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Life Contingencies

LIFE CONTINGENCIES Single Life Model Basic Random Variables X denotes the age-at-death random variable (x) denotes a life aged x (ie someone whos already survived to age x) denotes terminal age (unless otherwise stated, we assume = ) (no one survive past age )

T = T(x) denotes the continuous future lifetime of (x) random variable T=Xx|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.

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Page 1 of 52

Life Contingencies

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

(cumulative distribution function cdf) FX ( x ) = Pr( X x ) = x q0 (Recall that F = f , the probability density function (pdf)) (survival function sf)s X ( x ) = Pr( X > x ) = x p0 = 1 x q0 (Note that s = f )

(force of mortality fom) aka hazard rate or failure rate ( x) =f X ( x) d = [ln( s X ( x )] s X ( x) dx

Comments and Concepts 1. The force of mortality is the rate of mortality at a particular point in time. The expression ( x )dx represents the probability that a newborn that has survived to age x dies in the next instant dx. 2. 3.x

p0 = exp( ( s )ds )0

x

x

q0 = Pr( X x ) = f X ( s )ds = 0

x

x s

0

p0 ( s )ds

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Page 2 of 52

Life Contingencies

The (continuous) Future Lifetime Random Variable: T = T(x)Supp (T ) = (0, x )

(pdf): f T (t ) =

f X ( x + t) Pr( X > t )

(cdf): FT (t ) = Pr(T t )= t q x = (sf): sT (t ) = Pr(T > t )= t p x =

Pr( x < X x + t ) Pr( X > x + t ) = 1 Pr( X > x ) Pr( X > x )x +t

Pr( X > x + t ) = Pr( X > x )

p0 = 1 t q x x p0

Comments and Concepts 1. f T (t )= t p x ( x + t ) 2. t q x = 0 s p x ( x + s )ds 3.t t

p x = exp( ( x + s )ds ) = exp( 0

t

x +t

x

( s )ds ) Also, t p x = s p x ( x + s )dst

4. 1 q x = q x and 1 p x = p x 5. (survival factorization) 6. (deferred mortality)t |u n +t

p x = n p x t p x + n

qx = Pr(t < T ( x ) t + u ) =

t +u s

t

p x ( x + s )ds = t p x t + u p x = t + u qx t qx = t p x u qx + t

d [ t p x ] = f T (t ) = t p x ( x + t ) and so ( x + t ) = 7. dt

d [t px ] dt t px

8.

d [ t p x ]= t p x [ ( x ) ( x + t )] (Use 2nd equality in 3. above and the FTC) dx

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Page 3 of 52

Life Contingencies

9. (complete expectation of life for (x), aka mean residual lifetime)ex = E[T ( x )] = tt p x ( x + t )dt = t p x dt (The last equality is used often.)0 0 0

10. E[(T ( x )) 2 ] = 0 t 2 t p x ( x + t )dt = 02tt p x dt (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)e x :n | = E[T ( x ) n ] = tt p x ( x + t )dt + nn p x = 0 0 n n t 0

p x dt

12. (Recursion Formulas)e x = e x:n| + n p x e x + n and when n = 1 we get e x = t p x dt + p x e x +10 0 0 0 0 1 0 0 0 0

e x:n| = e x:m| + m p x e x + m:n m|

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

p x = 1 .01k

The median future lifetime of (x) is c50. 14. The mode of T is the point(s) at which the pdf f T (t ) is maximized.

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Page 4 of 52

Life Contingencies

The Curtate Future Lifetime Random Variable: K = K(x)Supp ( K ) = (0,1,2, L , x 1)

The probability distribution table for K is K 0 1 2M

Pr Pr(0 T < 1) = Pr(0 < T 1) = q x ( = 0| q x ) Pr(1 T < 2) = Pr(1 < T 2) = 1| q x Pr(2 T < 3) = Pr(2 < T 3) = 2| q xM

Note:

n|

q x = n p x n +1 p x = n +1 q x n q x = n p x q x + nk

(cdf): FK ( k ) = Pr( K k ) = n| q x (Notice that Pr( K k ) = Pr(T k + 1)= k +1 q x )n =0

Comments and Concepts 1. (curtate expectation of life for (x))e x = E [ K ( x )] = k k | q x = k p xk =0 k =1

(The last equality is used often. Notice the index starts at k = 1.) 2. E[( K ( x )) 2 ] = k 2 k | q x = (2k 1)k p x (Helps us get variance of K.)k =0 k =1

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)e x:n| = k p xk =1 n

4. (Recursion Formulas) e x = e x:n| + n p x e x + n and when n = 1 we get e x = p x (1 + e x +1 )e x:n| = e x:m| + m p x e x + m:n m|

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Page 5 of 52

Life Contingencies

Life Table Notationl0 = (arbitrary) number of newborns l x = l0 x p0 = (expected) number of survivors at age x (Note:n x

p0 =

lx ) l0

d x = l x l x +n = number of deaths between ages x and x+n

(1dx = dx ) Note: n d x = d x + d x +1 + L + d x +n 1 Formulas and concepts involving life table notation 1.n

px =

l x+n d l l and n q x = n x = x x +n = 1 n p x and lx lx lx d [l x ] d dx , and so [l x ] = l x (x ) dx lxn t

n|m

qx =

m

d x +n l x +n l x +n+m = lx lx

2. ( x ) =

3. 4.

n

d x = l x +t ( x + t )dt (follows since n q x = 0

n

0

p x ( x + t )dt )

n

Lx = the total number of years lived in the next n years by the l x peopleL x = l x +t dt = 0

alive at age xn x+n n x

l y dy = n l x + n + n d x E[T ( x) | T ( x) < n]

Note that E[T ( x) | T ( x) < n] =d [ n Lx ] = l x + n l x = n d x dx

n

0

t t p x ( x + t )dtn

qx

=

n

0

t l x + t ( x + t )dtn

dx

5.

6. Tx = Lx the total number of years lived in the future by the l x people alive at age x 7.n

Lx = Tx Tx + n and

d [Tx ] = l x dx

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Page 6 of 52

Life Contingencies

Life Table Notation (continued) 8. e x =0 0 Tx L and e x:n| = n x lx lx

9. E[(T ( x )) 2 ] =

2Y x where Y x = 0Tx +t dt lx

10. n m x =

n

dx is the n-year central mortality weight n Lx

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Page 7 of 52

Life Contingencies

Extending Discrete to Continuous (Fractional Age Assumptions) (x is an integer and all formulas are valid for 0 t 1 ) UDD (Uniform Distribution of Deaths: t d x = t d x )t

q x = t q x , and so t p x = 1 t q x , and f T (t ) = q x (a constant wrt t) qx qx and m x = 1 t qx 1 0.5q x

Then l x +t = l x t d x and ( x + t ) =

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 e x = e x + 0.5 . Also e x:n| = e x:n| + 0.5( n q x ) and Var (T ( x )) = Var( K ( x )) + For 0 s + t 1 , t q x + s =qx t qx and t p x + s ( x + s + t ) = 1 s qx 1 s qx0 0

1 12

Constant Force (Exponential Interpolation: ( x + t ) = ) For 0 s + t 1 , t p x + s = e t = ( p x ) tmx =

Balducci (Hyperbolic Interpolation)1 l x +t = 1 1 1 + t l l lx x x +1

( x + t) =

qx t qx and for 0 s + t 1 , t qx + s = = t ( x + s + t) 1 (1 s t ) qx 1 (1 t ) q x

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Page 8 of 52

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 x (t ) .

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Page 9 of 52

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: T ( xy ) = Min{T ( x ), T ( y )} (Joint-life status fails on the earlier of the deaths of (x) and (y))t

p xy = Pr(T ( xy ) > t ) = Pr(T ( x ) > t ) Pr(T ( y ) > t ) = t p x t p y q xy = 1 t p xy

t

e xy = t p xy dt0

0

e xy = k p xyk =1

t |u

q xy = t p xy t +u p xy

xy (t ) = x (t ) + y (t )

Last-Survivor Status: T ( xy ) = Max{T ( x ), T ( y )} (Last-survivor status fails on the latest of the deaths of (x) and (y)) For independent lives (x) and (y):t

qxy = Pr(T ( xy ) t ) = Pr(T ( x ) t ) Pr(T ( y ) t ) = t qx t q yp xy = 1 t q xy q xy = t +u q xy t q xyt

t

t |u

xy (t ) =

p x x (t )+ t p y y (t ) t p xy xy (t )t

p xy

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Page 10 of 52

Life Contingencies

Contingent Probabilities Notation:n 1 q x y = Pr((x) dies first and within the next n years)1 q x y = Pr((x) dies first) 2 q x y = Pr((y) dies second and within the next n years)

n

The notation x y indicates that the joint-life stat