Home Price, Time-on-Market, and Seller Heterogeneity Under Changing Market Conditions

Abstract

This paper develops a formal model to examine the effect of changing market conditions and individuals’ selling constraints on selling price and time-on-market. Using the concept of Relative Liquidity Constraint (RLC)—a stochastic variable that captures the randomness of future individual constraints and market conditions—the study presents the first ex ante analysis that extends the investigation of the issue of seller heterogeneity to the point of the buying decision, that is, from the perspective of the buyer’s (future seller’s) point of view. We show that seller constraint, as well as the uncertainty of such a constraint, significantly depresses the expected selling price and increases risk. Our closed-form formulas provide a set of simple quantitative tools that enable buyers and sellers to adjust the “market average” to their ex ante “individual expectations”.

Appendices

Appendix 1

One potential difficulty with application of Theorem 1 and 2 is that the time series return data, when broken down to sub-periods of different market conditions (Good, Bad, and Normal), are not long enough to estimate $u_\tau$, $\sigma _\tau$ for each sub-period when the holding period is long. One way to get around this problem is to use the common assumptions on holding-period return and risk: $u_\tau = \tau u$ and $\sigma _\tau = \tau \sigma ^2$, where τ is the total holding period until sale; u and σ ² are the single holding-period expected return and variance, respectively. Then we can rewrite Eqs. 20 and 21 as follows:

$$u_{L,\tau } = \tau u - \sqrt 3 \left( {L - 1} \right)\sqrt \tau \sigma ,$$

(43)

$$\sigma _{L,\tau }^2 = L^2 \tau \sigma ^2 .$$

(44)

Therefore, the average return and volatility per period can be expressed:

$$u_{L,\tau }^{{\text{period}}} = u - \frac{{\sqrt 3 \left( {L - 1} \right)\sigma }}{{\sqrt \tau }},$$

(45)

$$\sigma _{L,\tau }^{{\text{period}}} = L\sigma $$

(46)

The above two formula can be used to compute Table 1 with an assumed holding-period (5 years in our demonstration).

Similarly, from Eqs. 27 and 29 we can readily obtain:

$${\text{Var}}\left( {\tilde r_{\tilde L,\tau } } \right) = \left[ {\left( {E\left[ {\tilde L} \right]} \right)^2 + 4{\text{Var}}\left( {\tilde L} \right)} \right]\tau \sigma ^2 ,$$

(47)

$$E\left[ {\tilde r_{\tilde L,\tau } } \right] = \tau u - \sqrt 3 \left( {E\left[ {\tilde L} \right] - 1} \right)\sqrt \tau \sigma .$$

(48)

The average return and volatility per period can thus be written as:

$$u\left[ {\tilde r_{\tilde L,\tau } } \right]^{{\text{period}}} = u - \frac{{\sqrt 3 \left( {E\left[ {\tilde L} \right] - 1} \right)\sigma }}{{\sqrt \tau }},$$

(49)

$$\sigma \left( {\tilde r_{\tilde L,\tau } } \right)^{{\text{period}}} = \sqrt {\left[ {\left( {E\left[ {\tilde L} \right]} \right)^2 + 4{\text{Var}}\left( {\tilde L} \right)} \right]} \sigma .$$

(50)

The above two formula can be used to compute Table 2 with an assumed holding-period of 5 years.

Appendix 2

For demonstration purposes, we consider Scenario 3, where T _max < NST and the allowable expected TOM is a uniform distribution over [T _min, T _max]. From Eq. 23, we have:

$$E\left[ {\tilde L} \right] = \frac{{{\text{NST}}\left( {T_{\max } + T_{\min } } \right)}}{{2T_{\max } T_{\min } }}.$$

(23′)

As we know, the “corresponding” allowable expected TOM in Table 2 is equal to $\frac{{T_{\min } + T_{\max } }}{2}$, which is deterministic. As a result:

$$L = \frac{{{\text{NST}}}}{{\left[ {{{\left( {T_{\max } + T_{\min } } \right)} \mathord{\left/{\vphantom {{\left( {T_{\max } + T_{\min } } \right)} 2}} \right.\kern-\nulldelimiterspace} 2}} \right]}}$$

(51)

Since:

$$\begin{array}{*{20}l} {} \hfill & {\left( {T_{\max } - T_{\min } } \right)^2 > 0 \Rightarrow \frac{{T_{\max } + T_{\min } }}{2} > \frac{{2T_{\max } T_{\min } }}{{T_{\max } + T_{\min } }}} \hfill \\ \Rightarrow \hfill & {\frac{{{\text{NST}}\left( {T_{\max } + T_{\min } } \right)}}{{2T_{\max } T_{\min } }} > \frac{{{\text{NST}}}}{{\left[ {{{\left( {T_{\max } + T_{\min } } \right)} \mathord{\left/ {\vphantom {{\left( {T_{\max } + T_{\min } } \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} \right]}}} \hfill \\ \end{array} $$

From Eqs. 23′ and 51, we thus have:

$$E\left[ {\tilde L} \right] >L.$$

(52)

Equation 52 together with Theorems 1 and 2 suggest that the expected returns under uncertain $\tilde L$ are less than those with certain L even though the mean of “Possible distribution of expected TOM” in Table 2 is the same as the “Seller expected TOM” in Table 1, which is in fact consistent with the results in Tables 1 and 2.