Comparison of aleatory and epistemic uncertainty modelling, using @RISK Presentation at Palisade 2012 Risk Conference London, April 18-19, 2012

Hans Schjær-Jacobsen Professor, Director RD&I Copenhagen University College of Engineering Ballerup, Denmark +45 4480 5030 [email protected] www.ihk.dk

Agenda 1. 2. • • • 3. 4. 5.

Alternative representations of uncertainty Four dialogues on uncertainty Rectangular and triangular representations Non-monotonic functions Correlation Representation and calculation Net present value case Conclusions

Palisade 2012 Risk Conference

Copenhagen University College of Engineering

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Rectangular representation [a; b] and {µ; σ} Membership function

Probability distribution

Alternative interpretations

h

1

α

h = 1/(b-a) μ = (a+b)/2 σ2 = (b-a)2/12

α-cut

0 a Palisade 2012 Risk Conference

b Copenhagen University College of Engineering

0 3

Trapezoidal representation [a; c; d; b] and {µ; σ} Membership function

Probability distribution

Alternative interpretations

h

1

α

h = 2/(b-a+d-c) μ = h[(b3-d3)/(b-d)-(c3-a3)/(c-a)]/6 σ2 = [3(r+2s+t)4+6(r2+t2)(r+2s+t)2 -(r2-t2)2]/[12(r+2s+t)]2

α-cut

r

0 a Palisade 2012 Risk Conference

s

c

t

d

Copenhagen University College of Engineering

b

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Triangular representation [a; c; b] and {µ; σ} Membership function

Probability distribution

Alternative interpretations

h

1

α

h = 2/(b-a) μ = (a+b+c)/3 σ2 = (a2+b2+c2-ab-ac-bc)/18

α-cut

0 a Palisade 2012 Risk Conference

c Copenhagen University College of Engineering

b

0 5

A Dialogue on Uncertainty (1) Model Owner (MO): How should I represent the independent uncertain variable x in my model? Uncertainty Specialist (US): What do you know about x? MO: I know that x can attain any value between a and b (a < b). US: Do you know more about x? MO: Not really. US: Then I suggest that x is best represented by a rectangular possibility distribution (an interval [a; b]). MO: What is a possibility distribution? I am used to work with probability distributions. US: But you do’nt know the probability distribution of x. MO: No, but in that case I think it is reasonable to assume that x is best represented by a rectangular probability distribution [a; b]. Palisade 2012 Risk Conference

Copenhagen University College of Engineering

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Rectangular and triangular arguments of x(1-x)

2.4

1.2 Probability

Possibility

2.0

1.0

1.6

0.8 Probability

1.2

0.6

0.8

0.4

0.4

0.2

0.0

0.0 -0.2

0.0

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0.2

0.4

0.6

Copenhagen University College of Engineering

0.8

1.0

1.2

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Function x(1-x) with rectangular argument x = [0; 1] 35

1.2

Probability

Possibility

30

1.0

25 0.8 20 0.6 15 0.4 10 0.2

5

0 -0.05

0.00

0.05

0.10

0.15

0.20

0.0 0.30

0.25

x(1-x)

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A Dialogue on Uncertainty (2) Model Owner (MO): How should I represent the independent uncertain variable x in my model? Uncertainty Specialist (US): What do you know about x? MO: I know that the nominal value of x is c. Due to uncertainty x can attain any value between a and b (a < c < b). US: Do you know more about x? MO: Not really. US: Then I suggest that x is best represented by a triangular possibility distribution [a; c; b]. MO: What is a possibility distribution? I am used to work with probability distributions. US: But you do’nt know the probability distribution of x. MO: No, but in that case I think it is reasonable to assume that x is best represented by a triangular probability distribution [a; c; b]. Palisade 2012 Risk Conference

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Function x(1-x) with triangular argument x = [0; 0,2; 1] 35

1.2 Probability

Possibility

30

1.0

25 0.8 20 0.6 15 0.4 10

0.2

5

0 -0.05

0.00

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0.05

0.10

0.15

Copenhagen University College of Engineering

0.20

0.25

x(1-x)

0.0 0.30

10

A Dialogue on Uncertainty (3) Model Owner (MO): How should I represent the independent uncertain variables x1 and x2 in my model? Uncertainty Specialist (US): What do you know about the variables? MO: They can attain any value between a1 and b1 (a2 and b2). US: Do you know more? MO: Not really. US: Then I suggest that they are best represented by rectangular possibility distributions. MO: I am used to work with probability distributions. US: But you do’nt know the probability distributions. MO: No, but in that case I think that the variables are best represented by rectangular probability distributions. US: Do you know about the correlation between the variables? MO: No, I think I will run a series of correlation coefficients. Palisade 2012 Risk Conference

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Alternative interpretations of a rectangular distribution 1.2 Probability

0.125

μ = (a+b)/2 σ2 = (b-a)2/12

Possibility 1.0

0.100 0.8 0.075 0.6 h = 1/(b-a) 0.050

0.4

0.025

0.2

0.000

0.0 5

a=7

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b = 16

Copenhagen University College of Engineering

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12

Addition of two rectangular distributions, correlation in per cent 0.16

1.2 Probability

Possibility 1.0

0.12 0.8

0%

0.08

0.6

+100%

0.4 0.04 0.2

0.00

0.0 10

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15

20

25

Copenhagen University College of Engineering

30

Sum

35

13

Subtraction of two rectangular distributions, correlation in per cent 0.16

1.2 Possibility

Probability

1.0 0.12 0.8 0% 0.08

0.6 -100% 0.4

0.04 0.2

0.00

0.0 -10

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-5

0

Copenhagen University College of Engineering

5

Difference

10

14

Multiplication of two rectangular distributions, correlation in per cent 0.014

1.2 Probability

Possibility

0.012

1.0

0.010 0.8

0% 0.008

0.6

+100% 0.006

0.4 0.004 0.2

0.002

0.000

0.0 0

Palisade 2012 Risk Conference

50

100

150

200

Copenhagen University College of Engineering

250

Product

300

15

Division of two rectangular distributions, correlation in per cent 1.8

1.2 Possibility

Probability 1.6

1.0 1.4 0% 1.2

0.8 -100%

1.0 0.6 0.8

0.6

0.4

0.4 0.2 0.2 0.0

0.0 0.0

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0.5

1.0

1.5

Copenhagen University College of Engineering

2.0

Quotient

2.5

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A Dialogue on Uncertainty (4) Model Owner (MO): How should I represent the independent uncertain variables x1 and x2 in my model? Uncertainty Specialist (US): What do you know about the variables? MO: I know that the nominal values are c1 and c2. Due to uncertainty any value between a1 and b1 (a2 and < b2) may be attained. US: Do you know more? MO: Not really. US: Then I suggest that they are best represented by triangular possibility distributions. MO: I am used to work with probability distributions. US: But you do’nt know the probability distributions. MO: No, but in that case I think that the variables are best represented by triangular probability distributions. US: Do you know about the correlation between the variables? MO: No, I think I will run a series of correlation coefficients. Palisade 2012 Risk Conference

Copenhagen University College of Engineering

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Alternative interpretations of a triangular distribution 1.2 0.25

Probability

μ = (a+b+c)/3 σ2 = (a2+b2+c2-ab-ac-bc)/18

Possibility 1.0

0.20 0.8 0.15 0.6

0.10 α-cut

0.4

0.05

0.2 h = 2/(b-a)

0.00

0.0 5

a=7

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c = 10

Copenhagen University College of Engineering

b = 16

18

18

Addition of two triangular distributions, correlation in per cent 0.16

1.2 Probability

Possibility 1.0 0%

0.12 0.8

+100% 0.08

0.6

0.4 0.04 0.2

0.00

0.0 10

Palisade 2012 Risk Conference

15

20

25

Copenhagen University College of Engineering

30

Sum

35

19

Subtraction of two triangular distributions, correlation in per cent 0.16

1.2 Probability

Possibility 1.0

0% 0.12

0.8 -100% 0.08

0.6

0.4 0.04 0.2

0.00

0.0 -10

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-5

0

Copenhagen University College of Engineering

5

Difference

10

20

Multiplication of two triangular distributions, correlation in per cent 0.014

1.2 Probability

Possibility

0.012

0%

1.0

0.010 0.8

+100% 0.008

0.6 0.006 0.4 0.004 0.2

0.002

0.000

0.0 0

Palisade 2012 Risk Conference

50

100

150

200

Copenhagen University College of Engineering

250

Product

300

21

Division of two triangular distributions, correlation in per cent 1.8

1.2 Possibility

Probability

1.6

1.0 0%

1.4

1.2

0.8 -100%

1.0

0.6 0.8 0.6

0.4

0.4 0.2 0.2 0.0

0.0 0.0

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0.5

1.0

1.5

Copenhagen University College of Engineering

2.0

Quotient

2.5

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Two worlds of risk and uncertainty Uncertainty Imprecision Ignorance Lack of knowledge

Statistical nature Randomness Variability

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World

Representation and calculation

Possibility

Possibility distributions [a; …; b] Interval arithmetic Global optimisation

Probability

Probability distributions {µ; σ} Linear approximation Monte Carlo simulation

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Modelling by probability distributions The actual economic problem is modelled by a function Y of n independent and uncorrelated uncertain variables Y = Y(X1, X2,…, Xn). Linear approximation Y is approximated by means of a Taylor series Y

Y(μ1,…, μn) + ∂Y/∂X1·(X1-μ1) + ∂Y/∂X2·(X2-μ2) + … + ∂Y/∂Xn·(Xn-μn),

where ∂Y/∂Xi is the partial derivative of Y with respect to Xi calculated at (μ1,…, μn). The expected value is given by E(Y) = μ = Y(μ1,…, μn). The variance is approximated by VAR(Y) = σ2 (∂Y/∂X1)2·σ12 +…+ (∂Y/∂Xn)2·σn2. Palisade 2012 Risk Conference

Copenhagen University College of Engineering

Monte Carlo simulation

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Modelling by possibility distributions i.e. intervals, fuzzy intervals, etc. The actual economic problem is modelled by a function Y of n uncertain variables Y = Y(X1, X2,…, Xn). NB: Function can be arranged in different ways. In case of intervals Y is calculated by means of interval arithmetic (only applicable in the simple case) or global optimisation (applicable in the general case). In case of triple estimates Extreme values of Y are calculated as above. In case of fuzzy intervals As above, for all α-cuts. Palisade 2012 Risk Conference

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Independent stochastic variables

Intervals

Triple estimates

{μ; σ} = {μ1; σ1} # {μ2; σ2}

[a; b] = [a1; b1] # [a2; b2]

[a; c; b] = [a1; c1; b1] # [a2; c2; b2]

μ = μ1 + μ2;

a = a 1 + a 2;

σ2 = σ12 + σ22

b = b1 + b2

μ = μ1 - μ2;

a = a1 - b2;

σ2 = σ12 + σ22

b = b1 - a 2

μ = μ1·μ2;

a = min(a1a2, a1b2, b1a2, b1b2);

σ2 σ12·μ22 + σ22·μ12

b = max(a1a2, a1b2, b1a2, b1b2)

μ = μ1/μ2;

a = min(a1/b2, a1/a2, b1/b2, b1/a2,);

σ2 σ12/μ22 + σ22·μ12/μ24,

b = max(a1/b2, a1/a2, b1/b2, b1/a2),

Addition

Subtraction

Multiplication

Division

if μ2 ≠ 0

if 0

[a2; b2]

a = a1 + a2; c = c1 + c2; b = b 1 + b2 a = a1 - b2; c = c1 - c2; b = b1 - a 2 a = min(a1a2, a1b2, b1a2, b1b2); c = c1c2; b = max(a1a2, a1b2, b1a2, b1b2) a = min(a1/b2, a1/a2, b1/b2, b1/a2,); c = c1/c2; b = max(a1/b2, a1/a2, b1/b2, b1/a2), if 0

[a2; b2]

Table 1. Formulas for basic calculations with alternative representations of uncertain variables.

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Discounted cash flow case

Net present value over n periods NPV = a0 + a1·(1+r1)-1 + a2·(1+r1)-1·(1+r2)-1 + …+ an·(1+r1)-1·(1+r2)-1·…·(1+rn)-1, ai = Xi1·Xi2 + Xi3 + Xi4 +…+ Xim,

i = 0,…,n.

ai: net cash flow in i’th period ri: rate of interest in i’th period

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Interval analysis ($1000)

YEAR 0

YEAR 1

YEAR 2

YEAR 3

YEAR 4

[4.200; 5.200]

[12.400; 14.100]

[15.900; 18.100]

[13.800; 15.600]

Margin (%)

[44,50; 45,50] %

[45,00; 47,00] %

[45,50; 48,50] %

[44,00; 48,00] %

Direct cost

[-2.886; -2.289]

[-7.755; -6.572]

[-9.865; -8.188]

[-8.736; -7.176]

Margin

[-1.869; 2.366]

[5.580; 6.672]

[7.234; 8.779]

[6.072; 7.488]

[-1.000; -800]

[-975; -700]

[-800; -600]

[-800; -600]

[-950; -700]

[-1.375; -1.225]

[-675; -525]

[-675; -525]

Turnover

Marketing cost

[-1.050; -950]

Indirect production cost RD&E cost

[-3.050; -2.950]

[-1.700; -1.400]

[-350; -250]

[-150; -50]

[-150; -50]

Operating income

[-4.100; -3.900]

[-1.781; -534]

[2.880; 4.452]

[5.609; 7.604]

[4.447; 6.313]

Investment

[-5.100; -4.900]

[-2.200; -1.900]

Net cash flow NCF

[-9.200; -8.800]

[-3.981; -2.434]

[2.880; 4.452]

[5.609; 7.604]

[4.447; 7.013]

[8,50; 9,50] %

[9,00; 11,00] %

[9,50; 12,50] %

[10,50; 13,50] %

[-3.669, -2.223]

[2.369; 3.764]

[4.102; 5.871]

[2.865; 4.901]

Rate of interest r (%) Discounted cash flow DCF

[-9.200; -8.800]

Net present value NPV

[-3.532; 3.514]

[0; 700]

Table 2a. Discounted cash flow analysis by interval analysis (Interval Solver 2000, overall absolute and relative precision 10-6). Input variables in shaded cells.

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Uniform probability distributions ($1000)

YEAR 0

YEAR 1

YEAR 2

YEAR 3

YEAR 4

{4.700; 289}

{13.250; 491}

{17.000; 635}

{14.700; 520}

Margin (%)

{45,00; 0,29} %

{46,00; 0,58} %

{47,00; 0,87} %

{46,00; 1,15} %

Direct cost

{-2.585; 160}

{-7.155; 276}

{-9.010; 368}

{-7.938; 328}

Margin

{2.115; 131}

{6.095; 239}

{7.990; 333}

{6.762; 293}

{-900; 58}

{-838; 79}

{-700; 58}

{-700; 58}

{-825; 72}

{-1.300; 43}

{-600; 43}

{-600; 43}

Turnover

Marketing cost

{-1.000; 29}

Indirect production cost RD&E cost

{-3.000; 29}

{-1.550; 87}

{-300; 29}

{-100; 29}

{-100; 29}

Operating income

{-4.000; 41}

{-1.160; 182}

{3.657; 257}

{6.590; 342}

{5.362; 303}

Investment

{-5.000; 58}

{-2.050; 87}

Net cash flow NCF

{-9.000; 71}

{-3.210; 202}

{3.657; 257}

{6.590; 342}

{5.712; 364}

{9,00; 0,29} %

{10,00; 0,58} %

{11,00; 0,87} %

{12,00; 0,87} %

{-2.945; 184}

{3.050; 215}

{4.952; 262}

{3.832; 251}

Rate of interest r (%) Discounted cash flow DCF

{-9.000; 71}

Net present value NPV

{-111; 470}

{350; 202}

Table 2b. Discounted cash flow analysis by stochastic variables, formulas (11) - (19). Input variables (shaded cells) are derived from uniform probability distributions corresponding to the interval input variables in Table 2a, however converted to the form {μ; σ}.

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Net present values

Input

Interval/Fuzzy (double and triple estimates)

Monte Carlo: {-111; 468}

Stochastic

Uniform

[-3.532; 3.514]

{-111; 470}

Triangular

[-3.532; 1.317; 3.514]

{313; 355} Monte Carlo: {314; 354}

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Comparisons

Relative frequency

0,0010 0,0008 0,0006

Input: Triangular NPV: Normal µ = 313, σ = 355 (Table 3b)

1,2 1,0 0,8

Input: Uniform NPV: Normal µ = -111, σ = 470 (Table 2b)

0,6

0,0004

0,4

0,0002

Worst case (Table 2a & 3a)

0,2

Best case (Table 2a & 3a)

0,0000

0,0 -3000

-2000

-1000

0

1000

2000

3000

Net Present Value ($1000) Palisade 2012 Risk Conference

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Membership function

Most possible case (Table 3a)

0,0012

Conclusions

• Probability and possibility representations are different! • If no knowledge of probability distribution is available, use possibility representation • If additional statistical knowledge is available use probability distribution…

• … but be aware of under estimation of risk!

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Thank You!