# statistics the risk manager for Faulty Towels Hotel

Question

6)Basil, the risk manager for Faulty Towels Hotel, has been studying the pattern of property losses to the hotel over past years and has arrived at the following probability distribution:

Annual cost of losses ($) Probability

0 – 8,000 .15

8,001 – 16,000 .3

16,001 – 40,000 .25

40,001 – 80,000 .15

80,001 – 160,000 .08

160,001 – 400,000 .05

400,001 – 800,000 .02CRM/NOV11/1 Page 4 of 4

An insurance company has quoted an insurance premium of $60,000 to cover the risk.

REQUIRED (making any necessary assumptions and showing all workings):

a) Calculate the probability that losses will be $40,000 or below. (3 marks)

b) Calculate the expected $ loss per year. (4 marks)

c) Calculate, on the basis of expected monetary value (EMV), how much Basil would save if he did not purchase the insurance cover. (3 marks)

d) Now assume that an insurance policy with a $80,000 deductible can be purchased for an annual insurance premium of $20,000. Calculate the expected total cost of protection under this arrangement. (5 marks)

e) With reference to your calculations so far for this question, explain how Basil should choose between i) full retention of the risk, ii) the deductible arrangement and iii) full insurance. (5 marks)

f) Explain any factors, other than those mentioned in the question, which could also influence the decision. (5 marks)

6) Eddy Bartsto owns a refrigerated delivery vehicle (RDV) valued at £200,000.

He uses it to transport frozen foods between factories and supermarkets. The

RDV breaks down approximately twice per month and, each time it happens,

it costs around £90 in repairs and £750 in damaged cargo. Eddy is looking at

two contracts for controlling or financing the losses.

CRM/JUN11/1 Page 3 of 3

One contract (Contract A) is for an arrangement with a specialist RDV engineer who

is offering to perform maintenance work which would halve the number of

breakdowns. The cost of the contract for undertaking the work would be £10,000 per

year.

The other contract (Contract B) is an insurance contract which would cover the cost of

total destruction in the event that the RDV crashed and became a total write-off.

Eddy’s RDV is designed to a safety standard that the chances of becoming a total

write-off is approximately ‘once in every 20 years’. The cost of Contract B would

also be £10,000 per year.

REQUIRED (making any necessary assumptions and showing all workings)

a) Calculate the annual expected monetary value (EMV) of breakdowns for the

RDV.

(5 marks)

b) Calculate the annual EMV of total destruction of the RDV.

(5 marks)

c) Calculate the annual savings over EMV from entering into

i) Contract A (4 marks)and ii) Contract B (4 marks).

d) Offer your opinions on which contract is the best value for money and whether

Eddy should enter into either or both of them.

6)Eric is a risk manager for a firm which supplies CCTV (security) cameras to

industrial sites. The firm has supplied 5,000 cameras across the country to help

deter vandalism. Unfortunately, the cameras themselves are sometimes the

subject of vandalism and a source of loss. Eric has been keeping records of past

losses and has arrived at the following probability distribution for damage to a

SINGLE camera.

$ Amount of damage Probability

0 0.9

30 0.06

60 0.03

600 0.008

3,000 0.001

6,000 0.001

REQUIRED (Making any necessary assumptions and showing all workings)

a) Calculate the probability that a single camera will be vandalised. (1 mark)

b) Calculate the probability that damage to a single camera will exceed $60.

(1 mark) ((60X 0.03)+600×0.008+3000×0.001+6000×0.001)

c) Calculate the $ expected cost of damage to a single camera. (1 mark)

d) Calculate the $ expected value of total losses, conditional on some damage

being sustained. i.e. What is the expected value of the losses that do occur?

(2 marks)

e) Given that a camera has been vandalised, calculate the probability that the cost

will be $60 or greater. (2 marks)

f) Calculate the $ TOTAL expected loss for ALL the cameras. (2 marks)

g) Calculate the $ maximum probable loss (MPL) for a single camera on the basis

that any event with a chance of occurrence less than 0.002 can be ignored.

(2 marks)

h) Calculate the standard deviation of $ total losses, given that the standard

deviation of losses to a single camera is estimated at $218.49 and assuming that

loss exposures are independent (zero correlated). (4 marks)

i) Calculate the $ MPL for ALL the cameras, again using the assumption that

losses with a chance of occurrence less than 0.002 can be ignored ( as in part g)

of this question). HINT: 99.8% of the area under a normal distribution lies

below a point 2.89 standard deviations from the mean. (4 marks)

j) Calculate the ratio of the $ MPL to the $ expected loss (again, assuming that

losses with a chance of occurrence less than 0.002 can be ignored) for

i) a single camera (1 mark)

ii) 5,000 cameras. (1 mark)

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