Rushing to Overpay: Modeling and Measuring the REIT Premium

Abstract

We explore the questions of why Real Estate Investment Trusts (REITs) pay more for real estate than non-REIT buyers and by how much. First, we develop a search model where REITs optimally pay more for property because (1) they are willing, due to cost of capital advantages and, (2) they are occasionally rushed, due to external regulatory time constraints and internal incentives to deploy capital quickly. Second, using commercial real estate transactions, we find that the extant hedonic pricing models contain an unobserved explanatory variables bias leading to inflated estimates of the REIT premium. Third, using a repeat-sales methodology that controls for unobserved property characteristics, we derive more plausible estimates of the REIT premium. Consistent with our model, we also find the REIT-buyer premium depends on the size of the REIT advantage, the rush to deploy, and the relative presence of REITs in the market.

Appendices

Appendix A: Multiple Opportunity Extension

In our model presented in Section 2.2, we assumed REITs had one period to make a purchase and considered only one property during that period. Here, we consider a richer model in which REITs may be able to view multiple properties during the limited time in which they are eligible for extra benefit b. We find this easiest to illustrate in a continuous time framework. This model shares some features with the unemployment search model of Akin and Platt (2012); the reader can find additional exposition there.

Buyers

In this multiple-offer environment, a buyer encounters properties at Poisson rate λ; that is, encountering λ properties on average over one unit of time. Having found a property, the buyer observes the seller’s asking price for that property and chooses to either purchase the property or continue searching. As in the Section 2 model, suppose there are two prices, p _l and p _h , and the fraction of sellers offering the low price is φ.

As before, all buyers derive benefit x from owning the property; a REIT obtains an additional benefit b if it acquires a property within the first T units of time searching. Buyers and sellers both discount at rate ρ.

We denote the state of a buyer by the remaining time s until the extra REIT benefit is lost. Since expired REITs are identical to non-REITs, we represent both with state s = 0. All REITs enter the market with s = T, while all non-REITs enter with s = 0. Let V(s) denote the expected net present value of a REIT buyer who has s time remaining before losing the benefit.

A non-REIT’s decision can be represented by the following Bellman equation:

$$ \rho V(0) = \lambda \varphi \left( {x - {p_l} - V(0)} \right). $$

The right hand side indicates that properties arrive at rate λ. Yet only the low price is acceptable to non-REITs, so, conditional on encountering a property, they make a purchase with probability φ, which changes their payoff from V(0) to x–p _l .

A REIT with s >0 has the following Bellman equation:

$$ \rho V(s) = - {V^{\prime }}(s) + \lambda \varphi \left( {x + b - {p_l} - V(s)} \right) + \lambda \left( {1 - \varphi } \right)\left( {x + b - {p_h} - V(s)} \right). $$

Note the following changes, relative to those with no benefits. First, REITs are willing to pay either price. Second, when a purchase is made at price p, the buyer’s payoff changes from V(s) to x + b–p. Finally, the looming deadline is reflected in $- {V^{\prime }}(s)\kern-3pt$, since the state s deterministically falls until the buyer either makes a purchase or the benefit expires.

Sellers

Sellers face a stationary problem. If they decide to ask for the low price, they encounter buyers at rate λ, all of whom are willing to purchase at the low price. This results in the following Bellman equation:

$$ \rho {\Pi_l} = \lambda \left( {{p_l} - {\Pi_l}} \right), $$

so the net present value of profit is ${\Pi_l} = \frac{\lambda }{{\rho + \lambda }}{p_l}$ from this strategy.

Suppose instead that, having encountered a buyer, the seller were to ask the high price; then only fraction γ of buyers will make the purchase. This produces the following Bellman equation:

$$ \rho {\Pi_h} = \lambda \gamma \left( {{p_h} - {\Pi_h}} \right), $$

with a net present value of ${\Pi_h} = \frac{{\lambda \gamma }}{{\rho + \lambda \gamma }}{p_h}$. For both prices to occur in equilibrium, sellers must be indifferent between them. Thus${ }{\Pi_h} = {\Pi_l}$, which is to say:

$$ {p_l} = \frac{{\left( {\rho + \lambda } \right) \gamma }}{{\rho + \lambda \gamma }}{p_h}. $$

Steady State

Let H(s) denote the measure of buyers who have s or less time remaining, while ${H^{\prime }}(s)$ is the relative density of this distribution. To keep this distribution constant over time, it must obey the following three steady state conditions.

First, REITs enter the market at a constant Poisson rate N _r , so ${H^{\prime }}(T) = {N_r}$. Second, REITs encounter properties at rate λ and purchase them at either price. Thus, the density must fall at rate $\lambda {H^{\prime }}(s)$:

$$ {{H}^{{\prime \prime }}}(s) = \lambda {{H}^{\prime }}(s). $$

Third, the mass of buyers at state s = 0 increases at rate ${H^{\prime }}(0)$, due to expiring REITs, and rate N _n , due to newly entering non-REITs. At the same time, these buyers encounter properties at an acceptable price at rate λφH(0):

$$ {H^{\prime }}(0) + {N_n} = \lambda \varphi H(0). $$

The fraction of consumers willing to pay the high price is:

$$ \gamma = \frac{{H(T) - H(0)}}{{H(T)}}. $$

Equilibrium Solution

As before, the price p _l should make non-REITs indifferent about making the purchase. Thus, $x - {p_l} = V(0)$. Combining this withs the Bellman equation at s = 0 yields p _l = x and V(0) = 0.

The rest of the Bellman equation can be solved as a first-order differential equation, with boundary condition V(0) = 0. The result is:

$$ V(s) = \frac{\lambda }{{\lambda + \rho }}\left( {1 - {e^{{ - s\left( {\lambda + \rho } \right)}}}} \right)\left( {b + \left( {1 - \varphi } \right)\left( {x - {p_h}} \right)} \right). $$

The high price is chosen to make the REIT at s = T indifferent between making the purchase and continuing search, so ${p_h} = x + b - V(T)$ . By doing so, all buyers whose benefit has not expired will be willing to purchase at price p _h , since those with 0 < s < T have V(s) < V(T) and thus strictly benefit from such a purchase. This allows us to solve for p _h as:

$$ {p_h} = x + b\frac{{\rho + \lambda {e^{{ - T\left( {\lambda + \rho } \right)}}}}}{{\rho + \lambda \varphi + \lambda \left( {1 - \varphi } \right){e^{{ - T\left( {\lambda + \rho } \right)}}}}}. $$

Note that as long as 0 < φ < 1, then p _h > p _l .

The steady state distribution is a second-order differential equation with two boundary conditions. Its unique solution is:

$$ H(s) = \frac{{{N_n} + {N_r}\left( {1 + \varphi \left( {{e^{{s\lambda }}} - 1} \right)} \right){e^{{ - T\lambda }}}}}{{\lambda \varphi }}. $$

Next, the equal profit condition simplifies to $\left( {1 - \frac{{\rho H(0)}}{{\left( {\rho + \lambda } \right)\left( {H(T) - H(0)} \right) }}} \right){p_l} = {p_h}$. Substitution for prices and the steady state distribution results in:

$$ \varphi = \frac{1}{{\frac{{\lambda \left( {1 - {e^{{T\left( {\lambda + \rho } \right)}}}} \right)}}{{\left( {\lambda + \rho {e^{{T\left( {\lambda + \rho } \right)}}}} \right)}} + \frac{{b\left( {\lambda + \rho } \right)\left( {{e^{{T\lambda }}} - 1} \right){N_r}}}{{x\rho \left( {{N_r} + {e^{{T\lambda }}}{N_s}} \right)}}}}. $$

This value of φ can be substituted back into previous answers to calculate the final solution. Two consistency conditions must be checked to verify that the two-price equilibrium exists. The first is that 0 < φ <1, but it turns out that if φ <1, we automatically obtain φ <0. Thus, this requirement amounts to:

$$ \frac{b}{x}\frac{{\left( {{e^{{T\lambda }}} - 1} \right){N_r}}}{{\left( {{N_r} + {e^{{T\lambda }}}{N_s}} \right)}} > \frac{{\left( {\lambda + \rho } \right){e^{{T\left( {\lambda + \rho } \right)}}}}}{{\lambda + \rho {e^{{T\left( {\lambda + \rho } \right)}}}}}. $$

If this inequality does not hold, REITs are sufficiently scarce that sellers face too much delay in asking the higher price. Thus, a single-price equilibrium occurs. Also note that the transition between the two equilibria is continuous. For instance, as b falls, φ increases until it reaches 1; any further decline in b will violate the condition above, and hence maintain the single-price equilibrium (i.e., φ = 1).

The second consistency issue is to verify that asking a higher price is not profitable, e.g., $p(s) = x + b - V(s)$ for $s \in \left( {0,T} \right)$. Buyers with more than s time until the deadline would reject such a price, as would those who have passed the deadline, creating a smaller fraction of potential buyers compared to p _h . Indeed, the lower likelihood of acceptance outweighs the increase in price as long as 0 < φ <1. This can be verified by taking the derivative of:

$$ \Pi (s) = \frac{{\lambda \frac{{H(s) - H(0)}}{{H(T)}}}}{{\rho + \lambda \frac{{H(s) - H(0)}}{{H(T)}}}}p(s), $$

and verifying that it is positive at s = T. That is, charging a price just below p(T) = p _h will decrease profits.

The two-price equilibrium requires a larger benefit, b, or flow of REIT buyers, N _r , than the discrete model calibration in Section 5.3, but is still within the range of plausibility. For instance, suppose that the annual discount rate is ρ = 5 %, REITs constitute 20 % of the flow of buyers, their benefits expire in T = 1 year, and they encounter an average of λ = 2 properties in that time. Then a benefit b = 0.84x would sustain a two-price equilibrium, with 64.5 % of sellers offering the low price and REITs paying on average $\frac{{\rho \left( {{N_r} + {N_r}{e^{{ - \lambda T}}}} \right)}}{{{N_r}\left( {\lambda + \rho } \right)\left( {1 - {e^{{ - \lambda T}}}} \right)}} = 6.4\%$ more than non-REITs. Also, with these parameter values, only 13.5 % of REIT buyers would fail to encounter any properties (at either price) before the deadline.

The comparative statics on φ are quite intuitive. As N _r increases, sellers are more likely to encounter a REIT and thus more sellers ask for p _h (i.e., φ falls). A larger benefit b allows the sellers who target REITs to charge even more, which increases their expected profit. This in turn decreases φ because asking for p _l is relatively less attractive.

The comparative static with respect to λ, on the other hand, does not have a monotonic relationship. If properties arrive infrequently, few REITs will encounter a property while eligible for the extra benefits. Thus, sellers are not sufficiently likely to make a sale at price p _h to justify the long wait needed to encounter a REIT, and the single-price equilibrium occurs. On the other hand, if properties arrive frequently, REITs have no reason to rush into purchasing at price p _h , since they expect a number of more opportunities to find the lower price before their benefits expire. For intermediate values of λ, however, the two-price equilibrium can be sustained, and increases in λ cause φ to fall initially (as sellers become more willing to wait for REITs) and then rise (as REITs become more willing to wait for the low asking price). An increase in T has similar competing effects, but for parameters anywhere near those above the net effect is to reduce the REIT premium.

An increase in λ will always reduce the REIT premium in percentage terms. Intuitively, as REITs expect more offers before expiration, they can be more patient in their search and hold out for lower prices. This forces sellers to reduce p _h , lest everyone (except the few REITs near their deadline) turn them down.

The effect of λ on the REIT premium explains our empirical finding that the premium falls as property size increases. For instance, buyers of large properties may put forth greater effort to investigate the majority of what is on the market, effectively raising λ for larger x. Since less money is at stake for small properties, buyers expend less search effort, equivalent to a lower λ.

Figure 1 illustrates these comparative statics in a contour plot. The contours indicate the fraction of sellers, φ, asking the low price under the specified combination of b (relative to x) and λ. The white region indicates where the single-price equilibrium occurs; that is, all sellers ask the low price so φ = 1. All other parameters are fixed at their values used in the numerical example above. The plot looks nearly identical if b/x is replaced with N _r on the horizontal axis.

Fig. 1

Fraction of sellers asking for the low price (in the continuous time model) for various parameter values. Notes: Each contour indicates the fraction φ of sellers offering the low price for the indicated parameter values of benefit size b (relative to property value x) and property arrival rate λ. The point at (0.82,2) indicates the parameter values used in our numerical example. For instance, the boundary labeled 0.1 (the border between the two darkest regions) represents all parameter combinations for which 10 % of sellers ask the low price in a two-price equilibrium. In contrast, the white region represents parameters for which the single-price equilibrium occurs, with 100 % of sellers asking the low price

Appendix B: Repeat Sales Pricing Errors with Bootstrapped P-Values

In Section 4.2, we use a modified repeat-sales model to directly measure the REIT-buyer premium. That model has the advantage of only using observations where properties are sold more than once, eliminating the unobserved characteristics problem. At the same time, relying only on repeat transactions reduces the quantity of data, discarding useful information. Furthermore the residuals from the repeat-sales regression are slightly right skewed (third standardized moment = 0.347) and leptokurtic (fourth standardized moment = 5.591). A Kolmogorov-Smirnov test rejects the null of normally distributed residuals.³⁶

Here we present an alternative to the modified repeat-sales model that utilizes all of the data and measures the significance of the REIT-buyer premium without assuming normally distributed errors. The new methodology exploits the error term from our initial hedonic estimation, narrows the focus to residuals among properties that sold more than once, and then bootstraps the significance level of the test in the following three steps:

Residuals

Directly comparing the two raw prices is inappropriate because, while the property (e.g., location and size) are the same, market prices change over time and with the age of the property. Consequently, we compare residuals from the hedonic model (Eq. (10) excluding the REIT buyer variable). Note the hedonic model is not used to estimate a REIT premium, rather it controls for changes in year and age between the two transactions of the same property, where these controls are estimated with the full data set. The fitted value includes controls for all known property and transaction characteristics with the exception of the type of buyer (REIT versus non-REIT). We then identify all repeat sales and retain the residual on these 14,393 observations.

Test statistic

Where available, each REIT-buyer observation is paired with a twin observation or observations (same street address and city) where the buyer is not a REIT. Initially, we have 276 such REITBUYER/nonREITBUYER pairs. To make sure the two properties are actually twins without significant modifications, we further require that the twin properties be within 10 % of each other in square footage and that the raw price had not changed by more than 400 %.³⁷ These restrictions eliminate eight pairs, leaving 268 twins for our repeat transaction tests. Whereas in the paper’s body we criticize the hedonic model because of unobserved characteristics, here we use the model, along with its large sample, to adjust for changes in a property’s age and year between transactions. Then we eliminate this unobserved characteristics problem by looking at only repeat transactions. For each pair of transactions, we subtract the nonREITBUYER from the twin REITBUYER residual, which should be zero on average under the null hypothesis of no REIT price premium. Our test statistic, $\bar{D}$, is the mean of 268 differences in residuals and equals 0.0419.³⁸ That is, REITs tend to pay a premium, even when the same property is involved. However, size of the premium is only about 4.19 %, close to the 6.36 % found in the paper’s body using fewer, but higher quality, observations.

Bootstrapped p-value

The bootstrap recreates the test statistic, $\overline{D}$, one thousand times using the same process as the actual test, except neither of the pair of transactions has a REIT buyer. Specifically, we run 1,000 pseudo-tests where we randomly draw 268 pairs of non-REIT twins, randomly assign one to be subtracted from the other, calculate the differences between the residuals and then the mean of all 268 differences (a pseudo $\bar{D}$). The marginal significance level (p-value) for the tests of the actual REIT price premium is the fraction of realized pseudo $\bar{D}s$ greater than the actual $\bar{D}$ of 0.0419 found using REIT buyers. The mean of the one thousand pseudo $\bar{D}s$, −0.0027, is close to zero, as expected, and they range from −0.0813 to 0.0537. Figure 2 provides a histogram of the bootstrapped pseudo $\overline{D}s$, along with an arrow pointing to the actual $\overline D$ found using the REIT-purchased property and their twins.

Fig. 2

Histogram of mean differences between residuals for the 268 repeat sales involving REITBUYER/nonREITBUYER pairs and 1,000 bootstrapped mean differences between residuals for 268 repeat transaction involving nonREITBUYER/nonREITBUYER pairs

Only twelve out of 1,000 times is the mean difference in residuals above 0.0419. That is, we can reject the null hypothesis of no REIT premium with a p-value of 0.012 (compared to 0.0025 in Section 5.2). Thus, the alternative approach yields a slightly lower and slightly less significant REIT-buyer premium than the modified repeat-sales model used in the body of the paper.