Download Life Settlement Pricing

Patrick Brockett
University of Texas at
Austin
Patrick
Brockett
The University of
Texas at Austin
Joint work with Shuo-Li Chuang, Yinglu Deng
and Richard MacMinn
• Background
• What is longevity risk
• How important is it
• Where are we and where are we going
• Modeling Longevity Risk
• Capital Market Solutions to Individual
Longevity Risk
– Annuities
– Life Settlements
• Conclusions
What is Longevity Risk?
Life expectancy at birth 1900 to 2000
85
79
80
74
75
70
Age
65
women
60
55
50
45
40
1900
Source: U.S. Vital Statistics, 2004
1920
1940
1960
1980
2000
Age 65
Longevity risk is a real, underestimated,
uncertain and slow-burning risk
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Example mortality series : USA Data
Figure 1900-2004 Historical Mortality Rates
Thank you
Where Did The Mortality Improvements Come From?
.
Time period
% annual
improvement
% of improvement
by age
Reason
1900 to 1940
1
80% in those under
age 45
Better nutrition and public
health measures
1940 to 1960
2
Even across ages
Declining infectious diseases,
vaccines and antibiotics
1 to 1.5
65% in those over
age 45
Reduction in Cardio Vascular
events and decline in preemie
deaths
? Greatest at oldest
ages
Forces ↑ LE: Health Care
Genetics
Forces ↓ LE: Allocation of
Resources
Obesity Epidemic
1960 to present
2000 forward
??
Cutler and Meara, Sept 2001, Changes in the Age Distribution of Mortality Over the 20 th Century
We Care About This Topic Because
Longevity Risk is a Key Factor in Planning
on How to Translate Assets into Lifetime
Income Security
•
For individuals, governments and
corporations
E.g., for individuals, for example:
•
–
–
Conditional on Achieving Age 65, Probability of
Dying by 75 is 0.254 for Men, 0.189 for Women
Conditional on Age 65, Probability of Living to 90
is 0.181 for Men, 0.275 for Women
How do we fund this?
Dependency Ratio: The workers in the
Middle Pay for the Young and Old
Old Age Dependency: The Old get Very
Old
Also, Individual’s Mortality Perceptions
Are Biased (so planning is bad)
• Hurd / McGarry Compare Subjective Mortality
Probabilities in Health and Retirement Survey
with Actual Mortality Rates
• Men: In Survey, Average Estimated Survival
Probability to Age 75: 0.622, but “Actual” from
Mortality Table: 0.594. Similarly, for Women
These Numbers Are: 0.663 and 0.746.
• Survival to 85: Subjective from survey: 0.388
for Men (0.242 “actual”), 0.460 (0.438) for
Women
Cost of longevity risk to society
• Global pension liabilities = $23trn
• Roger Lowenstein* in While America Aged (2008)
discusses “how pension debts ruined General Motors,
stopped the New York subways, bankrupted San Diego,
and loom as the next financial crisis”
•
Author of When Genius Failed
• The impact of the longevity risk
– In the US, using up-to-date mortality tables increases
pension payments liability 8% for a male born in 1950
– In the U.K., true accounting for longevity doubles the
aggregate pension deficit from £46 billion to £100
billion of FTSE100 corporation pensions
– In the U.S., the new mortality assumptions for pension
contributions, increase pension liabilities by 5-10%
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So What is an individual to do?
• Buy annuities
• If you have real property: consider reverse
mortgage on the property
• If you have life insurance, consider selling it on
a secondary market known as the life
settlement market
• Individuals buying annuities is a useful risk
management strategy, but people just don’t do it
–
–
–
–
Fear of loss of control/trust
Bequest Motive
Uncertain future needs
Decision regret
• In the UK, people are forced by governmental
mandate to annuitize 75% of their pension funds
at retirement
– Still only a partial solution
– US (and other parts of the world) would have trouble
enforcing such a mandate
Decision Regret
Modeling Mortality
• Necessary for assessing future liabilities
• Long (but recent history)
• Pricing financial instruments that incorporate
longevity risk requires looking forward
mortality tables and models
Life Settlement Description
• A life settlement is a financial arrangement whereby the third party
(or investor) purchases a life insurance policy from the person who
originally purchased a life insurance policy.
• This third party pays the insured an amount greater than the cash
surrender value of the policy -- in effect, the trade-in value of the
policy as determined by the originating insurance company-- but
less than the face value (or the death benefit).
• It can be a win-win situation, as the investor can obtain a return on
their initial investment and premium payments once the death
benefit becomes payable and the owner of the policy obtains more
money than they otherwise could obtain by surrendering their
policy.
• It is estimated that in the past five years more than $40 billion of
face value has been sold in the life settlement market and the
market size will grow from $13 billion in 2004 to $161 billion over
the next few decades
Brief History
• Longevity risk traditionally viewed though its
impact on pensions, social security systems and
corporate defined benefit plan solvency, but the
life settlement market for individuals is even
more vulnerable to longevity risk.
• Illustrative of this is what happened to the viatical
settlement market, the precursor of life
settlement market.
• AIDS patients sold their life policies. Profitable to
investors, until 1996 when papers were
presented at the International AIDS Conference in
Vancouver, that gave evidence of a new drug
capable of substantially reducing, perhaps even
to zero, the level of HIV in its infectees.
Brief History
• The effect was evident in the collapsed value of viatical
settlement firms, and significant decrease in prices offered to
AIDS victims for insurance policies since it now might take
substantially longer to mature, their value plummeted.
• As market for viatical settlements plunged, companies, looked
for alternatives. They expanded life insurance purchases to
the elderly. Elderly with estimated low life expectancies
chosen because a low life expectancy meant a greater
possibility of profiting early. Today, this life settlement market
has increasing potential as baby boomers are enter old age
• To create distance between the association with AIDS and the
negative connotation that the term “viatical settlement” had,
companies chose the different title, “life settlement.”
Life Settlement Pricing Factors
• A life settlement can depend on the following
characteristics of the policy being settled:
• 1. The insurance carrier,
• 2. The face value (or benefit),
• 3. The age of the insured person,
• 4. The gender of the insured person,
• 5. The premium,
• 6. The issue date,
• 7. The estimated life expectancy of the insured,
• 8. The primary diagnosis of the insured’s illnesses if the
insured is in impaired health
• 9. The bidding or asking price of the policyholder.
Life settlement Pricing
• The main factor in the life settlement securities pricing
currently is the estimation of the life expectancy of the
insured (and the premium payments), but other
information may also be available.
• The life expectancy of the insured at the time of sale
(settlement) is often considered in the pricing as the
major random variable which influences the sales price
to the insured when he sells his life insurance policy to
the third party as a life settlement.
Life Settlement Structure
T
Life settlement Pricing
• A common method of pricing at the beginning
was “deterministic” pricing. One treats the
contract as one where the person lives to their
life expectancy, and then dies
– Viewed like a bond with a principal repayment
equal to the death benefit paid at the time of life
expectancy
– Negative coupons of size equal to the premiums
payable up to the time of life expectancy
Problem with deterministic pricing
Jensen’s inequality says for any convex function f and
any random variable Z, E[f(Z)] ≥ f(E[Z]).
Thus, since the discount function vT is convex in T,
according to Jensen’s inequality, using the expected life
time E[T] only and pricing the product like one would an
E[T]-year bond with a pay off of vE[T] , v being the
discount rate, results in incorrect pricing. This price vE[T]
is always smaller than the value of the E[vT] which is the
true expected net present value. Thus treating the life
settlement as a bond of duration E[T] is underprices the
value of the payoff and is not fair to the policyholder
Pricing Issues
This carries over into the time zero price of the life
settlement. If P is the premium to be paid at the
beginning of each year then the buyer pays an amount
equal to
1  vT
P[
iv
]
The time zero value of the life settlement, X is a convex
function of T, the future lifetime since vT is convex and
Again according to Jensen’s inequality, using the expected
life time E[T] only and pricing the product as X(E[T]) instead
of E[X(T)] results in systematically underpaying the insured.
Pricing Issues
• Moral: We should use the entire probability distribution for
pricing the life settlement, not just the expected lifetime of the
insured.
• Problem: While E[T] can be assessed by a medical underwriter,
usually the probability distribution of T is not known, cannot be
assessed effectively by the life underwriter, and reasonable
candidate distributions (e.g., impaired life mortality tables) for T will
not have the specified value of E[T] consistent with the medical
underwriter’s assessment.
• We may also have other information (e.g., variance, or survival
probability for a specified period, or relative mortality risk statistics
from the medical literature), and this should be incorporated
• The medical underwriter may be specifying median instead of
mean, or likelihood of death within a window of years may be
specified
Approaches to Modeling
• We want to use a “standard” or pre-specified
distribution (or life table) for T, but these will
not be consistent with medical underwriter.
• How do we adjust a standard mortality table
to obtain a new mortality table for the
individual that can reflect (be consistent with)
this known information?
An Information Theoretic Approach
• The statistical methodology we propose is based
upon information theory for adjusting mortality
tables to obtain exactly some known individual
characteristics, while obtaining a table that is as close
as possible to a standard one.
• In this way, the method provides more accurate
projection and evaluation for the life settlement
products, through incorporating more statistical
information of the insured’s future life time. One can
then price life settlements using the whole life
distribution rather than just life expectations
Information Theory Approach to Getting a
Mortality Model for Pricing
For distinguishing between two densities on the basis of an observation t, a
sufficient statistic is the log odds ratio in favor of the observation having come from
in favor of g. It is the amount of information contained in a observation t for
discriminating in favor of f over g. In a long sequence of observations from , the
long-run average or expected log odds ratio in favor of f is I(f|g) =
This reflects the expected amount of information for discriminating between f and g.
Note that I(f|g) ≧0 and = 0 if and only if f=g . Thus, the size of is a measure of the
closeness of the densities f and g.
For a given g, one can minimize I(f|g) over f to find the closest f. If we have
constraints, we can do a constrained optimization. E.g., if the mean is give as m,
then we have constraints:
Information Theory
• To phrase the problem mathematically, we desire to find
a vector of probabilities that solves the problem:
• If we have constraints E[ai(T)] = qi, i=1, …, k, then
Brockett, Charnes and Cooper (1980) show that the
problem has a unique solution, which is:
• The parameters can be obtained easily as the dual
variables in an unconstrained convex programming
problem:
Information Theoretic life
Settlement Adjustment
• In particular, if the mean life expectancy is
give as m, and a standard mortality table is
given with probability of death at exact year k
given as gk, then the problem becomes to find
the adjusted mortality table probabilities fk,
that satisfy
• Minimize I(f|g) over all possible f subject to
Solution
The solution is:
fk = gkexp(b0+b1k)
where the b’s are selected so the two constraints
are satisfied.
This gives the best fitting (least distinguishable
from the standard) mortality table that satisfies
the underwriter’s expectation. We can use this
for pricing.
Example (Life Policy from State Farm)
Female bought policy at age 40 in 1986, is being settled in 2006 at age 70
Medical underwriter says life expectancy is 8.5 years, but 2001 CSO table
on which the policy is based says life expectancy is 16.4 years
Even using a disabled life table without
adjustment underprices
Pricing
An Example
Present Value of Cash Flows
(excluding initial purchase price)
0.02
0.01
0.00
density
0.03
0.04
5%
10%
15%
20%
0
10000
20000
30000
realized values
40000
50000
How much difference does the standard table make?
Different Standards
Other types of information can be
incorporated – e.g., relative risk
Consider a person who has experienced a life event that has changed
their medical information. For example a spinal cord injury . We know
relative mortality impact. How do we adjust life table?
• We assume the level of injury (ip,cp,iq,cq)
influences mortality rate relative to standard
by increasing additively, e.g for incomplete
paraplegics, mortality is = mx+mip. The question
is how to estimate such that the adjusted
mortality table has the 7 year mortality ratios
as given in the previous table.
We also need the number of exposure units Ex in each age group (to get
average age at death)constraint
Getting a new adjusted mortality Table
Age at death constraint:
same for observed and
predicted
Solving these we obtain
(3)
Conclusion
• We have shown how to price life settlements
and how to assess the risk by Monte Carlo
simulations
• In adjusting a standard mortality table to
reflect known information, often the pricing
result is not very sensitive to the original
standard mortality table, but is sensitive to
the discount rate
• Other information can be incorporated as well
Thank you !
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