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- • Discrete Random Variables and Continuous Random Variables aretwo basic types of random variables where a random variable can bedefined as quantity whose future outcomes are not certain.• A Probability Distribution can be viewed in two ways: 1) ProbabilityFunction, and 2) Cumulative Distribution FunctionStudy Session 3, Reading 9
- • Properties of probability function:i. 0 ≤ p(x) ≤ 1ii. Sum of p(x) for all values of X is equal to 1.Study Session 3, Reading 9
- • The notation of a cdf for both Continuous and Discrete RandomVariables is:F(x) =P (X≤ x).• Properties of cdf:i. 0 ≤ F(x) ≤ 1ii. As the value of x increases, cdf either increases or remainsconstant.Study Session 3, Reading 9
- • A uniform discrete distribution is used for generating randomnumbers that provide random observations for probabilitydistributions.• A Binomial Distribution is used to find probability statementsabout an event when it has binary outcomes.• The basis for binomial distribution is the Bernoulli RandomVariable.Study Session 3, Reading 9
- • A Bernoulli trial is an experiment with two possible outcomes suchas up or down, success or failure.• Suppose the probability of success (denoted by 1) in trial is p andthe failure(denoted by 0) is 1-p, then the probability function for theBernoulli random variable X can be shown as:p (1) = P (X=1) = pP (0) = P(X=0) = 1- pStudy Session 3, Reading 9
- • If the Bernoulli Random Variable is represented as Yi where i=1,2,….., ∏ and Yi is the outcome of ith trial, then the binomial randomvariable X is: X = Y1 + Y2 + ... + Yn• A Bernoulli Random Variable is a binomial random variable withn = 1: Y ~ B(1, p)• The probability of x successes in n trials of a binomial randomvariable is given by:Study Session 3, Reading 9
- • A binomial tree can be stated as a graphical presentation of anasset pricing model.• A binomial tree can also be used to price options.Study Session 3, Reading 9
- Study Session 3, Reading 9• An example of a Binomial Tree is shown below:
- • Tracking error is defined as the total return on the portfolioless the return of the benchmark index.• The standard deviation of tracking error is called tracking risk.Study Session 3, Reading 9
- • If n is number of observations, p is the expected success rate of thefund manager keeping the tracking error within its limit, and x is thenumber of periods in which tracking error is within limit; then, theprobability of tracking error will be within limit F(x) is calculated as:= p (0) + p (1) + p (2) +…..p (n)where p (n) is equal toStudy Session 3, Reading 9
- • The major continuous distributions used in investment analysisare: 1) lognormal, and 2) normal.• Continuous Uniform Distributions are used as a technique togenerate random numbers that are used in Monte CarloSimulation.Study Session 3, Reading 9
- • The pdf for uniform random variable can be given as:Study Session 3, Reading 9
- • The cdf for a uniform distribution is:• For any continuous random variable (X), because the probability atend points a and b is 0:P(a≤X≤b) = P(a<X≤b) = P(a≤X<b) = P(a<X<b)Study Session 3, Reading 9
- • A normal distribution is widely used in modern portfoliotheory and risk management techniques.• The Central Limit Theorem has extended the role of thenormal distribution in regression analysis and statisticalinference.Study Session 3, Reading 9
- • A Normal Distribution can be graphically presented as:Study Session 3, Reading 9
- • (read as “X follows a normal distribution withmean µ and variance σ2 indicates that the normal distributioncan be described by its mean and variance.Study Session 3, Reading 9
- • A univariate distribution is a description of a single randomvariable.• A multivariate distribution is used to specify the probability ofa group of related random variables.Study Session 3, Reading 9
- • Three parameters that describe a multivariate normaldistribution for the returns of n securities are:–Average return on individual securities (n numbers)–Variance of the return on securities (n numbers)–Pair wise return correlations [n (n-1)/2 numbers]Study Session 3, Reading 9
- • The need of correlation in a multivariate normal distributiondistinguishes it from a univariate normal distribution.• To specify the normal distribution of a portfolio’s return, thefollowing three statistics are required: 1) means, 2) variances, and3) pairwise correlations.Study Session 3, Reading 9
- • The following tables gives the number of observations falling ina particular interval:Study Session 3, Reading 9% of observations Interval50689599
- The population mean (µ) is estimated using the sample meanand the population standard deviation (σ) is estimated fromthe sample standard deviation (s).Study Session 3, Reading 9
- • The following graph depicts number of observations lying in aparticular interval.Study Session 3, Reading 9
- • Standard Deviation is useful when comparing the dispersions ofdifferent normal distributions.Standard Normal Distribution• Standard normal distribution or unit normal distribution is a normaldistribution with µ = 0 and σ = 1.• Its skewness and excess kurtosis are 0.Study Session 3, Reading 9
- • A normal random variable can be standardized by using theexpression: Z = (X - µ) / σStudy Session 3, Reading 9
- • Normal Distributions are used in many investment decisions,financial risk management, and portfolio selections.• Various tools used for these applications are shortfall risk, Roy’sSafety First Ratio, Stress Testing, and Value at Risk.Study Session 3, Reading 9
- • Roy’s Safety First Ratio (SFR) can be calculated as:SFRatio = [E(Rp) – RL]/σpwhere E(Rp) is the portfolio’s expected return, RL is theminimum acceptable return, and σp is the portfolio’s standarddeviation.Study Session 3, Reading 9
- • The graph of lognormal distributions are highlighted below:Study Session 3, Reading 9
- • The mean and variance of a lognormal random variable arecalculated using the following formulas, where is variance and µis the mean of a normal distribution:o Mean µL = exp(µ + 0.50σ2)o Variance σL2 = exp(2 µ + σ2) x *exp(σ2) – 1]Study Session 3, Reading 9
- • The continuously compounded return from period t to t+1 iscalculated as:Rt,t+1 = ln(St+1 /S) = ln(1 + Rt,t+1)• The price relative can be calculated as:St+1/St = 1 + Rt, t+1Study Session 3, Reading 9
- • If one period continuous compounded returns are IID randomvariables, mean is µ and variance is σ2, then:E(r0,T) = E(rT-1,T) + E(rT-2,T-1) + . . . + E(r0,1) = µTStudy Session 3, Reading 9
- • Monte Carlo Simulation uses a probability distribution togenerate a large number of random samples.• It is used to model the complex financial systems.Study Session 3, Reading 9
- • Specifying the Monte Carlo Simulation involves three steps: 1)specifying the quantity of interest in terms of underlying value, 2)specifying the time grid, and 3) identifying distributionalassumptions for the underlying variables’ risk factors.Study Session 3, Reading 9
- • As compared to use of probability distribution in MonteCarlo Simulation, historical simulation uses samples from ahistorical record of returns.Study Session 3, Reading 9
- • A sample is a subset of the population. Statistics computed withsample information are estimates of the underlying population.•Two methods of sampling are:1) Simple Random Sampling, and2) Stratified Random Sampling.Study Session 3, Reading 10
- • A Time Series can be defined as a collection of observations atdiscrete and equal spaced intervals of time.• Cross sectional data are observations based on somecharacteristic like geographical regions, groups, companies,individuals etc.Study Session 3, Reading 10
- • According to the central limit theorem, if the data size of asample is large and the probability distribution of a populationhas a mean (µ) and a variance (σ2), the sampling distribution ofthe sample mean ( ), computed from n samples from thepopulation, is approximately normal and the mean will be equalto population mean and the variance will be the populationvariance divided by number of observation (i.e ( σ2/n)).Study Session 3, Reading 10
- • The Standard Deviation of a sample statistic is known as thestandard error of the statistic.• The Standard Error of a sample mean is the standarddeviation of the sampling distribution of the sample mean.• The standard error of the sample mean is applied in thecentral limit theorem.Study Session 3, Reading 10
- • The Standard Error of a Sample Mean is calculated as:(i) When the population standard deviation is known:(ii) When the population standard deviation is now known:Study Session 3, Reading 10
- • The sample variance (s2) is calculated as:Study Session 3, Reading 10
- • An estimator is a formula that is used to estimate the values ofsample statistics.• The value that is obtained from a sample observation by usingan estimator is called an estimate.Study Session 3, Reading 10
- •A desirable estimator should have three properties:1) Unbiasedness2) Efficiency3) Consistency.Study Session 3, Reading 10
- • Two types of estimates of a parameter are point estimates andinterval estimates.• Because of sampling error, the point estimate may not be equalto the population parameter in any sample. In this case, aninterval estimate is a better approach than a point estimate.Study Session 3, Reading 10
- • A Confidence Interval is a range that will contain the parameterintended to estimate, within a given probability, 1 – α (termed asdegree of confidence).• The structure of a 100(1- α)% confidence interval for aparameter is as given below:Point Estimate Reliability Factor x Standard ErrorStudy Session 3, Reading 10
- • When the population variance is not known, the Student’s t-distribution approach is used to calculate the confidence intervalfor the population mean, whether the sample size is large orsmall.• Student’s t-distribution can be described as a symmetricalprobability distribution that can defined by single parametercalled degrees of freedom (df).Study Session 3, Reading 10
- • A standard normal variable is denoted by z and zα denotes thepoint of a normal distribution in such a way that α of theprobability in the right tail.• A 100(1 – α)% confidence interval for the population mean (µ)and variance (σ2) and sampling is from a normal distribution canbe calculated as:Study Session 3, Reading 10
- • Can be calculated by the formula:Study Session 3, Reading 10
- • The reliability factors used to construct confidence intervalsbased on a standard normal distribution are:• 90% confidence intervals – use z0.05 = 1.65• 95% confidence intervals – use z0.025 = 1.96• 99% confidence intervals - use z0.005 = 2.58Study Session 3, Reading 10
- • Factors affecting width of confidence interval are: 1) choiceof statistic (t or z), 2) degree of confidence, and 3) sample size.• Some issues that are challenges in sampling are: 1) datamining bias, 2) sample selection bias, 3) look ahead bias, and4) time-period bias.Study Session 3, Reading 10
- • As the standard error of the sample mean is the samplestandard deviation divided by square root of sample size,increases in sample sizes decrease the standard error and as aresult the width of the confidence level decreases.Study Session 3, Reading 10
- • Statistical interference has two subdivisions: estimation andhypothesis testing.• Hypothesis testing is the process of making judgementsabout the population on the basis of sample data.Study Session 3, Reading 10
- • Three ways to formulate null and alternative hypothesis are:1. H0 : = 0 versus Ha : 0 (a not equal to the alternativehypothesis)2. H0 : 0 versus Ha : > 0 (a greater than the alternativehypothesis)3. H0 : 0 versus Ha : < 0 (a not equal to the alternativehypothesis)Study Session 3, Reading 10
- • A test statistic is used to decide whether or not to reject thenull hypothesis.• Rejection of a true null hypothesis is a Type I Error, and thenon-rejection of false null hypothesis is a Type II Error.• The significance level is the probability of a Type I Error.Study Session 3, Reading 11
- • A test statistic is calculated as:(Sample Statistic – Value of the population parameterunder H0) /Standard Error of sample statisticStudy Session 3, Reading 11
- • While testing a null hypothesis four outcomes are possible:–Rejecting a false hypothesis – No Error–Rejecting a true hypothesis - Type I Error–Rejecting a false null hypothesis – Type II Error–Not Rejecting a True null hypothesis – No errorStudy Session 3, Reading 11
- • If the calculated value of a test statistic is extreme or moreextreme than values determined by a specified significance level,the null hypothesis is rejected. This result is called statisticallysignificant.• The power of a test can be described as the probability ofcorrectly rejecting the null hypothesis when it is false.Study Session 3, Reading 11
- • A decision rule is the rejection points.•For a two tailed test, the rejection point is denoted as z /2.• For the test H0 : θ = θ0 versus Ha: θ ≠ θ0 , there are tworejection points, one positive and another negative. If z is thecalculated value of a test statistic, then the null hypothesis isrejected if z <negative value of z /2 or z>value of z /2.Study Session 3, Reading 11
- • For the test H0 : θ ≤ θ0 versus Ha : θ > θ0, the nullhypothesis is rejected if z > value of z.• For the test H0 : θ ≥ θ0 versus Ha : θ < θ0, the nullhypothesis is rejected if z < negative value of z.Study Session 3, Reading 11
- • The final step in hypothesis testing is making theinvestment or economic decision.• There are some non statistical considerations that shouldbe considered before making the final decision.Study Session 3, Reading 11
- • Statistical decision is rejecting or not rejecting the nullhypothesis.• Economic decision means considering the statistical decisionand all economic issues related to the decision.Study Session 3, Reading 11
- • The ρ - value can be defined as the smallest level of significanceat which null hypothesis can be rejected.• ρ - value is also called the marginal significance level.• The smaller the ρ - value, the higher are the chances of evidenceagainst the null hypothesis, and in favour of the alternativehypothesis.Study Session 3, Reading 11
- • In hypothesis testing, the ρ - value can be used as an alternativeto rejection points.• In the case that the ρ - value is less than the specified level ofsignificance, the null hypothesis is rejected.• The ρ - value provides more precise information on the strengthof the evidence, as the null hypothesis is rejected at smaller levelsof significance.Study Session 3, Reading 11
- • A t-test is conducting a hypothesis test by using a statistic thatfollows a t – distribution.• The t–distribution can be described as a probability distributiondefined by a single parameter – degrees of freedom (df).Study Session 3, Reading 11
- • If the population variance is not known, with any one of thefollowing conditions - the sample is large or the sample is small butpopulation sampled is normally distributed, the following teststatistic for a single population mean, µ, is used:• The denominator of a t-statistic is the sample mean standarderror i.e.Study Session 3, Reading 11
- • If the population sample is distributed normally and the knownvariance is 2, the following test statistic is used for hypothesis testsregarding single population mean (µ):• If the population variance is unknown, the following test statisticis used.Study Session 3, Reading 11
- • Two types of t–tests are used to test difference between themeans of two population means.• An assumption of these tests is that samples are independent.Study Session 3, Reading 11
- • The following hypotheses are used to test whether thepopulation mean are equal or whether one mean is largerthan another mean:1. H0 : µ1 - µ2 = 0 versus Ha : µ1 - µ2 0 (alternative is µ1 µ2)2. H0 : µ1 - µ2 ≤ 0 versus Ha : µ1 - µ2 > 0 (alternative is µ1 > µ2)3. H0 : µ1 - µ2 ≥ 0 versus Ha : µ1 - µ2 < 0 (alternative is µ1 < µ2)Study Session 3, Reading 11
- • t-test based on independent random samples when twopopulations are normally distributed and it is assumed thatunknown variances are equal:where is a pooled estimator of commonvariance and degrees of freedom is n1 + n2 - 2Study Session 3, Reading 11
- • If two populations are assumed to be normally distributed, butunknown population variances cannot be assumed to be equal,the following t-test based on independent variables is used:Study Session 3, Reading 11
- • Here, tables of t–distributions using modified degrees offreedom computed with the formula as given below are used:Study Session 3, Reading 11
- • When samples are dependent, the t-test on two means is calleda paired comparisons test.• The t-test is based on data arranged in paired observations.• Paired observations are those observations that have somethingin common.Study Session 3, Reading 11
- • When the populations are normally distributed, the populationvariance is unknown, and the data consists of paired observations,the following t-test is used:• is sample mean difference that can be calculated as:Study Session 3, Reading 11
- • is standard error of the mean difference and can be calculatedas:Here, sd is sample standard deviation which can be obtained bysquare root of variance calculated by the formula:Study Session 3, Reading 11
- Test Statistic for the Population Variance• If there are n independent observations of a normallydistributed population, the following chi-square test is used:• The sample variance s2 in the above formula can be calculatedas:Study Session 3, Reading 11
- • Not equal to Ha: Null hypothesis is rejected if the test statistic isgreater than the upper α/2 point ( /2) or less than the lower α/2point (denoted as /2) of chi-square distribution.• Greater than Ha: Null hypothesis is rejected if the test statistic isgreater than the upper α point.• Less than Ha: Null hypothesis is rejected in the case that the teststatistic result is less than lower α point.Study Session 3, Reading 11
- • The test statistic for the difference between the variance of twonormally distributed populations is computed as:• Larger of two ratios / or / is used as the test statistic. Sothe value of the test statistic is always greater than or equal to 1.Study Session 3, Reading 11
- • Parametric tests are those tests that are concerned with someparameter like variance or mean, and they are dependent uponcertain assumptions.• Non-parametric tests are not concerned with a parameter andthese tests make minimal assumptions about the population.Study Session 3, Reading 11
- • The Spearman Rank Correlation Coefficient is calculatedbased on ranks of two variables in the respective samples.• It is a number between -1 and +1 where -1 denotes aperfect inverse correlation, +1 indicates a perfect positivecorrelation, and 0 means that there is no correlation.Study Session 3, Reading 11
- • The following formula is used for calculating the SpearmanRank Correlation Coefficient (rs):Study Session 3, Reading 11
- • Technical Analysis is a type of security analysis thatgraphically represents price and volume data.•Technical Analysis can be defined as study of past trends andpatterns in order to predict future security prices.Study Session 3, Reading 12
- • It is supply and demand that determines the prices.• If there is change in supply and demand, prices will change.• Charts and other technical tools can be used to predict prices.Study Session 3, Reading 12
- • Technical analysis is based solely on price and volumewhereas fundamental analysis is based on data that is externalto market such as financial position of a company.• Data used by technical analysis is more concrete than dataused by fundamental analysis as fundamental analysis relies onobjective data.Study Session 3, Reading 12
- • Charts can be described as graphical display of price and volumedata.• Charts are used to get information about price behaviour of pastand make inferences about future prices.• Various types of charts that are used in technical analysis are linecharts, bar charts, candlestick charts and point and figure charts.Study Session 3, Reading 12
- • Line chart is simplest type of chart and it is agraphical display of price over a time period.Study Session 3, Reading 12
- • Bar charts have four bits of data for each time interval- high,low, opening and closing prices.Study Session 3, Reading 12CLOSELOWOPENHIGH
- • A candlestick chart is like a bar chart, providing open, high,low, close price during a period.Study Session 3, Reading 12
- • In line, bar and candlestick charts, the vertical axis can be a linearscale or logarithmic scale.• Volume is shown at the bottom of charts.• Time interval on the charts can be fixed according to requirementof analysis.• Relative strength analysis is used to compare the performance ofa particular asset with some benchmark.Study Session 3, Reading 12
- • Trend analysis is based on the fact that market participantsreact in herds and the trends observed as a result remain inplace for some time.Study Session 3, Reading 12
- • Support and resistance levels can be sloped lines or horizontallines.• According to change in polarity principle, after a support levelis breached it becomes a resistance level for the asset price.• Similarly according to polarity principle, if a resistance level isbreached it becomes support level.Study Session 3, Reading 12
- • Chart patterns can be defined as formations that appear inprice charts. These formations create different types ofrecognizable shapes.• Two categories in which chart patterns can be divided are –reversal patterns and continuation patters.Study Session 3, Reading 12
- • Price target for head and shoulder pattern is calculated asPrice target = Neckline – (Head – Neckline)Study Session 3, Reading 12
- • The three parts of inverse head and shoulder pattern are –left shoulder, head and right shoulder.• Price target for inverse head and shoulder patter is calculatedasPrice Target = Neckline + (Neckline – Head)Study Session 3, Reading 12
- • A rectangle pattern is formed if trendline connecting the highsand trendline connecting the lows are parallel.• Rectangles may be of two types – bullish rectangle and bearingrectangle.Study Session 3, Reading 12
- • A technical indicator may be defined as measure that is based onmarket sentiments, flow of funds or prices.• Technical indicators are used for making predictions about changein prices.• Technical indicators may be categorized into – price basedindicators, momentum oscillators, sentiment indicators and flow-of-funds indicators.Study Session 3, Reading 12
- • Momentum or rate of change (ROC) oscillator is calculated asgiven belowM = (V – Vx) x 100• In order to set oscillator to oscillate above and below 100 insteadof 0, the following formula is used:Study Session 3, Reading 12
- • The formula to calculate RSI is:whereStudy Session 3, Reading 12
- • The stochastic oscillator consists of two lines, %K and %Dthat are calculated as:Study Session 3, Reading 12
- • Short interest ratio is calculated asShort Interest Ratio = Short Interest/Average Daily TradingVolume• Short interest shows market sentiment and is a contrarianindicator.Study Session 3, Reading 12
- •Arms Index (TRIN) is calculated as:Study Session 3, Reading 12
- • Technical Analysts use observed cycles for prediction of futureprice movements.• Major cycles used by analysts are Kondratieff Wave, 18-yearCycle, Decimal Pattern and Presidential Cycle.Study Session 3, Reading 12
- • Kondratieff Wave theory• Kondratieff Wave theory• Decimal Pattern• Presidential Election Cycle theoryStudy Session 3, Reading 12
- • The sample size used for these cycles is very small andnumber of cycles occurred is also very small.• Another limitation is that data do not always fit into cycletheory even in the small number of cycles.Study Session 3, Reading 12
- • Elliott Wave Theory is used to forecast markets.• According to this theory markets moves in regular andrepeated repetitive wave or cycles and each wave can bebroken down into smaller and smaller sub waves.Study Session 3, Reading 12
- • Fibonacci sequence starts with numbers 0, 1, 1 andeach subsequent number is sum of preceding twonumbers.• According to Elliott Wave Theory, size of subsequentwave is generally a Fibonacci Ratio.Study Session 3, Reading 12
- • Intermarket analysis believes that all markets such as equities,bonds, currencies and commodities are interrelated andmovement in one market affects other markets.• Relative Strength Analysis is used for making allocation todifferent group of securities.Study Session 3, Reading 12

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