4. Life Insurance

[Pages:69]4. Life Insurance

4.1 Survival Distribution And Life Tables

Introduction

? X, Age-at-death

? T (x), time-until-death

? Life Table ? Engineers use life tables to study the reliability of complex mechanical and electronic systems. ? Biostatistician use life tables to compare the effectiveness of alternative treatments of serious disease. ? Demographers use life tables as tools in population projections. ? In this text, life tables are used to build models for insurance systems designed to assist individuals facing uncertainty about the time of their death.

1

Probability for the Age-at-Death

1 The Survival Function s(x) = 1 - FX(x) = 1 - Pr(X x) = Pr(X > x), x 0.

s(x) has traditionally been used as a starting point for further development in Actuarial science and demography. In statistics and probability, the d.f. usually plays this role.

Pr(x < X z) = FX(z) - FX(x) = s(x) - s(z).

2

Time-until-Death for a Person Age x

The conditional probability that a newborn will die between the ages x and z, given survival to age x, is

Pr(x < X z|X > x) = FX(z) - FX(x) = s(x) - s(z)

1 - FX(x)

s(x)

? The symbol (x) is used to denote a life-age-x

? The future lifetime of (x), X - x is denote by T (x).

The symbols of Actuarial science

? tqx = Pr[T (x) T ], t 0, tpx = 1 - tqx = Pr[T (x) > t], t 0. ? xp0 = s(x), x 0. ? qx = Pr[(x) will die within 1 year], px = Pr[(x) will attain age x + 1] 3

? (x) will die between ages x + t and x + t + u.

t|uqx = Pr[t < T (x) t + u] = t+uqx - tqx = tpx - t+upx.

If u = 1, the prefix is delete in t|uqx and denotes as t|qx.

?

tpx

=

x+tp0 xp0

=

s(x + t) s(x)

s(x + t) tqx = 1 - s(x)

?

s(x + t) - s(x + t + u)

t|uqx =

s(x)

s(x + t) s(x + t) - s(x + t + u)

=

s(x)

s(x + t)

= tpx ? uqx+t

4

Curtate-future-life time K(x) A discrete random variable associated with the future lifetime is the number of future years completed by (x) prior to death.

Pr[K(x) = k] = Pr[k K(x) < k + 1] = Pr[k < T (x) k + 1] = kpx - k+1px = kpxqx+k = k|qx

k

FK(x)(y) = h|qx = k+1qx,

h=0

y 0 and k is the greatest integer in y

5

Force of Mortality

Pr(x < X x + x|X > x) = FX(x + x) - FX(x) = fX(x)x

1 - FX(x)

1 - FX(x)

Definition of force of mortality

?(x)=^ fX(x)

-s (x) =

1 - FX(x) s(x)

In reliability theory, ?(x) is called failure rate or hazard rate or, hazard rate function.

6

Force of mortality can be used to specify the distribution of X.

n

npx = exp[- ?(x + s)ds].

0

n

np0 = exp[- ?(s)ds].

0 x

FX(x) = 1 - s(x) = 1 - exp[- u(s)ds]

0 x

FX(x) = fX(s) = exp[- ?(s)ds]?(x) = xp0?(x).

0

FT (x)(t) and fT (x)(t) denote, respectively, the d.f. and p.d.f. of T (x), the future lifetime of (x). Then we have

d

d

s(x + t)

fT (x)(t) = dttqx = dt 1 - s(x)

s(x + t) s (x + t)

=

-

s(x) s(x + t)

= tpx?(x + t), t 0

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