## Example 513 Potential Gains from Market Timing A Monte Carlo Simulation

Having discussed the plan of the Chua et al. study and illustrated the method for a hypothetical Monte Carlo simulation with four trials, we conclude our presentation of the study.

The hypothetical simulation in Example 5-12 had four trials, far too few to reach statistically precise conclusions. The simulation of Chua et al. incorporated 10,000 trials. Chua et al. specified bull- and bear-market prediction skill levels of 50, 60, 70, 80,90, and 100 percent. Table 5-9 presents a very small excerpt from their simulation results for the no transaction costs case (transaction costs were also examined). Reading across the row, the timer with 60 percent bull market and 80 percent bear market forecasting accuracy had a mean annual gain from market timing of -1.12 percent per year. On average, the buy-and-hold investor out-earned this skillful timer by 1.12 percentage points. There was substantial variability in gains across the simulation trials, however: The standard deviation of the gain was 14.77 percent, so in many trials (but not on average) the gain was positive. Row 3 (win/ loss) is the ratio of profitable switches between stocks and T-bills to unprofitable switches. This ratio was a favorable 1.2070 for the 60-80 timer. (When transaction costs were considered, however, fewer switches are profitable: The win-loss ratio was 0.5832 for the 60-80 timer.)

TABLE 5-9 Gains from Stock Market Timing (No Transaction Costs)

Bear Market Accuracy (%) Bull Market -

TABLE 5-9 Gains from Stock Market Timing (No Transaction Costs)

Bear Market Accuracy (%) Bull Market -

Accuracy (%) |
50 |
60 |
70 |
80 |
90 |
100 | |

60 |
Mean (%) |
-2.50 |
-1.99 |
-1.57 |
-1.12 |
-0.68 |
-0.22 |

S.D. (%) |
13.65 |
14.11 |
14.45 |
14.77 |
15.08 |
15.42 | |

Win/Loss |
0.7418 |
0.9062 |
1.0503 |
1.2070 |
1.3496 |
1.4986 |

Source: Chua, Woodward, and To (1987), Table II (excerpt)

Source: Chua, Woodward, and To (1987), Table II (excerpt)

The authors concluded that the cost of not being invested in the market during bull market years is high. Because a buy-and-hold investor never misses a bull market year, she has 100 percent forecast accuracy for bull markets (at the cost of 0 percent accuracy for bear markets). Given their definitions and assumptions, the authors also concluded that successful market timing requires a minimum accuracy of 80 percent in forecasting both bull and bear markets. Market timing is a continuing area of interest and study, and other perspectives exist. However, this example illustrates how Monte Carlo simulation is used to address important investment issues.

The analyst chooses the probability distributions in Monte Carlo simulation. By contrast, historical simulation samples from a historical record of returns (or other underlying variables) to simulate a process. The concept underlying historical simulation (also called back simulation) is that the historical record provides the most direct evidence on distributions (and that the past applies to the future). For example, refer back to Step 2 in the outline of Monte Carlo simulation above and suppose the time increment is one day. Further, suppose we base the simulation on the record of daily stock returns over the last five years. In one type of historical simulation, we randomly draw K returns from that record to generate one simulation trial. We put back the observations into the sample, and in the next trial we again randomly sample with replacement. The simulation results directly reflect frequencies in the data. A drawback of this approach is that any risk not represented in the time period selected (for example, a stock market crash) will not be reflected in the simulation. Compared with Monte Carlo simulation, historical simulation does not lend itself to "what if" analyses. Nevertheless, historic simulation is an established alternative simulation methodology.

Monte Carlo simulation is a complement to analytical methods. It provides only statistical estimates, not exact results. Analytical methods, where available, provide more insight into cause-and-effect relationships. For example, the Black-Scholes-Merton option pricing model for the value of a European call option is an analytical method, expressed as a formula. It is a much more efficient method for valuing such a call than is Monte Carlo simulation. As an analytical expression, the Black-Scholes-Merton model permits the analyst to quickly gauge the sensitivity of call value to changes in current stock price and the other variables that determine call value. In contrast, Monte Carlo simulations do not directly provide such precise insights. However, only some types of options can be priced with analytical expressions. As financial product innovations proceed, the field of applications for Monte Carlo simulation continues to grow.

5 SUMMARY

In this chapter, we have presented the most frequently used probability distributions in investment analysis and the Monte Carlo simulation.

• A probability distribution specifies the probabilities of the possible outcomes of a random variable.

• The two basic types of random variables are discrete random variables and continuous random variables. Discrete random variables take on at most a countable number of possible outcomes that we can list as xh x2, In contrast, we cannot describe the possible outcomes of a continuous random variable Z with a list z\, z2,... because the outcome (zi + z2)I2, not in the list, would always be possible.

• The probability function specifies the probability that the random variable will take on a specific value. The probability function is denoted p(x) for a discrete random variable and fix) for a continuous random variable. For any probability function p(x), 0 < p(x) < 1, and the sum of p(x) over all values of X equals 1.

• The cumulative distribution function, denoted F(x) for both continuous and discrete random variables, gives the probability that the random variable is less than or equal tox.

• The discrete uniform and the continuous uniform distributions are the distributions of equally likely outcomes.

• The binomial random variable is defined as the number of successes in n Bernoulli trials, where the probability of success, p, is constant for all trials and the trials are independent. A Bernoulli trial is an experiment with two outcomes, which can represent success or failure, an up move or a down move, or another binary (two-fold) outcome.

• A binomial random variable has an expected value or mean equal to np and variance equal to np( 1 - p).

• A binomial tree is the graphical representation of a model of asset price dynamics in which, at each period, the asset moves up with probability p or down with probability (1 - p). The binomial tree is a flexible method for modeling asset price movement and is widely used in pricing options.

• The normal distribution is a continuous symmetric probability distribution that is completely described by two parameters: its mean, p., and its variance, a2.

• A univariate distribution specifies the probabilities for a single random variable. A multivariate distribution specifies the probabilities for a group of related random variables.

• To specify the normal distribution for a portfolio when its component securities are normally distributed, we need the means, standard deviations, and all the distinct pairwise correlations of the securities. When we have those statistics, we have also specified a multivariate normal distribution for the securities.

• For a normal random variable, approximately 68 percent of all possible outcomes are within a one standard deviation interval about the mean, approximately 95 percent are within a two standard deviation interval about the mean, and approximately 99 percent are within a three standard deviation interval about the mean.

• A normal random variable, X, is standardized using the expression Z = (X - p)/cr, where p, and a are the mean and standard deviation of X. Generally, we use the sample mean X as an estimate of p and the sample standard deviation s as an estimate of <t in this expression.

• The standard normal random variable, denoted Z, has a mean equal to 0 and variance equal to 1. All questions about any normal random variable can be answered by referring to the cumulative distribution function of a standard normal random variable, denoted N(x) or N(z).

• Shortfall risk is the risk that portfolio value will fall below some minimum acceptable level over some time horizon.

• Roy's safety-first criterion, addressing shortfall risk, asserts that the optimal portfolio is the one that minimizes the probability that portfolio return falls below a threshold level. According to Roy's safety-first criterion, if returns are normally distributed, the safety-first optimal portfolio P is the one that maximizes the quantity [E(RP) - RL]/aP, where RL is the minimum acceptable level of return.

• A random variable follows a lognormal distribution if the natural logarithm of the random variable is normally distributed. The lognormal distribution is defined in terms of the mean and variance of its associated normal distribution. The lognormal distribution is bounded below by 0 and skewed to the right (it has a long right tail).

• The lognormal distribution is frequently used to model the probability distribution of asset prices because it is bounded below by zero.

• Continuous compounding views time as essentially continuous or unbroken; discrete compounding views time as advancing in discrete finite intervals.

• The continuously compounded return associated with a holding period is the natural log of 1 plus the holding period return, or equivalently, the natural log of ending price over beginning price.

• If continuously compounded returns are normally distributed, asset prices are log-normally distributed. This relationship is used to move back and forth between the distributions for return and price. Because of the central limit theorem, continuously compounded returns need not be normally distributed for asset prices to be reasonably well described by a lognormal distribution.

• Monte Carlo simulation involves the use of a computer to represent the operation of a complex financial system. A characteristic feature of Monte Carlo simulation is the generation of a large number of random samples from specified probability distribution^) to represent the operation of risk in the system. Monte Carlo simulation is used in planning, in financial risk management, and in valuing complex securities. Monte Carlo simulation is a complement to analytical methods but provides only statistical estimates, not exact results.

• Historical simulation is an established alternative to Monte Carlo simulation that in one implementation involves repeated sampling from a historical data series. Historical simulation is grounded in actual data but can reflect only risks represented in the sample historical data. Compared with Monte Carlo simulation, historical simulation does not lend itself to "what if" analyses.

PROBLEMS 1. A European put option on stock conveys the right to sell the stock at a prespecified price, called the exercise price, at the maturity date of the option. The value of this put at maturity is (Exercise price - Stock price) or $0, whichever is greater. Suppose the exercise price is $100 and the underlying stock trades in ticks of $0.01. At any time before maturity, the terminal value of the put is a random variable.

A. Describe the distinct possible outcomes for terminal put value. (Think of the put's maximum and minimum values and its minimum price increments.)

B. Is terminal put value, at a time before maturity, a discrete or continuous random variable?

C. Letting Y stand for terminal put value, express in standard notation the probability that terminal put value is less than or equal to $24. No calculations or formulas are necessary.

2. Suppose X, Y, and Z are discrete random variables with these sets of possible outcomes: X = {2,2.5,3}, Y= {0,1,2,3}, and Z = {10,11,12}. For each of the functions f(X), g(Y), and h(Z), state whether the function satisfies the conditions for a probability function.

B. g(0) = 0.25 g(l) = 0.50 g(2) = 0.125 ^(3) = 0.125

3. Define the term "binomial random variable." Describe the types of problems for which the binomial distribution is used.

4. Over the last 10 years, a company's annual earnings increased year over year seven times and decreased year over year three times. You decide to model the number of earnings increases for the next decade as a binomial random variable.

A. What is your estimate of the probability of success, defined as an increase in annual earnings?

For Parts B, C, and D of this problem, assume the estimated probability is the actual probability for the next decade.

B. What is the probability that earnings will increase in exactly 5 of the next 10 years?

C. Calculate the expected, number of yearly earnings increases during the next 10 years.

D. Calculate the variance and standard deviation of the number of yearly earnings increases during the next 10 years.

E. The expression for the probability function of a binomial random variable depends on two major assumptions. In the context of this problem, what must you assume about annual earnings increases to apply the binomial distribution in Part B? What reservations might you have about the validity of these assumptions?

5. You are examining the record of an investment newsletter writer who claims a 70 percent success rate in making investment recommendations that are profitable over a one-year time horizon. You have the one-year record of the newsletter's seven most recent recommendations. Four of those recommendations were profitable. If all the recommendations are independent and the newsletter writer's skill is as claimed, what is the probability of observing four or fewer profitable recommendations out of seven in total?

6. By definition, a down-and-out call option on stock becomes worthless and terminates if the price of the underlying stock moves down and touches a prespecified point during the life of the call. If the prespecified level is $75, for example, the call expires worthless if and when the stock price falls to $75. Describe, without a diagram, how a binomial tree can be used to value a down-and-out call option.

7. You are forecasting sales for a company in the fourth quarter of its fiscal year. Your low-end estimate of sales is €14 million, and your high-end estimate is €15 million. You decide to treat all outcomes for sales between these two values as equally likely, using a continuous uniform distribution.

A. What is the expected value of sales for the fourth quarter?

B. What is the probability that fourth-quarter sales will be less than or equal to €14,125,000?

8. State the approximate probability that a normal random variable will fall within the following intervals:

A. Mean plus or minus one standard deviation

B. Mean plus or minus two standard deviations

### C. Mean plus or minus three standard deviations

9. You are evaluating a diversified equity portfolio. The portfolio's mean monthly return is 0.56 percent, and its standard deviation of monthly returns is 8.86 percent.

A. Calculate a one standard deviation confidence interval for the return on this portfolio. Interpret this interval, with a normality assumption for returns.

B. Calculate an exact 95 percent confidence interval for portfolio return, assuming portfolio returns are described by a normal distribution.

C. Calculate an exact 99 percent confidence interval for portfolio return, assuming portfolio returns are described by a normal distribution.

10. Find the area under the normal curve up to z = 0.36; that is, find P(Z s 0.36). Interpret this value.

11. In futures markets, profits or losses on contracts are settled at the end of each trading day. This procedure is called marking to market or daily resettlement. By preventing a trader's losses from accumulating over many days, marking to market reduces the risk that traders will default on their obligations. A futures markets trader needs a liquidity pool to meet the daily mark to market. If liquidity is exhausted, the trader may be forced to unwind his position at an unfavorable time.

Suppose you are using financial futures contracts to hedge a risk in your portfolio. You have a liquidity pool (cash and cash equivalents) of \ dollars per contract and a time horizon of T trading days. For a given size liquidity pool, X, Kolb, Gay, and Hunter (1985) developed an expression for the probability stating that you will exhaust your liquidity pool within a T-day horizon as a result of the daily mark to market. Kolb et al. assumed that the expected change in futures price is 0 and that futures price changes are normally distributed. With cr representing the standard deviation of daily futures price changes, the standard deviation of price changes over a time horizon to day T is aVf, given continuous compounding. With that background, the Kolb et al. expression is

Probability of exhausting liquidity pool = 2[1 - N(x)]

where x = \/(aVf). Here xisa standardized value of X. N(x) is the standard normal cumulative distribution function. For some intuition about 1 - N(x) in the expression, note that the liquidity pool is exhausted if losses exceed the size of the liquidity pool at any time up to and including T; the probability of that event happening can be shown to be proportional to an area in the right tail of a standard normal distribution, 1 - N(x).

Using the Kolb et al. expression, answer the following questions:

A. Your hedging horizon is five days, and your liquidity pool is $2,000 per contract. You estimate that the standard deviation of daily price changes for the contract is $450. What is the probability that you will exhaust your liquidity pool in the five-day period?

B. Suppose your hedging horizon is 20 days, but all the other facts given in Part A remain the same. What is the probability that you will exhaust your liquidity pool in the 20-day period?

Use the information and table below to solve Problems 12 through 14.

As reported by Liang (1999), U.S. equity funds in three style categories had the following mean monthly returns, standard deviations of return, and Sharpe ratios during the period January 1994 to December 1996:

January 1994 to December 1996

Strategy |
Mean Return |
Standard Deviation |
Sharpe Ratio |

Large-cap growth |
1.15% |
2.89% |
0.26 |

Large-cap value |
1.08% |
2.20% |
0.31 |

Large-cap blend |
1.07% |
2.38% |
0.28 |

Source: Liang (1999), Table 5 (excerpt)

Source: Liang (1999), Table 5 (excerpt)

12. Basing your estimate of future-period monthly return parameters on the sample mean and standard deviation for the period January 1994 to December 1996, construct a 90 percent confidence interval for the monthly return on a large-cap blend fund. Assume fund returns are normally distributed.

13. Basing your estimate of future-period monthly return parameters on the sample mean and standard deviation for the period January 1994 to December 1996, calculate the probability that a large-cap growth fund will earn a monthly return of 0 percent or less. Assume fund returns are normally distributed.

14. Assuming fund returns are normally distributed, which fund category minimized the probability of earning less than the risk-free rate for the period January 1994 to December 1996?

15. A client has a portfolio of common stocks and fixed-income instruments with a current value of £1,350,000. She intends to liquidate £50,000 from the portfolio at the end of the year to purchase a partnership share in a business. Furthermore, the client would like to be able to withdraw the £50,000 without reducing the initial capital of £1,350,000. The following table shows four alternative asset allocations.

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