Quantitative Methods: Application
Moving Average Convergence/Divergence Oscillator
(MACD) the difference between the short term and long term moving average of a security's price 2 lines, MACD linetypically over 12 and 26 days, and signal line which is exponentially smoothed moving average of the MACD Line. typically 9 days when price and MACD move in the same direction is convergence when move opposite directions it is divergence. pg 297
Power of a test
- Probability of CORRECTLY rejecting the null hypothesis when it is false. -the higher the power the better it is for purposes of hypothesis testing Graph pg 255 Power of a test = 1-P(Type II Error) given a choice of tests the one with the highest power should be preferred. the test with the highest prob of rejecting the null hypothesis when it is false should be preferred -decreasing significance reduce prob of Type I error, however it is is increasing type II error -only way to decrease type II give a levelf of significance (prob of type 1 error) is to increase the sample size
Bollinger Bands
- consist of a simple moving average plus upper and lower bands that are calculated by adding and subtracting a specific number of standard deviations from the moving average Application -some short sellers or contrarian strategy will sell security when it reaches upper band and purchase when it touches lower. assumption here is that security will continue to trade in these bands pg294
Systematic Sample
- every kth member in the selection list is chosen. (or every 3rd car at a stoplight
Issues regarding appropriate sample size, data mining bias, sample selection bias, survivorship bias, look ahead bias and time period bias
- t stat gives a wider confidence interval -the higher the level of confidence increases the size of the confidence interval -larger standard error the wider the confidence interval -consider that with increasing size you may gain from a different population, or costs might not be worht is Biases Data mining- just searching for correlations and random variables show some significance -best way to test is to use out of sample data to corroborate Sample Selection Bias - exclusion due to unavailability of data survivorship bias- the database only includes funds that are currently existing instead of funds that failed look ahead bias - info that was not available on the test date time period bias - test is based on a certain period of time which maky make results period specific.
Point Estimate vs Confidence interval estimate
-a point estimate involves the use of sample data to calculate a single value (a statistic) that serves as an approximation of an unknown parameter. sample mean is a point estimate of the population mean for example
Discrete Random Variable
-a random variable is a variable whose outcome cannot be predicted (the number of cars that will cross a traffic light during a ten minute period -a discrete random variable is one that can take on a countable number of values. each specific probability of occurring which can be measured (common examples, toss of a coin, throwing a fair die, # of cars sold by a salesman in a week, number of bids put in for an item, number of people in a sample who prefer coke over pepsi)
Binomial Distribution and Bernoulli specifically here
-an experiment has only 2 possible outcomes, is mutually exclusive, and collectively exhaustive (no other outcomes are possible) Such an experiment is called a Bernoulli trial, if this is carried out n times the number of successes X, is called a bernoulli random variable, and follows bionomial distribution Bernoulli defined as X~B(n,p) n-number of trials p=prob of success so prob of x successes in n trials is giben by P(X=x)=nCx(p)^x*(1-p)^(n-x) p=prob of success x=# of successes 1-p=prob of failure note that nCx is computed as n!/(n-x)!x! -assumption of bernoulli trial ex 4 reading 9 You must assume that 1) the probability of an earnings increase (success) is constant from year to year and 2) earnings increases are independent trials. If current and past earnings help forecast next year's earnings, Assumption 2 is violated. If the company's business is subject to economic or industry cycles, neither assumption is likely to hold.
Stochastic Oscillator
-based on assumption that in an uptrend the stock price tends to close near the high of its recent range, while in a downtrend it tends to close around its recent low %K = 100 * (C-L14)/(H14-L14) C=last closing price L14 = Lowest price in last 14 days H14= highest price in last 14 days %D (signal line) = Avg of the last three %k values calculated differently when >80 it . usually indicates that the security is overbought and should be sold. A value lower than 20 indicates that the security is oversold and should be purchased when %K crossed %D from below it is a short term bullish singal, if it crosses %D from above it is a bearish symbol
simple random sample
-each member of the population has the same probability of being included in the sample.
Sampling Error
-error caused by observing a sample instead of the entire population. It is the difference between the sample statistic and the corresponding population parameter Sampling error of the mean = sample mean - population mean x̄-μ
Principles and assumptiosn
-people behave in irrational and emotional manner, and tend to behave similarly in similar circumstances -technicians believe that security price movements occur before fundamental developments occur or are reported.
Sampling Distribution
-prob distriubtuion under repeated sampling of the population -the distribution of these sample means is called sampling distribution of the mean
Discrete Uniform Distribution
-probability of each of the outcomes are the same (6 sided die. ***The CDF of these functions look like stairs adding up the y axis probability all the way to 1
Central Limit Theorem
-sampling distribution of the sample mean computed from sample size, n will be approximately normal with the population mean, and variance, when the sample size is greater than or equal to 30 **** It allows us to make accurate statements about the population mean and variance using the sample mean and variance REGARDLESS OF THE DISTRIBUTION OF THE POPULATION, as long as the sample size is adequate >=30 we assume normality (still divide by n though)
Multivariate distribtutions
-specify probs associated with a group of random variables taking into account the interrelationships that may exist between them -portfolio returns are said to have a multivariate normal distribution -for example 4 assets in a portfolio would be 4 means, 4 variances, and 6 correlations remember return correlations between each possible pair of stocks there will be n(n-1)/2 pairwise correlations in total -this need to determine correlations is what differentiates multivariate normal distributions from univariate normal distributions
Stratified Random Sampling
-strata must be mutually exclusive, each member assigned only to one stratum, so if 150 males out of 350 are in a scool you would want to sample atleast 43% males. within each stratum, observations are selected randomly
Continuous Patterns next cards
-used to confrim the resumption of the current market trend, aslo known as healthy market corrections
Question 20
20.An analyst stated that normal distributions are suitable for describing asset returns and that lognormal distributions are suitable for describing distributions of asset prices. The analyst's statement is correct in regard to: xA.both normal distributions and lognormal distributions. xB.normal distributions, but incorrect in regard to lognormal distributions. C.lognormal distributions, but incorrect in regard to normal distributions C is the correct answer here
Lognormal distribution
A random variable, Y, follows the lognormal distribution if its natural logarithm, ln Y, is normally distribtued 3 important features 1. it is bound by zero on the lower end 2. the upper end is unbounded 3. it is skewed to the right -Lognormal distribution is used to model asset prices (for example, holding period return can range between -100% and +infinity) - this would inherantly skew the data to the right, where as if we take the lognormal distribution, we can create a lower bound of zero Used in investment decision making. A further distinction is an underlying assumption that the values used to derive a lognormal distribution are normally distributed. Let me clarify with an example. An investor wants to know an expected future stock price. Since stocks grow at a compounded rate, she needs to use a growth factor. To calculate possible expected prices, she will take the current stock price and multiply it by various rates of return (which are mathematically derived exponential factors based on compounding) and which are assumed to be normally distributed. When the investor continuously compounds the returns, she creates a lognormal distribution which is always positive, even if some of the rates of return are negative, which will happen 50% of the time in a normal distribution. The future stock price will always be positive because stock prices cannot fall below $0!
Montecarlo example numer 18 in online textbook practice questions
A standard lookback call option on stock has a value at maturity equal to (Value of the stock at maturity - Minimum value of stock during the life of the option prior to maturity) or $0, whichever is greater. If the minimum value reached prior to maturity was $20.11 and the value of the stock at maturity is $23, for example, the call is worth $23 − $20.11 = $2.89. Briefly discuss how you might use Monte Carlo simulation in valuing a lookback call option. Answer In the text, we described how we could use Monte Carlo simulation to value an Asian option, a complex European-style option. Just as we can calculate the average value of the stock over a simulation trial to value an Asian option, we can also calculate the minimum value of the stock over a simulation trial. Then, for a given simulation trial, we can calculate the terminal value of the call, given the minimum value of the stock for the simulation trial. We can then discount back this terminal value to the present to get the value of the call today (t = 0). The average of these t = 0 values over all simulation trials is the Monte Carlo simulated value of the lookback call option
Calculating probabilities for z scores
A z table shoes the probability of a random variable having a standardized value lower than or equal to the given standardized value or P(Z<=z) -pg222 the table is given with tenths on one column and hundreths on the other ie. .1 .2 .3. 4 and on x .01 .02. 03 so if you wanted z score .33 you go to where they intesect. this value is equal to the probability that a random variable have a prob LESS THEN OR EQUAL TO do 1-zscore to find prob area greater then or equal to pg225 finding in between, you must subtract or add the probabilities depending, you've done this before
Shortfall risk
refers to the probability that a portfolio's value or return E(Rp) will fall below a praticular target value or return (Rt) over a given period (p and t are subscripts here) -Roy's Safety first criterion states an optimal portfolio minimizes the prbability that the actual portfolio return Rp will fall below the target return Minimize P(Rp<RT) Rp=portfolio return Rt=target return ***the minimum level is also called the threshold level, if portfolios returns are normal with a mean E(Rp) and a standard deviation σp (psubscript not prob) we can calculate the probability that returns will be lower than the threshold level Shortfall ratio or z-score = (E(Rp)-Rt)/σp -the higher the value of the SF ratio (or z score), the better the risk/return tradeoff the portfolio offers. A portfolio with a higher SF ratio also has a lower probability of attaining returns lower than the threshold level. because we are dividing by the standard deviation it is taking risk into account, so larger risk would create a lower shortfall ratio Conclusion: Portfiolios with higher SF ratios are preffered to those that have a lower SF ratio. Higher SF ratio portfolios have a lower probability of not meeting their target returns ***This is different than the sharpe ratio b/c Rt or target rate is typically higher then the risk free rate, unless you're target is the risk free rate which is unlikely for a good investor
When samples of the two populations whose means we are comparing are dependent, the paired comparisons test is used. Dependence can result from events that affect both populations. For example observed returns on two stocks over time are influenced by market and economic decisions
remember it must be normal
Hypothesis Testing for Variances of a normally dist pop
so far in the reading we have been testing hypotheses relating to means, sometimes we need to do for variances
Normal Distribution
states as X ~ N (μ,σ^2) -has a skewness of zero -kurtosis=3 and excess kurtosis equals 0 -probability of each random variable ranges further away from the mean and gets smaller and smaller, but never goes to zero, either tail extends to infinity -univariate distributions describe the distribution of a single random variable, up to this point we have only focused on univariate
LOS12a: Explain principles of technical analysis and its applications and underlying assumptions
technical analysis - security analysis technique that involves the examination of past market trends (using data such as prices and trading volumes) to predict future behavior of the overall market and of individual securities. Thought of as a study of collective investor sentiment. Technical analysis does not require in depth knowledge of the security begin analyzed, and can therefore be performed relatively quickly, while fundamental analysis takes longer Based on the following -Supply and demand determine prices in real time -changes in supply and demand cause changes in price -prices can be projected with charts and other technical tools Technical analysis is the only tool availbe to intrinsic values such as commodities and currencies. -sometimes an investor will use fundamental anlaysis and then technical analysis to determine when to best purchase the stock
Chi - Squared χ 2
three important features: 1. it is asymmetrical 2. it is bounded by zero, chi-square cannot be negative 3. it approaches normal as the degrees of freedom increase (skewed to the right) χ 2 =(n-1)s^2/ hypothesized value for population variance now we are just comparing χ 2 to critical chi squared values
Time series data cross-sectional data
time series data - data over a period of time spaced at uniform intervals cross sectional - observing many subjects at the same point in time
Confidence Interval
α = level of signification at 5% significance, that means 95% confident
LOS9p: DIstinguish between discretely and continuously compunded rates of return and calculte cont comp rate of return given a specific holding period returnX
EAR (Effective annual rate) EAR=e^(rcc) - 1 rcc=continuously compounded annual rate or rcc = ln(EAR+1) or rcc=ln(Vt/VO) remember HPR = EAR so can substitute (if talking in years and HPR = VT/Vo now if HPR is 2 years HPRt = e^(rcc*t) - 1
Continued
Essentially predicting a population parameter if you can prove that the test statistic is way farther away in standard deviations then the predicted parameter - then the parameter is wrong pg251 One Tailed vs Two Tailed Tests one-tailed test - > or < (can have equal sign attached) two tailed test are = or does not = the critical value is how many standard deviations away from the mean and are given Reject Ho when: Test statistic> positive criticalcritical value Fail to Ho: When test statistic<= positive ccritical value when left side test or negative crit and test stat Reject when t stat < negative critical value fail to reject when t stat >= neg crit value its easier two imagine this when looking at graphs pg 252
To compare equality of the variances of two normally distributed populations based on two independent random samples = F-Stat
F-test F=s^2/s^2 variance of each sample *Always put the larger variance in the numerator when calculating F-test Features 1. skewed to the right 2. bound by zero lon the left 3. it is defined by two seperate degrees of freedom
Momentum or Rate of Change Oscillator
M=(V-Vx)X100 M=momentum oscillator value V=last closing price Vx= closing price x days ago (Typically 10) used to identify changes in market sentiment. fluctuate in a rage or hover around a number typically b/w 0 and 100 Applications Oscillator range can be used to determine the strength of a trend -may signal a trend reversal when they reach historical highs or lows
Monte Carlo Simulation Los9q: Explain Monte Carlo simulation and describe its applications and limitations LOS9r: Compare Monte Carlo simulation and historical simulation
Monte Carlo simulation pg 232 -generates random numbers and operator inputs to synthetically create probability distributions for variables -investopedia -essentially take a stocks price and the next days price similar to above, once you find the standard deviation you -determines project variables and the range at which those variables can occur, associates a probability with each variable using a normal distribution (finds the mean and calculates a z score) then randomly establish relationships for correlated variables -apply random variables to run a sample size -chart the different options, and you can then determine the most likely result investopedia has a good chart -answers are only as good as the assumptions and model used -does not provide cause-and-effect relationships
Moving Averages
Moving average- average of the closing price over a given number of periods, moving averages smooth out short term fluctuations, and give a clearer picture of a market trend. simple moving average uses arithmetic mean exponential moving average attaches a greater weight to recent prices in computing the average Generally a stock that is in a downtrend tends to trade below its moving average sometimes more than one moving average is shown pg293 -there is a shorter term and longer term moving average. Golden cross- when the short term moving average intersects long term from below (bullish) -Dead-cross- short term moving average intersects long term moving average from above
Parametric vs Non-parametric test
Parametric - concerned with parameters, or defining features of distribution, it makes a definite set of assumptions non-parametric - worried about quantities other than parameters, assumptions made by parametric tests cannot be supported -when data is ranked (ordinaL) non-parametric methods are widely used.
Charts continued pg 282 Point and Figure Chart Scale VOlume
Point and Figure X is plotted as an increase in price while O is plotted as a decrease in price box size refers to minimal level of price change that will merit a O or an X reversal size - is typically a multiple of the box size, a reversal size of three means an analysts will move to the next column when the price reverses, or changes direction by three or more boxes Scale logarithmic scales are more appropriate when the range of data is larger, the length of time period depends on purpose, active traders typically prefer shorter time intervals Volume- used by technicians as a barometer of the strength of a trend. IF a security's price is increasing with increasing volumes, it shows that more and more investors are purchasing the stock at a higher price. This indicates that the trend is expected to continue as the two indicators confirm each other if a securit's price is rising with declining volumes, the two indicators are diverging, which suggests that the trend is losing its momentum as fewer investors are willing to buy at higher prices Daily price chart and volume chart (scale and volume are placed on top of each other
LOS9o: Explain the relationship between normal and lognormal distribtuions and why the lognormal distribution is used to model asset prices.
Quick refresher on logarithmic equations log is base 10 ln is base e
Relative Strength Index
RSI = 100 - (100/(1+RS)) RS = sum of (up changes for period under consideration)/ sum of (down changes for the period under consumption RSI lies between 0 and 100. A value above 70 typically represents an overbought situation while a value below 30 typically reflects an oversold situation. RSI is typically 14 day period
Relative Strength Analysis
Relative Strength Analysis is used to evaluate the relative performance of a security compared to a stated benchmark by plotting the ratio of the security's price to the benchmark index over time. An upward sloping line indicates outperformance whilile a downward sloping line suggest underperformance. pg 284. just graph index vs your stock or mutual fund or whatevs
Flags and Pennants
Show breaks in lines when pennant is at the expected price, essentially breaks up into triangles and rectangles and then theres a break in the lines (pg292)
LOS9m: Define the standard normal dist, explain how to standardize a random variable, and calculate and interpret probs using standard normal dist
Standard normal dist - prob statements for any kind of random variable can be made by referring to a singe normal curve **the stand norm dist has a mean of zero and a standard deviation of 1 To standardize a given observation we use a z-score z= (observed value -population mean)/ standard deviation z=(x-μ)/σ ****essentially the z score represents the number of standard deviations from the population mean a given observation lies so a z score of 2 means the observation lies 2 standard deviations above the mean
Continued Double Tops and Bottoms and Triple Tops and Bottoms
Stronger the rally preceding the head and shoulders, the more pronounced the expected reversal Double tops - when an uptrend prices reverses twice at approximately the same price level. the deeper the value and the longer the time period b/w the two tops, the more significant the formation is considered to be Double bottom - price expected to rise above the peak between the two bottoms by approx an amount equalt to the distance b/w bottoms and the peak Triple- Rare but when they occur they indicate mroe significant reversals then double tops and double bottoms
Technical vs Fundamental Analysis
Technical uses only trading data, which includes market price and volume information. Fundamental analysis uses external info (financial reports, macro analysis, etc) The data used by technical analysts is more concrete and reliable. Financial statements are subject to manipulation by management. -Fundamental analysis is more conceptual and determine long run intrinsic value of a security Technical analysis is more practical as it studies actual trading pattersns. ****Fundamental Analysts aim to forecast where a security Should trade, while technicians focus on prediciing the level at which it WILL trade Application of tech analysis is limited in markets that are subject to large manipulation, or rather illiquid
Hypothesis Tests concerning the mean
The decision to use critical values based on the z dist or the t dist depends on sample size, the distribution of the pop and whether the variance of the pop is known ( z test or t test)
When to use Lognormal and when to use normal function -lognormal essentially allows for a rightly skewed data to data to be normal by taking the ln of Vt/Vo instead of using return we use the ratio of the ending value of the investment to its beginning value 0<= Vt/Vo<= infinity Vt/Vo simplky equals 1 +(holding period return
The preceding description, although slightly complicated, was provided to help us arrive at what really matters for investors: when to use each method in making decisions. Lognormal, as we discussed, is extremely useful when analyzing stock prices. As long as the growth factor used is assumed to be normally distributed (as we assume with rate of return), then the lognormal distribution makes sense. Normal distribution cannot be used to model stock prices because it has a negative side and stock prices cannot fall below zero. **If a stock's continuously compunded return (growth variable) is nomrally distributed, then future stock price must be lognormally distributed
LOS9h: Calculate and interpret tracking error.
Tracking measure is a measure of how closely a portfolios returns match the returns of the index to which it is benchmarked. difference between total return on portfolio and total return on benchmark tracking error = gross return on port - total return on benchmark index alex has a 90% success rate calculate prob of 3 or less successful quarters out of 4 given a probability of success of .9 P(X<=3 = P(X=0) + P(X=1) + P(X=2) + P(X=3) so must use bernoulli formula for binomial distribution and then add the results
LOS12c: Explain uses of trend, support, resistance lines, and change in polarity
Trend analysis assumes investors tend to behave in herds and trends usually continue for an extended period of time. uptrend - higher highs and higher lows. each high lies above the previous high and each low lies above the previous low (retracement). in econ it means more ready buyers then are sellers downward- lower highs and lower lows narrow range or (sideways trend), typically options positions are more profitable than long or short positions on the security itself during a sideways trend.
Change in Polarity Principle (support and resistance levels)
Trend analysis involves the use of support and resistance levels. A support level is defined as the price at which there is sufficient buying interest, at this level investors believe security is attractive despite the recent price decline A resistance level is the price at which enough selling activity is generated to prevent any further increase in price. investors believe the security is overpriced Change in Polarity Principle - a key tent of trend analysis, once a price rises above the resistance level, it becomes the new support level, similarly once the price falls below a support level it becomes the new resistance level.
Type I and Type 2 Errors
Type I Rejecting the null when its actually true Type II : Failing to reject the null when it is actually false so the significance level represents the probability of making a type 1 error Sample sizew and choice of significance level together determine the probability of a Type II error. instead of concluding that the null is true we simply state the sample evidence is not enough to reject the null hypothesis (false or not false)
Desirable Properties of an Estimator
Unbiasedness- expected value is equal to the parameter being measured -efficience - one that has lowest variance of all unbiased estimators of the same parameter -consistency - one for which the probability of estimates close to the value of the population parameter increases as sample size increases. standard error falls as sample size increases.
Spearman Rank Correlation Coef
Used in non-parametric test - Nonparametric examples test concerning single mean - Wilcoxon signed rank test tests differences b/w means- Mann Whitney U Test Paired comparison tests (Mean differences), WIlcoxon signed rank test Sign test
Two Tailed Tests%
We assses wtheter the value of the population paramter is simply different from a given hypothesized value. ( either = or does not equal) that means the test stat can be greater then or equal to the mean, so a 5% significance level you have to divide the signficance level by 2. so it would be 2.5% on each tail, thus a critical value of 1.96 you would use the z table to find where the cumulative probabilty underneath the level of significance so here on z table you would go to where you find z of .975 which is 1 - .975=.025
Triangle Patterns
You draw these by connecting high and lows with a line over a period. Line that is horizontal if it is on top or bottom determines ascending or descending -when the range between highs and lows over a period narrows down on the price chart. eventually meets the lines connecting the lows forming a triangle (Very interesting but simple) see page 290 Ascending, Descending and Symmetrical triangles. When the two trend lines for highs and lows are both horizontal this is rectabnle patterns. -Rectangle patters say that investors are booking profits at the resistance level but reentering tat the support level, once price breaks out above the rectangle it will continue to rise ( this is a bullish rectangle example, there isbearish also) pg 291
LOS9l: Determine the probability that normally distributed random variables lies inside a given interval AKA CONFIDENCE INTERVALS
a confidence interval - range of values within which a certain population parameter is expected to lie in a specified percentage of the time. three most common confidence intervals 90% x̅ - 1.65s to x̅ + 1.65s 95% x̅ - 1.96s to x̅ + 1.96s 99% x̅ - 2.58s to x̅ + 2.58s s is the sample standard deviation remember 68 95 99.7 rule also 50% would be 2/3 of a standard deviation
Historical Simulation
assumes distribution of the random variable going forward depends on its distribution in the past -assumes future will be simlar to the past -no cuase and effect -a risk factor not represented in historical data will not be considered -does not allow for a what if analysis where as monte carlo does
Binomial Tree - pg 215
each part in the tree is one of two options. thus you find the total # of possible outcomes across the tree, such as up down down, or down up up, and find the probability by multiplying the probability of up or down along the way which is constant, that finds the probability, the multiply by the number of trials pg215 ** Quick note: skewness and the binomial distribution. -probability of success =.5 = symettric probability of success is less than .5 = skewed to the right more than .5 skewed to the left
expected value and variance of a binomial random variable:
expected value is simply p*n variance is given by σ^2 = n x p x (1-p)
Z score table reminder
for z = .36 the prob in the table .64 means that are under z=.36 64% of observations are below ore equal to z=.36 represented asP(Z ≤ 0.36). this is the are to the left of the z score
Example 2-8 pg 231 Volatility as used in option pricing model
given a list of stock prices over 5 days 1. estimate the volatility (annualize based on 250 days) 2. Identify the prob distribution for XYZ share prices if continually compounded daily returns follow the normal distribution 1. First we calculate continuously compunded daily returns Xt= ln(yt/yt-1), this means for every period we start with the previous as yt-1 -then we find (Xt-Xbar)^2 (the top of the variance equation -since we only have 4 full return periods (base year doesnt count) divide by n-1 which is 4 to get variance then find standard deviation -Annualilzed volatility = Daily standard deviation x (250)^.5 2. Since continuously compounded daily returns of XYZ stock follow the normal distribution XYZ prices should follow lognormal distribution
Hypothesis Testing
hypothesis - statement about the value of the population parameter. we conduct a hypothesis test on the sample information in order to be able to comment on the accuracy of the statement pertainign to the population paramter 1. null hypothesis - represents status quo, and we are interested in rejecting. null hypothesis will always include an equal sign, whether <= or >= = 2. alternate hypothesis - statement whose validity we are trying to evaluate example: in 49 games player averaged 36 points and stand deviation of 9 points. Determine accuracy of statment that his career average is greater then 30 poitns. Use 5% significance level (which is a critical value which is given 5% of observeled sample means lie more than 1.645 standard deviations above the mean the test statistic lies 4.67 standard deviations above the mean (the assumed 30 points) -the chance of a sample having a mean of 36 given a sample size of 49 and a standard deviation of 9 when the population mean equals 30 is less than 40% therefore we reject the null
Probability distribtuion
identifies the probability of each of the possible outcomes of a random variable 1. 0<p<1 2. sum of p(x) = 1 a prob function can be represented as P(X=x) X is the random variable while x represents different values it can take.
Identify approriate test stat and interpret results for a hypothesis test concerning the equality of population means of two at least approximately normally distributed populations, based on independent random sample with 1. equal or 2. unequal assumed variances Only need to identify the appropriate test and interpret results, do not need to know formulas (Z,stat, t stat, chisquared stat or F stat
if it is assumed the variances of the two populations are equal we use pooled variance where they are equal on equation sheet, don't need to know forumlats just when to apply. when variances ar assumed unequal we must account for each individually Test for pop mean when pop variances are assumed equal pg263 think of an example where we want to test equality of returns across two decades - here we would assume that variances are the same test for pop mean when pop variances are assumed unequal pg265 when variances are assumed unequal. such as difference in recovery rates for investors in financial sector and pharmaceutical sector debt at the 10% level of signficacne
Data sets can have both time series and cross sectional data Longitudinal data Panel data
longitudinal- data over time about multiple characteristics of the same observational unity (GDP growth rates, inflation Panel data - data collected over time about a single characteristic of multiple observational units
T-Dist
lower peak, and fatter tails for less then 30 as the degrees of freedom (or sample size increase the shape of the t dist approaches the shape of the standard normal curve -uses n-1 instead of n Practical interpretation for calculating confidence intervals with z or t score We can b XX% confidendt that the average population score for the actual SAT is b/w 1663 and 1836 -for standard error of t dist use
Continuous random variable
one for which the number of possible outcomes cannot be counted (infinite possible outcomes), and therefore probabilities cannot be attached to specific outcomes -for example, while the probability of a waiting period at a restaurant being between 25 and 30 minutes can be measured the measured. The probability of waiting for exactly 25 minutes and 15 seconds is 0 because time can take an infinite number of values like 100000ths of a second -return on stock is a continuous random variable -for continuous random variables we use a PDF - probability density function -probabilities of an outcome between a and b is the area under the curve thus why single points cannot be determined *** a cumulative distribution function (CDF) expresses the probability on a value less that or equal to a specific value of of
Explain and interpret the p - value
p value is the smallest level of significance at which the null hypothesis can be rejected, it represents the probaility of obtaining a critical value that would lead to a rejection of the null hypothesis Consider a two tailed tests where test statistic is 2.5 and critical values are 1.96 and -1.96, from z ttable we find prob of atatining a value greater than 2.5 standard deviations above the mean is .oo62. since this is two tailed p-value =.0124. this value tells us what we would reject the null hypothesis at the 5% significance level but not at the 1% level. so 1.24 is the lowest level of significance at which the null hypothesis can be rejected. pg258
Type of Tests Chart
pg 269 and on formula sheet
Technical Analysis Tools: Charts, Trend and Chart Paterns
pg 279, drawing charts on formula page Charts- used to illustrate historical price information Line Charts - graphical display of prices over time Bar chart - while a line chart has one data point for each value (horizontal=time) a bar chart presents four pieces of information, open, high, low and close. For each time interval the top of the line shows the highest price while the botom shows the lowest price. cross hatch to left indicates opening price, cross hatch to right = closing Candlestick chart pg 280- does the same thing as a bar chart but fills in to decide whether market closed high or low white = close>open closed high, black = closed<open or closed down
Describe Common Chart Patterns: pg 286 shoulder head neckline, shoulder
pg 286 left shoulder - strong rally with high volumes head - price starts to rise again and records a higher thant the one reached in left shoulder right shoulder - lower volumes Neckline is average line once a head and shoulders pattern has formed, prices are expected to decline VOlume is verty important in analyzing head and shoulder patterns, when one indicator is bullish (rising price) while another is bearish (lower volumes) it is known as divergence a downtrend precedes an inverse head and shoulders patern Selling price target uptrend Price Target = Neckline - (head-neckline) Selling price for downtrend is opposite neckline + (Neckline - head)
Standard error
population standard deviation/n^(1/2)
Calculated Statistical indices
put/call ratio - volume of put options traded divided by volume of call options traded. high pull call indicates a bearish market CBOE volatility index (VIX) - short term market volatility and is calculsted from the prices of options on stocks in the S&P 500, VIX rises when fearing a market decline -margin debt levels are strongly correlated with the movement in the market. margin debt reached its peak in 2007 for example Short term interest ratio = short interest/average daily trading volume -interpreted two ways, overall negative outlook on the security and one should expect the price to decline
