theory of interest and mathematics of life contingencies
TRANSCRIPT
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Theory of Interest
andMathematics of Life
ContingenciesReview Notes
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Table of Contents
1.Time Value of Money.................................... 2
1.1. Accumulation and Amount Function2
1.2. Simple and Compound Interest........ 2
1.3. Measures of Interest......................... 3
1.4. Force of Interest................................ 5
1.5. Miscellaneous Models....................... 6
2.Analysis of Annuities..................................... 7
2.1. Annuities............................................ 7
2.2. Perpetuities....................................... 8
2.3. General Annuities and Perpetuities.. 9
2.4. Inflation Rate................................... 112.5. Continuous Annuities...................... 11
2.6. Annuities Payable at a Different
Frequency than Interest is
Convertible...................................... 12
3.Cash Flow Analysis...................................... 15
3.1. Yield Rates....................................... 15
3.2. Existence and Uniqueness of the
Yield Rate......................................... 15
3.3. Interest Measurement of a Fund.... 15
3.4. Approximation Methods................. 16
3.5. Reinvestment Rates........................ 17
3.6. Miscellaneous Methods.................. 17
4.Loan Repayment.......................................... 19
4.1. Amortization Scheduling................. 19
4.2. Sinking Fund Method...................... 19
5.Analysis of Financial Instruments............... 21
5.1. Financial Instruments...................... 21
5.2. Callable Bonds................................. 21
6.Survival Models........................................... 23
6.1. Future Lifetime Random Variable... 23
6.2. Force of Mortality........................... 23
6.3. Mean and Variance of ................ 246.4. Curtate Future Lifetime Random
Variable........................................... 24
6.5. Fractional Ages................................ 25
6.6. Special Laws of Mortality................ 26
6.7. Life Tables........................................ 26
6.8. Select and Ultimate Life Tables...... 26
7.Analysis of Life Insurances.......................... 28
7.1. Life Insurances................................. 28
7.2. Relationship Between Insurances
Payable at Moment of Death and End
of Year of Death.............................. 29
7.3. Variance of ................................. 307.4. Varying Benefit Insurance............... 30
7.5. Commutation Notations................. 30
8.Analysis of Life Annuities............................ 32
8.1. Life Annuities................................... 32
8.2. Continuous Life Annuities............... 33
8.3. Life Annuities Payable at a Different
Frequency than Interest is
Convertible...................................... 34
8.4. Commutation Notations................. 34
9.Premiums and Reserves.............................. 35
9.1. Net Level Premiums........................ 35
9.2. Gross/Expense-Loaded Premiums.. 38
9.3. Benefit ReservesProspective
Approach......................................... 39
9.4. Benefit ReservesRetrospective
Approach......................................... 40
9.5. Other Reserve Formulas................. 41
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1. Time Value of Money1.1. Accumulation and Amount Function An initial amount of money invested on a
fund/portfolio is called the principal or
capital
An amount of money withdrawable fromthe fund/portfolio at time is calledthe accumulated value, denoted by
The difference between the accumulatedvalue and the principal is called the interest
A negative interest is called loss A unit measurement for time is called the
period
Consider an initial investment of 1 at time
o The accumulated value at time of thisinvestment is denoted by , knownas the accumulation function
Remarks:
1. 2. is normally increasing3. is not necessarily continuous Now consider an initial investment of
o The accumulated value at time of thisinvestment is denoted by , knownas the amount function
Remarks:
1. 2.
3. The properties of the amount function issimilar to that of the accumulation function
1.2. Simple and Compound Interest The simple interest rate model (SIRM) is
where the interest earned per period is
level and is not reinvested back into the
fund
Let be the interest earned on the period, then
For the SIRM with an initial value of 1 at o The accumulated value at is
o The accumulated value at is o Recursively, the accumulated value
at is
The SIRM takes the form of a step function, Ideally, the SIRM should be true Suppose interest accrues continuously, i.e.is continuous
o Note that The accumulated value at
time can be divided intothe interest part,
,
and the principal part
Accumulating the principal periods later plus theinterest yields the equality
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o Therefore, is linear
o If ,then
o Therefore, ;moreover,
The compound interest rate model (CIRM)is where the interest earned by the fund is
immediately reinvested back into the fund
For the CIRM with initial value of 1 at
o The accumulated value at
is
o The accumulated value at is o Recursively, the accumulated value
at is Similar to the SIRM, suppose interest
accrues continuously
o Note that If the initial value of 1 is
accumulated for
periods,
its accumulated value is
reinvested back into the
fund
Accumulating moreperiods yields the equality
o If , then o Therefore, ; moreover,
1.3. Measures of Interest1. Effective Rate of Interest Amount of interest earned by a deposit of 1
at the beginning of the period, during that
period, where the interest is credited at the
end of the period
We denote the ERI of the period by o For the SIRM
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o Therefore, for the SIRM, isconstant , as o For the CIRM
o Therefore, for the CIRM,
as
is constant
Suppose we know and we want toknow so as to achieve the specifiedvalue of , known as the present valueproblem
Consider the CIRM with ERI , then
Let , called the discountfactor, then 2. Effective Rate of Discount Amount of interest earned by a deposit of 1
done at the end of the period, during that
period, where interest is credited at the
start of the period
We denote the ERD of the period by o For the SIRM
o For the CIRM
o For the SIRM, , whileit is constant for the CIRM 3. Nominal Rate of Interest Suppose is the nominal rate of interest
per year/period,
payable/convertible/compounded every. / of a year/period, then the ERI per
./
of a year/period is
Thus, * and *4. Nominal Rate of Discount Suppose is the nominal rate of
discount per year/period,
payable/convertible/compounded every. / of a year/period, then the ERD per. /
of a year/period is
Thus, * and *
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Theorem:
For a CIRM
Remark:
1. Discount is also known as advance interest
Theorem: Equations of Value
Let be the net contributions at time ,and be the net returns at time , then
1.4.Force of Interest Suppose that we give (nominal) interest
continuously
o We define the force of interest asthe instantaneous rate of change
for our fund/portfolio
o We take , hence
*o If we have such that
*
o Therefore, Theorems:
Assuming we have a CIRM, with constantinterest rate , then
Suppose, however, that interest varies over
time, i.e. is the ERI of the period,then
If the force of interest varies, then { }
{ }Proof:
{
}
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1.5.Miscellaneous Models1. Fractional Periods
The SIRM maximizes the accumulated valuefor , while the CIRM maximizes theaccumulated value for
Therefore, the SIRM is preferred forfractional periods
Let o If is the ERI during the
period, then
2. Simple Discount Let be the ERI on the whole periods,
then Since , we can take , thus, , where is known as
the simple rate of discount
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2. Analysis of Annuities2.1.Annuities An annuity is a series of periodic payments The duration of an annuity is called its term Annuities may be, in nature, deterministic
(annuity-certain), or probabilistic
(contingent annuity)
Types of Annuity Certain
1. Annuity Immediate Consider an -year annuity that pays
amounts of 1 at the end of each year,
starting from the 1st up until the year,with ERI
o The present value of the annuity at , denoted by | , is given by|
o The accumulated value of the
annuity at , denoted by | , isgiven by
| |
2. Annuity Due Consider an -year annuity that pays
amounts of 1 at the beginning of each year
starting from the 1stup until the year, with ERI
o The present value of this annuity at , denoted by | , is given by| o The accumulated value of the
annuity at , denoted by | , isgiven by
| |
3. Deferred Annuities Consider an AI that pays amounts of 1,
starting at
up until
,
with ERI
o The present value of the annuity at deferred by periods,denoted by | , is given by
|| | |
|
o The accumulated value of thedeferred AI is nothing but |
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Similarly, consider an AD that pays amountsof 1, starting at up until
o The present value of the annuity at
deferred by
periods,
denoted by | , is given by|| | | | o The accumulated value of the
deferred AD is nothing but | Properties:
a. | | Proof:
|
| b. | | Proof:
|
|
c. | | Proof:
|
| 2.2.Perpetuities An annuity with an infinite number of
payments, with , is called a perpetuity Consider a perpetuity immediate, that is
| |
Next, consider a perpetuity due, that is | |
Consider an
-year perpetuity immediate
that pays amounts of 1 per year, that is || |
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Next, consider an -year perpetuity duethat pays amounts of 1 per year, that is
|| |
For a perpetuity immediate that pays
amounts of 1 per year, we define the
current value as
| | Similarly, for a perpetuity due, we define
the current value as
| | Remarks:
1. The accumulated value is the function for during or after the last payment
2. The present value is the function for before or during the first payment
3. The current value is the function for during the derivation of the annuity
2.3.General Annuities and Perpetuities Consider an -year AI paying an initial
amount of at , and increasing eachsucceeding payment by
o The present value is given by
|
o Let
| (| ) | |
o Specifically taking and The present value, denoted
by
|, is given by
| | | | | | |
The accumulated value,denoted by | , is givenby
| | | If we take
to approach
infinity such that we have a
PI with an initial payment of
1, increasing by 1
thereafter, we define the
present value, denoted by| , as
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| | |
|
o Now, we take and
The present value, denotedby | , is given by
| | |
| | | | The accumulated value,
denoted by | , is givenby
|
| | Now, for a n -year AD paying an initial
amount of at , and increasing eachsucceeding payment by
o The present value is given by
| |
+
| | o Thus, if we take and
The present value, denotedby | , is given by
| | The accumulated value,
denoted by | , is givenby | | If we take to approach
infinity such that we have a
PD with an initial payment
of 1, increasing by 1
thereafter, we define the
present value, denoted by| , as| | | |
o Then, if we take and The present value, denoted
by | , is given by | | The accumulated value,
denoted by | , is givenby | |
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2.4.Inflation Rate Inflation is the change in the price of the
commodity due to the law of supply and
demand
Inflation rate is determined by theconsumer price index
Consider an -year AI having an initialpayment of 1 and each payment thereafter
increases by a factor of ,with as the ERI per year
o The present value is defined as
( ) + ( ) *o If we let , where is the inflation
rate and the real/adjusted interestrate, then
| o For an annuity with inflation rate,
the accumulated value is nothing
but the present value accumulated
by the original interest
2.5.Continuous Annuities Suppose the function
is continuous
First, consider the -year annuity that paysat a rate of 1 per yearo The present value, denoted by | ,
is given by
|
o The accumulated value, denoted by
| , is given by | | Now, consider an -year annuity that pays
at a rate of
at time
o The present value, denoted by | , is given by |
| o The accumulated value, denoted by | , is given by
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| | |
| Lastly, we take the -year annuity with
payments at the rate of at time o The present value, denoted by | , is given by
|
| | | | |
o The accumulated value, denoted by | , is given by | | |
2.6.Annuities Payable at a Different Frequencythan Interest is Convertible
Recall that in evaluating annuities, we let be the ERI per period, in which case we takeone period to be the time between
periods of payments
We then divide the period into two: theInterest Conversion Period (ICP), and the
Payment Period (PP)
Case 1: ICP is less than PP
Let there be PPs per ICP, with as the ERIper ICP Consider the annuity that pays unit
amounts at time , divided equally at theend of each PP
o The present value, denoted by |,is given by
|
o The accumulated value, denoted by|, is given by
|
For the same annuity that pays at thebeginning of each PP
o The present value, denoted by |,is given by
| o The accumulated value, denoted by
|, is given by
| Next, we consider the annuity that pays at
the rate of at time , divided equally at theend of each respective PPs
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o The present value, denoted by|, is given by |
| || | | o The accumulated value, denoted by
|, is given by
| | For the same annuity that pays at the
beginning of each PP
o The present value, denoted by|, is given by
| |
o The accumulated value, denoted by|, is given by| |
Lastly, consider the annuity whosepayments increase by
per PPo The present value, denoted by
()|, is given by
. / | . / | |
()| |
o The accumulated value, denoted by()|, is given by
(
)|
|
Case 2: ICP is greater than PP
We make the outstanding assumption thatthere are exactly ICPs per period
Consider the annuity that pays amounts of1 every periods starting from the periods up until the period, with ERI
o The present value is given by
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||
||
o The accumulated value is given by || ||
Similarly, if we consider an annuity thatpays amounts of 1 every periods startingimmediately up until the period
o The present value is given by
|| || o The accumulated value is given by
|
|
||
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3. Cash Flow Analysis3.1.Yield Rates We first define the variables as the
investors net contribution at time , asthe investors net return at time
, and
as the investors outstanding balance at
time Let be the net present value of all thes under the ERI
Now, a yield rate of an investors portfolio
is an ERI for which
Notes:
, where is the ERI
for the investors portfolio/account
Remark:
1. The yield rate and the present value areinversely related
3.2.Existence and Uniqueness of the Yield Rate Without loss of generality, will be
used
Suppose that
, then
o By the Descartes Rule of Signs, ifwe have an odd number of
alternating signs in our s, thenwe are sure of the existence of a
yield rate
o By the same line of reasoning, ifthere is exactly one pair of
alternating signs in our s, thenwe are sure of the uniqueness of
the yield rate
Theorem:
Let be the outstanding balance of a fundat time , then if , and , then there exists a uniqueYR for the said fund
Proof:
Suppose and are the YRs of the fund, and , without loss of generality Define
as the fund balance at time
corresponding to and as the fundbalance at time corresponding to
Therefore, the yield rate is unique
3.3.Interest Measurement of a Fund Consider a company having an initial net
asset of at the start of the fiscal year Moreover, let be the net asset at the end
of the same year
o The accumulated value is given by
o Approximating the YR by using
simple interest
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o Define
|
* * o Let be the net interest earned by
the company during the whole year
3.4.Approximation Methods1. Linear Interpolation
We first initialize and , such that and , as stated by theIntermediate Value Theorem
We then create a line passing through( )and ( ) The zero of this line will approximate the
zero of Ideally,
must be small, so as to
attain a better approximate
Suppose we know and and wewant to find -| -
o One formula is derived as
o A similar formula is derived as
2. Newton-Raphson Method
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We first initialize , then compute for thefunction value , and its first derivativevalue,
The tangent line is defined as
, where
is solved by letting The whole process is repeated until isattained satisfying ||
The iteration formula is given by
3.5.Reinvestment Rates Consider an initial deposit of 1 at o Normally, ,
assuming that both principal and
interest is reinvested back
For some cases, the interest earned by theprimary fund is being reinvested at a
secondary fund with less interest rate
(which might be zero)
This new interest rate is called thereinvestment rate
We denote the interest rate credited by theprimary fund with , and the interest rateused by the second fund with , with ,usually
Consider a deposit of 1 at o The accumulated value is given by
|
Now, consider a unit annuity immediatepaying for periods
o The accumulated value is given by
| |
,
Lastly, consider an -period annuity dueo The accumulated value is given by
|
For the three payment patterns, thepresent value at is nothing but theaccumulated value at discounted forperiods using the yield rate
3.6.Miscellaneous Methods There are 2 underlying elements present in
cash flow analysis: time and money We make a distinction between interest
rates: time-weighted which depends solely
on time, and money-weighted which
depends on both time and money; an
example of this is the yield rate
Consider an asset manager that manages aportfolio with more than one investor
o At , is given by theinvestors
o At , if a new investorcontributes to , only the originalinvestors will get
o At , the new investor joins inthe shares of the original investors
in
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Methods for Fund Resource/Balance Allocation
1. Portfolio Method Uses the YR for one period, where the new
investor is treated like any other investor
2. Investment-Year Method Introduces new money rates, which are
given in an investment-year calendar (IYC)
o In using an IYC, a selected investoris subjected to a different set of
investment rates for a certain time
period before being subjected to
the portfolio rate (or, in the above
example, the interest rate of the
original investors)
o The calendar is used from left toright (until the PR), and then down
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4. Loan Repayment4.1.Amortization Scheduling Consider a loan for years with an initial
loan balance of | o We can repay the loan by means of
an -year unit AIo We want to find out what part ofour payments go to repaying the
original loan, and what of it goes to
paying the interest
Define as the outstanding loan balanceat time , as the payment at time , asthe principal repaid at time , and as theinterest paid at time
For this method of loan repayment, theinitial loan balance is set to be | which,eventually, would be zero at the finalperiod, with unit periodic payments
The amortization schedule is given asfollows:
- - - | |
|
TOTAL | | - The following formulas are used for
amortization scheduling:
Notes:
In making a schedule, we always computelast If , then it is called a balloon
payment, whereas if , thenit is called a drop payment
4.2.Sinking Fund Method Consider the equation | | Proof:
We define | as the annual payments of an-year AI with , and | as the annual payments of an -year AI with plus interest earned by aunit loan
|
|
In contrast to the amortization method, thesinking fund method assumes that the
borrower pays periodic interest while
investing on a fund (that usually earns at a
lower rate than the loan) so as to
accumulate the necessary principal to repay
his debt
We assume that the loan credits at a rate and the sinking fund earns at a rate We define as the interest paid at time ,as the sinking fund deposit at time ,
as the interest earned by the sinking fund at
time , as the sinking fund balance at
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5. Analysis of Financial Instruments5.1.Financial Instruments A stock is a share of a company/corporation A bond is a formal contract of indebtedness,
usually issued by the borrower
Types of Bonds
1. Accumulation Bond Usually defined by its accumulated value2. Coupon Bond Defined by face value at expiry plus periodic
payments of interest
Define as the face value of the bond(usually by hundreds), as the redemptionvalue of the bond, as the periodiccoupons, as the yield rate of the bond, as the coupon rate of the bond, and asthe purchase price
o The present value is given by |
|
| | | | | o This formula is called the
premium/discount formula
o Usually, we assume that if is not explicitly given
Types of Bonds with respect to and 1. Premium 2. Discount 3. Par
5.2.Callable Bonds For the usual bond, the investor can
only realize his return at maturity of
the bond
Callable bonds are defined as bondsredeemable/callable prior to maturity
For callable bonds, we price the bondsuch that the investors minimum YR is
satisfied, i.e. we get the possible prices
of the bond and get its minimum
Consider the scenario: suppose we havea bond with periodic coupons of thatwill mature at after 2 periods, and iscallable a period before maturity at
.
If an investor having a YR of at least
is
going to buy the bond, how much is she
willing to pay?
o The prices of the bond are given by |
o Note that, in general, the two pricesare not equal, and without loss of
generality,
o Suppose we price the bond at ;however, if the investor calls thebond at , we realize that
Obviously, ; moreover, , thus, we cant
price the bond at o In the other case where we price
the bond at , whatever thescenario is, i.e. the bond is called ormatured, then
|
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o Therefore, we take to be thebond price
In general, when ricing a callable bond, wetake all possible bond prices for each
admissible coupon date, then we take the
minimum
Next, consider a bond that matures at timeat par, and is callable at any coupon dateform time up until time at
Suppose we have an investor with a YR of atleast , whose bond with period couponsmay be matured with face value , or calledwith redemption value
|
| | o If the bond is selling at a premium,
i.e. , then | | So if we want to be
minimum, then we take the
minimum time, i.e. we price
at the earliest possible time
o If the bond is selling at a discount,i.e. , then | |
So if we want to beminimum, then we take the
maximum time, i.e. we
price at the latest possible
timeo If the bond is selling at par, i.e. , then | |
This means that the bond iscalled, which is just priced
as Therefore, in general (i.e. there are more
callable periods), we price the bond as
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6. Survival Models6.1.Future Lifetime Random Variable Consider a life aged , denoted by
o Let
be the continuous random
variable representing the future
lifetime of , then isdefined as the age-at-death random
variable for Associated with is the distribution
function , which is the probability thatdies within years is also known as the lifetime
distribution from age , denoted by Define the survival function
, which is theprobability that will survive to age , denoted by The survival function can be viewed as the
probability of surviving more yearsgiven an earlier survival of years; i.e.
|
Thus, the total survival probability can beexpressed as a product of more survival
probabilities
Properties of the Survival Function:
a. b. c. is a decreasing function
Remarks:
1. 2. Proof:
3. |, defined as the probability that
dies between ages and , orthe deferred mortality probability, is given
as
| 4.
6.2.Force of Mortality The force of mortality is the instantaneous
rate of mortality
is called the force of mortality,
denoted by
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can be expressed as , , or Using
| { }o If , then { }
The probability density function canbe expressed by and the force ofmortality
( )
6.3.Mean and Variance of Define as the expected future
lifetime of , the complete expectation oflife, or simply the mean of , denoted by
|
Define the variance of as
6.4.Curtate Future Lifetime Random Variable Let be the discrete random variable
representing the number of completed
future years of prior to death, which isthe integer part of the future lifetime for
Note that Define the probability mass function as
|
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Define as the curtate expectation of life
Define the variance of ,
+
6.5.Fractional Ages There are two assumptions when dealing
with probabilities with fractional ages: the
uniform distribution of deaths (UDD), and
constant force of mortality (CFM)
Let 1. Uniform Distribution of Deaths Linear interpolation is used for integer-age
probabilities
o For
o Thus, ; moreover,
For the force of mortality,
o Thus, ;moreover, the density/mass
function is simply The earlier derivations are used for
fractional-age probabilities
o For
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o Thus, 2. Constant Force of Mortality The force of mortality is constant between
integer ages, but differs for every interval
For integer-age probabilities, { }
{ }o Thus, ; moreover,
For the force of mortality, { }
For fractional-age probabilities,o Since is independent of ,
then o Thus, the density/mass function is
6.6.Special Laws of Mortality1. De Moivre (1724) , where is defined as
the limiting age
2. Gompertz (1825) , where is defined as the aging hazard
3. Makeham (1860) , where is defined as the accident
hazard
4. Weibull (1939) 6.7.Life Tables A life table (illustrative) is a roster of
mortality probabilities and other actuarial
indices for integer ages under a certain
rate/force
Define as the expected number ofsurvivors to age , and as the expectednumber of newborns
ca be expressed as the ratio of
over
; thus, can be expressed as Define as the expected number ofdeaths between ages and , which isgiven by
can be expressed as the ratio of over
6.8.Select and Ultimate Life Tables When the probability is defined by asurvival function appropriate for newborns,
under the single hypothesis that the
newborn has survived to age ceterisparibus, an aggregate table is used
When additional knowledge is availableabout , then special forces of mortality
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that incorporate those information as
considered to construct a select life table
The conditional probability of death in eachyear of duration is denoted by -
The impact of selection may diminish withtime, i.e. -
In general, - is the probability that aperson aged , selected at age , willdie within years, with the impact ofselection diminishing over time, i.e.
- The smallest integer satisfying the above is
called the select period of the policy and
beyond this is the ultimate period
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7. Analysis of Life Insurances7.1.Life Insurances There are two ways of paying the death
benefit (DB): DB paid at the moment of
death (MOD), and DB paid at the end of
year of death (EOYOD)
A single payment for the coverage that ismade at the beginning of policy issue is
called the net single premium (NSP),
expected present value (EPV), or the
actuarial present value (APV)
Define as the benefit function, as thediscount factor function, as thepresent value function, and
as
the random variable representing the
present value of the death benefit at policy
issue
Types of Insurances Payable at Moment of
Death (MOD)
1. Whole Life Insurance Insurance issued to with unit DB
payable at MOD, for any time
The EPV, denoted by , is given by 2. -year Term Insurance Insurance issued to with unit DB if MOD
occurs within the next years The EPV, denoted by
|, is given by
|
3. -year Pure Endowment This provides an endowment benefit (EB) of
1 only if survives to age The EPV, denoted by
|, or more
commonly, , is given by 4. -year Endowment This provides a DB of 1 if dies within
years and an EB of 1 if
survives to age
The EPV, denoted by | , is given by
| | | 5.
-year Deferred Whole Life Insurance
Similar to the Whole Life Insurance, but thecoverage is deferred years from policyissue
The EPV, denoted by |, is given by
| |
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Types of Insurances Payable at End of Year of
Death (EOYOD)
If death occurs between integer ages, thebenefit will be paid a year immediately after
the curtate lifetime The random variable will then be , in general1. Whole Life Insurance The EPV, denoted by is given by
2. -year Term Insurance The EPV, denoted by | , is given by
| 3. -year Pure Endowment The EPV, denoted by
|, or
, is given
by
4. -year Endowment The EPV, denoted by | , is given by
| | 5. -year Deferred Whole Life Insurance The EPV, denoted by |, is given by
| |
7.2.Relationship Between Insurances Payableat Moment of Death and End of Year of
Death
Define as the random variablerepresenting the fractional time betweenthe actual time of death and the curtate
Consider the whole life insurance ,assuming a UDD over each unit interval
Thus, be assuming UDD on a unit interval,the following identities are established:
1. 2. | | 3. | | |
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4. | | | | 5. | |7.3.Variance of Define the raw moments as ( )Theorem:
Without loss of generality, consider a wholelife insurance issued to payable at MOD
o Let be the force of interest attime , the benefit function, and
the discount function based on
o If , then ( ) , denoted by Corollary:
The form of the variance of , in general, isthe difference of the insurance measured at
double-force, and the square of the
insurance at single-force
7.4.Varying Benefit Insurance1. Increasing Insurance Suppose that the NSP is to be found for a
whole life insurance of 1 if dies duringthe 1styear, 2 if dies during the 2ndyear,and so on, if benefit is payable at (i) MOD,
and (ii) EOYOD
Case 1: MOD
The NSP for this case, denoted by , is
given by
Case 2: EOYOD
The NSP for this case, denoted by , isgiven by
2. Decreasing Insurance Suppose that the NSP is to be found for an
-year term insurance with DB of
,
where
is the number of completed years,
if DB is payable at (i) MOD, and (ii) EOYOD
Case 1: MOD
The NSP for this case, denoted by | ,
is given by
|
Case 2: EOYOD
The NSP for this case, denoted by | ,
is given by
|
7.5.Commutation Notations Consider an insurance that pays benefits at
EOYOD
o The EPV of this insurance can besolved using the elements found in
an illustrative life table
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Define the following symbols
Identities of Commutation Notations
1. Proof:
2. 3. Identities with EOYOD Insurances
1. Proof:
2. | Proof:
|
3. | 4. | Proof:
| 5. Proof:
6. | 7. |
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8. Analysis of Life Annuities8.1. Life Annuities A life annuity is a series of payments made
continuously or at equal intervals while a
given life survives
Payments may be due at the beginning ofthe payment intervalsan annuity-dueor
at the end of each payment interval an
annuity-immediate
Consider a life annuity-due with unit levelpayments
o Define as the present valuerandom variable of payments to be
made by the annuitant
o The general form of
is
|, the
present value of an annuity-due of
an annuitant, for the complete
years of life plus 1
o The EPV can be seen as the innerproduct of the valuated payments
of an annuity-certain and the
survival probabilities, or as the
inner product of the present value
random variable and the probability
mass function
Types of Discrete Life Annuities
1. Whole Life Annuity This annuity pays amounts of one for as
long as shall live Define | The EPV can be derived in two ways
|
The EPV, denoted by , is given by
2. -year Temporary Annuity This annuity pays amounts of one for as
long as shall live for years Define |
|
The EPV, denoted by|
, is given by
| | 3. -year Deferred Whole Life Annuity This annuity pays amounts of one for as
long as shall live after years fromissuance
Define (| | ) The EPV, denoted by
|, is given by
| 4. -year Certain and Life Annuity This annuity pays amounts of one for the 1st years, and then for as long as shall
live
Define | | The EPV, denoted by
|, is given by
| | |
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8.2. Continuous Life Annuities Consider a life annuity that pays amounts of
one continuously
o The general form of would thenbe
|, where
is the age-at-
death random variable
o The general form of the EPV wouldthen be the integral of the product
of the discount function and the
survival probabilities
Types of Continuous Life Annuities
1. Whole Life Annuity Define
|
The expected value of can be derived assuch: | |
The EPV, denoted by , is given by
The variance is given by
2. -year Term Life Annuity Define | | The EPV, denoted by | , is given by
| |
The variance is given by(| ) .| | /3. -year Deferred Whole Life Define (| | ) The EPV, denoted by |, is given by
| The variance is given by
(|) (|)4. -year Certain and Life Annuity Define | | The EPV, denoted by | , is given by
| | |
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8.3. Life Annuities Payable at a DifferentFrequency than Interest is Convertible
Consider an EOYOD whole life annuity-duethat pays amounts of
every
.
/
of a
periodo The EPV for this annuity with payments per year, denoted by, is given by
However, similar annuities are not included
in the life table as such; thus, the payments
must be expressed in terms of yearly life
annuities
Theorem:
Under UDD, ; thus, *
Identities of Life Annuities Payable at aDifferent Frequency than Interest is
Convertible
1. | . /2.
3. | | . /4. | . / . / |
5. | | | 8.4. Commutation Notations Since life annuities are based on life
insurances, the same symbols will be used
Identities for Life Annuities
1. Proof:
2. | Proof:
|
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9. Premiums and Reserves9.1. Net Level Premiums Define as the loss random variable, i.e.
the random variable representing the
difference of the present value of benefits
and the present value of premiums at time
Theorem: Equivalence Principle
The net level premium is derived from thezero expected loss at the initial time, i.e.
A fully continuous net level premium is a
combination of an insurance payable at
MOD and insurance premiums payablecontinuously
A fully discrete net level premium is acombination of an insurance payable at
EOYOD and insurance premiums payable at
the beginning of each year
A semi-continuous net level premium is acombination of a discrete/continuous
insurance and a continuous/discrete
insurance premium; the former
combination is more common in practice ly premiums are net level premiums
whose payments of insurance premiums
are payable times per year
Types of Net Level Premiums per Life Insurance
and Annuity
1. Ordinary Life A fully discrete NLP, denoted by
, is given
by
A semi-continuous NLP, denoted by ,
is given by
A fully continuous NLP, denoted by
,
is given by
An ly fully discrete NLP, denoted by, is given by
An ly semi-continuous NLP, denoted by, is given by
2. -year Term A fully discrete NLP, denoted by | , is
given by
| ||
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A semi-continuous NLP, denoted by .| /, is given by
.| / ||
A fully continuous NLP, denoted by .| /, is given by .| / ||
An ly fully discrete NLP, denoted by|
, is given by
| || An ly semi-continuous NLP, denoted by .| /, is given by
.| / ||
3. -year Pure Endowment A fully discrete NLP, denoted by | , is
given by
| ||
A semi-continuous NLP, denoted by
| , is given by |
||
A fully continuous NLP, denoted by | , is given by
| |
| An ly fully discrete NLP, denoted by|, is given by
| ||
An
ly semi-continuous NLP, denoted by
| , is given by |
|| 4. -year Endowment A fully discrete NLP, denoted by | is
given by
| || A semi-continuous NLP, denoted by(| ), is given by
(| ) || A fully continuous NLP, denoted by
(| ), is given by(| ) ||
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A semi-continuous NLP, denoted by | , is given by
| |
| A fully continuous NLP, denoted by
| , is given by |
|| An
ly fully discrete NLP, denoted by
|, is given by|
|| An ly semi-continuous NLP, denoted by
| , is given by | ||
8. -pay -year Endowment A fully discrete NLP, denoted by | , is
given by
| || A semi-continuous NLP, denoted by(| ), is given by
(| ) ||
A fully continuous NLP, denoted by(| ), is given by
(|
) ||
An ly fully discrete NLP, denoted by|, is given by| ||
An ly semi-continuous NLP, denoted by(| ), is given by
(| ) ||
9.2. Gross/Expense-Loaded Premiums Gross, or expense-loaded, premiums are
those which consider the expenses made
for the policy
Types of Policy Expenses
1. Commission to the Soliciting Agents Agents are compensated by combination of
salary and commission arrangement or by
straight commission basis
Uses a commission scale, although thecommission percentage varies by plan and
amount of benefit
2. Premium Tax A percentage of the premiums, net or gross3. Government Tax Value-added, license, or city tax4. Fees for Medical Examination and
Inspection Reports
Does not vary much by premium but theyvary in the policy size
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6. -pay -year Term .| /
| .
| / |
* | 7. -pay -year Pure Endowment
| | | | | 8. -pay -year Endowment (| ) (| (| )| ) | 9.4. Benefit Reserves Retrospective
Approach
Recall: Prospective Approach
The prospective approach looks forwardand calculates what is needed to cover
future obligations
The reserve at time is the actuarialpresent value of the insurance from age minus the actuarial present value ofthe future benefit premiums payablefrom age
The retrospective approach, on the otherhand, looks back at what funds have
accumulated and need to be kept for the
future
The reserve at time is the actuarialaccumulated value of the premiums from
age to minus the actuarialaccumulated value of the benefits paid
Without loss of generality, consider a fullydiscrete whole life insurance
o For the prospective method,
o However, for the retrospectivemethod,
| | | |
Theorem:
Without loss of generality, consider again afully discrete whole life insurance
o The net level benefit reserve fromthe prospective approach is equal
to the net level benefit reserve
from the retrospective approach
Proof:
.| | / ( )
( )
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Note:
The concept of the actuarial accumulatedvalue (AAV) is similar with discounting an
amount
, i.e. if the APV of an amount
from time is , then the AAV of anamount to time is 9.5. Other Reserve Formulas1. Premium Discount Formula This formula is analogous with that of
callable bonds
Without loss of generality, consider a fullydiscrete whole life insurance using theprospective method
2. Paid-up Insurance Formula If after paying premiums from age to for a fully continuous whole life
insurance (without loss of generality), the
policyholder decides to stop paying
premiums, the reserves can be given to the
policyholder as a whole life insurance with
death benefit , thus
3. Annuity Reserve Formula Without loss of generality, consider a fully
discrete whole life insurance using the
prospective method
4. Facklers Method in Reserve Valuation This is one of the first recursion formulas
developed for the computation of benefit
reserves
Without loss of generality, consider a fullydiscrete whole life insurance using the
prospective method
| *
( )