Optimal Mortgage Contract Choice Decision in the Presence of Pay Option Adjustable Rate Mortgage and the Balloon Mortgage

Abstract

The unprecedented run-up in global house prices of the 2000s was preceded by a revolution in U.S. mortgage markets in which borrowers faced a plethora of mortgages to choose from collectively known as nontraditional mortgages (NTMs), whose poor performance helped ignite the global financial crisis in 2007. This paper studies the choice of mortgage contracts in an expanded framework where the menu of contracts includes the pay option adjustable rate mortgage (PO-ARM), and the balloon mortgage (BM), alongside the traditional long horizon fixed rate mortgage (FRM) and the short horizon regular ARM. The inclusion of the PO-ARM is based on the fact it is the most controversial and perhaps the riskiest of the NTMs, whereas the BM has not been analyzed in the literature despite its different risk-sharing arrangement and long vintage. Our inclusive model relates the structural differences of these contracts to the horizon risk management problems and affordability constraints faced by the households that differ in terms of expected mobility. The numerical solutions of the model generates a number of interesting results suggesting that households select mortgage contracts to match their horizon, manage horizon risk and mitigate liquidity or affordability constraints they face. From a risk management and welfare perspectives, we find that the optimal contract for households with shorter horizons, specifically households who expect to move house once every 1 to 2 years, is the PO-ARM. Beyond 2 years the welfare advantage of the PO-ARM diminishes and BM becomes the more optimal contract up to 5-year horizon. Overall, the results suggest that households are neither as risk averse as the selection of the FRM would suggest, nor are they as risk-seeking as the selection of PO-ARM or regular ARM would suggest. The results also suggest that the exuberance demonstrated for NTMs by borrowers, especially PO-ARMs, may be both rational and irrational.

Appendices

Appendix 1

Table 7 Volume of Non-Traditional Mortgages (NTMs)

Appendix 2

Table 8 Delinquency rates on nontraditional mortgages

Appendix 3

Table 9 Mortgage payment schedule

Appendix 4

We assume the utility function as: $\mathrm{V}\left( {\mathrm{Z}-\mathrm{x}} \right)=1-{e^{{-\left( {1-\lambda } \right)R\left( {Z-x} \right)}}}$

Z is income, x is payment, λ is the probability of moving and R is a constant

We now solve

$\int\limits_{{-\infty}}^{\infty } {\left\{ {1-\exp \left( - \right.\left( {1-\lambda } \right)R\left[ {Z-B\frac{r}{{1-{{{\left( {1+r} \right)}}^{-T }}}}} \right]} \right\}f(r)dr}$ Where r ~ N (μ,σ ²), B is the mortgage outstanding.

$$ =\int\limits_{{-\infty}}^{\infty } {f(r)dr} -\exp \left( {-\left( {1-\lambda } \right)RZ} \right)\int\limits_{{-\infty}}^{\infty } {\exp \left[ {\left( {1-\lambda } \right)RB\frac{r}{{1-{{{\left( {1+r} \right)}}^{-T }}}}} \right]f(r)dr} $$

(12)

and

$\int\limits_{{-\infty}}^{\infty } {\exp \left[ {\left( {1-\lambda } \right)RB\frac{r}{{1-{{{\left( {1+r} \right)}}^{-T }}}}} \right]f(r)dr}$ in Eq. (12).

We define $g(r)=\frac{r}{{1-{{{\left( {1+r} \right)}}^{-T }}}}$, and use Taylor Expansion around r = k to the second differentiation to approximately estimate the function.

$g(r)=g(k)+g\prime (k)\left( {r-k} \right)+\frac{{g\prime \prime (k)}}{2}{{\left( {r-k} \right)}^2}$. This equation is then substituted into $\int\limits_{{-\infty}}^{\infty } {\exp \left[ {\left( {1-\lambda } \right)RB\frac{r}{{1-{{{\left( {1+r} \right)}}^{-T }}}}} \right]f(r)dr}$.

$$ \begin{array}{*{20}c} {\approx \int\limits_{{-\infty}}^{\infty } {\exp \left\{ {\left( {1-\lambda } \right)RB\left[ {g(k)+g\prime (k)\left( {r-k} \right)+\frac{{g\prime \prime (k)}}{2}{{{\left( {r-k} \right)}}^2}} \right]} \right\}f(r)dr} } \hfill \\ {=\exp \left( {\left( {1-\lambda } \right)RB\left[ {g(k)-\frac{{g\prime {(k)^2}}}{{2g\prime \prime (k)}}} \right]} \right.\int\limits_{{-\infty}}^{\infty } {\exp \left\{ {\left( {1-\lambda } \right)RB\frac{{g\prime \prime (k)}}{2}{{{\left[ {r-\left( {k-\frac{{g\prime (k)}}{{g\prime \prime (k)}}} \right)} \right]}}^2}} \right\}f(r)dr} } \hfill \\ \end{array} $$

(13)

To solve for $\int\limits_{{-\infty}}^{\infty } {\exp \left\{ {\left( {1-\lambda } \right)RB\frac{{g\prime \prime (k)}}{2}{{{\left( {r-\left( {k-\frac{{g\prime (k)}}{{g\prime \prime (k)}}} \right)} \right)}}^2}} \right\}f(r)dr}$ in Eq. (13), we define $v=k-\frac{{g\prime (k)}}{{g\prime \prime (k)}}$ and $t=\left( {1-\lambda } \right)RB\frac{{g\prime \prime (k)}}{2}$. We also have $f(r)=\frac{1}{{\sqrt{{2\pi }}\sigma }}{e^{{-\frac{{{{{\left( {r-\mu } \right)}}^2}}}{{2{\sigma^2}}}}}}$

$$ \begin{array}{*{20}c} {\int\limits_{{-\infty}}^{\infty } {{e^{{t{{{\left( {r-v} \right)}}^2}}}}\frac{1}{{\sqrt{{2\pi }}\sigma }}{e^{{-\frac{{{{{\left( {r-\mu } \right)}}^2}}}{{2{\sigma^2}}}}}}dr} } \hfill \\ {=\int\limits_{{-\infty}}^{\infty } {{e^{{t{r^2}-2trv+t{v^2}}}}\frac{1}{{\sqrt{{2\pi }}\sigma }}{e^{{-\frac{{{r^2}-2r\mu +{\mu^2}}}{{2{\sigma^2}}}}}}dr} } \hfill \\ {=\int\limits_{{-\infty}}^{\infty } {\frac{1}{{\sqrt{{2\pi }}\sigma }}{e^{{-\frac{{\left( {1-2t{\sigma^2}} \right){r^2}-2\left( {\mu -2tv > {\sigma^2}} \right)r+\left( {{\mu^2}-2t{v^2}{\sigma^2}} \right)}}{{2{\sigma^2}}}}}}dr} } \hfill \\ {={e^{{-\frac{{\left( {{\mu^2}-2t{v^2}{\sigma^2}} \right)}}{{2{\sigma^2}}}}}}\int\limits_{{-\infty}}^{\infty } {\frac{1}{{\sqrt{{2\pi }}\sigma }}{e^{{-\frac{{\left( {1-2t{\sigma^2}} \right){r^2}-2\left( {\mu -2tv > {\sigma^2}} \right)r+\frac{{{{{\left( {\mu -2tv > {\sigma^2}} \right)}}^2}}}{{\left( {1-2t{\sigma^2}} \right)}}}}{{2{\sigma^2}}}}}}{e^{{\frac{{\frac{{{{{\left( {\mu -2tv > {\sigma^2}} \right)}}^2}}}{{\left( {1-2t{\sigma^2}} \right)}}}}{{2{\sigma^2}}}}}}dr} } \hfill \\ {={e^{{-\frac{{\left( {{\mu^2}-2t{v^2}{\sigma^2}} \right)}}{{2{\sigma^2}}}}}}{e^{{\frac{{\frac{{{{{\left( {\mu -2tv > {\sigma^2}} \right)}}^2}}}{{\left( {1-2t{\sigma^2}} \right)}}}}{{2{\sigma^2}}}}}}\int\limits_{{-\infty}}^{\infty } {\frac{1}{{\sqrt{{2\pi }}\sigma }}{e^{{-\frac{{{{{\left[ {\sqrt{{1-2t{\sigma^2}}}r-\frac{{\left( {\mu -2tv > {\sigma^2}} \right)}}{{\sqrt{{1-2t{\sigma^2}}}}}} \right]}}^2}}}{{2{\sigma^3}}}}}}dr} } \hfill \\ \end{array} $$

$$ \begin{array}{*{20}c} {\mathrm{Let}=y=\sqrt{{1-2t{\sigma^2}}}r,\;\mathrm{then}\;dy=\sqrt{{1-2t{\sigma^2}}}dr} \hfill \\ {={e^{{-\frac{{\left( {{\mu^2}-2t{v^2}{\sigma^2}} \right)}}{{2{\sigma^2}}}}}}{e^{{\frac{{\frac{{{{{\left( {\mu -2tv > {\sigma^2}} \right)}}^2}}}{{\left( {1-2t{\sigma^2}} \right)}}}}{{2{\sigma^2}}}}}}\int\limits_{{-\infty}}^{\infty } {\frac{1}{{\sqrt{{2\pi }}\sigma }}{e^{{-\frac{{{{{\left[ {y-\frac{{\left( {\mu -2tv > {\sigma^2}} \right)}}{{\sqrt{{1-2t{\sigma^2}}}}}} \right]}}^2}}}{{2{\sigma^3}}}}}}\frac{1}{{\sqrt{{1-2t{\sigma^2}}}}}dy} } \hfill \\ {=\frac{1}{{\sqrt{{1-2t{\sigma^2}}}}}{e^{{-\frac{{\left( {{\mu^2}-2t{v^2}{\sigma^2}} \right)}}{{2{\sigma^2}}}}}}{e^{{\frac{{\frac{{{{{\left( {\mu -2tv > {\sigma^2}} \right)}}^2}}}{{\left( {1-2t{\sigma^2}} \right)}}}}{{2{\sigma^2}}}}}}} \hfill \\ {\int\limits_{{-\infty}}^{\infty } {\left\{ {1-\exp \left( - \right.\left( {1-\lambda } \right)R\left[ {Z-B\frac{r}{{1-{{{\left( {1+r} \right)}}^{-T }}}}} \right]} \right\}f(r)dr} } \hfill \\ {=1-\exp \left( {-\left( {1-\lambda } \right)RZ} \right)\int\limits_{{-\infty}}^{\infty } {\exp \left[ {\left( {1-\lambda } \right)RB\frac{r}{{1-{{{\left( {1+r} \right)}}^{-T }}}}} \right]f(r)dr} } \hfill \\ {=1-\exp \left( {-\left( {1-\lambda } \right)RZ} \right)\exp \left( {\left( {1-\lambda } \right)} \right.RB\left[ {g(k)-\frac{{g\prime {(k)^2}}}{{2g\prime \prime (k)}}} \right]\frac{1}{{\sqrt{{1-2t{\sigma^2}}}}}\exp \left( {-\frac{{\left( {{\mu^2}-2t{v^2}{\sigma^2}} \right)}}{{2{\sigma^2}}}} \right)\exp \left( {\frac{{{{{\left( {\mu -2tv > {\sigma^2}} \right)}}^2}}}{{2{\sigma^2}\left( {1-2t{\sigma^2}} \right)}}} \right)} \hfill \\ \end{array} $$

Where, $v=k-\frac{{g\prime (k)}}{{g\prime {\prime} (k)}}$, and $t=\left( {1-\lambda } \right)RB\frac{{g\prime {\prime} (k)}}{2}$, $\left| t \right|<1$.

Note,

Where

$$ \begin{array}{*{20}c} {g(r)=\frac{r}{{1-{{{\left( {1+r} \right)}}^{-T }}}}=\frac{{r{{{\left( {1+r} \right)}}^T}}}{{{{{\left( {1+r} \right)}}^T}-1}},} \hfill \\ {g\prime (r)=\left( {\frac{{{H_1}(r)}}{{{H_2}(r)}}} \right)\prime =\frac{{\left( {{H_1}(r)} \right)\prime {H_2}(r)-{H_1}(r)\left( {{H_2}(r)} \right)\prime }}{{{{{\left( {{H_2}(r)} \right)}}^2}}}} \hfill \\ \end{array} $$

Where

$$ \begin{array}{*{20}c} {{H_1}(r)=r{{{\left( {1+r} \right)}}^T}} \hfill \\ {{H_2}(r)={{{\left( {1+r} \right)}}^T}-1} \hfill \\ {{H_1}\prime (r)={{{\left( {1+r} \right)}}^T}+Tr{{{\left( {1+r} \right)}}^{T-1 }}} \hfill \\ {{H_2}\prime (r)=T{{{\left( {1+r} \right)}}^{T-1 }}} \hfill \\ \end{array} $$

$$ \begin{array}{*{20}c} {g\prime (r)=\frac{{\left[ {{{{\left( {1+r} \right)}}^T}+Tr{{{\left( {1+r} \right)}}^{T-1 }}} \right]\left[ {{{{\left( {1+r} \right)}}^T}-1} \right]-\left[ {r{{{\left( {1+r} \right)}}^T}} \right]\left[ {T{{{\left( {1+r} \right)}}^{T-1 }}} \right]}}{{{{{\left[ {{{{\left( {1+r} \right)}}^T}-1} \right]}}^2}}}} \hfill \\ {=\frac{{\left[ {{(1+r)^{2T }}-{{{\left( {1+r} \right)}}^T}-rT{{{\left( {1+r} \right)}}^{T-1 }}} \right]}}{{{{{\left[ {{{{\left( {1+r} \right)}}^T}-1} \right]}}^2}}}} \hfill \\ {g\prime \prime (r)=\left( {\frac{{{H_3}(r)}}{{{H_4}(r)}}} \right)\prime =\frac{{\left( {{H_3}(r)} \right)\prime {H_4}(r)-{H_3}(r)\left( {{H_4}(r)} \right)\prime }}{{{{{\left( {{H_4}(r)} \right)}}^2}}}} \hfill \\ \end{array} $$

Where

$$ \begin{array}{*{20}c} {{H_3}(r)=\left[ {{{{\left( {1+r} \right)}}^{2T }}-{{{\left( {1+r} \right)}}^T}-rT{{{\left( {1+r} \right)}}^{T-1 }}} \right]} \hfill \\ {{H_4}(r)={{{\left[ {{{{\left( {1+r} \right)}}^T}-1} \right]}}^2}} \hfill \\ {{H_3}\prime (r)=2T{{{\left( {1+r} \right)}}^{2T-1 }}-T{{{\left( {1+r} \right)}}^{T-1 }}-T{{{\left( {1+r} \right)}}^{T-1 }}-rT\left( {T-1} \right){{{\left( {1+r} \right)}}^{T-2 }}} \hfill \\ {{H_4}\prime (r)=2T\left[ {{{{\left( {1+r} \right)}}^T}-1} \right]{{{\left( {1+r} \right)}}^{T-1 }}} \hfill \\ \end{array} $$

$$ \begin{array}{*{20}c} {g\prime \prime (r)=\frac{{\left[ {2T{{{\left( {1+r} \right)}}^{2T-1 }}-T{{{\left( {1+r} \right)}}^{T-1 }}-T{{{\left( {1+r} \right)}}^{T-1 }}-rT\left( {T-1} \right){{{\left( {1+r} \right)}}^{T-2 }}} \right]{{{\left[ {{{{\left( {1+r} \right)}}^T}-1} \right]}}^2}}}{{{{{\left[ {{{{\left( {1+r} \right)}}^T}-1} \right]}}^2}}}} \hfill \\ {-\frac{{\left[ {{{{\left( {1+r} \right)}}^{2T }}-{{{\left( {1+r} \right)}}^T}-rT{{{\left( {1+r} \right)}}^{T-1 }}} \right]2T\left[ {{{{\left( {1+r} \right)}}^T}-1} \right]\left. {{{{\left( {1+r} \right)}}^{T-1 }}} \right]}}{{{{{\left[ {{{{\left( {1+r} \right)}}^T}-1} \right]}}^2}}}} \hfill \\ \end{array} $$

(II)

We also need to solve for the following expected utility when borrowers prepay.

$$ \begin{array}{*{20}c} {\int\limits_{{-\infty}}^{\infty } {\left\{ {1-\exp \left( - \right.\left( {1-\lambda } \right)R\left[ {Z-B\left( {1+r} \right)} \right]} \right\}f(r)dr} } \hfill \\ {=1-\exp \left[ {-\left( {1-\lambda } \right)R\left( {Z-B} \right)} \right]\exp \left( {\mu t+\frac{{{t^2}{\sigma^2}}}{2}} \right)} \hfill \\ \end{array} $$

Where $t=\left( {1-\lambda } \right)RB$, and $\left| t \right|<1$.