Time Value of Money
ordinary annuity END
annuity due BGN
 Perpetuity
 payment for infinite time period
 where PV = Present Value of the Perpetuity, A = the Amount of the periodic payment, and r = yield , discount rate or interest rate.
Discounted Cash Flow
 Holding Period Return
 true yield
 (ending value + dividend)/beginning value  1
 Bank discount yield
 percent discount form face value
 Effective annual yield
 for n compounding periods per year:

 for infinite time period:
Statistical Concepts
 measurement scales(NOIR)  Nominal(no context; weakest), Ordinal(accoding to characteristics), Interval(special meaning to difference to numberical values), Ratio(scale amounts; strongest)
 Correlation: degree of linear dependence between the variables
 1 in the case of an increasing linear relationship,
 ?1 in the case of a decreasing linear relationship,
 some value in between in all other cases
 the closer the coefficient is to either ?1 or 1, the stronger the correlation between the variables.
 Permutation: order matters nPr = n! / (nr)!
 Combination: order does not matter nCr = n! / r! (nr)!
 Chebyshev’s inequality : no more than 1/k2 of the values are more than k standard deviations away from the mean.
 Sharpe ratio: measure of the excess return (or Risk Premium) per unit of risk in an investment asset
 Roy's SafetyFirst criterion
P(Ra < Rm) = [E(R_{P})R_{L}]/σ_{P}
 Skewness
 measure of the asymmetry
 Positive skew: mode < median < mean
 negative skew: mode > median > mean
 Kurtosis
 measure of the "peakedness"; can be either leptokurtic or platykurtic
Probability Concepts
 Probability distribution: of all possible outcomes for a random variable
 discrete distribution: finite number of possible outcomes
 continuous distribution: infinite number of possible outcomes
 Probability function p(x): probability that a discrete random variable will take on the value x
 Probability density function f(x): probability a continuous random variable will take on a value within a range
 Cumulative distribution function F(x): probability a random variable will be less than or equal to a given value
 Binomial random variable: probability of exactly x successes in n trials
 Confidence interval: a range of values around an expected outcome within which we expect the actual outcome to occur some specified percent of the time
 90% confidence interval = Χ ± 1.65σ
 95% confidence interval = Χ ± 1.95σ
 99% confidence interval = Χ ± 2.58σ
 Degrees of freedom
 sufficiently high df is approximately normal
 higher degrees of freedom, thiner tails
 Standard normal distribution: μ=0 σ=1
where
plugin zvalue to get F(z) from ztable
C_{V} = σ / μ
 Monte Carlo Simulation: to estimate distribution of derivatives prices or of Net Present Values
 Continuous compounding = ln(1+HPR)
Sampling and Estimation
 Sampling: to make inferences about the parameters of a population
 timeseries data gathered from each time periods
 crosssectional data data from a single time period
 Stratified sampling: random picks within subgroups
 Central limit theorem
 sample mean for large sample sizes will be distributed normally
 as sample size increases, becomes more accurate in respect to population data
 holds for n > 30
 Test statistic: difference between population sample and hypothesized value
 Student's ttest
 used when sample size is small or variance unknown
used when H0: σ² = σ_{0}<super>2</super>
 Confidence interval (level of significance:probability of rejecting true H_{0})
 68% of observations fall in ±1σ
 95% of observations fall in ±1.96σ
 99% of observations fall in ±3σ
 Types of Bias
 datamining~ : repeatedly doing tests on same data sample
 sample selection~ : sample not really random
 survivorship~ : sampling only surviving firms
 lookahead~ : using information not available at the time to construct sample
 timeperiod~ : relationship exists only during the time period of sample data
Hypothesis Testing
 H_{0}: hypothesis set up to be nullified or refuted in order to support an alternate hypothesis.
 H_{1}: alternative hypothesis
 Type I error(Significance level): rejecting a null hypothesis when it is actually true; decreases as confidence interval(tradeoff) increases
 Type II error(1  Power of test): failing to reject a null hypothesis when the alternative hypothesis is the true; increases as confidence interval increases
 Volatility estimation
 unbiased~ : has an expected value equal to the true value of the population parameter
 consistent~ : more accurate the greater the sample size
 efficient~ : has the sampling distribution that is less than that of any other unbiased estimator
 Statistical significance omits transaction costs, taxes, risk factor from economical significance
Technical Analysis
 Assumptions
 stock values determined by supply & demand
 S&D driven by both rational & irrational behavior
 security prices move in trends
 changes observed in market price behavior
 Fundamental analysts > look for changes in stock values; stock prices adjust quickly to new information
 Technical analysts > look directly for signals & indicators of changes in S&D; stock prices move in trends that can persist for long periods
 + quick&easy, psychological reasons
  subjective judgement required for interpretation, no evidence in price trends, value neutralized
 Technical indicators
 Contrarian~ : do opposite of what majority are doing
 Smart money~ : mimic investors known for investment success
