MLC Review

Life Contingencies

Paris’s Exam MLC Seminar of 52 www.steveparisseminars.com

LIFE CONTINGENCIES

Single Life Model Basic Random Variables X – denotes the age-at-death random variable (x) – denotes a life aged x (ie someone who’s already survived to age x) ω – denotes terminal age (unless otherwise stated, we assume ∞=ω ) (no one survive past age ω ) T = T(x) – denotes the continuous future lifetime of (x) random variable T = X – x | X > x Note that X = T(0) K = K(x) – denotes the curtate future lifetime of (x) random variable This is a discretization of the T random variable.

Life Contingencies


The Age-at-Death Random Variable: X ( )ω,0)( =XSupp

(cumulative distribution function – cdf)

0)Pr()( qxXxF xX =≤= (Recall that fF =′ , the probability density function (pdf))

(survival function – sf)

00 1)Pr()( qpxXxs xxX −==>= (Note that fs −=′ ) (force of mortality – fom) aka hazard rate or failure rate

[ ])(ln()()()( xs

dxd

xsxfx X

X

X −==μ

Comments and Concepts 1. The force of mortality is the rate of mortality at a particular point in time. The expression dxx)(μ represents the probability that a newborn that has survived to age x dies in the next instant “dx”. 2. ))(exp(

00 ∫−=x

x dssp μ 3. ∫∫ ==≤=

x

s

x

Xx dsspdssfxXq0 000 )()()Pr( μ

Life Contingencies


The (continuous) Future Lifetime Random Variable: T = T(x) ( )xTSupp −= ω,0)(

(pdf):

)Pr()()(

tXtxftf X

T >+

=

(cdf):

)Pr()Pr(1

)Pr()Pr()Pr()(

xXtxX

xXtxXxqtTtF xtT >

+>−=

>+≤<

==≤=

(sf): xt

x

txxtT q

pp

xXtxXptTts −==

>+>

==>= + 1)Pr(

)Pr()Pr()(0

0

Comments and Concepts 1. )()( txptf xtT +⋅= μ

2. ∫ +⋅=

t

xsxt dssxpq0

)(μ

3. ∫∫+

−=+−=tx

x

t

xt dssdssxp ))(exp())(exp(0

μμ Also, ∫∞

+⋅=t xsxt dssxpp )(μ

4. xx qq =1 and xx pp =1

5. (survival factorization) nxtxnxtn ppp ++ ⋅=

6. (deferred mortality)

txuxtxtxutxutxt

ut

t xsxut qpqqppdssxputxTtq +++

+⋅=−=−=+⋅=+≤<= ∫ )())(Pr(| μ

7. [ ] )()( txptfpdtd

xtTxt +⋅=−−= μ and so xt

xt

p

pdtd

tx][

)(−

=+μ

8. [ ] [ ])()( txxpp

dxd

xtxt +−= μμ (Use 2nd equality in 3. above and the FTC)

Life Contingencies


9. (complete expectation of life for (x), aka mean residual lifetime)

∫ ∫∞ ∞

=+⋅⋅==0 0

0)()]([ dtpdttxptxTEe xtxtx μ (The last equality is used often.)

10. ∫ ∫

∞ ∞⋅=+⋅⋅=

0 0

22 2)(]))([( dtptdttxptxTE xtxt μ (Helps us get variance of T.)

11. (n-year temporary complete expectation of life for (x); this is the expected number of years lived by (x) between ages x and x+n)

∫∫ =⋅++⋅⋅=∧=n

xtxn

n

xtnx dtppndttxptnxTEe00

|:

0)(])([ μ

12. (Recursion Formulas)

nxxnnxx epee +⋅+=0

|:

00 and when n = 1 we get 1

01

0

0

+⋅+= ∫ xxxtx epdtpe

|:

0

|:

0

|:

0

mnmxxmmxnx epee −+⋅+=

13. The kth percentile of T(x), denoted ck, is found by solving kpxck

01.1−= The median future lifetime of (x) is c50.

14. The mode of T is the point(s) at which the pdf )(tfT is maximized.

Life Contingencies


The Curtate Future Lifetime Random Variable: K = K(x) ( )1,,2,1,0)( −−= xKSupp ωL

The probability distribution table for K is

K Pr 0 Pr(0 ≤ T < 1) = Pr(0 < T ≤ 1) = )( |0 xx qq = 1 Pr(1 ≤ T < 2) = Pr(1 < T ≤ 2) = xq|1 2 Pr(2 ≤ T < 3) = Pr(2 < T ≤ 3) = xq|2 M M

Note: nxxnxnxnxnxnxn qpqqppq +++ ⋅=−=−= 11|

(cdf): ∑=

=≤=k

nxnK qkKkF

0|)Pr()( (Notice that xk qkTkK 1)1Pr()Pr( +=+≤=≤ )

Comments and Concepts 1. (curtate expectation of life for (x))

∑ ∑∞

=

∞

=

=⋅==0 1

|)]([k k

xkxkx pqkxKEe

(The last equality is used often. Notice the index starts at k = 1.)

2. ∑ ∑∞

=

∞

=

⋅−=⋅=0 1

|22 )12(]))([(

k kxkxk pkqkxKE (Helps us get variance of K.)

3. (n-year temporary curtate expectation of life for (x); this is the expected

complete number of years lived by (x) between ages x and x+n)

∑=

=n

kxknx pe

1|:

4. (Recursion Formulas)

nxxnnxx epee +⋅+= |: and when n = 1 we get )1( 1++⋅= xxx epe

|:|:|: mnmxxmmxnx epee−+

⋅+=

Life Contingencies


Life Table Notation

0l = (arbitrary) number of newborns

00 pll xx ⋅= = (expected) number of survivors at age x (Note: 0

0 llp x

x = )

=−= +nxxxn lld number of deaths between ages x and x+n ( xx dd =1 ) Note: 11 −++ +++= nxxxxn dddd L Formulas and concepts involving life table notation 1.

x

nxxn l

lp += and xnx

nxx

x

xnxn p

lll

ldq −=

−== + 1 and

x

mnxnx

x

nxmxmn l

llldq ++++ −

==|

2. [ ]

x

x

l

ldxd

x−

=)(μ , and so [ ] )(xlldxd

xx μ⋅−=

3. ∫ +⋅= +

n

txxn dttxld0

)(μ (follows since ∫ +⋅=n

xtxn dttxpq0

)(μ )

4. =xn L the total number of years lived in the next n years by the xl people alive at age x

])(|)([0

nxTxTEdlndyldtlL xnnx

n nx

x ytxxn <⋅+⋅=== +

+

+∫ ∫

Note that xn

n

tx

xn

n

xt

d

dttxlt

q

dttxptnxTxTE ∫∫ +⋅⋅

=+⋅⋅

=<+00

)()(])(|)([

μμ

5. xnxnxxn dllL

dxd

=−−= +][

6. xx LT ∞= the total number of years lived in the future by the xl people alive

at age x

7. nxxxn TTL +−= and xx lTdxd

−=][

Life Contingencies


Life Table Notation (continued) 8.

x

xx

lTe =

0 and

x

xnnx

lLe =|:

0

9.

x

x

lYxTE 2]))([( 2 = where ∫

∞

+=0

dtTY txx

10.

xn

xnxn L

dm = is the n-year central mortality weight

Life Contingencies


Extending Discrete to Continuous (Fractional Age Assumptions) (x is an integer and all formulas are valid for 10 ≤≤ t ) UDD (Uniform Distribution of Deaths: xxt dtd ⋅= )

xxt qtq ⋅= , and so xxt qtp ⋅−= 1 , and xT qtf =)( (a constant wrt t) Then xxtx dtll ⋅−=+ and

x

x

qtqtx⋅−

=+1

)(μ and x

xx q

qm5.01−

=

Defining V(x) to be the random variable representing the fraction of the year lived in the year in which (x) dies, we can relate the random variables T and K; namely, T(x) = K(x) + V(x), and K and V are independent. Under UDD, we have V(x) ~ U(0,1). Then 5.0

0+= xx ee . Also )(5.0|:|:

0

xnnxnx qee += and 121))(())(( += xKVarxTVar

For 10 ≤+≤ ts , x

xsxt qs

qtq⋅−

⋅=+ 1

and x

xsxt qs

qtsxp

⋅−=++⋅+ 1

)(μ

Constant Force (Exponential Interpolation: μμ =+ )( tx ) For 10 ≤+≤ ts , t

xt

sxt pep )(== ⋅−+

μ

μ=xm Balducci (Hyperbolic Interpolation)

⎟⎟⎠

⎞⎜⎜⎝

⎛−+=

++ xxxtx llt

ll1111

1

x

x

qtqtx

⋅−−=+

)1(1)(μ and for 10 ≤+≤ ts , )(

)1(1tsxt

qtsqtq

x

xsxt ++⋅=

⋅−−−⋅

=+ μ

Life Contingencies


Select and Ultimate Mortality Select mortality rates are used for a period (usually 3 years or less) and are different than the ultimate (general population) rates. [x] denotes an x-year old for which select rates are used starting at age x [x] + k denotes an (x + k)-year old for which select rates are used starting at

age x [x + k] denotes an (x + k)-year old for which select rates are used starting at

age x + k (x + k) denotes an (x + k)-year old for which ultimate rates are used starting

at age x + k Comments: 1. [x] + k = (x + k) if k exceeds the select period

2. The force of mortality for [x] is denoted )(txμ .

Life Contingencies


Multiple-Life Models For the moment, we consider only two independent lives (x) and (y). The models can easily be extended to more than two lives. Joint-Life Status: )}(),({)( yTxTMinxyT = (Joint-life status fails on the earlier of the deaths of (x) and (y)) ytxtxyt pptyTtxTtxyTp ⋅=>⋅>=>= ))(Pr())(Pr())(Pr( xytxyt pq −= 1 ∫

∞=

0

0dtpe xytxy

∑∞

=

=1k

xykxy pe

xyutxytxyut ppq +−=| )()()( ttt yxxy μμμ += Last-Survivor Status: )}(),({)( yTxTMaxxyT = (Last-survivor status fails on the latest of the deaths of (x) and (y)) For independent lives (x) and (y): ytxtxyt qqtyTtxTtxyTq ⋅=≤⋅≤=≤= ))(Pr())(Pr())(Pr( xytxyt qp −= 1 xytxyutxyut qqq −= +|

xyt

xyxytyytxxtxy p

tptptpt

)()()()(

μμμμ

⋅−⋅+⋅=

Life Contingencies


Contingent Probabilities Notation: yxn q 1 = Pr((x) dies first and within the next n years)

yxq 1∞ = Pr((x) dies first)

2yxn q = Pr((y) dies second and within the next n years)

The notation yx

1 indicates that the joint-life status (xy) fails due to the failure

of the (x) status, ie the death of (x). Contingent Probability Formulas: ∫ ⋅=

n

xxytyxn dttpq0

)(1 μ

∫ ⋅⋅=n

yytxtyxn dttpqq0

)(2 μ Contingent Probability Relationships: 1. yxnyxnxn qqq 21 +=

2. 21

yxnyxnyn qqq +=

3. 11yxnyxnxyn qqq +=

4. 22

yxnyxnxyn qqq +=

Life Contingencies


Common Shock Model (In recent exams, this has been the only model tested in which the lifetimes of (x) and (y) are dependent.) Notation: Z = the common shock random variable. Always )1(~

λ=meanEXZ . (λ = the fom of the shock)

T*(x) = future lifetime of (x) random variable in the absence of the common shock ( *))(*Pr( xt ptxT => ) T*(y) = future lifetime of (y) random variable in the absence of the common shock ( *))(*Pr( yt ptyT => ) Z, T*(x), and T*(y) are all assumed mutually independent Define }),(*{)( ZxTMinxT = and define }),(*{)( ZyTMinyT = . (T(x) and T(y) are not independent, and }),(*),(*{)( ZyTxTMinxyT = ) Formulas: t

xtxtxt eptZptxTp λ−⋅=>⋅=>= ** )Pr())(Pr( t

ytytyt eptZptyTp λ−⋅=>⋅=>= ** )Pr())(Pr( t

ytxtytxtxyt epptZpptxyTp λ−⋅⋅=>⋅⋅=>= **** )Pr())(Pr( Often Tested Special Case of Common Shock Model (x) and (y) have constant forces, xμ and yμ , respectively

Given: )1(~)(*

x

meanEXxTμ

= and )1(~)(*y

meanEXyTμ

=

Formulas: t

xtxep ⋅−= μ* and t

ytyep ⋅−= μ*

txt

xep ⋅+−= )( λμ and tyt

yep ⋅+−= )( λμ t

xytyxep ⋅++−= )( λμμ

Special Probability: Pr((x) and (y) die within n years by the common shock) ( )∫∫ ⋅++−⋅++− −⋅

++=⋅=⋅=

n n

yx

tn

xytyxyx edtedtp

0

)()(

01 λμμλμμ

λμμλλλ

Life Contingencies


Insurance Present Value Random Variables Single Life

Insurance Payable At The End Of The Year Of Death (unless otherwise stated, benefit amount is 1 and age at issue is x) (PVRV = Present Value Random Variable) (SBP = Single Benefit Premium) (APV = Actuarial Present Value) 1. whole life insurance

The probability distribution table for the PVRV, xZ , is

xZ Probability v Pr(K = 0) = xq

2v Pr(K = 1) = xq|1 3v Pr(K = 2) = xq|2 M M

PVRV = xZ = 1+Kv SBP = APV = L++=== +

xxxK

x qvvqAvEZE |121][][

L++== xxxx qvqvAZE |1

4222 ][ . Therefore, 221 )()()( xxK

x AAvVarZVar −== + Comments and Concepts: (Applies to all insurances in this section.) (i) A2 means to perform the same calculation as with A , except use double the force of interest. We will generally have for insurance that if Z is the PVRV, then E[Z] = A and AZE 22 ][ = . This will not be the situation for annuities. (ii) We can calculate probabilities involving the random variable Z by rewriting the event in terms of the random variable K.

Life Contingencies


2. n-year term insurance (benefit is paid if death occurs within next n years)

The probability distribution table for the PVRV, |:1

nxZ , is

|:1

nxZ Probability v Pr(K = 0) = xq

2v Pr(K = 1) = xq|1 3v Pr(K = 2) = xq|2 M M nv Pr(K = 1−n ) = xn q|1−

0 Pr(K n≥ ) = xn p

PVRV = |:1

nxZ = ⎩⎨⎧

≥<+

nKnKv K

L

L

0

1

SBP = APV = xnn

xxnxnx qvqvvqAZE |1|12

|:|:11 ][ −+++== L

xnn

xxnxnx qvqvqvAZE |12

|142

|:22

|:11 ])[( −+++== L .

Therefore, 2|:|:

2|: )()( 111

nxnxnx AAZVar −= Comments and Concepts: The symbol |:

1nx is based on the contingent probability notation from

page 11 of these notes. Here, the “life” y is |n , an n-year certain period. Observe that n-year term insurance pays a death benefit when the status

|:1

nx fails. That is, the death benefit is paid on the death of (x) as long as this death occurs before the death of the n-year certain period (within n years).

Life Contingencies


3. n-year pure endowment (benefit is paid if participant survives n years)

The probability distribution table for the PVRV, 1|:nxZ , is

1|:nxZ Probability

0 Pr( nK < ) = xn q nv Pr( nK ≥ ) = xn p

PVRV = 1|:nxZ =

⎩⎨⎧

≥

<

nKvnK

nL

L0

SBP = APV = xnn

nxnx pvAZE ==11|:|: ][ (Another notation: xn

nnxxn pvAE ==1|: )

xnn

nxnx pvAZE 2|:

22|:

11 ])[( == . Therefore, 2|:|:

2|: )()( 111

nxnxnx AAZVar −= Comments and Concepts:

The symbol 1

|: nx is based on the contingent probability notation from page 11 of these notes. Observe that an n-year pure endowment pays

when the status 1

|: nx fails. That is, the benefit is paid on the death of the n-year certain period, |n , as long as this death occurs before the death of (x). This is equivalent to saying the benefit is paid after n years, as long as (x) survives that long.

Life Contingencies


4. n-year endowment insurance (benefit is paid at end of year of death if participant dies before age x + n, and benefit is paid at age x + n if participant survives to age x + n) The probability distribution table for the PVRV, |:nxZ , is

|:nxZ Probability v Pr(K = 0) = xq

2v Pr(K = 1) = xq|1 3v Pr(K = 2) = xq|2 M M nv Pr(K = 1−n ) = xn q|1− nv Pr(K n≥ ) = xn p

PVRV = |:nxZ = ⎪⎩

⎪⎨⎧

≥

<+

nKvnKv

n

K

L

L1

= |:1

nxZ + 1|:nxZ

SBP = APV = 11|:|:|:|: ][ nxnxnxnx AAAZE +==

11|:

2|:

2|:

22|: ])[( nxnxnxnx AAAZE +== . Therefore, 2

|:|:2

|: )()( nxnxnx AAZVar −= Comments and Concepts: (i) Observe that n-year endowment insurance pays when the joint life status |: nx fails. The benefit is guaranteed to be paid and will be paid at the earlier of the death of (x) and the death of |n .

(ii) On a random variable level, 2|:

2|:

2|: )()()( 11

nxnxnx ZZZ += since 011|:|: =⋅ nxnx ZZ .

So the expectation calculations above should be clear.

(iii) WARNING: )()()( 11|:|:|: nxnxnx ZVarZVarZVar +≠ It is easy to show that if

X and Y are random variables such that their product is 0, then YXYXCov μμ ⋅−=),( . Therefore the correct variance formula is

1111|:|:|:|:|: 2)()()( nxnxnxnxnx AAZVarZVarZVar −+=

(iv) With n = 1, we get vpqvpvqvA xxxxx =+=⋅+⋅= )(|1:

Life Contingencies


5. n-year deferred whole life insurance (benefit is paid at end of year of death if participant dies after age x + n, no benefit is paid if participant dies prior to age x + n) The probability distribution table for the PVRV, xn Z| , is

xn Z| Probability 0 Pr( nK < ) = xn q

1+nv Pr(K = n) = xn q| 2+nv Pr(K = n + 1) = xn q|1+

M M

PVRV = xn Z| = |:11

0nxxK ZZ

nKvnK

−=⎩⎨⎧

≥

<+ L

L

SBP = APV = |:||1][ nxxxnxn AAAZE −==

|:222

|2

|1])[( nxxxnxn AAAZE −== . Therefore, 2

|2|| )()( xnxnxn AAZVar −=

Comments and Concepts: (i) We also have the following APV formula: nxxn

nxn ApvA +=|

(ii) Remembering the meaning of A2 , we also get nxxn

nxn ApvA +⋅⋅= 222

| (iii) The ideas above can be extended to an n-year deferred, j-year term insurance, or an n-year deferred, j-year pure endowment, or a … If you are tested on one of these other insurance types, just use the basic principals learned by studying the above insurances.

Life Contingencies


6. whole life increasing insurance beginning at 1 The probability distribution table for the PVRV, xIZ )( , is

xIZ )( Probability v Pr(K = 0) = xq

22v Pr(K = 1) = xq|1 33v Pr(K = 2) = xq|2 M M

PVRV = xIZ )( = 1)1( +⋅+ KvK SBP = APV = L+++== xxxxx AAAIAIZE |2|1)(])[(

7. n-year term increasing insurance beginning at 1

The probability distribution table for the PVRV, |:1)( nxIZ , is

|:1)( nxIZ Probability

v Pr(K = 0) = xq 22v Pr(K = 1) = xq|1 33v Pr(K = 2) = xq|2 M M

nnv Pr(K = 1−n ) = xn q|1− 0 Pr(K n≥ ) = xn p

PVRV = |:1)( nxIZ =

⎩⎨⎧

≥<⋅+ +

nKnKvK K

L

L

0)1( 1

SBP = APV =

11123

12

|:|: 32)(])([ 11−+−++ ⋅⋅++⋅⋅+⋅⋅+⋅== nnxn

nxxxxxnxnx qpnvqpvqpvqvIAIZE L

Life Contingencies


8. n-year decreasing insurance to 1

The probability distribution table for the PVRV, |:1)( nxDZ , is

|:1)( nxDZ Probability

nv Pr(K = 0) = xq 2)1( vn − Pr(K = 1) = xq|1 3)2( vn − Pr(K = 2) = xq|2

M M nv Pr(K = 1−n ) = xn q|1−

0 Pr(K n≥ ) = xn p

PVRV = |:1)( nxDZ =

⎩⎨⎧

≥<⋅− +

nKnKvKn K

L

L

0)( 1

SBP = APV = |1:|2:|1:|:|:|:111111 )(])([ xnxnxnxnxnx AAAADADZE ++++== −− L

Remark: |:|:|:

111 )1()()( nxnxnx AnIADA +=+

Life Contingencies


Insurance Payable At The Moment Of Death 1. whole life insurance

PVRV = TT

x evZ ⋅−== δ SBP = APV = ∫

∞ ⋅− ⋅⋅==0

)(][ dttpeAZE xxtt

xx μδ

∫∞ ⋅− ⋅⋅==

0

222)(][ dttpeAZE xxt

txx μδ

Therefore, 22

)()( xxx AAZVar −= . Important Comment on Notation: We are using a bar over Z to denote a different random variable than the random variable Z. Sometimes a bar over a random variable indicates the mean of a random sample from the same distribution as the random variable. That’s not the case here. What we mean by using the bar notation is that the insurance is paid at the time of death, and thus the PVRV is a function of the complete future lifetime random variable, T.

2. n-year term insurance

PVRV = ⎩⎨⎧

≥<

=nT

nTvZ

T

nxL

L

0|:

1

SBP = APV = ∫ ∫ ⋅⋅=⋅== ⋅−n n

xxtt

Tt

nxnx dttpedttfvAZE0 0

|:|: )()(][ 11 μδ

∫ ⋅⋅== ⋅−n

xxtt

nxnx dttpeAZE0

2|:

22|: )(])[( 11 μδ

Therefore, 2

|:|:2

|: )()( 111nxnxnx AAZVar −= .

Life Contingencies


3. n-year endowment insurance

PVRV = 11|:|:|: nxnx

n

T

nx ZZnTvnTv

Z +=⎩⎨⎧

≥

<=

L

L

SBP = APV = 11|:|:|:|: ][ nxnxnxnx AAAZE +==

11|:

2|:

2|:

22|: ])[( nxnxnxnx AAAZE +==

Therefore, 2

|:|:2

|: )()( nxnxnx AAZVar −= . Comments and Concepts: (Comments (ii) – (iv) are analogues to the same numbered comments in the n-year endowment insurance payable at the end of the year of death case – “the discrete case”.) (i) Notice that the pure endowment random variable is always discrete, and so we do not have a 1

|:nxZ random variable. Subsequently, there are no actuarial symbols 1

|:nxA or 1|:

2nxA .

(ii) On a random variable level, 2|:

2|:

2|: )()()( 11

nxnxnx ZZZ += since 011|:|: =⋅ nxnx ZZ .

So the expectation calculations above should be clear.

(iii) WARNING: )()()( 11|:|:|: nxnxnx ZVarZVarZVar +≠ It is easy to show that if

X and Y are random variables such that their product is 0, then YXYXCov μμ ⋅−=),( . Therefore the correct variance formula is

1111|:|:|:|:|: 2)()()( nxnxnxnxnx AAZVarZVarZVar −+=

(iv) We do not get the nice formula in the n = 1 case that we got in the discrete case.

Life Contingencies


4. n-year deferred whole life insurance

PVRV = |:|1

0nxxTxn ZZ

nTvnT

Z −=⎩⎨⎧

≥

<=

L

L

SBP = APV = |:||

1][ nxxxnxn AAAZE −==

|:222

|2

|1])[( nxxxnxn AAAZE −==

Therefore, 2

|2|| )()( xnxnxn AAZVar −= .

Comments and Concepts: (These comments are analogues to the same numbered comments in the n-year deferred whole life insurance payable at the end of the year of death case – “the discrete case”.) (i) We also have the following APV formula: nxxn

nxn ApvA +=|

(ii) Remembering the meaning of A2 , we also get nxxn

nxn ApvA +⋅⋅=

222|

(iii) The ideas above can be extended to an n-year deferred, j-year term insurance, or an n-year deferred, j-year pure endowment, or a … If you are tested on one of these other insurance types, just use the basic principals learned by studying the above insurances.

Life Contingencies


Often Tested 1-Year Recursion Relationships

11|1:1

++ +=+= xxxxxxx AvpvqAvpAA

|1:1|1:1|1:|:1111

−+−+ +=+= nxxxnxxxnx AvpvqAvpAA

|1:1|1:1|1:|:1

−+−+ +=+= nxxxnxxxnx AvpvqAvpAA The next three are the continuous analogues to the previous three.

1|1:1

++= xxxx AvpAA

|1:1|1:|:111

−++= nxxxnx AvpAA

|1:1|1:|:1

−++= nxxxnx AvpAA Non-unit Insurance (benefit payable at time T is bT)

PVRV = Z = T

T vb (whole life)

Z = ⎩⎨⎧

≥<

nTnTvb T

T

L

L

0 (n-year term)

M M Z2 = T

T vb 22 (whole life)

Z2 = ⎩⎨⎧

≥<

nTnTvb T

T

L

L

0

22

(n-year term)

M M Notice that E[Z2] will not equal 2A unless the benefit is 1.

Life Contingencies


Annuity Present Value Random Variables Single Life

Life Annuities (Due) Payable At The Beginning Of The Year (all the following annuities have an annual payment of 1) 1. whole life annuity due

The probability distribution table for the PVRV, xY&& , is

xY&& Probability |1a&& Pr(K = 0) = xq

|2a&& Pr(K = 1) = xq|1 M M

PVRV = xY&& = dZ

dva x

K

K

−=

−=

+

+

11 1

|1&&

SBP = APV = LL&&&&&&&& +⋅++=+⋅+⋅== xxxxxx pvvpqaqaaYE 2

2|1|2|1 1][

Comments and Concepts:

(i) d

AdZEYEa xx

xx−

=⎥⎦⎤

⎢⎣⎡ −

==11][ &&&& (equivalently, xx adA &&⋅−= 1 )

(ii) ( )22

22 )(1)(1)( xxxx AAd

ZVard

YVar −⋅=⋅=&&

(iii) We can calculate probabilities involving the random variable Y by rewriting the event in terms of the random variable K.

Life Contingencies


2. n-year temporary life annuity due (pays 1 until the earlier of the death of the annuitant or an n-year certain period) The probability distribution table for the PVRV, |:nxY&& , is

|:nxY&& Probability |1a&& Pr(K = 0) = xq

|2a&& Pr(K = 1) = xq|1

|3a&& Pr(K = 2) = xq|2 M M

|na&& Pr(K = 1−n ) = xn q|1−

|na&& Pr(K n≥ ) = xn p

PVRV = |:nxY&& = ⎪⎩

⎪⎨⎧

≥

<+

nKa

nKa

n

K

L&&

L&&

|

|1 =

⎪⎪⎩

⎪⎪⎨

⎧

≥−

<− +

nKdv

nKdv

n

K

L

L

1

1 1

= dZ nx |:1−

SBP = APV =

xnn

xxxnnxnnxnxnx pvpvvppaqaqaaYE 11

22

||1||1|:|: 1][ −−

− ⋅++⋅++=⋅+⋅++⋅== L&&&&L&&&&&& Comments and Concepts:

(i) dA

dZ

EYEa nxnxnxnx

|:|:|:|:

11][

−=⎥

⎦

⎤⎢⎣

⎡ −== &&&& (equivalently, |:|: 1 nxnx adA &&⋅−= )

(ii) ( )2

|:|:2

2|:2|: )(1)(1)( nxnxnxnx AAd

ZVard

YVar −⋅=⋅=&&

(iii) We denote the actuarial accumulated value (AAV) of an n-year temporary annuity due by |:nxs&& and define it by the relationship

xnn

nxnx pvsa ⋅⋅= |:|:&&&& . That is,

1|:

|:|:|:|:

nx

nx

xn

nx

xnn

nxnx

A

a

E

a

pv

as

&&&&&&&& ==

⋅= .

(See page 44 for a more complete discussion on AAV’s.)

Life Contingencies


3. n-year deferred life annuity due The probability distribution table for the PVRV, xnY&&| , is

xnY&&| Probability 0 Pr(K < n) = xn q

|1| an && Pr(K = n) = xn q|

|2| an && Pr(K = n+1) = xn q|1+ M M

PVRV = xnY&&| = |1| nKn a

−+&& =

⎪⎩

⎪⎨⎧

≥−<

+nKaa

nK

nKL&&&&

L

||1

0 = |:nxx YY &&&& −

SBP = APV = L&&&&&&&& +⋅+⋅=−== +

+xn

nxn

nnxxxnxn pvpvaaaYE 1

1|:|| ][

Comments and Concepts: (i) nxxn

nxn apva +⋅⋅= &&&&|

(ii) nxxn

nnxxnnxx apvaaaa +⋅⋅+=+= &&&&&&&&&&

|:||: (iii) ( ) xnnxnxxn

nxn aaapv

dYE &&&&&&&& 2

|222

|2])[( +−⋅⋅⋅= ++ (unlikely to see this on the exam)

Life Contingencies


4. n-year certain and life annuity due (pays 1 until the later of the death of the annuitant or an n-year certain period) The probability distribution table for the PVRV,

|:nxY&& , is

|:nx

Y&& Probability

|na&& Pr(K < n) = xn q

|1+na&& Pr(K = n) = xn q|

|2+na&& Pr(K = n+1) = xn q|1+ M M

PVRV = |:nx

Y&& = xnn Ya &&&& ||+ = ⎪⎩

⎪⎨⎧

≥

<

+nKa

nKa

K

n

L&&

L&&

|1

|

SBP = APV = xnnnxnx

aaaYE &&&&&&&&|||:|:

][ +== Comments and Concepts: Observe that an n-year certain and life annuity due pays until the failure of the last-survivor status |: nx (here, the “life” y is |n , an n-year certain period). That is, the annuity pays until the later of the death of (x) and the death of |n , or equivalently, it pays as long as (x) is alive, with a minimum of n years.

Life Contingencies


Life Annuities (Immediate) Payable At The End Of The Year (all the following annuities have an annual payment of 1) Comments and Concepts: When dealing with annuities immediate, use the above formulas for annuities due and relationships between annuities immediate and annuities due. Examples:

1. If xY is the PVRV for a whole life annuity immediate, then 1−= xx YY && . Then SBP = APV = L&&&& +⋅+⋅=−=−== xxxxxx pvpvaYEYEa 2

21]1[][ and )()( xx YVarYVar &&= .

Important Special Formula: If i = 0, then )]([ xKEea xx ==

2. If |:nxY is the PVRV for an n-year temporary life annuity immediate,

then 1|1:|: −=+nxnx YY && . Then SBP = APV =

xnn

xxnxnxnxnx pvpvpvaYEYEa ⋅+⋅+⋅=−=−==++

L&&&&2

2|1:|1:|:|: 1]1[][ and

)()( |1:|: += nxnx YVarYVar &&

Important Special Formulas: (i) If i = 0, then |:|: nxnx ea = (ii) xnnxxn

nnxnx Eapvaa +−=⋅+−= 11 |:|:|:

&&&&

3. If xnY| is the PVRV for an n-year deferred life annuity immediate, then

xnxn YY &&|1| += . Then SBP = APV = xnxnxn aaYE &&|1|| ][ +== and

)()( |1| xnxn YVarYVar &&+=

Life Contingencies


Continuous Life Annuities (all the following annuities have an annual payment rate of 1) 1. whole life continuous annuity

PVRV = δδ

xT

TxZvaY −

=−

==11

|

SBP = APV = ∫∫

∞∞⋅=⋅⋅==

00| )(][ dtpvdttpaaYE xt

txxttxx μ

Comments and Concepts:

(i) δδ

xxxx

AZEYEa −=⎥

⎦

⎤⎢⎣

⎡ −==

11][ (equivalently, xx aA ⋅−= δ1 )

(ii) ( )22

22 )(1)(1)( xxxx AAZVarYVar −⋅=⋅=δδ

(iii) If i = 0, then xx ea

0= .

Important Comment on Notation: Just as with the case of insurance payable at the moment of death, we are using a bar over Y to denote a different random variable than the random variable Y. What we mean by using the bar notation is that the annuity is paid continuously, and thus the PVRV is a function of the complete future lifetime random variable, T.

Life Contingencies


2. n-year temporary life continuous annuity

PVRV = δ

|:

|

||:

1 nx

n

Tnx

ZnTa

nTaY −

=⎪⎩

⎪⎨⎧

≥

<=

L

L

SBP = APV = ∫∫ ⋅=⋅+⋅⋅==

n

xttn

xnnxxttnxnx dtpvpadttpaaYE00

|||:|: )(][ μ Comments and Concepts:

(i) δδ

|:|:|:|:

11][ nxnxnxnx

AZEYEa −=⎥

⎦

⎤⎢⎣

⎡ −== (equivalently, |:|: 1 nxnx aA ⋅−= δ )

(ii) ( )2

|:|:2

2|:2|: )(1)(1)( nxnxnxnx AAZVarYVar −⋅=⋅=δδ

(iii) If i = 0, then |:

0

|: nxnx ea = . (iv) We denote the actuarial accumulated value (AAV) of an n-year temporary continuous annuity by |:nxs and define it by the relationship

xnn

nxnx pvsa ⋅⋅= |:|: . Then 1|:

|:|:|:|:

nx

nx

xn

nx

xnn

nxnx

A

aE

apv

as ==⋅

= .

(See page 44 for a more complete discussion on AAV’s.)

Life Contingencies


3. n-year deferred whole life continuous annuity

PVRV = δ

xnxnxx

nTn

nTxn

ZZYYnTav

nT

nTaa

nTY −

=−=⎩⎨⎧

≥⋅

<=

⎩⎨⎧

≥−

<=

−

|:|:

||||

00

L

L

L

L

SBP = APV = ∫∫

∞∞− ⋅=⋅⋅⋅=−==

n xtt

n xxtntn

nxxxnxn dtpvdttpavaaaYE )(][ ||:|| μ Comments and Concepts: (i) nxxn

nxn apva +⋅⋅=|

(ii) nxxn

nnxxnnxx apvaaaa +⋅⋅+=+= |:||:

(iii) ( )nxnxxn

nxn aapvYE ++ −⋅⋅⋅=

222|

2])[(δ

(unlikely to see this on the exam)

4. n-year certain and life continuous annuity

PVRV = ⎪⎩

⎪⎨⎧

≥

<=+=

nTa

nTaYaY

T

nxnnnx

L

L

|

||||:

SBP = APV = xnnnxnx aaaYE |||:|: ][ +==

Life Contingencies


Life Annuities Due Payable mthly These annuities also have an annual payment of 1. Thus the symbol )(m

xaC &&⋅

represents the APV of a life annuity to (x) that pays mC at the beginning of

each of m periods per year for the life of (x). The total annual payment each year is C. Under the UDD assumption we have: )()()( mama x

mx βα −⋅= &&&& and ( )xnnx

mnx Emama −⋅−⋅= 1)()( |:

)(|: βα &&&&

where the values of )(mα and )(mβ will be given in the tables Often Tested 1-Year Recursion Relationships

( )11

|1:1|1:|:

1|1:

|1:1|:

1

1

11

++

−+

+

−+

+

+⋅⋅=⋅⋅+⋅=⋅⋅+=

⋅⋅+=

⋅⋅+=⋅⋅+=

xxxxxx

nxxxnx

xxxx

nxxnx

xxx

apvapvpvaapvaa

apvaa

apvaapva&&&&

&&&&

Life Contingencies


Insurance and Annuity Present Value Random Variables Multiple Life

Contingent Insurance Payable at the Moment of Death Type 1: Pays 1 on the death of (x) if (x) dies first SBP = APV = ∫

∞⋅⋅=

0)(1 dttpvA xxyt

tyx μ

Type 2: Pays 1 on the death of (x) if (x) dies second SBP = APV = ∫

∞⋅⋅⋅=

0)(2 dttpqvA xxtyt

tyx μ

Interchange the roles of (x) and (y) to get similar formulas for insurance payable on the death of (y).

Contingent insurance payable at the end of the year of death (Use summations instead of integrals)

Contingent Insurance Relationships

1. yxyxx AAA 21 +=

2. 21yxyxy AAA +=

3. 11yxyxxy AAA +=

4. 22yxyxxy AAA +=

There are similar relationships in the discrete case, i.e. no bars.

Life Contingencies


Continuous Insurance and Annuities for Joint-Life Status (Replace x in all single life formulas by xy) 1. ∫

∞⋅⋅==

0

)( )(][ dttpvvEA xyxyttxyT

xy μ

2. ∫∞

⋅⋅==0

2)(22)(][ dttpvvEA xyxyt

txyTxy μ

So 22)()( xyxyxy AAZVar −= = variance of PVRV for insurance that pays 1 at

the moment of the first death of (x) or (y)

3. ∫∞ −

=⋅=0

1δ

xyxyt

txy

Adtpva

4. ∫−

=⋅=n nxy

xytt

nxyAdtpva

0

|:|:

1δ

This is the APV of an n-year temporary life annuity that pays continuously at a rate of 1 per year for the joint lifetimes of (x) and (y). This annuity pays until failure of the |: nxy status, which fails at the first of the death of (x), the death of (y), and the death of |n . That is, the annuity pays until the first of the death of (x) or the death of (y), up to a maximum of n years.

5. ∫ ⋅⋅=n

xyxytt

nxy dttpvA0

|: )(1 μ

6. ( )22

2 )(1)( xyxyxy AAYVar −=δ

= variance of PVRV for a continuous annuity

that pays 1 per year for as long as the joint-life status (xy) survives, i.e. as long as both (x) and (y) are alive. M You get the idea.

Life Contingencies


Discrete Insurance and Annuities for Joint-Life Status (Replace x in all single life formulas by xy)

1. ∑∞

=

++ ⋅==0

|11)( ][

kxyk

kxyKxy qvvEA

2. ∑∞

=

++ ⋅==0

|)1(2)1)((22 ][

kxyk

kxyKxy qvvEA

So 22 )()( xyxyxy AAZVar −= = variance of PVRV for insurance that pays 1 at the end of the year of the first of the death of (x) or the death of (y)

3. xynn

n

kxyk

kxyn

nnxynxy pvqvpvAA ⋅+⋅=⋅+= ∑

−

=

+1

0|

1|:|:

1

4. dA

pva xy

kxyk

kxy

−=⋅= ∑

∞

=

1

0

&&

5. d

Apva nxy

n

kxyk

knxy

|:1

0|:

1−=⋅= ∑

−

=

&&

6. ( )2

|:|:2

2|: )(1)( nxynxynxy AAd

YVar −=&& = variance of PVRV for an n-year temporary

life annuity due that pays 1 at the beginning of each year that the joint-life status |: nxy survives, i.e. as long as both (x) and (y) are alive, up to a maximum of n years. M

You get the idea.

Life Contingencies


Insurance and Annuities for Last-Survivor Status Method 1: Replace x in all single life formulas by xy , or this is the same as replacing the subscript xy in the joint-life status formulas above by xy . Method 2: (This can make many computations much easier.) Make use of the following relationships. Often Used Relationships Between Joint-Life and Last-Survivor Statuses

Note: )()()()( yTxTxyTxyT +=+ and )()()()( yTxTxyTxyT ⋅=⋅

1. ytxtxytxyt pppp +=+

2. ytxtxytxyt qqqq +=+

3. yxxyxy eeee0000

+=+

4. yxxyxy eeee +=+

5. yxxyxy AAAA +=+

6. yxxyxy aaaa +=+

7. |:|:|:|: nynxnxynxy AAAA +=+

8. |:|:|:|:1111

nynxnxynxy AAAA +=+

9. |:|:|:|: nynxnxynxy aaaa +=+

10. ynxnxynxyn aaaa |||| +=+

11. ⎟⎠⎞

⎜⎝⎛ −⋅⎟

⎠⎞

⎜⎝⎛ −+= xyyxyx eeeeyTxTCovxyTxyTCov

0000))(),(())(),((

There are formulas in the continuous case too ( a ’s and A ’s), and there are other formulas, e.g. n-year deferred insurance, pure endowments, ….

Life Contingencies


Loss Random Variables and Reserves

We focus on the single life case with issue age x. The ideas can be extended to the multiple life case, as will be seen in some of the examples that we’ll do. Loss-at-issue Random Variable Suppose insurance is purchased with annual premiums of Q. Then the random variable representing the loss-at-issue will be of the form YQZL ⋅−= . Examples: 1a. Fully Continuous Whole Life Insurance of 1 with premiums of Q for life

δδδQZQZQZYQZL x

xxxx −⋅⎟

⎠⎞

⎜⎝⎛ +=

−⋅−=⋅−= 11

δδQAQaQALE xxx −⋅⎟

⎠⎞

⎜⎝⎛ +=⋅−= 1][

( ) ( )2222

)(11)( xxx AAQZVarQLVar −⋅⎟⎠⎞

⎜⎝⎛ +=⋅⎟

⎠⎞

⎜⎝⎛ +=

δδ

1b. Fully Discrete Whole Life Insurance of 1 with premiums of Q for life

dQZ

dQ

dZQZYQZL x

xxxx −⋅⎟

⎠⎞

⎜⎝⎛ +=

−⋅−=⋅−= 11

dQA

dQaQALE xxx −⋅⎟⎠⎞

⎜⎝⎛ +=⋅−= 1][ &&

( ) ( )2222

)(11)( xxx AAdQZVar

dQLVar −⋅⎟

⎠⎞

⎜⎝⎛ +=⋅⎟

⎠⎞

⎜⎝⎛ +=

2a. Fully Continuous n-year Endowment Insurance of 1 with premiums of Q

δδδQZQZQZYQZL nx

nxnxnxnx −⋅⎟

⎠⎞

⎜⎝⎛ +=

−⋅−=⋅−= |:

|:|:|:|: 11

δδQAQaQALE nxnxnx −⋅⎟

⎠⎞

⎜⎝⎛ +=⋅−= |:|:|: 1][

( ) ( )2|:|:

22

|:

2

)(11)( nxnxnx AAQZVarQLVar −⋅⎟⎠⎞

⎜⎝⎛ +=⋅⎟

⎠⎞

⎜⎝⎛ +=

δδ

Life Contingencies


2b. Fully Discrete n-year Endowment Insurance of 1 with premiums of Q

dQZ

dQ

dZ

QZYQZL nxnx

nxnxnx −⋅⎟⎠⎞

⎜⎝⎛ +=

−⋅−=⋅−= |:

|:|:|:|: 1

1&&

dQA

dQaQALE nxnxnx −⋅⎟⎠⎞

⎜⎝⎛ +=⋅−= |:|:|: 1][ &&

( ) ( )2|:|:

22

|:

2

)(11)( nxnxnx AAdQZVar

dQLVar −⋅⎟

⎠⎞

⎜⎝⎛ +=⋅⎟

⎠⎞

⎜⎝⎛ +=

You get the idea. There are many combinations of the type of insurance purchased and the method of paying premiums. We use one more example to illustrate the concept of the loss-at-issue random variable. 3. Insurance – n-year term with benefit of 1 payable at the moment of death

Premiums - paid at the beginning of each year for h (≤n) years (This is called h-payment, n-year term insurance. During the first h years, premiums are paid while living and the benefit is paid upon death. During the next n – h years, the insurance is already paid in full and so no premiums are paid, but the benefit is still paid upon death. After n years, the policy has expired and no benefit will be paid upon death.)

|:|: hxnx YQZL &&⋅−= |:|:

1][ hxnx aQALE &&⋅−= (If you’re asked about variance of this one, skip it. ☺)

Life Contingencies


Calculating Premiums Method 1: Percentiles The premium Q is found by solving a probability equation such as

05.0)0Pr( =>L (probability of a positive loss is 5%). This equation can be solved by writing the event first in terms of the random variable Z and then in terms of the random variable T (or K). Then use the distribution of T (or K) to solve the probability equation. We will illustrate this with examples.

Method 2: Equivalence Principle The premium when found using the equivalence principle is called the benefit premium.

The premium Q is found by solving the equation 0][ =LE . This is the same equation as the one obtained by setting the APV at issue of benefits equal to the APV at issue of premiums. Examples and Notation: 1. For fully discrete insurance of 1, benefit premiums are

x

xx a

AP&&

= = whole life insurance with premiums paid for life

|:

|:|:

11

nx

nxnx

aAP&&

= = n-year term insurance with at most n premiums

|:

|:|:

hx

nxnxh a

AP

&&= = h-payment n-year endowment insurance

M You get the idea. Important Fact: 11

|:|:|: nxnxnxPPP +=

Life Contingencies


2. For fully continuous insurance of 1, benefit premiums are

x

xx

aAAP =)( = whole life insurance with premiums paid for life

|:

|| )(

kx

xnxnk

aAAP = = k-payment n-year deferred whole life insurance

M You get the idea.

3. For semi-continuous insurance of 1, benefit premiums are

x

xx

aAAP&&

=)( = whole life insurance with premiums paid for life

|:

|:|:

11 )(

hx

nxnxh a

AAP&&

= = h-payment n-year term insurance

M You get the idea. Notice the notation suggests, as is correct, that the benefit is paid at the moment of death, and the premiums are paid at the beginning of each year.

4. mthly benefit premiums )12(

)12(

x

xx a

AP&&

= = fully discrete whole life insurance with premiums

of 12

)12(xP paid at the beginning of each month for life

)4(|:

|:|:

)4( )(nx

nxnx

aAAP&&

= = semi-continuous n-year endowment insurance

with premiums of 4

)( |:)4(

nxAP paid at the beginning

of each quarter until death of the insurance expires M You get the idea.

Life Contingencies


Special Relationships When Using Benefit Premiums Fully Continuous or Fully Discrete Whole Life or Endowment Insurance For fully continuous or fully discrete whole life insurance or endowment insurance, the variance of the loss-at-issue random variable had a factor of

⎟⎠⎞

⎜⎝⎛ +

δQ1 (continuous case) or ⎟

⎠⎞

⎜⎝⎛ +

dQ1 (discrete case).

If Q is the benefit premium, then we have

1. Fully Continuous Whole Life - x

xx

aAAPQ == )(

xxx

xx

x

x

AaaAa

aAQ

−=

⋅=

⋅+⋅

=⋅

+=⎟⎠⎞

⎜⎝⎛ +

11111

δδδ

δδ

2. Fully Continuous n-year Endowment Insurance - |:

|:|: )(

nx

nxnx

aAAPQ ==

|:|:|:

|:|:

|:

|:

11111

nxnxnx

nxnx

nx

nx

AaaAa

aAQ

−=

⋅=

⋅+⋅

=⋅

+=⎟⎠⎞

⎜⎝⎛ +

δδδ

δδ

3. Fully Discrete Whole Life -

x

xx a

APQ&&

==

xxx

xx

x

x

AadadAad

adA

dQ

−=

⋅=

⋅+⋅

=⋅

+=⎟⎠⎞

⎜⎝⎛ +

11111

&&&&

&&

&&

4. Fully Discrete n-year Endowment Insurance - |:

|:|:

nx

nxnx a

APQ

&&==

|:|:|:

|:|:

|:

|:

11111

nxnxnx

nxnx

nx

nx

Aadad

Aad

ad

A

dQ

−=

⋅=

⋅

+⋅=

⋅+=⎟

⎠⎞

⎜⎝⎛ +

&&&&

&&

&&

Life Contingencies


Prospective Loss At Time t Random Variable Notation: Lt denotes the prospective loss at time t random variable txtxt PVFPRVPVFBRVL ++ −= , where PVFBRVx+t = PVRV of Future Benefits from age x + t, and PVFPRVx+t = PVRV of Future Premiums from age x + t Note: LL =0 = loss-at-issue random variable. Terminal (End of Year) Reserves Notation: Vt denotes the tth year terminal (EOY) reserves ])(|[ txTLEV tt ≥= ( ])(|[ txKLEV tt ≥= in the discrete case) Prospective Calculation of Reserves: txtxt APVFPAPVFBV ++ −= , where APVFBx+t = APV of Future Benefits from age x + t, and APVFPx+t = APV of Future Premiums from age x + t Benefit reserves means the premiums are the benefit premiums. Benefit reserves can be calculated either prospectively or retrospectively. Retrospective Calculation of Benefit Reserves: [See Page 44 for a discussion of Actuarial Accumulated Value (AAV) of contingent payments.] txtxt AAVPBAAVPPV ++ −= , where AAVPPx+t = AAV of Past Premiums up to age x + t, and AAVPBx+t = AAV of Past Benefits up to age x + t

Life Contingencies


Examples of Prospective Terminal Benefit Reserves and Notation: (Insurance Benefit = 1) 1. Fully Discrete Whole Life Insurance

txxtxxt aPAV ++ −= &&

2. Fully Discrete n-year Pure Endowment

|:|:|:|:111

tntxnxtntxnxt aPAV−+−+ −= && (Important Fact: 11

|: =nxnV )

3. Fully Discrete n-year Endowment Insurance

11|:|:|:|:|:|: nxtnxttntxnxtntxnxt VVaPAV +=−=

−+−+&& (Important Fact: 1

|:=

nxnV )

4. Fully Discrete h-payment Whole Life Insurance

⎪⎩

⎪⎨⎧

≥

<−=

+

−++

htA

htaPAV

tx

thtxxhtxx

ht

L

L&&|:

5. Fully Continuous n-year Term Insurance

|:|:|:|: )()( 111

tntxnxtntxnxt aAPAAV −+−+ ⋅−=

6. Fully Continuous n-year Deferred Life Annuity

⎪⎩

⎪⎨⎧

≥

<⋅−=

+

−++−

nta

ntaaPaaV

tx

tntxxntxtnxnt

L

L|:|||

)()(

7. Semi-continuous Whole Life Insurance

txxtxxt aAPAAV ++ ⋅−= &&)()(

M You get the idea.

Life Contingencies


Special Benefit Reserve Formulas Whole Life and Endowment Insurance

1. 1

1

|:

|:)(1tx

txxtxxtx

x

txxt

P

PPaPP

aa

V−

=⋅−=−= +++ &&&&

&&

2. |:|:|:|:

|:|: )(1 tntxnxtntx

nx

tntxnxt aPP

aa

V−+−+

−+ ⋅−=−= &&&&

&&

3. )(

)()()]()([1)( 1

1

|:

|:

tx

txxtxxtx

x

txxt

APAPAPaAPAP

aaAV −

=⋅−=−= +++

4. |:|:|:|:

|:|: )]()([1)( tntxnxtntx

nx

tntxnxt aAPAP

aaAV −+−+

−+ ⋅−=−=

Terminal vs Initial Reserves and Retrospective Actuarial Calculations Calculate terminal reserves for the tth year by calculating reserves at the end of the tth year. In a prospective calculation, the benefit for the tth year will not be included, but the premium for the (t+1)st year is included. Calculate initial reserves for the tth year by calculating reserves at the beginning of the tth year. In a prospective calculation, the premium for the tth year will not be included. The initial reserves for the (t+1)st year will exceed the terminal reserves for the tth year by the amount of premium paid at time t. The retrospective calculation of reserves relies on being able to calculate the Actuarial Accumulated Value (AAV) of contingent payments. For a contingent payment at time k, the AAV at time n is calculated by actuarially accumulating the APV at time 0 of the contingent payment to time n. The APV at time 0 of a contingent payment is the interest discounted value of the expected payment. If a payment C is made at time k contingent on event E, then the APV at time 0 of the payment is kvEC ⋅⋅ )Pr( . The AAV at time n of

this payment is ( )xn

kn

xnn

k

xnn p

iECpv

vECE

APVAAV−+⋅⋅

=⋅⋅

=⋅=1)Pr()Pr(1

0 .

Life Contingencies


Special Variance Formulas for (Conditional) Loss At Time t Random Variable Fully Continuous or Fully Discrete Whole Life or Endowment Insurance For fully continuous or fully discrete whole life insurance or endowment insurance, the variance of the (conditional) loss at time t random variable is (See P. 33 of these notes.) 1. Fully Continuous Whole Life Insurance

( ) ( ) ( )222

222

)(1

1)()(1)(| txtxx

txtxx

t AAA

AAAPtxTLVar ++++ −⋅⎟⎠⎞

⎜⎝⎛−

=−⋅⎟⎟⎠

⎞⎜⎜⎝

⎛+=≥

δ

2. Fully Continuous n-year Endowment Insurance (t < n)

( ) ( )

( )2|:|:

22

|:

2|:|:

22

|:

)(1

1

)()(1)(|

tntxtntxnx

tntxtntxnx

t

AAA

AAAPtxTLVar

−+−+

−+−+

−⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

−⋅⎟⎟⎠

⎞⎜⎜⎝

⎛+=≥

δ

3. Fully Discrete Whole Life Insurance

( ) ( ) ( )222

222

)(1

1)(1)(| txtxx

txtxx

t AAA

AAdP

txKLVar ++++ −⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−

=−⋅⎟⎠⎞

⎜⎝⎛ +=≥

4. Fully Discrete n-year Endowment Insurance (t < n)

( ) ( ) ( )2|::|::

2

2

|:

2|::|::

2

2

|: )(1

1)(1)(|tntxtntx

nxtntxtntx

nxt AA

AAA

d

PtxKLVar

−+−+−+−+−⋅

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−=−⋅⎟

⎟⎠

⎞⎜⎜⎝

⎛+=≥

Life Contingencies


Recursive Relationship for Insurance Terminal Reserves (often tested) txtttxtt pvVQqvbV ++++ ⋅⋅+−⋅⋅= 11 , where Qt = premium at time t, and bt+1 = death benefit at time t+1 Recursive Relationship for Variance of (Conditional) Loss At Time t Discrete Random Variable ( )( ) )1)(|())(|( 1

2211 +≥⋅⋅+⋅⋅−=≥ ++++++ txKLVarpvqpVbvtxKLVar ttxtxtxttt

where bt+1 = death benefit at time t+1 Approximating Benefit Reserves at Fractional Durations (0 < s < 1) ))(1()())(1()())(1( 11 ttttttst QsVsVsVsQVsV −++−=++−= +++ where Qt = premium at time t Note: ))(1( tQs− is called the unearned premium

Life Contingencies


Expense Augmented Models Unless otherwise stated, assume expenses are paid BOY Exception: Settlement expenses are paid at the time benefit is paid Replace “benefits” by “benefits plus expenses” and replace “premiums” by “expense loaded premiums” in the random variable expressions discussed above. All formulas use the same concept as in the non-expense model. The following illustrates this. Recursion Relation for Expense Augmented Terminal Reserves txeattttxteat pvVQEqvsebV ++++ ⋅⋅+−+⋅⋅+= 11 )( , where eatV = expense augmented terminal reserves at time t se = settlement expenses Et = BOY expenses paid at time t Qt = expense loaded premium at time t (usually more in 1st year) Comments on Expense Loaded Premiums 1. Generally, part of the expense loaded premium will depend on the face

amount of the policy. The amount of the expense loaded premium that does not depend on the face amount of the policy is called the policy fee.

2. Letting Qt denote the expense loaded premium at time t, and letting Pt denote the benefit premium at time t, then et = Qt – Pt is called the expense loading at time t.

3. The expense loaded premium pays expenses, but not any profit. The contract premium is charged in order to expect a profit.

Separating Benefits and Expenses in an Expense Augmented Model expVVV tbenteat += , where bentV = reserves as before in the non-expense model (using the premiums Qt) expVt = expense reserves = APVF(expenses)x+t – APVF(expense loadings)x+t

Life Contingencies


Multiple Decrement Models

Two random variables: T(x) – future lifetime of (x) random variable J – mode of decrement random variable

Notation: )( j

xt q = Pr((x) departs within t years by decrement j) )( j

xt p has no meaning )(τ

xt q = Pr((x) departs within t years) = ∑j

jxt q

)(

)(τ

xt p = Pr((x) survives all decrements for t years) = )(1 τxt q−

)()( tj

xμ = force of mortality due to decrement j )()( tx

τμ = total force of mortality = ∑j

jx t)()(μ

Multiple Decrement Service Table Notation

)(τxl = the total number that have survived all decrements to age x

)( jxl = the total number at age x that will eventually depart by cause j

)(τxn d = the total number departing in next n years

)()( ττxx ld =∞

)( jxn d = the total number departing in next n years by cause j

)()( jx

jx ld =∞

Life Contingencies


Multiple Decrement Model Facts:

1. )(),( )()(, tpjtf j

xxtJT μτ ⋅=

2. )()( )()( tptf xxtTττ μ⋅=

3. )(

0

)()( )()( jx

jxxtJ qdttpjf ∞

∞=⋅= ∫ μτ

4. )(

)(

0

)()()( )( τ

ττττ μ

x

xtt

xxsxt lddsspq =⋅= ∫

5. )(

)(

0

)()()( )( ττ μ

x

jxt

t jxxs

jxt l

ddsspq =⋅= ∫

6. )(

)(

)()(

)()(

)()(

)(| τ

τ

ττ

ττ

ττ

τ

x

txu

txuxt

xtxut

xutxt

xut ld

qp

qq

pp

q +

+

+

+

=⎪⎩

⎪⎨

⎧

⋅

−

−

=

7. )(

)()()()(

| ττ

x

jtxuj

txuxtj

xut ldqpq +

+ =⋅=

8. ∫∞

==0

)()(0

][ dtpeTE xtxτ

τ

= expected time until decrement

9. )(|)1)(Pr())(Pr( τ

xk qkxTkkxK =+<≤==

10. )()()|()|Pr( )(

)(

| tttjftTjJ

x

jx

TJ τμμ

====

11. )(

)(

)|Pr( τxt

jxt

qqtTjJ =≤=

12. )(

),()|( ,

| jfjtf

jtfJ

JTJT =

13. dtjf

jtftjJTE

J

JT∫∞⋅==

0

,

)(),(

]|[ = expected time until departure, given the cause

is by decrement j

Life Contingencies


Associated Single Decrement Tables (Absolute Rates of Decrement) (Associated single decrement events are independent)

Probability Formulas:

)(

)(

)(

0

)()(][

)())(exp( jxt

jxt

jx

t jx

jxt p

pdtd

tdssp′

′−=⇒−=′ ∫ μμ

∫ ⋅′=′−=′t j

xj

xsj

xtj

xt dssppq0

)()()()( )(1 μ

∏ ′=j

jxtxt pp )()(τ

Calculating Total Probabilities:

If given associated single decrement probabilities (primes) then calculate total probabilities by

∏ ′=j

jxtxt pp )()(τ and )()( 1 ττ

xtxt pq −=

If given multiple decrement probabilities (no primes) then calculate total probabilities by

∑=j

jxtxt qq )()(τ and )()( 1 ττ

xtxt qp −=

Life Contingencies


Relating Multiple Decrement Probabilities (no primes) to Associated Single Decrement Probabilities (primes) Two Cases:

Case 1: MUDD ( )()( j

xj

xt qtq ⋅= ; UDD assumption in the multiple decrement model)

Important Formula: )(

)(

)( )()( tx

jx

qq

xsj

xs pp τ=′

Case 2: SUDD ( )()( j

xj

xt qtq ′⋅=′ ; UDD assumption in the associated single decrement model)

Important Formulas: ⎟⎠⎞

⎜⎝⎛ ′−⋅′= )2()1()1(

211 xxx qqq (2 decrement case)

⎟⎠⎞

⎜⎝⎛ ′⋅′+′+′−⋅′= )3()2()3()2()1()1(

31)(

211 xxxxxx qqqqqq (3 decrement case)

Interchange the roles of the superscripts to get probabilities of departing by causes other than cause 1. For example, interchanging the roles of 1 and 2 in the 3 decrement case gives us the probability

⎟⎠⎞

⎜⎝⎛ ′⋅′+′+′−⋅′= )3()1()3()1()2()2(

31)(

211 xxxxxx qqqqqq

It is unlikely that you will see more than 3 decrements.

Timing of Decrements

If some decrements happen at a fixed point in time, then calculate 1-year mortality probabilities in the associated single decrement table by using the

formula )(

)()(

jx

jxj

x NARd

q =′ , where )( jxNAR is the number at risk for decrement j, at

age x. Note that x

jxj

x ldq

)()( = always.

Life Contingencies


Asset Shares

This concept is based on a double decrement model, death and withdrawal. The death benefit at time h is denoted by DBh . If withdrawal occurs at time h, then a cash value, denoted CVh , is paid at that time. We let ASh denote the asset share at time h. Assume 00 =AS . There are two main formulas.

Recursion Formula:

ASpvqvseCVqvseDBEcGAS hhxw

hxhhd

hxhhhhh 1)()(

11)(

11 )()()1( ++++++++ ⋅⋅+⋅⋅++⋅⋅+=−−+ τ where G = contract premium ch = percentage of contract premium expense at time h Eh = other non-settlement expenses at time h h+1se = settlement expenses at time h+1 )(d

hxq + = probability that (x+h) dies by age x+h+1 )(w

hxq + = probability that (x+h) withdraws by age x+h+1

Time 0 Actuarial Present Value Formula: ASpvASAPV nxn

nn ⋅⋅= )(

0 )( τ = APV0(Premiums paid from time 0 through n-1, backing out non-settlement expenses) – APV0(Benefits + settlement expenses from time 1 through n)

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