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Insurance Abstract
Survival risk insurance is a method of transferring the financial
consequences associated with the risk that deaths occurring within
a specified period of time in a selected group of insured lives
will be less in number than the expected number of deaths or less
in amount than the expected amount of death benefits paid. More
particularly, one entity, the Coverage Recipient, can transfer a
financial risk that the actual number of deaths or the actual amount
of death benefits paid during a specified period relative to a selected
group of insured lives will be less than the expected deaths or
the expected amount of death benefits paid to another entity, the
Coverage Provider, for the payment of an appropriate premium based
on the method of this invention.
Insurance Claims
I claim:
1. A method for a Coverage Provider to provide survival risk insurance
to a Coverage Recipient, said method comprising the steps of: a)
selecting a group of insured lives such that: i. each one of said
insured lives is covered by an original life insurance policy; ii.
each one of said original life insurance policies is provided by
one or more original life insurance companies; iii. each one of
said original life insurance policies pays a death benefit, DB.sup.m,
to said Coverage Recipient upon the death of one of said insured
lives m; and iv. said insured lives belong to a given mortality
class as of a beginning date; b) calculating a single premium S1
for said survival risk insurance wherein said single premium S1
is greater than or equal to SPR, wherein 13 SPR = All Policies SPR
m and wherein: 14 SPR m = ( PV1 - PV3 ) + F + G 100 DB m + E + H
100 Y 100 DB m and wherein: 15 PV1 = v N Y 100 DB m and wherein:
16 PV3 = [ k = 0 - x - 1 v k + 1 p x k q x + k T1 + p x - x v -
x T2 ] i ln ( 1 + i ) DB m and wherein: i is an annual effective
interest rate representing the cost of use of money of the Coverage
Provider; F is a default risk attributable to said one or more original
life insurance companies issuing said original life insurance policies,
said F being a percentage of said death benefit DB.sup.m of insured
m; G is a default risk attributable to the Coverage Recipient wherein
said G is a percentage of the death benefit of insured m; T1 is
an adjustment factor to reflect any tax attributable to a death
benefit receivable by the Coverage Provider; T2 is an adjustment
factor to reflect tax attributable to any endowment receivable by
the Coverage Provider; N is the length of time between said beginning
date and said end date, wherein N has the units of years; E is an
expense factor set by said Coverage Provider wherein E is expressed
as a percentage of a death benefit DB.sup.m; H is a factor for desired
profit set by said Coverage Provider, wherein H is expressed as
a percentage of a death benefit DB.sup.m; Y is a percentage applied
to a death benefit DB.sup.m; .omega. is a maturity age of an original
life insurance policy; x is the age of said one of said insureds
m as of said beginning date; .sub.kp.sub.x is the probability that
a given one of said insureds will survive from said beginning date
for k years; .omega.-xp.sub.x is the probability that said one of
said insureds will survive from said beginning date to said maturity
age, .omega.; q.sub.x+k is the probability that said one of said
insureds will die between the beginning of year k and the end of
year k; and k is a number of years past said beginning date; .nu.
is an annual discount rate equal to 1/(1+i); and said step of calculating
said single premium is performed on a computer; and c) committing
said Coverage Provider to pay said Coverage Recipient a first benefit
B1 for said survival risk insurance wherein; 17 B1 = Y * All Policies
DB m and wherein said payment of said first benefit B1 is to be
due on said end date; d) committing said Coverage Recipient to pay
a set of premiums to said Coverage Provider in exchange for said
first benefit B1 wherein said set of premiums has a present value
as of said beginning date equal to said single premium S1.
2. The method of claim 1 wherein the set of premiums is one premium.
3. The method of claim 1 wherein said set of premiums comprises
annual premiums payable for a premium paying period.
4. The method of claim 1 wherein said end date is on or before
the end of the term of a loan, wherein said loan is from said Coverage
Recipient to at least one of said insured lives.
5. The method of claim 1 wherein said end date is chosen such that
the probability of death of said insureds as of said end date is
greater than or equal to 0.75.
6. The method of claim 1 wherein said first benefit B1 paid by
said Coverage Provider to said Coverage Recipient as a loan.
7. The method of claim 6 wherein said single premium S1 includes
a charge for interest on said loan.
8. The method of claim 1 wherein said single premium is first calculated
before said beginning date and then recalculated at least once after
said beginning date.
9. The method of claim 1 wherein at least one of said insured lives
is impaired.
10. The method of claim 1 which further comprises the step of said
Coverage Provider accepting the transfer of ownership of said original
life insurance policies from said Recipient such that said Coverage
Provider receives the death benefits DBm of said original life insurance
policies instead of said Recipient.
11. The method of claim 1 wherein said step of committing said
Coverage Provider to pay said Coverage Recipient said first benefit
B1 comprises said Coverage Provider signing a contract, said contract
stating at least a portion of the terms of said survival risk insurance.
12. The method of claim 1 wherein said step of committing said
Coverage Recipient to pay said set of premiums to said Coverage
Provider comprises said Coverage Provider signing a contract, said
contract stating at least a portion of the terms of said survival
risk insurance, and wherein said Coverage Recipient has made a conditional
commitment to abide by said contract if said Coverage Provider signs
said contract.
13. The method of claim 1 wherein said step of committing said
Coverage Recipient to pay said set of premiums to said Coverage
Provider comprises said Coverage Provider accepting the actual payment
of at least a portion of said set of premiums.
14. The method of claim 1 wherein said single premium S1 is greater
than or equal to said SPR plus an amount required to keep said original
life insurance policies in force from said beginning date to the
date when all of said original life insurance policies either pay
their respective death benefits or endow.
15. A method for a Coverage Provider to provide survival risk insurance
to a Coverage Recipient said method comprising the steps of: a)
selecting a group of insured lives such that: i. each one of said
insured lives is covered by an original life insurance policy; ii.
each one of said original life insurance policies is provided by
one or more original life insurance companies; iii. each one of
said original life insurance policies pays a death benefit, DB.sup.m,
to said Coverage Recipient upon the death of one of said insured
lives m; and iv. said insured lives belong to a mortality class
as of a beginning date; b) determining a schedule of benefit payments
BP to be paid by the Coverage Provider to the Coverage Recipient;
c) calculating a single premium S1 for said survival risk insurance
wherein said single premium S1 is greater than or equal to SPR,
wherein 18 SPR = All Policies SPR m and wherein: 19 SPR m = ( PV1
- PV3 ) + F + G 100 DB m + E + H 100 Y 100 DB m and wherein: PV1
equals the present value as of said beginning date of said schedule
of benefit payments BP; and wherein: 20 PV3 = [ k = 0 - x - 1 v
k + 1 p x k q x + k T1 + p x - x v - x T2 ] i ln ( 1 + i ) DB m
and wherein: i is an annual effective interest rate representing
the cost of use of money of the Coverage Provider; F is a default
risk attributable to said one or more original life insurance companies
issuing said original life insurance policies, said F being a percentage
of said death benefit DB.sup.m of insured m; G is a default risk
attributable to the Coverage Recipient wherein said G is a percentage
of the death benefit of insured m; T1 is an adjustment factor to
reflect any tax attributable to a death benefit receivable by the
Coverage Provider; T2 is an adjustment factor to reflect tax attributable
to any endowment receivable by the Coverage Provider; E is an expense
factor set by said Coverage Provider wherein E is expressed as a
percentage of a death benefit DB.sup.m; H is a factor for desired
profit set by said Coverage Provider, wherein H is expressed as
a percentage of a death benefit DB.sup.m; Y is a percentage applied
to a death benefit DB.sup.m; .omega. is a maturity age of an original
life insurance policy; x is the age of said one of said insureds
m as of said beginning date; .sub.kp.sub.x is the probability that
a given one of said insureds will survive from said beginning date
for k years; .omega.-xp.sub.x is the probability that said one of
said insureds will survive from said beginning date to said maturity
age, .omega.; q.sub.x+k is the probability that said one of said
insureds will die between the beginning of year k and the end of
year k; and k is a number of years past said beginning date; .nu.
is an annual discount rate equal to 1/(1+i); and said step of calculating
said single premium is performed on a computer; and d) committing
said Coverage Provider to pay said Coverage Recipient said schedule
of said benefit payments BP; e) committing said Coverage Recipient
to pay a set of premiums to said Coverage Provider in exchange for
said schedule of said benefit payments BP wherein said set of premiums
has a present value as of said beginning date equal to said single
premium S1; f) committing said Coverage Recipient to transfer ownership
of said original life insurance policies to said Coverage Provider
as of said beginning date such that said Coverage Provider becomes
the beneficiary of said original life insurance policies.
16. The method of claim 15 wherein said set of premiums is one
premium.
17. The method of claim 15 wherein said set of premiums comprises
annual premiums payable for a premium paying period.
18. The method of claim 15 wherein said end date is on or before
the end of the term of a loan, wherein said loan is from said Coverage
Recipient to at least one of said insured lives.
19. The method of claim 15 wherein said end date is chosen such
that the probability of death of said insureds as of said end date
is greater than or equal to 0.75.
20. The method of claim 15 wherein said at least a portion of said
schedule of benefit payments BP is paid by said Coverage Provider
to said Coverage Recipient as a loan.
21. The method of claim 20 wherein said single premium S1 includes
a charge for interest on said loan.
22. The method of claim 15 wherein said single premium S1 is first
calculated before said beginning date and then recalculated at least
once after said beginning date.
23. The method of claim 15 wherein at least one of said insured
lives is impaired.
24. The method of claim 15 wherein said schedule of benefit payments
BP comprises a single benefit payment, said single benefit payment
being equal to the sum of the benefits of said original life insurance
policies.
25. The method of claim 24 wherein said single benefit payment
is due as of said end date.
26. The method of claim 15 wherein said step of committing said
Coverage Provider to pay said Coverage Recipient said schedule of
benefit payments BP comprises said Coverage Provider signing a contract,
said contract stating at least a portion of the terms of said survival
risk insurance.
27. The method of claim 15 wherein said step of committing said
Coverage Recipient to pay said set of premiums to said Coverage
Provider comprises said Coverage Provider signing a contract, said
contract stating at least a portion of the terms of said survival
risk insurance, and wherein said Coverage Recipient has made a conditional
commitment to abide by said contract if said Coverage Provider signs
said contract.
28. The method of claim 15 wherein said step of committing said
Coverage Recipient to pay said set of premiums to said Coverage
Provider comprises said Coverage Provider accepting the actual payment
of at least a portion of said set of premiums.
29. The method of claim 15 wherein said single premium S1 is greater
than or equal to said SPR plus an amount required to keep said original
life insurance policies in force from said beginning date to the
date when all of said original life insurance policies either pay
their respective death benefits or endow.
Insurance Description
CROSS-REFERENCE TO RELATED APPLICATION
[0001] This application is a division of and claims the benefit
of the filing date of U.S. nonprovisional application entitled "Method
of Calculating Premium Payment to Cover the Risk Attributable to
Insureds Surviving a Specified Period", Ser. No. 10/743,201
filed Dec. 22, 2003 which is incorporated herein by reference.
[0002] Said U.S. nonprovisional application entitled "Method
of Calculating Premium Payment to Cover the Risk Attributable to
Insureds Surviving a Specified Period", in turn claims the
benefit of the filing date of U.S. provisional patent application
entitled "Method of Calculating Premium Payment to Cover the
Risk Attributable to Insureds Surviving a Specified Period",
Ser. No. 60/507,170, filed Sep. 30, 2003. Said provisional application
is incorporated herein by reference.
FIELD OF INVENTION
[0003] The invention relates to the field of financial products
and methods involving the provision of death benefits through life
insurance. As used in this invention, the expression "life
insurance" relates to the type of insurance policy that provides
a death benefit if the life or lives insured by the policy die while
the insurance policy is in force or effect.
BACKGROUND OF THE INVENTION
[0004] Life insurance companies, through the issuance of life insurance
policies, accept a transfer of the risk of adverse financial consequences
which would be created when an insured life dies. Typically, the
death of an individual creates adverse financial consequences for
those who depend on future income or work contributions lost as
a result of the individual's death. A life insurance policy is typically
purchased to provide a beneficiary with a death benefit payment.
The purpose of the death benefit payment is to provide the beneficiary
with the means to offset, at least in part, the financial strain
created by the unexpected or untimely early death of the insured.
[0005] Life insurance pricing (that is, the determination of the
premium charged by the insurance company for the acceptance of a
stated death benefit risk) is based on a number of factors for which
assumptions are made including: mortality, interest, expenses (including
taxes), and policy persistency. Policy persistency is the probability
that the owner of a life insurance policy will choose to keep the
policy in force by paying the premiums required as per the terms
of the insurance contract. A policyholder who does not persist is
said to have lapsed. An assumption with respect to lapse rates is
typically the way persistency is incorporated into pricing calculations.
Pricing assumptions are made prior to the issuance of a life insurance
policy and are made relative to an entire class of insured lives
and not with respect to individual insureds.
[0006] A number of different sets of pricing assumptions may be
used. Assumption sets may vary by mortality class, type of distribution
used, or by other characteristics commonly used to distinguish between
classes of insured lives in the insurance industry. For example,
different mortality assumptions may be applied to males versus females
or to non-smokers versus smokers. Those skilled in the art are well
aware of the fact that many other mortality class distinctions exist
or are possible.
[0007] Also, different expense assumptions may be applied in different
marketing situations. For example, insurance offered directly to
the consumer in a direct marketing distribution channel versus insurance
offered through a traditional agent based distribution channel will
have different expense assumptions applied to reflect the different
expected costs in these different marketing channels.
[0008] The interest rate assumptions used in pricing are important
because they provide an adjustment for the timing differences between
when cash goes into and out of the pricing calculations.
[0009] Persistency or lapse assumptions affect pricing calculations
by creating an expectation with respect to the occurrence of cash
flows which are dependent on the insurance policy being in force
or effect.
[0010] When an individual insured applies for life insurance, an
underwriting process is applied. The underwriting process determines,
the appropriate premium or underwriting class for the applicant
based on an evaluation of the applicant's mortality characteristics
and life expectancy. This is based in part on expertise the underwriter
derived from prior training or experience. Life expectancy is the
average number of years individuals in the same underwriting class
can be expected to live. Maximum life expectancy is the highest
age to which an individual in the underwriting class can be expected
to live.
[0011] A life insurance policy may provide for some change in assumptions
after the policy is issued. Such changes would modify the current
charges or credits provided for in the policy for a class of insureds.
These changes would result in a change in the overall cost of the
life insurance for each insured in the same class of insureds. Such
changes can only be justified by changes in experience after issue
for the whole class to which an insured belongs and can only be
applied to all insurance policies in the class. Typically the range
of change allowed in a life insurance policy is limited by minimum
or maximum guarantees made in the life insurance contract or policy
relative to each assumption that may be changed.
[0012] One type of life insurance policy contains an endowment
feature. This type of life insurance policy is called an endowment
policy. With respect to such life insurance, the insurance company
would pay an amount called the endowment benefit to the insured
on the endowment date, for example age 65, if the insured survived
to that date. Because of changes in the way life insurance is taxed
by the US federal government, endowment policies of this sort are,
typically, no longer offered as there are adverse tax consequences
associated with life insurance policies that endow prior to age
95.
[0013] The assumptions used in pricing a life insurance policy
include an assumption regarding the maximum life expectancy of the
lives insured. For many currently issued life insurance policies,
this maximum life expectancy has been assumed to be age 100. More
recent mortality tables used for life insurance pricing purposes
are beginning to recognize longer maximum life expectancies, for
example, age 120. These longer maximum life expectancies are made
possible by improvements in health care and a general improvement
in the health of the insured life population. Mortality tables developed
for purposes other than life insurance pricing may have found it
convenient to make assumptions regarding the maximum life expectancy
different from age 100. For example, annuity products may use mortality
tables for pricing purposes with a maximum life expectancy greater
than age 100.
[0014] This maximum life expectancy age is often referred to as
the maturity age for the life insurance policy. Any insured who
survives to the end of this period can be thought of as reaching
the ultimate endowment age. Typically, life insurance companies
will make the death benefit of the life insurance policy available
in some way to insureds who survive to the maturity age. One alternative
is to pay an amount equal to the death benefit at the maturity age
directly to the insured as an endowment benefit on the maturity
date. Another alternative is for the insurance company to hold such
a maturity age endowment benefit in an interest bearing account
until such time as the insured actually dies. Then the benefit is
paid as a death benefit. This later alternative is used to potentially
avoid the possible adverse tax consequences associated with paying
the death benefit to the insured before the insured actually died.
[0015] In the past, the value in a life insurance policy was determined
only by the contractual terms of the life insurance policy and confined
to the relationship the owner of the policy had with the insurance
company providing it. Recently, however, secondary markets for life
insurance policies have developed in which a life insurance policy
is purchased or the right to receive the death benefit is assigned
to a third party by the owner of the policy in exchange for a fee
or purchase price.
[0016] Examples of the operation of secondary markets can be seen
in the life settlement market and viaticals. These markets involve
life insurance policies on insured lives who become impaired after
their policies were originally issued. In these markets, life insurance
policies are purchased by third parties (that is, neither the owner
of the policy nor the insurance company issuing the policy) or assigned
to third parties for a fee or payment of some sort. For such payment
the third party receives the right to receive the policy death benefit
when the insured dies.
[0017] An impaired life is an insured life who develops an impairment
after the life insurance policy was originally issued which reduces
his or her life expectancy. An impairment is any medical condition
affecting the health status of the insured life which results in
a higher mortality rate for the insured life than was reflected
in the original mortality assumption made for the underwriting class
the insured was assigned to when the insurance policy was issued.
Because of the impairment acquired after original issuance of the
policy, the likelihood of an earlier than expected death claim is
increased. This situation may make life insurance policies covering
such impaired lives worth more than the cash surrender value provided
by the contractual terms of the policies.
[0018] The cash surrender value of a life insurance policy is the
amount of money the life insurance company that issued the policy
is willing to pay if the policy is lapsed or surrendered. A life
insurance policy is lapsed if the owner of the policy stops paying
the premiums required to keep the policy in force per the terms
of the life insurance contract. For a typical term life insurance
policy or whole life insurance policy, the policy lapses when the
owner stops paying the contractually required premiums. A whole
life insurance policy may have a cash value at the time it lapses
which can be surrendered and paid to the policy owner in cash or
applied by the policy owner under one of the nonforfeiture options
contained in the policy contract.
[0019] For a universal life or variable universal life insurance
policy with a flexible premium structure, the policy lapses when
the cash value of the policy becomes insufficient to cover the insurance
related charges specified in the policy contract. Typically this
occurs because the policy owner has not made premium payments prior
to the lapse sufficient to keep the policy cash value large enough
to cover said charges.
[0020] A life insurance policy can be surrendered by an owner who
voluntarily agrees to terminate the life insurance protection provided
by the policy in exchange for the payment of the policy's cash surrender
value. A life insurance policy's cash surrender value is the cash
value of the policy defined in the life insurance contract adjusted
for any amounts owed by the owner to the insurance company (for
example, policy loans) or any additional amounts owed by the insurance
company to the owner (for example, dividends). For universal and
variable universal life insurance forms of insurance there may also
be specifically stated surrender charges which are deducted to determine
the cash surrender value.
[0021] Life insurance benefits may also be assigned to third parties
in insurance marketing programs in which life insurance is used
as a funding mechanism by a benefit plan sponsor. The benefit plan
sponsor is a third party which pays for or in some other fashion
enables benefits related to the life or health of an individual
or individuals. The benefits provided by the benefit plan sponsor
consist of cash payments designed to fund health, retirement, or
death needs. Funding mechanisms which utilize life insurance benefits
rely either on the cash values built up within a set of life insurance
policies or the life insurance policies' death benefits to meet
funding requirements for a benefit plan.
[0022] Several benefit plans have been funded with the expectation
of full or partial funding cost recovery via anticipated or expected
death benefit proceeds from the life insurance policies. When death
benefit proceeds are used to reduce or recover the cost attributable
to benefit plans, it is important that the death proceeds be received
in a predictable manner based on a set of mortality assumptions
chosen by the benefit plan sponsor. Such chosen mortality assumptions,
however, are often inaccurate due to the fact that the actual mortality
experience for a selected group is difficult to determine because,
generally, such data is not publicized by the insurance companies
or reinsurance companies that collect the data.
[0023] Many benefit plans involving the use of life insurance policies
as a funding vehicle do not take into account changes in the health
status of the insured lives after the insurance policies were issued.
That is, they do not rely for their value on the insured life becoming
impaired. In order for life insurance policies to be an effective
funding tool in programs in which death proceeds are the funding
source, however, the death benefits actually received must be reasonably
close to the death benefits expected based on the mortality assumptions
used to set up the program.
[0024] It is well recognized that for the financial products created
by the use of life insurance death proceeds as a funding vehicle,
adverse financial consequences are created if the actual mortality
experience of the lives insured under the life insurance policies
being used is better than assumed. That is, adverse financial consequences
are created for the third parties if the insured lives, as a group,
live longer than expected. This can occur, for example, if a financial
product is created by the purchase of a pool of life insurance policies
insuring a group of insureds who have lives that are impaired at
the time of purchase but who ultimately as a group live longer than
expected. The investors providing the cash used to make these purchases
are expecting a return on the money they have invested. The return
the investors receive is derived from the death benefits that are
paid on the life insurance policies that were purchased with the
cash they provided. This return is expected to consist of the return
of their invested principal plus an investment return. The amount
of the investment return or income the investors receive is dependent
on the amount and timing of the actual death benefit proceeds received
from the group of policies purchased. A purchase price value is
calculated for the policies being purchased which is based on mortality
assumptions from which the timing and amount of expected future
death proceeds can be projected. In addition, other expense and
risk charge assumptions are typically made in order to determine
a final purchase price for such life insurance policies.
[0025] Since it is the death proceeds of the group of policies
purchased which provide the revenue to pay back the investors their
principal and a return on their investment, the actual death proceeds
must be reasonably close to the expected death proceeds in amount
and timing for the expected investment return to be realized. In
a life settlement transaction, the investors would experience adverse
financial consequences if the insured lives experienced better mortality
as a group than the mortality assumption used to determine the purchase
price for the policies.
[0026] Another example of when a set of insureds living longer
than expected would have adverse financial consequences for a third
party beneficiary, would be a situation in which a benefit plan's
funding was dependent on the actual death proceeds from a life insurance
policy or group of life insurance policies for which the insured
or insureds health was not impaired. The benefit plan sponsor anticipates
receiving life insurance policy benefits in an expected manner with
respect to timing and amount. Such expectation would be based on
the assumed level of mortality used in establishing the benefit
plan's funding. Death benefit proceeds received by the plan sponsor
later than expected, in lower amounts, or not at all would result
in adverse financial consequences to the benefit plan sponsor since
this would result in the benefit plan being under funded.
[0027] The financial risk that a third party beneficiary faces
from a set of insured lives living longer than expected is referred
to herein as a "survival risk". There is a need for a
method of insuring against the adverse financial consequences of
survival risk.
[0028] Given the variety of benefit programs in the insurance industry
which are based on the inherent tax advantages attributable to the
payment of death benefit proceeds and rely on the timely payment
of death benefit proceeds according to projections based on reasonable
mortality assumptions, it would be desirable to have an insurance
product under which the risk of survival and the adverse financial
consequences of survival longer than expected could be transferred
from one entity to another entity willing to accept that survival
risk for a premium.
SUMMARY OF THE INVENTION
[0029] The present invention comprises a method for insuring against
the adverse financial consequences of survival risk to a third party
and a means for calculating the premium for said insurance. Said
insurance is referred to herein as "survival risk insurance".
Said third party is referred to herein as the Coverage Recipient.
The party providing survival risk insurance is referred to herein
as the Coverage Provider.
[0030] The method for a Coverage Provider providing survival risk
insurance to a Coverage Recipient comprises the steps of:
[0031] selecting a group of insured lives such that the insured
lives belong to a mortality class as of a beginning date;
[0032] calculating an expected death benefit payable to the Coverage
Recipient due to expected deaths of the members of the group of
insured lives, said deaths occurring between the beginning date
and an end date;
[0033] committing the Coverage Provider to pay the Coverage Recipient
a benefit equal to a percentage of the positive difference between
the expected death benefit and an actual death benefit payable to
the Coverage Recipient due to actual deaths of members of the group
of insured lives, said deaths occurring between the beginning date
and the end date;
[0034] committing the Coverage Recipient to pay a premium to the
Coverage Provider in exchange for the benefit.
DETAILED DESCRIPTION OF THE INVENTION
[0035] Specific embodiments of the present invention are described
below. These examples are not meant to be limiting.
[0036] Survival risk insurance may comprise a Coverage Provider
offering to pay to a Coverage Recipient an amount equal to a percentage
of the death benefit of an original life insurance policy issued
to an insured by an original life insurance company at the end date
of a specified period of time if the death of the insured does not
occur during the specified period of time. The Coverage Provider
would receive the death benefit proceeds at a later time when the
death of the insured actually occurs or the original life insurance
policy endows. The original life insurance policy must be kept in
force by fulfilling the contractual obligation of the policy owner
to the original life insurance company.
[0037] Survival risk insurance would be evidenced by a survival
risk insurance policy issued by the Coverage Provider to the Coverage
Recipient. The specified period would begin on the date the survival
risk insurance policy was issued. This date is the beginning date.
[0038] The survival risk insurance policy premium payable by the
Coverage Recipient to the Coverage Provider would be calculated
using a method of premium calculation. The method may be expressed
in a formula. The results of the formula may be calculated using
a computer. More particularly, the results of multiple copies of
the formula may be calculated using a spreadsheet.
[0039] The formula to calculate the survival risk insurance policy
premium is a function of:
[0040] the probability of survival of the insured from the beginning
to the end of the specified period;
[0041] the probability of the death of the insured for each year
or fraction thereof after the specified period;
[0042] an interest rate representing the cost of use of money by
the Coverage Provider;
[0043] a default risk attributable to the original life insurance
company;
[0044] a default risk attributable to at least some entity maintaining
the original life insurance policy in force after the specified
period until maturity of the original life insurance policy by death
or endowment;
[0045] applicable expenses, such as taxes; and
[0046] profits.
[0047] Unless otherwise defined herein, all terms and expressions
used herein have their standard actuarial meaning as defined in
the Society of Actuaries' Textbook on Life Contingencies (Second
Edition) by Chester Wallace Jordan Jr. published by the Society
of Actuaries in 1967.
[0048] Description of Assumptions Used in Calculations Herein
[0049] Probability of death or survival: The probability of death
may be based on standard experience mortality tables, such as the
1990-95 Basic Select and Ultimate Mortality Table developed by the
Society of Actuaries, or the mortality tables based on mortality
experience and developed in-house by life insurance companies or
reinsurance companies. Such mortality tables would be modified with
respect to each insured life based on assumptions and methods commonly
applied by those skilled in the art to reasonably reflect the survival
risk of the lives in a select group of insured lives. These methods
involve collecting data applicable to the current health of an insured
life (including but not limited to the most recent medical reports
available on the health status of the insured life) in order to
determine a current mortality rating to be applied to a standard
experience mortality table.
[0050] Interest Rate: This is a hurdle or interest rate set by
the Coverage Provider representing the time value of money relative
to the Coverage Provider's needs. The interest rate used by the
Coverage Provider will reflect interest rates currently available
to the Coverage Provider for other uses of the Coverage Provider's
money. The time value of money reflected in an interest rate includes
a charge for the risk associated with the return of principal invested
or lent. Therefore, the interest rate used by the Coverage Provider
will reflect these environmental factors. Typically interest rates
can be in the range 4% to 25% for environmental factors within a
normal range. Interest rates are annual effective interest rates
unless otherwise indicated.
[0051] Default Risk Attributable to Original Life Insurance Company.
This is a risk associated with the probability that the original
life insurance company will ultimately pay the death benefit provided
for in the policy it issued to an insured life. The level of this
risk depends on the financial stability of the life insurance company,
historical trends, and financial strength ratings provided by various
insurance industry rating organizations (e.g. A. M. Best, Moody's,
Standard & Poors). A typical range for this factor is 0% to
5% of the death benefit.
[0052] Default Risk Attributable to Coverage Recipient. This is
a risk associated with the probability that the owner of the original
life insurance policy on the insured life will meet its contractual
obligations to keep the original policy in force beyond the expiry
of the specified period. The level of this risk depends on the steps
taken by the survival risk insurance policy owner to keep the original
life insurance policy in force. As used herein, "owner",
means the entity keeping the original life insurance policy in force.
The Coverage Recipient would be the owner if they purchased the
life insurance policy from the original insured. To minimize the
default risk attributable to the Coverage Recipient, for example,
the Coverage Recipient may purchase a Guaranteed Investment Contract
(GIC) or Annuity that will guarantee the payment of premiums on
the original life insurance policy beyond the expiry of the specified
period. A typical range for this factor is 0% to 10% of the death
benefit.
[0053] Tax Attributable to death or endowment proceeds: If a transfer
of ownership of the original life insurance policy is utilized between
the Coverage Recipient and the Coverage Provider, such transfer
of ownership of a life insurance policy may trigger a taxable income
to the Coverage Provider when the Coverage Provider ultimately receives
the death benefit proceeds of the original life insurance policy.
The survival of the insured to the maturity date of the original
life insurance policy (attained age 100 for a policy based on the
1980 CSO Valuation Mortality Tables, for example) may trigger an
endowment and result in taxable income to the Coverage Provider.
The range for this factor is 0 to 1.
[0054] Means for Calculating Premium
[0055] A means for calculating the premium for a survival risk
insurance policy comprises the following five steps.
[0056] Step I
[0057] For each life in a selected group of insured lives, determine
the present value of a survival risk benefit payable by a Coverage
Provider to a Coverage Recipient. The survival risk benefit for
each life is equal to a percentage of the life insurance benefit
of the insured life. The life insurance benefit is equal to the
death benefit or endowment benefit, depending upon whether or not
the insured lives to the endowment age specified in the life insurance
policy. The survival risk benefit is discounted from the end date
to the beginning date using an interest rate and assumed mortality.
The discounted survival risk benefit is defined as the present value
of the survival risk benefit. The present value of step I is referred
to herein as "PV1". It is also referred to herein as "the
present value of the Coverage Provider cost".
[0058] Step II
[0059] For each life in the selected group of insured lives, determine
the present value as of the end date of the expected life insurance
benefit for each life, assuming the life survives past the end date.
The life insurance benefit is discounted to the end date using the
probability that the insured with die, or the life insurance policy
will endow, on a given date and the interest rate. The present value
may be adjusted by tax factors. The present value of step II is
referred to herein as PV2.
[0060] Step III
[0061] For each life in the selected group of insured lives, determine
the present value of PV2 as of the beginning date. PV2 is discounted
using the interest rate and the expected probability that the life
will survive to the end date. The present value of step III is referred
to herein as PV3. It is also referred to herein as "the present
value of the Coverage Provider reimbursement".
[0062] Step IV
[0063] For each life in a selected group of insured lives, calculate
a single premium for the survival risk insurance policy. The single
premium is at least equal to PV1 minus PV3. The single premium may
also include factors attributable to the expense and profit of the
Coverage Provider, the default risk attributable to the insurance
company which issued the original of life insurance policy; and
the default risk attributable to the Coverage Recipient, other expenses
and profits for the Coverage Provider.
[0064] Alternatively, for each life in a selected group of insured
lives, calculate an annual premium for the survival risk insurance
policy. The annual premium would be paid over a premium paying period.
The annual premium is equal to the single premium divided by a life
annuity due factor. The life annuity due factor is the present value
of $1.00 payable at the beginning of each year of the premium paying
period. It is calculated based on an assumption of the probability
of the insured life's survival during the premium paying period.
It also is based on an interest rate set by the Coverage Provider.
Methods to calculate a life annuity due factor based on mortality
and interest assumptions are well known to life insurance actuaries
skilled in the art.
[0065] Step V
[0066] The single premiums or annual premiums for each insured
life are summed to give a total single premium or total annual premium
for the survival risk insurance policy.
[0067] The following describes formulas used to calculate the premium
as per above Steps I-V.
[0068] Definition of Terms Used in the Formulas:
[0069] x=The age of an insured life m at the beginning date. "m"
is the number of an insured life in a selected group of insured
lives. The term "insured" means "insured life"
unless otherwise indicated.
[0070] q.sub.x+j=Probability of death of the insured m between
the beginning of year j and the end of year j. j is a number of
years after the beginning date.
[0071] p.sub.x+j=Probability of survival of the m from the beginning
of year j to the end of year j.
[0072] i=annual effective interest rate representing the cost of
use of money of the Coverage Provider.
[0073] PV1=The amount determined in step I
[0074] PV2=The amount determined in step II
[0075] PV3=The amount determined in step III
[0076] DB.sup.m=The death benefit receivable by a beneficiary on
the death of insured m.
[0077] F=The default risk attributable to the insurance company
issuing the original life insurance policy on insured m. F is a
percentage of the death benefit of insured m.
[0078] G=The default risk attributable to the Coverage Recipient.
G is a percentage of the death benefit of insured m.
[0079] SPR.sup.m=Single premium chargeable by the Coverage Provider
with respect to a survival risk insurance policy on insured m.
[0080] SPR=Total single premium chargeable by the Coverage Provider
for all survival risk insurance policies on insured lives m in the
selected group.
[0081] s=The premium paying period for survival risk insurance
policy.
[0082] APR.sub.s.sup.m=Annual premium chargeable by the Coverage
Provider for s years with respect to a survival risk insurance policy
on insured m.
[0083] APR.sub.s=Total annual premium chargeable by the Coverage
Provider for s years for all survival risk insurance policies on
insured lives m in the selected group.
[0084] T1=An adjustment factor to reflect any tax attributable
to death benefit receivable by the Coverage Provider.
[0085] T2=An adjustment factor to reflect tax attributable to any
endowment receivable by the Coverage Provider.
[0086] N=Length of the specified period in years.
[0087] .sub.x:{overscore (s.vertline.)}=a life annuity due of $1.00
payable at the beginning of each year of a premium paying period.
[0088] E=Expense factor set by the Coverage Provider. E is expressed
as a percentage of the death benefit.
[0089] H=Factor for desired profit set by the Coverage Provider.
H is expressed as a percentage of the death benefit.
[0090] Y=A percentage applied to the death benefit proceeds.
[0091] .omega.=the maturity age of the life insurance policy of
insured m.
[0092] The following equations define certain parameters. 1 p x
+ j = 1 - q x + j v = 1 1 + i p x t = p x p x + 1 p x + t - 1
[0093] The following equations are used to calcuate the single
premium or annual premium of a survival risk insurance policy. 2
PV 1 = Y 100 DB m p x N v N PV 2 = { [ k = N - x - 1 v k + 1 - N
p x + N k - N q x + k T 1 + p x + N - x - N v - x - N T 2 ] i ln
( 1 + i ) } DB m PV 3 = p x N v N PV 2 SPR = All Policies SPR m
SPR m = ( PV 1 - PV 3 ) + F + G 100 DB m + E + H 100 Y 100 DB m
APR s m = SPR m a x : s _ APR s = All Policies APR m
[0094] The factors E, F, G, and H may be set by the Coverage Provider.
The factors may have values commonly used in the life insurance
industry or may be calculated using means commonly employed in the
life insurance industry.
[0095] Variations
[0096] A survival risk insurance policy may be modified in a number
of ways. The following describes some of these variations. Other
variations may be apparent to one skilled in the art.
[0097] Variation 1
[0098] The Coverage Provider may agree to absorb additional risk
by agreeing to pay the death benefit of the original life insurance
policies at the end of the specified period irrespective to whether
deaths occur during the specified period or after the specified
period. The Coverage Provider would then collect the death benefits
from the original life insurance policies regardless of when the
insureds' deaths occur.
[0099] The following formulas would be used to calculate PV1 and
PV3. 3 PV 1 = v N Y 100 DB m PV 3 = [ k = 0 - x - 1 v k + 1 p x
k q x + k T 1 + p x - x v - x T 2 ] i ln ( 1 + i ) DB m
[0100] Variation 2
[0101] The Coverage Provider may agree to pay a schedule of periodic
payments during the specified period and collect the death benefits
from original life insurance policies regardless of when the insured's
deaths occur. The Coverage Provider would receive the actual death
benefit proceeds for deaths occurring both during and after the
specified period. For example, the schedule of periodic payments
may be equal to the expected incremental death benefits calculated
by using the Coverage Recipient's assumed mortality rates.
[0102] PV1 would be calculated as the present value of such schedule
of payments using standard actuarial methods.
[0103] Variation 3
[0104] The Coverage Provider may undertake the responsibility to
maintain a positive cash value in the original life insurance policies
in order to keep the original policies in force. The survival risk
insurance policy premium would be increased to cover such additional
cost. The increase will be an amount such that, together with the
existing cash value of the original life insurance policy at the
time of purchase of the survival risk insurance policy and taking
into account the current and guaranteed expense and cost of insurance
assumptions of the original life insurance policy, it will be sufficient,
if paid as a premium, to keep the original policies in force. All
assumptions used will be those considered actuarially appropriate
for this purpose by the Coverage Provider. This variation may result
in a lower cost being associated with the Default Risk Attributable
to Coverage Recipient. Standard actuarial methods would be used
to adjust the given formulas.
[0105] Variation 4
[0106] The Coverage Provider may undertake to periodically redetermine
the mortality rate assumption based on the actual experience of
specific populations of insureds covered by survival risk insurance
policies. As a result, the Coverage Provider may reserve the right
to modify its premium depending on actual experience.
[0107] The Coverage Provider would recalculate the premium based
on the redetermined mortality rate assumption. The premium may be
increased or decreased.
[0108] Variation 5
[0109] The Coverage Provider may offer to pay an amount equal to
a percentage of the death benefit of an original life insurance
policy as a loan at the end of a specified period if the death of
the insured does not occur during the specified period. The Coverage
Provider will receive loan interest from the Coverage Recipient
in addition to the premium for the survival risk policy. The Coverage
Recipient will repay the loan upon receiving the death benefit when
the insured life actually dies.
[0110] This variation may allow T1 and T2 to be set equal to 1.0.
That is, no tax adjustment would be necessary.
[0111] Alternatively, loan interest may be taken into account in
calculating the premium payable to the Coverage Provider by the
Coverage Recipient so that the Coverage Recipient does not have
to pay loan interest as a separate payment. In such situations PV1
will be increased by an amount called LSPR.sup.m. LSPR.sup.m is
defined as follows: 4 LSPR m = p x N v N t = N - x - 1 v t + 1 -
N p x + N t - N q x + t L t + 1 - N
[0112] where L.sub.k is the loan interest payable in year k after
the end of the specified period.
[0113] L.sub.k may include partially accrued loan interest in the
year of the insured life's actual death. This depends on when in
the year of death the principal and the accrued loan interest is
paid to the Coverage Provider.
[0114] The default risk attributable to the Coverage Recipient
may be adjusted to reflect a change in default risk under this variation.
NUMERICAL EXAMPLES
Example 1
[0115] A premium is calculated for a survival risk insurance policy
on a single insured life. The insured life under an original life
insurance policy is a male nonsmoker who is age 70 at a beginning
date. The mortality rate assumed by the Coverage Provider for the
insured life is 900% of the 1990-95 Basic Select and Ultimate Mortality
Table developed by the Society of Actuaries.
[0116] .omega. equals age 100. T1 is set equal to 0.80. T2 is set
equal to 0.65. F is set equal to 1%. G is set equal to 2%. E is
set equal to 1%. H is set equal to 2%. The interest rate is set
equal to 8%. The percentage Y is set equal to 100%. DB.sup.m equals
$1,000,000. The specified period is 10 years. Annual premiums for
the survival risk insurance policy are to be paid for 3 years.
[0117] Step I: Calculate the value of PV1 defined by the formula:
5 PV 1 = Y 100 DB m p 70 10 v 10
[0118] Using standard actuarial methods given the mortality and
interest rate assumptions, .sub.10p.sub.70=0.11827 and .nu..sup.10=0.46319
(when i=8%). Therefore, the value of PV1=$1,000,000*0.11827*0.46319=$54,781.48.
[0119] Step II: Calculate the value of PV2 defined by the formula:
6 PV 2 = { [ k = 10 100 - x - 1 v k + 1 - 10 p 80 k - 10 q 70 +
k T 1 + p 80 20 v 20 T 2 ] i ln ( 1 + i ) } DB m
[0120] The following table shows the individual factors in the
summation expression in the formula above and the total for the
summation. In calculating the individual terms in the summation,
T1 is equal to 0.80 per the previously stated assumptions. .nu.=1/(1.08)=0.92593
was used to calculate the .nu..sup.t+1 term.
1TABLE 1 1990-95 q.sub.80+t Basic (=9 times S&U 1990-95 Sum
of Attained Mortality Mortality Previous Age k t Rates Rate) .sub.tp.sub.80
.sub.tp.sub.80 * q.sub.70+k .nu..sup.t+1 * .sub.tp.sub.80 * q.sub.70+k
* T1 Column 80 10 0 0.04008 0.36072 1.00000 0.36072 0.26720 81 11
1 0.05076 0.45684 0.63928 0.29205 0.20031 82 12 2 0.06112 0.55008
0.34723 0.19100 0.12130 83 13 3 0.07906 0.71154 0.15623 0.11116
0.06537 84 14 4 0.09117 0.82053 0.04506 0.03697 0.02013 85 15 5
0.10214 0.91926 0.00809 0.00744 0.00375 86 16 6 0.11477 1.00000
0.00065 0.00065 0.00030 87 17 7 N/A N/A 0 0 0 88 18 8 N/A N/A 0
0 0 89 19 9 N/A N/A 0 0 0 90 20 10 N/A N/A 0 0 0 91 21 11 N/A N/A
0 0 0 92 22 12 N/A N/A 0 0 0 93 23 13 N/A N/A 0 0 0 94 24 14 N/A
N/A 0 0 0 95 25 15 N/A N/A 0 0 0 96 26 16 N/A N/A 0 0 0 97 27 17
N/A N/A 0 0 0 98 28 18 N/A N/A 0 0 0 99 29 16 N/A N/A 0 0 0 100
30 20 N/A N/A 0 0 0 0.67836
[0121] For i=8% the value of the expression i/ln(1+i) is 1.039487.
Since .sub.20p.sub.80=0, the T2 expression in the formula is equal
to 0.
[0122] Therefore, the value of PV2=(0.67836+0)*1.039487*$1,000,000=$705,14-
6.40.
[0123] Step III: Calculate the value of PV3 defined by the formula:
PV3=.sub.10p.sub.70.multidot..nu..sup.10.multidot.PV2
[0124] Using previously given values, PV3=0.11827*0.46319*$705.146.40=$38,-
628.96.
[0125] Step IV: Calculate the single premium, SPR.sup.m, for the
survival risk insurance associated with this insured life's life
insurance policy, m, using the formula: 7 SPR m = ( PV 1 - PV 3
) + F + G 100 DB m + E + H 100 Y 100 DB m
[0126] and the annual premium payable for three years, APR.sub.3.sup.m,
for the survival risk insurance associated with this insured life
using the formula: 8 APR 3 m = SPR m a 70 : 3 _
[0127] Substituting the given assumptions and the previously calculated
values, SPR.sup.m=($54,781.48-$38,628.96)+30,000+30,000=$76,152.52.
This is approximately, 7.6% of the death benefit.
[0128] Then, given that .sub.70:{overscore (3.vertline.)}=2.61950
APR.sub.3.sup.m=$29,071.40. This can be expressed approximately
as 2.9% of the death benefit.
[0129] Step V: Calculate the total premium for the survival risk
insurance policy by summing the individual policy calculations using
the formulas: 9 SPR = All Policies SPR m APR s = All Policies APR
m
[0130] If there were 10 identical individual life insurance policies
in the selected group of policies then SPR=$761,525 and APR.sub.3=$290,714,
rounded to whole dollars.
Example 2
[0131] Calculate the premium for Variation 1 using the data from
Example 1, unless otherwise indicated.
[0132] Variation 1, as described above, involves the Coverage Provider
agreeing to pay the death benefits for the life insurance policies
in the selected group of policies at the end of the specified period
and receive all death proceeds irrespective of whether the deaths
occur during or after the specified period. This involves using
the following formulas in place of the formulas previously given
to calculate PV1 and PV3 (the term PV2 is not required for this
variation): 10 PV 1 = v 10 Y 100 DB m PV 3 = { [ k = 0 100 - 70
- 1 v k + 1 p 70 k q 70 + k T 1 + p 70 100 - 70 v 100 - 70 T 2 ]
i ln ( 1 + i ) } DB m
[0133] PV1 can be calculated to be equal to: $463,193.49.
[0134] The following table shows the individual factors in the
PV3 formula summation expression and the total for the summation.
2TABLE 2 1990-95 q.sub.70+t Basic (=9 times S&U 1990-95 Sum
of Attained Mortality Mortality Previous Age k t Rates Rate) .sub.tp.sub.70
.sub.tp.sub.70 * q.sub.70+t .nu..sup.t+1 * .sub.tp.sub.70 * q.sub.70+t
* T1 Column 70 0 0 0.00594 0.05346 1.00000 0.05346 0.03960 71 1
1 0.00937 0.08433 0.94654 0.07982 0.05475 72 2 2 0.01285 0.11565
0.86672 0.10024 0.06366 73 3 3 0.01628 0.14652 0.76648 0.11230 0.06604
74 4 4 0.01918 0.17262 0.65418 0.11292 0.06148 75 5 5 0.02217 0.19953
0.54125 0.10800 0.05444 76 6 6 0.02571 0.23139 0.43326 0.10025 0.04680
77 7 7 0.02944 0.26496 0.33301 0.08823 0.03814 78 8 8 0.03230 0.29070
0.24477 0.07116 0.02848 79 9 9 0.03542 0.31878 0.17362 0.05535 0.02051
80 10 10 0.04008 0.36072 0.11827 0.04266 0.01464 81 11 11 0.05076
0.45684 0.07561 0.03454 0.01097 82 12 12 0.06112 0.55008 0.04107
0.02259 0.00665 83 13 13 0.07906 0.71154 0.01848 0.01315 0.00358
84 14 14 0.09117 0.82053 0.00533 0.00437 0.00110 85 15 15 0.10214
0.91926 0.00096 0.00088 0.00021 86 16 16 0.11477 1.00000 0.00008
0 0.00002 87 17 17 N/A N/A 0 0 0 0.51107
[0135] As previously, for i=8% the value of the expression i/ln(1+i)
is 1.039487. And, since .sub.30p.sub.70=0, the T2 expression in
the formula is equal to 0.
[0136] Therefore, the value of PV3=(0.51107+0)*1.039487*$1,000,000=$531,25-
0.62. Therefore, SPR.sup.m=-$8,057.13. A negative value for SPR.sup.M
means the death benefits the Coverage Provider would receive under
this variation are worth more than the benefit that would be provided
by a survival risk insurance policy. Therefore, a Coverage Recipient
would conclude that this is an inappropriate variation to apply
Example 3
[0137] Calculate the premium for Variation 2 using the data from
Example 1, unless otherwise indicated.
[0138] Variation 2, as described above, assumes a schedule of payments
equal to $100,000 will be paid by Coverage Provider to the Coverage
Recipient at the end of each year of the specified period. Using
standard actuarial methods to determine the value of this schedule
of benefits under interest only discounting yields a value for PV1=$671,008.14.
Thus under variation 2, substituting this value of PV1 for the originally
calculated value and using the value for PV3 calculated under variation
1, SPR.sup.m=$199,757.52 and APR.sub.3.sup.m=$76,257.88.
Example 4
[0139] Calculate the premium for Variation 3 using the data from
Example 1, unless otherwise indicated.
[0140] Variation 3, as described above, assumes that the Coverage
Provider would allocate $40,000 to fulfill the premium payment obligation
to maintain the original life insurance policy in force. This $40,000
allocation would increase the value of PV1 by $40,000 but the Coverage
Recipient default risk would be reduced to zero and the term G,
therefore, which had been 2% would be set equal to 0. Therefore,
the value of SPR.sup.m would change from the original calculation
by a total of 40,000-20,000 or 20,000 and, under variation 3, SPR=$96,153.61
and APR.sub.3=$36,706.86.
Example 5
[0141] Calculate the premium for Variation 4 using the data from
Example 1, unless otherwise indicated.
[0142] Variation 4, as described above, allows the Coverage Provider
to periodically redetermine the mortality rate it uses to calculate
the survival risk cost based on its actual experience. We can assume
that the Coverage Provider determines at the end of the first year
of coverage provided under the survival risk insurance policy that
actual experience is better than originally assumed. The Coverage
Provider determines, for example, that mortality is only 80% of
the originally assumed mortality at the time of execution of the
agreement between Coverage Provider and Coverage Recipient. Recalculating
based on this reduced mortality assumption produces a higher value
for SPR.sup.m and APR.sub.3.sup.m. Therefore, the Coverage Provider
would charge an additional premium.
[0143] The following table provides values necessary to recompute
the Coverage Provider charge.
3TABLE 3 q.sub.80+t 1990-95 (=.8 .times. 9 Basic times S&U
1990-95 Sum of Attained Mortality Mortality Previous Age k t Rates
Rate) .sub.tp.sub.80 .sub.tp.sub.80 * q.sub.70+K .nu..sup.t+1 *
.sub.tp.sub.80 * q.sub.70+K * T1 Column 80 10 0 0.04008 0.28858
1.00000 0.28858 0.21376 81 11 1 0.05076 0.36547 0.71142 0.26000
0.17833 82 12 2 0.06112 0.44006 0.45142 0.19865 0.12616 83 13 3
0.07906 0.56923 0.25277 0.14388 0.08461 84 14 4 0.09117 0.65642
0.10888 0.07147 0.03891 85 15 5 0.10214 0.73541 0.03741 0.02751
0.01387 86 16 6 0.11477 0.82634 0.00990 0.00818 0.00382 87 17 7
0.12707 0.91490 0.00172 0.00157 0.00068 88 18 8 0.13936 1.00000
0.00000 0.00015 0.00006 89 19 9 N/A N/A N/A N/A N/A 0.66020
[0144] .sub.10p.sub.70=0.19017 under these revised mortality assumptions
and .nu..sup.10=0.46319 (when i=8%).
[0145] Therefore, the revised value of PV1=0.19017*0.46319*$1,000,000=$88,-
084.84. The revised value of PV2=(0.66020+0)*1.039487*$1,000,000=$686,269.-
32 Using previously given values, the revised value of PV3=0.19017*0.46319*$686,269.32=$60,449.92.
[0146] Given that under the revised mortality assumption {umlaut
over (.alpha.)}.sub.70:{overscore (3.vertline.)}=2.65163
[0147] SPR.sup.m=$87,634.92, APR.sub.3.sup.m=$33,049.45
[0148] Therefore, the additional single premium that should have
been charged at issue=$87,634.92-$76,153.61=$11,481.31. The additional
annual premium that should have been charged beginning at issue=$33,049.45-$30,811.46=$2,237.99
[0149] The Coverage Provider may charge an additional premium based
on standard actuarial practice including the time value and money.
Example 6
[0150] Calculate the premium for Variation 5 using the data from
Example 1, unless otherwise indicated.
[0151] Variation 5, as described above, assumes that an annual
loan interest rate equal to 5.00% will be charged by the Coverage
Provider. Under this variation, T1 and T2 are set equal to 1 since
there is assumed to be no tax consequences associated with the death
benefit amount being paid as a loan at the end of the specified
period. The values of SPR.sup.m and APR.sub.3.sup.m would be recalculated
with the new values for T1 and T2. Therefore under this variation
5, PV2=$881,433.00 and PV3=$48,286.21. This produces the following
values: SPR.sup.m=$66,496.36 and APR.sub.3.sup.m=$25,385.13. Given
a loan interest of 5%, then 5% of the $1,000,000 death benefit loaned,
or $50,000 would be paid by the coverage Recipient to the Coverage
Provider each year.
[0152] Alternatively, the anticipated loan interest could be included
in the calculation of the survival risk insurance premium by including
the term: 11 LSPR m = p 70 10 v 10 t = 10 100 - 70 - 1 v t + 1 -
10 p 70 + 10 t - 10 q 70 + t L t + 1 - 10
[0153] in the PV1 formula as noted in the description of variation
5.
[0154] The following table shows the steps in the calculation of
LSPR.sup.m:
4TABLE 4 L.sub.k Accumulated Loan Loan Balance interest due Value
of Attained at End of at End of LSPR.sup.m term Summation Age k
t Year Year .sub.tp.sub.80 * q.sub.70+t for value of k of LSPR.sup.m
80 1 10 1,050,000 50,000 0.36072 16,700.00 81 2 11 1,102,500 102,500
0.29205 25,664.54 82 3 12 1,157,625 157,625 0.19101 23,900.66 83
4 13 1,215,506 215,506 0.11116 17,608.12 84 5 14 1,276,282 276,282
0.03698 6,953.46 85 6 15 1,340,096 340,096 0.00744 1,594.53 86 7
16 1,407,100 407,100 0.00065 154.40 87 8 17 0.00000 0.0 92,576.71
[0155] Then, since .sub.10p.sub.70*.nu..sup.10=0.11827*0.46319=0.05478,
[0156] LSPR.sup.m=0.05478*92,576.71=$5,071.35.
[0157] PV1=54,781.38+5,071.35=$59,852.73
[0158] As in the first option for variation 5 above, the recalculated
value for PV2=$881,433.00 using T1=1.0 and T2=1.0.
[0159] Therefore, PV3=$48,286.21 as above and SPR.sup.m=$71,566.52.
[0160] Applications:
[0161] There are a number of applications of how a survival risk
insurance product could be used in sales or marketing situations.
These are shown for illustration and not meant to be limiting.
[0162] Application 1: Loan Maturity with Respect to Premium Financing
[0163] A life insurance policy owner may borrow from a lender to
pay premiums to fulfill the contractual premium payment obligations
with the life insurance company that issued the life insurance policy.
The lender will receive the repayment of the loan plus loan interest
at the time of the insured's death. The loan agreement will be structured
so that the death benefit proceeds will be large enough to pay to
the lender the outstanding loan balance including the accrued loan
interest at the time of the insured's death with any remaining death
benefit proceeds being paid as a lump sum to the insured's designated
beneficiary.
[0164] The cash value of the life insurance policy given the payment
of the above noted premium is expected, based on reasonable assumptions,
to grow, after all applicable deductions, sufficiently enough to
support the policy until death or maturity. However, the sufficiency
of the life insurance policy cash value to keep the policy in force
until death or maturity and provide a death benefit large enough
to repay the loan balance plus accrued loan interest is not guaranteed.
In addition, in the event that death does not occur during the maturity
period of loan, the lender would have to refinance the loan. Lenders
would prefer to be more secured with respect to recovering their
loan plus accrued loan interest. Loans are generally associated
with a fixed maturity period not a maturity period that is determined
by a contingent event such as the death of an insured life.
[0165] The present invention will enable a Coverage Provider to
absorb the lender's risk that insureds may survive an anticipated
fixed loan maturity period or any other specified period. The lender
would be the Coverage Recipient. A survival risk insurance policy
would be issued by the Coverage Provider which would allow the payoff
of the loan including accrued loan interest at the end of the fixed
period set by the lender. Such a policy would be issued for the
percentage of death benefit of the life insurance policy sufficient
to cover the loan plus accrued loan interest requirement. The specified
period for such a policy may be less than or equal to the maturity
period set by the lender.
[0166] As in the basic description of the present invention, the
Coverage Provider will be entitled to receive the death benefit
proceeds or a percentage of the death benefit proceeds of each life
insurance policy covered by survival risk insurance in situations
where the death of the insured life occurs after the end of the
specified period. The Coverage Provider would price the survival
risk insurance product on the assumption that it would receive,
at least, a percentage of the death proceeds equal to the amount
paid by the Coverage Provider to the lender at the end of the specified
period plus interest at an acceptable rate from the end of the specified
period until the actual death of the insured life.
[0167] In addition, the Coverage Provider may undertake the responsibility
to maintain the policy in-force until maturity as described in variation
3.
[0168] The following provides numerical example of how the present
invention would be applied to calculate a premium for a survival
risk insurance policy used in this sales or marketing concept.
[0169] This numerical example is based on an Age 65 Male Nonsmoker.
The Coverage Provider assumes mortality equal to 100% of the 1990-95
Basic Select and Ultimate Mortality Table. Other variables are set
as follows: T1=1.0, T2=1.0, F=1%, G=0%, E=0.5%, H=0.5%, i=8%, Y=100%,
DB.sup.m=$10,000,000. The specified period, which is equal to the
loan maturity period=20 years. An annual loan=$895,390 is required
to pay the policy premium for 6 years. The loan interest rate=6.00%.
The interest is not paid in cash but accrues. Therefore, the outstanding
loan balance at the end of each year of the loan equals the original
amount borrowed plus accrued loan interest. The loan plus accrued
interest will be repaid out of the death proceeds when the insured
dies. The amount of the annual premium, $895,390 payable over 6
years, is calculated under reasonable assumptions to support the
insurance policy with increasing death benefit to recover loan balance
plus $10,000,000.
[0170] Table 5 shows the calculation of the accrued loan balance
and the death benefit required at the end of each year to repay
the loan plus accrued loan interest and provide a death benefit
equal to the original death benefit.
5TABLE 5 Total Annual Original Required Attained Annual Loan Cumulative
Death Death Age t Loan Interest Loan Balance Benefit Benefit 65
1 895,390 53,723 949,113 10,000,000 10,949,113 66 2 895,390 110,670
1,955,174 10,000,000 11,955,174 67 3 895,390 171,034 3,021,597 10,000,000
13,021,597 68 4 895,390 235,019 4,152,007 10,000,000 14,152,007
69 5 895,390 302,844 5,350,240 10,000,000 15,350,240 70 6 895,390
374,738 6,620,368 10,000,000 16,620,368 71 7 397,222 7,017,590 10,000,000
17,017,590 72 8 421,055 7,438,646 10,000,000 17,438,646 73 9 446,319
7,884,965 10,000,000 17,884,965 74 10 473,098 8,358,062 10,000,000
18,358,062 75 11 501,484 8,859,546 10,000,000 18,859,546 76 12 531,573
9,391,119 10,000,000 19,391,119 77 13 563,467 9,954,586 10,000,000
19,954,586 78 14 597,275 10,551,861 10,000,000 20,551,861 79 15
633,112 11,184,973 10,000,000 21,184,973 80 16 671,098 11,856,071
10,000,000 21,856,071 81 17 711,364 12,567,436 10,000,000 22,567,436
82 18 754,046 13,321,482 10,000,000 23,321,482 83 19 799,289 14,120,771
10,000,000 24,120,771 84 20 847,246 14,968,017 10,000,000 24,968,017
85 21 898,081 15,866,098 10,000,000 25,866,098 86 22 951,966 16,818,064
10,000,000 26,818,064 87 23 1,009,084 17,827,148 10,000,000 27,827,148
88 24 1,069,629 18,896,776 10,000,000 28,896,776 89 25 1,133,807
20,030,583 10,000,000 30,030,583 90 26 1,201,835 21,232,418 10,000,000
31,232,418 91 27 1,273,945 22,506,363 10,000,000 32,506,363 92 28
1,350,382 23,856,745 10,000,000 33,856,745 93 29 1,431,405 25,288,150
10,000,000 35,288,150 94 30 1,517,289 26,805,439 10,000,000 36,805,439
95 31 1,608,326 28,413,765 10,000,000 38,413,765 96 32 1,704,826
30,118,591 10,000,000 40,118,591 97 33 1,807,115 31,925,706 10,000,000
41,925,706 98 34 1,915,542 33,841,249 10,000,000 43,841,249
[0171] Therefore DB.sup.m at the end of the 20.sup.th year is $24,968,017.
[0172] Given the following values, .sub.20p.sub.65=0.48884 and
.nu..sup.20=0.21455, PV1 can be calculated as follows: PV1=0.48884*0.21455*$24,968,017=$2,618,661.15.
[0173] Table 6 following provides additional information required
to complete the calculation of the survival risk premium for this
example.
[0174] Since the death benefit for the life insurance policy, DB.sup.m,
varies year to year, the formula for PV2 is modified for this example
as follows: 12 PV2 = { [ k = 20 100 - 65 - 1 v k + 1 - 20 p 85 k
- 20 q 65 + k T1 DB k m + 15 p 85 v 15 T2 ] .08 ln ( 1.08 ) }
[0175] For this example, the life insurance policy which is part
of this premium financing approach has sufficient funding to only
keep it in force through the end of policy year 33.
[0176] Therefore, it will lapse without value when the insured
life is attained age 98. Thus, all values beyond attained age 98
are equal to zero and not a factor in this calculation.
6TABLE 6 q.sub.65+t 1990-95 (=100% Basic of the S&U 1990-95
Summation Attained Mortality Mortality of Previous Age k t Rates
Rate) .sub.tp.sub.85 .sub.tp.sub.85 * q.sub.65+k .nu..sup.t+1 .sub.tp.sub.85
* q.sub.65+k * T1 * DB.sup.m.sub.k Column 85 20 0 0.10287 0.10287
1.00000 0.10287 2,463,745.83 86 21 1 0.11535 0.11535 0.89713 0.10348
2,379,320.19 87 22 2 0.12771 0.12771 0.79365 0.10136 2,238,974.76
88 23 3 0.13992 0.13992 0.69229 0.09687 2,057,415.09 89 24 4 0.15184
0.15184 0.59542 0.09041 1,847,812.01 90 25 5 0.16371 0.16371 0.50502
0.08268 1,627,206.52 91 26 6 0.17782 0.17782 0.42234 0.07510 1,424,439.45
92 27 7 0.19353 0.19353 0.34724 0.06720 1,229,225.77 93 28 8 0.21345
0.21345 0.28004 0.05977 1,055,182.72 94 29 9 0.23423 0.23423 0.22026
0.05159 879,548.35 95 30 10 0.25253 0.25253 0.16867 0.04259 701,745.52
96 31 11 0.26709 0.26709 0.12608 0.03367 536,479.47 97 32 12 0.27242
0.27242 0.09240 0.02517 388,057.49 98 33 13 0.28031 0.28031 0.06723
0.01885 281,290.59 19,110,443.77
[0177] Given 0.08/(ln(1.08))=1.03949, then
[0178] PV2=$19,110,443.77*1.039487=$19,865 057.86 and
[0179] PV3=0.48884*0.21455*$19,865,057.86=$2,083,459.62.
[0180] For purposes of calculating the terms involving the factors
E and F the death benefit at the end of policy year 20 which is
$24,968,017 is used. For purposes of calculating the terms involving
the factors G and H the death benefit at the end of policy year
33 which is $43,841,249 is used.
[0181] This makes SPR.sup.m=($2,618,661.15-$2,083,459.62)+$438,412.49+$249-
,680.17=$1,223,294.19
[0182] Application 2: Investment Returns Based on Death Benefits
[0183] A life insurance policy can be priced or funded to provide
life insurance for a specified term, L. A group of such policies
can be purchased by a purchasing entity on the lives of persons
on whom such purchasing entity can establish an insurable interest.
An entity has insurable interest in a life if that entity has a
financially interest in said life not dying. Premiums payable to
the life insurance company issuing the policies can be calculated
to support the policies for the L year term.
[0184] In this situation the purchasing entity purchased the life
insurance policies with a reasonable expectation that death benefit
proceeds would be received during the period L according to a scheduled
based on a mortality assumption. After a certain number of years,
P, the purchasing entity (generally a corporation or trust) may
determine that it would like to protect itself from the risk of
death benefit proceeds being received that were less than what it
originally assumed. By eliminating this risk the purchasing entity
would be guaranteed an investment return provided by the death benefit
proceeds on the life insurance policies it purchased and funded.
[0185] In order to provide itself with this protection the purchasing
entity can purchase survival risk insurance from a Coverage Provider
on each life on which it owns life insurance. The purchasing entity
would be the Coverage Recipient. The specified period for such a
survival risk insurance policy would be set equal to L-P. In order
to achieve the desired level of protection, or survival risk transfer,
the purchasing entity may only purchase a survival risk insurance
policy covering a percentage Y (less than 100%) of each original
life insurance policy's death benefit.
[0186] Such percentage Y will be determined such that the Coverage
Provider can attain its desired profit while the purchasing entity
or Coverage Recipient can ensure that its expected return will be
guaranteed not to fall below a predetermined rate. The predetermined
rate would be affected by the terms of survival risk insurance policy.
[0187] In this application the Coverage Provider will continue
the premium paying obligation to the life insurance company that
issued the life insurance policy on the insured life until the Coverage
Provider received sufficient death proceeds to satisfy its pricing
requirements.
[0188] The following provides a numerical example of how the present
invention would be applied to calculate a premium for a survival
risk insurance policy used in this application.
[0189] This numerical example is based on an Age 70 Male Nonsmoker.
The Coverage Recipient purchased 1,000 identical policies for a
single premium of $546,938,000 paid to an insurance company which
issued the life insurance policies. Said single premium was determined
to be sufficient to keep the policies in force for 15 years under
the Coverage Recipient's assumptions. The Coverage Recipient assumed
mortality equal to 125% of the 1990-95 Basic Select and Ultimate
Mortality Table for the 1,000 identical risks. The Coverage Provider
assumed mortality equal to 100% of the 1990-95 Basic Select and
Ultimate Mortality Table. Other variables are set as follows by
the Coverage Provider: T1=1.0, T2=1.0, F=2%, G=0%, E=2%, H=2%, i=6%,
Y=100%. The DB.sup.m for each policy equals $3,000,000.
[0190] The effective year of the contract between Coverage Provider
and the Coverage Recipient is 3 (that is, after 2 policy years)
and the specified period is equal to 13 which is 15-2. There is
an additional annual life insurance premium required per the original
life insurance policy to be paid by the Coverage Provider from policy
year 16 of the life insurance policies and thereafter. The additional
annual premium is $320,000. This premium is required to keep each
of the life insurance policies in force after policy year 16 so
that the Coverage Provider will be able to receive the death benefit
proceeds when the insured lives die. The total life insurance premium
in year 16 and thereafter is calculated as the product of $320,000
and the number of survivors expected in each year thereafter. This
is shown in Table 7.
7TABLE 7 Expected Attained Death Benefit # of Year Age Premium
Cash Value Proceeds Survivors IRR-1 IRR-2 0 546,938,000 0 0 1000
1 70 549,168,282 22,275,000 991 2 71 560,443,486 34,876,604 976
3 72 563,972,423 47,269,502 957 4 73 559,557,017 58,925,030 932
5 74 547,583,509 68,008,776 904 6 75 528,000,273 76,726,077 873
7 76 500,418,632 86,511,547 838 8 77 464,957,553 95,878,995 799
9 78 422,492,633 101,322,212 759 10 79 372,884,420 106,623,332 717
11 80 315,201,008 115,309,304 672 12 81 246,319,765 138,719,061
619 13 82 168,778,185 156,433,168 560 14 83 84,458,069 186,890,071
491 15 84 64,788 194,218,462 421 11.41% 8.88% 16 85 134,720,000
46,514,758 192,790,892 354 17 86 113,280,000 74,218,115 188,971,956
290 18 87 92,800,000 86,273,649 179,208,411 233 19 88 74,560,000
86,396,754 165,323,046 182 20 89 58,240,000 78,494,058 148,152,196
139 21 90 44,480,000 66,337,413 129,671,426 103 22 91 32,960,000
52,912,484 112,220,143 75 23 92 24,000,000 39,793,682 95,136,433
52 24 93 16,640,000 27,939,075 79,575,398 35 25 94 11,200,000 18,179,087
64,109,616 22 26 95 7,040,000 10,929,361 48,881,392 13 27 96 4,160,000
6,015,739 35,380,059 0 28 97 0 2,970,387 24,038,303 0 29 98 0 1,101,914
16,311,794 0 30 99 0 124,670 10,950,174 0
[0191] Using the methods of the present invention in a computer
implemented calculation, a value for SPR=$140,868,855 is determined.
This is the payment made by the Coverage Recipient to the Coverage
Provider at the beginning of policy year 3 (that is, the end of
policy year 2) to purchase survival risk insurance for the life
insurance policies purchased by the Coverage Recipient. This survival
risk insurance would be structured so that the Coverage Provider
guarantees that death benefit proceeds from the life insurance policies
purchased will at least equal the expected death benefit proceeds
per the Coverage Recipient's assumptions during the 15 year specified
period. As shown in the table, the Coverage Recipient's expected
return prior to purchasing survival risk insurance was 11.41% (column
IRR-1). With survival risk insurance the Coverage Recipient can
guarantee a return of 8.88% (column IRR-2).
[0192] Application 3: Specified Threshold of Probability of Death.
[0193] A specified period, N, will be a number of years set on
the basis of a threshold probability of the insured's death occurring
during such N year period. For example, N may be set such that the
probability of the death of the insured within the specified period
is 0.75. This means that under the mortality assumption used to
calculate the probability there is a 75% probability that the insured
will die before the end of the specified period. This also means
that there is a 25% probability that the insured will survive to
the end of the specified period.
[0194] For each life insurance policy within a group of life insurance
polices, a specified period N will be determined such that the probability
of the death of the insured life during that specified period is
equal to or nearly equal to the threshold probability of death.
From such group of life insurance policies, those life insurance
policies with the same specified period, N, will be selected. There
can be 2 or more selected groups each having the desired threshold
probability of death relative to different specified periods, N.
Mortality assumptions deemed appropriate for the lives being analyzed
by the entity creating such selected groups (selecting entity) will
be used.
[0195] To eliminate, reduce, or manage the survival risk associated
with any such selected group, the selecting entity may purchase
survival risk insurance from a Coverage Provider. The selecting
entity is the Coverage Recipient.
[0196] The survival risk insurance policy purchased by the Coverage
Recipient will pay amounts equal to the death benefits of the survivors
of the specified period, N, with respect to a selected group per
the present invention. The Coverage Provider will charge a premium
to the Coverage Recipient for the survival risk insurance.
[0197] The Coverage Provider will make its own mortality and other
assumptions in calculating the survival risk insurance premium.
In particular, the mortality rate assumptions used by the Coverage
provider may not be the same as the same mortality rate assumptions
used by the Coverage Recipient or any other entity.
[0198] The following provides a numerical example of how the present
invention would be applied to calculate a premium for a survival
risk insurance policy used in this sales or marketing concept.
[0199] For a male, age 70, nonsmoker, a Coverage Provider assumes
mortality rates equal to 900% of 1990-95 Basic Select and Ultimate
Mortality Table. Values for the other factors required for the present
invention are: T1=1.0, T2=1.0, F=1%, G=2%, E=1%, H=2%, i=8%, Y=100%,
and DB.sup.m=$1,000,000. The Coverage Provider decides to use a
threshold probability of 0.9999. Therefore, the specified period
N will equal the period N for which the probability the insured
will die during the period is 0.9999. The insured's probability
of survival during the period is 0.0001.
[0200] For the assumptions used in this example, the specified
period N would equal 16 years. Using a computer implementation of
the present invention as demonstrated in the preceding, a value
for SPR.sup.m=$60,022 can be calculated.
[0201] Definitions
[0202] The following definitions of terms used herein will be helpful
in understanding the present invention.
[0203] "Survival risk" is the financial risk that a third
party beneficiary faces from a set of insured lives living longer
than expected. While not being held to this explanation, survival
risk exists, in part, because the mortality assumptions used to
evaluate survival in a group of lives to the end of a specified
period are dependent upon the Law of Large Numbers. The Law of Large
Numbers states that the mean of a random sample drawn from a large
population will approach the mean of the population as the sample
size n increases. If a group's size is not large enough, then the
survival risk calculated for the group will be subject to "statistical
fluctuation" for small values of n.
[0204] The Weak Law of Large Numbers states that there exists an
integer n such that a sample of size n or larger will have a sample
mean which deviates from the population mean by <.epsilon. with
a probability.gtoreq.(1-*). Typically, this sample error is expressed
in the form of a 95% probability or confidence (when *=0.05) that
the sample mean will differ from the population mean by .+-..epsilon..
[0205] In addition, the survival risk exists, in part, because
the assumptions initially made in calculating it may have been wrong
or the mortality experience of the group may change after the calculation
was made.
[0206] The "beginning date" is the issue date of a survival
risk insurance policy. It is the date on which the risk transfer
of the survival risk is made. The beginning date is the start of
a specified period.
[0207] The "end date" is the end of the specified period
of time. For example, 20 years after the beginning date.
[0208] A "selected group" is a group of lives selected
based on a mortality rating.
[0209] A "positive difference between the expected death benefit
and an actual death benefit" references the death benefits
paid by the original life insurance company which issued the life
insurance policy on an insured life. Typically, the difference between
the death benefits expected to be paid by the original insurer to
the Coverage Recipient and those actually paid by the original insurer
to the Coverage Recipient during the specified period will be a
positive number or zero.
[0210] However, it is also possible that the expected death benefit
will be less than the actual death benefit and the difference will
be negative. In one embodiment, only positive differences will be
paid by a Coverage Provider to a Coverage Recipient under the terms
of a survival risk insurance policy. A negative difference does
not imply an explicit payment by the Coverage Recipient to the Coverage
Provider. Any potential negative difference would be reflected in
the premium the Coverage Provider charges for the survival risk
insurance policy.
[0211] "Transferring ownership" means the Coverage Recipient
assigning the original life insurance policies to the Coverage Provider
or in some other way assuring that the death benefits of these policies
will be paid to the Coverage Provider after the end of the specified
period.
[0212] The "designated beneficiary" is the entity entitled
to receive the death benefit proceeds of a life insurance policy.
The designated beneficiary is usually designated by the life insurance
policy owner.
[0213] "Premium" refers to the premium the Coverage Provider
would charge for the survival risk insurance policy it issues, unless
otherwise indicated.
[0214] An "in force death benefit" of a life insurance
policy is the amount, as of a specific date, that would be payable
if the life insured under such life insurance policy died on that
specified date. With respect to the present invention the specified
date is, typically, the end of the specified period.
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