Download Decision and Risk Analysis

DRA/K V
Decision and Risk Analysis
Financial Modelling &
Risk Analysis II
Kiriakos Vlahos
Spring 2000
DRA/K V
Session overview
• Probability distributions for Risk
Analysis
– Subjective
– Regression and Forecasting models
– Historic data
• Resampling
• Distribution fitting
• Sampling distributions
– Using histograms
– The inversion method
• Correlated random variables
• Comparing uncertain outcomes
– Dynatron case
DRA/K V
Using regression
models in risk analysis
Example:
Ferric regression model:
Cost = 11.75 + 7.93 * (1/Capacity)
Standard Error (SE) = 0.98
@RISK formula for cost:
Cost = 11.75 + 7.93 * (1/Capacity) +
RiskNormal(0,0.98)
Using historic data
Resampling
DRA/K V
Historical Data
Month
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Demand Average
10
5
6
10
8
7
5
5
5
3
2
6
5
6
3
5
4
5
4
4
3
3
3
4
@RISK funcion
RISKDUNIFORM(datarange)
At every iteration it picks one of the
historic values at random.
Historic data Distribution fitting
DRA/K V
2. Histogram
1. Historic data
Historical Data
Month
Demand
1
10
2
6
3
10
4
8
5
7
6
5
7
5
10
8
6
4
2
0
1
4. Fit theoretical
distribution
2
3
4
5
6
7
8
9
10
3.Cumulative function
1.2
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
2
4
6
8
10
0
2
4
6
8
10
5. Then use theoretical distribution in @RISK
Use statistical packages for distribution fitting
DRA/K V
Cumulative functions of
standard distributions
Cumulative function
Distribution function
Uniform
niform
orm(1,
1.0
00
0
0.8
00
0
0.6
00
0
0.4
00
0
0.2
00
0
0.0
00
0
1.0
1.1
1.3
1.5
1.6
1.8
2.0
0
7
3
0
7
3
1.0
1.1
1.3
1.5
1.6
1.8
2.0
0
7
3
0
7
3
0
Triangular
ng(1,3,
iang(1
6
1 .0 0 0
7
0 .8 0 0
8
0
.6 0 0
8
0
.4 0 0
9
0
.2 0 0
0
0
.0 0 0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0
0
0
0
0
0
0
1
1
2
2
3
.0
3
.5
4
.0
.5
.0
.5
.0
0
0
0
0
0
0
Normal
mal(5,1
ormal(
.0 0 0
8
.8 0 0
0
.6 0 0
3
.4 0 0
5
.2 0 0
8
.0 0 0
0
-2
0
2
4
1
6
9
.0
.2
.2
.5
1
.8
.0
0
7
.3
7
4
2
9
0.0
2.2
2.2
4.5
11.3
6.8
9.0
0
7
7
4
2
9
6
DRA/K V
Random sampling
• Probabilistic simulation depends on
creating samples of random variables
• In order to carry out random sampling
we need:
– a set of random numbers
– a distribution or cumulative function
for each of the random variables
– a mechanism for converting random
numbers into samples of the above
distributions
• Tables of random numbers
• Pseudo random number generators:
– e.g. Rj+1 = MOD(a Rj +c, m)
• The initial R is the seed
• Excel RAND() function
Inversion method
DRA/K V
Cumulative function
Pick random number
between 0 and 1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Read sample
0.0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Modelling correlated
variables
DRA/K V
Demand = risknormal(100,20)
Price = risknormal(100,20)
Sales = Demand * Price
for
R
5
8
1
PROBAILTY PROBAILTY
4
7
Min 2,000
Max: 20,500
St.d.: 2900
0
2
1
4
1
6
1
8
1
1
2
2
0
2
4
6
8
0
2
n
T ho
Assuming correlation of -0.8
for
R
0
0
0
0
Min 5,500
Max: 13,500
St.d.: 1300
0
0
4
1
6
1
8
1
1
0
1
0
0
2
0
2
2
0
4
6
0
8
0
0
0
0
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Always try to model correlation between random
variables
Expected value
DRA/K V
Production = 100
Demand = risknormal(100,20)
Sales = min(Production, Demand)
If we replace Demand with its expected
value then Sales equals 100. But the
expected value of Sales is less than 100.
In general:
E ( F ( x))  F ( E ( x))
i.e. replacing uncertain inputs with
their average values does not result
in the expected value of the output
unless the function is linear.
DRA/K V
Dynatron
• Decide about:
– The production level of
Dynatron toys
– the split into super and
standard
Dynatron - Decision
Alternatives
DRA/K V
Field Sales Representatives
Standard
Super
130,000
95,000
Total
225,000
Production Manager
Standard
Super
Total
70,000
80,000
150,000
Gassman
Standard
Super
115,000
85,000
Total
200,000
DRA/K V
Cost Accounting
in £
Factory price
Direct cost
Contribution
Indirect cost (12% of
contribution)
Profit per unit
Inventory cost per unit
(2% of direct costs per
month for 6 months)
Standard
4.30
2.50
1.80
0.22
Super
5.50
3.20
2.30
0.28
1.58
2.02
0.30
0.38
Additional production costs
Production Level
0 – 150,000
150,001 - 200,000
200,000 -
Investment cost (£)
0
15,000
70,000
DRA/K V
Base case model
DYNATRON SIMULATION
Prod
ProdQuan
Inv
Avail
Demand
Sales
Inv
Shortages
Dep Charge
Profit
ASSUMPTIONS
Unit Prof
Unit Inv
TotDemand
0
151
201
Std
80
12
92
Super
70
5
75
94
92
0
2
0
277
66
66
9
0
1.584
0.300
Depr
0
15
70
2.024
0.384
Profit = Revenue - Inventory cost
- Investment cost
Dynatron - Demand
uncertainty
DRA/K V
Median demand 150
Minimum 50 and maximum 300
1 in 4 chance that demand is at least 190
3 in 4 chances that demand is at least 125
Cumulative function
1
Probability
0.8
0.6
0.4
0.2
0
0
100
200
300
Demand
RiskCumul(50,300,{125,150,190},{0.25,0.5,0.75})
DRA/K V
Standard/super split
uncertainty
% of supers
Median 40 %
Minimum 30% and maximum 60%
75% chance to be 45% or less
25% to be 36% or less
Cumulative function
1
Probability
0.8
0.6
0.4
0.2
0
25% 30% 35% 40% 45% 50% 55% 60% 65%
% of super
RiskCumul(0.3,0.6,{0.36,0.4,0.45},{0.25,0.5,0.75})
Dynatron Simulaton Results
DRA/K V
n
fo
00
40
80
PROBAILTY PROBAILTY PROBAILTY
20
60
00
-5
1
1
2
5
2
3
0
3
4
0
5
0
0
0
5
0
5
0
0
0
0
0
0
0
0
or
G
00
40
80
20
60
00
-5
1
1
2
5
2
3
0
3
4
0
5
0
0
0
5
0
5
0
0
0
0
0
0
0
0
n
for
00
40
80
20
60
00
-5
1
1
2
5
2
3
0
3
4
0
5
0
0
0
5
0
5
0
0
0
0
0
0
0
0
DRA/K V
Comparing risky assets
Case 1
A>>B
B
A
Profit
Case 2
A
A>>B
B
Profit
Case 3
A>>B ?
B
A
Profit
DRA/K V
Risk-return tradeoff
Return
Efficient frontier
Dominated options
Risk
DRA/K V
Screening risky options
Cumulative Probability functions
1
A>>B
B
A
0
Return
1
if area (1) > area (2)
then project A >> B
Requires risk aversion
0
(2)
A
B
(1)
Return
DRA/K V
Dynatron Simulation Results
Cumulative probability distributions
and
0
0
0
o
b
of
u
L
e
eg
<
e
=
n
a x is
a lu e
F
0
D
0
0
1
1
5
5
2
2
0
3
3
0
4
5
0
0
0
5
0
5
0
0
0
0
0
0
0
0
Gassman Sales rep.
Expected
Profit
230
174
St. dev.
97
108
Prob. of
loss
0%
6.5%
Dynatron Simulation Results
DRA/K V
Cumulative probability distributions
rod
0
0
0
o
b
of
u
L
e
eg
<
e
=
n
a x is
a lu e
F
0
B
0
0
1
1
5
5
2
2
0
3
3
0
4
5
0
0
0
5
0
5
0
0
0
0
0
0
0
0
Gassman
Production
manager
Expected
Profit
230
232
St. dev.
97
67
Prob. of
loss
0%
0%
DRA/K V
Summary
• Integrating regression and forecasting
models with risk analysis
• Using historic data in risk analysis
• Resampling
• Distribution fitting
• Sampling distributions
– The inversion method
• Model correlation between random
variables!
• Comparing uncertain outcomes
– Screening options
– Risk return tradeoff
– Risk preferences