MLC Review
Transcript of MLC Review
Life Contingencies
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LIFE CONTINGENCIES
Single Life Model Basic Random Variables X – denotes the ageatdeath 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
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The AgeatDeath 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
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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
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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. (nyear 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
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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) = xq1 2 Pr(2 ≤ T < 3) = Pr(2 < T ≤ 3) = xq2 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. (nyear 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
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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
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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 nyear central mortality weight
Life Contingencies
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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
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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 xyear 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
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MultipleLife Models For the moment, we consider only two independent lives (x) and (y). The models can easily be extended to more than two lives. JointLife Status: )}(),({)( yTxTMinxyT = (Jointlife 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 += LastSurvivor Status: )}(),({)( yTxTMaxxyT = (Lastsurvivor 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
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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 jointlife 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
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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
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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) = xq1 3v Pr(K = 2) = xq2 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
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2. nyear 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) = xq1 3v Pr(K = 2) = xq2 M M nv Pr(K = 1−n ) = xn q1−
0 Pr(K n≥ ) = xn p
PVRV = :1
nxZ = ⎩⎨⎧
≥<+
nKnKv K
L
L
0
1
SBP = APV = xnn
xxnxnx qvqvvqAZE 112
::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 nyear certain period. Observe that nyear 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 nyear certain period (within n years).
Life Contingencies
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3. nyear 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 nyear pure endowment pays
when the status 1
: nx fails. That is, the benefit is paid on the death of the nyear 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
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4. nyear 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) = xq1 3v Pr(K = 2) = xq2 M M nv Pr(K = 1−n ) = xn q1− 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 nyear 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
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5. nyear 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 q1+
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 nyear deferred, jyear term insurance, or an nyear deferred, jyear 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
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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) = xq1 33v Pr(K = 2) = xq2 M M
PVRV = xIZ )( = 1)1( +⋅+ KvK SBP = APV = L+++== xxxxx AAAIAIZE 21)(])[(
7. nyear 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) = xq1 33v Pr(K = 2) = xq2 M M
nnv Pr(K = 1−n ) = xn q1− 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
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8. nyear 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) = xq1 3)2( vn − Pr(K = 2) = xq2
M M nv Pr(K = 1−n ) = xn q1−
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 +=+
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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. nyear 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
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3. nyear 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 nyear 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
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4. nyear 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 nyear 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 nyear deferred, jyear term insurance, or an nyear deferred, jyear 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.
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Often Tested 1Year Recursion Relationships
111:1
++ +=+= xxxxxxx AvpvqAvpAA
1:11:11::1111
−+−+ +=+= nxxxnxxxnx AvpvqAvpAA
1:11:11::1
−+−+ +=+= nxxxnxxxnx AvpvqAvpAA The next three are the continuous analogues to the previous three.
11:1
++= xxxx AvpAA
1:11::111
−++= nxxxnx AvpAA
1:11::1
−++= nxxxnx AvpAA Nonunit Insurance (benefit payable at time T is bT)
PVRV = Z = T
T vb (whole life)
Z = ⎩⎨⎧
≥<
nTnTvb T
T
L
L
0 (nyear term)
M M Z2 = T
T vb 22 (whole life)
Z2 = ⎩⎨⎧
≥<
nTnTvb T
T
L
L
0
22
(nyear term)
M M Notice that E[Z2] will not equal 2A unless the benefit is 1.
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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) = xq1 M M
PVRV = xY&& = dZ
dva x
K
K
−=
−=
+
+
11 1
1&&
SBP = APV = LL&&&&&&&& +⋅++=+⋅+⋅== xxxxxx pvvpqaqaaYE 2
2121 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.
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2. nyear temporary life annuity due (pays 1 until the earlier of the death of the annuitant or an nyear certain period) The probability distribution table for the PVRV, :nxY&& , is
:nxY&& Probability 1a&& Pr(K = 0) = xq
2a&& Pr(K = 1) = xq1
3a&& Pr(K = 2) = xq2 M M
na&& Pr(K = 1−n ) = xn q1−
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
11:: 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 nyear 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.)
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3. nyear 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 q1+ 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)
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4. nyear certain and life annuity due (pays 1 until the later of the death of the annuitant or an nyear 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 q1+ M M
PVRV = :nx
Y&& = xnn Ya &&&& + = ⎪⎩
⎪⎨⎧
≥
<
+nKa
nKa
K
n
L&&
L&&
1

SBP = APV = xnnnxnx
aaaYE &&&&&&&&::
][ +== Comments and Concepts: Observe that an nyear certain and life annuity due pays until the failure of the lastsurvivor status : nx (here, the “life” y is n , an nyear 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.
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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 nyear temporary life annuity immediate,
then 11:: −=+nxnx YY && . Then SBP = APV =
xnn
xxnxnxnxnx pvpvpvaYEYEa ⋅+⋅+⋅=−=−==++
L&&&&2
21:1::: 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 nyear deferred life annuity immediate, then
xnxn YY &&1 += . Then SBP = APV = xnxnxn aaYE &&1 ][ +== and
)()( 1 xnxn YVarYVar &&+=
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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.
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2. nyear 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 nyear 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.)
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3. nyear 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. nyear certain and life continuous annuity
PVRV = ⎪⎩
⎪⎨⎧
≥
<=+=
nTa
nTaYaY
T
nxnnnx
L
L

:
SBP = APV = xnnnxnx aaaYE :: ][ +==
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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 1Year Recursion Relationships
( )11
1:11::
11:
1:1:
1
1
11
++
−+
+
−+
+
+⋅⋅=⋅⋅+⋅=⋅⋅+=
⋅⋅+=
⋅⋅+=⋅⋅+=
xxxxxx
nxxxnx
xxxx
nxxnx
xxx
apvapvpvaapvaa
apvaa
apvaapva&&&&
&&&&
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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.
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Continuous Insurance and Annuities for JointLife 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 nyear 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 jointlife status (xy) survives, i.e. as long as both (x) and (y) are alive. M You get the idea.
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Discrete Insurance and Annuities for JointLife 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 nyear temporary
life annuity due that pays 1 at the beginning of each year that the jointlife 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.
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Insurance and Annuities for LastSurvivor Status Method 1: Replace x in all single life formulas by xy , or this is the same as replacing the subscript xy in the jointlife status formulas above by xy . Method 2: (This can make many computations much easier.) Make use of the following relationships. Often Used Relationships Between JointLife and LastSurvivor 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. nyear deferred insurance, pure endowments, ….
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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. Lossatissue Random Variable Suppose insurance is purchased with annual premiums of Q. Then the random variable representing the lossatissue 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 nyear Endowment Insurance of 1 with premiums of Q
δδδQZQZQZYQZL nx
nxnxnxnx −⋅⎟
⎠⎞
⎜⎝⎛ +=
−⋅−=⋅−= :
:::: 11
δδQAQaQALE nxnxnx −⋅⎟
⎠⎞
⎜⎝⎛ +=⋅−= ::: 1][
( ) ( )2::
22
:
2
)(11)( nxnxnx AAQZVarQLVar −⋅⎟⎠⎞
⎜⎝⎛ +=⋅⎟
⎠⎞
⎜⎝⎛ +=
δδ
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2b. Fully Discrete nyear 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 lossatissue random variable. 3. Insurance – nyear 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 hpayment, nyear 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. ☺)
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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&&
= = nyear term insurance with at most n premiums
:
::
hx
nxnxh a
AP
&&= = hpayment nyear endowment insurance
M You get the idea. Important Fact: 11
::: nxnxnxPPP +=
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2. For fully continuous insurance of 1, benefit premiums are
x
xx
aAAP =)( = whole life insurance with premiums paid for life
:
 )(
kx
xnxnk
aAAP = = kpayment nyear deferred whole life insurance
M You get the idea.
3. For semicontinuous insurance of 1, benefit premiums are
x
xx
aAAP&&
=)( = whole life insurance with premiums paid for life
:
::
11 )(
hx
nxnxh a
AAP&&
= = hpayment nyear 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&&
= = semicontinuous nyear 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.
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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 lossatissue 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 nyear 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 nyear Endowment Insurance  :
::
nx
nxnx a
APQ
&&==
:::
::
:
:
11111
nxnxnx
nxnx
nx
nx
Aadad
Aad
ad
A
dQ
−=
⋅=
⋅
+⋅=
⋅+=⎟
⎠⎞
⎜⎝⎛ +
&&&&
&&
&&
Life Contingencies
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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 = lossatissue 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
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Examples of Prospective Terminal Benefit Reserves and Notation: (Insurance Benefit = 1) 1. Fully Discrete Whole Life Insurance
txxtxxt aPAV ++ −= &&
2. Fully Discrete nyear Pure Endowment
::::111
tntxnxtntxnxt aPAV−+−+ −= && (Important Fact: 11
: =nxnV )
3. Fully Discrete nyear Endowment Insurance
11:::::: nxtnxttntxnxtntxnxt VVaPAV +=−=
−+−+&& (Important Fact: 1
:=
nxnV )
4. Fully Discrete hpayment Whole Life Insurance
⎪⎩
⎪⎨⎧
≥
<−=
+
−++
htA
htaPAV
tx
thtxxhtxx
ht
L
L&&:
5. Fully Continuous nyear Term Insurance
:::: )()( 111
tntxnxtntxnxt aAPAAV −+−+ ⋅−=
6. Fully Continuous nyear Deferred Life Annuity
⎪⎩
⎪⎨⎧
≥
<⋅−=
+
−++−
nta
ntaaPaaV
tx
tntxxntxtnxnt
L
L:
)()(
7. Semicontinuous Whole Life Insurance
txxtxxt aAPAAV ++ ⋅−= &&)()(
M You get the idea.
Life Contingencies
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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
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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 nyear 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 nyear Endowment Insurance (t < n)
( ) ( ) ( )2::::
2
2
:
2::::
2
2
: )(1
1)(1)(tntxtntx
nxtntxtntx
nxt AA
AAA
d
PtxKLVar
−+−+−+−+−⋅
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=−⋅⎟
⎟⎠
⎞⎜⎜⎝
⎛+=≥
Life Contingencies
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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
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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 nonexpense 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 nonexpense model (using the premiums Qt) expVt = expense reserves = APVF(expenses)x+t – APVF(expense loadings)x+t
Life Contingencies
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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
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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
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
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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 −=
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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
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 1year 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.
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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 nonsettlement 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 n1, backing out nonsettlement expenses) – APV0(Benefits + settlement expenses from time 1 through n)