Fin 500R: Topics in Quantitative Finance Tiana Li 9/30/2015

A longevity risk is any potential risk attached to the increasing life expectancy of pensioners and policy holders, which can eventually translate in higher than expected pay-out-ratios for many pension funds and insurance companies

Designed to enable elderly homeowner to consume some of the home equity but still maintain the ownership and residence of the home. The lender advances a lump sum or periodic payments to elderly homeowners. The loan accrues with interest and is settled using the sale proceeds of the property when borrowers die, sell or vacate their homes to live elsewhere.

 Lump-sum payment: the borrower receives a fixed amount of entire available principal limit at closing of the loan  Tenure payments : equal monthly payments are made as long as the borrower lives  Term payments: equal monthly payments are made for a fixed period of months selected by the borrower

This paper assumes that an insurance company sells three kind of products:  Life Insurance  Annuities  Reverse Mortgage

 Life Insurance -> Mortality Risk, Interest Risk  Annuities -> Longevity Risk, Interest Risk  Reverse Mortgage -> Longevity Risk Housing Pricing Risk Borrower Maintenance Risk Interest Risk

 Stochastic Mortality Model  Stochastic Interest Rate Model  The House Price Index Dynamic Model  The Proposed Generalized Immunization Approach

 Stochastic Mortality Model ---Cairns-Blake-Dowd (CBD) Model Annual Population Data Source: Surveillance, Epidemiology and End Results (SEER) programmed Annual Death Data Source: The Centers for Disease Control and Prevention and the National Center for Health Statistics

 Stochastic Mortality Model ---Cairns-Blake-Dowd (CBD) Model

 Stochastic Mortality Model ---Cairns-Blake-Dowd (CBD) Model Figure 1 shows that A1(t) is generally declining over time, and it corresponds to the characteristic that mortality rates exhibit improvement effects for all ages

 Stochastic Mortality Model ---Cairns-Blake-Dowd (CBD) Model Figure 2 shows that A2(t) is generating increasing over time. This suggests that the mortality improvements are more significant at lower ages than higher ages. ?

 Stochastic Interest Rate Model ---CIR Model aSpeed of mean reversion0.15 bLong-run mean of short rate0.05 r 。 Initial short rate0.01 Table 1 Parameters of the CIR model used in numerical example

 Stochastic House Price Index Dynamic Model ---Geometric Brownian Motion μSpeed of mean reversion0.15 Table 2 Parameters of the House Price Index Dynamic used in numerical example

Assume V is the value of the product portfolio and hedging assets

 The Proposed Generalized Immunization Approach Through Taylor’s formula, the change in V is approximated as

 The Proposed Generalized Immunization Approach

From the equations in last slice, by choosing a suitable W, we can make

 The Proposed Generalized Immunization Approach

Product or hedging assetAgeCoverageSum insuredCouponMaturityFace Value Whole-Life Annuity60Whole Life10, Term-Life Insurance5020 years1,000, Whole-Life Insurance50Whole Life1,000, Bond 1---3%10 years1,000,000 Bond 2---5%30 years1,000,000 Reverse Mortgage70Whole Life1,000, Table 3 Basic Assumptions for the Numerical Analysis

Product portfolio Hedging Strategy 1Hedged annuity by term-life and long-term bonds Hedging Strategy 2Hedged annuity by whole-life and long-term bonds Hedging Strategy 3Hedged annuity by reverse mortgage and long-term bonds Hedging Strategy 4Hedged annuity by reverse mortgage and long-term bonds with the house price being hedged by put option Hedging Strategy 5Hedged annuity by reverse mortgage and long-term bonds with the house price being perfectly hedged

 The Proposed Generalized Immunization Approach

Table 4 Partial derivatives computed by finite difference method AnnuityTerm Life Whole Life Bond 1Bond 2Reverse Mortgages V △V/△r △V/△q

Table 5 Portfolio mix without any Hedging Strategy ( x 10,000) AnnuityCash Unit Price-21.8 Holding Amount1,000 Total Value of Each Product-21,80021,800

Figure 3 Surplus distribution without any Hedging Strategy ( x 10,000)

Table 6 Portfolio mix for Hedging Strategy 1 ( x 10,000) AnnuityTerm LifeBond1Bond 2Cash Unit Price Holding Amount Total Value of Each Product -21, ,48019,

Figure 4 Surplus distribution for Hedging Strategy 1 ( x 10,000)

Table 6 Portfolio mix for Hedging Strategy 2 ( x 10,000) AnnuityWhole- Life Bond1Bond 2Cash Unit Price Holding Amount Total Value of Each Product -21, ,07519,

Figure 5 Surplus distribution for Hedging Strategy 2 ( x 10,000)

Table 6 Portfolio mix for Hedging Strategy 3 ( x 10,000) AnnuityReverse Mortgages Bond1Bond 2Cash Unit Price Holding Amount Total Value of Each Product -21,8002,2741,97619,560-2,010

Figure 6 Surplus distribution for Hedging Strategy 3 ( x 10,000)

Table 6 Portfolio mix for Hedging Strategy 4 ( x 10,000) AnnuityReverse Mortgages Bond1Bond 2Put Option Cash Unit Price Holding Amount Total Value of Each Product -21,8002,2361,95819, ,098

Figure 7 Surplus distribution for Hedging Strategy 5 ( x 10,000)

 The proposed generalized immunization approach can serve as an effective vehicle in controlling the aggregate risk of life insurance companies  Adding reverse mortgages to the product portfolio creates a better hedging effect and effectively reduces the total risk associated with the surplus of the life insurers.