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CFA1: Quantitative Methods

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.

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  • 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 71a583026c8839368732382b1af3c818.png 
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