1.2: Statistics Overview

Question 1

What type of stocks tend to perform best over very long investment horizons (historically)?

Answer 1

Small stocks, but they also have periods of significant losses

Question 2

What do investors demand to bear a give level of risk?

Answer 2

A risk premium (in terms of a higher expected return)

Question 3

What does a probability distribution do?

Answer 3

Assign a probability (pR) that each possible return R will occur

Question 4

What variance and standard deviation does a risk-free return have?

Answer 4

Zero

Question 5

What is the variance of a return?

Answer 5

A measure of how spread out the distribution of the return is

Question 6

What is a popular estimation approach when we don’t know the probability distribution?

Answer 6

To extrapolate from historical data. By counting the number of times realised return falls within a particular range, we can estimate the underlying probability distribution.

Question 7

What does the mean return capture?

Answer 7

How much we make on average

Question 8

What probability distribution do monthly stock returns follow?

Answer 8

Approximately a normal distribution

Question 9

What is the realised return?

Answer 9

The return that actually occurs over a particular time period

Question 10

Does the average return provide an estimate of the expected return?

Answer 10

Yes, it the probability distribution of returns is the same over time

Question 11

What is the standard error?

Answer 11

The estimation error of a statistical estimate; the standard deviation of the average return

Question 12

What are the necessary assumptions for how we calculate the standard error?

Answer 12

• The distribution of returns is identical each year
• Each year’s return is independent from prior years’ returns

Question 13

What is the excess return and what does it measure?

Answer 13

The difference between the average return for an investment and the average return for treasury bills (risk-free investment).

Measures the average risk premium investors earn for bearing the risk of the investment

Question 14

Is there a relationship historically between risk and return?

Answer 14

• For large portfolios, yes! Higher risk is rewarded with higher average return as a compensation
• For individual stocks, no.

Question 15

What is an observed relationship between size and risk?

Answer 15

Larger stocks have lower volatility overall.

Question 16

What are the two types of risk?

Answer 16

1. Common risk: perfectly correlated and affects all securities
2. Independent risk: uncorrelated and affects a particular security

Question 17

What is diversification?

Answer 17

The averaging out of independent risks in a large portfolio

Question 18

What does the risk of a portfolio depend on?

Answer 18

Whether the individual risks within it are common or independent. Independent risks are diversified in a large portfolio, while common risks are not

Question 19

What is firm-specific risk?

Answer 19

Only affects the company itself = independent risk. Also called idiosyncratic, unique, or diversifiable risk.

Question 20

What is systematic risk?

Answer 20

Affects the entire market = common risk. Also called undiversifiable, or market risk.

Question 21

How does volatility depend on the size of the portfolio?

Answer 21

When risk is diversified, the volatility declines with the size of the portfolio, until only the systematic risk remains

Question 22

What is the risk premium for diversifiable risk?

Answer 22

0.

Investors are not compensated for holding firm-specific risk, since they can “diversify it away”

Question 23

Why do investors require a risk premium for holding systematic risk?

Answer 23

Because they are risk averse and otherwise would be better off selling their stocks and investing in risk-free bonds

Question 24

What is the risk premium determined by?

Answer 24

The systematic risk of the security

Question 25

What are portfolio weights?

Answer 25

The fraction of the total investment in the portfolio held in each individual investment in the portfolio

Question 26

What is the total value of all the portfolio weights?

Answer 26

They always add up to 1. Individual weights can be negative (shorting or borrowing).

Question 27

What determines how much risk is eliminated in a portfolio?

Answer 27

• The degree to which the stocks face common risks
• The degree to which their prices move together

Question 28

What numbers is the correlation always between?

Answer 28

-1 and 1

Question 29

What is the variance of a portfolio?

Answer 29

The sum of the covariances of the returns of all pairs of stocks in the portfolio multiplied by each of their portfolio weights

Question 30

What can be said about the volatility of buying shares on margin (borrowing) compared to the volatility of the stock?

Answer 30

Your positions volatility will be higher than the volatility of the stock

Question 31

What is the formula for the standard error of the estimated expected return?

Answer 31

SE = SD(individual risk) / sqrt(number of observations)

Question 32

What is the formula for the 95% confidence interval for the expected return?

Answer 32

Historical average return +- (2*SE)

Question 33

What is the formula for the estimated covariance between the returns of two stocks?

Answer 33

Cov(Ri, Rj) = (1 / (n-1)) * (SUMOF (Ri -meanRi)*(Rj - meanRj)

Question 34

What is the formula for the correlation between the returns of two stock returns?

Answer 34

Corr(Ri, Rj) = Cov(Ri, Rj) / (SD(Ri) * SD(Rj) )

Question 35

How can we calculate the portfolio variance for a large portfolio?

Answer 35

Using the box method.

Each box is calculated by:
Weight(i) * Weight(j) * Cov(Ri, Rj)

Then, all boxes are summed up to get the portfolio variance