The Rent Term Premium for Cancellable Leases

Abstract

This study analyzes the rent term premium for leases that can be cancelled by the lessee. We model the lessor’s trade-off between leasing costs and the cost of cancellation options based on the recognition that many leases are cancellable by lessees, and lease markets involve significant transaction costs. We demonstrate that, regardless of the expected future rents, the rent term structure is upward-sloping when there is no leasing cost but U-shaped when the lessor faces moderate leasing costs. Residential leases in Japan, which are all cancellable by tenants, exhibit the term structure that is consistent with our calibrated model.

Appendices

Appendix A: Proofs

Proof of Lemma: To prove the lemma, we define two conditions:

• Condition (a) At least one of the initial leases has not been cancelled until period $T-1$; i.e., the market rent has not been low enough to induce cancellations of both leases: $\forall t\le T-2:\ {\hat{R}}_t^{T-t}> \min \left\{{\hat{R}}_0^T,{\hat{R}}_0^{T-1}\right\}$.

• Condition (b) The market rent for the final period is higher than the preceding rental income under the $T$ strategy: ${R}_{T-1}^1>{\hat{Y}}_{T-2}^T$ .

There are three cases with respect to the rent for the final period. We demonstrate that the rent for the $T-1$ strategy is strictly higher than the rent for the $T$ strategy in the first case and is equal to the rent for the $T$ strategy in the second and third cases.

First Case: Conditions $\left(\mathrm{a}\right)$ and $\left(\mathrm{b}\right)$ both hold. If the initial rent for the $T$ -period lease is greater than or equal to the initial rent for the $T-1$ -period lease $\left(\mathrm{i}.\mathrm{e}.,\ {\hat{R}}_0^T\ge {\hat{R}}_0^{T-1}\right)$ , then condition $\left(\mathrm{a}\right)$ implies that the initial $T-1$ period lease will never be canceled until maturity. The rent at time $T-2$ will be ${\hat{Y}}_{T-2}^{T-1}={\hat{R}}_0^{T-1}$ for the $T-1$ strategy and ${\hat{Y}}_{T-2}^T= \min \left\{{\hat{R}}_0^T,{\hat{R}}_t^{T-t};t\le T-2\right\}$ for the $T$ strategy. By condition $\left(\mathrm{b}\right)$, the rent under the $T$ strategy will be unchanged for the final period $\left({\hat{Y}}_{T-1}^T={\hat{Y}}_{T-2}^T\right)$, whereas under the $T-1$ strategy, the initial lease will expire at time $T-1$, and the rent during the final period will be adjusted upward to ${\hat{Y}}_{T-1}^{T-1}={R}_{T-1}^1$ . Condition $\left(\mathrm{b}\right)$ also implies that during this period, the rent for the $T$ strategy will be strictly lower than the rent for the $T-1$ strategy $\left({\hat{Y}}_{T-1}^T<{\hat{Y}}_{T-1}^{T-1}\right)$ .

Conversely, if the initial rent is lower for the T-period lease than for the $T-1$ -period lease $\left(\mathrm{i}.\mathrm{e}.,\ {\hat{R}}_0^T<{\hat{R}}_0^{T-1}\right)$, then conditions (a) and (b) imply that the initial $T$ period lease will never be canceled until maturity. Thus, the rent during the final period is the initial rent $\left({\hat{Y}}_{T-1}^T={\hat{R}}_0^T\right)$. In contrast, the $T-1$ -period lease may or may not be cancelled. If the initial lease has expired without a cancellation, the rent during the final period will be ${\hat{Y}}_{T-1}^{T-1}={R}_{T-1}^1$. Alternatively, if the initial lease has been cancelled, the rent during the final period will be the historical minimum of market rents: ${\hat{Y}}_{T-1}^{T-1}= \min \left\{{\hat{R}}_t^{T-t};t\le T-1\right\}$. In either case, conditions (a) and (b) imply that during this period, the rent for the T strategy will be strictly lower than the rent for the $T-1$ strategy $\left({\hat{Y}}_{T-1}^T<{\hat{Y}}_{T-1}^{T-1}\right)$.

Second Case: Condition (a) holds but condition (b) does not hold $\left({R}_{T-1}^1\le {\hat{Y}}_{T-2}^T\right)$ . The final rent is the same for both strategies $\left({\hat{Y}}_{T-1}^T={\hat{Y}}_{T-1}^{T-1}={R}_{T-1}^1\right)$ because the rent is adjusted to the low level of market rent that exists during the final period regardless of whether the initial T period rent is higher than the $T-1$ period rent.

Third Case: Condition (a) does not hold. Because both leases will have been cancelled at the same time before time $T-1$, both strategies will use the same lease during the final period. Thus, the final rent is the same for both strategies: ${\hat{Y}}_{T-1}^T={\hat{Y}}_{T-1}^{T-1}$.

In summary, the $\mathrm{T}$ strategy produces a strictly lower rent than the $T-1$ period strategy $\left({\hat{Y}}_{T-1}^T<{\hat{Y}}_{T-1}^{T-1}\right)$ if conditions $(a)$ and $(b)$ both hold. Otherwise, both strategies give the same rent $\left({\hat{Y}}_{T-1}^T={\hat{Y}}_{T-1}^{T-1}\right)$ .

Proof of Proposition: Suppose the initial rent for the $T$ -period lease is not higher than the initial rent for the $T-1$ -period lease $\left({\hat{R}}_0^T\le {\hat{R}}_0^{T-1}\right)$. The $T-1$ strategy will produce a weakly positive rent premium relative to the $T$ strategy until both leases are cancelled. In addition, the $T-1$ strategy will generate a strictly higher expected rent than the $T$ strategy for the final time period that is examined, as demonstrated by the lemma. Thus, the $T-1$ strategy would provide a strictly larger value to the lessor than the $T$ strategy; this result would violate the specified consistent pricing condition. Therefore, the initial rent for the $T$ -period lease must be higher than the initial rent for the $T-1$ -period lease $\left({\hat{R}}_0^T>{\hat{R}}_0^{T-1}\right)$. In other words, the term structure of cancellable leases must be upward sloping between $T-1$ and $T$. Because we can apply the same argument to any lease term $u\;\left(u=2,\dots, T\right)$, the term structure for all cancellable leases must be upward sloping.

Appendix B: Detail of Numerical Analysis

We numerically analyze the term structure by parameterizing the risk-free rate, rent volatility, the risk premium on the short-term rent, the expected rent growth, and most importantly, leasing costs. We construct an annual binomial-tree model for an 8 year period and generate 1000 paths of short-term lease rates. At each node of the tree, the short-term rate changes by a factor of either $\exp \left(\sigma \right)$ or $\exp \left(-\sigma \right)$, where σ ∈ [0, ∞) is a rent volatility measure. In the continuous-time limit, an instantaneous rate of rent growth during time period $\tau$ is normally distributed with the volatility $\sigma \sqrt{\tau }$. At the beginning of each year, a lessee chooses the rent that is represented by Equation (3).¹⁸ We introduce a proportional leasing cost $c{\widehat{Y}}_t^T$, which is equivalent to a $12c$ -month vacancy cost at the time of cancellation. The lost income during vacant period is typically the most significant cost for the lessor.¹⁹ We compute the landlord’s rental income for each rent path because the cancellation option is a path-dependent option. The present value of the T-year lease is computed as:

$$ {\hat{V}}^T={\displaystyle {\sum}_{t=0}^{T-1}{\mu}_t{\mathrm{E}}_0^Q\left[{\hat{Y}}_t^T\left(1-c{\mathbb{I}}_{\left\{{\hat{Y}}_t^T\ne {\hat{Y}}_{t-1}^T\right\}}\right)\right]}, $$

(B1)

where ${\mu}_t$ is the risk-free discount factor for t -time periods, ${\mathrm{E}}_0^Q\left[{\hat{Y}}_t^T\right]$ is the expected lease rate under the equivalent martingale measure Q, and ${\mathbb{I}}_{\left\{{\hat{Y}}_t^T\ne {\hat{Y}}_{t-1}^T\right\}}$ is an indicator function that takes the value of one if the lease is cancelled and zero otherwise.²⁰ After computing the present value ${V}^1$ for a roll-over short-term strategy, we determine the initial lease rate for a T-period lease ${\hat{R}}_0^T$ such that the consistent pricing condition holds: ${V}^1 = {\hat{V}}^T$ for T = 2, …, 8.

Appendix C: Parameter Values for Calibration

We determine parameter values as follows. We use 1 % for the risk-free rate because the average end-of-month yields of Japanese Government Bonds are 1.32 % for 10-year bonds and 0.65 % for 5-year bonds between 2001 and 2012. Regarding rent growth rates and volatility, we use the Recruit quality-adjusted rent index for the Kanto region between 2005 and 2012.²¹ In our lease sample, 39 % of general leases and 56 % of fixed-term leases are observed in this region. The average annual growth rate is 0.82 % and standard deviation is 1.97 %. Considering that there are smoothing issues in this aggregate index and that the volatility for an individual property is much larger, we use 1 % for the rent growth and 8 % for the rent volatility. Given that the rent volatility is relatively small, we assume that the risk premium on the short-term rent is 1 %. Regarding the lessor’s costs, we use the 2008 National Housing and Land Survey and compute the average vacancy rate for rental dwellings to be 19 %.²² Additional contracting costs are at most a few hundred dollars and ignorable. We use 19 % as the high vacancy rate and 5 % as the low vacancy rate.

Appendix D: The Difference in Property and Tenant Characteristics Between Fixed-Term Lease, General Lease, and Owner-Occupied Samples

In this appendix, we present the result of probit regressions regarding the difference between owner-occupied properties and leased properties (Model (a)) and between fixed-term lease properties and general lease properties (Model (b)). Tables 6 and 7 present results for property characteristics and tenant characteristics, respectively. In Model (a), most property and tenant characteristics exhibit statistically significant differences between owner-occupied properties and leased properties. In contrast, in Model (b), differences are not statistically significant for most characteristics. These results are consistent with the descriptive statistics presented in Tables 1 and 2.

Table 6 Difference in property characteristics

Table 7 Difference in tenant and owner characteristics