Decision Models -- Prof. Juran2 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. Juran3 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.
5 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. Juran12 It looks like Amanda has about a 48% chance of meeting her goal of $1 million in 30 years.
Decision Models -- Prof. Juran13 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. Juran14 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.)
Decision Models -- Prof. Juran15 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. Juran37 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. Juran38 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).
Decision Models -- Prof. Juran39 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. Juran40 (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. Juran43 Converting prices into returns:
Decision Models -- Prof. Juran44 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) :
Decision Models -- Prof. Juran51 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. Juran79 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.
Decision Models -- Prof. Juran80 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. Juran96 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. Juran97 Back to our example. Start the model with a spreadsheet similar to the parametric one. Notice the integers in column A.
Decision Models -- Prof. Juran98 Use the VLOOKUP function in B4:C6 to “look up” the paired scenario corresponding to the integer in A4:A6.