3. Quantitative Methods Flashcards
Nominal scale
Names are discretionary
Ordinal scale
Scale order with specific criteria
e.g.: lower to higher
Interval Scale
Scale order based in ranges
0-10 % vol / 11-20 % vol / 21-30% vol
Ratio Scale
Scale order based in ratios
e.g.: debt-to-equity
2 x / 3x / 4x
Absolute Frequency
Example: #1 case (even if out of 20 or 30)
Relative Frequency
Example:
1 out of 10 = 10%
1 out of 20 = 5%
Histogram
Graphic based in two axis:
X: interval / specific values
Y: frequency
Population Average (formula)
Sum of Observations / #Number of Observations
Median
- Ordenate observations
- Choose the one that divides the # of observations in two
- If there are 4 observations, take the measure between numbers #2 and #3
Mode
The most frequent observation
- Distributions may be unimodal, bimodal, trimodal etc
Geometric Mean
1 + Rg = Raiz cúbica de (1+R1)(1+R2)(1+R3)
Harmonic Mean
- Calculate the average price of a stock, given:
- total budget waste
- average price for each tranche
- # number of tranches
Quantile
Quartile = divided by 4 Quintile = divided by 5 Decile = divided by 10 Percentile = divided by 100
Measure of Location
“What is the 3rd. quartile for the following distribution?”
Location where y% of the observations lie below, in a given distribution
L (y) = (n+1) * y / 100, where
Given:
Ascending order
y = Position below which y% of observations lie
Instructions:
- Count the numbers
- If L(y) = 8.5, interpolate 50% of the difference between observations #8 and #10
Mean Absolut Deviation (formula)
Average of all absolute deviations from the mean
- Pega todos os desvios
- Tira a média deles
Population Variance (formula)
Var = [(Deviation - Avg)^2 + (Deviation - Avg)^2] / 2
Standard Deviation (formula)
StDev = Raiz da Variância
Sample Variance (difference in comparison to Population Var)
Instead of using N, divide by (n-1)
Skewness
Measure of symmetry
Positive: outliers on the right; outliers on higher values
Negative: outliers on the left; outliers on lower values
Kurtosis
Kurtosis = 3 (for normal distribution)
↑ Kurtosis = ↑ Mode
Coefficient of Variation (formula)
CV = StDev x / Avg Value of X
CV = Vol / X (it should be the benchmark)
↑ CV = ↑ Risk
Order of Average, Mode, Median given skewness
Positive (+) Skewness: Avg > Median > Mode
Due to high outliers
Negative (-) Skewness: Mode > Median > Avg
Due to low outliers
- Median always in the middle
- Average is push to where outliers are (lower numbers / higher numbers)
Correlation range
Between (-1) and (1)
Variance range
Unbounded
Correlation (formula)
Correl = Cov(A,B) / StDevA * StDevB
Odds (concept)
Odds: 1 to 7 means one win to 7 loses.
Probability of wins: 1/8
Portfolio Variance (formula)
Var (Rp) = ∑ Wi r Wj Cov (Ri, Rj)
Var (Rp) = W1²Var(R1) + W2²Var(R2) + 2W1W2Cov(R1,R2)
Std Dev = Raiz disso
Labeling (concept)
Labeling = ways to order
- Different ways of labeling
- You DO DISCOUNT redundant orders
Combination (concept and formula)
- Special case where labeling has two subgroups
C = n! / (n-r)! r!
Ex.: Divide 8 stocks between 4 buy and 2 sell
Combination: 8! / 4! 2!
Permutation (concept and formula)
- Case where ORDER MATTER, so you do NOT remove redundant orders
P = n! / (n-r)!
Ex.: How many ways to order the sell of 3 stocks out of 8 if the order matters
Expected value of X for a binomial distribution (formula)
For a binomial distribution
E (X) = n * p, where
p = probability of success of each trial n = n trials
Variance of X for a binomial distribution (formula)
Variance of (X) = np (1-p), where
p = probability of success of each trial n = n trials
Probability of X successes in n trials (binomial distribution formula)
P (x) = [ n! / (n-x)! x! ] * pˆx * (1-p)^n-x
Confidence of Interval @ 90% (Normal Distribution)
1.65 σ
Confidence of Interval @ 95% (Normal Distribution)
1.96 σ
Confidence of Interval @ 99% (Normal Distribution)
2.58 σ
Z (standardized) for a Normal Distribution (formula)
z = (Obs - μ) / σ
Roy’s Safety First Ratio (formula and concept)
Minimizes probability of the portfolio obtaining a return below target
SFR = [ E (Rp) - Rthreshold ] / σ P
↑ SFR = ↑ Better = Choose the largest
Lognormal distribution properties
- Positive skewness (outliers to the right)
- Used to price derivatives
Lognormal distribution formulas
a. S1 / Szero = 1 + HPR
b. ln (S1/Szero) = ln (1 + HPR) = Rcc, where
S1 = Price @ t = 1 S0 = Initial Price HPR = Return, disregarding time-period Rcc = Continuously Compounded Return
Find HPR, given Rcc
HP12C - Lognormal
Formula: Rcc = (e^HPR - 1)
a. Press Rcc (given)
b. Press G
c. Press e^x
d. Subtract 1
Find Rcc, given HPR
- Press HPR
- Add +1
- Press LN
- You will find Rcc
Monte Carlo Simulation
- Assume probabilities for random variables, given historical data
- Generate multiple random results
- Uses these results to calculate a distribution of possible values
Minimum Sample Size (#)
30 observations
Central Limit Theory
Randomly generated samples will tend to a normal distribution where:
a. x̄ ~ Pop. μ
b. Standard error (from mean) = σ / √N
c. Large Sample!!
Estimator Properties
CEU
- Consistent: lower standard error
- Efficient: lowest variance of all samples
- Unbiased: Expected Value Sample ~ EV Estimator
T-student Parameters
a. Only one parameter: Degrees of Freedom
b. Fatter tails (less observations around the mean)
c. ↑ DF = ↑ Sample = ↑ Obs. around x̄ = Approaches Normal
Effects from T-Student Fatter Tails
↑ DF = ↑ # Sample
The hypothesis test is more strict, as it is more difficult to reject H0
Survivorship Bias
Surviving funds’ returns are naturally higher, so the historical returns database is overrated
Look Ahead Bias
Price to Book Ratio shoould consider Today’s Price, even though Book Value is known for FY2019 only
Sample Selection Bias
It refers to the bias where theinformation is not totally tested/considered only because part of it is unavailable
Time Period Bias
Too short = not enough observations to find causality
Too long = there is causality, but it is dilluted so you can’t find it
Hypothesis testing steps
- State the Hypothesys
- Select the test (t or z)
- State the decision rule
- Sample
- Decide
Error Type #1
Reject H0 when H0 is true
Error Type #2
Not to reject H0 when H0 is false
Power of Test (formula)
Power of Test = 1 - (P Error Type II)
Probability of rejecting H0 when it is false
P-value (concept)
Lowest level of significance that allows me rejecting H0
If you expand the level of significance, the test will not work
Difference of Means Test (distribution to be used)
a. Two t-student (depends on whether variances are assumed to be equal/different)
Difference of Variances Test (distributions to be used)
Chi-squared
Equal Variances (distribution to be used)
F-distribution
Two-tailed test
H0 = 0
H1 ≠ 0
One-tailed test
H0 ≤ 0
H1 > 0
H0
H0 = 0
H0 ≤ 0
H0 ≥ 0
It always include “=”, even though it is an one-tailed test
Multivariate Distribution
A multivariate distribution is best defined as describing the behavior of:
2 or more DEPENDENT random variables
Sample Efficiency (Concept)
Variance of its sampling distribution is smaller than that of all other unbiased estimators of the parameter
According to the Central Limit Theorem, the distribution of the sample means is approximately normal if:
If the sample size is sufficiently large (i.e. greater than 30) the sampling distribution of the sample means will be approximately normal.
Unusual requirement in the U.S. (regarding the securities trading in the country)
The United States requires a signed management statement about the effectiveness of the firm’s internal controls is required by U.S. regulators for securities that trade in the U.S., but not elsewhere.
Estimate Pro Forma Earnings (Do / Don’ts)
There may be changes in KG, despesa de capital, novos ativos fixos, repagamento de dívida e recompra de stocks.
Fixed Ratios: Non-Cash Items as % of Sales
