Wrocław University of Technology, Faculty of Computer Science and Management, Study field: Management
EXERCISES „COMMERCIAL INSURANCE”
Issue: ”Insurance fundamentals, risk analysis, insurance premium”
EXERCISE No 1 (Fundamentals of insurance)
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What is insurance and what is an insurance event?
What economic and social benefits exist by insurance?
What are advantages and disadvantages of insurance?
What functions, principles, and forms does insurance have?
What is an indemnity and what is a deductible?
Explain how insurance subrogation supports the principle of indemnity?
What are basic requirements and conditions to start (launch) the operation as an insurer?
Explain the difference between pure risk and speculative risk.
What is the difference between peril and hazard? Define physical hazard, moral hazard, morale hazard, and
legal hazard.
10. What are the major differences between insurance and gambling and between insurance and hedging?
11. Explain methods of handling risk in insurance.
12. What is a specific of insurance business and sector?
13. What is an insurance contract and what legal characteristics does it have?
14. On what basis can the insurance company (insurer) deny payment of the claim?
15. What is an insurance premium and what structure does it have?
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EXERCISE No 2 (Risk)
Studying at the Wrocław University of Technology you are exposed to two types of risk: pure and speculative
one. Give examples of these risks?
Suggestion: pure risk means loses or no losses (loses = 0); speculative risk means loses, no losses or profits.
EXERCISE No 3 (Time value of money)
A software package for ABC Company’s management system will cost €175.000. The software implementation
project will provide net after-tax cash flow of €50.000 per year for five years. Assuming an interest (discount) rate
of 6 percent, what is the net present value (NPV) of this project? Does the NPV indicate this project is
acceptable?
Suggestion: use compounding formula FVn=PV0(1+i)n
EXERCISE No 4 (Insurance coverage bids)
Assume that you would like to purchase property insurance on a building. You are analysing two insurance
coverage bids. The bids are from comparable insurance companies, and the coverage amounts are the same.
The premium and deductibles, however, differ. Insurer A’s coverage requires an annual premium of €90.000
with a €5.000 per-claim deductible. Insurer B’s coverage requires an annual premium of €35.000 with a €10.000
per-claim deductible. You are wondering whether the additional €55.000 in premiums is warranted to obtain the
lower deductible. Using some of the loss forecasting methods, you predict that the following losses will occur
(table 1).
Table 1.
Expected number of losses
12
6
2
Expected size of losses
€5.000
€10.000
over €10.000
Which coverage bid should you select, based on the number of expected claims and the magnitude of these
claims? For simplicity, assume that premiums are paid at the start of the year, losses and deductibles are paid at
the end of the year, and 5 percent is the appropriate interest (discount) rate.
Suggestion: use compounding formula FVn=PV0(1+i)n and minimize the present value of the cash outflows.
EXERCISE No 5 (Technical interest rate in life insurance)
Life insurance enterprise calculated annual technical interest rate as 3 percent. For life-insurance policy claim of
€50.000 to be paid in 20 years calculate net premium paid by an insured. If real technical interest rate
decreases, will it be a loss or a profit for insurer?
Suggestion: Use compounding formula PV0= FVn (1+i)-n
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Wrocław University of Technology, Faculty of Computer Science and Management, Study field: Management
EXERCISE No 6 (Proportional liability of non-life insurance)
Using proportional liability principle of non-life insurance, calculate an indemnity (I) to a policy claim for partial
losses. The insurable value (V) is 12.000 zł, insurance amount (A) is 10.000 zł, and size of losses (L) is 10.000
zł. What indemnity will be paid when there is total damage (total losses).
Suggestion: use formula I/L=A/V.
EXERCISE No 7 (First risk liability of non-life insurance)
Using first risk liability principle of non-life insurance, calculate an indemnity (I) to a policy claim for partial losses.
Insurance amount (A) is 8.000 zł, and size of losses (L) is 5.000 zł. What indemnity will be paid when there is
total damage (total losses).
Suggestion: use formula I=L, when L≤A; and I=A, when L>A.
EXERCISE No 8 (Integral franchise deductible)
Using integral (conditional) franchise deductible instrument for reduction of non-life insurer’s liability calculate an
indemnity (I) to a policy claim for partial losses. Insurable value (V) is 10.000 zł, size of losses (L) is 5.000 zł,
and policy integral franchise (FI) deductible is set at 3 percent of insurable value. What indemnity will be paid
when there is total damage (total losses).
Suggestion: use formula I=0, when L≤(FI/100)V; and I=L, when L>(FI/100)V.
EXERCISE No 9 (Reduction franchise deductible)
Using reduction (unconditional) franchise deductible instrument for reduction of non-life insurer’s liability
calculate an indemnity (I) to a policy claim for partial losses. Insurable value (V) is 10.000 zł, size of losses (L) is
5.000 zł, and policy reduction franchise (FR) deductible is set at 3 percent of insurable value. What indemnity will
be paid when there is total damage (total losses).
Suggestion: use formula I=0, when L≤(FR/100)V; and I=L-(FR/100)V, when L>(FR/100)V.
EXERCISE No 10 (Deductible)
Using deductible (as insured’s participation in a risk) instrument for reduction of non-life insurer’s liability and
proportional insurer liability principle, calculate an indemnity (I) to a policy claim for partial losses. Insurable value
(V) is 12.000 zł, insurance amount (A) is 10.000 zł, size of losses (L) is 5.000 zł, and insured’s deductible (D) is
20 percent of of insurable value. What indemnity will be paid when there is total damage (total losses).
Suggestion: use formula I/Lr=A/V (and Lr=L--(D/100)V)
EXERCISE No 11 (Co-insurance)
The risk is insured by 2 non-life insurers (A and B). Using proportional principle for non-life co-insurance
calculate indemnities (IA, IB) paid by both insurers to a policy claim for partial losses. Insurance amount in A
policy (AA) is 100.000 zł, insurance amount in B policy (AB) is 25.000 zł, and the size of losses (L) is 75.000 zł,
and insured’s deductible (D) is 20 percent of of insurable value. What indemnities will be paid by A and B
insurers when there is total damage (total losses).
Suggestion: use formula IA=L (AA/(AA+AB)), IB=L (AB/(AA+AB)).
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EXERCISE No 12 (Life insurance premium)
An enterprise has portfolio of 100 life insurance policies for risk of workers accidents’ consequences. Insurance
amount (A) for each policy is €10.000. During the last 2 years there were 15 accidents with average losses and
claim of €2.000. Calculate a premium (P) for each policy.
Suggestion: use pure risk principle, equivalence rule for premiums and expected sizes of indemnities P=E(I),
and P=ipA, where i is intensiveness of losses, p is probability of claims after accidents.
EXERCISE No 13 (Non-life insurance premium)
A Home Appliance Shop Company has portfolio of 500 non-life insurance policies for risk of causes-of-loss in
basic form. The expected size and probability of losses is described as discrete random variable (table 1).
a) Calculate one policy insurance premium with application of loss expected value principle and safety load
() at 10 percent level;
b) Calculate one policy insurance premium with application of expected value principle with variance safety
load (a) at 1 percent level.
Table 1.
Loss size (L)
0
10.000 zł
20.000 zł
Probability of losses (L=l)
0.5
0.25
0.25
Suggestion: use law of large numbers (portfolio of a large size) and expected value principles: E(Z)=500·E(L);
P=(1+)E(Z) and P=E(Z)+aVar(Z).
Wrocław University of Technology, Faculty of Computer Science and Management, Study field: Management
EXERCISE No 14 (Non-life insurance premium with deductibles)
Non-life insurer possesses 200 policies for risk of causes-of-loss in basic form with the discrete random
probability distribution (table 1). The principle to calculate premium is standard deviation with safety load (b) of
20 percent.
Table 1.
Loss size (L)
0
100
200
300
400
500
Probability of losses (L=I)
0.5
0.1
0.1
0.1
0.1
0.1
Calculate premium (Pp) for one policy with the following assumption:
a) There is no liability limit or deduction;
b) There is upper liability limit at level 400 of loss size;
c) There is integral franchise deduction at level 100 of loss size;
d) There is reduction franchise deduction in amount of 100;
e) There is reduction franchise deduction in amount of 10 percent of loss size.
Suggestion: use law of large numbers (portfolio of a large size) and standard deviation principles:
E(Z)=200·E(L); P=E(Z)+b (Var(Z)).