Theory of Interest and Mathematics of Life Contingencies

8/13/2019 Theory of Interest and Mathematics of Life Contingencies

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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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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



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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 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

| *

( )

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Theory of Interest and Mathematics of Life Contingencies

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