Deterministic Risk Analysis Best Case, Worst Case, Most LikelyA quantitative risk analysis can be performed a couple of different ways. One way uses single-point estimates, or is deterministic in nature. Using this method, an analyst may assign values for discrete scenarios to see what the outcome might be in each. For example, in a financial model, an analyst commonly examines three different outcomes: worst case, best case, and most likely case, each defined as follows:

Worst case scenario All costs are the highest possible value, and sales revenues are the lowest of possible projections. The outcome is losing money.

Best case scenario All costs are the lowest possible value, and sales revenues are the highest of possible projections. The outcome is making a lot of money.

Most likely scenario Values are chosen in the middle for costs and revenue, and the outcome shows making a moderate amount of money.

There are several problems with this approach:

It considers only a few discrete outcomes, ignoring hundreds or thousands of others.

It gives equal weight to each outcome. That is, no attempt is made to assess the likelihood of each outcome.

Interdependence between inputs, impact of different inputs relative to the outcome, and other nuances are ignored, oversimplifying the model and reducing its accuracy.

Yet despite its drawbacks and inaccuracies, many organizations operate using this type of analysis.

Stochastic Risk Analysis Monte Carlo SimulationA better way to perform quantitative risk analysis is by using Monte Carlo simulation. In Monte Carlo simulation, uncertain inputs in a model are represented using ranges of possible values known as probability distributions. By using probability distributions, variables can have different probabilities of different outcomes occurring. Probability distributions are a much more realistic way of describing uncertainty in variables of a risk analysis. Common probability distributions include:

Normal Or bell curve. The user simply defines the mean or expected value and a standard deviation to describe the variation about the mean. Values in the middle near the mean are most likely to occur. It is symmetric and describes many natural phenomena such as peoples heights. Examples of variables described by normal distributions include inflation rates and energy prices.

Lognormal Values are positively skewed, not symmetric like a normal distribution. It is used to represent values that dont go below zero but have unlimited positive potential. Examples of variables described by lognormal distributions include real estate property values, stock prices, and oil reserves.

Uniform All values have an equal chance of occurring, and the user simply defines the minimum and maximum. Examples of variables that could be uniformly distributed include manufacturing costs or future sales revenues for a new product.

Triangular The user defines the minimum, most likely, and maximum values. Values around the most likely are more likely to occur. Variables that could be described by a triangular distribution include past sales history per unit of time and inventory levels.

PERT- The user defines the minimum, most likely, and maximum values, just like the triangular distribution. Values around the most likely are more likely to occur. However values between the most likely and extremes are more likely to occur than the triangular; that is, the extremes are not as emphasized. An example of the use of a PERT distribution is to describe the duration of a task in a project management model.

Discrete The user defines specific values that may occur and the likelihood of each. An example might be the results of a lawsuit: 20% chance of positive verdict, 30% change of negative verdict, 40% chance of settlement, and 10% chance of mistrial.

During a Monte Carlo simulation, values are sampled at random from the input probability distributions. Each set of samples is called an iteration, and the resulting outcome from that sample is recorded. Monte Carlo simulation does this hundreds or thousands of times, and the result is a probability distribution of possible outcomes. In this way, Monte Carlo simulation provides a much more comprehensive view of what may happen. It tells you not only what could happen, but how likely it is to happen.

Monte Carlo simulation provides a number of advantages over deterministic analysis:

Probabilistic Results. Results show not only what could happen, but how likely each outcome is.

Graphical Results. Because of the data a Monte Carlo simulation generates, its easy to create graphs of different outcomes and their chances of occurrence. This is important for communicating findings to other stakeholders.

Sensitivity Analysis. With just a few cases, deterministic analysis makes it difficult to see which variables impact the outcome the most. In Monte Carlo simulation, its easy to see which inputs had the biggest effect on bottom-line results.

Scenario Analysis. In deterministic models, its very difficult to model different combinations of values for different inputs to see the effects of truly different scenarios. Using Monte Carlo simulation, analysts can see exactly which inputs had which values together when certain outcomes occurred. This is invaluable for pursuing further analysis.

Correlation of Inputs. In Monte Carlo simulation, its possible to model interdependent relationships between input variables. Its important for accuracy to represent how, in reality, when some factors goes up, others go up or down accordingly.

Monte Carlo Simulation in Spreadsheets and Project SchedulesThe most common platform for performing quantitative risk analysis is the spreadsheet model. Many people still unnecessarily use deterministic risk analysis in spreadsheet models when they could easily add Monte Carlo simulation using @RISK in Excel. @RISK adds new functions to Excel for defining probability distributions and analyzing output results. @RISK is also available for Microsoft Project, assessing risks in project schedules and budgets.Multivariate Models: The Monte Carlo Analysis

by Robert Stammers,CFA (Contact Author | Biography)

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Filed Under: Financial Theory, Insurance, OptionsResearch analysts use multivariate models to forecast investment outcomes to understand the possibilities surrounding their investment exposures and to better mitigate risks. Monte Carlo analysis is one specific multivariate modeling technique that allows researchers to run multiple trials and define all potential outcomes of an event or investment. Running a Monte Carlo model creates a probability distribution or risk assessment for a given investment or event under review. By comparing results against risk tolerances, managers can decide whether to proceed with certain investments or projects. (To learn more about Monte Carlo basics, see Introduction To Monte Carlo Simulationand Monte Carlo Simulation With GBM.)

Multivariate ModelsMultivariate models can be thought of as complex, "What if?" scenarios. By changing the value of multiple variables, the modeler can ascertainhis or herimpact on the estimate being evaluated. These models are used by financial analysts to estimate cash flows and new product ideas. Portfolio managers and financial advisors use these models to determine the impact of investments on portfolio performance and risk. Insurance companies use these models to estimate the potential for claims and to price policies. Some of the best-known multivariate models are those used to value stock options. Multivariate models also help analysts determine the true drivers of value.

Monte Carlo AnalysisMonte Carlo analysis is named after the principality made famous by its casinos. With games of chance, all the possible outcomes and probabilities are known, but with most investments the set of future outcomes is unknown. It is up to the analyst to determinethe set of outcomesand the probability that they will occur. In Monte Carlo modeling, the analyst runs multiple trials (often thousands) to determine all the possible outcomes and the probability that they willtake place.

Monte Carlo analysisis useful for analysts becausemany investment and business decisions are made on the basis of one outcome. In other words, many analysts derive one possible scenario and then compare it to return hurdles to decide whether to proceed. Most pro forma estimates start with a base case. By inputting the highest probability assumption for each factor, an analystcan actuallyderive the highest probability outcome. However, making any decisions on the basis of a base case is problematic, and creating a forecast with only one outcome is insufficient because it says nothing about any other possible values that could occur. It also says nothing about the very real chance that the actual future value will be something other than the base case prediction. It is impossible to hedge or insure against a negative occurrence if the drivers and probabilities of these events are not calculated in advance. (To learn more about how to manage the risk in your portfolio, see our Risk and Diversification tutorial.)

Creating the ModelOnce designed, executing a Monte Carlo model requires a tool that will randomly select factor values that are bound by certain predetermined conditions. By running anumber oftrials with variables constrained by their own independent probability of occurrence, an analyst creates a distribution that includes all the possible outcomes and the probability that they will occur. There are many random number generators in the marketplace. The two most common tools for designing and executing Monte Carlo models are @Risk and Crystal Ball. Both of these can be used as add-ins for spreadsheets and allow random sampling to be incorporated into established spreadsheet models.

The art in developing an appropriate Monte Carlo model is to determine the correct constraints for each variable and the correct relationship between variables. For example, because portfolio diversification is based on the correlation between assets, any model developed to create expected portfolio values must include the correlation between investments. (To learn more, read The Importance of Diversification.)

In order to choose the correct distribution for a variable, one must understand each of the possible distributions available. For example, the most common one is a normal distribution, also known as a bell curve. In a normal distribution,all the occurrences are equally distributed (symmetrical) around the mean. The mean is the most probable event. Natural phenomena, people's heights and inflation are some examples of inputs that are normally distributed.

In the Monte Carlo analysis, a random-number generator picks a random value for each variable (within the constraints set by the model) and produces a probability distribution for all possible outcomes. The standard deviation of that probability is a statistic that denotes the likelihood that the actual outcome being estimated will be something other than the mean or most probable event. Assuming a probability distribution is normally distributed, approximately 68% of the valueswill fallwithinone standard deviation of the mean, about 95% of the valueswill fallwithin two standard deviations and about 99.7% will lie withinthree standard deviations of the mean. This is known as the "68-95-99.7 rule" or the "empirical rule".

ExamplesLet us take for example two separate, normally distributed probability distributions derived from random-factor analysis or from multiple scenarios of a Monte Carlo model.

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Figure 1

In both of the probability distributions (Figure 1), the expected value or base cases both equal 200. Without having performed scenario analysis, there would be no way to compare these two estimates and one could mistakenly conclude that they were equally beneficial. (To learn more, read Scenario Analysis Provides Glimpse of Portfolio Potential.)

In the two probability distributions, both have the same mean but one has a standard deviation of 100, while the other has a standard deviation of 200. This means that in the first scenario analysis there is a 68% chance that the outcome will be some number between 100 and 300, while in the second model there is a 68% chance that the outcome will bebetween 0 and 400. With all things being equal, the one with a standard deviation of 100 has the better risk-adjusted outcome. Here, by using Monte Carlo to derive the probability distributions, the analysis has given an investor a basis by which to compare the two initiatives.

Monte Carlo analysis can also help determine whether certain initiatives should be taken on by looking at the risk and return consequences of taking certain actions. Let us assume we want to place debt on our original investment.

Copyright 2008 Investopedia.com

Figure 2

The distributions in Figure 2 show the original outcome and the outcome after modeling the effects of leverage. Our new leveraged analysis shows an increase in the expected value from 200 to 400, but with anincreased financial risk of debt. Debt has increased the expected value by 200 but also the standard deviation. Before 1 standard deviation was a range from 100 to 300. Now with debt, 68% of values (1 standard deviation) fall between 0 and 400. By using scenario analysis an investor can now determinewhether the additional increase in return equals or outweighs the additional risk (variability of potential outcomes) that comes with taking on the new initiative.

ConclusionMonte Carlo analyses are not only conducted by finance professionals but also by many other businesses. It is a decision-making tool that integrates the concept that every decision will have some impact on overall risk. Every individual and institution has different risk/return tolerances. As such,it is important that the risk/return profile of any investment be calculated and compared to risk tolerances.

The probability distributions produced by a Monte Carlo model create a picture of risk. A picture is an easy way to convey the idea to others, such as superiors or prospective investors. Because of advances in software, very complex Monte Carlo models can be designed and executed by anyone with access to a personal computer.

by Robert Stammers,CFA (Contact Author | Biography)

Robert Stammers, CFA, uncovers and analyzes stock and option trading opportunities as a senior equity analyst at stock market education company BetterTrades. Previously, he was portfolio manager for a $1 billion enhanced real estate fund, a public timber fund and multiple pension fund separate accounts, while acting as a senior executive for several institutional fund managers. Mr. Stammers holds The CFA Institute's Chartered Financial Analyst designation, a Bachelor of Arts in economics from Connecticut College, and a Master of Business Administration with honors from Emory University. Visit Bettertrades.com to learn more timely strategies and tactics for creating cash flow with stocks and options. 2.20 - Discrete and Continuous Compounding

In discrete compounded rates of return, time moves forward in increments, with each increment having a rate of return (ending price / beginning price) equal to 1. Of course, the more frequent the compounding, the higher the rate of return. Take a security that is expected to return 12% annually:

With annual holding periods, 12% compounded once = (1.12)1 - 1 = 12%.

With quarterly holding periods, 3% compounded 4 times = (1.03)4 - 1 = 12.55%

With monthly holding periods, 1% compounded 12 times = (1.01)12 - 1 = 12.68%

With daily holding periods, (12/365) compounded 365 times = 12.7475%

With hourly holding periods, (12/(365*24) compounded (365*24) times = 12.7496%

With greater frequency of compounding (i.e. as holding periods become smaller and smaller) the effective rate gradually increases but in smaller and smaller amounts. Extending this further, we can reduce holding periods so that they are sliced smaller and smaller so they approach zero, at which point we have the continuously compounded rate of return. Discrete compounding relates to measurable holding periods and a finite number of holding periods. Continuous compounding relates to holding periods so small they cannot be measured, with frequency of compounding so large it goes to infinity.

The continuous rate associated with a holding period is found by taking the natural log of 1 + holding-period return) Say the holding period is one year and holding-period return is 12%:

ln (1.12) = 11.33% (approx.)

In other words, if 11.33% were continuously compounded, its effective rate of return would be about 12%.

Earlier we found that 12% compounded hourly comes to about 12.7496%. In fact, e (the transcendental number) raised to the 0.12 power yields 12.7497% (approximately).

As we've stated previously, actual calculations of natural logs are not likely for answering a question as they give an unfair advantage to those with higher function calculators. At the same time, an exam problem can test knowledge of a relationship without requiring the calculation. For example, a question could ask:

Q.A portfolio returned 5% over one year,if continuously compounded, this is equivalent to ____?

A. ln 5B. ln 1.05C. e5D. e1.05

The answer would be B based on the definition of continuous compounding. A financial function calculator or spreadsheet could yield the actual percentage of 4.879%, but wouldn't be necessary to answer the question correctly on the exam.

Monte Carlo SimulationAMonte Carlo Simulation refers to a computer-generated series of trials where the probabilities for both risk and reward are tested repeatedly in an effort to help define these parameters. These simulations are characterized by large numbers of trials - typically hundreds or even thousands of iterations, which is why it's typically described as "computer generated". Also know that Monte Carlo simulations rely on random numbers to generate a series of samples.

Monte Carlo simulations are used in a number of applications, often as a complement to other risk-assessment techniques in an effort to further define potential risk. For example, a pension-benefit administrator in charge of managing assets and liabilities for a large plan may use computer software with Monte Carlo simulation to help understand any potential downside risk over time, and how changes in investment policy (e.g. higher or lower allocations to certain asset classes, or the introduction of a new manager) may affect the plan. While traditional analysis focuses on returns, variances and correlations between assets, a Monte Carlo simulation can help introduce other pertinent economic variables (e.g. interest rates, GDP growth and foreign exchange rates) into the simulation.

Monte Carlo simulations are also important in pricing derivative securities for which there are no existing analytical methods. European- and Asian-style options are priced with Monte Carlo methods, as are certain mortgage-backed securities for which the embedded options (e.g. prepayment assumptions) are very complex.

A general outline for developing a Monte Carlo simulation involves the following steps (please note that we are oversimplifying a process that is often highly technical):

1. Identify all variables about which we are interested, the time horizon of the analysis and the distribution of all risk factors associated with each variable.

2. Draw K random numbers using a spreadsheet generator. Each random variable would then be standardized so we have Z1, Z2, Z3... ZK.

3. Simulate the possible values of the random variable by calculating its observed value with Z1, Z2, Z3... ZK.

4. Following a large number of iterations, estimate each variable and quantity of interest to complete one trial. Go back and complete additional trials to develop more accurate estimates.

Historical SimulationHistorical simulation, or back simulation, follows a similar process for large numbers of iterations, with historical simulation drawing from the previous record of that variable (e.g. past returns for a mutual fund). While both of these methods are very useful in developing a more meaningful and in-depth analysis of a complex system, it's important to recognize that they are basically statistical estimates; that is, they are not as analytical as (for example) the use of a correlation matrix to understand portfolio returns. Such simulations tend to work best when the input risk parameters are well defined.

Monte Carlo Simulation With GBM

by David Harper,CFA, FRM (Contact Author | Biography)

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Filed Under: Financial TheoryOne of the most common ways to estimate risk is the use of a Monte Carlo simulation (MCS). For example, to calculate the value at risk (VaR) of a portfolio, we can run a Monte Carlo simulation that attempts to predict the worst likely loss for a portfolio given a confidence interval over a specified time horizon - we always need to specify two conditions for VaR: confidence and horizon. (For related reading, seeThe Uses And Limits Of Volatility andIntroduction To Value At Risk (VAR) - Part 1 and Part 2.)

In this article, we will review a basic MCS applied to a stock price. We need a model to specify the behavior of the stock price, and we'll use one of the most common models in finance: geometric Brownian motion (GBM). Therefore, while Monte Carlo simulation can refer to a universe of different approaches to simulation, we will start here with the most basic.

Where to Start A Monte Carlo simulation is an attempt to predict the future many times over. At the end of the simulation, thousands or millions of "random trials" produce a distribution of outcomes that can be analyzed. The basics steps are:

1. Specify a model (e.g. geometric Brownian motion) 2. Generate random trials 3. Process the output

1. Specify a Model (e.g. GBM) In this article, we will use the geometric Brownian motion (GBM), which is technically a Markov process. This means that the stock price follows a random walk and is consistent with (at the very least) the weak form of the efficient market hypothesis (EMH): past price information is already incorporated and the next price movement is "conditionally independent" of past price movements. (For more on EMH, readWorking Through The Efficient Market Hypothesis and What Is Market Efficiency?)

The formula for GBM is found below, where "S" is the stock price, "m" (the Greek mu) is the expected return, "s" (Greek sigma) is the standard deviation of returns, "t" is time, and "e" (Greek epsilon) is the random variable:

If we rearrange the formula to solve just for the change in stock price, we see that GMB says the change in stock price is the stock price "S" multiplied by the two terms found inside the parenthesis below:

The first term is a "drift" and the second term is a "shock". For each time period, our model assumes the price will "drift" up by the expected return. But the drift will be shocked (added or subtracted) by a random shock. The random shock will be the standard deviation "s" multiplied by a random number "e". This is simply a way of scaling the standard deviation.

That is the essence of GBM, as illustrated in Figure 1. The stock price follows a series of steps, where each step is a drift plus/minus a random shock (itself a function of the stock's standard deviation):

Figure 1

2. Generate Random Trials Armed with a model specification, we then proceed to run random trials. To illustrate, we've used Microsoft Excel to run 40 trials. Keep in mind that this is an unrealistically small sample; most simulations or "sims" run at least several thousand trials.

In this case, let's assume that the stock begins on day zero with a price of $10. Here is a chart of the outcome where each time step (or interval) is one day and the series runs for ten days (in summary: forty trials with daily steps over ten days):

Figure 2: Geometric Brownian Motion

The result is forty simulated stock prices at the end of 10 days. None has happened to fall below $9, and one is above $11.

3. Process the Output The simulation produced a distribution of hypothetical future outcomes. We could do several things with the output. If, for example, we want to estimate VaR with 95% confidence, then we only need to locate the thirty-eighth-ranked outcome (the third-worst outcome). That's because 2/40 equals 5%, so the two worst outcomes are in the lowest 5%.

If we stack the illustrated outcomes into bins (each bin is one-third of $1, so three bins covers the interval from $9 to $10), we'll get the following histogram:

Figure3

Remember that our GBM model assumes normality: price returns are normally distributed with expected return (mean) "m" and standard deviation "s". Interestingly, our histogram isn't looking normal. In fact, with more trials, it will not tend toward normality. Instead, it will tend toward a lognormal distribution: a sharp drop off to the left of mean and a highly skewed "long tail" to the right of the mean. This often leads to a potentially confusing dynamic for first-time students:

Price returns are normally distributed.

Price levels are log-normally distributed.

Think about it this way: A stock can return up or down 5% or 10%, but after a certain period of time, the stock price cannot be negative. Further, price increases on the upside have a compounding effect, while price decreases on the downside reduce the base: lose 10% and you are left with less to lose the next time. Here is a chart of the lognormal distribution superimposed on our illustrated assumptions (e.g. starting price of $10):

Figure4

Summary A Monte Carlo simulation applies a selected model (a model that specifies the behavior of an instrument) to a large set of random trials in an attempt to produce a plausible set of possible future outcomes. In regard to simulating stock prices, the most common model is geometric Brownian motion (GBM). GBM assumes that a constant drift is accompanied by random shocks. While the period returns under GBM are normally distributed, the consequent multi-period (for example, ten days) price levels are lognormally distributed.

Check out David Harper's movie tutorial, Monte Carlo Simulation with Geometric Brownian Motion, to learn more on this topic.

by David Harper,CFA, FRM (Contact Author | Biography)

In addition to writing for Investopedia, David Harper, CFA, FRM, is the founder of The Bionic Turtle, a site that trains professionals in advanced and career-related finance, including financial certification. David was a founding co-editor of the Investopedia Advisor, where his original portfolios (core, growth and technology value) led to superior outperformance (+35% in the first year) with minimal risk and helped to successfully launch Advisor.

He is the principal of Investor Alternatives, a firm that conducts quantitative research, consulting (derivatives valuation), litigation support and financial education.

CFA Level 1 - Corporate Finance

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11.18 - Risk-Analysis Techniques

It is important to keep in mind that when a company analyzes a potential project, it is forecasting potential not actual cash flows for a project. As we all know, forecasts are based on assumptions that may be incorrect. It is therefore important for a company to perform a sensitivity analysis on its assumptions to get a better sense of the overall risk of the project the company is about to take.

There are three risk-analysis techniques that should be known for the exam:

1.Sensitivity analysis2.Scenario analysis3.Monte Carlo simulation

1.Sensitivity AnalysisSensitivity analysis is simply the method for determining how sensitive our NPV analysis is to changes in our variable assumptions. To begin a sensitivity analysis, we must first come up with a base-case scenario. This is typically the NPV using assumptions we believe are most accurate. From there, we can change various assumptions we had initially made based on other potential assumptions. NPV is then recalculated, and the sensitivity of the NPV based on the change in assumptions is determined. Depending on our confidence in our assumptions, we can determine how potentially risky a project can be.

2.Scenario AnalysisScenario analysis takes sensitivity analysis a step further. Rather than just looking at the sensitivity of our NPV analysis to changes in our variable assumptions, scenario analysis also looks at the probability distribution of the variables. Like sensitivity analysis, scenario analysis starts with the construction of a base case scenario. From there, other scenarios are considered, known as the "best-case scenario" and the "worst-case scenario". Probabilities are assigned to the scenarios and computed to arrive at an expected value. Given its simplicity, scenario analysis is one the most frequently used risk-analysis techniques.

3.Monte Carlo SimulationMonte Carlo simulationis considered to be the "best" method of sensitivity analysis. It comes up with infinite calculations (expected values) given a number of constraints. Constraints are added and the system generates random variables of inputs. From there, NPV is calculated. Rather than generating just a few iterations, the simulation repeats the process numerous times. From the numerous results, the expected value is then calculated

Stochastic Modeling

What Does Stochastic Modeling Mean?A method of financial modeling in which one or more variables within the model are random. Stochastic modeling is for the purpose of estimating the probability of outcomes within a forecast topredict what conditions might be like under different situations.The random variables are usually constrained by historical data, such as past market returns.

Investopedia explains Stochastic ModelingTheMonte CarloSimulationis an example of astochastic modelused in finance.When used in portfolio evaluation, multiple simulations of the performanceof theportfolioaredone based on the probability distributions of the individual stock returns.A statisticalanalysis of the results can then help determine the probability that the portfolio will providethe desired performance.

Quantitative Analysis

What Does Quantitative Analysis Mean?A business or financial analysis techniquethatseeks to understand behavior byusing complex mathematical and statistical modeling, measurement and research. By assigning a numericalvalue to variables, quantitative analyststry to replicate reality mathematically.

Quantitative analysis can be done for a number of reasonssuch as measurement,performance evaluation or valuation of a financial instrument. It can also be used topredict real worldevents such as changes in a share price.

Investopedia explains Quantitative AnalysisInbroadterms, quantitative analysis is simply a way of measuring things. Examples of quantitative analysisinclude everything from simplefinancial ratios such as earnings per share, to something as complicated as discounted cash flow, or option pricing.

Although quantitative analysis is a powerful tool for evaluating investments, it rarely tellsa complete story without the help of its opposite - qualitative analysis. In financial circles, quantitative analysts are affectionately referred to as "quants", "quant jockeys" or "rocket scientists".

Monte Carlo Simulation

What Does Monte Carlo Simulation Mean?A problem solving technique used to approximate the probability of certain outcomes by running multiple trial runs, called simulations, using random variables.

Investopedia explains Monte Carlo SimulationMonte Carlo simulation is named afterthe city in Monaco, where the primary attractions are casinos that have games of chance. Gambling games, like roulette, dice, and slot machines, exhibit random behavior.

Bell Curve

What Does Bell Curve Mean?The most common type of distribution for a variable.The term "bell curve" comes from the fact that the graph used to depict a normal distribution consists of a bell-shaped line.

The bell curve is also known as a normal distribution. The bell curve is less commonly referred to as a Gaussian distribution, after German mathematician and physicist Karl Gauss, who popularized the model in the scientific community by using it to analyze astronomical data.

Investopedia explains Bell CurveThe highest point in the curve, or the top of the bell, represents the most probable event. All possible occurrences are equally distributed around the most probable event, which creates a downward-sloping lineon each side of the peak.

Filed Under: Forex, Technical AnalysisRelated Terms Mean

Monte Carlo Simulation

Normal Distribution

Statistics

Variability

Venn Diagram

More Related Terms

Mean

What Does Mean Mean?The simple mathematical average of a set of two or more numbers. The mean for a given set of numbers can be computed in more than one way, including the arithmetic mean method, which uses the sum of the numbers in the series, and the geometric mean method. However, all of the primary methods for computing a simple average of a normal number series produce the same approximate result most of the time.

Investopedia explains MeanIf stock XYZ closed at $50, $51 and $54 over the past three days, the arithmetic mean would be the sum of those numbers divided by three, which is $51.67.

In contrast, the geometric mean would be computed asthird root of the numbers' product, or thethird root of 137,700, which approximately equals $51.64. While the two numbers are not exactly equal, most people consider arithmetic and geometric means to be equivalent for everyday purposes.

Arithmetic Mean

What Does Arithmetic Mean Mean?A mathematical representation of the typical value of a series of numbers, computed as the sum of all the numbers in the series divided by the count of all numbers in the series.

Arithmetic mean is commonly referred to as "average" or simply as "mean".

Investopedia explains Arithmetic MeanSuppose you wanted to know what the arithmetic mean of a stock's closing price was over the past week. If during the five-day week the stock closed at $14.50, $14.80, $15.20, $15.50, and then $14.00, its arithmetic mean closing price would be equal to the sum of the five numbers ($74.00) divided by five, or $14.80

Winsorized Mean

What Does Winsorized Mean Mean?A method of averaging that initially replaces the smallest and largest values with the observations closest to them. After replacing the values, a simple arithmetic averaging formula is used to calculate the winsorized mean.

Winsorized means are expressed in two ways. A "kth" winsorized mean refers to the replacement of the 'k' smallest and largest observations, where 'k' is an integer. A "X%" winsorized mean involves replacing a given percentage of values from both ends of the data.

Investopedia explains Winsorized MeanThe winsorized mean is less sensitive to outliers becauseit replaces them with less influential values. This method of averaging is similar to the trimmed mean; however, instead of eliminating data, observations are altered, allowing for a degree of influence.

Let's calculate thefirst winsorized mean for the following data set: 1, 5, 7, 8, 9, 10, 14. Because the winsorized mean is in the first order, we replace the smallest and largest values with their nearest observations. The data set now appears as follows: 5, 5, 7, 8, 9, 10, 10. Taking an arithmetic average of the new set produces a winsorized mean of 7.71 ( (5+5+7+8+9+10+10) / 7 ).

Geometric Mean

What Does Geometric Mean Mean?The average of a set of products, thecalculation of which iscommonly used to determine the performance results of an investment or portfolio.Technically defined as "the 'n'th root product of 'n' numbers", the formula for calculating geometric mean is most easily written as:

Where 'n' represents the number of returns in the series.

The geometric mean must be used when working with percentages (which are derived from values), whereas the standard arithmetic mean will work with the values themselves.

Investopedia explains Geometric MeanThe main benefit to using the geometric mean is that the actual amounts invested do not need to be known; the calculation focusesentirely on the return figures themselves and presents an "apples-to-apples" comparison when looking attwo investment options.

Average Price

What Does Average Price Mean?1. A representativemeasure of a range ofprices thatiscalculated by taking thesum of the values and dividing it by the number of prices being examined. The average price reduces the range into a singlevalue, which can then becompared to any point to determine if the value is higher or lower than what would be expected.

2. A bond's average price is calculated by adding its face value to the price paid for it and dividing the sum by two. The average price is sometimes used in determining a bond's yield to maturity where the average price replaces the purchase price in the yield to maturity calculation.

Investopedia explains Average Price1.In situations where there isa range of pricesit can be useful to calculate the average price to simplify a range of numbers into a single value. For example, if over a four-month periodyou paid $104, $105, $110, and $115 for your utilities, the average price or cost ofyour monthlyutilities would be $108.50.

2. Although the average price of a bond is not the most accurate method to find its YTM, it does give investors a rough and simple gauge to find out what a bond is worth.

Moving Average - MA

What Does Moving Average - MA Mean?An indicator frequently used in technical analysis showing the average value of a security's price over a set period. Moving averages are generally used to measure momentum and define areas of possible support and resistance.

Investopedia explains Moving Average - MAMoving averages are used to emphasize the direction of a trend and to smooth out price and volume fluctuations, or "noise", that can confuse interpretation. Typically, upward momentum is confirmed when a short-term average (e.g.15-day) crosses above a longer-term average (e.g. 50-day). Downward momentum is confirmed when a short-term average crosses below a long-term average.

Exponential Moving Average - EMA

What Does Exponential Moving Average - EMA Mean?A type of moving average that is similar to a simple moving average, except that more weight is given to the latest data. The exponential moving average is also known as "exponentially weighted moving average".

Investopedia explains Exponential Moving Average - EMAThis type of moving average reacts faster to recent price changes than a simple moving average. The 12- and 26-day EMAs are the most popularshort-term averages, and they are used to create indicators like the moving average convergence divergence (MACD) and the percentage price oscillator (PPO). In general, the 50- and 200-day EMAs are used as signals of long-term trends.

Double Exponential Moving Average - DEMA

What Does Double Exponential Moving Average - DEMA Mean?

Atechnical indicator developed by Patrick Mulloy that first appeared in the February, 1994 Technical Analysis of Stocks & Commodities. The DEMA is a calculation based on both a single exponential moving average (EMA) and a double EMA.

Investopedia explains Double Exponential Moving Average - DEMA The DEMA is a fast-acting moving average that is more responsive to market changes than a traditional moving average. It was developed in an attempt to create acalculation that eliminated some of the lag associated with traditional moving averages. The DEMA can be used as a stand-alone indicator and can be incorporated into other technical analysis tools whose logic are based on moving averages.

Simple Moving Average - SMA

What Does Simple Moving Average - SMA Mean?A simple, or arithmetic, moving average thatis calculated by adding the closing price of the security for a number of time periods and then dividing this total by the number of time periods. Short-term averages respond quickly to changes in the price of the underlying, while long-term averages are slow to react.

Investopedia explains Simple Moving Average - SMAIn other words, this is the average stock price over a certain period of time. Keep in mind that equal weighting is given to each daily price. As shown in the chart above, many traders watch for short-term averages to cross above longer-term averages to signal the beginning of an uptrend. As shownby the blue arrows,short-term averages (e.g. 15-period SMA) act as levels of support when the price experiences a pullback. Support levels become stronger and more significant as the number of time periods used in the calculations increases.

Generally, when you hear the term "moving average", it is in reference to a simple moving average. This can be important, especially when comparing to an exponential moving average(EMA).

Linearly Weighted Moving Average

What Does Linearly Weighted Moving Average Mean?A type of moving average that assigns a higher weighting to recent price data than does the common simple moving average. This average is calculated by taking each of the closing prices over a given time period and multiplyingthem by its certainposition in the data series. Once the position of the time periods have been accounted forthey aresummed together and divided by the sum of the number of time periods.

Investopedia explains Linearly Weighted Moving AverageFor example, in a 15-day linearly-weighted moving average, today's closing price is multiplied by 15, yesterday's by 14, and so on until day 1 in the period's range is reached. These results are then added together and divided by the sum of the multipliers (15 + 14 + 13 + ... + 3 + 2 + 1 = 120).

The linearly weighted moving average was one of the first responses to placing a greater importance on recent data. Thepopularity of this moving averagehas been diminished by the exponential moving average, but none the less it still proves to be very useful

What Does Weighted Mean?Figures or components thatare adjusted to reflect importance by value or proportion.

Investopedia explains WeightedCommonly used as a method to assign a value, based on proportion, to various securities within a given index.For example, the DJIA weighs each security based on the stock's price relative to the sum of all the stock prices. The Nasdaq on the other hand, is a market capitalization weighted index with each company weighting being proportionate to its market value.

Weighted Average

What Does Weighted Average Mean?An average in which each quantity to be averaged is assigned a weight. These weightings determine the relative importance of each quantity on the average. Weightings are the equivalent of having that many like items with the same value involved in the average.

Investopedia explains Weighted AverageTo demonstrate, let's take the value of letter tiles in the popular game Scrabble.

Value:10 8 5 4 3 2 1 0Occurrences: 22110 87682

To average these values, do a weighted average using the number of occurrences of each value as the weight. To calculate a weighted average:

1. Multiply each value by its weight. (Ans: 20, 16,5, 40, 24, 14, 68, and 0)2. Add up the products of value times weight to get the total value. (Ans: Sum=187)3. Add the weight themselves to get the total weight. (Ans: Sum=100)4. Divide the total value by the total weight. (Ans: 187/100 = 1.87 = average value of a Scrabble tile)

Price-Weighted Index

What Does Price-Weighted Index Mean?A stock indexin whicheach stock influences the index in proportion to its price per share. The value of the index is generated by adding the prices of each of the stocks in the index and dividingthem by the total number of stocks. Stocks with a higher price will begiven more weightand, therefore, will have a greater influence over theperformance of the index.

Investopedia explains Price-Weighted IndexFor example, assume that an index contains only two stocks, one priced at $1 and one priced at $10. The $10 stock is weightednine timeshigher than the $1 stock.Overall, this means that this indexis composed of 90%of the $10stocks and 10% of $1 stock.

In this case, a change in the value of the $1 stock will notaffect the index's valueby a large amount, because it makes up such a small percentage of the index.

A popular price-weighted stock marketindex is the Dow Jones Industrial Average. It includes a price-weighted average of 30 actively traded blue chip stocks.

Dow Jones Industrial Average - DJIA

What Does Dow Jones Industrial Average - DJIA Mean?The Dow Jones Industrial Average is a price-weighted average of 30 significant stocks traded on the New York Stock Exchange and the Nasdaq. The DJIA was invented by Charles Dow back in 1896.

Investopedia explains Dow Jones Industrial Average - DJIAOften referred to as "the Dow", the DJIA is the oldest and single most watched index in the world. The DJIA includes companies like General Electric, Disney, Exxon and Microsoft.

When the TV networks say "the market is up today", they are generally referring to the Dow.

Weighted Average Cost of Equity - WACE

What Does Weighted Average Cost of Equity - WACE Mean?A way to calculate the cost of a company's equitythat gives different weight to different aspects of the equities. Instead of lumping retained earnings, common stock, and preferred stock together, WACE provides a more accurate idea of a companies total cost of equity. Determining an accurate cost of equity for a firm is integral for the firm to be able tocalculateits cost of capital.

In turn, an accurate measure of the cost of capital is essential when a firm is trying to decide if a future project will be profitable or not.

Investopedia explains Weighted Average Cost of Equity - WACEHere is an example of how to calculate the WACE:

First, calculate the cost of new common stock, the cost of preferred stock and the cost of retained earnings. Lets assume we have already done this and the cost of common stock, preferred stock and retained earnings are 24%, 10% and 20% respectively. Now, you must calculate the portion of total equity that is occupied by each form of equity. Again, lets assume this is 50%, 25% and 25%, for common stock, preferred stock and retained earnings respectively.Finally, you multiply the cost of each form of equity by its respective portion of total equity and sum of the values-which results in the WACE. Our example results in a WACE of 19.5%.

WACE = (.24*.50) + (.10*.25) + (.20*.25) = 0.195 or 19.5%

Weighted Average Market Capitalization

What Does Weighted Average Market Capitalization Mean?A stock market index weighted by the market capitalization of each stock in the index. In such a weighting scheme, larger companies account for a greater portion of the index. Most indexes are constructed in this manner, with the best examplebeing the S&P 500.

Investopedia explains Weighted Average Market CapitalizationFor example, if a company's market capitalization is $1 million and the market capitalization of all stocks in the index is $100 million, then the company would be worth 1% of the index. The alternative to weighting by market cap is a price-weighted index such as the Dow Jones Industrial Average.

Capitalization-Weighted Index

What Does Capitalization-Weighted Index Mean?A type of market index whose individual components are weighted according to their market capitalization, so that larger components carry a larger percentage weighting.The value of a capitalization-weighted index can be computed by adding up the collective market capitalizations of its members and dividing it bythe number of securities in the index.

Also known as a "market-value weighted index".

Investopedia explains Capitalization-Weighted IndexMost of the broadly-used market indexes today are "cap-weighted" indexes, such as the S&P 500, Nasdaq, Wilshire, Hang-Seng and EAFE indexes. In a cap-weighted index, large price moves in the largest components can have a dramatic effect on the value of the index. Some investors feel that this overweighting toward the larger companies gives a distorted view of the market, but the fact that the largest companies also have the largest shareholder bases makes the case for having the higher relevancy in the index.

Standard & Poor's 500 Index - S&P 500

What Does Standard & Poor's 500 Index - S&P 500 Mean?An index of 500 stocks chosen for market size, liquidity and industry grouping, among other factors. The S&P 500 is designed to be a leading indicator of U.S. equities and is meant to reflect the risk/return characteristics of the large cap universe.

Companies included in the index are selected by the S&P Index Committee, a team of analysts and economists at Standard& Poor's.The S&P 500is a market value weighted index - each stock's weight is proportionate to its market value.

Investopedia explains Standard & Poor's 500 Index - S&P 500The S&P 500 is one of the most commonly used benchmarksfor the overall U.S.stock market. The Dow Jones Industrial Average (DJIA) was at one time the most renowned index for U.S. stocks, but because the DJIA contains only 30 companies, most people agree that the S&P 500 is a better representation of the U.S. market. In fact,many considerit to bethe definition of the market.

Other popularStandard & Poor'sindexes include the S&P 600,an index of small cap companies withmarket capitalizations between $300 million and $2 billion, and the S&P 400, an index of mid cap companies with market capitalizationsof $2 billion to $10 billion.

A number of financial productsbased on the S&P 500 areavailable to investors. These includeindex funds and exchange-traded funds. However, it would be difficult for individual investors to buy the index, as this would entail buying 500 different stocks.

Free-Float Methodology

What Does Free-Float Methodology Mean?A methodby which the market capitalizationof an index's underlying companies is calculated.Free-float methodology market capitalization is calculated by taking the equity's priceand multiplying it by the number of shares readily available in the market. Instead of using all of the shares outstanding like the full-market capitalization method, the free-float method excludes locked-in shares such as those held by promoters andgovernments.

Calculated as:

Investopedia explains Free-Float MethodologyThe free-float method is seen as a betterway of calculating market capitalizationbecause it provides a more accurate reflection of market movements. When using a free-float methodology, the resulting market capitalization is smaller than what would result from a full-market capitalization method.

Free-float methodology has been adoptedby most of the world's major indexes, including the Dow Jones Industrial Averageand the S&P 500.

Nasdaq

What Does Nasdaq Mean?Acomputerized system that facilitates trading and provides price quotations on more than 5,000 of the more actively traded over the counter stocks. Created in 1971, the Nasdaq was the world's first electronic stock market.

Stocks on the Nasdaq are traditionally listed under four or five letter ticker symbols. If the company is a transfer from the New York Stock Exchange, the symbol may be comprised of three letters.

Investopedia explains NasdaqThe term "Nasdaq" used to be capitalized "NASDAQ" as an acronym for National Association of Securities Dealers Automated Quotation. The acronym is no longer used and Nasdaq is now a proper noun.

The Nasdaq is traditionally home to many high-tech stocks, such as Microsoft, Intel, Dell and Cisco.

What is the difference between the Dow and the Nasdaq?

Because of the way people throw around the words "Dow" and "Nasdaq," both terms have become synonymous with "the market," giving people a hazy idea of what each term actually means. In this question, "the Dow" refers to the famous figure that peppers almost all business news reports: the Dow Jones Industrial Average (DJIA), an important index that many people watch to get an indication of how well the overall stock market is performing. The Dow, or the DJIA, is not exactly the same as Dow Jones and Company, the firm that publishes the Wall Street Journal. However,the editors of the Wall Street Journal are the people who maintain the DJIA, along with other Dow Jones indices. The Nasdaq is also a term that can refer to two different things: first, it is the National Association of Securities Dealers Automated Quotations System, which is the first electronic exchange, where investors can buy and sell stock. Second, when you hear people say that the "the Nasdaq is up today," they are referring to the Nasdaq Composite Index, which, like the DJIA, is a statistical measure of a portion of the market.

Both the Dow and the Nasdaq, then, refer to an index, or an average of a bunch of numbers derived from the price movements of certain stocks. The DJIA tracks the performance of 30 different companies that are considered major players in their industries. The Nasdaq Composite, on the other hand, tracks approximately 4,000 stocks, all of which are traded on the Nasdaq exchange. The DJIA is composed mainly of companies found on the NYSE, with only a couple of Nasdaq-listed stocks.

Remember, although both "the Dow" and the "Nasdaq" refer to market indices, only the Nasdaq also refers to an exchange where investors can buy and sell stock. Furthermore, an investor can't trade the Dow or the Nasdaq indexes because they each represent merely a mathematical average that people use to try and make sense of the stock market. You can, however, purchase index funds, which are a kind of mutual fund, or exchange traded funds, which are securities that track the indexes.

Blue Chip

What Does Blue Chip Mean?Anationally recognized,well-established and financially sound company. Blue chips generally sell high-quality,widely accepted products and services. Blue chip companies are known to weather downturns and operate profitably in the face of adverse economic conditions, which helps to contribute to their long record of stable and reliablegrowth.

Investopedia explains Blue ChipThe name "blue chip" came about because in the game of poker the blue chipshave the highest value.

Blue chip stocksare seen as aless volatile investment than owning shares in companieswithout blue chip statusbecauseblue chipshave aninstitutionalstatus in theeconomy. Investors may buy blue chip companies toprovide steady growth in their portfolios. The stock price of a blue chip usually closely follows the S&P 500.

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