Using Reverse Mortgages to Hedge Longevity and Financial Risks for Life

Insurers: A Generalized Immunization Approach

---Jennifer L. Wang, Ming-hua Hsieh and Yu-fen Chiu

Fin 500R: Topics in Quantitative Finance

Tiana Li9/30/2015

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

What is Longevity Risk

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.

What is Reverse Mortgages

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

Types of Reverse Mortgages

This paper assumes that an insurance company sells three kind of products:

Life Insurance

Annuities

Reverse Mortgage

What Products Insurance Company Sells

Life Insurance -> Mortality Risk, Interest Risk

Annuities -> Longevity Risk, Interest Risk

Reverse Mortgage ->

What risks Insurance Company Sells

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

Research Models

Stochastic Mortality Model

---Cairns-Blake-Dowd (CBD) Model

q(t,x)

Where q(t,x) is the realized mortality rate for age x

insured from time t to t+1

Research Models

Stochastic Mortality Model

---Cairns-Blake-Dowd (CBD) Model

Set , where the two stochastic trends and follow a

discretized diffusion process with a drift parameter

μ and a diffusion parameter C:

Research Models

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

Research Models

Stochastic Mortality Model

---Cairns-Blake-Dowd (CBD) Model

Research Models

Figure 1 Figure 2

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

Research Models

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.

Research Models

?

Stochastic Interest Rate Model

---CIR Model

It is a mean reversion process that the short rate is

reverting to the long-run value b.

Research Models

Stochastic Interest Rate Model

---CIR Model

Research Models

a Speed of mean reversion0.15

b Long-run mean of short rate 0.05

Volatility 0.06

r 。 Initial short rate0.01Table 1 Parameters of the CIR model used in numerical example

Stochastic House Price Index Dynamic Model

---Geometric Brownian Motion

Where the parameter μ is the expected rate of

return and the parameter σ is the volatility of the

house price.

Research Models

Stochastic House Price Index Dynamic Model

---Geometric Brownian Motion

Research Models

μ Speed of mean reversion0.15

Volatility 0.06

Table 2 Parameters of the House Price Index Dynamic used in numerical example

The Proposed Generalized Immunization

Approach

Research Models

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

The Proposed Generalized Immunization

Approach

Research Models

Through Taylor’s formula, the change in V is approximated as++

Where V(q,r,s) is the initial value, and is the V after shocks

The Proposed Generalized Immunization

Approach

Research Models

It is easy to see that the first-order and second-order partial derivatives of V are linear functions of the ’s, , =, =, ==, =, =

The Proposed Generalized Immunization

Approach

Research Models

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

The Proposed Generalized Immunization

Approach

Research Models

In the numerical examples, , , , are very small for all products, they are not set to be zero for immunization.

Numerical ExamplesProduct or hedging asset

Age Coverage Sum insured

Coupon Maturity Face Value

Whole-Life Annuity 60 Whole Life

10,000 - - -

Term-Life Insurance 50 20 years 1,000,000 - - -

Whole-Life Insurance 50 Whole Life

1,000,000 - - -

Bond 1 - - - 3% 10 years 1,000,000

Bond 2 - - - 5% 30 years 1,000,000

Reverse Mortgage 70 Whole Life

1,000,000 - - -

Table 3 Basic Assumptions for the Numerical Analysis

Numerical Examples

Table 3 Basic Assumptions for the Numerical Analysis

Product portfolio

Hedging Strategy 1 Hedged annuity by term-life and long-term bonds

Hedging Strategy 2 Hedged annuity by whole-life and long-term bonds

Hedging Strategy 3 Hedged annuity by reverse mortgage and long-term bonds

Hedging Strategy 4 Hedged annuity by reverse mortgage and long-term bonds with the house price being hedged by put option

Hedging Strategy 5 Hedged annuity by reverse mortgage and long-term bonds with the house price being perfectly hedged

The Proposed Generalized Immunization

Approach

Research Models

In the numerical examples, , , , are very small for all products, they are not set to be zero for immunization.

Numerical Examples

Table 4 Partial derivatives computed by finite difference method

Annuity

Term Life

Whole Life

Bond 1 Bond 2 Reverse Mortgages

V -21.8 -15.1 -33.4 98.8 120.2 75.8

△V/△r 38 25.5 66.4 -163.9 -218 -0.1

△V/△q -0.6 22.1 54.6 0 0 19.7

V/△ -76.1 -47.1 -138.3 285.3 431.7 -1.3

Numerical Examples

Table 5 Portfolio mix without any Hedging Strategy ( x 10,000)

Annuity Cash

Unit Price -21.8

Holding Amount 1,000

Total Value of Each Product -21,800 21,800

Numerical Examples

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

Numerical Examples

Table 6 Portfolio mix for Hedging Strategy 1 ( x 10,000)

Annuity Term Life Bond1 Bond 2 Cash

Unit Price -21.8 -15.1 98.8 120.2

Holding Amount

1000 26.1 25.1 162.5

Total Value of Each Product

-21,800 -394.1 2,480 19,532.5 181.7

Numerical Examples

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

Numerical Examples

Table 6 Portfolio mix for Hedging Strategy 2 ( x 10,000)

Annuity Whole-Life

Bond1 Bond 2 Cash

Unit Price -21.8 -33.4 98.8 120.2

Holding Amount

1000 10.6 21 165.8

Total Value of Each Product

-21,800 -354 2,075 19,929.2 150

Numerical Examples

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

Numerical Examples

Table 6 Portfolio mix for Hedging Strategy 3 ( x 10,000)

Annuity Reverse Mortgages

Bond1 Bond 2 Cash

Unit Price -21.8 75.8 98.8 120.2

Holding Amount

1000 30 20 163

Total Value of Each Product

-21,800 2,274 1,976 19,560 -2,010

Numerical Examples

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

Numerical Examples

Table 6 Portfolio mix for Hedging Strategy 4 ( x 10,000)

Annuity Reverse Mortgages

Bond1 Bond 2 Put Option

Cash

Unit Price -21.8 75.8 98.8 120.2 0.9

Holding Amount

1000 29.5 19.8 163.7 61

Total Value of Each Product

-21,800 2,236 1,958 19,649 55 -2,098

Numerical Examples

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

Conclusion

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.

Thank you!