Module 5.1: Probability models Portfolio Return & Risk

Question 1

What is the formula for expected return on a portfolio?

Answer 1

Question 2

What is the formula for weight of an asset within a portfolio?

Answer 2

Question 3

Define covariance, and give its range.

Answer 3

Covariance is a measure of how two assets move together.
The range of covariance is negative infinity to positive infinity.

Question 4

What is the formula for covariance?

Answer 4

Question 5

What does a positive covariance between two assets mean?

Answer 5

It means when one random variable is above its mean, the other random variable tends to be above its mean.

Question 6

What does a negative covariance between two assets mean?

Answer 6

That when one random variable is above its mean, the other random variable is typically below its mean.

Question 7

What is the formula for sample covariance?

Answer 7

Question 8

Does order matter in regard to covariance notation for each individual investments?

Answer 8

No.

Question 9

What is the formula for the number of unique covariance terms in a portfolio?

Answer 9

= n(n-1) / 2

Question 10

What is the formula for portfolio variance?

Answer 10

Question 11

What is the portfolio variance formula for a two asset portfolio?

Answer 11

Question 12

What is the formula for correlation coefficient, or correlation?

Answer 12

= Sample Covariance of XY / (sample Standard Deviation of X)(sample standard deviation of y)

Question 13

What is the formula for Sample Covariance of XY?

Answer 13

= Correlation XY(std X • std y)

Question 14

Define shortfall risk in relation to a portfolio.

Answer 14

Probability that a portfolio value or return will fall below a particular target.

Question 15

Assuming that returns are normally distributed, the portfolio with the LARGER safety first ratio using 0 as threshold return, will be the one with the _______ probability of negative returns.

Answer 15

Lower

Question 16

What is the formula for the safety first ratio?

Answer 16

Question 17

What is shortfall risk?

Answer 17

The probability that a portfolio value or return will fall below a particular target value or return over a given period.

Shortfall risk is a critical measure in portfolio management for assessing risk.

Question 18

What does Roy’s safety-first criterion state?

Answer 18

The optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level.

This criterion is used to prioritize safety in investment decisions.

Question 19

What is the minimum acceptable level in Roy’s safety-first criterion called?

Answer 19

Threshold level.

The threshold level is the return that is considered acceptable for the portfolio.

Question 20

Symbolically, how is Roy’s safety-first criterion expressed?

Answer 20

minimize P(Rp < RL) where Rp = portfolio return and RL = threshold level return.

This mathematical expression helps quantify the safety-first approach in portfolio selection.

Question 21

Fill in the blank: Shortfall risk is the probability that a portfolio value or return will fall below a particular _______.

Answer 21

target value or return.

Understanding this concept is essential for risk management in finance.

Question 22

What are the units of variance and covariance?

Answer 22

% Squared

Question 23

When variance and covariance are expressed as whole numbers in % Squared, what is the portfolio variance in?

Answer 23

% Squared

Question 24

What is the portfolio standard deviation expressed in when variance and covariance are in % Squared?

Answer 24

whole percentages

Question 25

What happens to portfolio variance and standard deviation when variance and covariance are expressed as decimals?

Answer 25

Both portfolio returns variance and standard deviation are in decimals

Question 26

Is the statement true or false? Lower covariance values always reduce portfolio variance.

Answer 26

True

Question 27

What is the formula for the correlation coefficient for two variables?

Answer 27

PAB = COVAB / (σA × σB)

Question 28

How can the correlation coefficient be used in the context of portfolio returns variance?

Answer 28

It can be substituted for covariance in the formula for portfolio returns variance

Question 29

What can be used to calculate portfolio returns variance instead of using covariances?

Answer 29

A correlation matrix

Question 30

What is the formula for the expected return of a portfolio composed of n assets?

Answer 30

E(Rp) = W1 E(R1) + W2 E(R2) + ... + Wn E(Rn) ## Footnote E(Rp) represents the expected return of the portfolio, Wi represents the weight of each asset, and E(Rj) represents the expected return of each asset.

Question 31

How is the weight of Asset i in a portfolio calculated?

Answer 31

Wi = Market value of investment in Asset i / Market value of the portfolio ## Footnote Wi represents the weight of Asset i, reflecting its proportion in the overall portfolio.

Question 32

Why is it important to analyze the relationship between two random variables in finance?

Answer 32

To understand how two assets' returns move in relation to each other ## Footnote This analysis helps investors gauge potential risks and returns associated with different assets.

Question 33

Fill in the blank: The expected return and variance for a portfolio of assets can be determined using the properties of the _______.

Answer 33

[individual assets] ## Footnote Understanding the characteristics of each asset is crucial for portfolio analysis.

Question 34

What is covariance?

Answer 34

A measure of how two assets move together

Question 35

What does covariance represent mathematically?

Answer 35

The expected value of the product of the deviations of two random variables from their expected values

Question 36

What is a common symbol for covariance between random variables X and Y?

Answer 36

Cov(X,Y)

Question 37

What is the formula for covariance of asset returns?

Answer 37

Cov(Ri,Rj) = E{[Ri — E(Ri)][Rj - E(Rj)]}

Question 38

What is the covariance of a random variable with itself?

Answer 38

Its variance; Cov(RA, RA) = Var(RA)

Question 39

What is the range of covariance?

Answer 39

From negative infinity to positive infinity

Question 40

What does a positive covariance indicate?

Answer 40

When one random variable is above its mean, the other tends to be above its mean

Question 41

What does a negative covariance indicate?

Answer 41

When one random variable is above its mean, the other tends to be below its mean

Question 42

What does a covariance matrix show?

Answer 42

The covariances between returns on a group of assets.

Question 43

What are the diagonal terms of a covariance matrix?

Answer 43

The variances of each asset's returns.

Question 44

What is the relationship between Cov(RA, RA) and Var(RA)?

Answer 44

Cov(RA, RA) = Var(RA).

Question 45

Is the covariance between two assets dependent on the order of the assets?

Answer 45

No, Cov(RA, RB) = Cov(RB, RA).

Question 46

How many unique off-diagonal covariance terms are there in a covariance matrix for n assets?

Answer 46

There are n(n-1)/2 unique off-diagonal covariance terms.

Question 47

Fill in the blank: In a covariance matrix, Cov(RA, RB) = Cov(RB, _______).

Answer 47

RA

Question 48

What is represented by Cov(RA, RB) in a covariance matrix?

Answer 48

The covariance between the returns on assets A and B.

Question 49

True or False: The covariance matrix is symmetric.

Answer 49

True.