READING 5 PORTFOLIO MATHEMATICS Flashcards
How is the weight of Asset i in a portfolio calculated?
A) Current yield / Asset price
B) Market value of Asset i / Total market value of portfolio
C) Return of Asset i / Return of portfolio
D) Price of Asset i / Total shares outstanding
B) Market value of Asset i / Total market value of portfolio
Weight is the proportion of total portfolio market value invested in Asset i.
What does covariance measure?
A) Risk of a single asset
B) Return relative to the market
C) How two variables move together
D) How far data is from the mean
C) How two variables move together
Covariance shows the directional relationship between the returns of two assets.
(AND CORRELATION MEASURES THE STRENGTH)
What is Cov(RA, RA)?
A) Zero
B) Variance of RA
C) Mean of RA
D) Covariance of RA and RB
B) Variance of RA
The covariance of a variable with itself is its variance.
What does a positive covariance between two assets indicate?
A) No relationship
B) They move opposite
C) They move together above their means
D) One asset leads the other
C) They move together above their means
A positive covariance means when one asset is above its mean, the other tends to be above its mean too.
What does a negative covariance indicate?
A) Assets move in opposite directions relative to their means
B) Assets move together
C) Assets are uncorrelated
D) Assets have same returns
A) Assets move in opposite directions relative to their means
Negative covariance means when one is above its mean, the other tends to be below.
What is the range of covariance values?
A) 0 to 1
B) -1 to +1
C) 0 to infinity
D) Negative infinity to positive infinity
D) Negative infinity to positive infinity
Covariance measures how two random variables move together and can take any real value from negative infinity to positive infinity.
In contrast, correlation is a standardized version of covariance and is always between -1 and +1.
In a covariance matrix, what are the diagonal elements?
A) Means
B) Covariances
C) Variances
D) Correlations
C) Variances
Diagonal terms represent each asset’s return variance.
Covariance between RA and RB is equal to:
A) Cov(RA, RB) ≠ Cov(RB, RA)
B) Cov(RA, RB) = Cov(RB, RA)
C) Cov(RA, RB) = 0
D) Cov(RA, RB) > Cov(RB, RA)
B) Cov(RA, RB) = Cov(RB, RA)
Covariance is symmetric, meaning the order does not matter.
Whether you calculate the covariance of Asset A with Asset B, or Asset B with Asset A, the result is the same.
Cov (RA,RB) = E[(RA−E(RA))(RB−E(RB))]
Since multiplication is commutative (meaning 𝑥 × 𝑦 = 𝑦 × 𝑥)
thus, the covariance of (RA,RB) is the same as (RB,RA).
How many unique covariance terms are there for n assets?
A) n
B) n²
C) n(n-1)/2
D) (n+1)/2
C) n(n-1)/2
When you have a group of n assets, you can think of the covariance matrix as an n × n table.
Diagonal elements = variances (example: Cov(RA,RA) = Var(RA))
Off-diagonal elements = covariances (example: Cov(RA,RB))
Since covariance is symmetric (Cov(RA,RB) = Cov(RB,RA)), you don’t count both Cov(RA,RB) and Cov(RB,RA) separately.
You only need the unique pairs.
The number of unique covariance terms is: n(n-1)/2
Example:
For 3 assets (A, B, C):
Pairs are: (A,B), (A,C), (B,C)
Total unique covariances = 3 × (3−1) ÷ 2 = 3
Which of the following is used to calculate portfolio variance?
A) Asset weights only
B) Asset variances and covariances
C) Only expected returns
D) Only covariances
B) Asset variances and covariances
Portfolio variance combines asset weights, variances, and covariances.
What does shortfall risk measure?
A) Chance portfolio beats benchmark
B) Probability return falls below a target
C) Volatility of returns
D) Beta of the portfolio
B) Probability return falls below a target
Shortfall risk is the probability that returns fall below a target level.
Roy’s safety-first criterion seeks to:
A) Maximize returns
B) Minimize standard deviation
C) Minimize probability of falling below a threshold return
D) Maximize probability of exceeding target
C) Minimize probability of falling below a threshold return
It minimizes the chance of portfolio returns falling below a minimum acceptable level.
What is the “threshold level” in Roy’s safety-first criterion?
A) Risk-free rate
B) Minimum acceptable return
C) Maximum drawdown
D) Target volatility
B) Minimum acceptable return
If two assets have positive covariance, the combination will likely have:
A) Higher portfolio risk
B) Lower portfolio risk
C) Zero portfolio risk
D) Undefined risk
A) Higher portfolio risk
Positive covariance increases portfolio risk.
When calculating expected return of a portfolio, the key formula uses:
A) Variances
B) Covariances
C) Weights and expected returns
D) Standard deviations
C) Weights and expected returns
Expected portfolio return is the weighted average of individual expected returns.
What is the expected return of a portfolio formula?
A) Σ (wi × ri)
B) Σ (wi × σi²)
C) Σ (wi × Cov(ri,rj))
D) Σ (wi / σi)
A) Σ (wi × ri)
It is the sum of weights times each asset’s expected return.
If Cov(RA, RB) is positive, what happens when RA increases above its mean?
A) RB decreases
B) RB is unrelated
C) RB tends to increase
D) RB becomes constant
C) RB tends to increase
Positive covariance implies they move together.
The covariance matrix for 3 assets has how many unique covariance terms (off-diagonal)?
A) 6
B) 3
C) 9
D) 1
B) 3
3(3-1)/2 = 3 unique off-diagonal covariances.
What are the properties of covariance?
A) Always positive
B) Symmetric and unbounded
C) Bounded between -1 and +1
D) Zero if variables are correlated
B) Symmetric and unbounded
Covariance is symmetric and (bounded) can take any value from negative to positive infinity.
Which risk measure considers “falling below a target”?
A) Variance
B) Shortfall risk
C) Standard deviation
D) Beta
B) Shortfall risk
Shortfall risk looks specifically at failing to meet a target return.
In portfolio variance, what role does correlation play?
A) None
B) It modifies covariance
C) It increases variance only
D) It is the variance
B) It modifies covariance
Correlation standardizes covariance and affects overall portfolio risk.
What does a covariance of zero imply between two assets?
A) Perfectly correlated
B) Move exactly opposite
C) No linear relationship
D) Perfect inverse correlation
Zero covariance means no linear relationship.
Covariance between Stock A and itself equals:
A) Zero
B) Correlation
C) Variance of Stock A
D) Covariance between Stock A and Stock B
C) Variance of Stock A
Cov(A, A) = Var(A).
Portfolio expected return is a function of:
A) Variances and covariances
B) Individual asset returns and weights
C) Betas only
D) Alphas only
B) Individual asset returns and weights
It’s based on asset expected returns weighted by portfolio allocations.
