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!
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