Download Risk Management Decision Making

C-2: Loss Simulation
Statistical Analysis in Risk
Management
– Two main approaches:


– Maximum probable loss (or MPY)


if $5 million is the maximum probable loss at the
_______percent level, then the firm’s losses will be less than
$_____million with probability 0.95.
Same concept as “Value at risk”
When to Use the Normal Distribution
– Most loss distributions are not normal
– From the __________ theorem, using the normal distribution
will nevertheless be appropriate when


– Example where it might be appropriate:


Using the Normal Distribution

Important property
– If Losses are normally distributed with


– Then

Probability (Loss <
) = 0.95

Probability (Loss <
) = 0.99
Using the Normal Distribution - An Example
– Worker compensation losses for Stallone Steel



sample mean loss per worker
= $_____
sample standard deviation per worker = $20,000
number of workers
= ________
– Assume total losses are normally distributed with


mean
= $3 million
standard deviation =
– Then maximum probable loss at the 95 percent level is

$3 million +
= $6.3 million
A Limitation of the Normal Distribution

Applies only to aggregate losses, not
_______losses

Thus, it cannot be used to analyze decisions
about per occurrence deductibles and limits
Monte Carlo Simulation
– Overcomes some of the shortcomings of the normal
distribution approach
– Overview:



Make assumptions about distributions for ________ and _______ of
individual losses
Randomly draw from each distribution and calculate the firm’s total
losses under alternative risk management strategies
Redo step two many times to obtain a distribution for total losses
A. Total Loss Profile
1. E(L) forecast
a. single best estimate ……….
b. variations from this number will occur, therefore
…
2. Example for a large company.(next slide)
mode, median
expected = $
Pr(L)
> $11,500,000 =
Pr(L)
> $14,000,000 =
Unlimited Loss Distribution
0.35
0.3
Probability
0.25
0.2
0.15
0.1
0.05
0
9
10
11
12
13
14
15
16
Total Losses (Millions)
17
18
19
3. Uses of Total Loss Profile
a. Evaluate
and
b.
c.
d. MPL (MPY)
loss limits
B. Monte Carlo Steps
1. Select frequency distribution
2. Select severity distribution
3. Draw from ________ distribution => N1 losses
4. Draw N1 severity values from severity distribution
5. Repeat steps____and ____ for 1000 or more iterations
Ni
S1
S2
…
S10
…
S23
…
S43
…
S70
Total
1
70
$
600
$ 18,400
Iteration Number
2
1,000
23
…
43
$ 94,000
$ _____
$
150
$
970
$
_____
$
2,600
$
$ 19,500
$
1,350
$ 32,150
3,750
NA
$182,000
$ 54,000
$
NA
$
$
500
NA
$
Rank Order the Total Losses
Iteration Percentile
Total Losses
1
0.1
$ 143,000
.
100
10
1,790,000
.
500
50
2,280,000
.
700
70
________
.
900
90
3,130,000
.
950
95
________
.
1,000
100
3,970,000
Horizontal Layering: From One Iteration
Layers for the 438th Iteration that produced 980 Severity Values
1,0005,00010000- 50,000GE
Draw LT 1,000
Total
4,999
9,999
49,999 99,999 100,000
1
625
625
…
98 ________
2,050
_________
…
251
999
4,000
_________
..
730
789
789
…
980
999
4,000
5,000 40,000 50,000 10,001
110,000
Total 920,000 450,000 414,000 180,000 119,000 47,000 2,130,000
D. Interpretation of Results
1. Look at summary statistics: mean, sigma, percentiles
2.
3.
Within Limits
,000
1 - 10
10 25
25 - 50
50 - 75
75 - 100
100 - 150
150 - 200
200 - 250
250 - 500
500 - 1,000
> 1,000
X BAR
$
$
$
$
$
$
$
$
$
$
$
$
612
326
128
65
60
26
15
23
9
1
Sigma
$
$
88
$
92
$
55
$
41
$
53
$
32
$
23
$
60
$
62
$
8
At Limits
X BAR
$
$ 2,655
$ 2,981
$ 3,109
$ 3,174
$ 3,234
$ 3,260
$ 3,275
$ 3,298
$ 3,307
$ 3,307
Sigma
$
$ 176
$ 239
$ 275
$ 298
$ 325
$ 340
$ 350
$ 370
$ 400
$ 404
E. Aggregates – Recap using text
Simulation Example - Assumptions
– Claim frequency follows a Poisson distribution

Important property: Poisson distribution gives the
probability of 0 claims, 1 claim, 2 claims, etc.
Simulation Example - Assumptions
– Claim severity follows a



expected value =
standard deviation =
note skewness
Simulation Example Assumptions
Fre que ncy Distribution w ith Ex pe cte d V a lue
Equa l to 30
0.25
PROBABILITY
0.2
0.15
0.1
0.05
54
48
42
36
30
24
18
12
6
0
0
Nu m b e r o f C laim s
S a m ple Fre que ncy Distribution w ith Unce rta in
Ex pe cte d V a lue (1000 tria ls)
0.25
PROBABILITY
0.2
0.15
0.1
0.05
54
48
42
36
30
24
18
12
6
0
0
Nu m b e r o f C laim s
S a m ple Loss S e ve rity Distribution
(1000 tria ls)
0.12
PROBABILITY
0.1
0.08
0.06
0.04
0.02
0
0.0075
0.6
1.2
1.8
L o s s in M illio n s
2.4
3
Simulation Example - Alternative
Strategies
Policy
1
2
3
$500,000
$1,000,000
none
Per Occurrence Policy Limit $5,000,000
$5,000,000
none
Aggregate Deductible
none
none
$6,000,000
Aggregate Policy Limit
none
none
$10,000,000
Premium
$780,000
$415,000
Per Occurrence Deductible
$165,000
Simulation Example - Results
$500,000 pe r Occurre nce Re te ntion
0.2
0.18
0.16
0.16
12
13
.5
13
.5
.5
10
9
V alu e s in M illio n s
$6 M illion Aggre ga te Annua l Re te ntion
$1 M illion pe r Occurre nce Re te ntion
0.2
0.18
0.16
0.16
P RO BABILITY
0.2
0.18
0.14
0.12
0.1
0.08
0.06
0.14
0.12
0.1
0.08
0.06
0.04
0.04
V alu e s in M illio n s
.5
10
9
5
7.
6
5
4.
3
0
.5
13
12
.5
10
9
5
7.
6
4.
3
5
1.
5
V alu e s in M illio n s
5
0
0
1.
0.02
0.02
0
P RO BABILITY
12
V alu e s in M illio n s
5
0
.5
13
12
.5
10
9
5
7.
6
5
4.
3
0
5
0.02
0
1.
0.04
0.02
7.
0.06
0.04
6
0.06
0.1
0.08
5
0.08
0.12
4.
0.1
0.14
3
0.12
5
0.14
1.
P RO BABILITY
0.2
0.18
0
P RO BABILITY
No Insura nce
Simulation Example - Results
Statistic
Policy 1:
Policy 2:
Mean value of retained losses
$______
$2,716
$2,925
$3,042
1,065
1,293
1,494
1,839
______
6,462
7,899
18,898
0.7%
0.1%
n.a.
____%
99.9%
92.7%
Standard deviation of retained losses
Maximum probable retained loss at 95% level
Maximum value of retained losses
4,254
11,325
Probability that losses exceed policy limits 1.1%
Probability that retained losses  $6 million 99.7%
5,003
12,125
Policy 3: No insurance
Premium
$780
$415
$165
$0
Mean total cost
3,194
3,131
3,090
3,042
5,418
6,165
6,462
Maximum probable total cost at 95% level 5,034