Question

3. Consider Brown Dog brewing company’s workers compensation loss distribution. Loss Probability $ 0 0.4 1,000 0.35 10,000 0.17 100,000 0.075 1,000,000 0.005

a. Calculate the expected loss and standard deviation of this loss distribution. In addition, produce a graph showing the discrete loss distribution (already shown in Excel as an example).

b. Assume that Brown dog Co. pools their workers compensation exposure with another brewery with the same loss outcome and probability. Assume these two breweries’ losses are uncorrelated or independent.

i. What is the expected loss and standard deviation for an individual brewery in this pool of two companies? (Please show/describe all possible outcomes. Some steps are already shown in the attached Excel spreadsheet)

ii. Produce a graph showing the loss distribution (please show in Excel. Use the Column Charts function in Excel and use the example from the previous question as a reference.)

c. Now assume that Brown Dog pools their workers compensation exposure with 9,999 other breweries with the same loss distribution as Brown Dog, and uncorrelated losses. What is the expected loss and standard deviation for an individual brewery in this pool of 10,000 companies? Use the short hand calculation and compare the standard deviation here with the standard deviation you calculated from part (a). What is the effect of risk pooling? You don’t need to produce a loss distribution graph, but take a guess on the shape (Hint: Central limit theorem point 2).

Answer 1

x	p
0	0.4
1000	0.35
10000	0.17
100000	0.075
1000000	0.005
	1

a)

x	p	x*p	x^2*p
0	0.4	0	0
1000	0.35	350	350000
10000	0.17	1700	17000000
100000	0.075	7500	750000000
1000000	0.005	5000	5000000000
	1	14550	5767350000

var(X) = E(X^2) - E(X)^2

= 5767350000- 14550^2

=

5555647500

sd =sqrt(Var(X)

= 74536.216

b)

Y = X1+ X2

E(Y) = E(X1+ X2) =E(X1)+ E(X2) = 2 E(X) = 2* 14550 = 29100

sd(Y) = sqrt(2) * sd(X) = 74536.216* sqrt(2) = 105410.1

c)

Y = X1+ X2+...X10000

E(Y) = 10000 * 14550

sd(Y) = sqrt(10000) * sd(X) = 74536.216 * 100 = 7453621.6

when we risk pool, our risk reduces significantly as sample size increases

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