Quantitative Methods

Front 1

Liquidity risk

Back 1

The risk of receiving less than fair value for an investment if it must be sold for cash quickly.

Front 2

Interest rate on security formula

Back 2

Risk free + default premium + liquidity premium + maturity risk premium

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Effective Annual Rate

Back 3

EAR = (1 + periodic rate)^m – 1

periodic rate = annual rate / m

m = compounding times per year

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PV of Perpetuity

Back 4

PV of Perpetuity= Payment / Interest

Front 5

IRR Method vs NPV method

Back 5

Select projects with higher IRRs and Higher NPVs.

IRR needs to be greater than the firm’s (investor’s) required rate of return

If conflicting, select the project with higher NPV

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holding period return

Back 6

Percentage change in the value of an investment over the period it is held.

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Time-weighted rate of return

Back 7

1 + time-weighted rate of return)^n =
(1+return p1)(1 + return p2)....(1 + return pn)

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Method of returns used in asset management industry

Back 8

The time-weighted rate of return is the preferred method of performance measurement, because it is not affected by the timing of cash inflows and outflows.

Front 9

bank discount yield (and formula)

Back 9

Pure discount instruments are quoted on a bank discount basis, which is based on the face value of the instrument instead of the purchase price

(Discount to face)/ Face x (360 / days remaining)

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Holding period yield formula

Back 10

(Price rcvd at maturity + dividends rcvd) / initial price of the instrument -1

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effective annual yield

Back 11

An annualized value, based on a 365-day year, that accounts for compound interest. It is calculated using the following equation:

EAY = (1+HPY)^(365 / t) – 1

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money market yield (or CD equivalent yield)

Back 12

annualized holding period yield, assuming a 360-day year.

HPY × (360/t)

also- Face / (Face- (BDY * (days /360))) -1

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Bond-equivalent yield

Back 13

2 × the semiannual discount rate

Annual Rate^(1/2) *2

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IRR for perpetuity Formula

Back 14

Perpetuity / Initial Investment = IRR

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descriptive statistic

Back 15

sed to summarize the important characteristics of large data sets

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Inferential statistics

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pertain to the procedures used to make forecasts, estimates, or judgments about a large set of data on the basis of the statistical characteristics of a smaller set

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Nominal scales

Back 17

contains the least information- classified or counted with no particular order.

ex:assigning the number 1 to a municipal bond fund, the number 2 to a corporate bond fund,

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Ordinal scales

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every observation is assigned to one of several categories. Then these categories are ordered with respect to a specified characteristic

ex: within small stocks, ranked from best to worst performer

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Interval scale

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provide relative ranking, like ordinal scales, plus the assurance that differences between scale values are equal.

ex: temperature

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Ratio scales

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Ratio scales provide ranking and equal differences between scale values, and they also have a true zero point as the origin

ex: money

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parameter

Back 21

A measure used to describe a characteristic of a population

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sample statistic

Back 22

used to measure a characteristic of a sample

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frequency distribution

Back 23

tabular presentation of statistical data that aids the analysis of large data sets

ex: sp returns over time broken into diff frequencies say -20 to -15%, -15 to -10% etc etc

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relative frequency

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the percentage of total observations falling within each interval

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cumulative relative frequency

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summming the absolute or relative frequencies starting at the lowest interval and progressing through the highest.

Last catagory ends with the bar all the way to the top

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unimodal

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Data set has one that appears most frequently (trimodal is 3 etc etc)

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geometric mean

Back 27

nth root of[ (1+r1) (1+r2)...(1+rn)

(used often for returns)

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harmonic mean

Back 28

N / [(1/x1) +(1/x2) +....(1/xN)]

Used for avg share purchase price observations (dollar cost averaging)

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Quantile calculation

Back 29

(N+1)*(%max of quintile)/100

ex 15 observations, 8th decile
16*(80/100) = 12.8

So 80% of the way between observations 12 and 13 is the cut off for being in the 8th decile

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mean absolute deviation

Back 30

the average of the absolute values of the deviations of individual observations from the arithmetic mean.

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population variance (σ^2)

Back 31

average of the squared deviations from the mean

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population standard deviation

Back 32

sqr root of variance

aka

sqr root of average of the squared deviations from the mean

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sample variance s^2

Back 33

squared devations from the mean / (N-1)

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sample standard deviation (s)

Back 34

square root of variance

aka

square root of(squared devations from the mean) / (N-1)

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Chebyshev’s inequality

Back 35

Any set of observations, (sample or population data) and regardless of the shape of the distribution, % of the observations that lie within k standard deviations of the mean is at least (1 – (1/k^2))

ex: % within 2 std devs is 1- (1/4) or 75%

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coefficient of variation

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standard devation / average value

measures the amount of dispersion in a distribution relative to the distribution’s mean

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Sharpe measure

Back 37

(return - risk free rate) / standard deviation of portfolio returns

(return - risk free rate)= excess returns

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positively skewed distribution

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Mean > mean >mode

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negatively skewed distribution

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Mode > Median > mode

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Kurtosis

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measure of the degree to which a distribution is more or less “peaked” than a normal distribution

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Leptokurtic

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distribution that is more peaked than a normal distribution

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platykurtic

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distribution that is less peaked, or flatter than a normal distribution

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Sample skewness

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equal to the sum of the cubed deviations from the mean divided by the cubed standard deviation and by the number of observations

numerator can be positive or negative, depending on whether observations above the mean or observations below the mean tend to be further from the mean on average

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Sample Kurtosis

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sum of the ^4 deviations from the mean divided by the ^4 standard deviation and by the number of observations

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interpret kurtosis

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excess kurtosis = sample kurtosis – 3

Sample >3 means more "peaked"

Sample <3 means more "flat

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random variable

Back 46

uncertain quantity/number.

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rolling a die: random variable, outcome, event, mutually exclusive event, exhaustive events

Back 47

The number that comes up is a random variable. If you roll a 4, that is an outcome. Rolling a 4 is an event, and rolling an even number is an event. Rolling a 4 and rolling a 6 are mutually exclusive events. Rolling an even number and rolling an odd number is a set of mutually exclusive and exhaustive events.

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empirical probability

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probability established by analyzing past data

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priori probability

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probability determined using a formal reasoning and inspection process

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subjective probability

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least formal method of developing probabilities and involves the use of personal judgment

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Unconditional probability

(a.k.a. marginal probability)

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refers to the probability of an event regardless of the past or future occurrence of other events.

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conditional probability

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the occurrence of one event affects the probability of the occurrence of another event.

P(A | B)= prob of a GIVEN B has already occured

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joint probability

multiplication rule of probability

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P(A&B) = P(A | B) × P(B)

P(A&B) is same as P(AB)

probability that they will both occur

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addition rule of probability

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probability that exaxtly one of two events will occur

P(A or B) = P(A) + P(B) – P(AB)

Think Venn Diagram

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total probability rule

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P(A) = P(A | B1)P(B1) + P(A | B2)P(B2) + … + P(A | BN)P(BN)

used to determine the unconditional probability of an event, given conditional probabilities:

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Joint Probability of any Number of Independent Events

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P(A or B) = P(A) + P(B) – P(AB), and P(A and B) = P(A) × P(B)

odds of 3 heads is 1/2 *1/2 *1/2

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Independent events

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events for which the occurrence of one has no influence on the occurrence of the others

P(A | B) = P(A), or equivalently, P(B | A) = P(B)

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Covariance

Back 58

measure of how two assets move together

It is the sum of the (product of the deviations of the two random variables from their respective expected values given an event) times (odds of that event)

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correlation coefficient

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Correlation(R1,R2) = Cov(R1,R2) / STD(R1)xSTD(R2)

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Properties of correlation

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Correlation measures the strength of the linear relationship between two random variables

ranges from -1 to 1

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Portfolio variance(2 asset)

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Var(portfolio) = (wA^2)(var(a)) + (wB^2)(var(b)) + 2(wA)(wB)Cov(AB)

w=weight

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Bayes’ formula

Back 62

New probability = Prior prob of event x (Prob of new info/Unconditional Prob)

P(I |O) = [ P(O|I) / P(O) ] * P(I)

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Labeling Formula

Back 63

situation where there are n items that can each receive one of k different labels.

The number of items that receives label 1 is n1 and the number that receive label 2 is n2, and so on, such that n1 + n2 + n3 + … + nk = n

Total ways that labels can be assigned is:
N! / (n1! x n2! x n3! x..... nk!)

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Permutation Formula
(when order matters)

Back 64

n! / (n-r)!

n= total choices and r =number in order

Front 65

discrete random variable

Back 65

the number of possible outcomes can be counted, and for each possible outcome, there is a measurable and positive probability

p(x) = 0 when x cannot occur, or p(x) > 0 if it can

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continuous random variable

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the number of possible outcomes is infinite, even if lower and upper bounds exist

p(x) = 0 even though x can occur

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cumulative distribution function

Back 67

defines the probability that a random variable, X, takes on a value equal to or less than a specific value, x

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discrete uniform random variable

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probabilities for all possible outcomes for a discrete random variable are equal

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binomial random variable

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the number of “successes” in a given number of trials, whereby the outcome can be either “success” or “failure.”

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Bernoulli random variable

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A binomial random variable for which the number of trials is 1

Probability of exactly x Successes in N trials is

[n! / {(n-x)!*x!}] * (p^x)*(1-p)^(n-x)

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Tracking error

Back 71

difference between the total return on a portfolio and the total return on the benchmark against which its performance is measured

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continuous uniform distribution formula

Back 72

Find prob between x1 and x2

x2-x1 / (upper bound of distribution - lower bound)

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standard normal distribution

Back 73

normal distribution that has been standardized so that it has a mean of zero and a standard deviation of 1

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z-value

Back 74

Represents the number of standard deviations a given observation is from the population mean

z = (observation - mean) / standard deviation

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Roy’s safety-first criterion

Back 75

optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level (called the threshold level)

minimize P(Rp < RL)

where:Rp = portfolio returnRL = threshold level return

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Roy’s safety-first criterion for normal distribution

Back 76

Maximize:
(Expected return - threshold return) / Standard deviation

Front 77

lognormal distribution

Back 77

generated by the function e^x, where x is normally distributed

The lognormal distribution is skewed to the right.The lognormal distribution is bounded from below by zero so that it is useful for modeling asset prices which never take negative values

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Discretely compounded returns

Back 78

the compound returns we are familiar with, given some discrete compounding period, such as semiannual or quarterly

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continuous compounding

Back 79

e^r -1 where r is annual rate if compounded continuosly

ln (s1 /s0) = ln(1 +Holding prd Returns)

ln(1+ return) = rcc

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Additive feature of compound returns

Back 80

Given investment results over a 2-year period, we can calculate the 2-year continuously compounded return and divide by two to get the annual rate.

If Rcc = 10%, the (effective) holding period return over two years is e(0.10)2 – 1 = 22.14%

Front 81

Monte Carlo simulation

Back 81

technique based on the repeated generation of one or more risk factors that affect security values, in order to generate a distribution of security values

limitations of MC simulation - it is complex and is no better than the assumptions that are used

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Simple random sampling

Back 82

method of selecting a sample in such a way that each item or person in the population being studied has the same likelihood of being included in the sample

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systematic sampling

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Selecting every nth member from a population.

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Sampling error

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difference between a sample statistic and its corresponding population parameter (mean of sample vs mean of pop for ex)

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sampling distribution

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a probability distribution of all possible sample statistics computed from a set of equal-size samples that were randomly drawn from the same population

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Stratified random sampling

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classification system to separate the population into smaller groups based on one or more distinguishing characteristics

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Time-series data

Back 87

Observations taken over a period of time at specific and equally spaced time intervals

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Cross-sectional data

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a sample of observations taken at a single point in time

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Longitudinal data

Back 89

observations over time of multiple characteristics of the same entity, such as unemployment, inflation, and GDP growth rates for a country over 10 years

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Panel data

Back 90

observations over time of the same characteristic for multiple entities, such as debt/equity ratios for 20 companies over the most recent 24 quarters.

Front 91

central limit theorem

Back 91

For simple random samples of size n from a population with a mean µ and a finite variance σ2...

The sampling distribution of the sample mean x approaches a normal probability distribution with mean µ and a variance equal to (variance / n) as the sample size becomes large.

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standard error of the sample mean

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standard deviation of the distribution of the sample mean

When the standard deviation of the population, σ, is known, the standard error of the sample mean is calculated as:

stddev (sample) = std dev (pop) / sqroot (n)

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desirable properties of an estimator

Back 93

unbiasedness, efficiency, and consistency

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Point estimates

Back 94

Single (sample) values used to estimate population parameters

he formula used to compute the point estimate is called the estimator. For example, the sample mean, xbar, is an estimator of the population mean µ and is computed using the familiar formula:

xbar = sum all X / n

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Student’s t- distribution

Back 95

a bell-shaped probability distribution that is symmetrical about its mean. It is the appropriate distribution to use when constructing confidence intervals based on small samples (n < 30) from populations with unknown variance and a normal, or approximately normal, distribution

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properties of t- distribution

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defined by a single parameter, the degrees of freedom (df), where the degrees of freedom are equal to the number of sample observations minus 1

It has more probability in the tails (“fatter tails”) than the normal distribution.

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confidence interval for the population mean

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xbar +/- Reliability factor (using z) * Standard dev / sqrroot (n)

estimates result in a range of values within which the actual value of a parameter will lie, given the probability of 1 - α

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Confidence Intervals for the Population Mean: Normal With Unknown Variance

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xbar +/- Reliability factor (using t) * Standard dev / sqrroot (n)

Front 99

normal / non normal
known / unknown variance

t or z

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normal / known : Z
normal unknown: t
non normal / known: Z (as long as n>=30)
non normal / uknown: t (as long as n>= 30

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Data mining

Back 100

when analysts repeatedly use the same database to search for patterns or trading rules until one that “works” is discovered.

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Sample selection bias

Back 101

when some data is systematically excluded from the analysis, usually because of the lack of availability.

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Survivorship bias

Back 102

most common form of sample selection bias, your sample only looks at ones that "survived"

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Look-ahead bias

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occurs when a study tests a relationship using sample data that was not available on the test date.

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Time-period bias

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the time period over which the data is gathered is either too short or too long.

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Hypothesis Testing Procedure

Back 105

1. State hypothesis

2. Select test statistic

3. Specify level of significance

4. State the decision rule regarding the hypothesis

5. Collect the sample and calc statistics

6.Make a decision regarding the hypothesis

7. Make a decision based on results of the test

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null hypothesis

Back 106

hypothesis that the researcher wants to reject. It is the hypothesis that is actually tested and is the basis for the selection of the test statistics

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alternative hypothesis

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designated Ha, is what is concluded if there is sufficient evidence to reject the null hypothesis

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two-tailed test

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A two-tailed test for the population mean may be structured as:H0: μ = μ0 versus Ha: μ ≠ μ0

a two-tailed test uses two critical values (or rejection points). Reject null if test statistic > (<) than upper (lower) critical value

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Hypothesis test formula

Back 109

sample mean / [sample std dev / (sqrrt (n)]

Compare this to your critical value

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Type I error

Back 110

the rejection of the null hypothesis when it is actually true.

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Type II error:

Back 111

the failure to reject the null hypothesis when it is actually false.

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p-value

Back 112

probability of obtaining a test statistic that would lead to a rejection of the null hypothesis, assuming the null hypothesis is true

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t statistic

Back 113

t n-1 = [sample mean - hypothesized mean] / [stadard dev sample / sqrt (n) ]

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z statistic

Back 114

z stat =[sample mean - hypothesized mean] / [stadard dev population / sqrt (n) ]

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t test two variables

Back 115

t = [mean 1 - mean2] / {[(var 1 /n1) +(var 2 /n2)]^1/2}

The degrees of freedom, df, is (n1 + n2 – 2).

Front 116

chi-square test

Back 116

used for hypothesis tests concerning the variance of a normally distributed population

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chi-square test statistic

Back 117

chisqrd (w/ n-1 DF) = (n-1)* sample variance / hypothesized value for the population variance

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Parametric test

Back 118

rely on assumptions regarding the distribution of the population and are specific to population parameters

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Nonparametric tests

Back 119

either do not consider a particular population parameter or have few assumptions about the population that is sampled

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Spearman rank correlation test

Back 120

Can be used when the data are not normally distributed. Consider the performance ranks of 20 mutual funds for two years. The ranks (1 through 20) are not normally distributed, so a standard t-test of the correlations is not appropriate

A large positive value of the Spearman rank correlations, such as 0.85, would indicate that a high (low) rank in one year is associated with a high (low) rank in the second year

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relative strength analysis

Back 121

calculates the ratios of an asset’s closing prices to benchmark values, such as a stock index or comparable asset, and draws a line chart of the ratios

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change in polarity

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this refers to a belief that breached resistance levels become support levels and that breached support levels become resistance levels

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Reversal patterns

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when a trend approaches a range of prices but fails to continue beyond that range.

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Bollinger bands

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constructed based on the standard deviation of closing prices over the last n periods. An analyst can draw high and low bands a chosen number of standard deviations (typically two) above and below the n-period moving average

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Oscillators

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group of tools technical analysts use to identify overbought or oversold markets. These indicators are based on market prices but scaled so that they “oscillate” around a given value, such as zero, or between two values such as zero and 100

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Rate of change oscillator

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ROC or momentum oscillator is calculated as 100 times the difference between the latest closing price and the closing price n periods earlier

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Relative Strength Index

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RSI is based on the ratio of total price increases to total price decreases over a selected number of periods. This ratio is then scaled to oscillate between 0 and 100

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Moving average convergence/divergence

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MACD oscillators are drawn using exponentially smoothed moving averages, which place greater weight on more recent observations. The “MACD line” is the difference between two exponentially smoothed moving averages of the price, and the “signal line” is an exponentially smoothed moving average of the MACD line

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Arms index or short-term trading index (TRIN)

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measure of funds flowing into advancing and declining stocks

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Kondratieff wave

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54-year cycles

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Elliott wave theory

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. Elliott wave theory is based on a belief that financial market prices can be described by an interconnected set of cycles. Can be few mins or centuries