ACTUARIAL SCIENCE FROM THE AIR
An Overview of Life Contingencies
I.
INTRODUCTION
To see the interconnection of streams and rivers in an unfamiliar landscape, take to the air.
From the ground individual streams may seem isolated; but from the air one sees their
confluence. So it is with mathematics. Much of mathematics describes the “real world” that
we touch and see at close range; but the wings of abstraction take us to a higher vantage
point, where we see theoretical connections.
Life Contingencies is a course in Actuarial Science where students encounter an abundance
of notation, explore a variety of concepts, and typically invest weeks of study before
perceiving certain important relationships. The purpose of this paper is to introduce some of
these concepts from the perspective of a simple unifying principle and to show interesting
relationships that can be presented in a single lecture.
Specifically, a number of topics in actuarial science, including life insurance and life
annuities, involve a sequence of possible payments. Certain contingencies determine
whether or not a given payment in the sequence is actually made. Part of the actuary’s job
is to assign to each payment the probability that it will actually be made. Assigning
different probabilities creates a variety of theoretically related examples.
II.
PRESENT VALUE
Insurance companies collect money in the present in order to pay benefits in the future. By
wisely investing the collected money, companies can reduce the prices of their insurance
products. Therefore, in setting prices, actuaries keep in mind the principle that invested
money grows over time.
To simplify the examples that follow, we assume that all investments grow at a constant
annual compound interest rate of 5%.
Example 1 (Present Value): To accumulate $1000 in 10 years at 5% annual compound
interest, invest $613.91 because
$613.91(1.05)10 = $1000
The amount $613.91 is called the present value of $1000 to be paid after 10 years.
Present Value Formula: The present value P of an amount X to be paid after n years (at
5% compound interest) satisfies the following equivalent equations
P(1.05)n = X
P = X/(1.05)n
Example 2 (Present Value of an Annuity): Suppose you are retiring and wish to receive a
yearly income. To receive $1000 at the end of each year for the next 10 years, invest
$7,721.73, the sum of the following 10 present values:
Year
1
2
3
4
5
6
7
8
9
10
Present Values for a Sequence of Ten $1000 Payments
Present Values
Accumulated Values
$952.38
$952.38 (1.05)1 =
$1,000
$907.03
$907.03 (1.05)2 =
$1,000
$863.84
$863.84 (1.05)3 =
$1,000
$822.70
$822.70 (1.05)4 =
$1,000
$783.53
$783.53 (1.05)5 =
$1,000
$746.22
$746.22 (1.05)6 =
$1,000
$710.68
$710.68 (1.05)7 =
$1,000
$676.84
$676.84 (1.05)8 =
$1,000
$644.61
$644.61 (1.05)9 =
$1,000
10
$613.91
$613.91 (1.05) =
$1,000
SUM = $7,721.73
SUM = $10,000
An insurance plan that provides guaranteed or certain payments is called an annuity
certain. By contrast, a life annuity offers regular payments only as long as the insured
survives. The table above gives the present values for the payments of a 10-year annuity
certain. These present values appear again in the following example:
Example 3 (Other Present Values):
Year
1
2
3
4
5
6
7
8
9
10
PresentValues
of $1000
$952.38
$907.03
$863.84
$822.70
$783.53
$746.22
$710.68
$676.84
$644.61
$613.91
PresentValues
of $1
$0.95238
$0.90703
$0.86384
$0.82270
$0.78353
$0.74622
$0.71068
$0.67684
$0.64461
$0.61391
PresentValues
of X dollars
$0.95238X
$0.90703X
$0.86384X
$0.82270X
$0.78353X
$0.74622X
$0.71068X
$0.67684X
$0.64461X
$0.61391X
Note that dividing the present values of $1000 by 1000 gives the present values of $1.
Similarly, the present values of X dollars are X times the present values of $1.
III.
LIFE TABLES
Life insurance pays upon death and a life annuity provides regular payments as long as the
insured survives. The cost to an insurance company of providing such benefits depends
upon when the insured dies. For example, the longer a person lives,
• the smaller the present value of his life insurance payment (decreasing the cost)
• the greater the number of his life annuity payments (increasing the cost)
In determining what to charge for insurance, a company needs to know the likelihood or
probability of death at various times. This information may be given in a life table.
According to the “Life Table for the Total Population: United States, 1979-1980” (Bowers,
et al. 1986. Actuarial Mathematics. The Society of Actuaries. Pages 55-58), about 5% of
people alive at age 75 would be expected to die in each of the next 15 years. (The actual
percents range between 4.3% and 5.4% and average out to 5%.) The assumption that the
same number of deaths occurs each year from a given population is called a uniform
distribution of deaths assumption. For simplicity, we make a uniform distribution of deaths
assumption in the following table by assuming that exactly 5% of 10,000 people alive at
age 75 will die in each of the next 10 years.
Then, for example, of 10,000 alive at age 75,
• 500 (= 5% of 10,000) will die in each of the next 10 years
• 2000 (= 4(500)) will die in the next 4 years
• 8000 (= 10,000 – 2000) will survive the next 4 years
• 0.80 = 8000/10000 = probability of surviving the next 4 years
Simplified Life Table for 10,000 People Alive at Age 75
Year n
1
2
3
4
5
6
7
8
9
10
In Year n
Probability of death
Number of deaths
0.05
500
0.05
500
0.05
500
0.05
500
0.05
500
0.05
500
0.05
500
0.05
500
0.05
500
0.05
500
Number alive
9500
9000
8500
8000
7500
7000
6500
6000
5500
5000
End of Year n
Probability of survival
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
Example 4 (Cost of a Future Benefit): Suppose each of two 75-year-old men wishes to
receive a benefit of $1000 if he is still living after 10 years. How much should each pay
now to receive that future benefit?
Solution 1 (Sharing Cost): By the life table, the probability of surviving 10 years is 0.50, so
that, on average, only one of the two men will survive 10 years and only one benefit of $1000
will be paid. The present value of this benefit is $613.91, and the men share the cost equally:
Cost for each = $613.91 / 2 = $306.96
Solution 2 (Expected Present Value): A 75-year-old man has a 50% chance of surviving
10 years. Therefore, the cost of paying him $1000 if he is alive after 10 years is 50% of the
present value of the payment:
Cost = [$1000/(1.05)10] 0.50 = [$613.91] 0.50 = $306.96
This second solution illustrates the following principle, which unifies the discussion of both
life insurance and life annuities:
Unifying Principle: The cost of a payment A to be made with probability p after n years
equals the present value of the payment times p:
Cost = [A/(1.05)n] p
This cost, called the actuarial present value of the payment A, is proportional to p.
IV.
LIFE CONTINGENCIES and APPLYING THE UNIFYING PRINCIPLE
The cost of a future benefit that is certain to be paid is the present value of that benefit.
However, life insurance and life annuities pay benefits that depend upon contingencies
related to the insured person’s time of death. These are called life contingencies.
Example 5: Suppose the estate of a 75-year-old man would like to receive a benefit of
$1000 to be paid in 7 years. What does the benefit cost under the following contingencies?
a) He dies in the seventh year
b) He dies within 7 years
c) He survives 7 years
Answer: In any case, by the Unifying Principle, the cost equals the present value of
$1000, to be paid in 7 years, multiplied by the probability p of payment:
Cost = [$1000/(1.06)7] p = $665.06 p
a) $665.06 (0.05) = $665.06 (probability of death in seventh year) = $33.25
b) $665.06 (0.35) = $665.06 (probability of death within seven years) = $232.77
c) $665.06 (0.65) = $665.06 (probability of surviving seven years) = $432.29
Example 6: Compute the costs of each of the following insurance products:
 10-year term life annuity
 pays $1000 at the end of each year of survival for the next 10 years
 10-year term life insurance
 pays $1000 at the end of the year of death if death occurs within 10 years
These insurance products are alike in that each offers a possible payment at the end of each
of the next ten years. Thus, their costs are computed similarly. By the Unifying Principle,
the cost of each is the sum of 10 actuarial present values.
Year n
1
2
3
4
5
6
7
8
9
10
Costs of 10-year Term Life Annuity and 10-year Term Life Insurance
Each with Potential Payments of $1000
Present value of
Life Annuity
Life Insurance
$1000 to be paid in n Probability of
Cost of
Probability of
Cost of
years
payment
payment
payment
payment
$952.38
0.95
$904.76
0.05
$47.62
$907.03
0.90
$816.33
0.05
$45.35
$863.84
0.85
$734.26
0.05
$43.19
$822.70
0.80
$658.16
0.05
$41.14
$783.53
0.75
$587.64
0.05
$39.18
$746.22
0.70
$522.35
0.05
$37.31
$710.68
0.65
$461.94
0.05
$35.53
$676.84
0.60
$406.10
0.05
$33.84
$644.61
0.55
$354.53
0.05
$32.23
$613.91
0.50
$306.96
0.05
$30.70
TOTALS: $7,722
COST = $5,753
COST = $386
Multiplying columns, we have
(Column 2)(Column 3) = Column 4, whose total equals the cost of the life annuity
(Column 2)(Column 5) = Column 6, whose total is the cost of the life insurance
Notice that the life annuity costs more than the life insurance. Why should we expect this?
Just as a 10-year term life annuity pays in each year of survival for 10 years, a 5-year term
life annuity pays in each year of survival for 5 years. However, a 5-year deferred annuity
makes no payments during the first 5 years.
Example 7: Compute the cost of the following two 5-year term life annuities:
 5-year term (undeferred)
 pays at the end of each year of survival for years 1-5
 probability of payment equals 0 for years 6-10
 5-year deferred, 5-year term
 probability of payment equals 0 for years 1-5
 pays at the end of each year of survival for years 6-10
n
1
2
3
4
5
6
7
8
9
10
Present value of
$1000 to be paid in n
years
$952.38
$907.03
$863.84
$822.70
$783.53
$746.22
$710.68
$676.84
$644.61
$613.91
TOTALS:
Cost of 5-year Term Life Annuities
Undeferred
5-year Deferred
Probability of
Cost of
Probability of
Cost of
payment
payment
payment
payment
0.95
$904.76
0
0.90
$816.33
0
0.85
$734.26
0
0.80
$658.16
0
0.75
$587.64
0
0
0.70
$522.35
0
0.65
$461.94
0
0.60
$406.10
0
0.55
$354.53
0
0.50
$306.96
COST = $3701
COST = $2052
As this table shows,
 The probability of an impossible payment is 0
 It is more costly to provide five immediate payments than five deferred payments
 A 10-year term life annuity equals a 5-year term life annuity combined with a 5-year term
5-year deferred life annuity, since
$5753.05 = $3701.16 + $2051.89
Example 8: What yearly premium X, paid at the beginning of each year for 10 years,
would cover the cost of 10 year term life insurance for a person aged 75?
Here, both the insured person and the insurance company face cost:
 the insured pays yearly premiums for ten years (or until death, if death occurs
within ten years)
 the company pays a death benefit if death occurs within ten years
Ignoring a company’s need to make a profit, let us assume simply that the cost to the
insured equals the cost to the company. In other words, the following are equal:
 Actuarial present value of the 10 premium payments of X
 Actuarial present value of the benefit payment
In the following table, we compute these actuarial present values and use their equality to
solve for X:
Yearly Premium X for 10-year Term Life Insurance Paying $1000
n
0
1
2
3
4
5
6
7
8
9
10
Present value of
$1000 paid in n
years
Life Insurance
Probability
of payment
$952.38
$907.03
$863.84
$822.70
$783.53
$746.22
$710.68
$676.84
$644.61
$613.91
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
0.05
Add the columns:
Cost of
payment
$47.62
$45.35
$43.19
$41.14
$39.18
$37.31
$35.53
$33.84
$32.23
$30.70
$389.09
Premium Payments
Present value
Probability
Cost of
of X dollars
of payment
premium
paid in n years
$X
1.00
$X
$0.95238X
0.95
$0.90476X
$0.90703X
0.90
$0.81633X
$0.86384X
0.85
$0.73426X
$0.82270X
0.80
$0.65816X
$0.78353X
0.75
$0.58764X
$0.74622X
0.70
$0.52235X
$0.71068X
0.65
$0.46194X
$0.67684X
0.60
$0.40610X
$0.64461X
0.55
$0.35453X
 Equal Totals 
6.44607X
Yearly Premium = X = $386.09/6.44607 = $59.90
Although at most one death benefit will be paid, there are ten possible payment times, with
payment at a given time contingent upon death in the preceding year. (This death occurs
with probability 0.05—see Column 3.) Thus, the cost of the death benefit is the sum of the
ten actuarial present values in Column 4, which are obtained by multiplying columns 2 and 3.
Similarly, up to ten premium payments will be made, each contingent upon survival to the
time of payment. (This survival becomes less likely with time—see Column 6.) Thus, the
cost to the insured is the sum of the ten actuarial present values in Column 7, which are
obtained by multiplying columns 5 and 6.
V.
Homework Problems
The following problems are designed to enhance understanding of present value and
actuarial present value, life tables, the unifying principle, and term insurance products.
Problem 1: In terms of the probabilities of payment, why does a 10-year term life annuity
cost considerably more than 10-year term life insurance?
(See Example 5.)
Problem 2: How does Example 6 illustrate the principle that an insurance product that
pays benefits sooner tends to cost more than a similar product that pays later.
In terms of present value, why is this principle correct?
Problem 3: How would each of the following affect the cost of life annuities?
• Change the annual interest rate from 5% to 3%
• Change the number of deaths in the life table from 500 to 600 per year
Problem 4: How would each of the following affect the cost of life insurance?
• Change the annual interest rate from 5% to 3%
• Change the number of deaths in the life table from 500 to 600 per year
Problem 5: Complete the following more realistic life table:
Year n
1
2
3
4
5
6
7
8
9
10
Life Table for 10,000 People Alive at Age 75
In Year n
End of Year n
Probability of
Number of
Number
Probability of
death
deaths
alive
survival
451
465
479
494
509
523
534
542
543
537
Problem 6: Assuming 6% annual compound interest, use the life table from Problem 5 to
compute the costs of a 10-year term life annuity and 10-year term life insurance. How do
your results compare with the results of Example 6?