session 9a

Session 9a

Decision Models -- Prof. Juran

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OverviewFinance Simulation Models• Forecasting

– Retirement Planning– Butterfly Strategy

• Risk Management– Introduction to VaR– Currency Risk

• Using Historical Data in Simulations– Parametric Approach– Resampling Approach


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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.


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1 2 3 4 5 6 7 8 9

A B C D E F G H I J Ann. Inv. $5,000 Mean 12% Reached goal? 0 Goal $1,000,000 Stdev 25%

Year Beginning Return Ending $500,957 Max Assets 0 0 $478,876 Final Assets 1 $5,000 -18.09% $4,095 2 $9,095 10.03% $10,008 3 $15,008 -29.95% $10,513 4 $15,513 3.44% $16,047

=IF(F4>B2,1,0)

=MAX(D6:D35) =D35

=B1 =B6*(1+C6) =D6+5000 =B7*(1+C7)


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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.


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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):

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A B C D E F G H I J KAnn. Inv. $5,000 Mean 12% Reached goal? 0Goal $1,000,000 Stdev 25%

Year Beginning Return Ending $500,957 Max Assets0 0 $478,876 Final Assets1 $5,000 -2.00% $4,9002 $9,900 13.45% $11,2313 $16,231 -6.75% $15,1354 $20,135 19.15% $23,9915 $28,991 34.26% $38,9246 $43,924 3.23% $45,3427 $50,342 21.07% $60,9498 $65,949 11.35% $73,4379 $78,437 11.07% $87,117

10 $92,117 34.29% $123,70311 $128,703 -1.52% $126,74612 $131,746 12.94% $148,79613 $153,796 10.45% $169,87514 $174,875 31.50% $229,965

Retirement Funds

$0

$200,000

$400,000

$600,000

$800,000

$1,000,000

$1,200,000

$1,400,000

$1,600,000

$1,800,000

$2,000,000

0 5 10 15 20 25 30


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It looks like Amanda has about a 48% chance of meeting her goal of $1 million in 30 years.


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Example 2: ButterflyThe 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.


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Avant Garde also sells 1-month call options on the S&J index, with current prices as follows:

Strike Option Bid Price Option Ask Price 580 $25.54 $25.64 585 $22.84 $22.94 590 $20.33 $20.43 595 $18.01 $18.11 600 $15.79 $15.89 605 $13.95 $14.05 610 $12.09 $12.19 615 $10.60 $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.)


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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.


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A B C D E F G H I JCash Out Cash In

605$ initial price = Index Profit605$ mean = Options Profit30$ stdev Total = Profit with Index + Options

605$ end priceDifference

(positive indicates butterfly strategy is better)

Strike Qty BoughtQty Sold Cash Out Cash In Option Bid Option Ask

index price 1 month payoff if bought payoff if sold

$580 $25.54 $25.64 $585 $22.84 $22.94 $590 $20.33 $20.43 $595 $18.01 $18.11 $600 $15.79 $15.89 $605 $13.95 $14.05 $610 $12.09 $12.19 $615 $10.60 $10.70

index (no calls)index (with calls)


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$580 0 0 $25.54 $25.64 $585 0 0 $22.84 $22.94 $590 0 0 $20.33 $20.43 $595 0 0 $18.01 $18.11 $600 100 0 $15.79 $15.89 $605 0 200 $13.95 $14.05 $610 100 0 $12.09 $12.19 $615 0 0 $10.60 $10.70


Put in quantities bought and sold, according to the analyst’s proposal


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$580 0 0 $0.00 $25.54 $25.64 $585 0 0 $0.00 $22.84 $22.94 $590 0 0 $0.00 $20.33 $20.43 $595 0 0 $0.00 $18.01 $18.11 $600 100 0 $1,589.00 $15.79 $15.89 $605 0 200 ($2,790.00) $13.95 $14.05 $610 100 0 $1,219.00 $12.09 $12.19 $615 0 0 $0.00 $10.60 $10.70


=B10*G10-C10*F10

Figure out how much cash is going out, in D10:D17


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$580 0 0 $0.00 $25.54 $25.64 605$ $585 0 0 $0.00 $22.84 $22.94 605$ $590 0 0 $0.00 $20.33 $20.43 605$ $595 0 0 $0.00 $18.01 $18.11 605$ $600 100 0 $1,589.00 $15.79 $15.89 605$ $605 0 200 ($2,790.00) $13.95 $14.05 605$ $610 100 0 $1,219.00 $12.09 $12.19 605$ $615 0 0 $0.00 $10.60 $10.70 605$


=$A$5

Cell A5 will be an assumption; the ending price of the option in one month. Put cell references to A5 into H10:H17.


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A B C D E F G H I J KCash Out Cash In






$580 0 0 $0.00 $25.54 $25.64 605$ $25.00$585 0 0 $0.00 $22.84 $22.94 605$ $20.00$590 0 0 $0.00 $20.33 $20.43 605$ $15.00$595 0 0 $0.00 $18.01 $18.11 605$ $10.00$600 100 0 $1,589.00 $15.79 $15.89 605$ $5.00$605 0 200 ($2,790.00) $13.95 $14.05 605$ $0.00$610 100 0 $1,219.00 $12.09 $12.19 605$ $0.00$615 0 0 $0.00 $10.60 $10.70 605$ $0.00


=MAX(0,H10-A10)

In I10:I17 enter a formula to calculate the payoff for options bought, as a function of the random ending price of the index.


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A B C D E F G H I J KCash Out Cash In






$580 0 0 $0.00 $25.54 $25.64 605$ $25.00 ($25.00)$585 0 0 $0.00 $22.84 $22.94 605$ $20.00 ($20.00)$590 0 0 $0.00 $20.33 $20.43 605$ $15.00 ($15.00)$595 0 0 $0.00 $18.01 $18.11 605$ $10.00 ($10.00)$600 100 0 $1,589.00 $15.79 $15.89 605$ $5.00 ($5.00)$605 0 200 ($2,790.00) $13.95 $14.05 605$ $0.00 $0.00$610 100 0 $1,219.00 $12.09 $12.19 605$ $0.00 $0.00$615 0 0 $0.00 $10.60 $10.70 605$ $0.00 $0.00


=MIN(0,A10-H10)

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.


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$580 0 0 $0.00 $25.54 $25.64 605$ $25.00 ($25.00)$585 0 0 $0.00 $22.84 $22.94 605$ $20.00 ($20.00)$590 0 0 $0.00 $20.33 $20.43 605$ $15.00 ($15.00)$595 0 0 $0.00 $18.01 $18.11 605$ $10.00 ($10.00)$600 100 0 $1,589.00 $15.79 $15.89 605$ $5.00 ($5.00)$605 0 200 ($2,790.00) $13.95 $14.05 605$ $0.00 $0.00$610 100 0 $1,219.00 $12.09 $12.19 605$ $0.00 $0.00$615 0 0 $0.00 $10.60 $10.70 605$ $0.00 $0.00

index (no calls) 165.29index (with calls) 165.29

=100000/A2=(100000-D3)/A3

In B19:B20, calculate how many shares of the index are being purchased.


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$580 0 0 $0.00 $0.00 $25.54 $25.64 611$ $31.00 ($31.00)$585 0 0 $0.00 $0.00 $22.84 $22.94 611$ $26.00 ($26.00)$590 0 0 $0.00 $0.00 $20.33 $20.43 611$ $21.00 ($21.00)$595 0 0 $0.00 $0.00 $18.01 $18.11 611$ $16.00 ($16.00)$600 100 0 $1,589.00 $1,100.00 $15.79 $15.89 611$ $11.00 ($11.00)$605 0 200 ($2,790.00) ($1,200.00) $13.95 $14.05 611$ $6.00 ($6.00)$610 100 0 $1,219.00 $100.00 $12.09 $12.19 611$ $1.00 ($1.00)$615 0 0 $0.00 $0.00 $10.60 $10.70 611$ $0.00 $0.00

index (no calls) 165.29index (with calls) 165.29

=SUMPRODUCT(B10:C10,I10:J10)

In E10:E17, calculate the amount of cash coming back in at the end of the month.


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A B C D E F G H ICash Out Cash In

605$ initial price 100,000$ 100,992$ $992 = Index Profit605$ mean = Options Profit30$ stdev Total = Profit with Index + Options



=B19*A2=B19*A5

=E2-D2

In D2:F2, calculate the P/L from the index.


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605$ initial price 100,000$ 100,992$ $992 = Index Profit605$ mean 18$ -$ ($18) = Options Profit30$ stdev Total = Profit with Index + Options



=SUM(D10:D17) =SUM(E10:E17)

=E3-D3

In D3:F3, calculate the P/L from the options.


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In D4:F4, calculate the total P/L.

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605$ initial price 100,000$ 100,992$ $992 = Index Profit605$ mean 18$ -$ ($18) = Options Profit30$ stdev Total 100,000$ 100,974$ $974 = Profit with Index + Options


(positive indicates butterfly strategy is better)=(B20*A2)+D3=(B20*A5)+E3

=E4-D4


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In F6 calculate the difference between the two strategies (with and without the options).

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605$ initial price 100,000$ 100,992$ $992 = Index Profit605$ mean 18$ -$ ($18) = Options Profit30$ stdev Total 100,000$ 100,974$ $974 = Profit with Index + Options

611$ end price($18) Difference

(positive indicates butterfly strategy is better)=F4-F2


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605$ initial price 100,000$ 99,339$ ($661) = Index Profit605$ mean 18$ 100$ $82 = Options Profit30$ stdev Total 100,000$ 99,421$ ($579) = Profit with Index + Options

601$ $82.12 Difference




$580 0 0 $0.00 $0.00 $25.54 $25.64 $601.00 $21.00 ($21.00)$585 0 0 $0.00 $0.00 $22.84 $22.94 $601.00 $16.00 ($16.00)$590 0 0 $0.00 $0.00 $20.33 $20.43 $601.00 $11.00 ($11.00)$595 0 0 $0.00 $0.00 $18.01 $18.11 $601.00 $6.00 ($6.00)$600 100 0 $1,589.00 $100.00 $15.79 $15.89 $601.00 $1.00 ($1.00)$605 0 200 ($2,790.00) $0.00 $13.95 $14.05 $601.00 $0.00 $0.00$610 100 0 $1,219.00 $0.00 $12.09 $12.19 $601.00 $0.00 $0.00$615 0 0 $0.00 $0.00 $10.60 $10.70 $601.00 $0.00 $0.00

index (no calls) 165.29 0index (with calls) 165.26 0

=B17*G17-C17*F17 =SUMPRODUCT(B17:C17,I17:J17) =$A$5 =MAX(0,H17-A17) =MIN(0,A17-H17)=100000/A2 =(100000-D3)/A3

=F4-F2


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232425262728293031323334353637383940414243444546474849505152535455

A B CDifference

$82590591592593594595596597598599600601602603604605606607608609610611612613614615616617618619620

=F6

An old Excel trick:DataTable


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Select A24:B55, then Data Table


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232425262728293031323334353637383940414243444546474849505152535455

A BDifference

$82590 (17.55)$ 591 (17.58)$ 592 (17.61)$ 593 (17.64)$ 594 (17.67)$ 595 (17.70)$ 596 (17.73)$ 597 (17.76)$ 598 (17.79)$ 599 (17.82)$ 600 (17.85)$ 601 82.12$ 602 182.09$ 603 282.06$ 604 382.03$ 605 482.00$ 606 381.97$ 607 281.94$ 608 181.91$ 609 81.88$ 610 (18.15)$ 611 (18.18)$ 612 (18.21)$ 613 (18.24)$ 614 (18.27)$ 615 (18.30)$ 616 (18.33)$ 617 (18.36)$ 618 (18.39)$ 619 (18.42)$ 620 (18.45)$


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Benefits from the Butterfly Strategy

$(100)

$-

$100

$200

$300

$400

$500

$600

$590 $595 $600 $605 $610 $615 $620

Ending Index Price

But

terf

ly B

enef

it


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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.


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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 700,337,2$000,000,1*4940.71000,000,1*4845.1

1000,000,1*6533.01


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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.


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(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?


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POUNDS/$US FRANCS/US$ KRONER/US$ EURO/US$Jan 0.6146 1.5808 7.9640 0.9847Feb 0.6192 1.6540 8.2770 1.0276Mar 0.6310 1.6568 8.3115 1.0309Apr 0.6258 1.6587 8.4640 1.0460May 0.6428 1.7135 8.9050 1.0965Jun 0.6705 1.6878 8.9400 1.0745Jul 0.6607 1.6323 8.5880 1.0498

Aug 0.6670 1.6758 8.8850 1.0837Sep 0.6847 1.7230 9.0108 1.1120Oct 0.6814 1.7322 9.1269 1.1356Nov 0.6919 1.7765 9.2020 1.1650Dec 0.6957 1.7285 9.2475 1.1409Jan 0.6677 1.6075 8.7600 1.0565Feb 0.6768 1.6330 8.7550 1.0656Mar 0.6871 1.6557 8.8650 1.0763Apr 0.7042 1.7317 9.1610 1.1333May 0.6974 1.7255 9.0540 1.1189Jun 0.7062 1.7992 9.4538 1.1832Jul 0.7058 1.8003 9.4030 1.1827

Aug 0.6978 1.7158 9.0980 1.1373Sep 0.6923 1.7075 8.9380 1.1276Oct 0.6764 1.6196 8.8244 1.0918Nov 0.6840 1.6295 8.8200 1.1057Dec 0.7034 1.6550 8.9790 1.1240Jan 0.6920 1.6424 8.8775 1.1073Feb 0.7063 1.7179 9.1050 1.1610Mar 0.7047 1.7060 8.8875 1.1558Apr 0.6941 1.6607 8.7450 1.1356May 0.6839 1.6010 8.3500 1.1035Jun 0.6705 1.6878 8.9400 1.0745Jul 0.6533 1.4845 7.4940 1.0108


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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:

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125Ja

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Feb

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POUNDS/$USFRANCS/US$KRONER/US$EURO/US$


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A B C D E F G H IJan $US/POUND FRANCS/US$ KRONER/US$ EURO/US$Feb 0.7430% 4.6306% 3.9302% 4.3572%Mar 1.8992% 0.1693% 0.4168% 0.3196%Apr -0.8198% 0.1147% 1.8348% 1.4644%May 2.7124% 3.3038% 5.2103% 4.8246%Jun 4.3111% -1.4999% 0.3930% -2.0092%Jul -1.4536% -3.2883% -3.9374% -2.2990%Aug 0.9538% 2.6650% 3.4583% 3.2293%Sep 2.6498% 2.8166% 1.4159% 2.6131%Oct -0.4770% 0.5340% 1.2885% 2.1236%Nov 1.5430% 2.5574% 0.8228% 2.5862%Dec 0.5496% -2.7019% 0.4945% -2.0650%Jan -4.0329% -7.0003% -5.2717% -7.3957%Feb 1.3672% 1.5863% -0.0571% 0.8632%Mar 1.5255% 1.3901% 1.2564% 1.0010%Apr 2.4859% 4.5902% 3.3390% 5.2924%May -0.9763% -0.3580% -1.1680% -1.2644%Jun 1.2712% 4.2712% 4.4157% 5.7383%Jul -0.0565% 0.0611% -0.5374% -0.0355%Aug -1.1305% -4.6937% -3.2436% -3.8440%Sep -0.7893% -0.4837% -1.7586% -0.8457%Oct -2.3064% -5.1479% -1.2710% -3.1772%Nov 1.1286% 0.6113% -0.0499% 1.2716%Dec 2.8346% 1.5649% 1.8027% 1.6522%Jan -1.6193% -0.7613% -1.1304% -1.4838%Feb 2.0695% 4.5969% 2.5627% 4.8531%Mar -0.2255% -0.6927% -2.3888% -0.4508%

=(data!E3-data!E2)/data!E2

Converting prices into returns:


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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) :

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A B C D E F G H IPOUNDS/$USFRANCS/US$ KRONER/US$ EURO/US$

mean 0.1578% 0.0243% 0.0841% 0.0571%stdev 2.0683% 3.4421% 3.4732% 3.4733%t-stat 0.4180 0.0387 0.1326 0.0901p-value 0.6789 0.9694 0.8954 0.9288

=AVERAGE(E9:E38)=STDEV(E9:E38)=(E2)/(E3/SQRT(COUNT(E9:E38)))=TDIST(ABS(E4),COUNT(E9:E38),2)


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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


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Distribution fitting: Checking to see which Crystal Ball distribution best fits the data (in this case the British pound’s return against the dollar).


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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.


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A B C D E F G H IFrancS Kroner Pounds

Receivable in one month (millions) 1.0000 1.0000 1.0000 Pound Franc KronerCurrent rate in US$ 0.6736 0.1334 1.5307 Pound 1Volatility (stdev of yield) 0.0344 0.0347 0.0207 Franc 0.6661 1

Kroner 0.6301 0.8909 1Euro 0.7527 0.8754 0.7883

Return Price Return Price Return PriceJan 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307Feb 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307Mar 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307Apr 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307

May 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307Jun 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307Jul 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307

Aug 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307Sep 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307Oct 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307Nov 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307Dec 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307

Revenue in one year ($million) 0.6736 0.1334 1.5307Total 2.3377

PoundKronerFrancS

Correlations

=B3*(1+B10)=C10*(1+B11)

=C23+F23+I23=F21*C2


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We start by creating the “January” cell for each currency. The Swiss franc:


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The Norwegian kroner:


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The British pound:


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Kroner 0.6301 0.8909 1Euro 0.7527 0.8754 0.7883


May 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307Jun 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307Jul 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307

PoundKronerFrancS

Correlations


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For more than a few correlated green cells, it’s more efficient to use the matrix view.

You can specify bivariate correlations in the Define Assumption window.


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Back inside the Swiss franc (after defining two other green cells):


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Kroner 0.6301 0.8909 1Euro 0.7527 0.8754 0.7883


May 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307Jun 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307Jul 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307

Aug 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307Sep 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307Oct 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307Nov 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307Dec 0.0000 0.6736 0.0000 0.1334 0.0000 1.5307

Revenue in one year ($million) 0.6736 0.1334 1.5307

PoundKronerFrancS

Correlations


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VaR approach: Click the right grabber and then enter 95 in the certainty box.

2.3377 – 2.0412 = 0.2965 ($296,500)


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“Round dollar amount” approach:

2.3377 – 0.2500 = 2.0877

Chances of losing $250k or more = 1 – 0.9146 = 0.0854


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W e update the model to include the return on the Euro versus the dollar, including the appropriate correlations:

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A B C D E F G H I J K L MFrancS Kroner Pounds Euro

Receivable in one m onth (m illions) 1.0000 1.0000 1.0000 0.0000 Pound Franc Kroner Euro Put O ptionsCurrent rate in US$ 0.6736 0.1334 1.5307 0.9893 Pound 1 Units Purchased 1.3Volatility (stdev of yield) 0.0344 0.0347 0.0207 0.0347 Franc 0.6661 1 Strike 0.98

Kroner 0.6301 0.8909 1 Cost 0.0432Euro 0.7527 0.8754 0.7883 Payout 0

Return Price Return Price Return Price Return PriceJan -0.0267 0.6556 -0.0084 0.1323 -0.0147 1.5082 0.0008 1.000797Feb 0.0681 0.7003 0.0717 0.1418 0.0556 1.5921 0.0696 1.070441Mar -0.0035 0.6978 0.0522 0.1492 -0.0055 1.5834 -0.0155 1.053869Apr 0.0441 0.7286 0.0352 0.1544 0.0036 1.5891 0.0387 1.094608

May 0.0239 0.746 -0.0254 0.1505 0.0218 1.6237 0.0132 1.109103Jun -0.0119 0.7371 -0.0327 0.1456 -0.0053 1.6151 0.0119 1.122303Jul -0.0173 0.7243 0.0120 0.1473 0.0104 1.6318 0.0213 1.146221

Aug -0.0176 0.7116 -0.0128 0.1454 -0.0042 1.625 -0.0238 1.118907Sep 0.0260 0.7301 0.0389 0.1511 0.0242 1.6643 0.0525 1.177609O ct -0.0837 0.669 -0.1016 0.1357 -0.0222 1.6274 -0.0678 1.097722Nov 0.0129 0.6776 -0.0034 0.1353 -0.0166 1.6005 0.0083 1.106847Dec 0.0008 0.6781 0.0034 0.1357 0.0087 1.6143 0.0249 1.134432

Revenue in one year ($m illion) 0.6781 0.1357 1.6143 -0.0562Total 2.3720

FrancS

Correlations

EuroPoundKroner


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Here’s one way to model the cash flow associated with the Euro put options:

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K L M N O

Euro Put OptionsUnits Purchased 1.3Strike 0.98Cost 0.0432Payout 0

Return Price0.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 10.0000 1

-0.0562

Euro

=L20*(1+K21)

=M3*(M6-M5)

=MAX(0,M4-L21)


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+Smaller standard deviation+Truncated lower tail−Lower expected value


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+VaR is $196,800 (better than $296,500)


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+Chance of $250k loss 0.0111 (better than 0.0854)


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0.000

0.500

1.000

1.500

2.000

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

Expe

cted

Rev

enue

($m

illio

ns)

Std Deviation ($millions)

Clifford & Kearns

Total Unhedged Revenue

Total Hedged Revenue


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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.


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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 goodness-of-fit test.


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The Parametric Approach1. 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.


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A B C D E F G HPortfolio Weights 0.75 0.25

S&P 500 T-Bill Portfolio Returns Start End-3.13% 0.56% -2.20% 100.00$ 97.80$ -3.13% 0.56% -2.20% 97.80$ 95.64$ -3.13% 0.56% -2.20% 95.64$ 93.53$

Historical Data S&P 500 T-BillMonth 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%

=SUMPRODUCT($B$1:$C$1,B6:C6)

=F4*(1+D4)


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83


84


85


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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%


=F4*(1+D4)


87


88


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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%


=F4*(1+D4)


90


91


92


93


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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%


=F4*(1+D4)


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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%


=F4*(1+D4)


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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).


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Back to our example. Start the model with a spreadsheet similar to the parametric one. Notice the integers in column A.

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A B C D E F GPortfolio Weights 0.75 0.25

Random Months S&P 500 T-Bill Portfolio Returns Start End5 -5.47% 0.53% -3.97% 100.00$ 96.03$ 5 -5.47% 0.53% -3.97% 96.03$ 92.21$ 4 -8.89% 0.50% -6.54% 92.21$ 86.18$


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%


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Use the VLOOKUP function in B4:C6 to “look up” the paired scenario corresponding to the integer in A4:A6.

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Random Months S&P 500 T-Bill Portfolio Returns Start End5 -5.47% 0.53% -3.97% 100.00$ 96.03$ 5 -5.47% 0.53% -3.97% 96.03$ 92.21$ 4 -8.89% 0.50% -6.54% 92.21$ 86.18$


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%

=VLOOKUP(A6,$A$10:$C$24,2,0)

=VLOOKUP(A6,$A$10:$C$24,3,0)


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100

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Random Months S&P 500 T-Bill Portfolio Returns Start End2 5.86% 0.62% 4.55% 100.00$ 104.55$ 1 -7.43% 0.60% -5.43% 104.55$ 98.88$ 8 5.09% 0.53% 3.95% 98.88$ 102.78$


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%

=VLOOKUP(A6,$A$10:$C$24,2,0)

=VLOOKUP(A6,$A$10:$C$24,3,0)


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SummaryFinance Simulation Models• Forecasting

– Retirement Planning– Butterfly Strategy

• Risk Management– Introduction to VaR– Currency Risk

• Using Historical Data in Simulations– Parametric Approach– Resampling Approach

session 9a

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