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CFA Lvl 1 > Quantitative Methods > Flashcards

Quantitative Methods Flashcards

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1
Q

Numerical Data (e.g. Discrete, Continuous)

A

Values that can be counted.

We can perform mathematical operations only on numerical data.

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2
Q

Categorical Data (e.g. Nominal, Ordinal)

A

consist of labels that can be used to classify a set of data into groups. Categorical data may be nominal or ordinal.

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3
Q

Discrete Data

A

Countable data , such as the months, days, or hours in a year

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4
Q

Continuous Data

A

Can take any fractional value (e.g., the annual percentage return on an investment).

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5
Q

Nominal Data

A

Data that cannot be placed in a logical order

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6
Q

Ordinal Data

A

Can be ranked in logical order

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7
Q

Structured Data

A

Data that can be organised in a defined way

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8
Q

Time series

A

A set of observations taken periodically e.g. at equal intervals over time

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9
Q

Cross-sectional data

A

Refers to a set of comparable observations all taken at one specific point in time.

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10
Q

Panel Data

A

Time series and cross-sectional data combined

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11
Q

Unstructured Data

A

A mix of data with no defined structure

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12
Q

One-dimensional array

A

represents a single variable (e.g. a time series)

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13
Q

Two-dimensional array

A

Represents two variables (e.g. panel data)

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14
Q

Contingency table

A

A two-dimensional array that displays the joint frequencies of two variables

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15
Q

Confusion matrix

A

A contingency table (two variables) that displays predicted and actual occurrences of an event

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16
Q

Relationship between geometric and arithmetic mean

A

The geometric mean is always less than or equal to the arithmetic mean, and the difference increases as the dispersion of the observations increases

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17
Q

Trimmed mean

A

Estimate the mean without the effects of a given percentage of outliers.

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18
Q

Winsorized mean

A

Decrease the effect of outliers on the mean.

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19
Q

Harmonic mean

A

Calculate the average share cost from periodic purchases in a fixed dollar amount.

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20
Q

Empirical probability

A

established by analysing past data (outcomes)

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21
Q

Priori probability

A

determined using reasoning and inspection process (not data) e.g. looking at a coin and deciding there is a 50/50 chance of each outcome.

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22
Q

Subjective probability

A

Established using personal judgement

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23
Q

Unconditional probability (marginal probability)

A

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

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24
Q

Conditional probability

A

where the occurrence of one event affects the probability of the occurrence of another event. e.g. Prob (A I B)

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25
Q

Multiplication rule of probability

A

P (AB) = P (A I B) * P (B)

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26
Q

Addition rule of probability

A

P (A or B) = P (A) + P (B) - P (AB)

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27
Q

Probability distribution

A

The probabilities of all the possible outcomes for a random variable

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28
Q

A discrete random variable

A

when the number of possible outcomes in a probability can be counted and there is a measurable/positive probability. e.g. the number of days it may rain in a month.

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29
Q

A continuous random variable

A

When the number of possible outcomes is infinite, even if upper and lower bands exist. e.g. the amount of rainfall per month.

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30
Q

The probability function , p(x)

A

gives the probability that a discrete random variable will equal X

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31
Q

A cumulative probability function (cdf) , F(x)

A

gives the probability that a random variable will be less than or equal to a given value.

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32
Q

Binomial Random Variable - E(X) = np

A

Binomial Random Variable - Var(X) = np(1-p)

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33
Q

For a continuous random variable X, the probability of any single value of X is

A

0

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34
Q

The normal distribution has the following key properties:

A
  • It is completely described by its mean, μ, and variance, σ2, stated as X ~ N(μ, σ2).
    In words, this says that “X is normally distributed with mean μ and variance σ2.”
  • Skewness = 0 (symmetrical),
    meaning that the normal distribution is symmetric about its mean, so that P(X ≤ μ) = P(μ ≤ X) = 0.5, and mean = median = mode.
  • Kurtosis = 3;
    this is a measure of how flat the distribution is. Recall that excess kurtosis is measured relative to 3, the kurtosis of the normal distribution.
  • A linear combination of normally distributed random variables is also normally distributed.
  • The probabilities of outcomes further above and below the mean get smaller and smaller but do not go to zero (the tails get very thin but extend infinitely).
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35
Q

univariate distribution

A

 the distribution of a single random variable

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36
Q

A multivariate distribution

A

the distribution of two or more random variables (takes into account correlation coefficients)

  • specifies the probabilities associated with a group of random variables and is meaningful only when the behavior of each random variable in the group is in some way dependent on the behavior of the others.
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37
Q

Number of correlations in a portfolio

A

0.5n*(n-1)

n = no. of assets in portfolio / variables

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38
Q

Normal distribution: +/-1 s.d. from the mean

A

68% confidence interval

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39
Q

Normal distribution: +/-1.65 s.d. from the mean

A

90% confidence interval

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40
Q

Normal distribution: +/-1.96 s.d. from the mean

A

95% confidence interval

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41
Q

Normal distribution: +/-2.58 s.d. from the mean

A

99% confidence interval

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42
Q

“standardizing a random variable” (finding z)

A

measuring how far it lies from the arithmetic mean

z = the no. of standard deviations the variable is from the mean

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43
Q

How to calculate z

how many standard deviations a variable is from the mean

A

z = ( x - pop. mean ) / s.d.

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44
Q

shortfall risk

A

probability that a portfolio return or value will be below a target return or value

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45
Q

Roy’s safety first ratio (SF ratio)

A

no. of standard deviations the target return is from the expected return/value

The larger the SF ratio, the lower the probability of falling below the minimum threshold.

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46
Q

For a standard normal distribution, F(0) is:

A

0.5

By the symmetry of the z-distribution and F(0) = 0.5. Half the distribution lies on each side of the mean. (LOS 4.j)

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47
Q

Holding period of return –> Continuously compounded rate

A

ln ( 1 + holding period of return)

ln = natural log

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48
Q

Continuously compounded rate –> Holding period of return

A

e^ continuously compounded rate -1

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49
Q

t-distribution

A
  • Symmetrical.
  • Defined by degrees of freedom (df), where the degrees of freedom = no. of sample observations - 1 (n – 1), for sample means.
  • More probability in the tails (“fatter tails”) than the normal distribution.
  • As the degrees of freedom (the sample size) gets larger, the shape of the t-distribution more closely approaches a standard normal distribution.

NOT PLATYKURTIC - is less peaked than normal dist. but has fatter tails.

For t-distribution, the lower the degrees of freedom, the fatter the tails and the greater the probability of extreme outcomes.

50
Q

chi-square distribution (x^2)

used to test variance of a normally distributed population

A
  • Distribution of the sum of squared values of n independent standard normal random variables (all positive values)
  • Asymmetric
  • Degrees of freedom = n-1
  • As degrees of freedom increases, approaches normal distribution
51
Q

Degrees of freedom, k (in context of distribution charts)

A

the number of values a random variable can vary from the mean

52
Q

F-distribution

used to test variance of two population variances

A
  • Quotient of two chi-square distributions with m and n degrees of freedom (all positive values)
  • Asymmetric
  • As degrees of freedom increase, approaches normal distribution
53
Q

F-stat formula

F-distribution

A

F-stat = ( x^2 / m ) / ( x^2 / n )

= (chi-square for sample 1 / m ) / (chi-square for sample 1 / n )

54
Q

Monte Carlo Simulation

used to estimate a distribution of asset prices

A

Generating 1000s of simulations of the asset using its variables, then calculate the mean/variance of the outcomes and price the asset accordingly

55
Q

Use of Monte Carlo Simulation

A
  • Value complex securities.
  • Simulate the profits/losses from a trading strategy.
  • Calculate estimates of value at risk (VaR) to determine the riskiness of a portfolio of assets and liabilities.
  • Simulate pension fund assets and liabilities over time to examine the variability of the difference between the two.
  • Value portfolios of assets that have abnormal returns distributions.
56
Q

Binomial random variable

A

When there are only two possible outcomes of a given event.

57
Q

Sampling error

A

The difference between a sample statistic and its corresponding population parameter

sampling error of the mean = sample mean – population mean = x – µ

58
Q

The standard error of the sample mean (when population is known)

A

standard deviation of the distribution of the sample means

σx = σp / √n

59
Q

Effect on the standard deviation of the sample, if the sample (n) increases?

A

n ↑ σs ↓

60
Q

Desirable characteristics of an estimator (sample statistic)

A

Unbiased
Efficient
Consistent

61
Q

Simple random sampling

A

Selecting a sample where each item in the population is has the same probability of being chosen

62
Q

Stratified random sampling

A

randomly selecting samples proportionally from sub-groups. Sub-groups are formed based on one or more defining characteristics

63
Q

Cluster sampling

A

similar to strat. random sampling - but subgroups are not necessarily based on the data.

1 stage: sample is chosen from random subgroups (clusters)
2 stage: sample is chosen from each subgroup (cluster)

64
Q

Central limit theorem

A

For a population with a mean (µ) and a variance (σ^2) - ta sample distribution of 30+ will reflect the distribution of the population

65
Q

Confidence interval

A

A range of values in which the population mean is expected to lie within a given probability

66
Q

Reliability factor for 90% confidence interval

A

1.645

67
Q

Reliability factor for 95% confidence interval

A

1.96

68
Q

Reliability factor for 99% confidence interval

A

2.575

69
Q

Confidence interval for a single item selected from the population

A

population mean (µ) +/- reliability factor * σ

70
Q

Confidence interval for a point estimate (values used to estimate population parameters) selected from a sample

A

Mean of sample +/ reliability factor * standard error

71
Q

Confidence interval for a sample mean

A

population mean (µ) +/- reliability factor * standard erro

72
Q

Which test statistic should be used for a normal distribution with a known variance?

A

z-statistic

73
Q

Which test statistic should be used for a normal distribution with an unknown variance?

A

t-statistic

74
Q

Which test statistic should be used for a non-normal distribution with a known variance?

A

z-statistic

NB not available with a small sample n<30

75
Q

Which test statistic should be used for a non-normal distribution with an unknown variance?

A

t-statistic

NB not available with a small sample n<30

76
Q

Jackknife method of estimating standard error of the sample mean

A

Calculate the s.d. of multiple sample means (each sample with one observation removed from the sample).

  • Computationally simple
  • Used when population is small
  • Removes bias from statistical estimates
77
Q

Bootstrap method of estimating standard error of the sample mean

A

Calculate the s.d. of multiple sample means (each sample possible).

78
Q

Two issues of the idea that larger samples increase accuracy of understanding the population

A
  • May contain wrong observations (from other populations)

- Additional cost

79
Q

Data snooping

A

Using a sample of observations to form an opinion - leads to ‘data snooping bias’

80
Q

Sample selection bias

A

When certain observations are systematically excluded from the analysis (usually due to lack of available data)

81
Q

Survivorship bias

A

Only including active/live data. e.g. only including active funds in an analysis of fund performance.

82
Q

Time-period bias

A

Using data within a time period that is either too long or too short

83
Q

Look-ahead bias

A

When a study tests a relationship with data that was not available on the test date.

84
Q

Stratified random sampling is most often used to preserve the distribution of risk factors when creating a portfolio to track an index of:

A

Corporate bonds

risk factors e.g. ‘stratas’ can be more easily identified - which forms the basis of the sample

85
Q

If random variable Y follows a lognormal distribution then the natural log of Y must be:

A

normally distributed.

86
Q

Steps involved in hypothesis testing:

A
  • Hypothesis
  • Test statistic
  • Level of significance
  • Decision rule for hypothesis
  • Collect sample and calculate statistics
  • Make decision on hypothesis
  • Make decision on test results
87
Q

Null hypothesis (Ho)

A
  • Always includes ‘=’ sign
  • Two tailed test
  • The test the researcher wants to reject
88
Q

Alternative hypothesis (Ha)

A
  • What is concluded if null hypothesis is wrong
89
Q

General decision rule for a two-tailed test:

A

Reject Ho (null hypothesis) if:
test statistic > upper critical value, or
test statistic < lower critical value

(in one of the outer tails)

90
Q

test statistic equation

A

(sample statistic - hypothesized value) / SE of sample statistic

91
Q

Type I error

A

Rejecting the null hypothesis when it is true

92
Q

Type II error

A

Failing to reject null hypothesis when it is false

determined by sample size and choice of significance level

93
Q

Probability of making a Type I error

wrongly rejecting null hypothesis

A

The significance level (α)

94
Q

Probability of correctly rejecting null hypothesis?

A

The power of the test

1 - the prob. of making a type 2 error

95
Q

What is the decision rule for rejecting or failing to reject the null hypothesis based on?

A

the distribution of the test statistic

96
Q

Statistical significance

A

refers to the use of a sample to carry out a statistical test meant to reveal any significant deviation from the stated null hypothesis.

97
Q

Economic significance

A

the degree to which is the statistical significance is economically viable

98
Q

p-value

A

Probability of obtaining a test statistic that would lead to a rejection of the null hypothesis (assuming the null hypothesis is true)

The smallest level of significance where the null can be rejected

99
Q

When is it appropriate to use a z-test as the appropriate hypothesis test of the population mean?

A

Normal distribution and known variance

100
Q

When is it appropriate to use a t-test as the appropriate hypothesis test of the population mean?

A

Unknown variance

101
Q

Critical z-values for 10% level of significance

A

Two-tailed test: +/-1.65

One-tailed test: +1.28 or -1.28

102
Q

Critical z-values for 5% level of significance

A

Two-tailed test: +/-1.96

One-tailed test: +1.65 or -1.65

103
Q

Critical z-values for 1% level of significance

A

Two-tailed test: +/-2.58

One-tailed test: +2.33 or -2.33

104
Q

Difference in means test

A

Two populations that are independent and normally distributed

105
Q

Paired comparisons test

A

Two populations that are dependent of each other and normally distributed

106
Q

How to test for the variance of a normally distributed population

A

The chi-squared test

107
Q

How to test whether the variances of two normal populations are equal

A

The F -test

108
Q

Parametric tests

A

based on assumptions about population distribution and parameters (e.g. mean = 3, variance = 100)

109
Q

Non-parametric tests

A

based on minimal/no assumptions of population and test things other than parameter values (e.g. rank correlation tests, runs tests,)

110
Q

How to test whether two characteristics in a sample of data are independent of each other?

A

The X^2 test

111
Q

The appropriate test statistic for a test of the equality of variances for two normally distributed random variables, based on two independent random samples, is:

A

the F-test.

112
Q

The appropriate test statistic to test the hypothesis that the variance of a normally distributed population is equal to 13 is:

A

the χ2 test.

A test of the population variance is a chi-square test.

113
Q

The test statistic for a Spearman rank correlation test for a sample size greater than 30 follows:

A

a t-distribution.

The test statistic for the Spearman rank correlation test follows a t-distribution.

114
Q

Assumptions of Linear Regression

A
  • Linear relationship between the dependent and independent variables
  • Variance of the residual term is constant (homoskedasticity)
  • Residual terms independently and normally distributed
115
Q

Coefficient of Variation (R^2)

A

= SSR / SST
measures the percentage of total variation in Y variable explained by the variation in X

For simple regression R^2 = correlation^2 XY

116
Q

Factorial function

A

The factorial function, denoted n!, tells how many different ways n items can be arranged where all the items are included.

117
Q

Coefficient of Variation

A

σ/µ

118
Q

For a test of the equality of two variances

A

F-statistic.

119
Q

unbiased estimator

A

the expected value equals the parameter it is intended to estimate.

120
Q

A consistent estimator

A

the probability of estimates close to the value of the population parameter increases as sample size increases.

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