Session 9a
Overview
Finance Simulation Models
• Forecasting
– Retirement Planning
– Butterfly Strategy
• Risk Management
– Introduction to VaR
– Currency Risk
• Using Historical Data in Simulations
– Parametric Approach
– Resampling Approach
Decision Models -- Prof. Juran
2
Example 1: Retirement Planning
Amanda has 30 years to save for her retirement. At the beginning
of each year, she puts $5000 into her retirement account. At any
point in time, all of Amanda's retirement funds are tied up in the
stock market. Suppose the annual return on stocks follows a
normal distribution with mean 12% and standard deviation 25%.
What is the probability that at the end of 30 years, Amanda will
have reached her goal of having $1,000,000 for retirement?
Assume that if Amanda reaches her goal before 30 years, she will
stop investing.
Decision Models -- Prof. Juran
3
A
1 Ann. Inv.
2 Goal
3
4
Year
5
0
6
1
7
2
8
3
9
4
B
$5,000
$1,000,000
Beginning
$5,000
$9,095
$15,008
$15,513
C
D
Mean
Stdev
Return
=B1
-18.09%
=D6+5000
10.03%
-29.95%
3.44%
Decision Models -- Prof. Juran
Ending
0
$4,095
$10,008
$10,513
$16,047
E
12%
25%
F
$500,957
$478,876
=B6*(1+C6)
G
Reached goal?
H
0
I
J
=IF(F4>B2,1,0)
=MAX(D6:D35)
=D35 Max Assets
Final Assets
=B7*(1+C7)
4
The annual investment activities (columns A-D, beginning in row 5)
actually extend down to row 35, to include 30 years of simulated
returns.
The range C6:C35 will be random numbers, generated by Crystal Ball.
We could track Amanda’s simulated investment performance either
with cell F5 (simply =D35, the final amount in Amanda’s retirement
account), or with F4 (the maximum amount over 30 years). Using F4
allows us to assume that she would stop investing if she ever reached
$1,000,000 at any time during the 30 years, which is the assumption
given in the problem statement.
Cell H1 is either 1 (she made it to $1 million) or 0 (she didn’t). Over
many trials, the average of this cell will be out estimate of the
probability that Amanda does accumulate $1 million. This will be a
Crystal Ball forecast cell.
Decision Models -- Prof. Juran
5
Decision Models -- Prof. Juran
6
Decision Models -- Prof. Juran
7
Decision Models -- Prof. Juran
8
Decision Models -- Prof. Juran
9
As an added touch, we create a graph showing the amount of money in
Amanda’s retirement account during the simulation (this adds little to our
understanding, but it’s fun to watch):
A
1 Ann. Inv.
2 Goal
3
4
Year
5
0
6
1
7
2
8
3
9
4
10
5
11
6
12
7
13
8
14
9
15
10
16
11
17
12
18
13
19
14
B
$5,000
$1,000,000
C
Beginning
Return
$5,000
$9,900
$16,231
$20,135
$28,991
$43,924
$50,342
$65,949
$78,437
$92,117
$128,703
$131,746
$153,796
$174,875
-2.00%
13.45%
-6.75%
19.15%
34.26%
3.23%
21.07%
11.35%
11.07%
34.29%
-1.52%
12.94%
10.45%
31.50%
D
Mean
Stdev
Ending
0
$4,900
$11,231
$15,135
$23,991
$38,924
$45,342
$60,949
$73,437
$87,117
$123,703
$126,746
$148,796
$169,875
$229,965
Decision Models -- Prof. Juran
E
12%
25%
F
G
Reached goal?
$500,957
$478,876
H
0
I
J
K
Max Assets
Final Assets
Retirement Funds
$2,000,000
$1,800,000
$1,600,000
$1,400,000
$1,200,000
$1,000,000
$800,000
$600,000
$400,000
$200,000
$0
0
5
10
15
20
25
30
10
Decision Models -- Prof. Juran
11
It looks like Amanda has about a 48% chance of meeting her
goal of $1 million in 30 years.
Decision Models -- Prof. Juran
12
Example 2: Butterfly
The S&J index is a measure of overall equity value in
the software publishing industry.
Shares of a “tracking” mutual fund (a fund that tracks
this index) are available from Avant Garde
Investments, Inc. Shares in the mutual fund are
currently available at a price of $605.
Decision Models -- Prof. Juran
13
Avant Garde also sells 1-month call options on the S&J index, with current prices
as follows:
Strike
580
585
590
595
600
605
610
615
Option Bid Price
$25.54
$22.84
$20.33
$18.01
$15.79
$13.95
$12.09
$10.60
Option Ask Price
$25.64
$22.94
$20.43
$18.11
$15.89
$14.05
$12.19
$10.70
(A call option gives its holder the right to purchase one share on the expiration
date at the strike price. For example, if we buy one call option at the 600 strike
price, and the S&J is at 620 on the expiration date, we can exercise the option and
buy one share at 600 and immediately sell it for a $20 gross profit. The net profit
would be $20.00 – $15.89 = $4.11, which is a ($4.11 / $15.89) = 25.9% gain.)
Decision Models -- Prof. Juran
14
We are considering investing $100,000 in the S&J index over the next
month, based on our estimation that the S&J’s level one month from
now is a log-normally distributed random variable with a mean of 605
and a one month standard deviation of 30.
An analyst proposes that in addition to investing the $100,000 in the
S&J index, we take some positions in call options. He suggests selling
200 options contracts (1 option contract is an option to purchase 100
shares) at the 605 strike price, and buying 100 option contracts each of
the 600 and 610 strike prices.
What do you think of this scheme? Does it have any advantage over
simply investing all the money in the index? Assume that there are no
transaction costs.
Decision Models -- Prof. Juran
15
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
$
$
$
$
B
C
D
Cash Out
E
Cash In
F
605 initial price
605
mean
30
stdev
Total
605 end price
G
H
I
J
= Index Profit
= Options Profit
= Profit with Index + Options
Difference
(positive indicates butterfly strategy is better)
Strike
Qty
Qty Bought Sold
Cash Out
$580
$585
$590
$595
$600
$605
$610
$615
Cash In
Option Bid
Option Ask
$25.54
$22.84
$20.33
$18.01
$15.79
$13.95
$12.09
$10.60
$25.64
$22.94
$20.43
$18.11
$15.89
$14.05
$12.19
$10.70
index
price 1
month
payoff if bought payoff if sold
index (no calls)
index (with calls)
Decision Models -- Prof. Juran
16
Put in quantities bought and sold, according to the
analyst’s proposal
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
$
$
$
$
B
C
D
Cash Out
E
Cash In
F
605 initial price
605
mean
30
stdev
Total
605 end price
G
H
I
J
= Index Profit
= Options Profit
= Profit with Index + Options
Difference
(positive indicates butterfly strategy is better)
Strike
$580
$585
$590
$595
$600
$605
$610
$615
Qty
Qty Bought Sold
0
0
0
0
100
0
100
0
Cash Out
0
0
0
0
0
200
0
0
Cash In
Option Bid
Option Ask
$25.54
$22.84
$20.33
$18.01
$15.79
$13.95
$12.09
$10.60
$25.64
$22.94
$20.43
$18.11
$15.89
$14.05
$12.19
$10.70
index
price 1
month
payoff if bought payoff if sold
index (no calls)
index (with calls)
Decision Models -- Prof. Juran
17
Figure out how much cash is going out, in D10:D17
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
$
$
$
$
B
C
D
Cash Out
E
Cash In
F
605 initial price
605
mean
30
stdev
Total
605 end price
G
H
I
J
= Index Profit
= Options Profit
= Profit with Index + Options
Difference
(positive indicates butterfly strategy is better)
Strike
$580
$585
$590
$595
$600
$605
$610
$615
Qty
Qty Bought Sold
0
0
0
0
100
0
100
0
0
0
0
0
0
200
0
0
Cash Out
$0.00
$0.00
$0.00
$0.00
$1,589.00
($2,790.00)
$1,219.00
$0.00
Cash In
Option Bid
=B10*G10-C10*F10
$25.54
$22.84
$20.33
$18.01
$15.79
$13.95
$12.09
$10.60
Option Ask
index
price 1
month
payoff if bought payoff if sold
$25.64
$22.94
$20.43
$18.11
$15.89
$14.05
$12.19
$10.70
index (no calls)
index (with calls)
Decision Models -- Prof. Juran
18
Cell A5 will be an assumption; the ending price of the option
in one month. Put cell references to A5 into H10:H17.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
$
$
$
$
B
C
D
Cash Out
E
Cash In
F
605 initial price
605
mean
30
stdev
Total
605 end price
G
H
I
J
= Index Profit
= Options Profit
= Profit with Index + Options
Difference
(positive indicates butterfly strategy is better)
Strike
$580
$585
$590
$595
$600
$605
$610
$615
Qty
Qty Bought Sold
0
0
0
0
100
0
100
0
0
0
0
0
0
200
0
0
Cash Out
$0.00
$0.00
$0.00
$0.00
$1,589.00
($2,790.00)
$1,219.00
$0.00
Cash In
Option Bid
Option Ask
$25.54
$22.84
$20.33
$18.01
$15.79
$13.95
$12.09
$10.60
$25.64
$22.94
$20.43
$18.11
$15.89
$14.05
$12.19
$10.70
index
price 1
month
$
$
$
$
$
$
$
$
605
605
605
605
605
605
605
605
payoff if bought payoff if sold
=$A$5
index (no calls)
index (with calls)
Decision Models -- Prof. Juran
19
In I10:I17 enter a formula to calculate the payoff for options
bought, as a function of the random ending price of the
index.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
$
$
$
$
B
C
D
Cash Out
E
Cash In
F
605 initial price
605
mean
30
stdev
Total
605 end price
G
H
I
J
K
= Index Profit
= Options Profit
= Profit with Index + Options
Difference
(positive indicates butterfly strategy is better)
Strike
$580
$585
$590
$595
$600
$605
$610
$615
Qty
Qty Bought Sold
0
0
0
0
100
0
100
0
0
0
0
0
0
200
0
0
Cash Out
$0.00
$0.00
$0.00
$0.00
$1,589.00
($2,790.00)
$1,219.00
$0.00
Cash In
Option Bid
Option Ask
$25.54
$22.84
$20.33
$18.01
$15.79
$13.95
$12.09
$10.60
$25.64
$22.94
$20.43
$18.11
$15.89
$14.05
$12.19
$10.70
index
price 1
month
$
$
$
$
$
$
$
$
605
605
605
605
605
605
605
605
payoff if bought payoff if sold
=MAX(0,H10-A10)
$25.00
$20.00
$15.00
$10.00
$5.00
$0.00
$0.00
$0.00
index (no calls)
index (with calls)
Decision Models -- Prof. Juran
20
Similarly, in J10:J17 enter a formula to calculate the payoff
for options sold, as a function of the random ending price of
the index.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
$
$
$
$
B
C
D
Cash Out
E
Cash In
F
605 initial price
605
mean
30
stdev
Total
605 end price
G
H
I
J
K
= Index Profit
= Options Profit
= Profit with Index + Options
Difference
(positive indicates butterfly strategy is better)
Strike
$580
$585
$590
$595
$600
$605
$610
$615
Qty
Qty Bought Sold
0
0
0
0
100
0
100
0
0
0
0
0
0
200
0
0
Cash Out
$0.00
$0.00
$0.00
$0.00
$1,589.00
($2,790.00)
$1,219.00
$0.00
Cash In
Option Bid
Option Ask
$25.54
$22.84
$20.33
$18.01
$15.79
$13.95
$12.09
$10.60
$25.64
$22.94
$20.43
$18.11
$15.89
$14.05
$12.19
$10.70
index
price 1
month
$
$
$
$
$
$
$
$
605
605
605
605
605
605
605
605
payoff if bought payoff if sold
$25.00
$20.00
$15.00
$10.00
$5.00
$0.00
$0.00
$0.00
($25.00)
($20.00)
($15.00)
($10.00)
($5.00)
$0.00
$0.00
$0.00
=MIN(0,A10-H10)
index (no calls)
index (with calls)
Decision Models -- Prof. Juran
21
In B19:B20, calculate how many shares of the index are
being purchased.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
$
$
$
$
B
C
D
Cash Out
E
Cash In
F
605 initial price
605
mean
30
stdev
Total
605 end price
G
H
I
J
= Index Profit
= Options Profit
= Profit with Index + Options
Difference
(positive indicates butterfly strategy is better)
Strike
Qty
Qty Bought Sold
$580
$585
$590
$595
$600
$605
$610
$615
0
0
0
0
100
0
100
0
index (no calls)
index (with calls)
165.29
165.29
0
0
0
0
0
200
0
0
Cash Out
$0.00
$0.00
$0.00
$0.00
$1,589.00
($2,790.00)
$1,219.00
$0.00
Cash In
Option Bid
Option Ask
$25.54
$22.84
$20.33
$18.01
$15.79
$13.95
$12.09
$10.60
$25.64
$22.94
$20.43
$18.11
$15.89
$14.05
$12.19
$10.70
index
price 1
month
$
$
$
$
$
$
$
$
605
605
605
605
605
605
605
605
payoff if bought payoff if sold
$25.00
$20.00
$15.00
$10.00
$5.00
$0.00
$0.00
$0.00
($25.00)
($20.00)
($15.00)
($10.00)
($5.00)
$0.00
$0.00
$0.00
=100000/A2
=(100000-D3)/A3
Decision Models -- Prof. Juran
22
In E10:E17, calculate the amount of cash coming back
in at the end of the month.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
$
$
$
$
B
C
D
Cash Out
E
Cash In
605 initial price
605
mean
30
stdev
Total
611 end price
F
G
H
I
J
= Index Profit
= Options Profit
= Profit with Index + Options
Difference
(positive indicates butterfly strategy is better)
Strike
Qty
Qty Bought Sold
$580
$585
$590
$595
$600
$605
$610
$615
0
0
0
0
100
0
100
0
index (no calls)
index (with calls)
165.29
165.29
0
0
0
0
0
200
0
0
Cash Out
Cash In
$0.00
$0.00
$0.00
$0.00
$1,589.00
($2,790.00)
$1,219.00
$0.00
$0.00
$0.00
$0.00
$0.00
$1,100.00
($1,200.00)
$100.00
$0.00
Decision Models -- Prof. Juran
index
price 1
Option Bid
Option Ask
month payoff if bought payoff if sold
=SUMPRODUCT(B10:C10,I10:J10)
$25.54
$25.64
$
611
$31.00
($31.00)
$
611
$26.00
($26.00)
$22.84
$22.94
$
611
$21.00
($21.00)
$20.33
$20.43
$
611
$16.00
($16.00)
$18.01
$18.11
$
611
$11.00
($11.00)
$15.79
$15.89
$
611
$6.00
($6.00)
$13.95
$14.05
$
611
$1.00
($1.00)
$12.09
$12.19
$
611
$0.00
$0.00
$10.60
$10.70
23
In D2:F2, calculate the P/L from the index.
A
1
2
3
4
5
6
7
$
$
$
$
B
C
605 initial price
605
mean
30
stdev
Total
611 end price
D
Cash Out
$ 100,000
=B19*A2
E
Cash In
$ 100,992
=B19*A5
F
$992
=E2-D2
G
H
I
= Index Profit
= Options Profit
= Profit with Index + Options
Difference
(positive indicates butterfly strategy is better)
Decision Models -- Prof. Juran
24
In D3:F3, calculate the P/L from the options.
A
1
2
3
4
5
6
7
$
$
$
$
B
C
D
Cash Out
$ 100,000
$
18
605 initial price
605
mean
30
stdev
Total
=SUM(D10:D17)
611 end price
E
Cash In
$ 100,992
$
-
F
$992
($18)
=SUM(E10:E17)
G
H
I
= Index Profit
=E3-D3
= Options Profit
= Profit with Index + Options
Difference
(positive indicates butterfly strategy is better)
Decision Models -- Prof. Juran
25
In D4:F4, calculate the total P/L.
A
1
2
3
4
5
6
7
8
$
$
$
$
B
C
605 initial price
605
mean
30
stdev
Total
611 end price
Decision Models -- Prof. Juran
D
Cash Out
$ 100,000
$
18
$ 100,000
E
Cash In
$ 100,992
$
$ 100,974
=(B20*A2)+D3
F
$992
($18)
$974
G
H
I
= Index Profit
= Options Profit
=E4-D4
= Profit with Index + Options
Difference
=(B20*A5)+E3
(positive indicates butterfly strategy is better)
26
In F6 calculate the difference between the two strategies
(with and without the options).
A
1
2
3
4
5
6
7
$
$
$
$
B
C
605 initial price
605
mean
30
stdev
Total
611 end price
Decision Models -- Prof. Juran
D
Cash Out
$ 100,000
$
18
$ 100,000
E
Cash In
$ 100,992
$
$ 100,974
F
$992
($18)
$974
G
H
I
= Index Profit
= Options Profit
= Profit with Index + Options
=F4-F2
($18)
Difference
(positive indicates butterfly strategy is better)
27
Decision Models -- Prof. Juran
28
Decision Models -- Prof. Juran
29
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
$
$
$
$
B
C
605 initial price
605
mean
30
stdev
Total
601
D
Cash Out
$ 100,000
$
18
$ 100,000
E
Cash In
$ 99,339
$
100
$ 99,421
F
($661)
$82
($579)
G
H
I
J
= Index Profit
= Options Profit
= Profit with Index + Options
=F4-F2
$82.12
Difference
(positive indicates butterfly strategy is better)
Cash Out
Cash In
Option Bid
Option Ask
index
price 1
month
$0.00
$0.00
$0.00
$0.00
$1,589.00
($2,790.00)
$1,219.00
$0.00
$0.00
$0.00
$0.00
$0.00
$100.00
$0.00
$0.00
$0.00
$25.54
$22.84
$20.33
$18.01
$15.79
$13.95
$12.09
$10.60
$25.64
$22.94
$20.43
$18.11
$15.89
$14.05
$12.19
$10.70
$601.00
$601.00
$601.00
$601.00
$601.00
$601.00
$601.00
$601.00
Qty
Qty Bought Sold
Strike
$580
$585
$590
$595
$600
$605
$610
$615
0
0
0
0
100
0
100
0
=100000/A2
index (no calls)
index (with calls)
0
0
0
0
0
200
0
0
payoff if bought payoff if sold
$21.00
$16.00
$11.00
$6.00
$1.00
$0.00
$0.00
$0.00
($21.00)
($16.00)
($11.00)
($6.00)
($1.00)
$0.00
$0.00
$0.00
=(100000-D3)/A3
165.29
165.26
0
0
=B17*G17-C17*F17
Decision Models -- Prof. Juran
=SUMPRODUCT(B17:C17,I17:J17)
=$A$5
=MAX(0,H17-A17)
=MIN(0,A17-H17)
30
Decision Models -- Prof. Juran
31
Decision Models -- Prof. Juran
32
A
An old Excel trick:
DataTable
Decision Models -- Prof. Juran
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
B
Difference
$82
C
=F6
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
33
Select A24:B55, then Data Table
Decision Models -- Prof. Juran
34
A
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
Decision Models -- Prof. Juran
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
B
Difference
$82
$ (17.55)
$ (17.58)
$ (17.61)
$ (17.64)
$ (17.67)
$ (17.70)
$ (17.73)
$ (17.76)
$ (17.79)
$ (17.82)
$ (17.85)
$ 82.12
$ 182.09
$ 282.06
$ 382.03
$ 482.00
$ 381.97
$ 281.94
$ 181.91
$ 81.88
$ (18.15)
$ (18.18)
$ (18.21)
$ (18.24)
$ (18.27)
$ (18.30)
$ (18.33)
$ (18.36)
$ (18.39)
$ (18.42)
$ (18.45)
35
Benefits from the Butterfly Strategy
$600
Butterfly Benefit
$500
$400
$300
$200
$100
$$590
$595
$600
$605
$610
$615
$620
$(100)
Ending Index Price
Decision Models -- Prof. Juran
36
3. Evaluation of Hedging Strategies
It is July 1, 2002, and international entrepreneurs Clifford & Kearns
(C&K) are concerned about volatility in the exchange rates between
U.S. dollars and certain European currencies.
C&K have incurred costs in dollars to develop, produce, and distribute
merchandise to Norway, Switzerland, and Great Britain, for which they
expect to realize revenues in 12 months.
Decision Models -- Prof. Juran
37
Specifically, they expect to earn 1 million units each of British pounds,
Swiss francs, and Norwegian kroner. Based on current exchange rates,
this should result in $2,337,700 in revenue (see current rates below).
POUNDS/$US FRANCS/US$ KRONER/US$
0.6533
1.4845
7.4940
Revenue

1
1
1
* 1,000 ,000 
* 1,000 ,000 
* 1,000 ,000  $2 ,337 ,700
0.6533
1.4845
7.4940
Decision Models -- Prof. Juran
38
Unfortunately, it is possible that one or more of these currencies
could devalue against the dollar in that one year, causing C&K to
realize a smaller total revenue (in dollars) than expected. C&K has
turned to their investment bank, Nuccio, Noto, and Rizzi (NNR)
for advice.
NNR has recommended buying 1.3 million 1-year Euro put options
with a strike price of $0.98, for $0.0432 each. NNR claims that this
hedging strategy will substantially decrease the risk of a large loss
due to exchange rate fluctuations.
Decision Models -- Prof. Juran
39
(a)
Create a simulation model to study the “unhedged”
distribution of revenue for C&K, using the historical exchange rate
data in Exhibit 2. Make a histogram and report summary statistics.
What is the 5% value at risk (VAR) for C&K’s revenue from these
three countries over the next 12 months? What is the probability that
C&K’s revenue will be less than $2,087,700 (i.e., a $250,000 loss or
worse)?
(b)
Create a simulation model to study the “hedged” distribution
of revenue for C&K. Make a histogram and report summary statistics
with the policy recommended by NNR. What is the 5% VAR for
C&K’s revenue from these three countries over the next 12 months?
What is the probability that C&K’s revenue will be less than
$2,087,700?
Decision Models -- Prof. Juran
40
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
POUNDS/$US FRANCS/US$ KRONER/US$ EURO/US$
0.6146
1.5808
7.9640
0.9847
0.6192
1.6540
8.2770
1.0276
0.6310
1.6568
8.3115
1.0309
0.6258
1.6587
8.4640
1.0460
0.6428
1.7135
8.9050
1.0965
0.6705
1.6878
8.9400
1.0745
0.6607
1.6323
8.5880
1.0498
0.6670
1.6758
8.8850
1.0837
0.6847
1.7230
9.0108
1.1120
0.6814
1.7322
9.1269
1.1356
0.6919
1.7765
9.2020
1.1650
0.6957
1.7285
9.2475
1.1409
0.6677
1.6075
8.7600
1.0565
0.6768
1.6330
8.7550
1.0656
0.6871
1.6557
8.8650
1.0763
0.7042
1.7317
9.1610
1.1333
0.6974
1.7255
9.0540
1.1189
0.7062
1.7992
9.4538
1.1832
0.7058
1.8003
9.4030
1.1827
0.6978
1.7158
9.0980
1.1373
0.6923
1.7075
8.9380
1.1276
0.6764
1.6196
8.8244
1.0918
0.6840
1.6295
8.8200
1.1057
0.7034
1.6550
8.9790
1.1240
0.6920
1.6424
8.8775
1.1073
0.7063
1.7179
9.1050
1.1610
0.7047
1.7060
8.8875
1.1558
0.6941
1.6607
8.7450
1.1356
0.6839
1.6010
8.3500
1.1035
0.6705
1.6878
8.9400
1.0745
0.6533
1.4845
7.4940
1.0108
Decision Models -- Prof. Juran
41
Here is a time-series graph showing the movements of all four relevant currencies
against the dollar. We observe that they move more or less together:
125
120
115
110
105
100
95
POUNDS/$US
90
FRANCS/US$
KRONER/US$
85
EURO/US$
80
Decision Models -- Prof. Juran
Jul
Jun
May
Apr
Mar
Feb
Jan
Dec
Nov
Oct
Sep
Aug
Jul
Jun
May
Apr
Mar
Feb
Jan
Dec
Nov
Oct
Sep
Aug
Jul
Jun
May
Apr
Mar
Feb
Jan
75
42
Converting prices into returns:
A
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Feb
Mar
B
C
D
E
$US/POUND FRANCS/US$ KRONER/US$ EURO/US$
0.7430%
4.6306%
3.9302%
4.3572%
1.8992%
0.1693%
0.4168%
0.3196%
-0.8198%
0.1147%
1.8348%
1.4644%
2.7124%
3.3038%
5.2103%
4.8246%
4.3111%
-1.4999%
0.3930%
-2.0092%
-1.4536%
-3.2883%
-3.9374%
-2.2990%
0.9538%
2.6650%
3.4583%
3.2293%
2.6498%
2.8166%
1.4159%
2.6131%
-0.4770%
0.5340%
1.2885%
2.1236%
1.5430%
2.5574%
0.8228%
2.5862%
0.5496%
-2.7019%
0.4945%
-2.0650%
-4.0329%
-7.0003%
-5.2717%
-7.3957%
1.3672%
1.5863%
-0.0571%
0.8632%
1.5255%
1.3901%
1.2564%
1.0010%
2.4859%
4.5902%
3.3390%
5.2924%
-0.9763%
-0.3580%
-1.1680%
-1.2644%
1.2712%
4.2712%
4.4157%
5.7383%
-0.0565%
0.0611%
-0.5374%
-0.0355%
-1.1305%
-4.6937%
-3.2436%
-3.8440%
-0.7893%
-0.4837%
-1.7586%
-0.8457%
-2.3064%
-5.1479%
-1.2710%
-3.1772%
1.1286%
0.6113%
-0.0499%
1.2716%
2.8346%
1.5649%
1.8027%
1.6522%
-1.6193%
-0.7613%
-1.1304%
-1.4838%
2.0695%
4.5969%
2.5627%
4.8531%
-0.2255%
-0.6927%
-2.3888%
-0.4508%
Decision Models -- Prof. Juran
F
G
H
I
=(data!E3-data!E2)/data!E2
43
Here are summary statistics for each of the currencies’ returns against
the dollar, including a t-test to see if the means are significantly
different from zero (they are not) :
A
1
2
3
4
5
mean
stdev
t-stat
p-value
B
C
D
E
POUNDS/$USFRANCS/US$ KRONER/US$ EURO/US$
0.1578%
0.0243%
0.0841%
0.0571%
2.0683%
3.4421%
3.4732%
3.4733%
0.4180
0.0387
0.1326
0.0901
0.6789
0.9694
0.8954
0.9288
Decision Models -- Prof. Juran
F
G
H
I
=AVERAGE(E9:E38)
=STDEV(E9:E38)
=(E2)/(E3/SQRT(COUNT(E9:E38)))
=TDIST(ABS(E4),COUNT(E9:E38),2)
44
Correlation analysis suggests that the returns on these currencies (including the Euro) are all closely and
positively related to each other:
$US/POUND FRANCS/US$ KRONER/US$ EURO/US$
$US/POUND
1
FRANCS/US$
0.6661
1
KRONER/US$
0.6301
0.8909
1
EURO/US$
0.7527
0.8754
0.7883
1
Decision Models -- Prof. Juran
45
Distribution fitting: Checking to see which Crystal Ball distribution best
fits the data (in this case the British pound’s return against the dollar).
Decision Models -- Prof. Juran
46
Decision Models -- Prof. Juran
47
Decision Models -- Prof. Juran
48
Decision Models -- Prof. Juran
49
Decision Models -- Prof. Juran
50
It turns out that all four of our variables can be modeled reasonably
well by normal distributions; normal is always either the best fit or
the second best fit.
We’ll use normal distributions with means of zero and standard
deviations estimated from our sample data.
Decision Models -- Prof. Juran
51
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Receivable in one month (millions)
Current rate in US$
Volatility (stdev of yield)
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Revenue in one year ($million)
Total
Decision Models -- Prof. Juran
B
FrancS
1.0000
0.6736
0.0344
D
C
Kroner Pounds
1.0000 1.0000
0.1334 1.5307
0.0347 0.0207
FrancS
Return Price
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
2.3377
E
F
Pound
Franc
Kroner
Euro
Kroner
Return Price
=B3*(1+B10)
0.0000
=C10*(1+B11) 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.6736
=C23+F23+I23
0.1334
I
H
G
Correlations
Pound Franc Kroner
1
1
0.6661
1
0.6301 0.8909
0.7527 0.8754 0.7883
Pound
Return Price
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
=F21*C2
1.5307
52
We start by creating the “January” cell for each currency. The Swiss franc:
Decision Models -- Prof. Juran
53
The Norwegian kroner:
Decision Models -- Prof. Juran
54
The British pound:
Decision Models -- Prof. Juran
55
A
1
2 Receivable in one month (millions)
3 Current rate in US$
4 Volatility (stdev of yield)
5
6
7
8
9
10
Jan
11
Feb
12
Mar
13
Apr
14
May
15
Jun
16
Jul
Decision Models -- Prof. Juran
B
FrancS
1.0000
0.6736
0.0344
C
D
Kroner Pounds
1.0000 1.0000
0.1334 1.5307
0.0347 0.0207
FrancS
Return Price
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
E
F
Pound
Franc
Kroner
Euro
Kroner
Return Price
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
G
H
I
Correlations
Pound Franc Kroner
1
0.6661
1
0.6301 0.8909
1
0.7527 0.8754 0.7883
Pound
Return Price
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
56
You can specify bivariate correlations in
the Define Assumption window.
For more than a few correlated green cells,
it’s more efficient to use the matrix view.
Decision Models -- Prof. Juran
57
Back inside the Swiss franc (after defining two other green cells):
Decision Models -- Prof. Juran
58
Decision Models -- Prof. Juran
59
Decision Models -- Prof. Juran
60
Decision Models -- Prof. Juran
61
Decision Models -- Prof. Juran
62
Decision Models -- Prof. Juran
63
Decision Models -- Prof. Juran
64
Decision Models -- Prof. Juran
65
Decision Models -- Prof. Juran
66
Decision Models -- Prof. Juran
67
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
Receivable in one month (millions)
Current rate in US$
Volatility (stdev of yield)
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Revenue in one year ($million)
Decision Models -- Prof. Juran
B
FrancS
1.0000
0.6736
0.0344
C
D
Kroner Pounds
1.0000 1.0000
0.1334 1.5307
0.0347 0.0207
E
F
Pound
Franc
Kroner
Euro
G
H
I
Correlations
Pound Franc Kroner
1
0.6661
1
0.6301 0.8909
1
0.7527 0.8754 0.7883
FrancS
Return Price
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
0.0000 0.6736
Kroner
Return Price
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
0.0000 0.1334
Pound
Return Price
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.0000 1.5307
0.6736
0.1334
1.5307
68
Decision Models -- Prof. Juran
69
VaR approach: Click the right grabber and then enter 95 in the certainty box.
2.3377 – 2.0412 = 0.2965
Decision Models -- Prof. Juran
($296,500)
70
“Round dollar amount” approach:
2.3377 – 0.2500 = 2.0877
Chances of losing $250k or more = 1 – 0.9146 = 0.0854
Decision Models -- Prof. Juran
71
We update the model to include the return on the Euro versus the dollar,
including the appropriate correlations:
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
Receivable in one month (millions)
Current rate in US$
Volatility (stdev of yield)
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Revenue in one year ($million)
Total
B
FrancS
1.0000
0.6736
0.0344
C
D
Kroner Pounds
1.0000 1.0000
0.1334 1.5307
0.0347 0.0207
E
Euro
0.0000
0.9893
0.0347
F
G
Pound
Franc
Kroner
Euro
H
I
J
Correlations
Pound Franc Kroner
1
0.6661
1
0.6301 0.8909
1
0.7527 0.8754 0.7883
FrancS
Return Price
-0.0267 0.6556
0.0681 0.7003
-0.0035 0.6978
0.0441 0.7286
0.0239 0.746
-0.0119 0.7371
-0.0173 0.7243
-0.0176 0.7116
0.0260 0.7301
-0.0837 0.669
0.0129 0.6776
0.0008 0.6781
Kroner
Return Price
-0.0084 0.1323
0.0717 0.1418
0.0522 0.1492
0.0352 0.1544
-0.0254 0.1505
-0.0327 0.1456
0.0120 0.1473
-0.0128 0.1454
0.0389 0.1511
-0.1016 0.1357
-0.0034 0.1353
0.0034 0.1357
Pound
Return Price
-0.0147 1.5082
0.0556 1.5921
-0.0055 1.5834
0.0036 1.5891
0.0218 1.6237
-0.0053 1.6151
0.0104 1.6318
-0.0042 1.625
0.0242 1.6643
-0.0222 1.6274
-0.0166 1.6005
0.0087 1.6143
0.6781
0.1357
1.6143
K
L
M
Euro Put Options
Units Purchased 1.3
Strike
0.98
Cost
0.0432
Payout
0
Euro
Return
Price
0.0008 1.000797
0.0696 1.070441
-0.0155 1.053869
0.0387 1.094608
0.0132 1.109103
0.0119 1.122303
0.0213 1.146221
-0.0238 1.118907
0.0525 1.177609
-0.0678 1.097722
0.0083 1.106847
0.0249 1.134432
-0.0562
2.3720
Decision Models -- Prof. Juran
72
Here’s one way to model the cash flow associated with the Euro put options:
K
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
L
Euro Put Options
Units Purchased
Strike
Cost
Payout
Return
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
Euro
Price
1
1
1
1
1
1
1
1
1
1
1
1
-0.0562
Decision Models -- Prof. Juran
M
1.3
0.98
0.0432
0
N
O
=MAX(0,M4-L21)
=L20*(1+K21)
=M3*(M6-M5)
73
+Smaller standard deviation
+Truncated lower tail
−Lower expected value
Decision Models -- Prof. Juran
74
+VaR is $196,800
(better than $296,500)
Decision Models -- Prof. Juran
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+Chance of $250k loss 0.0111
(better than 0.0854)
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Clifford & Kearns
Expected Revenue ($millions)
2.000
Total Unhedged Revenue
1.500
Total Hedged Revenue
1.000
0.500
0.000
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
Std Deviation ($millions)
Decision Models -- Prof. Juran
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Using Historical Data in Crystal Ball
There are two basic approaches to using historical data in a simulation, which we
will refer to here as the parametric approach and the resampling approach.
Each has advantages and disadvantages, and the modeler will use one or the
other depending on the circumstances.
Decision Models -- Prof. Juran
78
The Parametric Approach
“Fit” the data to some theoretical distribution (such as normal or
exponential) and estimate the parameters appropriate to the
distribution (such as mean and standard deviation for a normal
distribution, or lambda for an exponential distribution).
Advantage: Simplicity (a random variable can be described with a
few parameters instead of all the data).
Disadvantage: Need assurance that the theoretical distribution we
choose is in fact a good “fit” to the data.
This gives rise to a special kind of hypothesis test, called a goodnessof-fit test.
Decision Models -- Prof. Juran
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The Parametric Approach
1. find which theoretical distribution best fits each
variable,
2. estimate the proper parameters for each, and
3. specify a correlation coefficient for the
relationship between the two variables.
Decision Models -- Prof. Juran
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1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A
Portfolio Weights
B
0.75
C
0.25
D
S&P 500
-3.13%
-3.13%
-3.13%
T-Bill
0.56%
0.56%
0.56%
Portfolio Returns
-2.20%
-2.20%
-2.20%
Historical Data
S&P 500
T-Bill
Month
Total Return Total Return
1
-7.43%
0.60%
2
5.86%
0.62%
3
0.30%
0.57%
4
-8.89%
0.50%
5
-5.47%
0.53%
6
-4.82%
0.58%
Decision Models -- Prof. Juran
E
F
G
Start
$100.00
$ 97.80
$ 95.64
End
$97.80
$95.64
$93.53
H
=F4*(1+D4)
=SUMPRODUCT($B$1:$C$1,B6:C6)
81
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83
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84
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85
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
A
Portfolio Weights
B
0.75
C
0.25
D
S&P 500
-3.13%
-3.13%
-3.13%
T-Bill
0.56%
0.56%
0.56%
Portfolio Returns
-2.20%
-2.20%
-2.20%
Historical Data
S&P 500
T-Bill
Month
Total Return Total Return
1
-7.43%
0.60%
2
5.86%
0.62%
3
0.30%
0.57%
4
-8.89%
0.50%
5
-5.47%
0.53%
6
-4.82%
0.58%
7
7.52%
0.52%
8
5.09%
0.53%
9
3.47%
0.54%
10
-0.97%
0.46%
11
5.36%
0.46%
12
5.84%
0.42%
13
4.19%
0.38%
14
1.41%
0.33%
15
3.82%
0.30%
Decision Models -- Prof. Juran
E
F
G
Start
$100.00
$ 97.80
$ 95.64
End
$97.80
$95.64
$93.53
H
=F4*(1+D4)
=SUMPRODUCT($B$1:$C$1,B6:C6)
86
Decision Models -- Prof. Juran
87
Decision Models -- Prof. Juran
88
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
A
Portfolio Weights
B
0.75
C
0.25
D
S&P 500
-3.13%
-3.13%
-3.13%
T-Bill
0.56%
0.56%
0.56%
Portfolio Returns
-2.20%
-2.20%
-2.20%
Historical Data
S&P 500
T-Bill
Month
Total Return Total Return
1
-7.43%
0.60%
2
5.86%
0.62%
3
0.30%
0.57%
4
-8.89%
0.50%
5
-5.47%
0.53%
6
-4.82%
0.58%
7
7.52%
0.52%
8
5.09%
0.53%
9
3.47%
0.54%
10
-0.97%
0.46%
11
5.36%
0.46%
12
5.84%
0.42%
13
4.19%
0.38%
14
1.41%
0.33%
15
3.82%
0.30%
Decision Models -- Prof. Juran
E
F
G
Start
$100.00
$ 97.80
$ 95.64
End
$97.80
$95.64
$93.53
H
=F4*(1+D4)
=SUMPRODUCT($B$1:$C$1,B6:C6)
89
Decision Models -- Prof. Juran
90
Decision Models -- Prof. Juran
91
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92
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1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
A
Portfolio Weights
B
0.75
C
0.25
D
S&P 500
-3.13%
-3.13%
-3.13%
T-Bill
0.56%
0.56%
0.56%
Portfolio Returns
-2.20%
-2.20%
-2.20%
Historical Data
S&P 500
T-Bill
Month
Total Return Total Return
1
-7.43%
0.60%
2
5.86%
0.62%
3
0.30%
0.57%
4
-8.89%
0.50%
5
-5.47%
0.53%
6
-4.82%
0.58%
7
7.52%
0.52%
8
5.09%
0.53%
9
3.47%
0.54%
10
-0.97%
0.46%
11
5.36%
0.46%
12
5.84%
0.42%
13
4.19%
0.38%
14
1.41%
0.33%
15
3.82%
0.30%
Decision Models -- Prof. Juran
E
F
G
Start
$100.00
$ 97.80
$ 95.64
End
$97.80
$95.64
$93.53
H
=F4*(1+D4)
=SUMPRODUCT($B$1:$C$1,B6:C6)
94
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
A
Portfolio Weights
B
0.75
C
0.25
D
S&P 500
-3.13%
-3.13%
-3.13%
T-Bill
0.56%
0.56%
0.56%
Portfolio Returns
-2.20%
-2.20%
-2.20%
Historical Data
S&P 500
T-Bill
Month
Total Return Total Return
1
-7.43%
0.60%
2
5.86%
0.62%
3
0.30%
0.57%
4
-8.89%
0.50%
5
-5.47%
0.53%
6
-4.82%
0.58%
7
7.52%
0.52%
8
5.09%
0.53%
9
3.47%
0.54%
10
-0.97%
0.46%
11
5.36%
0.46%
12
5.84%
0.42%
13
4.19%
0.38%
14
1.41%
0.33%
15
3.82%
0.30%
Decision Models -- Prof. Juran
E
F
G
Start
$100.00
$ 97.80
$ 95.64
End
$97.80
$95.64
$93.53
H
=F4*(1+D4)
=SUMPRODUCT($B$1:$C$1,B6:C6)
95
The Resampling Approach
In this approach, we make no assumptions about any
theoretical distributions that may or may not actually fit our
data; we use the data themselves as the basis for our
simulation.
Advantages: Avoids the problem of Type II errors in the
Chi-square test. Also spares us from dealing explicitly with
correlation.
Disadvantage: our model may have to include a large set of
data (as opposed to the few parameters we used in the
parametric approach).
Decision Models -- Prof. Juran
96
Back to our example. Start the model with a spreadsheet
similar to the parametric one. Notice the integers in
column A.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A
Portfolio Weights
B
0.75
C
0.25
D
Random Months
5
5
4
S&P 500
-5.47%
-5.47%
-8.89%
T-Bill
0.53%
0.53%
0.50%
Portfolio Returns
-3.97%
-3.97%
-6.54%
Historical Data
Month
1
2
3
4
5
6
E
F
G
Start
$ 100.00
$ 96.03
$ 92.21
End
$ 96.03
$ 92.21
$ 86.18
S&P 500
T-Bill
Total Return Total Return
-7.43%
0.60%
5.86%
0.62%
0.30%
0.57%
-8.89%
0.50%
-5.47%
0.53%
-4.82%
0.58%
Decision Models -- Prof. Juran
97
Use the VLOOKUP function in B4:C6 to “look up” the paired scenario
corresponding to the integer in A4:A6.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A
Portfolio Weights
B
0.75
Random Months
5
5
4
S&P 500
-5.47%
-5.47%
-8.89%
Historical Data
Month
1
2
3
4
5
6
C
0.25
D
E
F
G
T-Bill
Portfolio Returns
Start
End
0.53%
-3.97%
$ 100.00 $ 96.03
=VLOOKUP(A6,$A$10:$C$24,3,0)
0.53%
-3.97%
$ 96.03 $ 92.21
0.50%
-6.54%
$ 92.21 $ 86.18
=VLOOKUP(A6,$A$10:$C$24,2,0)
S&P 500
T-Bill
Total Return Total Return
-7.43%
0.60%
5.86%
0.62%
0.30%
0.57%
-8.89%
0.50%
-5.47%
0.53%
-4.82%
0.58%
Decision Models -- Prof. Juran
98
Decision Models -- Prof. Juran
99
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A
Portfolio Weights
B
0.75
Random Months
2
1
8
S&P 500
5.86%
-7.43%
5.09%
Historical Data
Month
1
2
3
4
5
6
C
0.25
D
E
F
G
T-Bill
Portfolio Returns
Start
End
0.62%
4.55%
$
100.00
$
104.55
=VLOOKUP(A6,$A$10:$C$24,3,0)
0.60%
-5.43%
$ 104.55 $ 98.88
0.53%
3.95%
$ 98.88 $ 102.78
=VLOOKUP(A6,$A$10:$C$24,2,0)
S&P 500
T-Bill
Total Return Total Return
-7.43%
0.60%
5.86%
0.62%
0.30%
0.57%
-8.89%
0.50%
-5.47%
0.53%
-4.82%
0.58%
Decision Models -- Prof. Juran
100
Summary
Finance Simulation Models
• Forecasting
– Retirement Planning
– Butterfly Strategy
• Risk Management
– Introduction to VaR
– Currency Risk
• Using Historical Data in Simulations
– Parametric Approach
– Resampling Approach
Decision Models -- Prof. Juran
101