Quantitative Methods Flashcards by Kevin Li

What are the 3 types of risk for a security?

Default Risk
Liquidity Risk - risk of receiving less than fair value if sold
Maturity Risk

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What are the 4 types of measurement scales?

Nominal scales - observations are counted with no particular order
Ordinal scales
Interval scale - provide relative ranking and assurance differences in the scale are equal
Ratio scales - provide ranking and equal difference and have a true zero point

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Compare the arithmetic mean to the geometric mean when measuring investment returns

Arithmetic mean is best measure of next year’s performance. Geometric mean is best when calculating returns over multiple periods or when measuring compounding growth rates

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How is the harmonic mean calculated?

N/sum(1/X_i)

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What are the relative sizes of the arithmetic, geometric, and harmonic mean for a data set with unequal values?

harmonic < geometric < arithmetic

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What is the formula for calculating a percentile?

L = (n+1)(y/100) for the yth percentile and n observations

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What is Chebyshev’s inequality?

For any set of observations, the % of observations that lie within k standard deviations is 1 - 1/k^2 for all k > 1

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What is the coefficient of variation?

standard deviation/mean

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What does leptokurtic mean?

Distribution is more peaked than a normal

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What does platykurtic mean?

Distribution is flatter than a normal

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What is an empirical probability?

Probability established by analyzing past data

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What is an a priori probability?

Probability that is determined using formal reasoning and inspection

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How are odds for an event calculated?

p/(1-p)

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What is likelihood?

It’s equivalent to conditional probability

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What is the equation for covariance?

Cov(X,Y) = E[(X - E[X])(Y - E[Y])]

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What does it mean if two variables are spuriously correlated?

The variables are correlated by chance or because they are both correlated with a 3rd variable

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What is the equation for calculating “labeling”?

For n items that receive k labels, the total number of ways the labels can be assigned is n!/(n_1! * n_2! * n_k!)

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What are the mean and variance of a binomial distribution?

E[X] = np
Var[X] = np(1-p)

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What 3 parameters are necessary to define a multivariate normal distribution of n asset returns?

n # of means
n # of variances
0.5n(n-1) pair-wise correlations

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What is the 90% confidence interval for the mean of X?

-1.65 to 1.65

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What is the 95% confidence interval for the mean of X?

-1.96 to 1.96

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What is the 99% confidence interval for the mean of X?

-2.58 to 2.58

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What is shortfall risk?

The probability that a portfolio value/return will fall below a target

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What is Roy’s safety-first criterion?

Maximize the ratio where (E[R_p] - R_L)/sigma_p

What is the effective annual rate for a continuously compounded rate?

EAR = e^R - 1

What are the limitations of Monte Carlo?

They're complex and are no better than the assumptions about the distributions

What is systematic sampling?

Example: selecting every nth member from a population

What is cross sectional data?

Sample of observations taken at a single point in time

What is longitudinal data?

Observations over time of multiple characteristics of the same entity, like unemployment, inflation, and GDP

What is panel data?

Observations over time of the same characteristic for multiple entities

What does the central limit theorem state?

The sampling distribution of the sample mean approaches a normal with mean u and variance equal to sigma^2/n

What are the 3 desirable properties of an estimator?

1. Unbiased - expected value of estimator is equal to the parameter 2. Efficient - variance of sampling distribution is smaller than other unbiased estimators 3. Consistent - Accuracy of estimate increases as sample size increases

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

t-statistic

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

If sample size is small, none. If sample size is large, z-statistic

What test statistic should be used for a nonnormal distribution with unknown variance?

If sample size is small, none. If sample size is large, t-statistic

What is data-mining bias?

When data is repeatedly sampled until a pattern is found

What is sample selection bias?

When data is systematically excluded from the analysis

What is survivorship bias?

A type of sample selection bias where only survivors are sampled

What is look-ahead bias?

Using sample data that was not available on the test date

What is time-period bias?

When the sampled time period is too short or too long

What are the 7 steps of hypothesis testing?

1. State hypothesis 2. Select test statistic 3. Specify level of significance 4. State decision rule 5. Collect sample and calculate sample statistics 6. Make decision regarding hypothesis 7. Make decision based on results of the test

What is a Type I error?

Rejecting the null when it is actually true

What is a Type II error?

Failing to reject the null when it is false

What is the power of a test?

Probability of correctly rejecting the null when it is false. Also equal to 1 - P(Type II error)

What is the expected return of a portfolio?

sum(w_i * E(R_i))

What is the variance of a portfolio with 2 assets?

(w_a)^2(Var_A) + (w_b)^2(Var_B) + 2(w_a)(w_b)Cov(A,B)

What is the equation for correlation?

Cov(X,Y)/(sigma_X * sigma_Y)

Given return on assets A and B and their joint probability distribution, how do you calculate the expected return of A? The covariance of A and B?

E(RA) = P(RA1,RB1)RA1 + P(RA2,RB2)RA2 + P(RA3,RB3)RA3

What are the 2 ways t-tests are used to test differences between means of two populations?

If the variances are unknown but equal, the variances can be pooled. If both variances are unknown and unequal, the individual sample variances must be used.

When is a paired comparisons or mean differences test used?

When two populations are dependent

What is a chi-squared test used for?

To determine if the alternative hypothesis variance of a normal distribution is statistically different the null hypothesis variance

What is a F-test used for?

To determine if the variances of two independent normal distributions are statistically different

How is the F-statistic calculated? How many degrees of freedom are used?

F = s1^2/(s2^2) with s1^2 always being the larger variance. There are n1 -1 and n2 -1 degrees of freedom

How is the test statistic calculated to measure whether two variables are correlated (null = correlation is zero)? What distribution does the test statistic follow?

r*root(n-2)/root(1-r^2) where r = sample correlation and n = sample size. This follows a t-distribution with n - 2 degrees of freedom

What two qualities characterize a parametric test?

The test is concerned with population parameters like the mean or variance. The population distribution that the sample is taken from is assumed

What characterizes nonparametric tests?

They either do not consider a parameter or make no assumptions about the population distribution.

Name 3 situations in which nonparametric tests are more appropriate.

1. Assumptions can't be made about the distribution. For example, measuring the mean when the population is nonnormal and the sample size is small. 2. When data is ordinal (made up of ranking) rather than values. 3. The hypothesis does not involve parameters of a distribution

What is the Spearman rank correlation test?

Can be used when data is not normally distributed. A higher number means high (low) ranks are correlated with high (low) ranks in the next year