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135 Cards in this Set
- Front
- Back
Valuation Methods
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Time Value of Money
Expected Value Arbitrage [#1 assumes no Risk, other two include Risk in the valuation process.] |
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Basic Time Value of Money
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Invest P0 at rate K for 1 year:
F1 = P0 * (1 + K), or P0 + P0* K. |
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Basic TVM Formula
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FN = PV * (1 + K)^N
where: PV is the present value, K is the interest or growth rate, N is the number of periods, and FN is the future value. [Given any 3, be able to calc the 4th.] |
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Two Kinds of Annuities
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Ordinary annuity
Annuity Due |
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Ordinary annuity
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equal periodic cash flows at the end of each period
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Annuity Due
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equal periodic cash flows at the beginning of each period
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Internal Rate of Return
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Find the interest rate, given PV, FV, PMT, and N periods.
For simple problems, use TVM keys to find i. For unequal cash flows, use IRR key. |
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Time-Weighted ROR
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the best measure of the portfolio manager performance because it is not affected by clients’ actions (adding or subtracting funds)
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Money-Weighted ROR
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is affected by clients’ actions, but may be a good measure for each client
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Continuous Compounded RoR (Formula)
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Log (1 + Discrete RoR)
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Three Means Required at Lv I
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Arithmetic Mean
Geometric Mean Harmonic Mean |
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Geometric Mean
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Form value relative: add 1.0 to each annual return expressed as a decimal.
Find the product of the value relatives. Take the nth root of the product, step 2, then subtract 1. |
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Measurement Scales
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Nominal
Ordinal Interval Ratio |
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Nominal
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names but does not order
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Ordinal
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– orders data but no + or -.
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Interval
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+ and – allowed but no division since no zero point.
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Ratio
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includes a true zero, all operations are allowed.
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Two Themes of Statistical Measures
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central tendency, ex. Mean,
Median, and Mode. dispersion, ex. for a single variable: Variance, Skewness, and Kurtosis, and for two variables: Covariance |
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Harmonic Mean (Formula)
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N / ∑1/D
Where N is number of values and D is the denominator or divisor |
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Chebyshev’s Inequality (Formula)
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The proportion of data values of x being within k standard deviations of the mean, where k > 1 is:
1 – 1/k2 Note this inequality holds regardless of the shape of the probability distribution. |
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Coefficient of Variation
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CV = s/avg
Risk per unit of average return. |
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Shortcomings of the Sharpe Ratio
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Negative Sharpe Ratio is possible in bear markets. Authors suggest consider some other decision criterion.
Sx may not be the best risk measure. It is OK for symmetric distributions (Normal), but not OK for skewed distributions. |
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Covariance (Formula)
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is a measure of the joint variation of two random variables.
∑ [X – Xmean] * [Y – Ymean] * pi If equal joint probabilities, then: for a population, use 1 / N for pi, but for a sample, use 1 / (N – 1) for pi. |
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Correlation Analysis
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a basic tool in measuring how two variables vary in relation to each other
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Correlation Coefficient (Formula)
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R(xy) = Covariance(xy) / [Std Dev(x) * Std Dev(y)]
+1: perfect linear relationship 0: no linear relationship –1: perfect negative linear relationship |
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Four Moments of a Distribution
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Mean or Expected Value
Variance Skewness Kurtosis |
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Kurtosis Platykurtic
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less peaked than the Normal Distribution, or flatter. Platy means broad.
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Kurtosis Mesokurtic
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as peaked as the Normal. Meso means medium
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Kurtosis Leptokurtic
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more peaked than the Normal. Lepto means slender.
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Safety-First Ratio (Formula)
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(Exp Rtn – Min Rtn)/Standard Deviation
Similar to the Sharpe Ratio where the Minimum Acceptable Return is the Risk Free Rate. |
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Mutually Exclusive
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no more than one of the events can occur at the same time
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Collectively Exhaustive
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the group of events represents all possible outcomes.
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Empirical Probability
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determined by observing the relative frequency when the activity takes place.
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A Priori
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based on logical analysis.
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Subjective
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determined based on personal judgment.
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Posterior
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a revised probability based on some new information about the event.
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Dutch Book Theorem
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Inconsistent implied probabilities involving two stocks can create a profitable arbitrage opportunity.
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Bayes’ Theorem
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P(Event | Information) =P(Information | Event ) * P(Event)/P(Information)
where: P(Event) is the prior probability. P(Event | Information) is the posterior probability, after the information is available. |
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Total Probability Rule
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P(negative ratio) = P(negative ratio|earnings decline) * P(earnings decline) + P(negative ratio|No earnings decline) * P(No earnings decline)
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Monte Carlo simulation
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Assume a probability distribution for some critical variables.
Repeat the model of the process many, many times and observe the distribution of results. |
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Binomial Distribution
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Each outcome is in one of two mutually exclusive categories.
Probability of an outcome remains the same from trial to trial, called Bernoulli Trials. Each trial is independent. N = number of trials, P = prob of result occurring |
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Sampling Methods
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Population
Simple Random Stratified |
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Parameter Statistic
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some characteristic of a population, ex. population mean, (often hard to determine in large populations).
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Sample Statistic
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a measurable characteristic computed from a sample.
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Sampling Error
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difference between the parameter and the sample statistic.
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Central Limit Theorem
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The sampling distribution of sample means is approximately normally distributed regardless of the shape of the population distribution.
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Estimator vs. Estimate
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Estimator is a formula, ex. sample mean.
The formula produces a resulting number, called the estimate. |
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Desirable Properties of a Point Estimator
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Unbiased
Efficient Consistent |
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Confidence Interval
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an interval where there is a (1 – α) probability that the population parameter is within the interval.
Usually α is 5%, so the probability of the interval is 95%. |
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Two Key Properties of a Probability Density Function
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The probability density function must be ≥ 0 for all values of the random variable.
The sum or integral over the range of possible random variables must equal 1.0. |
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Normal Distribution
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Two parameters are needed to specify this distribution: mean and standard deviation.
You must memorize the probabilities of: Mean ± 1 standard deviation. Mean ± 2 standard deviations. (Mean ± 3, optional.) |
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Reliability Factor
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Reliability Factor is the number of standard deviations to go out from the mean in each direction to construct a (1 – α) confidence interval.
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Standard Normal, or Z value
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(X – avg) / std dev of x
Standardized normal distribution has Mean = 0 and Standard Deviation = 1. |
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t-Distribution
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Similar to a normal distribution, but has:
fatter tails than a normal, and requires three parameters: mean, stnd dev, and Degrees of Freedom. Degrees of freedom represent the number of independent observations used. |
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Number of Degrees of Freedom
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N – 1 for hypothesis tests involving confidence intervals.
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Lognormal Distribution
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Used in valuation of options (Lv II) and other pricing models.
If the rate-of-return of a security is modeled well using the Normal Distribution, then the price of the security is often modeled using the Lognormal. |
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What is a Hypothesis?
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a statement about: the value of a population parameter, that is to be tested.
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Null Hypothesis
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a statement about a parameter of the distribution
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Alternative Hypothesis
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a statement about the same parameter, but logically opposite of the Null Hypothesis.
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Hypothesis Testing Process
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The Null Hypothesis is considered true
unless the sample gives strong evidence that the null should be rejected, and the alternative hypothesis should be accepted. |
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Accept vs. Reject
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Inside the (1 – α) confidence interval, then Accept the Null Hypothesis.
Outside the (1– α) confidence interval, then Reject the Null Hypothesis and accept the alternative hypothesis. |
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Wrong Decisions
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Type I: Reject the null when it is true.
Type II: Accept the null when it is false. |
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P-Values
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P-value is the smallest level of significance at which the null hypothesis is on the borderline of being rejected.
Large p-value → null likely true. Small p-value → null likely false. |
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One-Sided t-Test
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Follow the multi step process for Two-sided t-Test but use 5% in the tail, rather than the traditional 2.5%.
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2 Sided Z-Test
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Mean not known, but population standard deviation is known.
Rather than t-test, use a z-test (and z-table). |
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Real Risk Free Interest Rate
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nominal risk-free interest rate - Inflation Premium
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Required Rate of Return for an Investment
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minimum expected rate of return an investor is willing to receive in order to accept the investment
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Opportunity Cost
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The highest return that an investor forgoes by choosing whether to invest or not.
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Nominal Interest Rate
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A started annual rate. Does not get compounded more than once per year.
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Effective Annual Interest Rate (Definition)
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May have more than 1 interest period and therefore will compound during the year.
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Effective Annual Interest Rate (Formula)
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(1+ (i/n))^N -1
where: i:nominal interest rate (annual rate) n: frequency |
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Effective Annual Interest Rate (Continuous Formula)
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e^(r*t) - 1
where r: nominal rate t: years |
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Perpetual Ordinary Annuity (Formula)
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PMT/K
where PMT: payment K: Interest Rate |
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Discounted Present Value of a Perpetual Annuity Due (Formula)
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PMT/K * (1+K)
where PMT: payment K: Interest Rate |
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Perpetual Ordinary Annuity-Due to Present Value (Formula)
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P * (K/1+K)
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Future Value of Future Compound Cash Flows (Formula)
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NPV * (1+K)^N
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Annuity Due (Numerous Periods Formula)
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AD/(1+K)^(N-1)
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Holding Period Rate of Return (Formula)
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[[P(t)-P(t-1)] + CF(t)] / P(t-1)
where |
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Money Weighted Rate of Return (Definition)
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depends on both the amount invested and the rate of return during various sub-periods.
can be more significantly affected by the actions of clients due to deposits/withdrawals. can be a good measure for client performance. |
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Time Weighted Rate of Return (Definition)
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depends on the RoR during various sub-periods.
better measure the actions of the Portfolio Manager than the actions of the client. |
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Time Weighted Rate of Return (Formula)
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[(1+RoR(n))^(1/2)] -1
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Bank Discount Yield (BDY) Formula
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(FV-DV/FV) * 360/DTM
where: FV: Face Value DV: Discount Value DTM: Days to Maturity |
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Money Market Yield (MMY) Formula
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(FV-DV/DV) * 360/DTM
where: FV: Face Value DV: Discount Value DTM: Days to Maturity |
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Effective Annual Yield (EAY) Formula
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(1+HPY)^(365/DTM) -1
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Descriptive Statistics
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the study of how data can be summarized effectively to describe the important aspects of large data sets
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Statistical Inference
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involves making forecasts, estimates, or judgments about a Larger set of data form a small set actually observed
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Parameter
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a characteristic of a Population
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Statistic
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a characteristic of a Sample
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Ratio Scale
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measurement scale with includes zero.
Rates of Returns are measure on this scale. |
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Frequency Distribution
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is a display of data values summarized into a moderate to small number of intervals
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Geometric Mean (formula)
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Add 1 to each value, multiply.
[N(1) +1 * N(2) + 1]^(1/N) -1 |
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Harmonic Mean (formula)
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Total Number Sample/[1/N(1) + 1/N(2)]
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Determining Percentile Values
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N+1/100 * P
where P = the Particular Percentile |
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Range
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Highest Value - Lowest Value
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Population Variance (Formula)
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(Sum[X(i)-WM]^2)/N
Where: X(i)=Value, WM=Weighted Mean, N=Number of Values in Population |
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Sample Variance (Formula)
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Population Variance * N/(N-1)
or (Sum[X(i)-WM]^2)/[N] * N/(N-1) Where: X(i)=Value, WM=Weighted Mean, N=Number of Values in Population |
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Standard Deviation (Formula)
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Sum[(Value - WAM)^2 * Probability]
Where WAM=Weighted Average Mean |
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Mean Absolute Deviation (MAD) Formula
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Sum[(Value - WAM) * Probablity)
Where: WAM= Weighted Average Mean |
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Chebyshev's Inequality
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At least 75% of all data values in a distribution fall within 2 standard deviations from the mean
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Co Efficient of Variation Formula
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SD/RoR
where: SD=Standard Deviation, RoR=Rate of Return |
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Sharpe Ratio (Formula)
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(Avg. RoR - RfR)/SD
where: RfR=Risk Free Rate SD=Standard Deviation, RoR=Rate of Return |
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Harmonic Mean (Formula)
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Sum of: N/[1/P(n1) +1/P(n2)]
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Relationship between Harmonic Mean and Weighted Average Mean
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They are the same value.
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Advantage of Mean Absolute Deviation (MAD)
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gives relatively less weight to the more extreme data values than Standard Deviation
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Co Efficient of Variation Reciprocal Formula
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RoR/SD
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Sharpe Ratio Definition
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the difference between the mean rate of return and the risk-free rate of return divided by the standard deviation.
Measures excess mean reate of return relative to risk (Standard deviation) If positive, The Higher value, the better if negative, the higher absolute value is better. |
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Correlation Co Efficient Formula
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CV(xy)/[SD(x) * SD(y)]
Where: CV(xy)=Coveriance of x and y scenarios, SD(x)=Standard Deviation of x, SD(y)= Standard Deviation of y |
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Interpretation of Correlation Co Efficient
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If =1 then perfect linear relationship between x & y
if >0 && <1 then positive linear relationship between x & y if =0, no indication of linear relationship. if <0 && >-1 then negative linear relationship between x & y if ==-1 then perfect linear relationship between x & y |
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Safety First Ratio Formula
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For a Portfolio:
RoR-LR/SD where: RoR=Rate of Return, LR=min. acceptable rate of return, SD=Standard Deveation Note: Very similar to the Sharp Ratio if risk free rate=minimum expected rate of return |
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Odds Against (Formula)
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1-P(e)/P(e)
where P(e) is the probability of the event |
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Mutually Exclusive
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no more than 1 of the events can occur at the same time.
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Collectively exhaustive
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at least one of the events must occur
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A-Priori Probability
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based on logical analysis of the relevant activity and the particular event
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Empirical Probability
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determined by observing the relative frequency of occurrence of that event when the relevant activity takes place
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Subjective Probability
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determined by personal judgement
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Conditional Probability (of B given A) (Formula)
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P(A&B)/P(A)
if P(B) !=0 |
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Bayes' Theorem P(B|A) (Formula)
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[P(A|B) * P(B)]/P(A)
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P(A&B) (Formula)
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P(A|B) * P(A)
or P(B|A) * P(B) |
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Conditional Probability
P(B|A) |
P(A&B)/P(A)
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Combination Formula
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N!/[(N-R)! * R!]
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Permutation Formula
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N!/(N-R)!
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Normal Distribution Probabilities
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1 SD= 68%, 16% on each tail
2 SD=95%, 2.5% on each tail 3 SD=99.7, .03% on each tail |
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Observation to Z-Value (Formula)
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[X-Mean]/SD
where: X=observation, SD=Standard Deviation |
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Normal Distribution Confidence Interval Formula (Individual Values)
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Population Mean +/- Z * Pop-SD
where: Z=Confidence Interval, Pop-SD=Population Standard Deviation |
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Normal Distribution Confidence Interval Formula (Sample Values)
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Sample Mean +/- Z * (Pop-SD/[N^1/2])
where: Z=Confidence Interval, Pop-SD=Population Standard Deviation, N=Sample Size |
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T-Distribution
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Fatter Tails, and includes degrees of freedom vs Normal Distribution
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Degrees of Freedom Formula
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N-1
where: N=Sample Size |
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t-Distribution Confidence Interval Formula (Sample Values)
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Population Mean +/- (N-1) * (SD/[N^1/2]
where: SD=Sample Standard Deviation |
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Chi-Squared Distribution
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N Degrees of Freedom if independent values are calculated.
N-1 Degrees of Freedom if sample is calculated if >30 degrees of freedom, Normal Distribution is a good approximation Starts at 0 and is skewed to the right to positive infinity. |
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Type 2 Error
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when a null hypothesis is incorrectly accepted when it is actually false
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Type 1 Error
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when the null hypothesis is incorrectly rejected when it is actually true
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Power Hypothesis Test (Formula)
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1-Probability of Type 2 Error
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If P-Value >= Selected Significance Level
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Accept Null Hypothesis
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If P-Value < Selected Significance Level
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Reject Null Hypothesis
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P-Values in Hypothesis Testing
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the smallest level of significance at which the null hypothesis can be rejected
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