Profit Testing
Lecture: Week 13+
Lecture: Week 13+ (Math 3631)
Profit Testing
Spring 2017 - Valdez
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Chapter summary
Chapter summary
To illustrate the ideas of a profit test
Purposes of profit testing
Profit testing for a term insurance policy
Chapter 12 (Dickson, et al.)
Lecture: Week 13+ (Math 3631)
Profit Testing
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Introduction
Profit testing
A profit test is an examination of the cash flows, and the eventual
profits, that emerge over time for an insurance contract.
Very flexible since it can accommodate for example non-constant
interest rates, policyholder behavior/lapses.
It offers the actuary the ability to “explore the risk and return for a
wide range of traditional and modern contracts”.
Some possible applications:
For setting premiums and/or reserves
For profitability analysis
For stress testing of profits (what if conditions are better or worse?)
For accounting how much of surplus or profit can be distributed in the
form of dividends
Lecture: Week 13+ (Math 3631)
Profit Testing
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Numerical illustration
Numerical illustration
Consider a 3-year term insurance policy issued to age 55. You are given:
The gross annual premium is 140, payable at the beginning of each
year.
The death benefit is 10,000, to be paid at the end of the year of
death.
Mortality: q55+t = 0.010 + 0.002 t, for t = 0, 1, 2
Pre-contract expense is 10 and is paid at time 0, or just immediately
prior to policy issue.
Expenses after issue are 1.5 each year paid at time premium is
received.
The reserves at time 0, 1 and 2 are, respectively, 0, 50 and 85.
Cash flows will be accumulated at an annual effective rate of 5%.
Profits are to be discounted at an annual effective rate of 10%.
Lecture: Week 13+ (Math 3631)
Profit Testing
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Numerical illustration
Emerging profit
The following table provides details of the calculations of the profit
emerging for this policy:
t
0
1
2
3
tV
0
50
85
P
140
140
140
Et
10.0
1.5
1.5
1.5
Lecture: Week 13+ (Math 3631)
It
q55+t−1
EDBt
Et V
6.925
9.425
11.175
0.010
0.012
0.014
100
120
140
49.50
83.98
0.00
Profit Testing
Prt
-10.00
-4.08
-6.05
94.68
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Numerical illustration
Details of the calculations
We explain the details of the calculations of the previous table below:
t clearly refers to the year.
tV
is the start of the year reserve and are given here in the problem.
P is clearly the gross annual premium.
Et is the expense.
It = 0.05 ∗ (t V + P − Et )
q55+t−1 is the applicable mortality rate which is given.
EDBt = 10000 ∗ q55+t−1 is the expected death benefit at the end of
the year.
Et V = (1 − q55+t−1 ) ∗ t V is the expected reserve for the end of the
year.
Lecture: Week 13+ (Math 3631)
Profit Testing
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Numerical illustration
The profit vector
The expected profit emerging for a policy in force at the start of the year
is defined to be
Pr0 = −E0 − 0 V
at time issue (t = 0), and for subsequent years (t > 0), we have
Prt =
t−1 V
+ P − Et + It − EDBt − Et V
The profit vector is defined to be the vector of these profits and sometimes
expressed as
Pr = (Pr0 , Pr1 , Pr2 , Pr3 )0 = (−10, −4.08, −6.05, 94.68)0
Lecture: Week 13+ (Math 3631)
Profit Testing
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Numerical illustration
Profit signature
We denote the profit signature by πt and is defined to be the profit
emerging unconditionally so that we have:
π0 = Pr0
at time issue (t = 0), and for subsequent years (t > 0), we have
πt =
t−1p55 Prt
The profit signature is sometimes expressed as
π = (π0 , π1 , π2 , π3 )0 = (−10, −4.08, −5.99, 92.60)0
Lecture: Week 13+ (Math 3631)
Profit Testing
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Numerical illustration
- continued
The following table provides details of the calculations of the profit
signature:
t
0
1
2
3
Lecture: Week 13+ (Math 3631)
Prt
-10.00
-4.08
-6.05
94.68
t−1p55
1
0.99
0.97812
Profit Testing
πt
-10.00
-4.08
-5.99
92.60
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Numerical illustration
Net present value
Assuming a risk discount rate of 10%, as given in this problem, the net
present value for this policy is given by
NPV =
3
X
πt
t=0
1
1.1
t
It refers to the actuarial present value of expected profits at issue. For our
example, it is easy to verify that
3
2
NPV = −10 − 4.08v.1 − 5.99 ∗ v.1
+ 92.60 ∗ v.1
= 50.91225,
where v.1 = 1/1.10 and 10% is called the risk discount rate or sometimes,
hurdle rate.
Lecture: Week 13+ (Math 3631)
Profit Testing
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Numerical illustration
Profit margin
The profit margin is defined to be the ratio
Profit Margin =
NPV
APV(expected profits)
=
APV(gross premiums)
APV(gross premiums)
where the APV calculations are at issue. Back to our example, we have
2
APV(gross premiums) = 140 × [1 + v.05 (0.990) + v.05
(0.990)(0.988)]
= 396.2057
so that the profit margin for our policy is
Profit Margin =
Lecture: Week 13+ (Math 3631)
50.91225
= 12.85%
396.2057
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Numerical illustration
Other profit measures
The internal rate of return, IRR, is the risk discount rate that results in a
zero NPV. For our example, it can be shown (though you need a
spreadsheet or a software to compute) that the IRR is 88%, pretty good
IRR.
The discounted payback period is the first year the accumulation of the
NPV reaches non-negative, that is, the first year k for which
NPV(k) =
k
X
πt vrt ≥ 0,
t=0
where r is the risk discount rate. It refers to the earliest time period
insurer recovers the large loss typically incurred at issue. In our example, it
is clear that the discounted payback period is 3 years.
Lecture: Week 13+ (Math 3631)
Profit Testing
Spring 2017 - Valdez
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Numerical illustration
Numerical illustration
Numerical illustration
WA Question 5, Fall 2016
Lecture: Week 13+ (Math 3631)
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