percent. In summary, the exposure of expected returns of offices to macroeconomic variables appears comparable to that of the other property types. Given that the growth in office rents is more correlated with expected returns and that this growth rate is generally more volatile, it is not surprising that office cap rates are unable to forecast future returns. 7 Conclusions This paper empirically analyzes the fluctuations in returns and rental growth rates for apartments, office buildings, retail properties, and industrial properties. We find that for apartments as well as retail and industrial properties, the cap rate forecasts time variation in expected returns but does not forecast expected rental growth rates. For these property types, the time variation in expected returns generates economically significant movements in corresponding property prices. For offices, by contrast, the cap rate neither forecasts expected returns nor rent growth rates. Commercial real estate markets offer a natural setting in which to demonstrate that the predictability of expected returns by the dividend-price ratio, that is, the cap rate, is sensitive to the assumption that the growth rate of cash flows, in our case rents, is unforecastable as suggested earlier by Campbell and Shiller (1988b) and more recently argued by Lettau and Ludvigson (2004) and Menzly, Santos, and Veronesi (2004). We demonstrate that while the expected returns of the four commercial property types have similar exposures to macroeconomic variables, their rental growth rates differ in terms of their correlations with expected returns as well as in their volatilities. As a result, the cap rate for offices, whose rental growth rate is the most highly correlated with expected returns as well as also 30
  33. 33. being the most volatile, does not forecast expected returns even though these returns are themselves time-varying. Investigating the economic sources underlying the cyclical variation of office rental growth rates is an interesting issue for future research. Since under certain circumstances cap rates cannot capture the variation in expected returns, it is natural to ask whether there is a variable better suited for this task. To answer this question, an extension of the Menzly, Santos, and Veronesi (2004) model to the case of commercial real estate would be an interesting problem to pursue in future work. We also find that the expected returns of commercial real estate and common stock have similar correlations with macroeconomic variables. In addition to the evidence that expected returns of commercial real estate are time-varying, this finding suggests that commercial real estate may not provide an effective hedge against fluctuation in the stock market and underlying economic conditions. While this paper deals exclusively with the implications of our findings on the pricing of commercial real estate properties, the portfolio choice problem involving commercial real estate is also very interesting and is left for future research. Some work incorporating real estate already exists in this area (Piazzesi, Schneider, and Tuzel (2003) and Lustig and Van Nieuwerburgh (2004)), but its focus is residential real estate. Future research in this area should further explore the role of commercial real estate and its stochastic properties in a portfolio setting. 31
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