L3 - Review of Mathematics and Statistics

Question 1

What is a Population in statistics?

Answer 1

• All the possible outcome
• Something we want to understand but cannot directly observe
• We usually impose some structure on r to understand the return generating process.

•E.g., we may assume r = ε, where ε ~ N( μ, σ²).

where r = asset return

Question 2

What is a Sample?

Answer 2

–The outcomes we get to observe (historical data in this case)

–Not comprehensive, but is the best we can have

–Usually use X(bar) as an estimate for μ, although we never get to observe the true value of μ.

Question 3

How to calculate expected return?

Answer 3

• also the population mean and expected value
• usually cant observe the population mean however so we use the sample mean instead

Question 4

What do use for a proxy for the population mean?

Answer 4

Question 5

How do you calculate variance?

Answer 5

Question 6

Properties of Variance?

Answer 6

• • non-negative
• variance of a constant is 0
• If you add a constant to all values in a data set variance is still the same
• Var(aX)= a²Var(X)
• Var(aX+b)= a²Var(X)

Question 7

How do you calculate sample variance?

Answer 7

• denote sample variance with S² and population with σ²

Question 8

Why do we use both standard deviation and variance?

Answer 8

• standard deviation is useful to have even though it gives that same information as variance - as it is in the same units as the mean (uniting them both)

Question 9

How do you calculate risk using the population?

Answer 9

Question 10

How do you calculate the risk of a portfolio of a sample?

Answer 10

Question 11

How do you calculate Covariance?

Answer 11

• Cov(X,a)=0
• Cov(X,X)=Var(X)
• Cov(X,Y)=Cov(Y,X)
• Cov(aX,bY)=abCov(X,Y)
• Cov(X+a,Y+b)=Cov(X,Y)

Question 12

How do you calculate the Variance of the Sum of two correlated random variables?

Answer 12

Question 13

How do you calculate the Sample Covariance?

Answer 13

Question 14

What is a flaw in covariance?

Answer 14

• Covariances are hard to interpret. Larger covariance does not necessarily imply stronger co-movement.
• One major flaw of the covariance: its magnitude depends on the measurement units of X and Y, not its degree of covariance.
• For example, suppose Cov(X,Y) = 35 when X is measured in centimetre and Y is measured in a kilogram.
• If we measure X in meter, we get Cov(X,Y) = 0.35
• If we measure X in inches, we get Cov(X,Y) = 13.78

•The related and more commonly-used correlation coefficient remedies this disadvantage.

Question 15

How do you calculate Correlation?

Answer 15

• we use the correlation coefficient as it takes the measurement units out of the calculation

Question 16

What is the Correlation Coefficient Flaw?

Answer 16

So even though they are related and there is an obvious relationship between the two variables - correlation is still 0

Question 17

important formulas for linear regression?

Answer 17

Question 18

How do you calculate the holding period return?

Answer 18

• numerator = profit/loss
• first part = capital gain
• second part = dividend yield

Question 19

How do you calculate the rate of return on a long-term asset?

Answer 19

• solving for r

Question 20

Who laid the groundwork for Modern Portfolio theory?

Answer 20

• •Harry Markowitz (Nobel Winner, 1990)
  
  • –He noticed the lacks of analysis of risk in theory.
  • –He applied statistical concepts to study the effects of asset risk and return, and to solve investment problems.
  • –His theories totally changed the practices of professional investment.
• •Modern Portfolio Theory: Maximize portfolio expected return for a given amount of portfolio risk, or equivalently minimize risk for a given level of expected return.
  
  • –Prove mathematically the best possible diversification strategy

Question 21

What are the characteristics of a probability distribution?

Answer 21

Characteristics of probability distributions:

1) Mean

–Most likely value (expected value)

2)Variance or standard deviation (volatility)

–Degree of deviation from the mean value

3)Skewness (depends on the third moment E(X³))

−Degree of asymmetry in the distribution

4)Kurtosis (depends on the fourth moment E(X⁴))

–Degree of fatness in the tail area –> important because shows how more likely extreme positive and negative values are

Question 22

What does the degree of skewness signify for risk?

Answer 22

• extreme negative returns are high when the left tail is long

Question 23

Why is kurotosis important for analysing a portfolios returns?

Answer 23

important because shows how more likely extreme positive and negative values (returns) are

Question 24

Why may standard deviation not be a good risk measure?

Answer 24

• Investors are concerned about downside risk.
• Standard deviation includes both the above-average returns (upside risk) and the below-average returns (downside risk). If returns are skewed, standard deviation is not the only relevant measure of risk. –> no measure of skew or kurtosis
• Holding expected return and standard deviation constant, investor would prefer positive skewed distribution.

Question 25

what factors may affect your risk aversion?

Answer 25

• Wealth
• Gender
• Marital Status
• Age
• Experience
• Other factors…