Quantitative Methods Flashcards
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
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
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
How is the harmonic mean calculated?
N/sum(1/X_i)
What are the relative sizes of the arithmetic, geometric, and harmonic mean for a data set with unequal values?
harmonic < geometric < arithmetic
What is the formula for calculating a percentile?
L = (n+1)(y/100) for the yth percentile and n observations
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
What is the coefficient of variation?
standard deviation/mean
What does leptokurtic mean?
Distribution is more peaked than a normal
What does platykurtic mean?
Distribution is flatter than a normal
What is an empirical probability?
Probability established by analyzing past data
What is an a priori probability?
Probability that is determined using formal reasoning and inspection
How are odds for an event calculated?
p/(1-p)
What is likelihood?
It’s equivalent to conditional probability
What is the equation for covariance?
Cov(X,Y) = E[(X - E[X])(Y - E[Y])]
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
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!)
What are the mean and variance of a binomial distribution?
E[X] = np Var[X] = np(1-p)
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
What is the 90% confidence interval for the mean of X?
-1.65 to 1.65
What is the 95% confidence interval for the mean of X?
-1.96 to 1.96
What is the 99% confidence interval for the mean of X?
-2.58 to 2.58
What is shortfall risk?
The probability that a portfolio value/return will fall below a target
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?
- Unbiased - expected value of estimator is equal to the parameter
- Efficient - variance of sampling distribution is smaller than other unbiased estimators
- 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?
- State hypothesis
- Select test statistic
- Specify level of significance
- State decision rule
- Collect sample and calculate sample statistics
- Make decision regarding hypothesis
- 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.
- Assumptions can’t be made about the distribution. For example, measuring the mean when the population is nonnormal and the sample size is small.
- When data is ordinal (made up of ranking) rather than values.
- 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
