Transcript APPLICATIONS OF PARAMETERIZATION OF VARIABLES FOR
APPLICATIONS OF PARAMETERIZATION OF VARIABLES FOR MONTE-CARLO RISK ANALYSIS Teaching Note (MS-Excel)
597
WHY ?
• Monte-Carlo risk analysis requires having a defined probability distribution for each risk variable • In most cases the probability distribution is not readily available • Need to derive an appropriate distribution from raw data 598
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STEPS TO FOLLOW:
Identify the risk variable and nature of risk Obtain historical data on the variable Transfer raw data into spreadsheet Convert nominal values into real values Calculate correlations among variables, if needed Run a regression to identify a trend over years Obtain residuals from regression Express residuals as a percentage deviation from the trend Rank the percentage deviations Group percentage deviations into ranges Specify frequency of occurrence for each range Calculate the expected value Make adjustments to frequencies, so that the expected value equals to the deterministic value of risk variable (check for the adjusted expected value) Transfer the derived probability distribution into risk analysis software 599
1. IDENTIFY THE RISK VARIABLE AND NATURE OF RISK • A financial/economic model of the project has to be complete • Sensitivity analysis suggests candidates to be included as “risk variables” • A “risk variable” must be both
risky
(have a great impact on the project) and
uncertain
(not predictable) • Sensitivity analysis helps to identify the risky variables • It is the task of analyst to understand the underlying reasons for uncertainty of variable 600
QUESTIONS TO UNDERSTAND RISK
• What are the fundamental reasons for movements of the variable over time?
• Can the causes of risk be predicted?
• Are there any related variables, which move in the same or opposite direction at the same time?
• Is it possible to avoid the risk or reduce it somehow?
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2. OBTAIN HISTORICAL DATA ON THE VARIABLE • Once the risk variable is identified and justified to be included into risk analysis • Need to obtain a reliable set of data on the variable over time • As many observations as possible • If data on the variable itself is not available – use data on a related variable (fluctuations in the price of natural gas can be reasonably approximated by movements of the oil prices) 602
EXAMPLE: DERIVATION OF A PROBABILITY DISTRIBUTION FOR NATURAL GAS PRICE • Natural gas is the major input for production of urea in a fertilizer plant project • Price of input was identified as a very risky variable, having a strong impact on the project’s returns • Project purchases natural gas as a price-taker • Natural gas prices follow the international gas prices • Prices can not be fully predicted – risk analysis is needed 603
• Data on the domestic and international gas prices were not available • It is believed that the crude oil prices can be used as a proxy for fluctuations in the prices of natural gas • Historic records of the crude oil prices supplied by the OPEC were obtained from “
OPEC Annual Statistical Bulletin 2000
” {www.opec.org} • Crude oil prices are expressed in nominal US dollar 604
3. TRANSFER RAW DATA INTO SPREADSHEET • All data records must be transferred into an electronic form • Data is on the crude oil prices in nominal terms, 1976–1999 ($/barrel) • There are 24 observations • Prices are annual averages • The prices are nominal, inclusive of inflation • The relevant inflation is the us dollar inflation • Inflation effect must be removed
1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 Year 1976 1977 1978 1979 1980 1981 1982 1983 1984 Nominal Oil Price, $/barrel
11.5
12.4
12.7
17.3
28.6
32.5
32.4
29.0
28.2
27.0
13.5
17.7
14.2
17.3
22.3
18.6
18.4
16.3
15.5
16.9
20.3
18.7
12.3
17.5
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4. CONVERT NOMINAL VALUES INTO REAL VALUES
Producer Price Index, USA,1995=100
• Since the oil prices are quoted in us dollar, use the us inflation index • The relevant inflation measure is the us producer price index, base 1995=100 • Data on the US producer price index were obtained from “
IMF Financial Statistics Yearbook 2000”.
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 Year 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989
93.2
93.4
93.9
95.3
96.5
100.0
102.3
102.3
99.7
100.6
49.0
52.0
56.0
63.1
72.0
78.6
80.1
81.1
83.1
82.7
80.3
82.4
85.7
90.0
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Year 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 Nominal Oil Price, $/barrel
11.5
12.4
12.7
17.3
28.6
32.5
32.4
29.0
28.2
27.0
13.5
17.7
14.2
17.3
22.3
18.6
18.4
16.3
15.5
16.9
20.3
18.7
12.3
17.5
Producer Price Index, USA, 1995=100
80.3
82.4
85.7
90.0
93.2
93.4
93.9
95.3
96.5
100.0
49.0
52.0
56.0
63.1
72.0
78.6
80.1
81.1
83.1
82.7
102.3
102.3
99.7
100.6
REAL PRICE = NOMINAL PRICE PRICE INDEX x 100 Real Oil Price, $/barrel
23.5
23.8
22.7
27.3
39.8
41.4
40.4
35.8
33.9
32.7
16.8
21.5
16.6
19.2
23.9
19.9
19.6
17.1
16.1
16.9
19.8
18.3
12.3
17.4
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5. CALCULATE CORRELATIONS BETWEEN VARIABLES • If variables tend to move together over time – there is a correlation • Coefficient of correlation can be easily estimated from two sets of data • Both data sets must be expressed in real terms • Example: correlation between the price of crude oil (input) and price of urea fertilizer (output) • Real price of urea was obtained from nominal price in the same manner as real oil price 608
CORRELATION BETWEEN THE Real Oil Price, PRICE OF CRUDE OIL AND PRICE OF $/barrel UREA FERTILIZER
19.9
19.6
17.1
16.1
16.9
19.8
18.3
12.3
17.4
23.5
23.8
22.7
27.3
39.8
41.4
40.4
35.8
33.9
32.7
16.8
21.5
16.6
19.2
23.9
Use ms-excel formula “CORREL“ to estimate the correlation coefficient between two sets of data:
=CORREL(OIL,UREA) = 0.544
Real Urea Price, $/Mt
234.7
269.2
267.9
296.4
326.4
225.2
177.9
172.6
219.6
129.4
87.5
120.2
153.4
101.4
167.7
154.2
122.3
115.4
180.6
207.2
164.6
91.8
67.9
68.8
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6. RUN A REGRESSION TO IDENTIFY A TREND OVER YEARS • There is a trend in the real price of oil • Generally, trend can be increasing, decreasing or constant over years • If plotted, the trend can be seen visually on the chart • Trend represents “predicted” values • The difference between the actual price and predicted price is called “residual” value, which is not explained by trend • Residuals represent the random factors affecting the real price of oil • Residuals represent the risk 610
25 20 15 10 5 0
REAL PRICE OF CRUDE OIL: ACTUAL VS. PREDICTED
45 40 35 y = -0.7859x + 33.859
R 2 = 0.4159
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RESIDUAL Real Price of Oil (1976-99)
RANDOM FACTORS
ACTUAL REAL PRICE IN 1984 PREDICTED
TREND
Year RESIDUAL = ACTUAL – PREDICTED
CALCULATED FOR EVERY YEAR 611
• Regression is needed • Running a regression is easy • Use an “add-in” in excel, called “data analysis” • To start:
TOOLS=> DATA ANALYSIS => REGRESSION
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• SELECT “REGRESSION” AND PRESS “OK” • Fill in the required fields in the regression box and press “OK” • The regression will estimate the predicted values and residuals for every year 613
REAL PRICE OF OIL, 1976-99 YEARS, 1976-99 NEW WORKSHEET PLY [OIL] RESIDUALS • Fill-in the regression box as shown above • Do not change other settings • When done, a new worksheet called “oil” will appear 614
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OBTAIN RESIDUALS FROM REGRESSION
RESIDUAL OUTPUT
Observation
1 2 3 4 5 6 7 8 9 18 19 20 21 22 23 24 10 11 12 13 14 15 16 17
Predicted Y
33.1
32.3
31.5
30.7
29.9
29.1
28.4
27.6
26.8
26.0
25.2
24.4
23.6
22.9
22.1
21.3
20.5
19.7
18.9
18.1
17.4
16.6
15.8
15.0
Residuals
-9.6
-8.5
-8.8
-3.4
9.8
12.2
12.1
8.2
7.1
6.7
-8.4
-2.9
-7.0
-3.6
1.8
-1.3
-0.9
-2.6
-2.8
-1.3
2.5
1.7
-3.5
2.4
• New worksheet “oil” will contain the regression statistics and residual output • Residuals are estimated in the units of variable, $/barrel • Need to express residuals as a percentage deviation from the trend (from predicted value) 615
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Predicted Y Residuals
33.1
32.3
-9.6
-8.5
31.5
30.7
29.9
29.1
28.4
-8.8
-3.4
9.8
12.2
12.1
27.6
26.8
26.0
25.2
8.2
7.1
6.7
-8.4
24.4
23.6
22.9
22.1
21.3
20.5
19.7
18.9
18.1
17.4
16.6
15.8
15.0
-2.9
-7.0
-3.6
1.8
-1.3
-0.9
-2.6
-2.8
-1.3
2.5
1.7
-3.5
2.4
EXPRESS RESIDUALS AS A PERCENTAGE DEVIATION FROM THE TREND
• USE A SIMPLE FORMULA:
=RESIDUAL/(PREDICTED/100)/100
For example (1 st observation ):
% Deviation from Trend
-28.98% -26.20% -28.01% -11.00% 32.91% 41.92% 42.55% 29.87% = -9.6/33.1
= -0.2898
26.69% 25.62% -33.17% -11.92% -29.72% • Express the result as a percentage • Percentage represents a deviation from the trend -15.85% 8.22% -6.34% -4.20% -13.07% -14.97% -7.06% 14.29% 10.21% -21.96% 15.80% 616
9. RANK THE PERCENTAGE DEVIATIONS • • • Residuals in percentage form represent the deviations from the trend The percentage deviations must be ranked from the lowest to highest 1.
2.
3.
Use a built-in “sort” function in excel: Highlight all percentage deviations Open “
DATA
” => “
SORT…
” Fill-in the sorting box 617
• Fill-in as follows: SORT BY: % DEVIATION FROM TREND ASCENDING HEADER ROW • When done, press “OK” 618
10. GROUP PERCENTAGE DEVIATIONS INTO RANGES Ranked % Deviation
-33.17% -29.72% -28.98% -28.01% -26.20% -21.96% -15.85% -14.97% -13.07% -11.92% -11.00% -7.06% -6.34% -4.20% 8.22% 10.21% 14.29% 15.80% 25.62% 26.69% 29.87% 32.91% 41.92% 42.55% -35% to -30% -30% to -20% -20% to -10% -10% to 0% 0% to 10% 10% to 20% 20% to 30% 30% to 40% 40% to 45% • Ranked percentage deviations show the minimum and maximum deviations from trend over the years • They can be grouped into ranges, for simplicity • In each range, there will be a few observations 619
11. SPECIFY FREQUENCY OF OCCURRENCE FOR EACH RANGE • Frequency of occurrence is the number of observations in each range • Total number of observations must be 24 • Express frequencies as probability of occurrence • Total probability must be always 100% • Probability of occurrence – is really the derived probability distribution • If the expected value of this distribution is equal zero – then, probability distribution is ready for use • If the expected value of this distribution is equal zero – then, further adjustments must be made 620
Ranked % Deviation
-33.17% -29.72% -28.98% -28.01% -26.20% -21.96% -15.85% -14.97% -13.07% -11.92% -11.00% -7.06% -6.34% -4.20% 8.22% 10.21% 14.29% 15.80% 25.62% 26.69% 29.87% 32.91% 41.92% 42.55% -35% to -30% -30% to -20% -20% to -10% -10% to 0% 0% to 10% 10% to 20% 20% to 30% 30% to 40% 40% to 45%
Frequency
1 5 5 3 1 3 3
% Occurrence
4.17% 20.83% 20.83% 12.50% 4.17% 12.50% 12.50% 1 2 4.17% 8.33%
Total: 24 100%
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12. CALCULATE THE EXPECTED VALUE
• • • • Expected value is a weighted average of mid-point of all ranges and their probability of occurrence 1.
To calculate: Find the mid-point of each range 2.
3.
Multiply each mid-point by its probability of occurrence Sum up the results The expected value of probability distribution must be equal zero, to remain unbiased If the estimated expected value is not zero, further adjustments are needed 622
From To
-35.0% -30.0% -30.0% -20.0% -20.0% -10.0% -10.0% 0.0% 0.0% 10.0% 10.0% 20.0% 30.0% 40.0% 20.0% 30.0% 40.0% 45.0%
Mid-point
-32.5% -25.0% -15.0% -5.0% 5.0% 15.0% 25.0% 35.0% 42.5%
Frequency
1 5 5 3 1 3 3 1 2
Total: 24 % Occurrence
4.17% 20.83% 20.83% 12.50% 4.17% 12.50% 12.50% 4.17% 8.3%
100.00% Mid-point X % Occurrence
-1.35% -5.21% -3.13% -0.63% 0.21% 1.88% 3.13% 1.46% 3.54%
Expected Value (weighted average): -0.1042%
• Expected value is simply a weighted average of mid-point of all ranges and their probability of occurrence • Expected value here is not equal to zero 623
13. MAKE ADJUSTMENTS TO FREQUENCIES • To adjust the expected value of probability distribution to zero, use Excel’s “SOLVER” add-in To start: “
TOOLS
” => “
SOLVER…
” 624
BY CHANGING CELLS: (ALL FREQUENCIES)
Frequency
1 5 5 3 1 3 3 1 2
Total: 24
SET TARGET CELL = EXPECTED VALUE CELL EQUAL TO: VALUE OF 0 Subject to constraints: press “
ADD
” And take cell with total frequencies and set this cell = 24 • When completed, press “SOLVE” 625
From To
-35.0% -30.0% -30.0% -20.0% -20.0% -10.0% -10.0% 0.0% 10.0% 20.0% 30.0% 40.0% 0.0% 10.0% 20.0% 30.0% 40.0% 45.0%
Mid-point
-32.5% -25.0% -15.0% -5.0% 5.0% 15.0% 25.0% 35.0% 42.5%
Frequency
0.95
5.00
5.00
3.00
1.01
3.01
3.01
1.01
2.01
% Occurrence
3.97% 20.84% 20.84% 12.52% 4.19% 12.53% 12.53% 4.21% 8.38%
Mid-point X % Occurrence
-1.29% -5.21% -3.13% -0.63% 0.21% 1.88% 3.13% 1.47% 3.56%
Total: 24 100.0% Expected Value (weighted average): 0.0%
• Expected value is equal to zero • Probability distribution is ready 626
14. Transfer the derived probability distribution into risk analysis software • We have obtained the following “step” distribution for the disturbance to the real price of crude oil:
From
-35.0% -30.0% -20.0% -10.0% 0.0% 10.0% 20.0% 30.0% 40.0%
To
-30.0% -20.0% -10.0% 0.0% 10.0% 20.0% 30.0% 40.0% 45.0%
% Occurrence
3.97% 20.84% 20.84% 12.52% 4.19% 12.53% 12.53% 4.21% 8.38%
100.0%
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• Using the “Crystal Ball” risk analysis software will depict this probability distribution as: 628
FINAL NOTE
• In most cases, probability distribution is applied not on the value of a variable itself • Probability distribution is applied on the disturbance to this variable • Disturbance, on the average, is expected to be zero • Spreadsheet may need to be modified to include the disturbance 629
CORRECT WAY TO MODEL ANNUAL DISTURBANCE: YEAR Domestic Price Index
= Link to Parameter (120D$/ton, assumed to remain constant)
Year 0 1.000
Year 1 1.037
Year 2 1.075
Year 3 1.115
Disturbance to REAL Price of urea EXPORTS REAL Price of urea EXPORTS (D$/ton) Unadjusted REAL Price of urea EXPORTS (D$/ton) Adjusted NOMINAL Price of urea EXPORTS (D$/ton) 0.0% 120 120 120 0.0% 120 120 123 0.0% 120 120 127 0.0% 120 120 130
= Real Price YearX (Unadj.) * (1+Disturbance YearX ) = 120 * (1 + 0.0%) = Real Price YearX (Adj.) * Domestic Inflation Index YearX 127 = 120 * 1.075 [for Year 2]
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