CFA - QM Flashcards
Nominal Risk free Rate =
Real Risk free rate + expected inflation
EAR =
Effective Annual Rate - (1 + I)^T - 1
Nominal rate of interest =
Nominal risk free rate + risk premium
Annuity due
Beginning of each period
Ordinary annuities
end of each compounding period
PV perpetuity =
PMT / r
Numerical data
values that can be counted or measured
Dentro de Numerical data, temos discrete data que seria
countable (days, months, etc)
Dento de Numerical data, temos Contiunous data que seria
can take any fractional value
Categorical data
Labels that can be used to classify a set of data into grupos
Dento de Categorical Data, temos Nominal data que seria
cannot be placed in order logically
Dento de Categorical Data, temos Ordinary data que seria
can be ranked in a logical order
Time series
set of observations taken periodically
Cross-sectional data
set of comparable observations all taken at one specific point in time
Structural data
organized in a defined way
Unstructured data
no defined structured
Marginal Frequency
total of frenquecies for a row ou colum
Joint frequencies
displays of two variables
Confusion Matrix
is a contigency table that displays predicted and actual occurences of an event
Arithmetic mean =
( x1 + x2 + x3 + xn) / n
Geometric Mean =
=[(1+R 1 )(1+R 2)…(1+R n)] 1/n −1
Weighted Mean =
W1 x R1 + W2 x R2 + W3 X R3 + Wn x Rn
Weighted Mean often used to calculate …
Portfolio return
Harmonic Mean =
N / (1/xi) -> Sendo N=numero total e Xi=valores
Trimmed Mean
exclude a stated percentage of most extreme observations
Winsorized mean
substitute values for the most extreme observations
Median
midpoint of a data set
Mode
value occruing most frequently in a data set
Quantile =
(n+1) x Y/100
Mean Absolute Deviation (MAD) =
MAD= ∑∣x i − xˉ ∣ / n
Sample Variance =
s^2 = ∑ (x i − xˉ)^2 / n − 1
Sample Standard Deviation =
s = raiz quadrada de: ∑ (x i − xˉ)^2 / n − 1
Sample Variance nos mostra…
a variação dos dados em relação ao ponto central
Sample Standard Deviation nos mostra
a variação média
Coefficient of Variation (CV) =
Standard Deviation (s) / average value of x
Coefficient of Variation nos mostra…
quantas unidades de risco por quantas unidades de retorno - (Measures the amount of dispersion in a distribution relative to the distributions mean
Coefficient of Varition é melhor: um coeficiente alto ou baixo?
Baixo, pq indica pouca variação
Target Downside Deviation =
s = raiz quadrada de: ∑ (x i − Bˉ)^2 / n − 1
Target Downside Deviation calulate…
risk based on outcomes both above and below the mean
Skew measures…
the dregree to wich a distribuiton lacks symmetry
Skew = 0; Median, Mode and mean is
Mean = Median = Mode
Skew positiva
Mean > Median > Mode
Kurtosis measures
the degree to which a distribution is more or less peaked than a normal distribution (concentação nas caldas)
Skew negativa
Mean < Median < Mode
Kurtosis é igual a qual número se a distribuição for simétrica
igual 3
Sample Covariance =
Cov(X,Y)=∑(Xi−¯X)(Yi−¯Y) / n−1
Sample Covariance measure…
of how two variables move together
Correlation is
a standardized measure of the linear relationship between two variables
Correlation =
Corr(X,Y) = Cov[X,Y] / ( StdDev(X) ∙ StdDev(Y) )
Spurious Correlation refers
to correlation that is either the result of chance or present due to changes in both variables over time that is caused by their association with third variable. Exemplo: morte de pinguim e queimada na amazonia.
Random Variable is
an uncertain quantity/number
Probability of an event, a probabilidade é de xx até xx
de 0 até 1
Mutually exclusive events are
events that cannot both happen at the same time
Exhaustive events are
those tha include all possible outcomes
Probability Empirical is based…
on analysis of data
Probability of a priori is based
on reasing, not experience
Probability Subjective is based on…
personal perception
Probability Unconditional is …..
probabilidade de um evento acontecer, sem ter nenhuma condição de outro evento P(A)
Probability Conditional is ….
probabilidade de um evento acontecer, por conta de outro evento P(A | B)
Joint probability is …
de dois eventos acontecerem ao mesmo tempo P(AB)
Rules of Probability, Quando usar + e - e quando usar X. Sobre and e or
Para and -> P (AB) = P (A|B) X P (B) / Para or -> P (A or B) = P(A) + P(B) - P(AB)
Bayer´s Formula =
Prob (HG | DI) = P (DI | HG) X P(HG) / P (DI)
Portifolio expected return =
E(rp) = W1xE(R1) + W2xE(R2) + WnxE(Rn) -> W = peso do ativo e E(R) = retorno esperado
Variance =
σ^2 = P(xi) x [xi - E(x)]^2 -> E(x) seria o valor esperado com base no peso e retorno esperado de todos ativos
Standard Deviation =
σ = raiz de (P(xi) x [xi - E(x)]^2 )
Covariance (Ra,Rb) =
{[Ra - E(Ra)] x [Rb - E(Rb)]} x P
Portifolio Standard Deviation or Portfolio Variance =
Portfolio variance = w1^2σ1^2 + w2^2σ2^2 + 2w1w2Cov1,2
Labeling, qual é a fórmula e qunado usar?
= n! / (n1!) x (n2!) x …. -> “there are n items that can each receive one K different labels, seria omo quero classificar. Ex: quantas classificações consigo fazer para ações de diferentes segmentos
Combination Fórmula, qual é a fórmula e qunado usar?
= n! / ((n-r) x r! ) -> neste caso a ordem importa (dica memorização, “combination = combinações, ou seja, quantas combinações consigo fazer considerando que a ordem não importa). Ex: número de possiblidade de venda das ações.
Permutation Fórmula, qual é a fórmula e qunado usar?
= n! / (n-r)! -> neste caso a ordem importa. Ex: ordem de venda de ações diferentes
Probability Distribution gives …
the probabilities of all possible outcomes for a random variable.
Discrete Distribution is ….
a countable number of possibles outcomes
Continuous Distribution is …
an infinite number of possible outcomes
Probability fuction gives the probability that a discrete random variable will take ….
on the value x
Continuous Uniform Distribution defined ….
over a range that spans between some lower limit a, and some upper limit b.
Cumulative distribution fuction gives the probability that a random variable will be ….
less than or equal to a given value
Binomial Random Variable da a probabilidade de … e formula?
sucesso ou falhar em n tentativa (dica memorização, bi pode dar duas coisas). Formula = (n! / ((n-x)! x!) . p^x . (1-p)^n-x (dica memorização da formula, primeira parte é a combination formula)
Binominal tree, qual é a formula para calcular o Down factor?
D = 1 / up
Para o confidence Interval (Normal Distribution), qual valor para 68%, 90%, 95% e 99%
68% = 1,00 ; 90% = 1,65; 95% = 1,96 ; 99% = 2,58
Para normal distribution, no intervalo de confiança de 68%, quantos desvios padrões tem?
1
Para normal distribution, no intervalo de confiança de 95%, quantos desvios padrões tem?
2
Standard Normal Distribution, a formula é ….
z = x - u / σ
Shortfall risk is the probability that a …..
portfolio return or value will be below a target return or value
Sobre target return ou treshhold, usar a formula?
Sf Ratio = [ E(Rp) - RL ] / σp -> E(Rp) = valor esperado; RL = minimo acieto; σp = desvio padrão
Em uma Lognormal Distribution the skewed será? (Positiva ou negativa / Para direita ou esquerda?
Será positiva e para direita, sempre.
Continuous Compounding Rate =
ln (1 + HPR)
Quando falamos de continuous compounding ratem com EAY with continuous compounding a formula é?
=e^i - 1
T-students, principais caracteristicas
small samples (n<30), unkonwn variance, symmetrical (bell shaped), fatter tails than a normal distribution.
Para T-students degrees of fredoom é
df = n -1
Chi-Square Distribution princiapis caracteristicas
asymetric
F-Distribution is quotient of two
chi-squared distributions with m and n degrees of freedom.
F-Distribution is symmetric ou asymmetric?
asymetric
T-students nos da o resultado de
population mean, difference between means of two populations, difference between paired observations, population correlation
Chi-square nos da o resultado de
value of variance of normal population
F-Distribution nos da o resultado de
equality of variances for two normal population
Monte Carlo Simulation can be used to estimate a
distribution of derivatives prices or of NPVs
Passo a Passo de Monte Carlo
1) Specify the parameters of the distribution; 2) Use computer random generation of variables; 3) Value the derivative using those values; 4)Repeat steps 2 and 3, 1000s of times; 4) Calculate mean/variance of all values
Sampling Error is the difference between
a sample statistic and true population parameter.
Sampling Distribution is the distribution of all …
possible values of a statistic for samples of size n
Simple random sampling, every population member has an equal or different probability of being select?
equal
Nonprobability sampling, use judment of researcher and it is
low cost/readily available data, to select sample items
Stratified Random Sampling you need to create..
subgrupos da população com base nas caracteristicas principais
Cluster Sampling create …
subsets, each of which is representative of an aoveral population.
Convenience sampling you use…
readily available low cost dat for prelimary investigation
Judmental sampling you use..
select observations from population based on analyst judment
Central limit Theorem, a teoria é sobre?
quanto maior a amostra (sample), mais próximo de uma distribuição normal fica.
Standard Error of the sample mean, as duas formulas usadas são:
σxi = σ / raiz n ; sxi = s / raiz n
About Desirable Estimator Properties, there is Unbiased, Efficient and Consistent. Explicar os 3
Unbiased = expected value equal to parameter; Efficient = sampling distribution has smallest variance of all unbiased estimators; Consistent = a larger sample
Normal Distribution and you Known variance. Qual tabela usa para Small Sample e para Large Sample
Para Small Sample = Z-statistic; Para Large = Z-Statistic
Normal Distribution and you Unknown variance. Qual tabela usa para Small Sample e para Large Sample
Para Small Sample = t-statistic; Para Large = t-Statistic
NonNormal Distribution and you Unknown variance. Qual tabela usa para Small Sample e para Large Sample
Para Small Sample = N.a ; Para Large = t-Statistic
NonNormal Distribution and you known variance. Qual tabela usa para Small Sample e para Large Sample
Para Small Sample = N.a; Para Large = Z-Statistic
Two alternative methods to estimate the standard error of the sample mean is Jacknife and BootStrap. Explicar os dois
Jacknife: calculate multiple sample mean, each with one observation;
BootStrap: take many samples of size n, calculate their sample mean
Data-Snooping bias:
from repeatedly doing tests on same data sample
Sample Selection bias:
sample not really random
Survivership bias:
sampling only surviving firms
Look=ahead bias:
using information not available at the time to construct sample
Time-period bias:
relatioship existis only during the time period of sample data
Self-selection bias:
backfill bias
Para que serve Alternative Hypothesis (Ha)?
supported if the researcher rejects the null hypothesis
Para hypothesis testing, o passo a passo é (1 até 7):
1) State the hypothesis-relation to be tested; 2) Select a test statistic; 3) Specify the level of Significance; 4) State the decision rule for the hypothesis; 5) Collect the sample and calculate statistics; 6) Make a decision about the hypothesis; 7) Make a decision based on the test results
Para que serve Rull Hypothesis (Ho)?
é para ver se a hipotese nula é verdadeira ou falsa
Em hypothesis testing, the test statistic is calculated from sample data and compared to
critical values to test Ho
Em hypothesis testing, the test statistic if exceeds the critical value or is outside the range of critical values, the researcher rejects or accept the Ho?
Rejects.
Em hypothesis testing, type I Error is
rejecting Ho when it is actually true
Em hypothesis testing, type II Error is
failing to reject when Ho it is false
Em hypothesis testing, significance leve is probability of Type I Error or II Error?
Type I Error
Em hypothesis testing, two independent normal populations. Assume population variance are equal and use prob. estimate of variance using both sample. Use t-test, f-test or Chi-square?
T-test and reject if test statistic is outside critical values.
Power test is 1 - prob. of type I Error or II Error?
Type Error II
Em hypothesis testing, p-value is the smallest level of significance at the null can be rejected or accepeted?
Rejected.
Em hypothesis testing test of weather the variance of a normal population equals σ^2.Use t-test, f-test or Chi-square?
Uses Chi-square, two tailed test, reject if outside the critical values
Em hypothesis testing, two dependent normal populations. Numerator is avarage difference between paired observations and denominator is standard deviation of the differences.Use t-test, f-test or Chi-square?
T-test and reject if test statistic is outside critical values.
Em hypothesis testing test of weather the variance of two normal population are equal.Use t-test, f-test or Chi-square?
Uses F-test, VarA / VarB, put larger variance in numerator for one tail test
Em hypothesis testing test of weather the population correlation coefficient equals zero.Use t-test, f-test or Chi-square?
Uses t-test, two tailed, with n-2 degres of fredom
Em hypothesis testing, parametric tests are based on assumptions about …
population distributions and population parametres (t-test, z-test, f-test)
Em hypothesis testing, nonparametric tests are based on assumptions about …
population distributions and test things other than parameter values.
Formula de parametric test of correlation =
t-stat = (r . raiz de n-2) / raiz de 1 - r^2
formula de test of independence from contingency table =
expected if independent = (total for row i x total for colum j) / total for all colums and rows ; Ou pode usar (o - E)^2 / E
Steps in hypothesis testing (1 até 7)
1 Stating the hypotheses.
2 Identifying the test statistic and its probability distribution.
3 Specifying the significance level.
4 Stating the decision rule.
5 Collecting the data and performing the calculations.
6 Making the statistical decision.
7 Making the economic or investment decision.
Null Hypothesis (Ho)
sempre inclui =< ou =>. É para ver se é V ou F
Alternative Hipothesis (Ha)
supported if the researcher rejects the null hipothesis
Em hypothesis testing. O test statistic é comparado com o critical value do test Ho. If the test statistic exceeds the critical value . The researcher should reject Ho or not?
Yes, because is outside the range of critical values.
P-Value
Is the smallest level of significance a which the null can be rejected. Ex: if the p-valeu is given as 0.0213 -> You can reject the null at 5% significance but not at 1%
Em hypothesis testing. When do you use two-tailed test?
Use when testing to see if a population parameter is different from a specified value. (Ex: Ho: u = 0 ; Ha: u <> 0 )
Em hypothesis testing. When do you use one-tailed test?
Use when testing to see if a parameter is above or below a specified value. (Ex: Ho: u <= 0 ; Ha: u > 0 )
Em hypothesis testing. Type I Error
rejecting Ho when it is true
Em hypothesis testing. Type I Error Type II Error
Failing to reject Ho when it is false
Em hypothesis testing. Significa Level is ?
Probability of Type I Error
Em hypothesis testing. Power test is
1 - Probability of Type II Error
Em hypothesis testing. Difference in means test. Two INDEPENDENT NORMAL populations, variances are equal. What Type of test should you use?
T-test, reject if test statistic is outside of the critical value
Em hypothesis testing. Difference in means test. Two DEPENDENT NORMAL populations, numerator is avarage difference between paired observations and denominator is standard deviation of the differences. What Type of test should you use?
T-test, reject if test statistic is outside of the critical value
