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## Time Value of Money

ordinary annuity END

annuity due BGN • Internal Rate of Return
• set NPV = 0
• IRR -> CPT • 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. • Risk Free 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

• Weighted average • Geometric mean • Harmonic mean • 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.  • Bayes formula • Permutation: order matters nPr = n! / (n-r)!
• Combination: order does not matter nCr = n! / r! (n-r)!
• Mean Absolute Deviation • Standard deviation • 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 Safety-First criterion

P(Ra < Rm) = [E(RP)-RL]/σP

• 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
• z-value standardization where plug-in z-value to get F(z) from z-table

• Covariance • Coefficient of Variation

CV = 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
• time-series data- gathered from each time periods
• cross-sectional 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
• Z-test • Standard error • Student's t-test
• used when sample size is small or variance unknown • Chi square test

used when H0: σ² = σ0<super>2</super>

• Confidence interval (level of significance:probability of rejecting true H0)
• 68% of observations fall in ±1σ
• 95% of observations fall in ±1.96σ
• 99% of observations fall in ±3σ
• Types of Bias
• data-mining~ : repeatedly doing tests on same data sample
• sample selection~ : sample not really random
• survivorship~ : sampling only surviving firms
• look-ahead~ : using information not available at the time to construct sample
• time-period~ : relationship exists only during the time period of sample data

### Hypothesis Testing

• H0: hypothesis set up to be nullified or refuted in order to support an alternate hypothesis.
• H1: 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