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MONTE CARLO SIMULATION IN FINANCE

Jamal Munshi, Sonoma State Univesity, November 2002
All rights reserved

ABSTRACT

Monte Carlo simulation is used to help students visualize concepts in Finance that are not obvious when traditional methods of instruction are used. Two models are described. The first model displays the distribution of the NPV of a project when sales projections are not known with certainty. The second model uses simulation rather than the Black Scholes stochastic model for the valuation of call and put options on underlying assets whose value in the future, when the option may be exercised, is uncertain.

BACKGROUND AND MOTIVATION

A key element in Finance is the notion of risk. It measures the uncertainty in projecting values that financial variables such as returns on investment may assume in the future. Risk is operationalized in financial theory as the standard deviation computed either from historical data or from subjective probabilities. Using the normality assumption and stochastic calculus, we then proceed to derive the Gaussian parameters of the decisional variables from the ones that are projected. For example, project unit sales and price and derive NPV (net present value); and use the NPV distribution to compute the probability of negative NPV in order to make the investment decision.

The problem with such an approach is two-fold. First, the procedure usually requires the use of crippling assumptions that tend to make the results unrealistic and a hard sell to the sharper students. And second, the equations themselves are complex and obfuscating; far from being an educational tool they reveal nothing about the process and leave most students with the notion that the concepts being taught are too complex to understand.

These concerns may be alleviated by using simulation rather than stochastic calculus to teach concepts of risk and return in Finance. In this paper we offer two examples to show how simple simulation models may be used to teach Finance. In the first example a simulation model is built using the DataDesk statistical package to describe a capital budgeting problem with uncertain cash flows. In the second example, the DataDesk program is used to develop a model for the valuation of contingent claims such as stock options. In each example, we show that the simulation models are easier for students to grasp because the components of the model are kept simple.

Other areas of Finance where simulation may be used include portfolio theory, the capital asset pricing model, tests of the efficient market hypothesis, and studies of market microstructure.

THE CAPITAL BUDGETING PROBLEM

Capital budgeting plays a crucial role in financial economics. It deals with the investment decision and it uses the usual axiom in Finance that the value of an investment is the present value of future cash flows that the asset is expected to generate. A project is therefore evaluated by subtracting the required investment in capital (both productive assets and working capital) required today from the present value of the expected cash flows that the project will generate. This difference, called the net present value or NPV is then used to make the investment decision. If the NPV is positive, then invest else do not invest . The time horizon for the decision is an assumed finite value at the end of which all remaining assets are expected to be liquidated at a projected liquidation value (Pinches 1992).

Some of the cash flows generated may have to be set aside if additional working capital is needed for operations in the following year. The remaining cash flows, called net cash flows or NCF, are assumed to be re-invested so that during the life of the project, they earn returns at a known re-investment rate r. These re-investment rates are used to compute the future value of the NCF stream at the end the life of the project. To compute the present value of the NCF stream, this future value must be discounted back to the present using a discount rate, k, that is appropriate for the perceived riskiness of the project. In general k will be different from and higher than the re-investment rate (Mao 1969). The present value is then subtracted from the investment capital required to compute the NPV. In making the above computations all cash flows are assumed to be known with certainty.

A numerical example may clarify the salient points. Suppose that an investment opportunity exists that will require an initial investment in fixed assets of $200 which at the end of the 5-year project life will have a salvage value of $90. It is expected to generate sales of $100 the first year and thereafter, the sales volume is expected to grow at an annual rate of 15%. Since it is typical for revenues to dip in the last year (in preparation for shutdown) we will assume that revenue in year 5 will be the same as that in year 2.

The following operating data are projected. Variable costs are 60% of sales. Fixed costs are $10 per year not including depreciation. Working capital requirements are estimated to be 14% of sales. We expect to be able to re-invest operating cash flows at our corporate cost of capital of 6%. This project represents a risky venture and we will use a required return or discount rate of 8% to evaluate its wealth effect on our shareholders. Should this investment be made? The investment required is $214 ($200 plus 14% of $100 as working capital). The future value of the NCF stream after year 5 is $317 if each NCF is re-invested at 6%. The present value discounted at 8% is $214.60. Since the NPV is positive (14 cents) the decision is to invest.

UNCERTAINTY IN SALES PROJECTION

The essential concepts of capital budgeting are first introduced to students using the scenario described above. We then extend the generality and usefulness of the model by allowing the projected sales vector to have a degree of uncertainty. The uncertain sales projection is modeled with a Gaussian distributions having the expected value of sales as its mean and a standard deviation that is in proportion to the degree of uncertainty. It is assumed that uncertainty in sales projection is the only source of risk, that is other factors such as rates of return, the cost structure, and salvage value are known with certainty. The problem can then be stated as follows: given the distributions of the sales projections and a fixed set of operating and financial parameters what is the distribution of the NPV?

To derive a strictly stochastic algebraic solution we may use the relations mean (y+a) = mean(y) + a mean (ay) = a*mean(y) variance (y+a) = variance(y) variance (ay) = a2*variance(y).

The so called "portfolio" equation may be used to compute the variance of a sum of stochastic variables as portfolio variance = SUMi(SUMj(wiwjsigmaij)) where sigmaij is the pairwise covariances between the present value of the cash flow (PV) of the ith year and the PV of the jth year and is equal to the variance of the PV of the ith year when j=i. The terms wi, wj represent the relative weights of the PV of the ith and jth years in the portfolio. The term `SUMi indicates summation over all values of i and similarly `SUMj is summation over all values of j (Pinches 1992) .

It is evident that we could slog through this algebra and come up with an algebraic solution especially if the number of years in the project time horizon is kept at a manageable number. The number of terms in the portfolio equation increases with the square of the number of cash flows to be combined. For n=5 years there will be 5 variances and (25-5)/2 or 10 covariances with a total of 15 terms. But for n=10 years there will be 10 variances and (100-10)/2 or 45 covariances for a total of 50 terms. This problem is sometimes avoided by making the assumption that the sales distributions are independent. This simplifying, and perhaps crippling, assumption gets rid of the covariance terms leaving us only with n variance terms where n is the number of years in the time horizon of the project evaluation.

The problem is that by now we are so deep into computational details that we have lost sight of conceptual underpinnings of capital budgeting under uncertainty. Even more egregious, we have lost the attention and the interest of most undergraduate business students. Not only have we failed to convey to them the simple concepts involved, we have actually succeeded in convincing them that these concepts are difficult and possibly beyond their ability to understand.

A MONTE CARLO SIMULATION OF THE NPV PROBLEM

A Monte Carlo simulation of such problems differs in one respect from simulation of dynamic processes such as bank queues and factory operations. In dynamic simulation, the passage of each tick of time causes events to occur some of which may be generated by stochastic processes. These events "flow" through the simulation interacting according to the model specifications. Monte Carlo simulations are not dynamic but static and each pick from the random number generator is not an event in flowing time but a trial of many trials that will be used to describe the distribution of the target variable (Morris 1987).

To set up the 5-year NPV problem we need generators that will produce numbers from a given distribution for all five years of sales during each trial. Once these 5 numbers are available, we may compute the NPV for that trial using the simplified procedure used when sales are known with certainty. This procedure is repeated as many times as desired until a sufficiently large sample of NPV values is obtained. It is then possible to observe the distributional properties of NPV under conditions of uncertainty.

A convenient tool for setting up such a simulation is an interactive statistical program that has a variety of stochastic number generators and dynamic and interactive computation and graphical display capability. Using these criteria, the DataDesk program from Data Description inc. for the Apple Macintosh computer was selected as the appropriate simulation tool.

DESCRIPTION OF THE DATADESK MODEL

The numerical example described above is modeled to demonstrate the procedure and its effectiveness. For this demonstration, we have arbitrarily used a sample size of 100 trials, a time horizon of n=5 years, Gaussian sales distributions, and fixed operating and financial parameters. The operating and financial parameters may be made stochastic variables and distributions other than the Gaussian may be specified. For each of five years we generate 100 standard normal numbers (drawn from a Normal population with a mean of zero and a standard deviation of 1). Sales for each year is then computed as its expected value plus a risk multiplier times the standard normal values. Since the standard normal numbers were picked independently, an additional mechanism is used to induce degrees of dependence or correlation among the sales projections.

DataDesk has a feature called "slider variables" that allows users to set up parameters that may be dynamically controlled by dragging the mouse across a scale (Velleman 1993). In the model, all operating and financial parameters are slider controlled as are those that impart uncertainty to sales projections and the degree of dependence of sales in any given year to that realized in the previous year.

RESULTS OF THE SIMULATION The distribution, mean, standard deviation, and 95% confidence interval of the NPV are shown below for various degrees of uncertainty and dependence. The effect of some of the other parameters is also shown.

The RISK variable is used to set the degree of uncertainty in sales projections and the CORR variable attenuates the dependence between successive years of sales.The GROWTH parameter sets the percentage rate at which sales are expected to grow for the first n-1 years. Revenue in the nth year are expected to be the same as that in year-2. The WCS parameter sets working capital management policy and is equal to the percentage of sales that is to be held as non-productive working capital. The re-investment and discount rates are shown below as `r and `k respectively.

In the actual model, students are able to watch the histograms and computed sample statistics change dynamically as they drag the slider variables to different locations on the scale. Within several minutes they develop a `feel for the effect of these parameters. More important, the concepts used to build these models are so simple that with a little help they are able to modify the model and build their own models to introduce additional levels of complexity.

Figures 1 through 4 show some of the ways that data may be displayed using DataDesk. Figure 1 shows sample statistics and confidence intervals of NPV under various conditions. Figures 2 and 3 are box-lots and are useful for viewing a the series of uncertain sales and net cash flows. The uncertainty in sales projection is set as an increasing function of time.

FIGURE 1:SOME VALUES OF NPV COMPUTED

  • RISK=15,CORR=75%,GROWTH=15%,WCS=14%
  • Mean NPV 0.58522615, StdDev 16.961188
  • With 95.00% Confidence NPV within -2.7802415 and 3.9506938

Impact of working capital management

  • RISK=15,CORR=75%,GROWTH=15%,WCS=7%
  • Mean NPV 3.1756781, StdDev 17.239772
  • With 95.00% Confidence NPV within -0.24506675 and 6.5964230

Impact of reduced dependency in sales

  • RISK=15,CORR=25%,GROWTH=15%,WCS=14%
  • Mean NPV 1.9813802, StdDev 15.277067
  • With 95.00% Confidence NPV within -1.0499214 and 5.0126818

Impact of increased uncertainty in sales projections

  • RISK=25,CORR=75%,GROWTH=15%,WCS=14%
  • Mean NPV -0.35074655, StdDev 28.268646
  • With 95.00% Confidence NPV within -5.9598592 and 5.2583661

Impact of growth rate

  • RISK=15,CORR=75%,GROWTH=20%,WCS=14%
  • Mean NPV 11.993128, StdDev 17.778327
  • With 95.00% Confidence NPV within 8.4655219 and 15.520733

FIGURE 2: BOX PLOT OF NCF STREAM

FIGURE 3: BOX PLOT OF SALES PROJECTIONS

FIGURE 4: HISTOGRAM OF NPV

STOCK OPTION PRICING

A prominent component of Financial theory is the pricing of contingent claims. These include the option to buy (`call option) or sell (`put option) underlying assets such as stocks, at a fixed price called the exercise price. The option is good only for a certain time after which it expires. The question is: what is the value of the option?

During the time that the option is held the price of the underlying asset is assumed to vary within a stationary Gaussian distribution with given mean (Expected Value) and standard deviation. The holder of the call option benefits fully when the price is higher than the exercise price but limits her losses to the price of the option by simply refusing to exercise the option to purchase the underlying asset when the price is below the exercise price.

The call option derives its value from this asymmetry and its value can be derived by using the properties of the Gaussian distribution and by making some simplifying assumption (Black and Scholes 1973). The algebraic result, the so called "Black Scholes Option Valuation Equation" can be found in any Finance textbook (Pinches 1992) and is presented below:

Vc=PoN(d1)-XN(d2)/ert where Vc = value of the call option Po = the expected value of the stock X = the exercise price of the option t = time to expiration r = the riskless rate of return e = the exponential function N(d1) = Gaussian probability of realizing a value z Consider a numerical example (Pinches 1992 page 279). Po=100, X=90, t=6 months (or .5 years), r=10%, s=0.28

step 1: compute d1, d2, N(d1), and N(d2) d1=ln(100/90)+(.1+.5*(.28)2)*.5/{.28*sqrt(.5) = 0.8837 d2=0.8837-.28*sqrt(.5) =0.6857 N(d1)=0.8115 (with interpolation) n(d2)=0.7535 (with interpolation)

step 2: compute Vc Vc=81.15 - (90/e0.1*.5)*0.7535 =16.64

Using this equation to teach the mechanism and fundamentals of option valuation is a daunting task. First, the derivation of the equation requires math skills that business students cannot be expected to have. Second, the equation itself reveals nothing about the process and provides no insight. And third, the solution is very involved, requires the use of tables, and is prone to errors. Not surprisingly many textbooks and teachers of beginning Finance simply skip option valuation altogether.

Yet the valuation mechanism itself is conceptually and graphically simple to explain - the holder of the call option deriving the value of the up-side variance while holding the down-side loss to the cost of the option. Using Monte Carlo simulation, it is just as simple to model. The model is not only easier to set up and use and easier to understand, it actually provides more information than the equation. For example, we can view the actual distribution of the call option price and compute all its parameters instead of just its mean.

In DataDesk we begin by first setting up a z-distribution of a sufficient sample size (I chose 200 for this demonstration) and then constructing a Price distribution by adding the desired multiple of the z-distribution to the expected value of the price. We use the DataDesk `slider mechanism to set up parameters that the user may change dynamically to set the expected value of the price, the degree of uncertainty (imparting a higher standard deviation to the price distribution), the call and put strike prices, the time to expiration, and the riskless rate.

The value of the call is then the value of the assets price distribution to the right of the call exercise price adjusted for the time value of money using the riskless rate. Similarly, the value of the put is the value of the assets price distribution to the left of the put exercise price also time adjusted. Using the model, students are able to understand the concepts used in the valuation of contingent claims and to develop an intuition for the behavior of this value. For example, with the model it is easy to see why a call option to buy a stock at an exercise price equal to the expected value of the stock would have a positive non-zero value as long as there is uncertainty in the asset price and why this value will increase as the riskiness of the asset increases.

The numerical problem described above cannot be exactly reproduced since the model cannot relate prices to returns. Some results from a similar problem is shown below in Figures 5, 6, 7, and 8. Figures 5,6, and 7 are histograms of the price of the underlying asset, the call option, and the put option. Figure 8 shows the mean and standard deviation of each of these variables. The standard deviation of the option, not computable using the Black-Scholes model, is readily computed in a simulation. The standard deviation is large and indicative of the riskiness of the options market.

FIGURE 5: HISTOGRAM OF THE PRICE OF THE ASSET

FIGURE 6: HISTOGRAM OF THE CALL OPTION VALUE

FIGURE 7: HISTOGRAM OF THE PUT OPTION VALUE


FIGURE 8: TABLE OF SUMMARY STATISTICS

Summary statistic Mean Stdev
Asset Price 100 28
Call option value (X=90) 16.33 18.17
Put option value (X=110) 15.75 19.25


CONCLUSION

Not long ago books of logarithm tables adorned most engineering offices. Lookup tables, like nomographs, are computational devices from another era. Our ancestors were forced to resort to such cleverness because they did not have the sort of computational machinery we do. It is becoming apparent that at least in some cases, the quest to reach a deterministic algebraic equation may belong in the same category of computational thinking as log tables. They make sense in a world without computers but not necessarily in ours.

REFERENCES

Black, Fischer and Myron Scholes 1973, The Pricing of Options and Corporate Liabilities, Journal of Political Economy v83 May-June, 1973, pp 637-654

Mao, James 1969, Quantitative Analysis of Financial Decisions, The Macmillan Company, London, 1969

Morris, James R. 1987, The Dow Jones-Irwin Guide to Financial Modeling pp 275-292, Dow Jones-Irwin, Homewood, IL, 1987

Pinches, George E. 1992, Essentials of Financial Management, Harper Collins Publishers 1992 pp 280-286

Velleman, Paul 1993, Data Analysis With DataDesk, WH Freeman NY, 1993

ENDNOTES

DataDesk is a trademark of Data Description, Inc. Macintosh is a trademark of Apple Computer, Inc.