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Discrete Random Variable. Example?
the number of possible outcomes can be counted for each possible outcome there is a measure of positive probability.
Number of days it rains in a given month
Two Key properties of a Probability Function.
Each probablility is greater than or equal to zero, and less than or equal to 1.
Additionally the sume of all probabilities is equal to 1.
Continuous Random Variable Definition and Example.
Number of possible outcomes is infinitie, even if lower and upper bounds exist.
Example is dayily rainfall between zero and 100 inches is becaue the amount of rainfall can take infinite number of values. Rainfall can be measured in inches, half inches, quarter inches, etc.
Major Difference Between Continous Distribution and Discrete Distribution?
when probability of x = 0 for discrete, there's no probability of it happening. For continous it can happen, however it is one value wtihin infinitie, so it technically can't happen. You must put probability of occurence of a range of items.
Binomial Random Variable? Bernoulli Random Variable? What is the Binomial Probability Equation?
Number of Successes in a given number of trials where the outcome is defined by either success or failur. P = probability of success is constant for each trial, and is independent.
Bernoulli trial is the number of trials being 1.
Variance and expected value of a BBinomial Random Variable?
- Expected Value = N * P
- Variance of X=N*P(1-P)
Tracking Error? Example?
Difference between the total return on a portfolio and the total return on the benchmark against which its performance is measured.
Comparing a portfolio of US stocks with a comparable U.S. Stock Index
Continuous Uniform Distribution. 3 Rules. Equation given probablilites.
Devfined over a range that spans between a lower limit and upper limit which are parameters of distribution.
- 1. A is less than or equal to Xi < X2 is less than or equal to B
- 2. Anything outside the Limits of A and B has a probability of 0
- 3. is the equation below.
Probability of anything outside of parameters is zero.
Way to get probability is X2-Xi/ b-a, B being upper limit A being lower limit.
Normal Distribution has following properties
- - Is completely described by it's mean and variance
- -Skewness =0, mean=median=mode
- -Kurtosis =3
- -Linear combination of normally distruibted random variables is normally distributed
- -Probabilities of outcomes above and below mean get smaller but do not go to zero.
Multivariate Distribution? Can continuous and discrete variables have multivariate?
specifies the probabilty associated with a group of random variables and is meaningful only when the behavior of each random variable is in some way dependent upon the behavior of others.
Both continuous and sicrete can have multivariate.
What feature distinguishes a multivariate distribtuion from a univariate normal distribution?
Correlation. Indicates strength of linear relationship between two random variables.
How do you determine # of Corelations?
3 Confidence Intervals of Most Interest
- 90% = +-1.645
Standard Normal Distribution Definition? Z-Value? Standarization formula.
Standard normal Distribvbution is haveing a mean of zero and stand deviation of 1
Z-value is how many standar deviations away from the mean.
Formula is Observation-Mean/Standard Dev
Roy's Safety-first criterion? Threshold level?
the criterion states that the optimal portfolio minimzes probability that return of portfolio falls between minimum acceptable level. This level is called threshold level.
Minimize P (Rp<Rl)
- Rp= Portfolio Return
- RL=Threshold level return
- Very simliar to Sharpe Ratio except it's excess return over threshold rather than risk free rate.
- If normally distruibuted the SFRatio is =[E(Rp)-Rl]/Standard Deviation of Portfolio
Lognormal Distribution and Differences from Normal Distribution?
Lognormal is generated by the function e^x where x is normally distributed.
- -Lognormal distribution is skewed right
- -Lognormal distribution is bounded from below by zero so that it is useful for modeling asset prices which never take negative values
Discrete Compounding. Continuous compounding, formula.
Discrete compounding is compounding at a discrete compounding period such as semi annual or monthly.
Continous compounding is a constant compounding of an asset
The equation is e^.06-1 or whatever the rate is substituted for .06. Inversley, to get the annual rate for a continuous we would use ln(1+continous compound rate) to get the annual rate.
For a particular holding period return we see: ln(s1/s0) = ln(1+HPR)=Continuous rate of return
Over multiple periods T, HPRt= e^(continuous compounding rate*2) -1
Monte Carlo simulation? Steps? Uses? Negatives?
technique based on repeated generation of oneo r more risk factors affeting security value, in order to generate distribution of security values.
- 1. Specifidy probability distributions of stock prices and relevant interest rates as well as parameters(mean variance, skew)
- 2. Randomly generate stock price and interest rates
- 3. Value options for each pair of risk factor values
- 4. Calculate mean option value and use as estimate of option value
- -Value complex Securities
- -Simulate profits/loss from trading strategy
- -Calculate estimates of VAR to determine the riskiness of protfolio of assets and liabs.
- -Simulate pension fund assets and liabs over time to examine variability of difference
- -value portfolios of assets that have non-normal return distributions
- -Provide answers no better than assumptions of distributions of risk factors/pricing
- -Simulation is not analytical, but statistica, cannot provide insights that analytics can.
Difference between historical simulation vs. Monte Carlo
Negatives of history
Historical is based on actual changes in value or risk factors that occured over a prior period. Rather than model the distribution of risk factors, you use changes over a period of time.
- -Suffers bc past is not always indicative of future.
- -Evants that occur infrequently may not be represented in the time frame
- -it can't address what if questions monte carlo simulation can.
Systematic Sampling? Simple Random Sampling?
Simple Random is method of selcting in a way that each item in population being studied has same lielihood of being included. Like picking out of a hat.
Simple Random Sampling is selecting every nth member of a population
Sampling Error is?
Error between sample means variance and standard deviation and population.
Population mean-Sample mean = mean Sampling error
Sampling Distribution? Difference between distribution of actual prices.
Distribution of sampling is the picking 100 bonds out of a group of 1000. Repeating the process comes up with many different estimates of the population mean return, and creates a sampling distriubtion. The actual distribution of the 1000 bonds is different.
Stratified Random Sampling?
Pooling into different groups based on characteristics and picking a certain number from each pool depending on how much weight the pool is relative to entire portfolio
Difference between Time-series data and Cross-Sectional Data?
Time series is observations over a period time, at specific time intervals. Example is Microsoft Monthly Returns from 2008-2010.
Cross Sectional data are sample observations taken at a single poiont in time. The sample of reported earnings per share of all NASDAQ companies as of December 31st.
Longituadal Data vs. panel Data
Longitudal is observations over time of multiple characteristics of same enitty, IE, Unemployment, inflation, and GDP.
Panel is obervations over time of same characteristic for multiple entities.
Central Limit Theorum? Important Properties of CLT?
Central Limit Theorum states for simple random samples of size n from a population, with a mean and a finite variance, sample mean approaches population mean and a variance equal to (Variance of population/N) as it grows larger. Regardless of population distribution, inferences can be made as long as greater or equal to 30.
-If the sample size is equal to greather than 30, the sampling distribution of the sample means will be normal.
- -The mean of the population and mean of distribution of all possible sample means are equal
- -variance of distribution of sample means is (population variance/sample size) or (standar Dev)/Root of N
How do you get standard error of sample mean without using population Variance?
Sample Standard Deviaton/ Root of n
You want an estimator Estimator to be what 3 things? Define?
Unbiased, you want the estimator to be unbiased, meaning the expected value of the estimator is equal to the paramter trying to estimate. For example expected value of sample mean, to be population mean.
Efficient, meaning the variance of its sampling distribution is smaller than all other unbiased estimators of paramter trying to estimate.
Consistent, meaning the accuracy of paramter estimate increases as sample size increases.
Point Estimates are? Confidence Intervale?
Point Estimate is single sample values used to estimate population paramters. For example point estimate of a population couuld be the sample mean which has an estimator of sum of xi.../n
Confidence interval is range of values in which population apramter exped to lie.
T-Distriubtion? Properties, and degrees of freedom?
T-Distribution is a bell shaped probability dist. that is symmetrical. Use to construct confidenc intervals based on samples smaller than 30 from populations with unkonw variance and normal distribution or when population variance is uknown and sample size is large enough that CLTheorum assure sampling distribution is normal.
- -Defined by degrees of freedom = n-1
- -as degree of freedom gets larger, distribution gets closer to normal.
-Has more probability with fatter tails than normal distribution
When compared to normal distribution, t-distribution is ?
flatter with fatter tails
What is Confidence Interval for a known Population Standard deviation and normal distriubiton?
Mean+-(alpha)*(Standard Dev/Root of N)
What is Confidence Interval for an unknown Population Standard deviation and normal distriubiton?
Mean+-(T Confidence interval/2)*(S/root of N)
T Confidence Interval is one tail on chart, so if looking for 90% confidence, must look up (1-.9)/2=.05
Confidence Interval for a Pupulation mean when variance is unkown given a large sample from any type of distribution?
If distribution is nonnormal but population varince know, we can use z-stat as long as over 30 observations
if dstriubtion si noonormal and poulation variance is unkown, t statistic can be used as long as sample is larger than 30. * z-Stat can also be used but not as conservative.
What are the two exceptions that can be used to combat the Larger the sample the size the better claim?
1. Cost of More observations could be more, and not always apporpriate
2. Possibility that observations can come from a different distrubution with different population parameter, which can throw off the population parameter estimates
What is Data Mining and data mining bias?
Data mining occurs when analysts repeatedly use the same database to search for patters or trading rules until one that "works" is discovered.
Data Mining Bias is results where statistical signifcance of the pattern is overestimated bc results were found through data mining.
Sample selcetion bias?
Occurs when data is systematically excluded from analysis usually bc of lack of availiability. Making sample non random
most common form of sample selection. Only looking at survivors, and dropping one's that ceased.
Look ahead bias?
Occurs when a study tests a relationship using sample data not avialbe on the test date.
Time Period Bias?
if time period over which data is gathered is either too short or too long
What are the steps to the Hypothesis testing procedure?
Specify Hypothesis - Select appropriate test statistic- specify level of significance - state decision rule regarding the hypothesis - Collect the sample and calculate sample statistics- make a decision regarding the hypothesis - make a decision based on the results of the test
What is a null hypothesis?
it is the hypothesis that the researcher wants to reject. it is the hypothesis that is actually tested and is the basis for the selction of test statistic.
is what is concluded if there is sufficient evidence to reject null hypothesis.
What is the test statistic?
Difference between sample statistic and hypothesized value, scaled by standard error of sample statistic.
Equation is = Sample statistic-Hypothesized Value/Standard error of sample statistic
What is Type 1 Error? Type 2?
Type 1 error is the rejection of the null hypothesis when it is actually true. Alpha is the probability that we will reject a true null hypothesis.
Type 2: faliure to rejct null hypothesis when it is actually false
Power of a test is what?
Correctly Rejecting a Null Hypothesis when it is false
What is P-value?
P-Value is probability of obtaining a test statistic that would lead to a rejection of the null hypothesis.
When do we use the t-test? What is the equation to get the t-statistic?
- Use ttest if population variance is unkown, and either of following conidstions exist:
- Sample is larger than 30
- Sample is less than 30 but distribtuion of population is normal or approximatley normal
= (sample mean-hypothesized population mean)/(Standard deviation of sample/root of N)
When do we use the Z-test? What is the equation to get the Z-statistic?
Z-test is only used if the population is normally distriubed with known variance.
Z test is same as (mean of sample-hypothesized mean)/(Standard Deviation of population/Root of N)
Level of significance and corresponding z-values. For two tailed test and one tailed.
- 1 Tailed Test
What do we look at when looking at test results concerning the means of two normally distributed populations?
First off we only accept to look at if they are independant and if normally distributed.
We measure wehn population variances are unkown, but assumed to be either equal ro uneqaul.
Know that when Sample means are further apart, the larger the t-statistic and we reject equality.
Difference of means and significance of the mean of the differences between paired observations.
First shows whether or not means are same or not. Second shows whether differences of means averaged are same.
How do we test differences in variance? What is distribution look like? Equation?
Chi Squared Test. Distribution is Asymetrical Skew right. (Sample size-1)*Variance/Hypothesized value for population variance
F-distributed test statistic? Assumptions?
F-test concerned wtih equality of variances of two populations. Normally distrubted and independant assumptions.
The Test Statistic is the ratio of Sample Variances. * Always Put the larger value in numerator.
Rely on assumptions regarding distribtuio9n of population parameter. For example must test N to see if over 30t o be inline with central limit theorum.
All tests like Z-Test, T-test, Chi Square, F-Tests
Non paramteric tests?
either do not consider a particular population paramter or have few assupmtions about the population that is sampled.
- 1. when assumptions do not allow for t-tes or z-test/ paramteric tests are not met
- 2.data is ranked on an ordinal scal rather than values
- 3.hypothesis does not involve paramter of distribution, like whether variable is normally distributed.
Spearman rank correlation test?
used when data are not normally distributed.and involved with ranking. Like if mutual of 20 over two year. ranked 1-20. If corelation of spearman rank is .85 that means a high rank is assocaited with a high rank in the second year, and a negative rank would mean a good rank in first year indicates a bad rank in second year.
Technical Analyis?Fundamental Analysis? Key Assumptions? Advantages?Disadvantages?
- Technical Analysis - Study of collective market sentiment expressed in buying and selling of assets. Uses only firms share price and tradiung volume data to project price.
- Funadmental Analysis - Looking at firms financial statements to project price.
- Key Assumptions:
- 1.Marekt prices reflect both rational and irrational investor behavior. (Assumption emplies efficient market hypothesis does not hold)
- 2.beleive investor behavior is reflected in trends and patterns that repat and can be identified to forecast price
- Advantages of technical analysis?
- -Can be applied to prices of assets that do not produce future c/f, such as commodities
- -Can be used to detect financiel statement fraud
- -limited to markets where price and volume reflect supply and deman. If mareket is illiquid, and subject to market manipulation, doesn't work
For a candelstick Chart, what is the difference between a clear/white and a filled segment?
Clear/white means that closing price is higher than opening price, and filled is closing price is lower than opening price.
support level, and resistance level? Head and shoulders pattern? Size of breakdown trend after head and shoulder? Double Top?
Support level, buying is expected to emerge that prevents further price decreases,
Resistance level, selling is expected to emerge to prevent price increase.
Head and shoulders = price approaches a range of values but fails to maintain above a certain price.
Size of breakdown of head and shoulder if price declines beyond neckline for head and shoulder, downt end is projected to continue from that price by the size of the head and shoulder pattern.
When the price reaches a level that the buying pressure becomes weak, and the price drops.
What is a continuation pattern? What are triangles?
Continuation pattern is a pause in a trend rather than a reversal. Triangles form when symmetrical triangle higher lows lower highs, ascending higher lows resistance level, or descending lower highs and support level. Sugest buying and selling pressure become roughly equal temporarily.
What are bolinger bands? Move closer when?
Band constructed based on standard deviaton of closing prices over last n periods.
Move closer when less volatility.
Oscillators?rate of change oscillator?Relative Strength Indix ?Moving average convergence/divergence ?Stochastic oscillator ?
Indicators based on market prices but scaled so they oscillate around a given value. Helps to indicate market overbough or over sold. Also help to identify convergence and divergence of oscillator and market prices.
ROC (rate of change oscillator) - 100 times the difference between last closign price and lclosing price n period away oscillates around 0. or Ratio of current price to past price oscillates around 100
Relative Strength Indix - based on total ratio of total price increases to total price decreases over periods. Ratio scaled 0 to 100, greater than 70 in overbought, less than 30 in over sold.
Moving average convergence/divergence - Exponetially smoothed moving avergaes placing more wight on recent observation. MACD line is difference between two exponetilaly smoothed lines. Signal line is an exponetilly smoothed moving average of MACD l ine. MACD line crossing above smoother signla line viewed as a buy signal and vice versa.
Stochastic oscillator - calculated from lastest closing price and highest and lowes prices reached in a recent period. %k is difference betwen lates price and recent low as a percen between difference between recent high and low. Poin
used to discern the view of potential buyers and sellers.
- -Opinon polls
- -Put/call ratio - ratio of put options to call options. As ratio increases, Bearish Market, vice versa.
- -Volatility Index - by CBOEx - Indicates volatility of options. High levels of VIX suggest investors fear declines. and bullish market
- -Margin Debt - As Margin Debts increase, people are trying to buy moreoptions, and bearish sign.
- -short interest ratio - number of shares investors have borrowed and sole short/average trading volume.
Arms index or Short term trading index (TRIN)
-Mutual fund cash postion
-new equity issuance
-Measur eof funds flowing into advancing and declining stocks.
- TRIN= (#of advancing issues/Number of declining issues)/(volume of advancing issues/Volume of declineing issues)
- Greater than 1means volume in declining stock.
-Mutual Fund cash position - as market is bulish, mutual funds put money into investments leaving no cash, and contrarian theory to beleive that it's leading to a bearish market.
-new equity issuance means that selling when market is peaking
4 year cycle, 10 year cycle, 54 year cycle.
- Presidential cycle
- CDecennial pattern
- Kondratieff Wave
Elliot Wave Theory
- -Financial market prices can be described by interconnected set of cycles.
- -Uptrend is 5 moves up then down, hit peak then 3 up/down. Downtrend is 5 moves down, hit bottoum 3 up/downs.
- -Numbers move according to fibonaccia number(0,1,1,2,3,5...). .618 and 1.618 are ratios that are used to predict.