# Insurance Case

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## Content

McCardle
Fall 2015

Management 402
Evaluating Single-Premium Insurance Policies
You are an investment adviser and your client, a white American woman who just today reached the age
of 65, is considering purchase of a single-premium insurance policy. The fundamental feature of this
policy is that the premium is paid all at once when the policy is written, and this one-payment feature
does not run afoul of any IRS tax codes. In her case, each dollar of premium generates 3.6914 dollars of
insurance to be paid at her death. She seeks to evaluate this insurance policy as an investment by
comparing it with an alternative strategy of investment (and reinvestment) in tax-free municipal bonds.
Her objective is to maximize the expected amount of money for her estate at the time of her death.
Some of the data you need in order to help her make her decision can be found in mortality tables issued
by the National Center for Health Statistics. Table 6 of Vital Statistics of the United States -- Mortality
is included in amended form as Exhibit 1. It shows, for example, that of 100,000 new-born white
females, 87,633 reached their 65th birthday. Of these, 999 died prior to their 66th birthday. Similar
calculations can be carried out to age 120, at which point you may assume that all of the women are
deceased. The data in the table is available on the network.
Let R be the random variable representing the residual life of your client, where residual life is the
number of years till her death. To simplify calculations and the use of the table, assume that all deaths
occur half way through the year. That is, if she dies between today and her 66th birthday, R= 0.5. If she
dies between her 68th and 69th birthday, R= 3.5. The probability distribution for R can be computed
from Exhibit 1.
Assume that the annual rate on tax-free municipal bonds is 6.50% and that all of the dividends
associated with these bonds can be reinvested at this rate. Assume also that the bonds mature at the
moment of death, that is, half way into each year. Thus, in two and a half years they would be worth
(1.065)2.5 = 1.1705 dollars per dollar invested.
The insurance policy is the superior investment if she lives only a short time whereas the alternative
strategy of investing in municipal bonds is preferable if she lives a very long time -- say 20 years or
longer. Today, you must help her decide which investment strategy to pursue, prior to knowing the
exact date of her demise. To aid in the comparison, ignore any potential tax effects and assume that the
probability of default is zero for both investments.
Answer the following questions. We will discuss the solutions in class on the date listed on the syllabus.
This assignment is to be completed individually -- you may discuss it within your group, but each
student should solve and submit their own solution.

1

McCardle
Fall 2015

Management 402
Evaluating Single-Premium Insurance Policies

1. Compute the probability distribution for R from the data in Exhibit 1. What are the mean and
standard deviation of R?
2. Assuming she invests one dollar in the municipal bonds today, what is the expected amount of
money on hand at the time of her death? What is its standard deviation?
3. What is the probability that the payout on the bonds exceeds the payout on the insurance policy?
4. Based on her criteria of maximizing the expected amount of money on hand at the time of her death,
which investment is preferable, the bonds or the insurance policy?
5. Defining rates of return for random cash flows can be problematic, but let's try anyway. The
insurance company pays 3.6914 at the time of death. Define the implicit rate of return for the
insurance policy as that rate i you would have to earn on the tax-free municipal bonds to match this
payout in expectation. What is the implicit rate of return on the insurance policy?

This case was written by Professor Steven Lippman, UCLA.

2

McCardle
Fall 2015

EXHIBIT 1
Table 6. Life table for white females: United States, 2003
Amended with mortality statistics for 2003, ages 100+

Age
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40

Number
surviving
to
age
100,000
99,495
99,458
99,431
99,411
99,394
99,381
99,370
99,357
99,346
99,334
99,321
99,310
99,298
99,282
99,265
99,240
99,204
99,163
99,114
99,068
99,024
98,977
98,930
98,885
98,837
98,791
98,743
98,694
98,643
98,590
98,536
98,476
98,414
98,343
98,271
98,186
98,097
97,997
97,892
97,773

Age
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80

Number
surviving
to
age
97,642
97,500
97,342
97,175
96,991
96,794
96,582
96,355
96,116
95,851
95,567
95,266
94,937
94,595
94,206
93,791
93,308
92,832
92,263
91,642
90,951
90,247
89,445
88,589
87,633
86,634
85,555
84,382
83,109
81,764
80,258
78,682
76,976
75,156
73,195
71,092
68,828
66,399
63,839
61,105

3

Age
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120

Number
surviving
to
age
58,150
55,069
51,855
48,350
44,940
41,439
37,876
34,285
30,704
27,176
23,744
20,455
17,350
14,470
11,847
9,506
7,464
5,723
4,277
3,110
2,013
1,257
768
455
252
137
77
38
19
10
5
3
2
1
1
1
0
0
0
0

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