MATH3510-Actuarial Mathematics 1-Lecture Notes release PDF

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MATH3510: Actuarial Mathematics 1 Outline Lecture Notes Georgios Aivaliotis and Jonty Carruthers (Student) School of Mathematics University of Leeds December 14, 2016

Contents 1 Survival Models 1.1 Survival, distribution, and density functions . . . 1.2 The force of mortality . . . . . . . . . . . . . . . 1.3 Parametric classes and the future lifetime random 1.4 Deferred mortality probabilities . . . . . . . . . . 1.5 Curtate and expected future lifetime . . . . . . . 1.6 Temporary expected future lifetime . . . . . . . .

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2 2 4 6 8 9 11

2 Life 2.1 2.2 2.3 2.4

Tables and Selection Introduction to life tables . . . Uniform distribution of deaths . Constant force of mortality . . Select and ultimate life tables .

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14 14 17 19 20

3 Life 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10

Insurance Benefits Introduction to life insurance functions . . Term and Whole life insurance . . . . . . Endowment and deferred insurances . . . Insurance functions for De Moivre’s model Examples - continuous insurances . . . . . Discrete time insurance functions . . . . . Recursive Relationships . . . . . . . . . . Benefit payments for mthly periods . . . . Continuous and discrete relationships . . . Variable insurance benefits . . . . . . . .

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23 23 24 27 28 30 32 35 38 40 42

4 Life 4.1 4.2 4.3 4.4 4.5 4.6

Annuities Introduction to life annuities . . . . . . Temporary and deferred life annuities Guaranteed annuities . . . . . . . . . . The UDD assumption for life annuities Life annuities payable continuously . . Increasing Annuities . . . . . . . .

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45 45 47 50 53 55 58

5 Premium Calculation and Policy Values 5.1 The Equivalence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Equivalence principle (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Policy Values and Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Illustrative Life Table

1

Chapter 1 Survival Models 1.1

Survival, distribution, and density functions

When considering a life insurance policy the most important factor is determining how long the inidividual is likely to survive. In this chapter, survival models, including those by Makeham and De Moivre, are introduced to provide a method of computing survival and mortality probabilities for varying ages. An introduction to the notation used in actuarial science is also provided, parts of which may have already been seen in MATH2775 Survival Analysis. The future lifetime and curtate future lifetime random variables are defined and it is these that will form a vital part of the insurance and annuity functions defined in later chapters. Let X be a continuous and non-negative random variable, then X is an appropriate random variable to model the age at which an individual dies and is therefore called the age at death random variable. Definition: The survival distribution function of X is the probability that a newborn survives to at least age x and is defined as S0 (x) = Pr(X > x) (1.1) This function is also known as the decumulative distribution function but more commonly it is simply referred to as the survival function. For a function to be a legitimate survival function it must satisfy all three of the following properties • S0 (0) = 1, • limx→∞ S0 (x) = 0, • S0 (x) is a non-increasing function of x. Example 1: Consider the proposed survival function 1  x 2 S0 (x) = 1 − 100

0 ≤ x ≤ 100

Determine whether this function is a legitimate survival function. Solution: We need to check that S0 (x) satisfies the three conditions previously given. By substituting for x = 0 it can be seen that S0 (0) = 1, and similarly for x = 100, S0 (100) = 0. To determine whether the function is non-increasing we consider the first derivative. Differentiating once with respect to x gives   1 x −2 −1 1 − 100 S ′0 (x) = 200 2

CHAPTER 1. SURVIVAL MODELS

3

Since S0′ (x) is non-positive for all x in the given domain, S0 (x) must be non-increasing. All three properties are satisfied so we conclude that S0 (x) is a legitimate survival function. Note that in the previous example there is a slight alteration to the second property and the limit is taken as x tends to 100 as opposed to infinity, this is becasue the survival function is only defined for x ≤ 100. In survival models a limiting age is often defined since it is not necessary to consider unrealistically large values of x. The limiting age is denoted by ω and is usually between 100 and 120, the limiting age in the previous example is ω = 100. Definition: The cumulative distribution function of X is the probability that a newborn dies before age x and is given by F0 (x) = Pr(X ≤ x) (1.2)

This function is more often referred to as the distribution function and as with the survival function there are three properties that any legitimate distribution function must satisfy. These properties are • F0 (0) = 0, • limx→∞ F0 (x) = 1, • F0 (x) is a non-decreasing function.

Since surviving beyond age x and dying before age x are complementary events, there exists a simple relationship between the survival and distribution functions, namely, F0 (x) = 1 − S0 (x)

(1.3)

The final function used to calculate mortality probabilities is the probability density function (p.d.f). Definition: Assume that for the r.v. X there exists a continuously differentiable survival function S0 (x), then X has a continuous distribution function F0 (x) and the probability density function may be defined as f0 (x) =

d F (x) dx 0

a standard result that should be recognised from previous courses in probability. Given the p.d.f there also exist integral expressions for the survival and distribution functions S0 (x) =

R∞ x

f0 (s) ds

F0 (x) =

Rx 0

f0 (s) ds

Occasionally a question may ask to find the p.d.f given only the survival function, in which case an obvious approach may be to first calculate the distribution function and then differentiate to find the p.d.f. However, by using equation (1.3) we can obtain the following expression for the density in terms of the survival function.

f0 (x) =

d d d F0 (x) = (1 − S0 (x)) = − S0 (x) dx dx dx

(1.4)

The following example demonstrates how the functions defined above can be used to calculate mortality probabilities.

CHAPTER 1. SURVIVAL MODELS

4

Example: Consider the survival function S0 (x) =

1 10

1

[100 − x] 2

for 0 ≤ x ≤ 100

What is the probability that a newborn dies between the ages of 65 and 75? Solution: The required probability is Pr(65 ≤ X ≤ 75) = Pr(X ≤ 75) − Pr(X ≤ 65) = F0 (75) − F0 (65) = (1 − S0 (75)) − (1 − S0 (65)) = S0 (65) − S0 (75) = 0.5 − 0.40839 = 0.0916.

1.2



The force of mortality

The force of mortality is an important concept in actuarial science and the modelling of future lifetimes, it considers the probability of death within an infinitesimal interval that we shall denote by ∆t. The force of mortality may also be referred to as the force of failure or the hazard rate. Definition: For a life aged x the force of mortality is defined as 1 Pr (X ≤ x + ∆x|X > x) ∆x→0 ∆x

µx = lim

(1.5)

Let µ(x) be a non-negative real-valued function, µ(x) may be considered as a force of mortality if and only if the following properties are satisfied • for all x ≥ 0, µ(x) ≥ 0 R∞ • 0 µ(x) dx = ∞

From the definition of the force of mortality, we can obtain an equation linking it with both the survival function and p.d.f. The following derivation uses the formal definition of a derivative seen in second year analysis courses in addition to equation (1.4) from Section 1. 1 Pr (X ≤ x + ∆x|X > x) ∆x→0 ∆x 1 Pr(X > x) − Pr(X > x + ∆x) = lim ∆x→0 ∆x Pr(X > x) 1 S0 (x) − S0 (x + ∆x) = lim S0 (x) ∆x→0 ∆x S0 (x + ∆x) − S0 (x) 1 =− lim S0 (x) ∆x→0 ∆x 1 d S0 (x) =− S0 (x) dx

µx = lim

=⇒

µx =

f0 (x) S0 (x)

(1.6)

By an application of the chain rule, an alternative equation for the force of mortality is µx = −

1 d d S0 (x) = − log (S0 (x)) dx S0 (x) dx

(1.7)

CHAPTER 1. SURVIVAL MODELS

5

Example: The distribution function for a survival model is  x  16 F0 (x) = 1 − 1 − 120

Find an expression for the force of mortality µx .

Solution: The respective survival and density functions are  x  16 S0 (x) = 1 − 120

and

f0 (x) =

5 1  x − 6 1− 120 720

Therefore, the force of mortality may written as 1 1 1  x −1 = = µx = 1− 120 6(120 − x) 720 720 − 6x

 Note that it is easier to solve this example using equation (1.6) instead of (1.7) so it is important to learn both. Equation (1.7) is only particularly useful when the survival function is written as an exponent, however, it can also be used to derive one of the most important formulae in the course. Integrating equation (1.7) between 0 and y gives Z y µx dx = − [log (S0 (y)) − log (S0 (0))] 0

but since S0 (0) = 1, it follows that log (S0 (0)) = 0 and therefore  Z y  Z y µx dx µx dx = −log (S0 (y)) ⇐⇒ S0 (y) = exp − 0

0

In keeping with the notation previously used, a simple exchange of variables gives  Z x  S0 (x) = exp − µs ds

(1.8)

0

We now give an example demonstrating an application of this equation. Example: A survival model is defined with force of mortality µs = ksn for constants k and n such that s ≥ n, k > 0 and n ≥ 1. Find an expression for the survival function S0 (x). Solution: Substituting the expression for the force of mortality into equation (1.8) gives  Z x  n S0 (x) = exp − ks ds 0   k  n+1x s = exp − 0 n+1   kxn+1 = exp − n+1  The force of mortality in the previous example is in fact the force of mortality for the Weibull model, one of a number of parametric models defined in the following section.

CHAPTER 1. SURVIVAL MODELS

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1.3 Parametric classes and the future lifetime random variable De Moivre’s Law De Moivre’s law is a simple law of mortality based on a linear survival function where the age at death random variable X follows a uniform distribution between 0 and the limiting age ω, that is X ∼ U(0, ω). Recall that for Z ∼ U(a, b), the p.d.f is 1 f (z) = b−a Under De Moivre’s model the p.d.f is therefore f0 (x) =

1 ω

Using this, along with relations defined in previous sections, expressions can be obtained for the survival function, distribution function, and force of mortality. Z x Z x 1 x f0 (s) ds = F0 (x) = ds =⇒ F0 (x) = ω ω 0 0 S0 (x) =

Z

ω

f0 (z) dz =

Z

x

x

µx =

Example: Given that µx =

ω

h z iz=ω 1 dz = ω z=x ω

1 ω f0 (x) = . S0 (x) ω ω − x

1 ,0 100−x

=⇒

=⇒ µx =

S0 (x) = 1 −

x ω

1 ω−x

≤ x ≤ 100, find an expression for the survival function S0 (x).

Solution: There are two ways to approach this question, either we can recognise that the given force of mortality is in the required form for De Moivre’s model with a limiting age ω = 100 and hence x , or, a safer approach is to find the expression explicitly using equation (1.8). S0 (x) = 1 − 100 x

 1 dz = exp ([log(100 − z)]z=x z=0 ) 100 − z 0 = exp (log(100 − x) − log(100))   x  = exp log 1 − 100 x = 1− 100

 Z S0 (x) = exp −

 The table on the following page provides a list of the key parametric models used in actuarial science along with the corresponding survival function, force of mortality, and any limitations on the parameters involved. Makeham’s law is another commonly used model since the force of mortality is comprised of two components, a constant force that is independent of current age, and a force that increases exponentially with age. Note that a special case of Makeham’s law is when A=0 and the model reduces to Gompertz’s law.

CHAPTER 1. SURVIVAL MODELS Law/distribution

µx

De Moivre (Uniform)

1 ω−x

Exponential

µ

Gompertz

Bcx

Makeham

A + Bcx

Weibull

kxn

7 S0 (x) 1−

x ω

exp(−µx)   x B exp − logc (c − 1)   B exp −Ax − logc (cx − 1)  n+1  exp −kx n+1

limitations 0≤x≤ω µ>0 B > 0, c > 1 A ≥ −B, c > 1, B ≥ 0 x ≥ n ≥ 1, k > 0

The future lifetime random variable So far we have only considered modeling future lifetime by considering the age at which the individual dies, however, given an individual aged x, it is sometimes more appropriate to consider how much longer they will survive for. We now introduce the notion of the future lifetime random variable, revise our expressions for the survival and distribution functions and include the notation commonly used throughout actuarial science. Definition: If X is the age at death random variable and x is the current age of an individual, the future lifetime can be defined by the random variable Tx = X − x

(1.9)

Consider an individual currently aged x who survives for another t years, the probability of this is given by the survival function Sx (t) = Pr(Tx > t) Similarly, the probability that the individual dies within t years is given by the distribution function Fx (t) = Pr(Tx ≤ t) When we consider Sx (t) we make the underlying assumption that the life has survived from birth through to age x before surviving from age x to x + t. This assumption allows us to write Sx (t) in terms of the age at death random variable X and results in a very important equation relating S0 (x) and Sx (t). Pr(X > x + t) Sx (t) = Pr(Tx > t) = Pr(X > x + t|X > x) = Pr(X > x) =⇒

Sx (t) =

S0 (x + t) S0 (x)

(1.10)

The actuarial notation used to define the survival and distribution functions is as follows, this notation will be used in place of the existing notation throughout the remainder of the course. Sx (t) = t px

Fx (t) = t qx

In actuarial notation, the equation linking the survival and distribution functions may therefore be written as t p x + t qx = 1

CHAPTER 1. SURVIVAL MODELS

8

It is also common practice that in the case where t = 1 the first subscript is omitted and we simply write px and qx for the survival and distribution functions. The p.d.f of the future lifetime random variable is given in terms of the distribution and survival functions in a similar way as before. fx (t) =

d d Fx (t) = − Sx (t) dt dt

(1.11)

The force of mortality may also be defined for the future lifetime random variable as µx+t = lim

∆t→0

1 Pr(t < Tx < t + ∆t|Tx > t) ∆t

Recall that equations (1.6) and (1.7) give two different expressions for the force of mortality of a life aged x. Equivalent expressions also exist for the force of mortality at any age x+t, t > 0, the derivations of which are similar to before but instead begin with the definition of µx+t as opposed to µx . For this reason the derivations are not included but the results are as follows µx+t =

fx (t) Sx (t)

and

d µx+t = − log (Sx (t)) dt

Finally we state without proof the analogue to equation (1.8) for calculating the survival function Sx (t) given only the force of mortality. The proof may be attempted as an exercise and is a simple application of equation (1.10) and (1.8).   Z t µx+s ds (1.12) Sx (t) = exp − 0

1.4

Deferred mortality probabilities

Sometimes we may wish to consider the probability that an individual currently aged x survives for some period of length t but dies in the subsequent u years, that is, the individual dies between the ages x + t and x + t + u. This is called a deferred mortality probability and is a key part in the formulation of the insurance functions that will be introduced in chapter 3. In actuarial notation the deferred mortality probability is given by t|u qx

= Pr(t < Tx ≤ t + u) = Pr(Tx > t) − Pr(Tx > t + u) = t px − t+u px

(1.13)

There also exists a second equation for the deferred mortality probability, but before we show this we need to consider the multiplicative property of the survival function. Suppose we are interested in the probability that a life currently aged x survives for another 2 years but only know the probability of survival for each year separately. The probability of surviving for 2 years may be written as the probability of surviving the first year, multiplied by the probability of surviving the second year. In actuarial notation this may be written as 2 px

= px · px+1

It is important to note that in general, the multiplicative rule only holds for the survival function and not the distribution fun...