# Housing Market Bubbles and Mortgage Contract Design: Implications for Mortgage Lenders and Households

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. US and Canadian Mortgage Contract Design

## 3. Immunizing Risk to Mortgage Lenders: Basic Results

_{H}is the market value of residential household assets associated with the mortgage; L

_{H}is the market value of household mortgage liabilities; S

_{H}is the household equity position. For lenders: A

_{L}is the market value of the lender mortgage assets; L

_{L}is the market value of mortgage lender liabilities used to fund mortgage assets; S

_{L}is the lender equity position.

_{H}= A

_{L}acts to transfer mortgage value changes between households and lenders. This balance sheet transfer is an essential avenue for mortgage funding risk transmission. Considering only interest rate risk and ignoring house price risk, a univariate Taylor series expansion is applied to the aggregate surplus value function S = S

_{H}+ S

_{L}= A

_{H}− L

_{H}+ A

_{L}− L

_{L}(where S[y]) to give:

**Definition**

**1.**

_{S}= (S

_{H}/S) CON

_{SH}+ (S

_{L}/S) CON

_{SL}is the modified convexity of system surplus. The mortgage funding system interest rate immunization conditions follow: (1) Set the duration of aggregate surplus equal to zero; and (2) have a positive convexity of aggregate surplus.

_{S}= (S

_{H}/S) DUR

_{SH}+ (S

_{L}/S) DUR

_{SL}= 0 - reveals that interest rate risk immunization for a mortgage funding system using mortgage contracts with term to maturity equal to the amortization period cannot be achieved unless interest-bearing assets are included on the household balance sheet. More precisely:

**Proposition**

**1.**

_{S}= ((S

_{H}/S) DUR

_{SH}) + ((S

_{L}/S) DUR

_{SL}= 0

_{H}= A

_{L}, the aggregate zero duration of mortgage funding system surplus requires:

_{H}) are identical to the mortgage assets held by mortgage lenders (A

_{L}), it follows that DUR

_{H}

^{L}= DUR

_{L}

^{A}.

_{H}

^{A}= 0 and zero duration of mortgage funding system surplus cannot be achieved.5 Of course, the level of interest rates, over time, will be capitalized into house prices. However, this impact is gradual while the valuation impact of interest rate changes on mortgage lender liabilities used to fund mortgages will be comparatively rapid.

**Proposition**

**2.**

_{L}= A

_{L}− L

_{L}and the zero duration of surplus condition for lenders is: $\frac{1}{{S}_{L}}\frac{{dS}_{L}}{dy}=0=\frac{{A}_{L}}{{S}_{L}}{{DUR}_{L}}^{A}-\frac{{L}_{L}}{{S}_{L}}{{DUR}_{L}}^{L}$.

_{H}= F

_{H}+ U

_{H}be the funded and unfunded portions of L

_{H}, respectively. Once the amortization period (N) and the interest rate are specified, the value of the annual fixed rate payment (M) on the mortgage can be determined. Given this, U

_{H}is the unpaid balance on the mortgage that needs to be refinanced at time T*, the maturity date of the short-term mortgage:

_{H}represents the unpaid balance due at time T*, changes in interest rates will alter the intrinsic value of this future liability for households. However, because the unpaid balance will be financed at future market interest rates, from the perspective of mortgage lenders U

_{H}is a floating rate contract where DUR

_{H}

^{U}= 0. The imposition of a yield maintenance prepayment penalty that is at least equal to loss of interest prevents T* < N households from capturing any significant gain by refinancing prior to the maturity of the mortgage.

_{H}

^{#}/S) DUR

_{H}

^{A}

^{#}/(A

_{H}/S) DUR

_{H}

^{A}+ (U

_{H}/S) DUR

_{H}

^{U}produces the following:

**Proposition**

**3.**

_{H}= A

_{L}in the T* < N case, it follows that zero duration of system surplus requires:

_{H}

^{A}

^{#}introducing an element that facilitates achievement of the zero duration of mortgage funding system surplus condition. By mitigating the impact of interest changes on the mortgage funding system, this result demonstrates the benefits of T* < N mortgage contracts for preventing contagion from collapse of housing bubbles. This is apparent by comparing the results of Propositions 1 and 3.

**Corollary**

**1.**

## 4. House Prices, Prepayment and Mortgage Default

**Definition**

**2.**

_{H}[h,y], involves two stylized formulations of the household equity function associated with the T* < N and T = N mortgage funding systems. Theoretically the less complex case is for mortgages with T* < N, no prepayment and full recourse. In this case:

**Definition**

**3.**

_{H}

_{,0}is the market value of the residential house asset (other assets not included in this case) and L

_{H}

_{,0}the market value of the mortgage payment cash flows, both at time t = 0; and, M

_{i}is the fixed rate mortgage payment determined at t = i. To avoid presentation of complicated and unrevealing equilibrium conditions, it is helpful to make simplifying assumptions. Poitras and Zanotti (2016, p. 326) proceed by assuming PDUR

_{Hh}

^{L}= 0, i.e., that an instantaneous change in the house price growth rate will not change the value of mortgage liabilities held by households. While clearly inapplicable in the long run, this is a valid assumption for small time intervals. Using this assumption, the first order term in the Taylor series provides:

**Proposition**

**4.**

_{H}) condition:

_{H}= F

_{H}+ U

_{H}. This follows because:

_{Lh}

^{L}= 0, for much the same reason as the assumption PDUR

_{Hh}

^{L}= 0, Poitras and Zanotti (2016, p. 326) produce:

**Proposition**

**5.**

**Corollary**

**2.**

_{H}+ S

_{L}with the contribution from mortgage lender surplus being relatively small due to the smaller exposure arising from the use of shorter term to maturity mortgages. The implication is that, in a mortgage funding system with T* < N mortgage contracts, lower mortgage loan to house price value ratios at mortgage origination, yield maintenance prepayment penalties and the full recourse provision supported by substantial household non-housing assets are critical mortgage contract features for preventing potentially drastic consequences from collapse in a housing market bubble.

_{H}), the borrower prepayment option (PPO); and, for the residential house asset, the borrower mortgage default option (DO). The S

_{H}from the household balance sheet can now be decomposed as:

_{H}= (A

_{H}* + DO) − (L

_{H}* − PPO) = A

_{H}− L

_{H}

_{H}

^{*}and L

_{H}

^{*}are the market value of the residential house asset and the fixed rate annuity portion of the household mortgage liability, respectively. Even though DO is embedded in the mortgage contract (L

_{H}= L

_{H}

^{*}− PPO − DO), the no recourse provision makes the value of this option dependent on house price changes. Calculating the partial durations and convexities of these option payoffs is decidedly less tractable, though there are some helpful studies that address related issues, e.g., Sharp et al. (2008). Given this, Poitras and Zanotti (2016, p. 326) use the following:

**Definition**

**4.**

_{H}* → 0 (−1) as h → +∞ (−∞) with 0 < DO < L*, it is not possible to say that DO will be exercised when S

_{H}< 0 or L

_{H}* > A

_{H}*. No recourse default is triggered when mortgage payments cease prior to T and the residential property is surrendered for the unpaid balance on the mortgage, the payout from lenders to households at that date being: L

_{H}* − A

_{H}*. This value is undetermined at origination as the default decision date is determined strategically, depending on other factors, not limited to having S

_{H}< 0 or L

_{H}* > A

_{H}*. Similarly, ∂PPO/∂L

_{H}* → 1 (0) as y → 0 (+∞) with 0 < PPO < L

_{H}*. However, transaction costs associated with PPO exercise could delay exercise if there is an expectation that rates will continue to fall. Upon exercise, the yield maintenance payout from lenders to households is equal to loss of interest calculated as the annuity value of the difference in mortgage payments over the remaining amortization period. Because ∂DO/∂T > 0 and ∂PPO/∂T > 0, it follows:

**Corollary**

**3.**

**Proposition**

**6.**

_{H}

_{,0}* is the market value of the residential house asset (with default option value not included) and L

_{H}

_{,0}* is the market value of the mortgage payment cash flows (with prepayment option not included), both at time t = 0. These partial derivatives are then used to determine the zero duration of household surplus for the stylized T = N household equity function as:

**Corollary**

**4.**

_{H}= A

_{L}needs to be accurately priced. Such valuation is required to ensure S

_{L}is sufficient to offset the loss when PPO or DO is exercised. Unfortunately, obtaining an accurate contingency price at T = N mortgage contract origination is decidedly difficult, especially when viewed from the perspective of a future housing bubble forming and collapsing over the 30-year T = N horizon of the conventional US mortgage contract. In practice, market incentives associated with an originate-to-distribute method of mortgage issuance has the potential for PPO and DO being unpriced or, at best, grossly underpriced. In contrast, instead of attempting to embed a difficult-to-price prepayment option premium into the mortgage price at origination, yield maintenance penalties involve payment of the option premium upon exercise, with the premium equal to loss of interest on the remaining term to maturity. This leads to the following:

**Corollary**

**5.**

_{H}+ S

_{L}. Changes in mortgage lender surplus would be associated with duration gap exposure with house price changes impacting household surplus. In contrast, when the PPO contingency is included to insulate the household surplus, the burden of surplus changes associated with interest rate immunization falls on S = S

_{L}. In this case, the zero change of mortgage lender surplus condition becomes the zero change of mortgage funding system surplus for the stylized T = N mortgage contract.

**Proposition**

**7.**

_{L}= (A

_{L}* − DO − PPO) − L

_{L}. While the market value of mortgage lender liabilities increases when interest rates decrease, the associated gain in lender assets is strangled by exercise of the prepayment option and the gain is captured on household balance sheets. Conversely, when interest rates increase, the loss on lender assets will not be fully balanced by the reduction of lender liabilities if there is a duration gap. In addition, mortgage lenders are now exposed to downward movement in house prices that, if large enough, will trigger exercise of the DO default option. This potential future loss of lender surplus is not offset by gains from house price increases that are realized on the household balance sheet. Consequently, the avenue for sufficient surplus transfer from households to lenders is accurate mortgage pricing at origination to reflect the fair value of the embedded contingencies.

**Corollary**

**6.**

## 5. Conclusions: Housing Market Bubbles and Mortgage Contract Design

## Funding

## Conflicts of Interest

## References

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1 | In addition to theoretical studies on the definition and measurement of >housing market bubbles=, there is rapidly accumulating evidence on bubbles emerging in various countries. Included in studies on defining and measuring bubbles are: Shi (2017); Bono et al. (2017); Giglio et al. (2016); Phillips et al. (2015) and Himmelberg et al. (2005). Recent empirical evidence for housing market bubbles in various countries includes: Gomez-Gonzalez et al. (2018); Vogiazas and Alexiou (2017) and Engsted et al. (2016) for OECD countries; Shi (2017) for certain regional markets in the US; Dermani et al. (2016) for Sweden; Shi et al. (2016) for Australia; Shi and Kabir (2018) for New Zealand; Kim and Lim (2016) for Korea; Glaeser et al. (2017) for China; Huang and Shen (2017) for Hong Kong; Boelhouwer (2017) for Netherlands; and, Yip et al. (2017) for Malaysia. In contrast, Giglio et al. (2016) found no evidence of the violation of the transversality condition in the UK and Singapore housing markets indicating no support for the presence of a bubble. |

2 | The “no recourse” provision refers to the lender having ‘no recourse’ in the event of default against any other assets (or income) of the household than the housing asset securing the mortgage. In other words, the household borrower can ‘put’ the house back to the lender in exchange for the unpaid balance on the mortgage. A ‘full recourse’ provision allows the lender in the event of default to make a claim against other household assets and future income for the difference between the unpaid balance on the mortgage and the market price received when the housing asset is sold by the lender following default. |

3 | A typical Canadian mortgage contract offered by the chartered banks specifies a prepayment penalty that is loss of interest or 3 months payment, whichever is greater. However, the typical contract permits a once a year payment, without penalty, of 10% of the initial principal amount. Payments beyond 10% incur a penalty on the entire amount of the prepayment. The 10% penalty free prepayment amount cannot be carried forward to the next year. It is also possible to “double-up” on monthly payments, i.e., to make a payment that is twice the amount of a regular monthly payment with a corresponding reduction in remaining principle, subject to some restrictions, in addition to the 10% annual prepayment. |

4 | Though subject to less government regulation than in the US, there is regulation of mortgage lenders in Canada by the Office of the Superintendent of Financial Institutions (OSFI), an independent agency of the Government of Canada, created in 1987, reporting to the Minister of Finance. OSFI is responsible for supervising and regulating banks, insurers and some federally registered pension plans. In addition, the Canada Mortgage and Housing Corporation (CMHC) also plays a regulatory role as the provider of mortgage insurance and, since 2001, has been a conduit for Canada Mortgage Bonds issued through the CMHC-run entity the Canada Housing Trust. Through CMHC and OSFI, the federal government has authority over the maximum amortization period of mortgages qualifying for CMHC insurance. |

5 | This conclusion assumes that mortgage lender liabilities are not a completely floating rate where DUR _{L}^{L} = 0. Assuming for simplicity that DUR_{L}^{L} = DUR_{H}^{A} = 0, the mechanism of system surplus adjustment for the zero duration of surplus result in Proposition 1 follows: market value losses (gains) on mortgage lender assets due to interest rate increases (decreases) will be offset by gains (losses) on household liabilities, resulting in no change in the aggregate mortgage funding system equity from changes in mortgage market value. This result does not imply that the surpluses of households and lenders will both be unchanged. Rather, only that surplus gains for households (lenders) will be exactly equal to surplus losses for lenders (households) for an instantaneous increase (decrease) in interest rates. These gains and losses will offset over an interest rate cycle. If −DUR_{L}^{L} > 0 and DUR_{H}^{A} = 0, then lenders will have an offsetting gain (loss) of surplus from an interest rate increase (decrease) from the change in the market value of liabilities. This implies that for no recourse T = N mortgages without contingencies, interest rate decreases (increases) are riskier (less risky) from a system surplus perspective. This is because, when interest rates decrease, households are locked into long term mortgages with higher fixed rates with no offsetting gain from interest-bearing household assets. At the same time, the surplus gain to lenders having mortgage assets that pay higher rates is partially offset with a smaller increase in the value of liabilities. Aggregate mortgage funding surplus has fallen. When interest rates increase, households (lenders) have surplus gain (loss) from mortgage liabilities (assets) with lower fixed rates. Lenders have a smaller surplus gain from liabilities that are funded at lower rates with shorter term to maturity than the mortgage. Aggregate mortgage funding surplus has risen. |

6 | To see the importance of the prepayment option in undermining the simplified case, consider the inability of mortgage lender surplus to adequately deal with the ‘sub-prime’ bubble collapse. In the simplified case, the surplus available would have been supplemented by surplus gains associated with falling interest rates. However, exercise of the prepayment option allowed households to reduce the fixed rate on mortgages as interest rates fell significantly following collapse of the ‘dot.com’ bubble (the ‘Greenspan put’), eliminating much of the gain to mortgage lender surplus from falling rates. When interest rates rose significantly, this surplus was unavailable. In contrast, the prepayment option played an insignificant roll in the duration gap problem created by increasing interest rates that was a central issue in the collapse of the S&L industry in the US during the 1980s. In this case, the regime of persistently lower interest rates permitted by Regulation Q undermined the ability of lenders to accumulate surplus during a period of falling interest rates and, due to the bias toward savings deposits as S&L liabilities, there was little surplus gain from the reduction in the value of lender liabilities. |

7 | This solution requires the following results: $\frac{\partial {A}_{H}}{\partial y}=0\frac{\partial DO}{\partial h}=\frac{\partial DO}{\partial {A}_{H}}\frac{\partial {A}_{H}}{\partial h}=\frac{\partial DO}{\partial {A}_{H}^{*}}\frac{\partial {A}_{H,0}^{*}\left(1+h\right)}{\partial h}=\frac{\partial DO}{\partial {A}_{H}^{*}}{A}_{H,O}^{*}$. |

8 | While Canadian mortgage lenders would also be impacted by an interest rate increase, the loss of interest from a 1% move in interest rates on a 30 year term to maturity mortgage is about 3.5 times the loss from a 5 year term to maturity. |

9 | The Federal Home Loan Mortgage Corporation (2017, p. 98) reports that 78% of mortgage purchased were 30-year fixed rate compared to 75% in 2015 and 2014. In 2016 a further 16% were 15-year fixed rate and 4% were 20-year fixed rate. By comparison, in Canada less than 1% of fixed rate mortgages have terms to maturity greater than 10 years with about half of all mortgages having a 4–5-year term to maturity. |

Prepayment Penalty | Default Recourse | Main Funding Source | |

US Contract | No Penalty: Not Assumable | Depends on State Usually No Recourse | Government Sponsored Enterprises (GNMA, FNMA, Freddie Mac) |

Canadian Contract | Loss of Interest: Assumable | Full Recourse | Depository Institutions |

Benchmark Term | Benchmark Amortization | Floating Rate Benchmark | |

US Contract | 30 Year | 30 Year | ARM: Fixed Reset Date |

Canadian Contract | 5 Year | 25 Year | Variable Rate: Prime Rate Reset |

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## Share and Cite

**MDPI and ACS Style**

Poitras, G.; Zanotti, G. Housing Market Bubbles and Mortgage Contract Design: Implications for Mortgage Lenders and Households. *J. Risk Financial Manag.* **2018**, *11*, 42.
https://doi.org/10.3390/jrfm11030042

**AMA Style**

Poitras G, Zanotti G. Housing Market Bubbles and Mortgage Contract Design: Implications for Mortgage Lenders and Households. *Journal of Risk and Financial Management*. 2018; 11(3):42.
https://doi.org/10.3390/jrfm11030042

**Chicago/Turabian Style**

Poitras, Geoffrey, and Giovanna Zanotti. 2018. "Housing Market Bubbles and Mortgage Contract Design: Implications for Mortgage Lenders and Households" *Journal of Risk and Financial Management* 11, no. 3: 42.
https://doi.org/10.3390/jrfm11030042