Introduction
Non-life insurance
Life insurance
Safety loadings
Premium calculation
Jiřı́ Valecký
Technical University of Ostrava
Faculty of Economics
department of Finance
summer semester 2013/2014
Jiřı́ Valecký
Premium calculation
Introduction
Non-life insurance
Life insurance
Safety loadings
Content
1
Introduction
Fundamentals
Insurer’s expenses
2
Non-life insurance
Equivalence principles
Calculation principles
3
Life insurance
Equivalence principle
Premium calculations
term life insurance
endowment insurance
whole life insurance
4
Safety loadings
Jiřı́ Valecký
Premium calculation
Introduction
Non-life insurance
Life insurance
Safety loadings
Fundamentals
Insurer’s expenses
Introduction
Insurers offer a policy with certain benefits under given
conditions and sell it at stated price, premium.
If the contract is accepted and premium is paid by the
customer, the customer becomes the policyholder.
The aim
This lecture is focused on premium calculations.
Jiřı́ Valecký
Premium calculation
Introduction
Non-life insurance
Life insurance
Safety loadings
Fundamentals
Insurer’s expenses
Fundamentals
Figure: Premium calculation
Jiřı́ Valecký
Premium calculation
Introduction
Non-life insurance
Life insurance
Safety loadings
Fundamentals
Insurer’s expenses
Fundamentals
Step 1
The choice of statistical bases to construct the probability
distribution of random (PV of the) benefits.
Step 2
The choice of annual interest rate (or term structure of interest
rates) - discounting is omitted if duration is short.
Step 3
Calculation of other expenses not related to the benefits.
Step 4
Calculation of minimal profit margin for insurer.
Jiřı́ Valecký
Premium calculation
Introduction
Non-life insurance
Life insurance
Safety loadings
Fundamentals
Insurer’s expenses
Insurer’s expenses
Initial expenses
occur when the policy is issued (commissions to agents for
selling and underwritting expenses).
Renewal expenses
occur each time when premium is payable (also cover fixed
costs, for example staff salaries etc.).
Termination expenses
occur when policy expires (on the death, on the maturity
date, etc.); are associated with the paperwork to finalize
and pay a claim.
Jiřı́ Valecký
Premium calculation
Introduction
Non-life insurance
Life insurance
Safety loadings
Equivalence principles
Calculation principles
Non-life insurance
Equivalence principle:
P = E [S] ,
where P is premium and E [S] is expected risk (or mean
aggregated claims in portfolio).
P > E [S] ⇒ P − E [S] = m,
where m is risk (safety) loadings.
Jiřı́ Valecký
Premium calculation
Introduction
Non-life insurance
Life insurance
Safety loadings
Equivalence principles
Calculation principles
Calculation principles
The expected value principle (EVP)
P = E [S] + αE [S] = (1 + α) E [S] for α > 0,
where α is relative security loading (or pure premium loading
factor) on the pure premium E [S].
The standard deviation principle (SDP)
P = E [S] + αSD [S]
The variance principle (VP)
P = E [S] + αVar [S]
Jiřı́ Valecký
Premium calculation
Introduction
Non-life insurance
Life insurance
Safety loadings
Equivalence principles
Calculation principles
Calculation principles
The quantile principle (QP)
Premium P is set for a risk as a certain percentile, p, of the
distribution of the risk S, i.e. P satisfies
Pr (S ≤ P) = p
The zero utility principle (ZUP)
Let W be initial wealth and let’s have utility function u satisfying
u 0 (x) ≥ 0, u 00 (x) ≤ 0 for x > 0 (it implies that u is concave) and
u is not exponential function. P is stated to keep zero gain in
expected utility by insuring the risk, thus
u (W ) = E [u (W + P − S)] .
Jiřı́ Valecký
Premium calculation
Introduction
Non-life insurance
Life insurance
Safety loadings
Equivalence principle
Premium calculations
LIfe insurance
Equivalence principle:
P = PV (E [S]) ,
where PV is present value.
Remark
Premium can be paid as a lump sum (single premium) or as a
regular series of payments (annually, monthly,...).
Jiřı́ Valecký
Premium calculation
Introduction
Non-life insurance
Life insurance
Safety loadings
Equivalence principle
Premium calculations
Premium calculations
Single premium for term life insurance

C (1 + i)−1




 C (1 + i)−2
S=
...

−r



 C (1 + i)
0
if the insured dies in the first year
if the insured dies in the second year
...
if the insured dies in the rth year
if the insured is alive in the rth year
P = C(1 + i)−1 0|1 qx + C(1 + i)−2 1|1 qx + . . . + C(1 + i)−r r −1|1 qx ,
where r −1|1 qx is the probability that the insured, age x, dies
between time r − 1 and r .
Jiřı́ Valecký
Premium calculation
Introduction
Non-life insurance
Life insurance
Safety loadings
Equivalence principle
Premium calculations
Premium calculations
Single premium for endowment insurance

0



0
S=
...



C (1 + i)−r
if the insured is alive in the first year
if the insured is alive in the second year
...
if the insured is alive in the rth year
P = PV (E [S]) = C (1 + i)−r 1 − 0|r qx ,
where 1 − 0|r qx is probability that the insured will survive r
year.
Jiřı́ Valecký
Premium calculation
Introduction
Non-life insurance
Life insurance
Safety loadings
Equivalence principle
Premium calculations
Premium calculations
Annual premium for endowment insurance
P ä n| = PV (E [Y ]) ⇒ P =
PV (E [Y ])
,
ä n|
where ä n| is present value of an anuity-certain of 1 payable
annually in advance for n years, thus
ä n| = 1 + v + v 2 + ... + v n =
(1 + i)n − 1
(1 + i) ,
(1 + i)n i
where v n = (1 + i)−n is a discount factor.
Jiřı́ Valecký
Premium calculation
Introduction
Non-life insurance
Life insurance
Safety loadings
Equivalence principle
Premium calculations
Premium calculations
Single premium for whole life insurance


Aa n| (1 + i)−1




 Aa n| (1 + i)−2
S=
...

−r


 Aa n| (1 + i)

 0
if the insured dies in the first year
if the insured dies in the second year
...
if the insured dies in the rth year
if the insured is alive in the rth year
P = PV (E [S]) = Aa n| (1 + i)−1 0|1 qx +
Aa n| (1 + i)−2 1|1 qx + . . . + Aa n| (1 + i)−r r −1|1 qx
Jiřı́ Valecký
Premium calculation
Introduction
Non-life insurance
Life insurance
Safety loadings
Equivalence principle
Premium calculations
Premium calculations
Single premium for whole life insurance
a n| = v + v 2 + ... + v n =
(1 + i)n − 1
(1 + i)n i
is present value of an annuity-certain of 1 payable annually in
arrear for n years.
Annual premium for whole life insurance
P ä n| = PV (E [Y ]) ⇒ P =
Jiřı́ Valecký
PV (E [Y ])
ä n|
Premium calculation
Introduction
Non-life insurance
Life insurance
Safety loadings
Setting the safety loadings
Explicit safety loading approach
the value of α is chosen (SDP, VP, etc.);
common in practice of non-life insurance.
Implicit safety loading approach
no explicit safety loading parameter in the formula within
life insurance;
it is included in the premium calculation by setting
0
0
r −1|1 qx > r −1|1 qx and discount rate i < i;
for instance:
−r
1 − 0|r qx0 ,
P = PV (E [S]) = C 1 + i 0
Jiřı́ Valecký
Premium calculation
Introduction
Non-life insurance
Life insurance
Safety loadings
For further study
D. C. M. Dickson, M. Hardy, and H. R. Waters, Actuarial
mathematics for life contingent risks.
Cambridge: Cambridge University Press, 2009.
Y.-K. Tse, Nonlife actuarial models : theory, methods and
evaluation.
Cambridge: Cambridge University Press, 2009.
Jiřı́ Valecký
Premium calculation