Short-Term and Long-Term Discount Rates For Real Estate Investment Trusts

Abstract

How should we discount cash flows generated by real estate projects up to 30 years in the future? In this paper, we develop a methodology to estimate the term structure of discount rates. We provide empirical estimates of the yield curve for the REIT market, and three subsectors: Equity, Mortgage, and Hybrids. We explore three empirical hypotheses related to the behavior of the yield curve: First, we examine whether the term structure for REITs should be upward or downward sloping. Second, because REIT betas tend to be counter-cyclical, we examine the hypothesis that discount rates implied by our term structure model should be higher than those implied by the traditional CAPM. Third, in the spirit of Campbell 1991, we decompose the variance of REIT prices into variation due to the cost of capital and variation due to expected cash flows and find some unexpected results.

Appendices

APPENDIX A A Simple Multiperiod Discount Rate Model

Suppose the sequence of discount rates implied by the CAPM may be modeled by a first-order autoregressive process: ${\mu}_{t+1}=\left(1-\varphi \right)\bar{\mu}+\varphi {\mu}_t+{\epsilon}_{t+1}$ . Then, the future sequence (μ_t + 1, μ_t + 2, … , μ_t + τ ) may be represented in matrix form as

$$ \kern1em \left[\begin{array}{cc}\begin{array}{cc}1& 0\\ {}-\varphi & 1\end{array}& \begin{array}{ccc}0& \dots & \kern0.75em 0\\ {}0& \dots & \kern0.75em 0\end{array}\\ {}\begin{array}{cc}& .\\ {}0& \dots \\ {}0& \dots \end{array}& \begin{array}{ccc}.& .& \\ {}-\varphi & 1& 0\\ {}0& -\varphi & 1\end{array}\end{array}\right]\kern0.5em \left(\begin{array}{c}\begin{array}{c}{\mu}_{t+1}\\ {}.\end{array}\\ {}\begin{array}{c}.\\ {}.\\ {}{\mu}_{t+\tau}\end{array}\end{array}\right)=\left(1-\varphi \right)\bar{\mu}\left(\begin{array}{c}\begin{array}{c}1\\ {}.\end{array}\\ {}\begin{array}{c}.\\ {}.\\ {}1\end{array}\end{array}\right)+\left(\begin{array}{c}\begin{array}{c}\varphi {\mu}_t\ \\ {}0\end{array}\\ {}\begin{array}{c}.\\ {}.\\ {}0\end{array}\end{array}\right)+\left(\begin{array}{c}\begin{array}{c}{\epsilon}_{t+1}\\ {}.\end{array}\\ {}\begin{array}{c}.\\ {}.\\ {}{\epsilon}_{t+\tau}\end{array}\end{array}\right) $$

Alternatively, in compact form, this system of equations is: $\Phi\ \mu =\left(1-\varphi \right)\bar{\mu}\upharpoonleft +\varphi {\mu}_te1+\epsilon$, where ↿ is a column vector of 1 s, and e1 is a column vector with 1 in the first row and 0 in the remaining rows. This specification implies closed form solutions for the moments of the cumulative discount rate ${\sum}_{j=1}^{\tau }{\mu}_j$ . Specifically, the conditional mean is $\left(1-\varphi \right)\bar{\mu}{\upharpoonleft}^{\prime }{\varPhi}^{-1}\upharpoonleft +\varphi {\mu}_t{\upharpoonleft}^{\prime }{\varPhi}^{-1}e1$ and conditional variance σ²↿^′Φ⁻¹(Φ⁻¹)^′↿. Using a result from time series analysis (Ali 1977), we show next that these moments may be computed without inverting the Φ matrix. Define the vector Z^′ ≡ (z(τ), z(τ − 1), …, z(1) ) = ↿^′Φ⁻¹ and note that each element may be computed recursively from the previous one: z(j) = φz(j − 1) + 1 for j = 2, 3, …, τ, and starting value z(1)=1 . Hence, the conditional mean is ${E}_t\ \left[\kern0.5em {\sum}_{j=1}^{\tau }{\mu}_j\right]=\kern0.5em \left[\ \left(1-\varphi \right)\bar{\mu}{\sum}_{j=1}^{\tau }z(j)\right]+\varphi {\mu}_tz\left(\tau \right),$ and for the conditional variance we have ${Var}_t\ \left[\kern0.5em {\sum}_{j=1}^{\tau }{\mu}_j\right]={\sigma}^2\ \left[\ \left(1-\varphi \right)\bar{\mu}{\sum}_{j=1}^{\tau }z{(j)}^2\ \right].$

By definition, the discounted value of a single cash flow C_t + τ expected τ periods from the current period t is: $V{\left(\tau \right)}_t={E}_t\left[\ {e}^{-{\sum}_{j=1}^{\tau }{\mu}_{t+j}}\ {C}_{t+\tau}\right]$. For simplicity, in this example we assume the expected cash flow is independent of the discount rate. Then, by the properties of log-Normal random variables, the conditional expected value is $V{\left(\tau \right)}_t={e}^{-\tau\ \pi {\left(\tau \right)}_t}\ {E}_t\left[{C}_{t+\tau}\right]$ where π(τ)_t is the geometric average: ${\left(\tau \right)}_t=\frac{1}{\tau}\left\{\ {E}_t\ \left[{\sum}_{j=1}^{\tau }{\mu}_{t+j}\right]-\frac{1}{2}{Var}_t\ \left[\ {\sum}_{j=1}^{\tau }{\mu}_{t+j}\right]\kern0.5em \right\}$ . Using the closed form solutions developed above, we obtain a model for the term structure of discount rates for each maturity τ:

$$ \pi {\left(\tau \right)}_t=\frac{1}{\tau}\left\{\left[\left(1-\varphi \right)\bar{\mu}{\sum}_{j=1}^{\tau }z(j)+\varphi {\mu}_t\ z\left(\tau \right)\right]-\frac{1}{2}{\sigma}^2{\sum}_{j=1}^{\tau }z{(j)}^2\kern0.5em \right\} $$

(A1)

Estimation of Market Beta

B.1 Prior Distribution Model for Market Beta

We follow the empirical literature (e.g., Petkova and Zhang 2005, and Cosemans et al. 2016) and model time-varying beta as a linear function of firm size (SIZE), book-to-market (BM), Operating leverage (OP_LEV), and Financial leverage (FIN_LEV). The literature also indicates that the relationship between beta and firm characteristics changes with the economic environments. We use the default spread (DEF) to measure general business conditions. The default spread, measured as the difference in yield to maturity between Baa rated corporate bonds and Aaa rated bonds, is lower during economic expansions and higher in recessions. Thus, our prior information for beta is a conditional model specified as follows:

$$ {\displaystyle \begin{array}{l}{\beta}_{it\mid t-1}^{\ast }={\delta}_{0,i}+{\delta}_{1,i}\; DE{F}_{t-1}+{\gamma}_1\; SIZ{E}_{i,t-1}+{\gamma}_2\;B{M}_{i,t-1}+{\gamma}_3\; OP\_ LE{V}_{i,t-1}\\ {}\kern2.879999em +{\gamma}_4\; FIN\_ LE{V}_{i,t-1}+{\gamma}_5\; SIZ{E}_{i,t-1}\ast DE{F}_{t-1}+{\gamma}_6\;B{M}_{i,t-1}\ast DE{F}_{t-1}\end{array}} $$

(B1)

We assume that the intercept term δ_{0, i}and slope coefficient on the business cycle variable DEF, δ_{1, i}, vary across our sample of REITs to capture unobserved variation across firms. The slope coefficient vector γ '  = (γ₁, γ₂, γ₃, γ₄, γ₅, γ₆) does not carry the i subscript as we assume that the relationship between beta and firm characteristics is constant across firms. This specification should yield more efficient parameter estimates.

Time Series Specification for Time-Varying Beta

We define r_{i t, s} as the daily return for portfolio i for day s, in excess of the riskless rate, and r_{m t, s} as the corresponding excess market portfolio return. The time index s covers a six months period ending on the last trading day of month t and beginning 125 trading days earlier. Consider the rolling window regression of portfolio return i on a constant and the market return:

$$ {r}_{i\;t,s}={\alpha}_{i,t}+{\beta}_{i,t}{r}_{m\;t,s}+{\varsigma}_{i\;t,s} $$

(B2)

The intercept α_i is the risk-adjusted return, and the slope β_i measures systematic, or market, risk. The idiosyncratic return ς_{i t, s} is modeled as a sequence of identically, independently distributed random variables with zero mean and variance V[ς_i].¹¹

Assuming actual and idiosyncratic returns are normally distributed, the estimated rolling window (RW) betas ${\hat{\beta}}_{i,t}^{RW}$ are also normal with mean β_{i, t} and variance $V\left[{\hat{\beta}}_{i,t}^{RW}\right]={\hat{\sigma}}_{\varsigma}^2{\left(\;\sum \limits_s{r}_{m,s}^2\right)}^{-1}$, where ${\hat{\sigma}}_{\varsigma}^2$ is the residual error variance estimate for the month t regression.

Bayes Conditional Beta Model

The Bayesian beta model proposed by Cosemans et al. (2016) is based on the combination of two estimates: rolling window, and market beta based on fundamental firm characteristics (FC). In the spirit of Bayesian analysis we take the prior distribution to be normal with mean ${\beta}_{i\;t\mid t-1}^{\ast}$(Equation B1) and variance $V\left[{\hat{\beta}}_{i,t-1}^{FC}\right]$, where ${\hat{\beta}}_{i,t-1}^{FC}$ is the fundamentals beta estimator. Then, the posterior beta for portfolio i at the end of month t is

$$ {\hat{\beta}}_{i\;t\mid t}^{\ast }={w}_{i,t}{\hat{\beta}}_{i,t-1}^{FC}+\left(1-{w}_{i,t}\right){\hat{\beta}}_{i,t}^{RW} $$

(B3)

where the corresponding shrinkage weight w_{i, t} is based on the variance ratio

$$ {w}_{i,t}=\frac{V\left[{\hat{\beta}}_{i,t}^{RW}\right]}{\kern0.24em V\left[{\hat{\beta}}_{i,t-1}^{FC}\right]+V\left[{\hat{\beta}}_{i,t}^{RW}\right]\kern0.24em } $$

(B4)

We discuss the methodology to obtain ${\hat{\beta}}_{i,t-1}^{FC}$ and its variance below; but first we note that Equation (B3) is standard Bayesian econometrics. The posterior mean ${\hat{\beta}}_{i\;t\mid t}^{\ast }$ is a weighted average of the prior beta, conditional on firm characteristics, and the rolling window beta. Equation (B4) shows that if the variance $V\left[{\hat{\beta}}_{i,t-1}^{FC}\right]$ is relatively small, then firm characteristics play a dominant role in determining the posterior beta. Conversely, precise estimates from the rolling window beta, low variance $V\left[{\hat{\beta}}_{i,t}^{RW}\right]$, lead to a larger role for time series information.

To obtain ${\hat{\beta}}_{i,t-1}^{FC}$ and its variance, we use a panel time-series regression for each period ending in month t. We define the excess returns column vector ℜ_iwith s^th element R_{i, s} − R_{f, s}, (s = t, t-1, t-2, . . ., t–T + 1). Define Π 1 as a matrix with s^th row given by (1, DEF_s − 1)[R_{m, s} − R_{f, s}], and let Π 2_i be a matrix of macro and firm specific characteristics for portfolio i with s^th row given by$\left( SIZ{E}_{i,s-1},B{M}_{i,s-1},O{P}_{LE{V}_{i,s-1}}, FI{N}_{LE{V}_{i,s-1}}, SIZ{E}_{i,s-1}\ast DE{F}_{s-1},B{M}_{i,s-1}\ast DE{F}_{s-1}\right)\left[{R}_{m,s}-{R}_{f,s}\right]$ . Then the panel regression model to estimate prior distribution parameters is

$$ \left(\begin{array}{c}{\Re}_1\\ {}{\Re}_2\\ {}.\\ {}.\\ {}{\Re}_N\end{array}\right)=\left(\begin{array}{c}{\alpha}_1^{\ast}\\ {}{\alpha}_2^{\ast}\\ {}.\\ {}.\\ {}{\alpha}_N^{\ast}\end{array}\right)+\kern0.36em \left[\begin{array}{cccccc}\varPi\;1& 0& 0& .& 0& \varPi {2}_1\\ {}0& \varPi\;1& 0& .& 0& \varPi {2}_2\\ {}0& .& .& .& 0& .\\ {}.& .& .& .& .& .\\ {}0& & & 0& \varPi\;1& \varPi {2}_N\end{array}\right]\kern0.24em \left(\begin{array}{c}{\delta}_1\\ {}{\delta}_2\\ {}\begin{array}{l}.\\ {}.\end{array}\\ {}{\delta}_N\\ {}\gamma \end{array}\right)+\left(\begin{array}{c}{\zeta}_1\\ {}{\zeta}_2\\ {}.\\ {}.\\ {}{\zeta}_N\end{array}\right) $$

(B5)

where ${\alpha}_i^{\ast }$ is a column vector of constants. The idiosyncratic returns ζ_i t are identically and independently distributed with zero mean and variance ${\sigma}_{\zeta}^2$; we assume also that these error terms are independent across firms. However, we note that by construction (see equation B5) actual REIT returns are correlated across firms.

The firm characteristic beta may be easily obtained as a linear combination of the least squares estimators of the slope coefficients in the panel regression, ($\hat{\gamma}$, ${\hat{\delta}}_i$) and two data (row) vectors for the month prior to the current month t ($\pi {1}_{t-1}^{\hbox{'}}$, $\pi {2}_{i,t-1}^{\hbox{'}}$):

$$ {\hat{\beta}}_{i,t-1}^{FC}=\pi {1}_{t-1}^{\prime }{\hat{\delta}}_i+\pi {2}_{i,t-1}^{\prime}\hat{\gamma} $$

(B6)

where

$\pi {1}_{t-1}^{\hbox{'}}=\left(1, DE{F}_{t-1}\right)$, and $\pi {2}_{i,t-1}^{\hbox{'}}=\Big( SIZ{E}_{i,t-1},B{M}_{i,t-1}, OP\_ LE{V}_{i,s-1}, FIN\_ LE{V}_{i,s-1},$

SIZE_{i, t − 1} ∗ DEF_t − 1, BM_{i, t − 1} ∗ DEF_t − 1). Using standard arguments as in Fama and French (1997), this beta estimator has mean ${\beta}_{it\mid t-1}^{\ast }$. For the variance, we know from Equation (B6) that $\pi\;{2}_{i,t-1}^{\hbox{'}}V\left[\hat{\gamma}\right]\;\pi\;{2}_{i,t-1}+2\;\pi\;{1}_{t-1}^{\hbox{'}} Cov\left[{\hat{\delta}}_i,{\hat{\gamma}}^{\hbox{'}}\right]\;\pi\;{2}_{i,t-1}$. The individual components $V\left[{\hat{\delta}}_i\right],\kern0.36em V\left[\hat{\gamma}\right]$ and $Cov\left[{\hat{\delta}}_i,{\hat{\gamma}}^{\hbox{'}}\right]$ may be obtained from the panel regression error variance ${\hat{\sigma}}_{\zeta}^2{\left[{\varPi}^{\hbox{'}}\;\varPi \right]}^{-1}$, where Π is the data matrix on the right hand side of equation (B5), and ${\hat{\sigma}}_{\zeta}^2$ is the sample variance of the error terms.

The assumption that the error terms in Equation (B5) are independent across firms may seem a bit restrictive. But without this assumption the algebra becomes intractable. We show next that it causes little harm. Consider the panel regression in (B5), written in more compact form as $\mathfrak{R}=\varPi \Delta +\zeta$ . To allow for cross-sectional correlation between firms one may assume that the variance-covariance matrix of the error terms ζ is ${\sigma}_{\zeta}^2\Omega$, where the off-diagonal elements in Ω are non-zero.

Let $\hat{\Delta}={\left[{\varPi}^{\prime}\varPi \right]}^{-1}{\varPi}^{\prime }\ \mathfrak{R}$ be the OLS estimator. Then, $E\left(\ \hat{\Delta}\right)=\Delta +{\left[{\varPi}^{\prime}\varPi \right]}^{-1}{\varPi}^{\prime }\ E\left(\zeta\ \right)$ . This estimator is unbiased because the expected error term has zero mean. The firm characteristic beta, ${\hat{\beta}}_{i,t-1}^{FC}=\pi {1}_{t-1}^{\prime }{\hat{\delta}}_i+\pi {2}_{i,t-1}^{\prime}\hat{\gamma}$, is a linear combination of $\hat{\Delta}$, therefore it is also unbiased.

For the variance of ${\hat{\beta}}_{i,t-1}^{FC}$ we have $V\left(\ \hat{\Delta}\ |\ \Omega\ \right)=\kern0.5em {\left[{\varPi}^{\prime}\varPi \right]}^{-1}{\varPi}^{\prime }\ \left(\ {\sigma}_{\zeta}^2\Omega\ \right)\varPi\ {\left[{\varPi}^{\prime}\varPi \right]}^{-1}$. But to compute the weights in (B3) we use instead the OLS variance $V\left(\ \hat{\Delta}\right)={\sigma}_{\zeta}^2\ {\left[{\varPi}^{\prime}\varPi \right]}^{-1}\kern0.5em$ . The Gauss-Markov theorem suggests that the difference in variances is positive definite:

$$ V\left(\ \hat{\Delta}\ |\ \Omega\ \right)-V\left(\ \hat{\Delta}\right)=\kern0.5em {\sigma}_{\zeta}^2\ {\left[{\varPi}^{\prime}\varPi \right]}^{-1}{\varPi}^{\prime }\ \left(\ \Omega -I\right)\varPi\ {\left[{\varPi}^{\prime}\varPi \right]}^{-1}>0 $$

In turn, this inequality implies a bigger value for the variance $V\left(\ {\hat{\beta}}_{i,t-1}^{FC}|\ \Omega \right)$ in Equation (B3) – relative to the OLS variance:

$$ V\left(\ {\hat{\beta}}_{i,t-1}^{FC}\ |\ \Omega\ \right)-V\left(\ {\hat{\beta}}_{i,t-1}^{FC}\right)= $$

$$ =\kern0.5em {\sigma}_{\zeta}^2\kern0.5em \left\{\ \left(\ \pi {1}_{t-1}^{\prime}\kern1em \pi {2}_{i,t-1}^{\prime }\ \right)\kern0.5em {\left[{\varPi}^{\prime}\varPi \right]}^{-1}{\varPi}^{\prime }\ \left(\ \Omega -I\right)\varPi\ {\left[{\varPi}^{\prime}\varPi \right]}^{-1}\left(\begin{array}{c}\pi {1}_{t-1}\\ {}\ \pi {2}_{t-1}\ \end{array}\right)\right\}\kern0.75em >0 $$

This result shows that our estimates, based on OLS, place more weight on the firm characteristic beta rather than the running regression beta. But this is likely to be a second order effect. Both betas in equation (B3) are unbiased and a weighted average should also share that property.