Week 5

Question 1

What does COCO do for any set of normally distributed data?

Answer 1

COCO transforms any set of normally distributed data into standard normally distributed data. COCO makes sure the Z terms are all independent, the sample variance/the population variance = the sum of all Z^2 values (except the initial Z0) over N-1, and that Z0 = (sample mean-population mean)/(population variance/N^0.5)). It also shows that the sample variance and sample mean are statistically independent.

Question 2

What is a robust statistic?

Answer 2

One that is insensitive to general non-normality of the parent population. This means a departure from normality must retain the same sign.

Question 3

Is the T-statistic robust?

Answer 3

The T-statistic is not robust if the data has skewness or non-normal kurtosis if the sample size is too small. However, if the sample size is large enough (100+ for large caps or 200+ for small), the T-Stat will still follow a t-distribution(standard normal for large sample size).

Question 4

What will occur for a T test if the data used is not independent?

Answer 4

The scaled sample mean will not be standard normal, and hence will no longer generate a scaled chi-square, making the t-statistic for the mean invalid.
This occurs because the variance of the sum of the two samples will have cross product terms, which will be positive if the dependence is positive(increasing variance) and negative otherwise(decreasing variance).
This will change the variance assumed by the t-statistic, and hence the standard error. As such the t stat will be smaller (overstated t stat) if there is positive correlation, or larger (understated t stat) if there is negative correlation.

Question 5

How can we fix a T-stat with dependent data?

Answer 5

multiply our t-stat by the square root of the yule factor = sqrt(1-correlation)/(1+correlation).
If the correlation is small we can just multiply by 1-correlation.

Question 6

What does parametric and non-parametric mean? How does a T statistic relate to this?

Answer 6

Parametric means with a measure, non-parametric means without a measure. This could refer to functional form or distribution.
The T statistic is distributionally non-parametric in large samples (because it doesn’t have to be normally distributed), and distributionally parametric in small samples (because it does have to be normally distributed).
By default parametric refers to distributional parametric.

Question 7

What are some examples of functionally and non-functionally parametric tests?

Answer 7

PPMCC, and Ordinary least squares are examples of functionally parametric tests. SROCC is an example of a non-functionally parametric test.

Question 8

What are the ingredients of an optimization? How are they typically done?

Answer 8

An objective function to optimize (e.g x squared), choice variables (what can be changed, e.g x), and constraints (anything x cannot be).
Typically this will be done using a hill climber function to maximize or minimize the objective function.

Question 9

What is an example of an optimization problem in finance where the choice variables are not graphed?

Answer 9

The minimum variance frontier, graphs the returns and standard deviations, though the choice variables are the portfolio weights of the stocks.

Question 10

What is the covariance of two different portfolios given by if we have their holdings weight and the covariance matrix?

Answer 10

The transverse of the holdings matrix * the covariance matrix * the holdings matrix of the second portfolio.

Question 11

How do we typically get the beta of a portfolio relative to a benchmark?

Answer 11

covariance of the portfolio and the benchmark divided by the variance of the benchmark.

Question 12

Why can constraints be bad for analytical calculus?

Answer 12

Some constraints can prevent us from using analytical calculus, forcing us to use numeric calculus instead.

Question 13

What is the markowitz frontier problem?

Answer 13

Given a mean return vector and covariance matrix. Find the vector of proportional holdings in the risk assets that minimzes the portfolio risk for a given level of portfolio expected return, with full investment (sum of weights = 1) and possible short selling. This can be done using any two frontier portfolios with distinct means, as long as they are both fully invested.

Question 14

Where to the Tobin frontier and Markowitz frontier meet?

Answer 14

At the tangency portfolio.

Question 15

What is the Tobin frontier?

Answer 15

Given a mean returns, and covariance matrix of the vector of future returns to some risk assets, find the vector of proportional holdings that minimized portfolio risk for a given level of expected return. This does not require the weights to be 1 (balancing is done with riskless borrowing/lending at the risk free rate.

Question 16

What is the upper part of the Tobin frontier known as? Is there another part?

Answer 16

The capital market line, It will be tangential to the Markowitz frontier, meeting at the tangency portfolio.
The other part is downard sloping, and goes under the Markowitz frontier.

Question 17

What is CAPM? What does it assume?

Answer 17

CAPM asserts that the expected return on an asset is the riskless rate plus a risk premium. This risk premium is the model’s compensation for non-diversifiable risk. Any idiosyncratic risk a tock holds is removed from the CAPM return, as such CAPM assumes a well diversified portfolio, leaving only a stock’s systematic risk for the risk premium.
The actual equation is CAPM return = risk free rate + beta(systematic risk) * market risk premium.

Question 18

What is Idiosyncratic risk?

Answer 18

Risk that is removed with proper diversification.

Question 19

What is the total risk of an asset given by?

Answer 19

Variance of the asset = systematic risk (beta^2 of the asset(relative systematic risk) * variance of the market + non systematic risk.

Question 20

How can we estimate beta of a stock/portfolio?

Answer 20

Using regression, we can find the covariance and variance of the asset, allowing us to find the beta.

Question 21

Is CAPM a market model and vice versa? Is CAPM a model of risk?

Answer 21

CAPM is not a market model and neither is a market model CAPM. The market model however can be used to estimate CAPM beta. CAPM is not a model of risk, but instead a model of consensus expected returns.

Question 22

What is the roll critique?

Answer 22

A critique of the ability to test CAPM. It states that the returns to this global mean-variance efficient portfolio of all risky assets (the market) is unobservable in practice. The CAPM model is also logically tautological, because any sample of observations on individual returns, regardless of the generating process, will always produce an infinite number of ex post mean-variance efficient portfolios. For each of these, if betas are calculated against one of these portfolios will satisfy the linearity relationship exactly, regardless of whether the true market portfolio is actually mean-variance efficient.
The roll critique only holds for fully-invested reference portfolios.

Question 23

What is the security market line? What does it look like?

Answer 23

The security market line plots beta against returns, this will be perfectly straight if and only if it is plotted against a portfolio along the Markowitz frontier, passing through beta = 1 where it will have the reference portfolio return, and when beta = 0, where the return is the risk free rate.
If betas are not calculated against the Markowitz frontier the security market line will not be straight.

Question 24

If betas are calculated relative to a reference portfolio on the minimum variance of the Markowitz frontier what can we do? Does CAPM satisfy this?

Answer 24

In this case every portfolio in the mean-stdeviation space is projected horizontally onto a perfectly linear security market line, this allows us to infer a portfolio’s beta from its return without needing to see the security market line.
CAPM usually satisfies these properties because the world market portfolio of all risky assets is mean variance efficient. This allows the CAPM betas to be inferred from the Markowitz mean-stdeviation plot.

Question 25

What is a common problem with CAPM for tests with regards to backward looking and forward looking returns?

Answer 25

CAPM is about forward-looking expected returns but most tests are done using backward-looking realized returns, creating problems for tests.

Question 26

What do Ex ante and ex post mean? Which do both lead to with Markowitz/Tobin frontiers and tangency portfolios?

Answer 26

Ex ante means before the event, so making use of future predictions. ex post means using previous returns. A Markowitz/Tobin frontier and Tangency portfolio lead to the security market line known as CAPM with ex ante returns, or a perfectly straight security market line if betas are calculated relative to the tangency potfolio in the event of ex post (random points if not to tangency portfolio in ex post).

Question 27

What are the main conclusions of the roll critique?

Answer 27

1. CAPM cannot be tested with past returns because the linearity is determined by the reference portfolio and 2. the world market portfolio of risky assets is unobservable, as such the correct reference portfolio and the theoretical world CAPM works in therefore never appears for testing.