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Optimal Risky Portfolios

Professor Beason. This set of notes contains information about: Optima...
Course

Investments: Debt, Equity And Derivatives (FIN3144)

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Academic year: 2022/2023
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Virginia Polytechnic Institute and State University

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Optimal Risky Portfolios

Introduction:

● Optimal risky portfolios refer to a portfolio of assets that provides the highest expected return for a given level of risk. ● The goal of constructing an optimal risky portfolio is to maximize the expected return while minimizing the risk of the portfolio. ● This can be achieved through a process known as mean-variance optimization, which involves selecting the optimal combination of assets based on their expected returns and risk.

Mean-Variance Optimization:

● Mean-variance optimization is a mathematical technique that helps investors identify the optimal combination of assets for their portfolio based on their expected returns and risk. ● The expected return of an asset is the expected outcome of an investment in that asset. It is calculated as the sum of the probability of each possible outcome multiplied by its corresponding return. ● The risk of an asset is usually measured by its variance or standard deviation. The variance is a measure of how far an asset's returns are spread out from its expected return, while the standard deviation is a measure of how far the returns are spread out from the mean in terms of standard deviations. ● The mean-variance optimization process involves finding the optimal combination of assets that provides the highest expected return for a given level of risk, or the lowest level of risk for a given expected return.

Efficient Frontier:

● The efficient frontier is a graphical representation of the mean-variance optimization process. It is a curve that shows the highest expected return that can be achieved for a given level of risk, or the lowest level of risk that can be achieved for a given expected return.

● The efficient frontier is determined by the set of portfolios that lie on the boundary between the set of feasible portfolios (i. portfolios that can be constructed using the available assets) and the set of infeasible portfolios (i. portfolios that cannot be constructed using the available assets). ● The optimal risky portfolio is the portfolio that lies on the efficient frontier and provides the highest expected return for a given level of risk, or the lowest level of risk for a given expected return.

Practice Problems:

  1. An investor is considering two risky assets, A and B, with the following characteristics: ● Asset A: Expected return = 10%, Standard deviation = 15% ● Asset B: Expected return = 5%, Standard deviation = 10%

The investor has a risk tolerance of 10%. Which of the following portfolios is the optimal risky portfolio?

a) 100% Asset A b) 50% Asset A, 50% Asset B

c) 100% Asset B

Answer: b) 50% Asset A, 50% Asset B

Explanation: The optimal risky portfolio is the portfolio that provides the highest expected return for a given level of risk, or the lowest level of risk for a given expected return. In this case, the investor has a risk tolerance of 10%, so the optimal risky portfolio should have a risk level that is no higher than 10%. Portfolio a) has a risk level of 15%, which is higher than the investor's risk tolerance, while portfolio c) has a risk level of 10%, which is equal to the investor's risk tolerance. Portfolio b) has a risk level of 12%, which is lower than the investor's risk tolerance and therefore the optimal risky portfolio.

  1. An investor is considering three risky assets, C, D, and E, with the following characteristics:

● Diversification is a risk management technique that involves investing in a variety of assets in order to reduce the overall risk of the portfolio. ● By diversifying a portfolio, an investor can minimize the impact of any particular asset's poor performance on the overall portfolio. ● Diversification can be achieved by investing in a variety of asset classes (e. stocks, bonds, real estate, etc.), sectors, industries, and geographical regions.

Practice Problems:

  1. An investor is considering three risky assets, F, G, and H, with the following characteristics: ● Asset F: Expected return = 12%, Standard deviation = 20% ● Asset G: Expected return = 8%, Standard deviation = 15% ● Asset H: Expected return = 4%, Standard deviation = 10%

The investor has a risk tolerance of 15% and wants to construct a diversified portfolio. Which of the following portfolios is the most diversified?

a) 100% Asset F b) 50% Asset F, 50% Asset G c) 50% Asset F, 25% Asset G, 25% Asset H

d) 50% Asset G, 50% Asset H

Answer: c) 50% Asset F, 25% Asset G, 25% Asset H

Explanation: A diversified portfolio is one that is spread out across a variety of assets, sectors, industries, and geographical regions in order to reduce overall risk. Portfolio a) is not diversified because it is composed entirely of a single asset. Portfolios b) and d) are somewhat diversified, but portfolio c) is the most diversified because it is composed of three different assets in different proportions.

  1. An investor is considering two risky assets, I and J, with the following characteristics: ● Asset I: Expected return = 15%, Standard deviation = 20% ● Asset J: Expected return = 10%, Standard deviation = 15%

The investor has a risk tolerance of 15% and wants to construct a divers ified portfolio. Which of the following portfolios is the most diversified?

a) 100% Asset I b) 50% Asset I, 50% Asset J

c) 100% Asset J

Answer: b) 50% Asset I, 50% Asset J

Explanation: A diversified portfolio is one that is spread out across a variety of assets, sectors, industries, and geographical regions in order to reduce overall risk. Portfolios a) and c) are not diversified because they are composed entirely of a single asset. Portfolio b) is the most diversified because it is composed of two different assets in equal proportions.

Risk Aversion:

● Risk aversion is the preference for a lower level of risk over a higher level of risk, given a choice between two investments with the same expected return. ● Risk aversion is an important consideration for investors when constructing their portfolios, as it affects their willingness to take on risk in exchange for potential returns. ● Risk aversion can be measured using the concept of risk premium, which is the difference between the expected return of a risky asset and the risk-free rate. The risk-free rate is the return that can be expected from an investment with zero risk, such as a U. Treasury bond.

Practice Problems:

  1. An investor is considering two risky assets, K and L, with the following characteristics: ● Asset K: Expected return = 12%, Standard deviation = 20% ● Asset L: Expected return = 8%, Standard deviation = 15%

The risk-free rate is currently 2%. Which of the following statements is true?

Required rate of return = Risk-free rate + Beta (Expected return of the market - Risk-free rate)

Where:

● Risk-free rate is the return that can be expected from an investment with zero risk, such as a U. Treasury bond. ● Beta is a measure of the volatility of an asset compared to the overall market. A beta of 1 indicates that the asset's returns are closely correlated with the market, while a beta greater than 1 indicates that the asset is more volatile than the market, and a beta less than 1 indicates that the asset is less volatile than the market. ● Expected return of the market is the expected return of a broad-based market index, such as the S&P 500.

Practice Problems:

  1. An investor is considering two assets, P and Q, with the following characteristics: ● Asset P: Beta = 1, Expected return of the market = 8% ● Asset Q: Beta = 0, Expected return of the market = 8%

The risk-free rate is currently 2%. Which of the following assets has the higher required rate of return?

a) Asset P

b) Asset Q

Answer: a) Asset P

Explanation: The required rate of return for an asset is determined using the CAPM formula:

Required rate of return = Risk-free rate + Beta (Expected return of the market - Risk-free rate)

In this case, the required rate of return for Asset P is 2% + 1 (8% - 2%) = 11%, and the required rate of return for Asset Q is 2% + 0 (8% - 2%) = 6%. Therefore, Asset P has the higher required rate of return.

  1. An investor is considering three assets, R, S, and T, with the following characteristics: ● Asset R: Beta = 1, Expected return of the market = 9% ● Asset S: Beta = 1, Expected return of the market = 9% ● Asset T: Beta = 0, Expected return of the market = 9%

The risk-free rate is currently 3%. Which of the following assets has the lower required rate of return?

a) Asset R b) Asset S

c) Asset T

Answer: c) Asset T

Explanation: The required rate of return for an asset is determined using the CAPM formula:

Required rate of return = Risk-free rate + Beta (Expected return of the market - Risk-free rate)

In this case, the required rate of return for Asset R is 3% + 1 (9% - 3%) = 10%, the required rate of return for Asset S is 3% + 1 (9% - 3%) = 9%, and the required rate of return for Asset T is 3% + 0 (9% - 3%) = 8%. Therefore, Asset T has the lower required rate of return.

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Optimal Risky Portfolios

Course: Investments: Debt, Equity And Derivatives (FIN3144)

11 Documents
Students shared 11 documents in this course

University: Virginia Polytechnic Institute and State University

Was this document helpful?
Optimal Risky Portfolios
Introduction:
●Optimal risky portfolios refer to a portfolio of assets that provides the highest
expected return for a given level of risk.
●The goal of constructing an optimal risky portfolio is to maximize the expected
return while minimizing the risk of the portfolio.
●This can be achieved through a process known as mean-variance optimization,
which involves selecting the optimal combination of assets based on their
expected returns and risk.
Mean-Variance Optimization:
●Mean-variance optimization is a mathematical technique that helps investors
identify the optimal combination of assets for their portfolio based on their
expected returns and risk.
●The expected return of an asset is the expected outcome of an investment in that
asset. It is calculated as the sum of the probability of each possible outcome
multiplied by its corresponding return.
●The risk of an asset is usually measured by its variance or standard deviation.
The variance is a measure of how far an asset's returns are spread out from its
expected return, while the standard deviation is a measure of how far the returns
are spread out from the mean in terms of standard deviations.
●The mean-variance optimization process involves finding the optimal
combination of assets that provides the highest expected return for a given level
of risk, or the lowest level of risk for a given expected return.
Efficient Frontier:
●The efficient frontier is a graphical representation of the mean-variance
optimization process. It is a curve that shows the highest expected return that
can be achieved for a given level of risk, or the lowest level of risk that can be
achieved for a given expected return.

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