MATEMATIKA KEUANGAN - Matematika IPB

Capital Redemption Policies (Kebijakan Pengembalian Modal)
MATEMATIKA KEUANGAN
Pertemuan ke-12, Mei 2012
Departemen Matematika FMIPA IPB

Capital Redemption Policies (Kebijakan Pengembalian Modal)
Introduction and Premium Calculation
Capital Redemption Policy
A capital redemption policy is a contract which, in return for
a single payment or a series of payments of stated
amount, provides a specified sum of money at the end of a
fixed period.
The payment made by the policyholder are known as
premium and the sum payable at the end of the fixed
period is called the sum asured.
The date on which the sum assured is payable is called the
maturity date.

Capital Redemption Policies (Kebijakan Pengembalian Modal)
Introduction and Premium Calculation
Capital Redemption Policy
The premiums paid by the policyholder are invested by the
insurance company to produce the sum assured at the
maturity date.
In calculating the premiuns to be charged, the company
must make:
1 appropriate assumptions about the rate (or rates) of interest
(net of tax) at which it will be able to invest the premiums
received over the duration of the policy and
2 due allowance for the expences which it expects to incur in
relation to the policy.

Capital Redemption Policies (Kebijakan Pengembalian Modal)
Introduction and Premium Calculation
Capital Redemption Policy
The essential requirement is that at the maturity date the
accumulated amount of the premiums received, less any
expenses incurred, should suffice to pay the sum assured.
An explicit profit margin may also be required, but it is
more usual for the issuing office to charge implicit profit
margins, by making rather conservative assumptions about
interest and expenses. If there is no explicit profit margin,
the equation of value for the policy is:
P = B + E
where
P = Value of premiums to be received by the company.
B = Value of benefits payable by the company.
E = Value of expenses associated with the policy, incurred
by the company.

Capital Redemption Policies (Kebijakan Pengembalian Modal)
Introduction and Premium Calculation
Example
Example 6.1.1
A capital redemption policy with sum assured £10 000 and term
15 years has level annual premiums, payable in advance
throughout the duration of the policy. The company issuing the
policy calculated the annual premium on the basis of an interest
rate of 8% per annum with allowance for
(a) initial expenses of £100 plus 10% of the first annual
premium and
(b) renewal expences of 4% of the second and each
subsequent annual premium.
Find the annual premium for the policy.

Capital Redemption Policies (Kebijakan Pengembalian Modal)
Introduction and Premium Calculation
Solution 6.1.1
Let P ′ be the annual premium. After meeting its expences the
company invests (0.9P ′ − 100) from the first premium and
0.96P ′ from each subsequent premium. Thus, to provide the
sum assured at the end of 15 years, we must have
(0.9P ′ − 100)(1 + i) 15 14
+ 0.96P ′ (1 + i) 15−t = 10000 at i = 8%.
t=1
This equation may be expressed in terms of standard functions
in several ways. For example, we may write

Capital Redemption Policies (Kebijakan Pengembalian Modal)
Introduction and Premium Calculation
Solution 6.1.1
0.9P ′ (1 + i) 15 + 0.96P ′ ¨s 14| = 10000 + 100(1 + i) 15
0.9P ′ ¨s 15| + 0.06P ′ ¨s 14| = 10000 + 100(1 + i) 15
0.96P ′ ¨s 15| − 0.06P ′ (1 + i) 15 = 10000 + 100(1 + i) 15
0.96P ′ (s 16| − 1) − 0.06P ′ (1 + i) 15 = 10000 + 100(1 + i) 15
P ′ (0.96(s 16| − 1) − 0.06(1 + i) 15 ) = 10000 + 100(1 + i) 15
From the above equations we obtain the premium as
P ′ =
10000 + 100(1 + i) 15
at i = 8%
0.96(s16| − 1) − 0.06(1 + i) 15
= $368.99

Capital Redemption Policies (Kebijakan Pengembalian Modal)
Introduction and Premium Calculation
Example
Example 6.1.2
Consider the capital redemption policy described in the
previous example. Suppose, however, that, in order to allow for
an increases in expenses over the duration of the policy, the
company calculated the annual premium on the basis of the
same interest rate and initial expenses as above but allowed for
renewal expenses (associated with the second and each
subsequent premium) of
(a) £5 plus
(b) an increasing percentage of each premium-the amount
increasing linearly from 2.5% of the second premium to 5%
of the final premium.
Find the annual premium.

Capital Redemption Policies (Kebijakan Pengembalian Modal)
Introduction and Premium Calculation
Solution 6.1.2
For t = 1, 2, . . . , 14, let λt denote the increasing percentage of
the premium due at time t which is adsorbed by expenses.
Then
λt − 2.5 =
5 − 2.5
(t − 1)
14 − 1
λt = 2.3077 + 0.1923t.
Let P ′ be the annual premium. The equation of value for the
policy may expressed as
(0.9P ′ − 100)(1 + i) 15 +
14
t=1
P ′ − 5 −
2.3077 + 0.1923t
P
100
′
(1 + i) 15−t = 10000.

Capital Redemption Policies (Kebijakan Pengembalian Modal)
Introduction and Premium Calculation
Solution 6.1.2
Thus
(0.9P ′ (1 + i) 15 + 0.976923P ′ ¨s 14| − 0.001923P ′
14
= 10000 + 5¨s 14| + 100(1 + i) 15 .
The summation on the left-hand side of the last equation is
simply (I¨s) 14| , which equals
¨s 14| − 14
d
t=1
t(1 + i) 15−t

Capital Redemption Policies (Kebijakan Pengembalian Modal)
Introduction and Premium Calculation
Solution 6.1.2
Hence the equation for P ′ is
P ′
0.9(1 + i) 15
¨s 14| − 14
+ 0.976923¨s 14| − 0.001923
d
= 10000 + 5¨s 14| + 100(1 + i) 15 at i = 8%.
Finally we obtain
P ′ = 10447.977/28.0881 = $371.97
This premium is slightly greater than that found in example
6.1.1. The renewal expenses increase from £14.30 to £23.60,
in comparison with the level renewal expenses of £14.76 for the
first example.