The Anatomy of Public and Private Real Estate Return Premia

Abstract

Market-wide, stock market specific, and real estate market specific risk – what kind of risk and to which extent drives the returns of listed real estate? Based on a structural asset pricing model calibrated to the empirical data in the U.S., we show that at least two thirds of the risk premium of listed real estate are driven by the same factors as direct real estate. Our results shed new light on the risk-characteristics of listed real estate returns and are of high interest for academics, regulators, and portfolio managers alike.

Appendix: Details of the Model

In this section, we outline the details of our structural model of listed and direct real estate. The model is a variant of Koijen et al. (2017). The derivations below closely follow their discussion.

Economy

Macroeconomic activity, s_t+ 1, is the central state variable of the structural model. It captures business cycle activity, i.e. the state of the economy, through an autoregressive process and can be described by the following process:

$$s_{t + 1}=\rho_{s}s_{t}+\sigma_{s}\epsilon_{t + 1}^{s}. $$

High values of s_t+ 1 correspond to strong economic activity. The parameter ρ _s implies that business cycle activity is to some degree persistent. The innovation term, $\epsilon _{t + 1}^{s}$, is the first priced source of risk in the model.

We model the returns of three assets. The first is the aggregate stock market (M), the second is listed real estate (L), and the third is direct real estate (D), so that the real dividend growth of asset i = {M, L, D} is described by:

$$\triangle d_{t + 1}^{i}=\gamma_{0i}+\gamma_{1i}s_{t}+\sigma_{mi}\epsilon_{t + 1}^{m}+\sigma_{di}\epsilon_{t + 1}^{d},\:\forall i=\left\{ M,\:L,\:D\right\} . $$

Shock $\epsilon _{t + 1}^{m}$ is an aggregate stock market dividend shock, the second source of risk for the structural model. Shock 𝜖t+ 1d is an aggregate direct real estate rent shock and the third source of risk. The parameter γ_1i is the sensitivity of dividend growth to business cycle activity. Listed real estate is leveraged; thus, we set γ_1L = 2 × γ_1D and σ _{d L} = 2 × σ _{d D} to capture a leverage of two against direct real estate. Furthermore, listed real estate vehicles are value stocks. In line with this observation, we impose γ_1L > γ_1M, since Koijen et al. (2017) show that value stocks are more sensitive to recession risk than growth stocks. The coefficient σ _{m i} represents stock market risk and is zero for direct real estate (σ _{m D} = 0). We can utilize this coefficient to model stock-market spillovers in listed real estate by σ _{m L} > 0.

Investors’ preferences are captured by a stochastic discount factor (SDF) following the log process:

$$-m_{t + 1}=y+\Lambda^{\prime}\epsilon_{t + 1}, $$

where the vector $\epsilon _{t + 1}=\left (\epsilon _{t + 1}^{m},\epsilon _{t + 1}^{d},\epsilon _{t + 1}^{s}\right )^{\prime }$captures the shocks, and $\tilde {y}=y-\frac {1}{2}\Lambda ^{\prime }\Lambda$ is the real interest rate. This model has three positively priced sources of risk: aggregate stock market dividend risk (Λ _m > 0), aggregate direct real estate rent risk (Λ _d > 0), and business cycle risk (Λ _s > 0), which are summarized via the vector:

$$\Lambda=\left( \begin{array}{c} \Lambda_{m}\\ \Lambda_{d}\\ \Lambda_{s} \end{array}\right). $$

Asset Prices

The log return of asset i follows:

$$r_{t + 1}^{i}=\kappa_{0i}+\kappa_{1i}pd_{t + 1}^{i}+\triangle d_{t + 1}^{i}-pd_{t}, $$

where $pd_{t + 1}^{i}$ is the log price-dividend ratio, and κ_0i, κ_1i are constants (see below). The log-price-dividend ratio is linear in the state of the economy:

$$pd_{t + 1}^{i}=A_{i}+B_{i}s_{t + 1}, $$

where

$$\begin{array}{@{}rcl@{}} B_{i}&=&\frac{\gamma_{1i}}{1-\kappa_{1i}\rho_{s}},\\ A_{i}\!&=&\!\frac{-y+\gamma_{0i}+\kappa_{0i}+\frac{1}{2}\sigma_{mi}^{2}+\frac{1}{2}\sigma_{di}^{2}+\frac{1}{2}\kappa_{1i}^{2}B_{i}{\sigma_{s}^{2}}-\Lambda_{m}\sigma_{mi}\,-\,\Lambda_{d}\sigma_{di}\,-\,\Lambda_{s}\kappa_{1i}B_{i}\sigma_{s}}{1-\kappa_{1i}}. \end{array} $$

Decomposition of Risk Premia

The risk premium for our three assets can be computed by the covariance to the SDF:

$$\begin{array}{@{}rcl@{}} E_{t}\left( r_{t + 1}^{i}-y\right)+\frac{1}{2}V_{t}\left( r_{t + 1}^{i}\right)&=&Cov_{t}\left( -m_{t + 1},r_{t + 1}^{i}\right)\\ &=&Cov_{t}\left( {\Lambda}^{\prime}\epsilon_{t + 1},\kappa_{1i}B_{i}\epsilon_{t + 1}^{s}+\sigma_{mi}\epsilon_{t + 1}^{m}+\sigma_{di}\epsilon_{t + 1}^{d}\right)\\ &=&{\Lambda}_{m}\sigma_{mi}+{\Lambda}_{d}\sigma_{di}+{\Lambda}_{s}\kappa_{1i}B_{i}. \end{array} $$

The first component of the constant risk premium compensates investors for aggregate stock market dividend risk (Λ _m σ _{m i} ). For direct real estate σ _{m D} = 0; thus, this term is zero. The second component of the constant risk premium compensates investors for aggregate real estate rent risk (Λ _d σ _{d i} ). For the aggregate stock market σ _{d M} = 0; thus, this term is zero. The third constant risk premium term compensates for business cycle risk (Λ _s κ_1iB _i σ _s ).

Proof

The logreturn for any asset i can be approximated by e.g. Campbell et al. (1997):

$$\begin{array}{@{}rcl@{}} r_{t + 1}&=&\kappa_{0}+\kappa_{1}pd_{t + 1}+\triangle d_{t + 1}-pd_{t},\\ \kappa_{0}&=&ln\left( exp\left( \bar{pd}\right)+ 1\right)-\kappa_{1}\bar{pd},\\ \kappa_{1}&=&\frac{exp\left( \bar{pd}\right)}{exp\left( \bar{pd}\right)+ 1}, \end{array} $$

where we drop the subscripts i for convenience. The log price-dividend ratio is assumed to be linear in the state of the economy:

$$pd_{t + 1}=A+Bs_{t + 1}. $$

The coefficients A and B are found by solving the asset pricing equation:

$$E_{t}\left( M_{t + 1}R_{t + 1}\right)= 1, $$

$$1=E_{t}\left( exp\left( m_{t + 1}+r_{t + 1}\right)\right), $$

$$\begin{array}{@{}rcl@{}} 0&=&E_{t}\left( m_{t + 1}\right)+\frac{1}{2}V_{t}\left( m_{t + 1}\right)+E_{t}\left( r_{t + 1}\right)+\frac{1}{2}V_{t}\left( r_{t + 1}\right)+Cov_{t}\left( m_{t + 1},r_{t + 1}\right),\\ &=&-y+\kappa_{0}+\gamma_{0}+\gamma_{1}s_{t}+\left( \kappa_{1}-1\right)A+\left( \kappa_{1}\rho_{s}-1\right)Bs_{t}+\frac{1}{2}{\sigma_{m}^{2}}+\frac{1}{2}{\sigma_{d}^{2}}\\ &&\;\;+\frac{1}{2}{\kappa_{1}^{2}}B^{2}{\sigma_{s}^{2}}-\Lambda_{m}\sigma_{m}-\Lambda_{d}\sigma_{d}-\Lambda_{s}\kappa_{1}B\sigma_{s}. \end{array} $$

Collecting all s _t terms and all others results in the following system of two equations:

$$\begin{array}{@{}rcl@{}} 0 \!&=&\!\gamma_{1}s_{t}\,+\,\left( \kappa_{1}\rho_{s}\,-\,1\right)Bs_{t},\\ 0&=&\,-\,y\,+\,\kappa_{0}\,+\,\gamma_{0}\,+\,\left( \kappa_{1}\,-\,1\right)A\,+\,\frac{1}{2}{\sigma_{m}^{2}}\,+\,\frac{1}{2}{\sigma_{d}^{2}}\,+\,\frac{1}{2}{\kappa_{1}^{2}}B^{2}{\sigma_{s}^{2}}\,-\,\Lambda_{m}\sigma_{m}\,-\,\Lambda_{d}\sigma_{d}\,-\,\Lambda_{s}\kappa_{1}B\sigma_{s}, \end{array} $$

which can be solved as:

$$B=\frac{\gamma_{1}}{1-\kappa_{1}\rho_{s}}, $$

$$A=\frac{-y+\gamma_{0}+\kappa_{0}+\frac{1}{2}{\sigma_{m}^{2}}+\frac{1}{2}{\sigma_{d}^{2}}+\frac{1}{2}{\kappa_{1}^{2}}B^{2}{\sigma_{s}^{2}}-\Lambda_{m}\sigma_{m}-\Lambda_{d}\sigma_{d}-\Lambda_{s}\kappa_{1}B\sigma_{s}}{1-\kappa_{1}}. $$

Parameter Values

This section provides details on how we calibrate our structural model. We simulate the model using a monthly time interval 10,000 times and afterwards, we convert the data to an annual time interval with a sample length of 27 years. The persistence parameter for the state of the economy (ρ _s = 0.936) is taken from Koijen et al. (2017). On an annual basis, this is equal to a modest persistence of 0.45. The volatility parameter (σ _s = 0.01) is an arbitrary normalization.

The stock market dividend volatility parameter (σ _{m M} = 0.04) and the direct real estate rent volatility parameter (σ _{d D} = 0.02) closely match the empirical annual dividend volatilities of 13.2% and 5.3%. The respective volatilities implied by our calibration are 13.9% ($= 4\%\times \sqrt {12}$) and 6.9% ($= 2\%\times \sqrt {12}$). The listed real estate dividend and rent volatility parameters depend on our model assumptions, i.e. (i) σ _{m L} will be one half of the value of σ _{m M} if there are stock market spillovers and (ii) zero if there are no stock market spillovers. The parameter σ _{d L} is two times σ _{d D} to reflect leverage of listed real estate.

We impose two restrictions on the recession risk sensitivity parameters, (i) γ_1L = 2 × γ_1D to reflect leverage of listed real estate and (ii) γ_1L > γ_1M, since listed real estate are value stocks and should be more exposed to recession risk than common stocks. The parameter values γ_1D = 0.10, γ_1L = 0.20, and γ_1M = 0.14 satisfy these criteria.

After defining the parameters for the business cycle state variable, dividend growth and rent growth, we choose in a final step SDF risk prices, i.e. the parameter vector Λ, to generate risk premia which are as close as possible to the observed average returns of common stocks, listed real estate, and direct real estate. Since we measure returns in excess of the risk-free rate, the real interest rate ($\tilde {y}$) does not affect our results on risk premia and is set to 2% p.a. in all simulations.

Table 5 Parameter values