Profit Test Modelling in Life Assurance Using Spreadsheets
PROFIT TEST
MODELLING IN LIFE
ASSURANCE USING
SPREADSHEETS
PART ONE
Erik Alm
Peter Millington
2004
1
Profit Test Modelling in Life Assurance Using Spreadsheets
Profit test modelling in life assurance
1. Introduction
The aim of this brief is to demonstrate the development of profit test models in life
assurance using spreadsheets. We will work through several illustrative examples
step-by-step and the degree of complexity will increase from one example to the
next. We will start with a single policy excluding loadings and gradually work our way
to an entire portfolio of policies where we also include investment return, expenses,
loadings and maturity benefits. In part two we will build more advanced models
where we will include surrenders, paid-ups and mortality risk.
It is very common in life assurance that the life office has high initial expenses when
writing a new policy. This cost could be commission to the sales agent but could also
be internal costs for underwriting or for IT systems. This leads to a negative cash flow
or negative result for the life office at the inception of a life policy.
The premiums charged by a life office are calculated in such a way that the present
value of the premiums should be equal to or exceed the present value of the future
benefits and expenses. If not, the policy is written at an expected loss which, if it is
done consistently, would threaten the solvency of the life office.
Letting the present value of the premiums being equal to the present value of the
benefits and expenses is not enough. In order for the life office to be able to write
new business, it needs to put up risk capital. For a proprietary company, this is done
by shareholders who expect return on their share capital. For a mutual company this
is done by the existing policyholders, who expect the surplus accumulated in the
company not to be diluted by the writing of new policies that do not contribute to this
surplus.
The life office usually sets internal rules determining the minimum profit to be
emerging from a new life policy or new block of life policies written. One such rule
could be that the present value of the premiums should exceed the present value of
the benefits by a certain percentage. Another way of expressing profit requirements
for a life policy is to state when the initial expenses are repaid at a stated internal
discount rate. We will mainly work with this type of profit requirement in this brief.
2. Unit-linked assurance
2.1. A policy
Let us first start with a unit-linked policy as an example.
In a typical unit-linked assurance policy the financial risk is born by the policyholder.
This differs from traditional assurance where the products typically provide
guaranteed benefits at maturity, death or surrender. In unit-linked assurance,
premiums are invested in a fund of the choice of the policyholder after deductions for
expenses and mortality. The premium reserve will thus (in most cases) be defined
retrospectively, using the investment return earned by the fund.
It should be noted that there are unit-linked policies sold which provide some
guarantees. One typical guarantee is a guaranteed benefit of at least the sum of
premium paid. In our discussion, we will assume that no guarantees are provided.
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Profit Test Modelling in Life Assurance Using Spreadsheets
Let
Vt = premium reserve at year t
Pt = premium paid at the beginning of the year t
it = investment return of the fund during the period
Assuming zero initial expenses and no mortality risk the premium reserve at time t is
expressed as:
Vt = (Vt −1 + Pt ) * (1 + it )
(1)
Expression (1) shows the development of the fund in the time-discrete case. The
keen student could here and in later examples construct the corresponding formula in
the time-continuous case.
This type of policy could, just as well as a traditional policy, be studied analytically.
We will however instead mainly study it in a more straightforward way in a
spreadsheet environment. The main reason for this is that the assumptions used do
often not lead to a nice analytical formula, like the Makeham formula. Another
important advantage of this method is that the student could much easier see what
happens during the lifetime of the policy rather than just seeing one figure being the
result of a long formula. Also, it is easier in a spreadsheet to create ‘what if’
situations, i.e. to test the effect of changes in assumptions and to see how these
changes affect the policy in different periods.
We will look at some principle problems and also give some practical tips on how this
type of study is best done in a spreadsheet environment.
We will generally assume a level premium is paid, i.e.
Pt = P , 0<t=d,
where d is the duration of the policy and t denotes time.
We choose a policy where annual premiums of 100 units are paid for ten years and
express this as in the following:
Pt = 100 , 0<t=10
We will often use only P to denote a level premium.
Let us look at a very simple table illustrating this in a table taken from an Excel (or Lotus or MoSeS)
spreadsheet.
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Profit Test Modelling in Life Assurance Using Spreadsheets
Policy duration
Premium
10 years
100 per year
Premium
100
100
100
100
100
100
100
100
100
100
Year
1
2
3
4
5
6
7
8
9
10
One important thing to note here is that all parameters should be stated explicitly in any Excel
spreadsheet used for this type of calculations. It should always be very easy to change the parameters
and study the effects of such change. If
Pt = 50
for 0<t=5
we should then easily get:
Policy duration
Premium
5 years
50 per year
Premium
50
50
50
50
50
0
0
0
0
0
Year
1
2
3
4
5
6
7
8
9
10
Let us assume that the investment return is zero. At the end of the policy a maturity
benefit is paid, consisting of the sum of the premiums:
Policy duration
Premium payment
Premium
10 years
10 years
100 per year
Year
1
2
3
4
5
6
7
8
9
10
2004
Premium
100
100
100
100
100
100
100
100
100
100
Maturity benefit
-1000
4
Profit Test Modelling in Life Assurance Using Spreadsheets
We show the maturity benefit as negative, since it represents an outgo for the life
office.
Let us also include the cash flow to the life office:
Policy duration
Premium payment
Premium
10 years
10 years
100 per year
Year
1
2
3
4
5
6
7
8
9
10
Premium
100
100
100
100
100
100
100
100
100
100
Maturity benefit
-1000
Cash flow
100
100
100
100
100
100
100
100
100
-900
The premiums the life office receives years 1 to 10 must be reserved, since it will be
needed to pay for the maturity benefit in year 10. The development of the premium
reserve is given by (assuming zero investment return):
Vt = Vt −1 + Pt
or
Vt = t ∗ P
The maturity benefit C d at time d is given by
C d = Vd = d ∗ P
C t = 0 for t ? d.
Policy duration
Premium payment
Premium
Year
1
2
3
4
5
6
7
8
9
10
10 years
10 years
100 per year
Premium
100
100
100
100
100
100
100
100
100
100
Maturity benefit
-1000
Cash flow
100
100
100
100
100
100
100
100
100
-900
Reserve
100
200
300
400
500
600
700
800
900
0
We here use a minus sign for the maturity benefit, since it enters the cash flow as negative.
The reserve increases with the premiums paid and decreases with the maturity
benefit paid out and we note that it is zero after the maturity of the policy just as we
would expect.
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Profit Test Modelling in Life Assurance Using Spreadsheets
For unit-linked business, the reserve (at least in simple cases) consists of savings
belonging to the policyholder, where the policyholder bears the financial risks
connected with the reserve. In such case, the reserve is often called the fund and the
cash flow to and from the fund is not included when one studies the cash flow from
the life office's point of view. The fund is like a bank account and is treated as such in
US GAAP, the general accepted accounting principles in the US.
In our simplified example, the cash flow is given by
CFt = Vt −1 + Pt − C t − Vt = 0
or in spreadsheet format:
Policy duration
Premium payment
Premium
Year
1
2
3
4
5
6
7
8
9
10
10 years
10 years
100 per year
Premium
100
100
100
100
100
100
100
100
100
100
Maturity benefit
Fund
100
200
300
400
500
600
700
800
900
0
-1000
Cash flow
0
0
0
0
0
0
0
0
0
0
In order to avoid having different formulae for year one and subsequent years, we include the value of
the fund at the beginning of the year and the value of the fund at the end of the year. We have divided
the table into two parts, one showing the development of the fund and one showing the cash flow to the
life office.
Policy duration
Premium payment
Premium
Year
1
2
3
4
5
6
7
8
9
10
10 years
10 years
100 per year
Fund in
0
100
200
300
400
500
600
700
800
900
Premium
100
100
100
100
100
100
100
100
100
100
Maturity benefit
-1000
Fund out
-100
-200
-300
-400
-500
-600
-700
-800
-900
0
Cash flow
0
0
0
0
0
0
0
0
0
0
For practical reasons, we use the convention that the fund out is shown with a minus sign (it is positive
for the client but is a liability for the life office).
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Profit Test Modelling in Life Assurance Using Spreadsheets
2.2. Investment return
We have up to now assumed that the money in the fund will earn no return. In real
life, this money is invested in financial assets, in equities or bonds or both. The
investment return includes dividends on shares and realised or unrealised gains on
shares or bonds. We will assume that the investment return is fixed at a rate of it per
time period t, even though the fund value might change continuously.
The fund value and cash flows are given by:
Vt = (Vt −1 + Pt ) ∗ (1 + it )
CFt = (Vt −1 + Pt ) ∗ (1 + it ) − C t − Vt or
CFt = Vt −1 + Pt + (Vt −1 + Pt ) ∗ it − C t − Vt
Assuming a level premium, the maturity benefit at maturity date d is expressed as
d
C d = Vd = P ∗ ∑ (1 + i ) t
t =1
Important to remember is that the investment return varies over time, depending on
the development of the assets in the fund. Let us now assume that the fund will earn
5% annual return (after taxes and after internal fund expenses).
Policy duration
Premium payment
Expected increase in unit value
Premium
Year
1
2
3
4
5
6
7
8
9
10
Fund in
0
105
215
331
453
580
714
855
1 003
1 158
10
10
5%
100
Premium
100
100
100
100
100
100
100
100
100
100
years
years
annually
per year
Interest
5
10
16
22
28
34
41
48
55
63
Maturity benefit
0
0
0
0
0
0
0
0
0
-1 321
Fund out
-105
-215
-331
-453
-580
-714
-855
-1 003
-1 158
0
Cash flow
0
0
0
0
0
0
0
0
0
0
(In unit–linked business, the value of the fund is often expressed as a number of
units, multiplied by the value of a unit. When the underlying assets increase in value,
the number of units remains constant while the value of a unit increases. When new
premium is added to the fund, the number of unit increases.)
Problem: What interest rate is needed in order to provide a maturity benefit of 2000?
Answer: 12.3%
This problem could be solved by analytical methods, but a more practical and faster way is to use the
problem solving methods of Excel: Goal Seeker or Solver. (Tools, Goal Seeker).
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Profit Test Modelling in Life Assurance Using Spreadsheets
2.3. Initial commission
Up to now, the cash flow for the life office has been zero. We shall now start to look
at this cash flow.
The most important cost for the life office in writing business is the initial expenses,
and especially the commission paid to the sales agents, being ties agents or brokers.
This commission is often paid up-front (i.e. directly after a sale is made) and is often
calculated as a percentage (or per mille) of the total premium volume of the contract.
We call the percentage a. The initial commission is thus given by
I 1 = α ∗ d ∗ P , I t = 0, t ≠ 1
This initial commission enters as negative cash flow year one. The cash flow formula
is now given by:
CFt = Vt −1 + Pt + (Vt −1 + Pt ) ∗ it − C t − I t − Vt
The above expression consists of two parts. The first part is inflow and outflow of the
premium reserve. This part does in reality not affect the life office as such but rather
the client fund. The second part is the initial commission. Looking at cash flow that
affects the life office separately, we have:
CFt = − I t
Let us assume that the commission is 40 per mille of the total premium. For our
contract, the total premium is 10*100 = 1000 and the commission is thus 40.
Policy duration
Premium payment
Expected increase in unit value
Premium
Initial commission
10
10
5%
100
4
years
years
annually
per year
% of total premium
Year
Fund in
Premium
Interest
1
2
3
4
5
6
7
8
9
10
0
105
215
331
453
580
714
855
1 003
1 158
100
100
100
100
100
100
100
100
100
100
5
10
16
22
28
34
41
48
55
63
Maturity
benefit
0
0
0
0
0
0
0
0
0
-1 321
Fund out
-105
-215
-331
-453
-580
-714
-855
-1 003
-1 158
0
Commission
-40
0
0
0
0
0
0
0
0
0
Cash
flow
-40
0
0
0
0
0
0
0
0
0
In our tables, we show commission and other expenses as negative, since they mean outflow for the life
office.
2.4. Premium charges
We have an outflow from the life office in the form of the commission. The life office
will need to cover these expenses and this is done by introducing some charges that
the policyholder has to pay. One way of doing this is to charge a percentage of each
premium paid to the life office. Let us introduce such a charge and let that charge ?
be the same as the commission, i.e. 4%.
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Profit Test Modelling in Life Assurance Using Spreadsheets
Introducing premium charges, the development of the premium reserve is given by:
Vt = (Vt −1 + P ∗ (1 − γ )) ∗ (1 + it ) − C t
= Vt + P − P ∗ γ + (Vt + P ∗ γ ) ∗ i − C t
The maturity benefit is
d
C d = Vd = P ∗ (1 − γ ) ∗ ∑ (1 + i ) t
t =1
The cash flow to the life office is
CFt = P ∗ γ − I t
Policy duration
Premium payment
Expected increase in unit value
Premium
Initial commission
Premium charge
10
10
5%
100
4
4
years
years
annually
per year
% of total premium
% of each premium
Year
Fund
in
Premium
Charge
Interest
1
2
3
4
5
6
7
8
9
10
0
101
207
318
435
558
687
822
964
1 114
100
100
100
100
100
100
100
100
100
100
-4
-4
-4
-4
-4
-4
-4
-4
-4
-4
5
10
15
21
27
33
39
46
53
61
2004
Maturity
benefit
0
0
0
0
0
0
0
0
0
-1 270
Fund
out
Charg
e
Commi
ssion
Cash
flow
-101
-207
-318
-435
-558
-687
-822
-964
-1 114
0
4
4
4
4
4
4
4
4
4
4
-40
0
0
0
0
0
0
0
0
0
-36
4
4
4
4
4
4
4
4
4
9
Profit Test Modelling in Life Assurance Using Spreadsheets
A profit testing study in a spreadsheet environment is normally done vertically the way we have done it
up to now. We will however do it horizontally for the remainder of this brief.
Policy duration
Premium payment
Expected increase in unit value
Premium
Initial commission
Premium charge
1
0
100
-4
5
0
-101
4
-40
-36
Year
Fund in
Premium
Charge
Interest
Maturity
Fund out
Charge
Comm
Cash flow
2
101
100
-4
10
0
-207
4
0
4
10
10
5%
100
4
4
3
207
100
-4
15
0
-318
4
0
4
years
years
annually
per year
% of total premium
% of each premium
4
318
100
-4
21
0
-435
4
0
4
5
435
100
-4
27
0
-558
4
0
4
6
558
100
-4
33
0
-687
4
0
4
7
687
100
-4
39
0
-822
4
0
4
8
822
100
-4
46
0
-964
4
0
4
9
964
100
-4
53
0
-1 114
4
0
4
10
1 114
100
-4
61
-1 270
0
4
0
4
The premium charge is shown twice, as an expense for the policyholder and as an income for the life
office.
Let us also look at the accumulated cash flow at time t where t <= d. This is given by:
t
AccCFt = ∑ CFx
x =1
Year
Fund in
Premium
Charge
Interest
Maturity
Fund out
Charge
Comm
Cash flow
Accumulated
cash flow
1
0
100
-4
5
0
-101
4
-40
-36
-36
2
101
100
-4
10
0
-207
4
0
4
-32
3
207
100
-4
15
0
-318
4
0
4
-28
4
318
100
-4
21
0
-435
4
0
4
-24
5
435
100
-4
27
0
-558
4
0
4
-20
6
558
100
-4
33
0
-687
4
0
4
-16
7
687
100
-4
39
0
-822
4
0
4
-12
8
822
100
-4
46
0
-964
4
0
4
-8
9
964
100
-4
53
0
-1 114
4
0
4
-4
10
1 114
100
-4
61
-1 270
0
4
0
4
0
2.5. Net present value
We note in the previous table that the accumulated cash flow amounts to zero at
maturity date d, which seems to show that income and outgo for the life office are
equal. The timing of the two is however not equal. The life office has an initial outgo
while the income comes later and the life office will need to borrow externally or use
internal funds to finance this outgo. These funds are not free and the life office must
therefore include the effect of this cost in its calculations. The most common way to
do this is to calculate present values (the future cash flows are discounted to the
present time.)
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Profit Test Modelling in Life Assurance Using Spreadsheets
The general formula for calculation of Net Present Value as per the beginning of year
1 is
d
d −1
k =1
k =0
NPV ( X 0 ... X n ) = ∑ X k ∗ v k −1 = ∑ X k +1 ∗ v k
where
v=
1
is the discount factor and r is the discount rate.
1+ r
X k = cash flow at time k (i.e. at beginning of year k)
One may use the NPV function of Excel to do this calculation.
One must decide on an appropriate discount interest rate. This discount rate should
take into account the cost of money for the life office. If the commission is financed
through new equity in the company, the cost of money is the return the shareholders
want on this new equity (including tax). This might be 15%. If the life office has idle
funds which would otherwise be invested, the discount rate should take into account
the income which would have been received in such an alternative investment, where
one should include the risk involved with investing funds into initial commissions. If
the initial commission investment is funded through reinsurance, the cost of this
reinsurance could be used for the discount rate.
We will here assume a discount rate of 10%, giving us a discount factor v=0.90909.
Policy duration
Premium payment
Expected increase in unit value
Premium
Initial commission
Premium charge
10
10
5%
100
4
4
Year
Fund in
Premium
Charge
Interest
Maturity
Fund out
Charge
Comm
Cash flow
Accumulated
cash flow
Discount
factor
Discounted
cash flow
Accumulated
discounted
cash flow
years
years
annually
per year
% of total premium
% of each premium
Discount rate
NPV
10%
-13
1
0
100
-4
5
0
-101
4
-40
-36
-36
2
101
100
-4
10
0
-207
4
0
4
-32
3
207
100
-4
15
0
-318
4
0
4
-28
4
318
100
-4
21
0
-435
4
0
4
-24
5
435
100
-4
27
0
-558
4
0
4
-20
6
558
100
-4
33
0
-687
4
0
4
-16
7
687
100
-4
39
0
-822
4
0
4
-12
8
822
100
-4
46
0
-964
4
0
4
-8
9
964
100
-4
53
0
-1 114
4
0
4
-4
10
1 114
100
-4
61
-1 270
0
4
0
4
0
1
0.909
0.826
0.751
0.683
0.621
0.564
0.513
0.467
0.424
-36
4
3
3
3
2
2
2
2
2
-36
-32
-29
-26
-23
-21
-19
-17
-15
-13
We can see that the Accumulated discounted cash flow is equal to –13 at the
maturity age. This is the NPV of the cash flow valued at the discount rate of 10%. We
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Profit Test Modelling in Life Assurance Using Spreadsheets
define this as our profit and our profit goal is that the profit should be positive (or at
least not negative).
The profit could also be calculated directly by using the NPV function in Excel. Please note that the
Excel formula assumes that all payments are made in arrears i.e.at the end of the period in question,
while we here assume that all payments (except the maturity) are made at the beginning of the period in
question. The value calculated by Excel must therefore be multiplied by (1+r), in our case 110% in order
to arrive at the right answer. Using the NPV formula in Excel gives the answer –12, which multiplied by
110% gives –13 as can be found in the lower right hand corner of the table above.
We see that we must use a premium charge greater than 4% in order to break-even,
i.e. a profit of zero. We can calculate the premium that is required for a break-even
situation by setting the NPV of future premium charges equal to the initial
commission.
This gives:
n −1
∑γ ∗ P ∗ v
t
t =0
= I t = α ∗ d ∗ P (n=d).
where n=10, d=10, i=5% and a=4%,
We get
9
∑γ ∗ P ∗ v
t
= α ∗ d ∗ P or
t =0
9
γ ∗ ∑ v t = α ∗ d = 0 .4
t =0
1 − v 10
= 0.4
1− v
γ ∗ 6.76 = 0.4
γ∗
The premium charge that will give break-even is
γ = 0.059
The same answer could have been found by once again using the Goal Seek or Solver.
2.6. Portfolios, model points
We have up to now looked at a 10-year policy. Let us look at a 5-year policy,
assuming an initial commission of 5.9% of the total premium.
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Profit Test Modelling in Life Assurance Using Spreadsheets
Policy duration
Premium payment
Expected increase in unit value
Premium
Initial commission
Premium charge
Year
Fund in
Premium
Charge
Interest
Maturity
Fund out
Charge
Comm
Cash flow
Accumulated
cash flow
Discount
factor
Discounted
cash flow
Accumulated
discounted
cash flow
5
5
5%
100
4
5.9
years
years
annually
per year
% of total premium
% of each premium
Discount rate
NPV
10%
5
1
0
100
-6
5
0
-99
6
-20
-14
-14
2
99
100
-6
10
0
-203
6
0
6
-8
3
203
100
-6
15
0
-312
6
0
6
-2
4
312
100
-6
21
0
-427
6
0
6
4
5
427
100
-6
26
-548
0
6
0
6
10
6
0
0
0
0
0
0
0
0
0
10
7
0
0
0
0
0
0
0
0
0
10
8
0
0
0
0
0
0
0
0
0
10
9
0
0
0
0
0
0
0
0
0
10
10
0
0
0
0
0
0
0
0
0
10
1
0.909
0.826
0.751
0.683
0.621
0.564
0.513
0.467
0.424
-14
5
5
4
4
0
0
0
0
0
-14
-9
-4
1
5
5
5
5
5
5
The table above shows that the 5-year policy gives a profit of 5. If we instead
calculate the profit of a 15-year policy, we would make a loss of 11. For a 20-year
policy, we make a loss of 25. The initial commission formula gets more expensive for
long term policies. Let us therefore assume that the agent gets commission for only
the first 20 premiums, even if the policy duration is longer. This is a common way to
construct sales commission scales. The initial commission is given by:
I 1 = α ∗ min(20; d ) ∗ P
Let us now assume that the maturity age is 65 years, x is the age of the assured, (i.e.
d=65-x) and that the minimum initial age is 20. Let us also assume that the
distribution of initial age will be even over the age band 20-64 years. We could then
calculate the profitability of each initial age and sum the result over all ages as:
NPV = P ∗
 65− x t −1

 ( ∑ v ∗ γ ) − α ∗ min(20;65 − x) 
∑
x = 20 t =1

64
One could in principle solve for ? from the above expression by setting the total to
zero to get the break-even situation.
 65− x t −1

 ( ∑ v ∗ γ ) − α ∗ min(20;65 − x)  = 0
∑
x = 20 t =1

64
NPV =
A rearrangement of the terms gives
64
α ∗ ∑ min(20;65 − x) =
x = 20
64 65 − x
∑ ∑v
t −1
∗γ
x = 20 t =1
and then
2004
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Profit Test Modelling in Life Assurance Using Spreadsheets
64
γ =
α ∗ ∑ min(20;65 − x)
x = 20
64 65 − x
∑ ∑v
t −1
x = 20 t =1
This could however be a bit complicated to handle. Another problem is that, by using
this complex formula, one can not differentiate the profitable from the non-profitable
policies. One gets a much better view of the situation by studying the different
policies one by one. We study therefore the expression for calculating the profit for a
cohort of policies
65 − x
NPV = (
∑v
t −1
∗ γ ∗ P) − α ∗ P ∗ min(20;65 − x) for x = 20, 21,…,64.
t =1
This is straightforward but could be cumbersome. One common way to simplify the
calculations is to use model points. The profits of a 25-year and a 26-year policy are
rather equal and the 25-year policy could represent both a 24-year and a 26-year
policy. We therefore choose a number of model policies that will represent the rest.
Using this principle and letting each 5-year age bands be represented by its middle
point, we thus study
NPV = P ∗ γ ∗ (
65 − x
∑v
t −1
) − P ∗ α ∗ min(20;65 − x) for x = 22, 27,…,62.
t =1
This gives:
Maturity age
Expected increase in unit value
Premium
Initial commission
Premium charge
Age
62
57
52
47
42
37
32
27
22
Policy
duration
3
8
13
18
23
28
33
38
43
Total
65
5%
100
4
6
years
Discount rate
annually
per year
Max commission years
% of total premium max
% of each premium
10%
20
80%
Profit
4
3
-5
-18
-21
-19
-17
-16
-15
-103
This calculation could be done by testing the policy durations one at a time. A quicker way is to use the
Data Table function in Excel which gives all values at the same time. Please note that tables are
dynamically updated if this function is not turned off (Tools, Calculation, Automatic except tables), why
having large tables might lead to heavy update times.
We find:
2004
14
Profit Test Modelling in Life Assurance Using Spreadsheets
65 − x


100 ∗ 5.9% ∗ ( ∑ v t −1 ) − 100 ∗ α ∗ min(20;65 − x)  = −103
x = 22 , 27... 
t =1

62
∑
NPV =
(a)
The result is not good, but it is hard to see how bad it is. We want to know how much
we need to increase the premium charge in order to go break-even. We want to find
a k such that the profit is equal to zero, i.e.:
65 − x


NPV = ∑ 100 ∗ (5.9% + k ) ∗ ( ∑ v t −1 ) − 100 ∗ α ∗ min(20;65 − x)  = 0
x = 22 , 27... 
t =1

62
(b)
Inserting expression (a) in (b) gives
65 − x
64
65 − x



t −1 
∗
+
k
∗
v
v t )  = 103
−
∗
∗
100
(
5
.
9
%
)
(
)
100
5
.
9
%
(



∑
∑
∑
∑
x = 22 , 27... 
t =1
t =1
 x =20

62
This then gives
65 − x


∗
k
∗
v t −1 )  ≈
100
(

∑
∑
t =1
x = 20... 

64
65 −1


k
v t −1  = 103
∗
∗
100

∑
∑
x = 22 , 27... 
t =1

62
Further
k≈
103
64

65 − x
∑ 100 ∗ ( ∑ v
x = 20
t =1
t −1

)

≈
103
64
∑ NPV ( P)
x = 20
We therefore also include the net present value of the premiums paid for each policy
in our table.
Age
62
57
52
47
42
37
32
27
22
Policy
duration
3
8
13
18
23
28
33
38
43
Total
Profit
NPV of
premium
4
3
-5
-18
-21
-19
-17
-16
-15
274
587
781
902
977
1 024
1 053
1 071
1 082
7 750
-103
This gives
k=
103
= 0.0133
7750
The loss is thus -1.33% of the NPV of the total premium. Let us therefore increase
the premium charge with 1.4% to 7.4%:
2004
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Profit Test Modelling in Life Assurance Using Spreadsheets
Maturity age
Expected increase in unit value
Premium
Initial commission
Premium charge
Age
Policy
duration
62
57
52
47
42
37
32
27
22
65
5%
100
4
7.4
years
Discount rate
annually
per year
Max commission years
% of total premium max
% of each premium
Profit
10%
20
80%
NPV of
premium
3
8
13
18
23
28
33
38
43
8
11
6
-5
-8
-4
-2
-1
0
5
Total
274
587
781
902
977
1 024
1 053
1 071
1 082
7 750
We find that the portfolio has a break-even point with a premium charge of 7.4%
Let us now assume that we expect to sell more of some policies and less of others.
Most of our new clients are expected to be around 35 years and few are 20 or 60
years. We include this in our calculation by weighting the different policies by their
expected sales figures:
NPV = P ∗
 65− x t −1

W
∗ γ ) − α ∗ min(20;65 − x) 
∑
x (∑v
x = 20
 t =1

64
Assume that our portfolio has an average duration of 23 years and has an age
distribution as shown in the table below:
Age
Policy
duration
62
57
52
47
42
37
32
27
22
3
8
13
18
23
28
33
38
43
Total
2004
Number of
policies
100
200
300
400
500
400
300
200
100
3 000
16
Profit Test Modelling in Life Assurance Using Spreadsheets
This gives the following results:
Policy
duration
Number of
policies
3
8
13
18
23
28
33
38
43
Total
Profit per
policy
100
200
300
400
500
400
300
200
100
3 000
8
11
6
-5
-8
-4
-2
-1
0
NPV
premium per
policy
274
587
781
902
977
1 024
1 053
1 071
1 082
Total profit
Total NPV of
premium
824
2 285
1 746
-2 096
-3 845
-1 698
-631
-155
5
-3 565
27 355
117 368
234 411
360 862
488 577
409 489
315 791
214 118
108 174
2 276 146
The figures in the column Profit per policy are rounded to the nearest integer. When calculating the total
profit, non-rounded figures are used.
We have here more of the non-profitable policies and less of the profitable policies.
The NPV of the loss is only 0.16% of the NPV of the total premium, why an increase
of the premium charge of 0.2% should be enough to make the portfolio profitable. We
increase the premium charge to 7.6%.
Maturity age
Expected increase in unit value
Premium
Initial commission
Premium charge
Total
65
5%
100
4
7.6
years
Discount rate
annually
per year
Max commission years
% of total premium max
% of each premium
Policy
duration
Number of
policies
Profit per
policy
3
8
13
18
23
28
33
38
43
100
200
300
400
500
400
300
200
100
3 000
9
13
7
-3
-6
-2
0
1
2
NPV
premium per
policy
274
587
781
902
977
1 024
1 053
1 071
1 082
10%
20
80%
Total profit
Total NPV of
premium
879
2 520
2 215
-1 374
-2 868
-879
0
273
221
987
27 355
117 368
234 411
360 862
488 577
409 489
315 791
214 118
108 174
2 276 146
As shown in the previous table, we have arrived at a small profit of 987.
In the real world, you might not know the actual age distribution of the portfolio. It is
therefore often a good idea to test different reasonably realistic age distributions in
the portfolio and choose the least favourable. In our case, we assume that we will sell
either the evenly distributed portfolio or the one with the weight on duration 23 and
we choose the latter one and thus the premium charge of 7.6%.
We discussed in section 2.5. the choice of discount rate. The result that we arrive at
is dependent on the discount rate chosen.
Problem: How would the profitability be with a discount rate of 12%
2004
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Profit Test Modelling in Life Assurance Using Spreadsheets
Answer: There will be a loss of 0.66% of NPV of total premiums. A high discount
makes it more expensive to have high initial costs.
2.7. Fixed costs
Up to now, we have only included the commissions to the sales agents as expenses.
These commissions are defined to be proportional to premium volume, why it does
not matter if we have sold small or large policies.
Let us now assume that we have an initial fixed expense of 10 for each new policy.
The introduction of this new expense leads to a need of increase in charges. One
possibility could be to introduce a policy charge of the same amount as the policy
expense. Another would be to increase the premium charge. We will choose the
latter alternative. We therefore now want to determine how much we need to
increase the premium charge to offset this expense.
With a fixed cost, large policies will be more profitable than small policies. We will
investigate the effect on a portfolio of policies with different premium. The example
below shows the case for one policy with a premium of 100.
Policy duration
Expected increase in unit value
Premium
Initial commission
Internal initial expenses
Premium charge
Year
Fund in
Premium
Charge
Interest
Maturity
10
5%
100
4
10
7.6
years
Discount rate
annually
NPV of profit
per year
NPV of premium
% of total premium max
per policy
Max commission years
% of each premium
1
0
100
-8
5
2
97
100
-8
9
3
199
100
-8
15
4
306
100
-8
20
5
418
100
-8
26
6
536
100
-8
31
7
660
100
-8
38
8
790
100
-8
44
9
926
100
-8
51
0
0
0
0
0
0
0
0
-97
8
-40
-199
8
0
-306
8
0
-418
8
0
-536
8
0
-660
8
0
-790
8
0
-926
8
0
0
-1
070
8
0
-10
-42
0
8
0
8
0
8
0
8
0
8
0
8
0
8
0
8
0
8
-42
-35
-27
-20
-12
-4
3
11
18
26
1
0.909
0.826
0.751
0.683
0.621
0.564
0.513
0.467
0.424
-42
7
6
6
5
5
4
4
4
3
-42
-35
-29
-23
-18
-14
-9
-5
-2
1
Fund out
Charge
Comm
Internal
expenses
Cash flow
Accumulated
cash flow
Discount
factor
Discounted
cash flow
Accumulated
discounted
cash flow
2004
10%
-9
68
80%
20
10
1 070
100
-8
58
-1
220
0
8
0
18
Profit Test Modelling in Life Assurance Using Spreadsheets
If we do this calculation for different policy premiums and durations, we get:
Policy duration
Expected increase in unit value
Premium
Initial commission
Internal initial expenses
Premium charge
x
5%
100
4
y
7.6
years
Discount rate
annually
NPV of profit
per year
NPV of premium
% of total premium max
per policy
Max commission years
% of each premium
10%
-9
68
80%
20
Profit
Duration
3
8
13
18
23
28
33
38
43
Annual premium
10
-9
-9
-9
-10
-11
-10
-10
-10
-10
40
-6
-5
-7
-11
-12
-11
-10
-9
-9
100
-1
3
-3
-13
-16
-12
-10
-9
-8
250
12
21
8
-19
-24
-15
-10
-7
-4
1000
78
116
64
-44
-67
-32
-10
4
12
If all policy durations and premiums were evenly distributed, we could just sum up a
total and get –156. Let us now however assume that the policies are expected to be
distributed as follows:
3
8
13
18
23
28
33
38
43
10
0.80%
1.60%
2.40%
3.20%
4.00%
3.20%
2.40%
1.60%
0.80%
40
1.60%
3.20%
4.80%
6.40%
8.00%
6.40%
4.80%
3.20%
1.60%
100
1.00%
2.00%
3.00%
4.00%
5.00%
4.00%
3.00%
2.00%
1.00%
250
0.40%
0.80%
1.20%
1.60%
2.00%
1.60%
1.20%
0.80%
0.40%
1000
0.20%
0.40%
0.60%
0.80%
1.00%
0.80%
0.60%
0.40%
0.20%
As before, we multiply the result for each type of policy with the probability weight of
that policy in order to arrive at the portfolio probability. We thus multiply the profit
matrix with the distribution matrix and arrive at the following result.
Profit
3
8
13
18
23
28
33
38
43
10
-0.07
-0.14
-0.22
-0.33
-0.42
-0.33
-0.24
-0.16
-0.08
40
-0.10
-0.16
-0.34
-0.73
-0.98
-0.70
-0.48
-0.30
-0.15
100
-0.01
0.05
-0.08
-0.54
-0.79
-0.49
-0.30
-0.17
-0.08
250
0.05
0.17
0.10
-0.30
-0.49
-0.25
-0.12
-0.05
-0.02
1000
0.16
0.46
0.38
-0.35
-0.67
-0.26
-0.06
0.01
0.02
Total –9.53
For the premium, we correspondingly multiply the premium per policy with the weight:
2004
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Profit Test Modelling in Life Assurance Using Spreadsheets
Premium
3
8
13
18
23
28
33
38
43
10
0.22
0.94
1.88
2.89
3.91
3.28
2.53
1.71
0.87
40
1.75
7.51
15.00
23.10
31.27
26.21
20.21
13.70
6.92
100
2.74
11.74
23.44
36.09
48.86
40.95
31.58
21.41
10.82
250
2.74
11.74
23.44
36.09
48.86
40.95
31.58
21.41
10.82
1000
5.47
23.47
46.88
72.17
97.72
81.90
63.16
42.82
21.63
Total 1074
The profit in relation to the premium is -9.53/1074 = –0.9%. Let us try with a premium
charge of 8.5%. We get a profit very close to zero as expected. We have thus found
our break-even point.
2004
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