Long-Term Actuarial Mathematics I
Chapter 2 –
Survival Models
MATH 3630 | Fall 2020
1
Chapter 2 Topics
❑ Future lifetime random variable,
❑ Force of mortality
❑ Actuarial notation
❑ Mean and variance of
❑ Curtate future lifetime random variable
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The Future Lifetime Random Variable
❑ Life insurance provides a death benefit in exchange for policyholder premiums
❑ For pricing and valuation purposes, it is essential to estimate the timing of the death benefit
❑ Therefore, mortality must be modeled as a random variable, where probabilities of death can
be calculated at specific ages
❑ Let denote a life aged , where ≥ 0
❑ The random variable that defines the future lifetime of () is denoted by
▪ Therefore, + represents the age-at-death random variable for (x)
❑ The resulting distribution, survival, and density functions are as follows:
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Distribution Function: = Pr ≤
Survival Function: = Pr > = 1 −
Density Function: =
= −
The Future Lifetime Random Variable
❑ We’ll now explore the relationships between the collection of random variables ≥0
❑ Throughout this course, we require the following events to be equivalent:
≤ = 0 ≤ + |0 >
❑ This is an important relationship and implies the following:
Pr[ ≤ ] = Pr[0 ≤ + |0 > ]
❑ As a consequence, we have the following concept:
0 + = 0() ∙ ()
❑ More generally, this leads to:
+ = () ∙ +()
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Survival Function - Conditions & Assumptions
❑ Any survival function for a lifetime distribution must satisfy the following conditions to be
considered valid:
1. 0 = 1
2. lim
→∞
= 0
3. −
❑ The textbook also makes the following assumptions for survival distributions used in this
course:
1. > 0
2. lim
→∞
∙ = 0
3. lim
→∞
2 ∙ = 0
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Example
Assume the survival function for a newborn is given by:
0 = 8( + 2)
−3
1. Verify that this is a valid survival function
2. Find the density function associated with the future lifetime random variable 0
3. Find the probability that a newborn dies between the ages of 1 and 2
4. Find the survival function corresponding to the random variable 10
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Example
Assume the survival function for a newborn is given by:
0 = 1 − 1 −
120
1
6
< 120
1. Calculate the probability a newborn life survives beyond age 30
2. Calculate the probability that (30) dies before age 50
3. Calculate the probability that (40) survives beyond age 65
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The Force of Mortality
❑ The force of mortality is instrumental for modeling the future lifetime random variable and
can be defined as the conditional instantaneous measure of death at age x:
= lim
→0+
1

Pr[0 ≤ + |0 > ]
❑ For very small , we can use the following approximation:
≈ Pr 0 ≤ + 0 >
▪ For very small , we can interpret as the probability that (x) dies before attaining age +
❑ The force of mortality can also be related to the survival function:
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=
−1
0
0 =
0()
0()
and = (−׬0
+)
Mortality Laws
❑ Because we can relate and , the full distribution of can be determined if we know
the force of mortality
❑ There are several important distributions for future lifetime, distinguishable by the force of
mortality
Gompertz’s Law: =
> 0, > 0
Makeham’s Law: = +
, > 0, > 0
de Moivre: =
1
−
0 ≤ <
Exponential: = 0 ≤ ≤ ∞
❑ Under Gompertz’ law, the force of mortality increases exponentially with age
▪ Makeham’s Law adds a constant term, , designed to reflect risk of accidental death
❑ The Exponential and de Moivre distributions are unrealistic and primarily used for
mathematical convenience.
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Example
Assume the survival function for a newborn is given by:
0 = 8( + 2)
−3
1. Find the force of mortality, .
2. Sketch the shape of .
3. Does this distribution look appropriate for modeling the future lifetime of a human?
4. What would we expect the force of mortality to look like for a human?
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Actuarial Notation
❑ The content in prior slides used standard statistics notation to describe the properties of the
future lifetime random variable.
❑ Actuaries have also developed their own notation, known as International Actuarial Notation
(IAN), to describe many of the same concepts.
= Pr > =
probability that (x) survives to at least age +
= Pr ≤ =
probability that (x) dies before age +
| = Pr < ≤ + = − +
probability that (x) dies between the ages of + and + +
❑ The subscript, , may be dropped if its value is equal to 1.
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Important Relationships
❑ The following relationships can be expressed based on the definitions we’ve learned
previously in this chapter:
▪ + = 1
▪ | = ∙ + = − +
▪ + = ∙ +
▪ = exp −׬0
+
▪ = ׬0
∙ +
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Example
Consider an individual aged 70 who is subject to the following force of mortality:
70+ = ቊ
0.01, ≤ 5
0.02, > 5
Calculate 2070 for this individual.
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Mean of the Future Lifetime Random Variable
❑ We can also consider other properties of the future lifetime random variable.
❑ One quantity that is often of interest is the mean of the future lifetime random variable,
[], which is called the complete expectation of life.
❑ The actuarial symbol for this is written as
∘
.
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E =
∘
= න
0
∞
∙ ()
= න
0
∞
∙ ∙ +
= න
0
∞
Variance of Future Lifetime Random Variable
❑ In order to calculate the variance, the second moment of the future lifetime random variable
must also be defined:
❑ The variance is then calculated as:
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E
2 = න
0
∞
2 ∙ ()
= න
0
∞
2 ∙ ∙ +
= 2න
0
∞
∙
=
2 −
∘
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Term Expectation of Life
❑ We are often interested in the future lifetime given a cap of n years (why?)
❑ The resulting random variable is defined as min(, )
❑ The mean can be derived using a similar approach to the mean of
min(, ) =
∘
: = න
0
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Example
Assume the survival function for a newborn is given by:
0 = 1 − 1 −

120
1
6
< 120
Find
∘
30
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Curtate Future Lifetime
❑ We are often interested in the integral number of years lived in the future by an individual
❑ This discrete random variable is called the curtate future lifetime and is denoted by
❑ We define the probability mass function of as follows:
❑ We can think of Kx as the number of complete years lived by (x) in the future, and relate to
Tx using the floor function:
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Pr = = Pr ≤ T < + 1
= |
= +
= උ ۂ
Curtate Future Lifetime – Mean
❑ We can also calculate the expected value of , referred to as the curtate expectation of life:
❑ Analogous to the continuous version of the future lifetime random variable, we also have the
random variable min(, ) for which we calculate the expected value as:
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E = = ෍
=0
∞
∙ =
= ෍
=0
∞
∙ ( − +1)
= ෍
=1
∞
: = ෍
=1
n
Example
You are given the following probabilities of survival and death:
▪ 351 = 0.612
▪ 250 = 0.145
▪ 52 = 0.150
▪ 252 = 0.680
▪ 54 = 1.000
Calculate the following quantities:
a. 350
b. 450
c. 51
d. 50
e. 2|250
f. 50
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Example
You are given the following table of values
Calculate 360
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x
60 15.96
61 15.27
62 14.60
63 13.94