CFA 1_Quant Methods Flashcards
Nominal risk-free rate
real risk-free rate + expected inflation rate
Effective Annual Rate (EAR)
EAR = (1 + periodic rate)^n
periodic rate = stated annual rate / n = number of compounding periods per year
FV and PV equation
FV = PV*(1 + I/Y)^n
I/Y = annual rate of return / n
n = number of compounding periods per year
PV = FV / (1 + I/Y)^n
Annuity due (TI BA II Plus)
2nd > BGN > 2nd > Set
OR take FV/PV of ordinary annuity and multiply by (1 + I/Y)
FVAd = FVAo * (1 + I/Y)
PVAd = PVAo * (1 + I/Y)
Don’t forget to change back to END!
PV of a perpetuity
PMT / I/Y
PV of uneven CF series
Cpt PV of each cashflow, OR, use the CF function in your calculator and cpt the NPV.
Determining principal component of a PMT
- Interest component = beginning balance (PV in period) * periodic coupon interest rate
- Principal component = PMT - interest component
- Beginning balance of next period = prior beginning balance - principal component (principal component amortizes the principal balance)
NB: PMT is the same each year… but the PV changes. It is the PV on which you base the interest component calculation.
Principal component is PMT - interest component. Try making an amortisation table.
Computing PV if series of PMTs doesn’t start until a future period
- Cpt PV of future series of payments
- Discount that PV by the number of years until that PMT series started. BUT, be careful you are using the right “N”. If it starts 5 years from today, N should be 4, because that PV comes at the “end” of year 4/beginning of year 5.
Holding period return (HPR)
The % change in the value of an investment over the period it is held.
HPR = (ending value - beginning value + CF received) / beginning value
HPR = ((ending value + CF received) / beginning value) -1
Money-weighted return
The IRR talking into account all cash inflows/outflows.Use CF function then Cpt IRR. If funds are contributed to portfolio just before a period of poor performance, the money-weighted RoR will be lower than the time-weighted.If funds are contributed to portfolio just before a period of good performance, the money-weighted RoR will be higher than the time-weighted.
Time-weighted rate of return
Measures compound growth unaffected by timing of cash inflows/outflows.
- Disaggregate period into subperiods based on major cash flow events
- Cpt HPRs of these subperiods
HPR = ((ending value + CF received) / beginning value) -1
- Multiply HPRs of each subperiod. If more than a year, must use geometric mean.
(1 + time-weighted RoR)^2 = (1 + HPRa)*(1 + HPRb)
time-weighted RoR = [(1 + HPRa)*(1 + HPRb)]^.5
Bank discount yield (BDY)
How T-Bills are quoted. Discount based on FACE VALUE of bond, rather than market price.
rBD: bank discount return
rBD = (D / F) * (360 / t)D = dollar discount, the difference b/w face value and purchase price
F = face value (par) of bond
t = number of days remaining until maturity
360 = bank convention of # days in year (market convention is 365)
BDY is not representative of return earned by an investor.
Holding period yield (HPY)
Same as HPR. Just using specific terms. The actual return an investor will receive if money market instrument is held until maturity.
HPY = (P1 - P0 + D1) / P0HPY = ((P1 + D1) / P0) - 1
Effective annual yield (EAY)
Annualized HPY on basis of 365-day year, incorporating effects of compounding.
EAY = (1 + HPY)^365/t - 1
HPY = (P1 - P0 + D1) / P0
HPY = ((P1 + D1) / P0) - 1
Money market yield (CD equivalent yield)
Annualized HPY based on price using 360 days (bank days). Does not account for effects of compounding—assumes simple interest.
rMM: money mkt return
rMM = HPY * (360 / # days)
Given the bank discount yield (rBD):
rMM = (360 * rBD) / (360 - (t * rBD))
Bond-equivalent yield (BEY)
2 * semiannual discount rate
HPY * (365/days to maturity)
Measurement scales (four)
Nominal: no particular order
Ordinal: relative ranking, ordered by assigned categories
Interval: relative ranking and equal distances (eg celcius)
Ratio: ranking and equal distance and true zero point (money)
NOIR
Mean (Arithmetic)
Sum of observation values / Number of observations
Arithmetic mean is best estimate of true mean and value of NEXT observation (eg next year’s return; use geometric mean for estimating multi-year returns). The sum of actual deviations from the arithmetic mean is ZERO.
Mean (Weighted)
Sum of (weight * observed value) of all observations
Mean (Geometric)
(1+observed value A * 1+observed value B * …)^[1/n], then - 1
ie the nth root of (observed value A * observed value B * …)
When calculating for returns, must add 1 to each value under the radical and then subtract 1 from the result.
Used for multiple periods or when measuring compound growth rates. Geometric mean return shows the rate that would have to be compounded over n periods to get the same observed returns. Preferred number to estimate multi-year returns (eg next 3 years).
Mean (Harmonic)
N / (1/observed value A + 1/observed value B)
Used for stuff like avg cost of shares over time.
Median
Midpoint of data set. If even number of observations, then it is the mean of the two middle observations.
Mode
Most occurring value. Can have unimodal, bimodal, trimodal, etc, if more than one value appears frequently.
Quantile
Value at or below which a states proportion of data lies.Ly = (n + 1) * (y/100)
y = percent
Eg, what is third quartile for 10 observations? It’s the point below which 75% of observations lie.
(10 + 1) * (75/100) = 8.25
Once you have the quartile, eg 2.8 (between funds 2 and 3) out of 13 funds, you can use linear interpolation to get the value of the quartile.
Approx. value of quartile ≈ X2 + (Computer quartile - y/100) × (X3 – X2),
Range
Difference between largest and smallest value in data set.range = max value - min value
Range is a measure of dispersion (variability around the central tendency).
Mean absolute deviation
MAD = [Absolute value of (Observed value A - arithmetic mean) + (Observed value B - arithmetic mean) + …] / n
Note that the differences are not squared here (unlike for variance calculations).
The average of the absolute values of the deviations of observations from the arithmetic mean. Measure of dispersion.
Absolute values are used because the sum of actual deviations from the arithmetic mean is zero.
Population variance and standard deviation
Average of sum of squared deviations from the means (SD^2)
Variance = [(Observed value A - arithmetic mean)^2 + (Observed value B - arithmetic mean)^2 + …] / n
Sample variance and SD is divided instead by (n -1).
SD is the square root.
Sample variance and standard deviation
[(Observed value A - arithmetic mean)^2 + (Observed value B - arithmetic mean)^2 + …] / (n - 1)
Same as population variance except for (n - 1) instead of n
SD is square root.
Chebyshev’s inequality
Describes percentage of observations that fall within a number of SD’s from the mean. Applies to ALL distributions.
Percent = 1 - (1 / number of SDs^2)
Coefficient of variation
CV = SD of x / Avg value of x
Measurement of relative dispersion. Measures amount of dispersion relative to the mean. Enables comparison of dispersions in different data set.
Sharpe ratio
Measures excess returns per unit of risk. Bigger is better.
Limitations: if two portfolios have negative Sharpe ratios, the higher doesn’t imply superior performance. Increasing risk moves Sharpe ratio closer to 0.
Sharpe ratio = (R(portfolio) - Rf) / SD of portfolio
Skewness
Extent to which a distribution is asymmetrical.
Positive skew: higher frequency in right (upper) tail.
Negative skew: higher frequency in left (lower) tail.
Kurtosis
Degree to which distribution is more or less “peaked” than normal.
Leptokurtic: more peaked (higher graph at median). Will have more returns clustered around mean and more returns with large deviations from the mean (not good for risk management). MOST EQUITY RETURN SERIES ARE LEPTOKURTIC.
Platykurtic: less peaked (flatter)
Mesokurtic: normal distribution
Sample skewness
Sum of the cubed deviations from the mean divided by the cubed SD and then multiplied by reciprocal of number of observations.
Sk = (1/n) * [sum of(deviation from arithmetic mean^3)] / SD^3
Right skewed distribution has positive sample skewness; left skewed distribution has negative.
Sample kurtosis replaces ^3 with ^4.
Excess kurtosis = sample kurtosis - 3
Sample kurtosis
Same as sample skewness equation, but instead of ^3 it is ^4.
Sk = (1/n) * [sum of(deviation from arithmetic mean^4)] / SD^4
Excess kurtosis = sample kurtosis - 3
Stating the odds
The odds an event will occur = probability / (1 - probability)
Eg. You have a probability of 0.125 (one-eighth). 0.125 / (1 - 0.125) = (1/8) / (7/8) = 1/7. The odds of the event occurring are one-to-seven. NB if answer is a whole number, that means whole number to 1, dumbass.
So you could work backwards too. If one-to-seven odds, then 1 / (1 + 7) = .0125.
Defining properties of probability
- Probability of any event is between 0 and 1
2. If set of events is mutually exclusive and exhaustive, those probabilities sum to 1.
Unconditional/conditional probability
Unconditional: probability of an event regardless of past or future occurrence
Conditional: probability dependent on occurrence of an event. Key word is “given”. P(A | B) means probability of A given B. Conditional probability of an occurrence is called “likelihood”.
Multiplication rule of probability
To determine joint probability of a conditional and unconditional event.
P(AB) = P(A | B) * P(B)
The joint probability of A and P is equal to the conditional probability of A given B, times the unconditional probability of B.
NB, the word “and” means to multiply.Can rearrange to solve for conditional probability.
P(A | B) = P(AB) / P(B)
Addition rule of probability
Used to determine probability that at least one of two events will occur.
P(A or B) = P(A) + P(B) - P(AB)
Subtracting P(AB) avoids double counting. For mutually exclusive events, P(AB) is 0. NB: the word “or” means to add.
For mutually exclusive events, P(A or B) = P(A) + P(B)
Total probability rule
Used to determine the unconditional probability of an event, given conditional probabilities
P(A) = P(A | B1)P(B1) + P(A | B2)P(B2) + …
Independent versus dependent events
Independent events: those for which the occurrence of one event has no influence on occurrence of others. Eg the a priori probabilities of dice tosses or coin flips.To determine joint probability of any number of independent events
P(AB) = P(A) * P(B) * …
Dependent: occurrence of one event is influenced by occurrence of another.
Joint probability of any number of independent events
Or to determine joint probability of any number of independent events (events for which occurrence of one has no influence on occurrence of others).
Just multiply out. EG what is probability to get heads on 3 coin flips?P(heads) = .5 * .5 * .5 = ..125
Expected value
The weighted avg (weighted by probability) of possible outcomes of a random variable. Expected value = (P(x1) * x1) + (P(x2) * x2)
Eg, a dice toss
EV = (1/6)1 + (1/6) 2 + …
Calculating variance from probability
1) Calculate expected value
Expected value = (P(x1) * x1) + (P(x2) * x2) …
2) Variance is sum of the probability weighted squared differences between each outcome and the expected value. Variance is SD squared.
Variance = P(x1) * (x1 - EV)^2 + …
NB: the difference is not squared when calculating covariance
Covariance of two assets
Measures how a random variable moves with another random variable. Ranges from negative to positive infinity.
Cov(Ra,Rb) = E{[Ra - E(Ra)] * [Rb - E(Rb)]}
However, we will calculate covariance from a joint probability model using a probability-weighted avg of the products of the deviation from the mean for each outcome.
1) Calculate expected value (same as for calculating variance)
Expected value = (P(x1) * x1) + (P(x2) * x2) …
2) Covariance is the sum each scenario of the probability weighted, multiplied deviations from the expected value.
Cov(Ra, RB) = Sum for each scenario of [P(S) * (Ra - E(Ra)) * (Rb - E(Rb))]
NB, unlike variance, covariance does not square the differences.
Correlation coefficient
Relationship between covariances, SDs, and correlations. Ranges from -1 to +1. 1 is perfect positive correlation. 0 is no linear relationship. Expressed as rho (p).
Corr(Ra, Rb) = Cov(Ra, Rb) / (SD(Ra) * SD(Rb))
Correlation coefficient = Covariance / SDa*SDb
Therefore:
Cov(Ra, Rb) = Corr(Ra, Rb) * (SD(Ra) * SD(Rb))
Portfolio expected value
Sum of expected values weighted by the market weight of that variable.
Portfolio variance (of a 2 asset porttfolio)
Var(Rp) = [Wa^2 * SD^2 * (Ra)] + [Wb^2 * SD^2 * (Rb)] + [2 * Wa * Wb * SDa * SDb * Correlation coefficient (a,b)]
Var(Rp) = [Wa^2 * SD^2 * (Ra)] + [Wb^2 * SD^2 * (Rb)] + [2 * Wa * Wb * Covariance(a,b)]
Bayes’ formula (updated probability)
updated probability = [probability of new info for an event / unconditional probability of new info] * prior probability of event
Can use to compute P(B | A) from P(B), P(A | B), and P(A | Bconditional)
P(B | A) = [P(A | B) * P(B)] / P(A)
Labeling
When n items can receive one of k different labels. Applies to three or more subgroups of predetermined size.
n! / (n1! * n2! * n3!* …)
If the number of labels = n (k = n), then
n! / 1 = n!
Combination formula (binomial formula)
An extension of labeling. Used when you have 2 labels (k = 2). Applies when you have only two groups of predetermined size. Look for word “choose” or “combination”. Order doesn’t matter.
nCr = n! / [(n - r)! * r!]
n = total set of items
r = number of items to select from set
nCr is the number of combinations of selecting r items from a set of n items when order is not important.
Contrast with permutation formula where order matters:n! / (n - r)!
Permutation formula
A permutation is a specific ordering of a group of objects. Answers how many different groups of size r in specific order can be chosen from n objects. Applies only to two groups of predetermined size. Look for word “order”.
nPr = n! / (n - r)!
n = total set of items
r = number of items to select from set
Contrast with combination formula where order doesn’t matter:nCr = n! / [(n - r)! * r!]
Discrete vs. Continuous random variable
Discrete: one for which there is a finite number of possible outcomes, and each outcome is measurable and has positive probability > 0. Eg the number of days it will rain in a month. p(x) = 0 when x cannot occur, and p(x) > 0 when x can occur.
Discrete uniform random variable: A discrete variable which the probabilities for all possible outcomes are equal
Continuous: number of possible outcomes is infinite, even if lower/upper bounds exist. Eg, the amount it will rain in a month (can be measured on infinitesimal level). p(x) = 0 even when x can occur. Time is usually a continuous random variable.
For a continuous random variable X, the probability of any single value of X is zero. p(x) = 0 asshole!
Properties of probability (determining whether a function satisfied conditions for probability)
1) Answer must be between 0 and 1 (therefore must be positive)
2) Sum of probabilities of all possible outcomes must = 1
Cumulative distribution function (cdf)
F(x) = P(X <= x)
Defines the probability a random variable X taxes a value equal or less than a specific value x. Represents the sum of probabilities for outcomes UP TO and INCLUDING a specified outcome.
So to compute cdf, just add up all the probabilities up to and including the # of x in the function.
E.g. X = {1, 2, 3, 4}, p(x) = x / 10
F(3) = 1/10 + 2/10 + 3/10 = 0.6.
Discrete uniform probability distribution function
Discrete uniform random variable: A discrete variable which the probabilities for all possible outcomes are equal.
F(x) = n*p(x)
X = {2, 4, 6, 8, 10}, p(x) = 0.2
F(6) = P(X <= 8) = 4(.2) = .8
k is 4 here because there are 4 outcomes in range of 2 to 8.
Expected value same as before: Sum of (P(X) * X)
Binomial probability function
Binomial random variable: the number of successes in a given number of trials, where probability of success is constant and trials are independent. One for which the number of trials is 1 is a Bernoulli random variable.
Bin prob function:
p(x) = (n! / (n-x)! * x!) * p^x * (1 - p)^(n-x)
where n = number of trials, x = successes, and (n! / (n-x)! * x!) = number of ways to choose x from n [NB this is the same as the combination formula!].
Note: Watch out for problems that have just one “success” and one “failure” rate and then probability! Eg, 10% of grads stay with same company, in sample of 6 grads what is probability 2 will stay. These are solved with the above formula!
Expected value of a binomial random variable = n*p
Eg. 10% stay, in sample of six grads how many will stay?
Variance of a binomial random variable = np(1 - p)
Continuous uniform distribution
Has an upper and lower limit, outside of which P = 0.The probability of outcomes of a range within the distribution is
(Range 2 - Range 1) / (Upper limit - lower limit)
Normal distribution
1) X is normally distributed with mean mu and variance theta^2
2) Skewness = 0, both sides symmetrical. Mean = median = mode.
3) Kurtosis = 3, measure of how flat distribution is
4) Tails extend to infinity and never reach zeo
5) a linear combination of normally distributed random variables is normally distributed as well (if return of each stock in portfolio is normally distributed, the portfolio itself will be as well).
Tracking error
The difference between the total return on a portfolio and the total return on the benchmark against which performance is measured.
Univariate vs Multivariate distributions
Univariate: distribution of a single random variable
Multivariate: Specifies probabilities associated with a group of random variables and is meaningful only when behavior of each random variable in the group is in some way dependent upon the behavior of the others. Multivariate distributions between two discrete random variables are described using joint probability tables.
Correlation is the feature that distinguishes a multivariate distribution from a univariate normal distribution.
A multivariate distribution will for “n” assets have:
n means
n variances
0.5 * n * (n - 1) correlations
Confidence interval
The range of values around which we expect the actual outcome to be a specified % of the time.
90% CI is -1.65 to 1.65 SDs
95% CI is -1.96 to 1.96 SDs
99% CI is -2.58 to 2.58 SDs
To calculate the range of confidence interval of a normal distribution (in which mean is mu and variance theta^2):
Avg return +/- CI*(SD of returns) = two answers
CI = Sample mean +/- (z-reliability factor) * (std error)
Std error = SD / (sample size)^0.5
Standard normal distribution and z-score
A normal distribution that has been standardized so that it has a mean of zero and an SD of 1. In this case F(0) = 0.5 because half of the distribution lies on each side of mean.
z-value: represents the number of SDs a given observation is away from the population mean.
Standardization: process of converting an observed value for a random variable to a z-value.
z = (observation - population mean) / SD
z = x - mu / theta
Using z-score and z-tables to find cumulative probabilities
The z-score represents the number of SDs a given observation is away from the population mean.
A z-table lists the percentages of cumulative probabilities by z-score. The left hand rows are the base z-score (eg 1.6), and the top column are the fine z-scores to the tenth (eg .06, therefore 1.66).
The positive z-table percents show everything that is BELOW that z-score. The negative z-table percents indicate everything that is ABOVE that z-score. But because a normal distribution is symmetrical, we can take the positive table and just “1 - x” to get what would be above. Also do this is you need to translate from positive table to negative equivalent.
If asked how much lies between 0 and x, then HALF the number in the table, because the number represents the range from the negative to positive of x.
Roy’s safety-first criterion
Shortfall risk: probability that portfolio value/return will fall below a particular target.
Roy’s safety-first criterion: optimal portfolio minimizes probability that return of portfolio falls below minimum acceptable threshold level.
minimize P(Rp < Rl)
where Rp = portfolio return, and Rl = threshold level.
SFRatio:
SFRatio = [E(Rp) - Rl] / SDp
The larger the SFRatio, the better.
Compare to the Sharpe ratio:
[E(Rp) - Rf] / SDp
Lognormal distribution
A lognormal distribution is generated by e^x, where x is normally distributed. Because ln of e^x is x, the logarithms of lognormally distributed random variables are normally distributed. Therefore, lognormal.
Lognormal distribution is (1) skewed to the right, and (2) bounded by zero to the left. For a lognormal distribution, probability cannot be negative.
Because it can’t go below zero (unlike normal distributions which extend to infinity), it allows for computing price relativity (end of period value / beginning of period value)
Continuous compounding
Discrete compounding: compounding over a discrete period, ie semiannual.
Continuously compounded EAR or HPR = e^(annual cc rate * number of years) - 1
NB number of years only applies if for multiple periods.
BA II PLUS: [2nd] [e^x]
Continuously compounded rate from normal EAR or HPR = ln (1 + EAR or HPR) / number of periods if applicable
Or, thanks to price relativity,
= ln (ending price / beginning price) / number of periods if applicable
BA II PLUS: LN
Continuously compounded rates are ADDITIVE for multiple periods.
Monte Carlo simulation
Technique based on repeated generation of one or more risk factors to generate a distribution of security values. Uses hypothesized parameter values ad a random number generator. Limitations are that it’s statistical and complex.
Historical simulation: based on actual changes. Set of all changes in relevant risk factors over some prior period is used.
Simple random sampling
Simple random sampling: Method of selecting a sample in such a way that each item or person in the population being studied has sample likelihood of being included.
Systematic sampling: selecting every nth member from a population
Sampling error: difference between a sample statistic (eg mean) and its corresponding population parameter (eg population mean).
Sampling distribution: the probability distribution of all sample statistics computed from a set of equal-size samples randomly drawn from same population (compared with other samples of same population). Sampling distribution of the mean is the distribution of these estimates.
Stratified random sampling
Uses classification system to separate population into smaller groups based on one or more distinguishing characteristics. From each subgroup a random sample is taken and then the results are pooled. Size of sample for each stratum is based on the size of the stratum relative to the population.
By using stratified random sampling we can be assured we will get representation from a given subgroup, whereas we used simple random sampling we might not.
Time-series vs. Cross-sectional data
Time-series data: consist of observations taken over a period of time
Cross-sectional data: sample of observations taken at a single point in time
Longitudinal data: observations over time of multiple characteristics of same entity, eg inflation rates over 10 years.
Central limit theorem
Central limit theorem: for simple random samples of size “n” from a population with mean “mu” and finite variance of “SD^2”, the sampling distribution of the sample mean approaches a normal probability distribution with mean mu and a variance equal to SD^2 / n as sample size becomes large.
1) If sample size n is sufficiently large (> 30), the sampling distribution of the sample means will be approximately normal.
2) Mean of the population, my, and mean of the distribution of all possible sample means are equal
3) Variance of the distribution of sample means is population variance / sample size
Standard error of the sample mean
The SD of the distribution of the sample means. When SD of the population is known, standard error is calculated as:
Standard error of sample mean = SDpopulation / size of sample^0.5
If SDpopulation is not known (often the case), can use SDsample.
As sample size gets larger, the standard error gets smaller because the sample mean approaches the true mean of the population.
Desired properties of an estimator (formula used to compute a point estimate, ie a single sample value used to estimate population parameters)
1) Unbiasdness: EV of sample mean is same as population mean
2) Efficiency: variance of sample is smaller than all other unbiased estimators of parameter
3) Consistency: accuracy of parameter estimate increases as sample size increases
Point estimate
Point estimates: single sample values used to estimate population parameters. Formula used to compute the point estimate is called the estimator.
Confidence interval
CI: Range of values in which the population parameter is expected to lie, given the probability of 1 - alpha. CIs can be interpreted from a probalistic or practical perspective.
alpha = level of significance for the CI
1 - alpha = degree of confidence
Now,
CI = point estimate +/- (reliability factor * standard error)
where point estimate = value of sample statistic. typically is the sample mean.
reliability factor = number that depends on sampling distribution of point estimate and probability that point estimate falls in confidence interval, (1 - alpha)
standard error = standard error of point estimate
Student’s t-distribution
A bell-shaped probability distribution that is symmetrical about its mean. Most appropriate to use for small samples (n < 30).
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 (n - 1) for sample means.
Compared to normal distribution has fatter tails (more probability in tails), until df increases large enough. This makes hypothesis testing with the t-distribution harder than with the z-distribution because fatter tales means more outlines.
t-distribution is used when population variance is unknown.
Confidence interval for the population mean, population variance is known
Population variance is known. z-test.
CI = point estimate +/- (reliability factor * standard error)
CI = sample mean +/- (Za/2)*(SD/n^0.5)
where, Za/2: reliability factor, the z-score that leaves a/2 of probability in the upper tail. To cpt, use the commonly used set below.
and, SD/n^0.5: standard error
Reliability factors:
Za/2 = 1.645 for 90% CIs (significance level of 10%, 5% under each tail (ie SD))
Za/2 = 1.960 for 95% CIs (significance level of 5%, 2.5% under each tail (ie SD))
Za/2 = 2.575 for 99% CIs (significance level of 1%, 0.5% under each tail (ie SD))
Confidence interval for the population mean: normal with unknown population variance
If distribution of population is normal with unknown population variance, can use the t-distribution. CIs using t-reliability factors will be more conservative with those using z-reliability factors because of larger tails.
CI = sample mean +/- (ta/2)*(SD/n^0.5)
where ta/2: t-reliability factor. Must calculate from table of t-distribution.
and, SD/n^0.5: standard error
Problem is calculating ta/2. First remember that degrees of freedom (df) is n -1. So, subtract 1 from the sample size and you have the left side of the t-distribution table. Second we need to get top column (level of significance for one-tailed test). For a 95% CI, the significance level is 5%, or 2.5% under each tail. So on the t-table columns choose 0.025. Then you will cross-sect at the t-reliability factor.
Rememba:
Reliability factors (z not t):
Za/2 = 1.645 for 90% CIs (significance level of 10%, 5% under each tail (ie SD))
Za/2 = 1.960 for 95% CIs (significance level of 5%, 2.5% under each tail (ie SD))
Za/2 = 2.575 for 99% CIs (significance level of 1%, 0.5% under each tail (ie SD))
Confidence interval for population mean when population variance is unknown, given a large sample from any type of distribution
Use z-statistic for NORMAL distribution with KNOWN population variance.
Use t-statistic for NORMAL distribution with UNKNOWN population variance. (z-stat however may be used but is less conservative)
Use Z-statistic for NONNORMAL distribution with KNOWN population variance when (n >= 30).
Use t-statistic for NONNORMAL distribution with UNKNOWN population variance when (n >= 30).
We cannot create a CI if the sample size is less than 30 and distribution is nonnormal.
Use z-stat with known population variance, and t-stat with unknown population variance.
Biases in sample selection
Data mining bias: Overestimating statistical significance of pattern found through data mining. Data mining is repeatedly using same database to search for patterns until one is discovered.
Sample selection bias: When some data is systematically excluded from the analysis, usually due to lack of availability. Renders observed sample nonrandom.
Survivorship bias: Remaining sample is overestimated because poor performers have been dropped over time.
Look-ahead bias: When a study tests a relationship using sample data that wasn’t available on the test date (eg used estimates).
Time-period bias: If time period over which data is gathered is either too short or too long
Reliability factors
Za/2 = 1.645 for 90% CIs (significance level of 10%, 5% under each tail (ie SD))
Za/2 = 1.960 for 95% CIs (significance level of 5%, 2.5% under each tail (ie SD))
Za/2 = 2.575 for 99% CIs (significance level of 1%, 0.5% under each tail (ie SD))
Hypothesis testing procedure
1) State hypothesis: null and alternative. Often it will be Ho = 0, and Ha =/= 0.
2) Select appropriate test statistic: either z-stat or t-stat, depends on population size and whether pop variance is known (z-stat) or unknown (t-stat, but z-stat okay if large sample).
3) Specify level of significance: hope this is given in problem
4) State decision rule regarding the hypothesis: select the critical value based off the level of significance (90% = 1.645 and 10% level of sig, 95% = 1.960 and 5% level of sig, 99% = 2.575 and 1% level of sig)
Decision rule is to reject the null hypothesis if the z-statistic is outside the range of critical values for two-tailed test, or greater than the critical value for 1-tailed test.
5) Collect sample and calculate sample statistics
z or t-stat = (sample mean - hypothesized mean) / std error
std error = SD / sample size^0.5
6) Make a decision regarding hypothesis:P if calculated z-stat or t-stat is outside the critical value range then reject the null hypothesis.
7) Make a decision based on results of the test
Null vs, Alternative hypothesis
Null: hypothesis researcher wants to reject. Hypothesis that is actually tested. Always includes “equal to” condition (and may also include greater or less than). Often will be like not equal to zero.
Alternative: hypothesis concluded upon if there is enough evidence to reject null
One tailed test: will use >= for null and < for alternative (or vice versa).
Two tailed test: will use = for null and =/= for alternative (or vice versa).
Two-tailed test of hypotheses
Two-tailed test: used when research question is whether return is different from zero. Allows for deviation on both sides of hypothesized value (zero). Most tests are two-tailed. Often null hypothesis will be whether returns are equal to 0. Used when there is only = (not >= or upper critical z-value (+ reliability factor), or < lower critical z-value (- reliability factor)
One tailed test: will use >= for null and < for alternative (or vice versa).
Two tailed test: will use = for null and =/= for alternative (or vice versa).
Choose z-value (reliability factor) based on significant level.
1) Specify null and alternative hypotheses, define level of significance (2x SD, see table below) to get the critical values
2) Compute standard error;
SD / (size of sample)^0.5
3) Divide the mean return by the standard error to get the test statistic
(mean return - hypothesized value) / standard error
NB: test statistic = (sample statistic - hypothesized value) / standard error
Here the hypothesized value was 0.
4) Compare the test statistic to the critical values
Decision rule is to reject the null hypothesis if the z-statistic is outside the range of critical values for two-tailed test, or greater than the critical value for 1-tailed test.
Rememba: Reliability factors (z-values):
Za/2 = 1.645 for 90% CIs (significance level of 10%, 5% under each tail (ie SD))
Za/2 = 1.960 for 95% CIs (significance level of 5%, 2.5% under each tail (ie SD))
Za/2 = 2.575 for 99% CIs (significance level of 1%, 0.5% under each tail (ie SD))
One-tailed test of hypotheses
A one-tailed test is used when researcher wants to test whether return on stock is GREATER than zero. Often null hypothesis will be whether returns are less than or equal to 0. Used when there is >= or related to 1.645 instead of 1.960.
One tailed test: will use >= for null and < for alternative (or vice versa).
Two tailed test: will use = for null and =/= for alternative (or vice versa).
1) Specify null and alternative hypotheses, define level of significance (2x SD /2, see table below) to get the critical values
2) Compute standard error;
SD / (size of sample)^0.5
3) Divide the mean return by the standard error to get the test statistic
(mean return - hypothesized value) / standard error
NB: test statistic = (sample statistic - hypothesized value) / standard error
Here the hypothesized value was 0.
4) Compare the test statistic to the critical values
Decision rule is to reject the null hypothesis if the z-statistic is outside the range of critical values for two-tailed test, or greater than the critical value for 1-tailed test.
Rememba: Reliability factors (z-values):
Za/2 = 1.645 for 90% CIs (significance level of 10%, 5% under each tail (ie SD))
Za/2 = 1.960 for 95% CIs (significance level of 5%, 2.5% under each tail (ie SD))
Za/2 = 2.575 for 99% CIs (significance level of 1%, 0.5% under each tail (ie SD))
Test statistic
Hypothesis testing involves two statistics: the test statistic and the critical value of the test statistic, which the test statistic is compared to.
test statistic = (sample statistic - hypothesized value) / standard error
Type I vs, Type II errors
Type I: rejection of null hypothesis when it is actually true (incorrect rejection)
Significance level is the probability of making a Type I error, and is designated by alpha (a). A significance level of a = 0.5 means that there is a 5% chance of rejecting a true null hypothesis.
p-value: Probability of obtaining a test statistic that would lead to Type I error.
Type II: failure to reject null hypothesis when it is actually false (incorrect support)
Power of a test: probability of correctly rejecting null hypothesis when it is false (not making a Type II error). Power = 1 - P(Type II error).
Decreasing the significance level (probability of a Type I error) will increase the chance of a Type II error. Likewise, increasing the power of a test (probability of not making Type II error) will increase the chance of a Type I error.
Confidence intervals and hypothesis tests
CI is a range of values within which the researcher believes the true population parameter may lie.
CI = sample mean +/- (Za/2)*(SD/n^0.5)
CI = {[sample statistic - (critical value)*(standard error)] <= …
Rememba: Reliability factors (z-values):
Za/2 = 1.645 for 90% CIs (significance level of 10%, 5% under each tail (ie SD))
Za/2 = 1.960 for 95% CIs (significance level of 5%, 2.5% under each tail (ie SD))
Za/2 = 2.575 for 99% CIs (significance level of 1%, 0.5% under each tail (ie SD))
p-value
Probability of obtaining a test statistic that would lead to a rejection of the null hypothesis when the null hypothesis is true (Type I error).
p-value is the probability that lies ABOVE the test statistic for the upper and/or lower tail. Take the test statistic and compare it to the z-table to determine the probability of getting a value less than or equal to that. Then subtract that z-score from 1 to get the probability of being greater than that.
If a 2-tailed test then double it.
t-Test
t-test is used when population variance is UNKNOWN, and either sample is LARGE, or, if sample is small (<30) the distribution is normal. Small, non-normal samples cannot be reliably tested.
As in tests of hypotheses above,
t-stat with n-1 degrees of freedom = (sample mean - hypothesized mean) / standard error
std error = SD / n^0.5
z-Test
Used when population is normally distributed with a KNOWN population variance. Can be used if variance is unknown if sample size is large and normal. Small, non-normal samples cannot be reliably tested.
z-stat = (sample mean - hypothesized mean) / std error
Hypothesis tests concerning difference of means of two populations, based on two independent samples (test of differences in means)
Assumption that two samples taken are independent, and the two populations are normally distributed. Population variances are unknown. Use a t-test.
Two methods of t-tests:
1) Assume population variances are EQUAL, and pool the sample observations (pooled variance).
2) Assume population variances are UNEQUAL, and use sample variances for both populations.
Numerator of both is difference of sample means (since we assume them to be independent).
Hypothesis tests concerning difference of means of two populations, based on two dependent samples (test of significance of the mean of differences between paired observations)
When we have two dependent samples (eg two firms whose returns are dependent on same economic events)
Instead of the difference in means used for two independent samples, here we use the difference in returns (paired comparisons test). Is it significantly different from zero?
The statistic is the avg difference in paired observations divided by the standard error of the differences between observations.
chi-square (X^2) test
Used for hypothesis tests concerning variance of a single normally distributed population (for testing an actual and hypothesized variance). chi-square distribution is asymmetrical and bounded by 0 (can’t go left of 0), and approaches normal distribution in shape as df increase. Versus an F-test that tests the equality of two variances.
chi-square stat = [(sample size - 1) * sample variance] / hypothesized population variance
NB: (sample size - 1) = degrees of freedom
1) State hypothesis
2) Select appropriate test statistic (see above)
3) Specify level of significance
4) State decision rule regarding hypothesis: take degrees of freedom and probabilities (derived from level of significance) and use table of chi-square values to find critical values.
eg for two tailed test:
critical value a < chi-stat < critical value b
If chi-stat falls within these then we SUPPORT the null.
5) Collect sample and calculate sample statistics (with the test statistic above)
6) Make decision regarding hypothesis: if the test statistics falls outside the critical values, can reject the null hypothesis. If it falls within the values, have to support null hypothesis.
F-test
Tests EQUALITY of variances of two populations. Assumes samples are independent and populations are normally distributed. Versus a chi-square test that tests variance of a single normally distributed population. Bound by zero on the left (can’t go to the left of zero). Right-skewed.
F-stat = variance of sample taken from pop1 / variance of sample taken from pop2
Always put the larger variable in the numerator.
Look up the critical values on the F table with the degrees of freedom of each population (sample size - 1). Compare the F-stat to those critical values.
Parametric vs. Nonparametric tests
Parametric: reply on assumptions regarding distribution of population and are specific to population parameters. eg, z-test relies on a mean and SD to define the normal distribution.
Nonparametric tests: either don’t consider a particular population parameter or have few assumptions about population that is sampled. Used when there is concern about quantities other than parameters of distribution, or when assumptions of parametric tests can’t be supported. Also used when data is not suitable for parametric tests (eg ranked observations).
Nonparametric tests are used when:
1) Assumptions about distribution of variable that supports parametric test are not met. Eg not normal distribution and small sample size.
2) When data are ranks (ordinal measurement scale) rather than values
3) Hypothesis does not involve parameters of distribution. Eg a runs test to determine whether data is random.
Spearman rank correlation test: used when data are not normally distributed.
Technical analysis
Only uses firm’s share price and trading volume data.
Assumes market prices reflect both rational and irrational investor behavior—therefore efficient markets hypothesis does NOT hold.
Based on idea that prices are determined by interaction of supply and demand.
Advantages: price and volume data is observable. Can be applied to price assets that do not produce future cash flows, eg commodities. Also can uncover fraud (eg through trading volume).
Limited in illiquid markets and those markets subject to outside manipulation (currency markets through central banks).
Breakout/breakdown
Signals the end of a prior trend.
Breakout: When stock price crosses a down trendline by a significant amount.
Breakdown: when price crosses an up trendline by a significant amount.
Change in polarity
Breached resistance levels become support levels, and breached support levels become resistance levels.
A trendline is thought to represent a level of support or resistance.
Bollinger bands
Moving lines that graph the range of standard deviation over n periods. Illustrate volatility. The bands get wider in higher price volatility, and narrower in lower price volatility.
Prices above the top band may indicate an overbought (too high) market, and prices below the bottom band may indicate an oversold (too low) market.
Oscillators
Tools based on market prices and scaled to oscillate around a given value, or between two values. High oscillator values indicate market is overbought (too high), while low values indicate it is oversold (too low).
Rate of change oscillator: aka momentum oscillator. calculated as 100 times difference between latest closing price and closing price n periods earlier. Buy when oscillator changes from negative to positive during an upswing.
Relative strength index: ratio of price increases to price decreases over n periods.
Moving avg convergence/divergence: drawn using exponentially smoothed moving avgs of stock price, which place greater weight on more recent observations.
Stochastic oscillator: calculated from latest closing price and highest and lowest prices reached in a recent period.
Sentiment indicators
Bullish/bearish.
Put/call ratio: put volume / call volume. If ratio is high, indicates strong bearish sentiment. If ratio is low, indicates strong bullish sentiment.
Volatility index (VIX): reported by Chicago Board Options Exchange. measures volatility of S&P 500. High levels of VIX suggest investors fear decline.
Margin debt: reported by brokers. Increases in margin debt suggest bullish sentiment. Decreases suggest bearish and declining prices.
Short interest ratio: short interest (# of shares investors have borrowed and sold short) / avg daily trading volume. High ratio indicates bearish sentiment. But also implies that there will be future buying demand when borrowers must return their borrowed shares.
Flow of funds (in the financial markets)
Arms index (Short-term trading index, TRIN): measures funds flowing into advancing and declining stocks. Index value > 1 indicates majority of volume is in declining stocks. < 1 indicates volume is in advancing stocks.
TRIN = (Advancing issues / Declining issues) / (Volume of advancing issues / Volume of declining issues)
Margin debt: increasing margin debt may indicate investors want to buy more stocks. Decreasing margin debt indicates increased selling.
Mutual fund cash position: mutual funds’ cash / total assets. During uptrends managers want to invest cash quickly. As a result, cash positions increase when market is falling and decrease when market is rising.
New equity issuance (IPOs) and secondary offerings: add to supply of stocks. because issuers tend to issue when stock prices are though to be high, this may indicate a peak in the market.
Elliot wave theory
The Elliott Wave Principle posits that collective investor psychology, or crowd psychology, moves between optimism (5 wave dominant trend) and pessimism (3 wave corrective trend) in natural sequences. These mood swings create patterns evidenced in the price movements of markets at every degree of trend or time scale.
The size of these waves is thought to correspond with Fibonacci ratios. Fibonacci numbers are found by starting with 0 and 1, then adding each of the previous two numbers to produce the rest.
The ratio of large, consecutive Fibonacci numbers is 1.618.
Other waves include the Kondratiev wave (54 year cycle), and 4 year Presidential cycle.
Intermarket analysis
Analysis of interrelationships among market values of major asset classes.
Relative strength ratio
Stock price / benchmark value
Liquidity premium
The liquidity premium compensates investors for the risk of loss relative to an investment’s fair value if the investment needs to be converted to cash quickly.
