Transparency in the Mortgage Market

Abstract

This paper studies the impact of transparency in the mortgage market on the underlying real estate market. We show that geographic transparency in the secondary mortgage market, which implies geographic risk based pricing in the primary market, can limit risk-sharing and make house prices more volatile. Ex ante, regions prefer opaque markets to enable insurance opportunities. We discuss the implications for risk based pricing and house price volatility more generally. In addition, we investigate the specific conditions under which competitive lenders would optimally choose to provide opaque lending, thus reducing volatility in the real estate market. We show that in general the opaque competitive equilibrium is not stable, and lenders have incentive to switch to transparent lending if one of the geographic regions has experienced a negative income shock. We propose market and regulatory mechanisms that make the opaque competitive equilibrium stable and insurance opportunities possible.

Appendix: 2 periods, 2 states

Transparent Mortgage Markets

The lender gets ${L_{0}^{j}}{R}$ if the borrower repays, and p ₁ ^j H if the borrower defaults.

Assume city A starts with the low income shock and city B starts with the high income shock: ${y_{0}^{A}}=y_{L}$, ${y_{0}^{B}}=y_{H}$.

If we denote the ratio $\frac {{p_{t}^{A}}}{{p_{t}^{B}}}$ with θ _t , then the income shock effects imply that θ ₁ < θ ₀. Without loss of generality let assume that θ ₀ = 1, then the income shock effects are depicted in the fact that θ ₁ < 1.

The probability city A will have a low shock next period is given by:

$P\left \{ {y_{1}^{A}}=y_{L} | {y_{0}^{A}}= y_{L} \right \} =\frac {1+\rho }{2}$

Where ρ ∈ [−1, 1] is the auto-correlation for income. We assume income follows a two-state Markov Chain:

$${y_{t}^{j}}\sim\left(\begin{array}{cc} \frac{1+\rho}{2} & \frac{1-\rho}{2} \\ \frac{1-\rho}{2} & \frac{1+\rho}{2}\end{array}\right).$$

For simplicity we assume that the spatial correlation in income shocks is perfectly negative ρ _{A, B} = −1, so whenever city A has a bad shock, city B will have a good shock vice-versa.

In a transparent market, the lender’s expected profit to city j is:

$$\begin{array}{@{}rcl@{}} \mathbb{E}\left[ {\pi_{t}^{j}} \right] &=& -{L_{t}^{j}} + \eta \mathbb{E}_{t} \min\left[ {L_{t}^{j}} R , p_{t+1}^{j} H \right]\\ &=& -{L_{t}^{j}} + \eta {L_{t}^{j}} R \cdot P\left\{ {L_{t}^{j}} R \leq p_{t+1}^{j} H \right\} +\eta H \mathbb{E}_{t}\left[ p_{t+1}^{j} | {L_{t}^{j}} R > p_{t+1}^{j} H \right] \end{array} $$

The zero expected profit condition implies:

${L_{0}^{A}} = \eta \left (\frac {1-\rho }{2} \right ) {L_{0}^{A}} R + \eta \left (\frac {1+\rho }{2} \right ) {p_{1}^{A}}H$ ⇔ ${\kern 11pt} {L_{0}^{A}} \left (1- \eta \left (\frac {1-\rho }{2} \right ) R \right ) = \eta \left (\frac {1+\rho }{2} \right ) H {p_{1}^{A}}$ ⇔ ${\kern 11pt} {L_{0}^{A}} = \frac { \eta \left (\frac {1+\rho }{2} \right ) H } { \left (1- \eta \left (\frac {1-\rho }{2} \right ) R \right ) } {p_{1}^{A}}$

Likewise,

$${L_{0}^{B}} = \eta \left( \frac{1+\rho}{2} \right) {L_{0}^{B}} R + \eta \left( \frac{1-\rho}{2} \right) {p_{1}^{B}} H$$

$${L_{0}^{B}} = \frac{ \eta \left( \frac{1-\rho}{2} \right) H } { \left(1- \eta \left( \frac{1+\rho}{2} \right) R \right) } {p_{1}^{B}} $$

In an opaque market, the lender’s zero profit condition is:

$L_{0} = \frac {1}{2} \eta L_{0} R \cdot \left (\frac {1-\rho }{2}\right ) + \frac {1}{2} \eta \left (\frac {1+\rho }{2} \right ) H {p_{1}^{A}}\\ {\kern 11pt}+ \frac {1}{2} \eta \left (\frac {1+\rho }{2} \right ) L_{0} R + \frac {1}{2} \eta \left (\frac {1-\rho }{2} \right ) H {p_{1}^{B}}$ ⇔ ${\kern 11pt} L_{0} = \frac {1}{2} \eta L_{0} R\\ {\kern 11pt}+\frac {1}{2} \eta H \left [\left (\frac {1+\rho }{2} \right ) {p_{1}^{A}} {\kern 11pt}+ \left (\frac {1-\rho }{2} \right ) {p_{1}^{B}}\right ]$ ⇔ ${\kern 11pt} L_{0} \left (1 - \frac {1}{2} \eta R \right ) = \frac {1}{2} \eta H \left [\left (\frac {1+\rho }{2} \right ) {p_{1}^{A}} + \left (\frac {1-\rho }{2} \right ) {p_{1}^{B}}\right ]$ ⇔ ${\kern 11pt} L_{0} = \frac { \frac {1}{2} \eta H } { \left (1 - \frac {1}{2} \eta R \right ) } p_{1}$

Where $p_{1}= \left [\left (\frac {1+\rho }{2} \right ) {p_{1}^{A}} + \left (\frac {1-\rho }{2} \right ) {p_{1}^{B}}\right ]$, and we have ${p_{1}^{A}}<p_{1}<{p_{1}^{B}}$, also notice that in the opaque market case, each city receives the same loan L ₀.

Proposition 1

If ρ > 0, then

if ηR > 1:

if ηR < 1: ${L_{0}^{A}} > L_{0} > {L_{0}^{B}}$ If income is negatively correlated ρ < 0, then signs are reversed. But this is not the case we are interested in.

The case we study has ρ > 0 and ηR > 1. Plugging this into the equilibrium price function: ${p_{0}^{j}} = \frac {1}{\gamma } \left (\alpha + {y_{0}^{j}} + {L_{0}^{j}} - H \right )$ Since the loan to city A under transparency is smaller than the loan to city A under opacity ${L_{0}^{A}}<L_{0}$ , the transparent price is lower than the opaque price: $p_{0}^{A,trans} = \frac {1}{\gamma } \left (\alpha + {y_{0}^{j}} + {L_{0}^{A}} - H \right ) < \frac {1}{\gamma } \left (\alpha + {y_{0}^{j}} + L_{0} - H \right ) = p_{0}^{A,opaque}$ Likewise: $p_{0}^{B,trans} = \frac {1}{\gamma } \left (\alpha + {y_{0}^{j}} + {L_{0}^{B}} - H \right ) > \frac {1}{\gamma } \left (\alpha + {y_{0}^{j}} + L_{0} - H \right ) = p_{0}^{B,opaque}$

NOTE: we have assumed that city A starts with a bad income shock at time 0 and city B starts with a good income shock. Ex ante with probability $\frac {1}{2}$ we have ${y_{0}^{A}}=y_{L}$ and ${y_{0}^{B}}=y_{H}$ , and with probability $\frac {1}{2}$ we have ${y_{0}^{A}}=y_{H}$ and ${y_{0}^{B}}=y_{L}$ . However, ex ante neither city knows which state of the world they will start in they will prefer opacity to have smoother house prices.

The lesson from this model is that a geographically transparent mortgage market has more volatile house prices which are more strongly correlated to local risks.