# Multivariate Risk-Neutral Pricing of Reverse Mortgages under the Bayesian Framework

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## Abstract

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## 1. Introduction

## 2. Pricing Mechanism

## 3. Bayesian Modelling

## 4. Analysis of Modelling Results

_{0}= 550,00011. We consider an individual aged 65 or 75 entering into a reverse mortgage contract with a loan-to-value (LTV) ratio of 10%, 20%, …, or 100%, i.e., L

_{0}= 55,000, 110,000, …, or 550,000, and l = 1%, 2%, …, or 10% p.a12. These are a total of 200 cases in which we examine the calculated market prices of the reverse mortgage based on the different terms being offered.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**Sample autocorrelations between successive Markov chain Monte Carlo (MCMC) samples of future survival probabilities, house price growth rates and interest rates.

## References

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1 | The average balance in superannuation of those Australians who were aged 60 to 64 was merely AUD$214,897 in 2015–2016, but the amount required to achieve a “comfortable retirement” is estimated to be AUD$640,000 for couples and AUD$545,000 for singles (https://www.superannuation.asn.au/). |

2 | The reverse mortgage industry revenue in Australia has grown by about 0.9% p.a. in the last five years (https://www.ibisworld.com.au/). |

3 | For the numerical results in Section 4, we have empirically verified the minimisation of the information criterion by testing several other randomly picked values of risk-neutral probabilities and checking whether the numerically optimised values do represent the minimum. Another theoretical way to verify the minimisation is to compute the bordered Hessian matrix and apply the second derivative test for an extrema of a Lagrange expression. However, this approach is impractical for our analysis as the dimension of the matrix is too large. |

4 | For convenience, we assume that the repayment is settled at the end of the year of death. |

5 | We assume that the reverse mortgage portfolio is very large and so sampling error due to individual uncertainty (non-systematic longevity risk) is negligible. |

6 | In Bayesian modelling, all the unknown parameters are treated as random variables, and so in theory there is no identifiability issue and parameter constraints are not needed. But we realise that our simulation algorithm converges much more slowly if there is no constraint. Hence, we set the two constraints ${\sum}_{x}{\beta}_{x}}=1$ and k _{0} = c, along the line of the initial Lee–Carter model. |

7 | The variances ${\sigma}_{\alpha}^{2}$ and ${\sigma}_{\beta}^{2}$ are set as the sample variances of the estimated ${\alpha}_{x}$ and ${\beta}_{x}$ over age times 10; $a$ is set to be 2.1; $b$ is set to be 1.1 times the sample variance of the estimated ${\kappa}_{t}-{\kappa}_{t-1}$ over time; ${\mu}_{0}$ and ${\sigma}_{\mu}^{2}$ are computed from the sample mean and its standard error of the estimated ${\kappa}_{t}-{\kappa}_{t-1}$. These values are largely chosen to represent vague prior knowledge. |

8 | Due to the existence of autocorrelations and cross-correlations, the widely used assumption of the geometric Brownian motion is not suitable here. |

9 | The inverse covariance matrix of $\varphi $’s is assumed as Wishart with a degree of freedom of 4$p$ + 4. The inverse covariance matrix of $({\nu}_{t},{\omega}_{t}{)}^{\prime}$ is assumed as Wishart with a degree of freedom of 4. The small degrees of freedom are selected to represent vague prior knowledge. |

10 | It is now called the Actuaries Institute and the website is www.actuaries.asn.au. |

11 | The national median home price in Australia is AUD$552,141 as of 31 August 2018. (https://edge.alluremedia.com.au/uploads/businessinsider/2018/09/CoreLogic-home-value-index-August-2018-table.jpg). |

12 | The analysis can readily be extended to a floating loan interest rate, which can be set as the risk-free rate plus a spread. Accordingly, there would be some offsetting between the loan interest rate and the discounting. There may then be less interest rate risk and the calculated reverse mortgage price may be lower. We leave this interesting option for future research. |

**Figure 2.**Bayesian Lee–Carter mortality index for 1975 to 2014 (solid line) and projected values with 95% prediction intervals for 2015 to 2050 (dashed and dotted lines).

**Figure 3.**Simulated density functions of future survival probabilities of the cohort aged 65 (at the start of 2019) for 10, 20 and 30 years.

**Figure 4.**Australian quarterly residential property price index values (weighted average of Sydney, Melbourne, Brisbane, Adelaide, Perth, Hobart, Darwin and Canberra) and growth rates, and 3-month bank accepted bills yields, from September 2003 to June 2018.

**Figure 5.**Sample autocorrelations and cross-correlations of house price growth rates and interest rates.

**Figure 7.**Simulated density functions of future house price growth rates and interest rates after 10, 20 and 30 years (from the start of 2019).

**Table 1.**Expected present values of the reverse mortgage under different loan interest rates and loan-to-value (LTV) ratios for a homeowner aged 65.

**Table 2.**Expected present values of the reverse mortgage under different loan interest rates and LTV ratios for a homeowner aged 75.

Bank | Interest Rate (p.a.) |
---|---|

Bankwest | 6.42% |

Commonwealth Bank | 6.52% |

Heartland Seniors Finance | 6.54% |

Heritage Bank | 6.29% |

P&N Bank | 6.24% |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, J.; Kogure, A.; Liu, J.
Multivariate Risk-Neutral Pricing of Reverse Mortgages under the Bayesian Framework. *Risks* **2019**, *7*, 11.
https://doi.org/10.3390/risks7010011

**AMA Style**

Li J, Kogure A, Liu J.
Multivariate Risk-Neutral Pricing of Reverse Mortgages under the Bayesian Framework. *Risks*. 2019; 7(1):11.
https://doi.org/10.3390/risks7010011

**Chicago/Turabian Style**

Li, Jackie, Atsuyuki Kogure, and Jia Liu.
2019. "Multivariate Risk-Neutral Pricing of Reverse Mortgages under the Bayesian Framework" *Risks* 7, no. 1: 11.
https://doi.org/10.3390/risks7010011