Conditional Expectation and Asset Return

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Questions and Answers

What is the relationship established by the linearity property of conditional expectation?

The average of random variables is always greater than zero.

Conditioning on a smaller sigma-algebra increases the expected value.

Expected values are invariant under linear transformations.

The expected value of a sum is equal to the sum of the expected values. (correct)

Which inequality reflects Jensen's inequality when considering a convex function?

E[φ(f (ω))|Σ] ≥ φ(g(ω)) (correct)

E[φ(f )] < E[φ(g)]

E[φ(f (ω))|Σ] ≤ φ(E[f |Σ])

E[φ(f (ω))|Σ] = φ(g(ω))

In mean-variance portfolio theory, how is total return calculated?

Total return equals price appreciation only.

Total return is determined solely by dividends received during the period.

Total return is equal to the total amount invested minus losses.

Total return is the end price of an asset divided by initial investment. (correct)

What does the tower property of conditional expectation imply?

The expected value conditionally reduces complexity with nested sigma-algebras. Signup and view all the answers

When evaluating the risk-return relationship, what does a higher return typically indicate?

Higher risk is associated with higher expected returns. Signup and view all the answers

How do covariance and correlation differ in financial analysis?

Covariance provides direction, while correlation provides magnitude only. Signup and view all the answers

In the context of portfolio theory, what is often considered a bounded measurable function?

A function that maintains values within a set range. Signup and view all the answers

Which statement best describes CAPM theory in financial markets?

CAPM establishes a linear relationship between risk and expected return. Signup and view all the answers

What is the primary use of the critical value $p^*$ in portfolio theory?

To determine the optimal weight of an asset in a portfolio. Signup and view all the answers

What formula represents the portfolio return when combining two assets?

r = pr1 + (1 - p)r2 Signup and view all the answers

Which of the following statements is true regarding the feasible set of portfolios?

The feasible set is a solid (simply connected) region if there are at least three assets. Signup and view all the answers

How is the variance of the portfolio return represented mathematically?

σ² = p²σ12 + (1 - p)²σ22 + 2p(1 - p)ρσ1σ2 Signup and view all the answers

How does covariance relate to portfolio risk management?

It is used to determine how asset returns move together. Signup and view all the answers

What does ρ represent in the context of the portfolio?

The correlation between the returns of the two assets Signup and view all the answers

What is indicated by a positive correlation coefficient between two assets?

The assets are likely to move in the same direction. Signup and view all the answers

In the context of CAPM theory, what does the beta ($eta$) of a portfolio measure?

The sensitivity of the portfolio's returns to market returns. Signup and view all the answers

What does the equation σ = pσ1 + (1 - p)σ2 signify when ρ = +1?

The assets are perfectly correlated and follow a straight line relationship in the (r̄, σ) plane Signup and view all the answers

Which component does NOT affect the portfolio's variance in the formula provided?

The returns of the first asset (r1) Signup and view all the answers

Which of the following best describes a convex feasible set?

A set where any line segment between two points in the set remains within the set. Signup and view all the answers

When calculating the expected return of the portfolio, which aspect is crucial to consider?

The individual expected returns and their allocation weights Signup and view all the answers

What happens to the risk of a portfolio as more assets are added, given their return correlations?

Risk decreases if new assets are less correlated with existing ones. Signup and view all the answers

What relationship is illustrated by the mean-standard deviation diagram referenced?

The trade-off between risk and return of a portfolio Signup and view all the answers

What do the parameters $ ho$, $ ho > 0$, and $ ho < 0$ indicate in relation to asset returns?

Positive and negative correlation between asset returns. Signup and view all the answers

In the context of portfolio theory, why is the covariance important?

It measures the degree to which two assets move together, impacting combined risk Signup and view all the answers

Study Notes

Linearity of Conditional Expectation

The conditional expectation of a linear combination of random variables is equal to the linear combination of the conditional expectations of the individual random variables.
This property holds for any constants c1 and c2.

Jensen's Inequality for Conditional Expectation

Jensen's inequality states that the conditional expectation of a convex function of a random variable is greater than or equal to the convex function of the conditional expectation of the random variable.
It means that for a convex function φ(x) and a random variable f, the following inequality holds: E[φ(f(ω))|Σ] ≥ φ(E[f(ω)|Σ]).
This inequality also holds for the unconditional expectations: E[φ(f)] ≥ E[φ(g)], where g = E[f |Σ].

Asset Return

In financial markets, the amount received from an investment is often uncertain.
The total return is calculated as the amount received divided by the initial outlay.
An example of an asset with certain initial outlay and uncertain return is a zero-coupon bond held to maturity.

Portfolio Diagram

Portfolio diagrams represent assets on a mean-standard deviation diagram.
They show the relationship between the expected return and the standard deviation of the asset's return.
The exact location of a portfolio on the diagram cannot be determined solely from the locations of the original assets due to the covariance.
By combining two assets, the weights can be represented as w1=p and w2=1-p, where p is the percentage invested in the first asset.
To determine the average return of the portfolio, the weighted average of the individual returns can be applied: r̄ = pr̄1 + (1 − p)r̄2.

Portfolio Variance

The variance of the portfolio return can be expressed using the variances of the individual assets and the correlation between their returns.
The formula for variance is: σ 2 = E[(r − r̄)2] = p2σ12 + (1 − p)2σ22 + 2p(1 − p)ρσ1σ2.

Correlation

The correlation between the return of two assets, denoted by ρ, is a measure of their linear dependence.
The correlation can be calculated using the covariances and standard deviations of returns: ρ = Cov(r1, r2) / (σ1σ2).

Accessible Region in (r̄, σ) Plane

When the correlation between the two assets is +1 or -1, the portfolio return lies on a straight line connecting the points representing the individual assets on the mean-standard deviation diagram.
This line defines the boundary of the accessible region in the (r̄, σ) plane.
In the case of perfect correlation (+1), the equation for the variance is given by σ = pσ1 + (1 − p)σ2, which represents a straight line.

Minimizing Portfolio Variance

The variance of the portfolio return can be minimized by choosing the right weight p.
There exists a critical value of p, p∗, that minimizes the variance.
This critical value can be found using the formula: p∗ = (σ2(σ2 − ρσ1))/(σ1 − 2ρσ1σ2 + σ2).

Feasible Set

For n assets, the feasible set in the mean-standard deviation diagram consists of all possible portfolio combinations achievable with different combinations of weights for the assets.
This set is solid and simply connected for n ≥ 3.
For n ≥ 3, the feasible set is a solid region, meaning every point inside is also a feasible portfolio.
The feasible set is convex to the left, indicating that it is always possible to find a portfolio with a lower standard deviation (risk) for the same level of expected return.

Standard Deviation of Portfolio Return

An example is given with two assets, a standard deviation of 0.2 for one asset and 0.3 for the other, with a correlation of 0.15.
The standard deviation of the portfolio return is illustrated with a graph.
The graph shows how the standard deviation varies with the weighting of the two assets.
This example demonstrates how the correlation between asset returns impacts the risk and return characteristics of a portfolio.

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Description

This quiz covers concepts related to the linearity of conditional expectation, Jensen's inequality, and the calculation of asset returns in financial markets. Understand the properties of conditional expectations and their implications in finance.