Topic 10: Portfolio Theory & CAPM

Question 1

How do portfolio weights indicate the allocation of investments in a portfolio, and what is the formula for calculating the weight of an individual investment?

Answer 1

Portfolio weights show the fraction or percentage of the total portfolio value allocated to each security.

The weight of the i-th investment is calculated as:

x = Value of Investment i/Total Value of Portfolio

All weights add up to 1

Question 2

How are the expected return and the realized return of a portfolio with multiple securities calculated?

Answer 2

The expected return for the portfolio as:

E[Rp] = x1E[R1] + x2E[R2] +… xnE[Rn]

• where xi are the weights and E[R] are the expected returns for the individual securities

The realized return of a portfolio is calculated as:

Rp= x1R1 + x2R2 +… xnRn

where xi are the weights and R are the realized returns of the individual securities

Question 3

How is the variance of a portfolio with two stocks calculated?

Answer 3

The variance of a portfolio with two stocks (𝑅𝑝=𝑥1𝑅1+𝑥2𝑅2) is calculated using:

Var(𝑅𝑝)=𝑥1^2SD(𝑅1)^2+𝑥2^2 SD(𝑅2)^2+2𝑥1𝑥2Corr(𝑅1,𝑅2)SD(𝑅1)SD(𝑅2)

Question 4

How is the variance of an equally weighted portfolio calculated, and what does it imply about diversification?

Answer 4

In an equally weighted portfolio each security has an equal investment, so with so with n securities x1=x2 = … = xn = 1/n

The variance is calculated as:

Var(Rp)= (1/n) x Avg variance of the individual securities + (1 - 1/n) x Average covariance between the securities

This formula shows that as the number of securities increases, the impact of individual security variance decreases, emphasizing the benefits of diversification.

Question 5

How does decreasing correlation between securities in a portfolio affect diversification and volatility?

Answer 5

As correlation decreases, expected returns remain unchanged, but the diversification effect strengthens. With the same expected return and volatility, more negatively correlated securities are better. A correlation of -1 expands the Efficient Frontier, reducing volatility and increasing diversification.

Question 6

How does the Tangent Portfolio differ from the Minimum Variance Portfolio, and why is the Tangent Portfolio optimal in terms of the Sharpe ratio?

Answer 6

Minimum Variance Portfolio: The point on the Efficient Frontier with the lowest risk.

Tangent Portfolio: The portfolio with the highest Sharpe ratio, offering the best return per unit of risk.

The minimum variance point offers the lowest risk but not the best risk-return tradeoff. The Tangent Portfolio provides the optimal balance of risk and return by maximizing the return per unit of risk, thus offering a better overall tradeoff.

Question 7

What defines an inefficient portfolio, and when is it considered dominated by another portfolio?

Answer 7

An inefficient portfolio is dominated by another portfolio when it is possible to find another portfolio that is better in terms of expected return or volatility and no worse in the other parameter. In other words, an inefficient portfolio does not provide the best return for its level of risk.

Question 8

What happens to the Efficient Frontier when adding a third stock to a portfolio, and how does this affect diversification and expected return?

Answer 8

Adding a third stock to the portfolio creates an entire region of risk and return possibilities rather than a single curve. The efficient portfolios, offering the highest possible expected return for a given level of volatility, form the Efficient Frontier.

When adding stock C (especially with negative correlation), the frontier expands, providing higher expected returns (E[R]) for the same level of risk, enhancing diversification.

Question 9

How does adding more assets to a portfolio improve the set of available risk and return combinations?

Answer 9

Adding more assets improves the risk-return combinations due to increased diversification possibilities.

Despite a restriction on too much diversification, the mix of assets with more stocks (e.g., 10 stocks) is better than fewer stocks (e.g., 3 stocks) because it enhances diversification and reduces volatility, providing better risk-return possibilities.

It is advised not to hand-pick stocks, even if some are less preferred, as increased diversification is beneficial for the portfolio.

Question 10

How do risk-free saving and borrowing strategies affect portfolio diversification and risk management?

Answer 10

Diversification: Including all risky investments in the Efficient Frontier maximizes diversification.

Risk-Free Investment: Allocating a percentage of the portfolio to risk-free assets like treasury bills reduces risk.

Aggressive Investing: Aggressive investors might borrow at a risk-free rate to invest in risky assets, leveraging up expected returns and increasing risk and return, as seen by increased volatility on the Efficient Frontier.

Risk Preferences: Adjust strategies according to individual risk preferences.

Question 11

How do the expected return and volatility of a portfolio change when combining a risk-free asset with a risky portfolio?

Answer 11

Expected Return: The expected return is equal to the risk-free rate plus a fraction of the portfolio’s risk premium:

ExpectedReturn
=𝑟𝑓+𝑥(𝐸[𝑅𝑝]−𝑟𝑓)

Volatility: The volatility is a fraction of the portfolio’s volatility, based on the amount invested in the risky portfolio:

Volatility
=𝑥SD(𝑅𝑝)

Risk-Free Asset: The risk-free asset has a variance and covariance of 0, so the portfolio’s risk level depends only on the risky assets.

Question 12

What is the Sharpe Ratio, and how is it used to determine the best portfolio of risky assets?

Answer 12

The Sharpe Ratio of portfolio
𝑃 is calculated as:

SharpeRatio
=𝐸[𝑅𝑝]−𝑟𝑓/SD(𝑅𝑝)

Interpretation: Combining a portfolio of risky assets with a risk-free asset forms a straight line on a plot of expected return vs. volatility.

Optimal Portfolio: The best portfolio of risky assets will produce a straight line with the steepest slope, representing the highest Sharpe Ratio.

Tangent Portfolio: The line tangent to the Efficient Frontier of risky assets has the highest Sharpe Ratio, with the tangency point indicating the expected return and volatility of the Tangent Portfolio.

Question 13

What is the Tangent Portfolio and why is it significant?

Answer 13

The Tangent Portfolio has the highest Sharpe ratio, offering the best risk-return tradeoff.

It lies on the Capital Market Line (CML), connecting risk-free investments and the Tangent Portfolio, providing optimal risk-return.

Question 14

Why is the Tangent Portfolio considered the optimal portfolio?

Answer 14

The Tangent Portfolio has the highest Sharpe ratio, providing the biggest reward per unit of volatility. All other risky portfolios lie below the tangent line.

Question 15

Why is finding the efficient (tangent) portfolio not feasible without assumptions, and what do the assumptions imply?

Answer 15

Finding the efficient portfolio requires knowing all expected returns, volatilities, and correlations of every investment, which isn’t feasible. CAPM assumptions imply the tangent portfolio is the market portfolio.

Question 16

Why is the market portfolio considered efficient under CAPM assumptions?

Answer 16

When CAPM assumptions hold, rational investors buy an efficient mix of assets, making the market portfolio efficient. The Capital Market Line (CML) is tangent to the Efficient Frontier at the market portfolio.

Question 17

What is the formula for expected return according to CAPM, and what does Beta represent?

Answer 17

E[R] = rf + B * risk premium(Emkt - rf)

Beta measures the stocks volatility due to market risk and its sensitivity to market movements

Law of One Price states that investments with similar risk should have the same expected return

Question 18

Is there a clear relationship between an individual security’s volatility and its expected return?

Answer 18

No

Question 19

What does the Security Market Line (SML) represent, and what does it show according to CAPM?

Answer 19

The SML shows where individual securities should lie when expected returns are plotted against betas.

It represents the expected return for each security as a function of its beta with the market. According to CAPM, the market portfolio is efficient, so all stocks and portfolios should lie on the SML.

Question 20

What are the key conclusions of the CAPM based on its assumptions?

Answer 20

The market portfolio is the efficient portfolio on the Efficient Frontier and offers the highest Sharpe ratio. It lies on the Capital Market Line (CML), providing the best risk-return combination by combining with the risk-free asset.

The risk premium for any investment is proportional to its beta with the market, reflecting the excess return for taking on additional systematic risk.

Question 21

How do volatility and correlation affect an equally weighted portfolio with
𝑛 stocks, and what happens as
𝑛 increases?

Answer 21

Volatility (SD): Decreases as
𝑛 increases, due to diversification.

Diversification: Reduces unsystematic risk, but only diversifies independent, not common risk.

Effect: As 𝑛 gets very large, diversification benefits taper off.

Question 22

How does the volatility of an equally weighted portfolio change with the number of stocks, and what risk remains?

Answer 22

Declining Volatility: Volatility decreases as the number of stocks in the portfolio increases.

Residual Market Risk: Even in a very large portfolio, market risk remains, despite the elimination of diversifiable/independent risk.

Question 23

Where is the worst risk-return tradeoff on the efficient frontier, and why?

Answer 23

Location: Below the minimum variance point.

Reason: Portfolios here have higher risk (variance) but offer lower returns than the minimum variance portfolio.

Inefficiency: These are inefficient portfolios because better returns can be achieved with the same or lower risk by moving up the efficient frontier.

Question 24

What does the Capital Asset Pricing Model (CAPM) determine, and what factors does it consider?

Answer 24

CAPM Purpose: Determines the expected return of an individual asset based on its systematic risk (market risk) compared to the overall market.

Expected Return (E[R]): The return you expect to earn from the investment.

Risk-Free Rate (r_f): The return from a risk-free investment, like government bonds.

Market Return (E[R_m]): The average return of the market (e.g., S&P 500).

Beta (β): Measures how much the investment’s returns move compared to the market.

β = 1: The investment is as risky as the market.

β > 1: The investment is riskier than the market.

β < 1: The investment is less risky than the market.

Question 25

What are the key concepts of Portfolio Theory?

Answer 25

Optimal Portfolio: Construct an optimal portfolio of risky assets to maximize expected return for a given level of risk (or minimize risk for a given level of return).

Diversification: Combine different assets in a portfolio to reduce risk.

Efficient Frontier: A curve representing portfolios offering the highest return for each level of risk.

Risk and Return: Balance risk (volatility) and return by choosing the right mix of assets.

Correlation: Reduce risk through diversification by choosing assets that are more negatively correlated.

Question 26

How can combining stocks into a portfolio reduce risk?

Answer 26

Combining stocks into a portfolio reduces risk through diversification. By selecting assets with low or negative covariances, investors can create a portfolio that minimizes risk and volatility.

Question 27

How does covariance influence the overall volatility of a portfolio?

Answer 27

Covariance affects portfolio volatility by indicating how different assets’ returns move relative to each other. Positive covariance can increase volatility, while negative covariance can reduce it, providing a stabilizing effect.

Question 28

What is the formula for calculating covariance between two returns 𝑅𝑖 and 𝑅𝑗?

Answer 28

The covariance between returns
𝑅𝑖 and 𝑅𝑗 is calculated using:

Cov(𝑅𝑖,𝑅𝑗)=𝐸[(𝑅𝑖−𝐸[𝑅𝑖])(𝑅𝑗−𝐸[𝑅𝑗])]

Question 29

How can you estimate covariance from a sample of historical data?

Answer 29

Covariance for historical returns is estimated using:

Cov(𝑅𝑖,𝑅𝑗)=1/(𝑇−1)∑(𝑅𝑖−𝑅i)(𝑅𝑗−𝑅j)

where 𝑅𝑖 and 𝑅𝑗 are the mean returns, and 𝑇 is the number of observations, adjusted for loss of degrees of freedom.

Note: the second R is using the average return based on historical data instead of realized return

Question 30

What does covariance measure in the context of two securities?

Answer 30

Covariance measures how two securities move in relation to each other. Positive covariance means they move in the same direction, while negative covariance means they move in opposite directions.

Question 31

Why is correlation needed in addition to covariance?

Answer 31

Correlation is needed because it standardizes covariance, making it easier to interpret the strength and direction of the relationship between two securities.

Question 32

What does a bigger deviation from expected return indicate about variance and volatility?

Answer 32

A bigger deviation from expected return indicates greater variance and volatility, implying higher risk.

Question 33

How do the signs of covariance relate to the movements of two securities?

Answer 33

If 𝑅𝑖<𝐸[𝑅𝑖] and 𝑅𝑗<𝐸[𝑅𝑗], or if 𝑅𝑖>𝐸[𝑅𝑖] and 𝑅𝑗>𝐸[𝑅𝑗]

Positive covariance (Both returns deviate in the same direction either both above or both below expected returns)

If 𝑅𝑖<𝐸[𝑅𝑖] and 𝑅𝑗>𝐸[𝑅𝑗], or if
𝑅𝑖>𝐸[𝑅𝑖] and 𝑅𝑗<𝐸[𝑅𝑗]

Negative covariance (Returns deviate in opposite directions (one above and one below expected returns)

Question 34

What is correlation, how is it calculated and what do the values -1, 0, and +1 signify?

Answer 34

Measures the strength of the relationship between returns, while controlling for the stocks volatilities

Correlation is calculated using:

Corr(𝑅𝑖,𝑅𝑗)=Cov(𝑅𝑖,𝑅𝑗)/SD(𝑅𝑖)×SD(𝑅𝑗)

Interpretation:

+1: Securities move entirely together (e.g., if A increases 1%, B also increases 1%).

-1: Securities move exactly in opposite directions.

0: No correlation; the securities’ movements are independent.

Question 35

What happens when you vary the weights of a two-stock portfolio?

Answer 35

Varying the weights of a two-stock portfolio (with 𝑋𝐴+𝑋𝐵=1) allows us to find different combinations of expected return and volatility, helping to optimize the portfolio’s risk-return tradeoff.

Question 36

How does the shape of the Efficient Frontier inform investment decisions and portfolio optimization?

Answer 36

Efficient Frontier: Plots risk (standard deviation) against return (expected return), with efficient portfolios lying on the curve for the best return at each risk level.

Minimum Variance Point: The lowest risk point on the frontier, with weights optimized to minimize variance.

Other Points: Each return level has a unique set of weights that minimizes risk.

Frontier Shape: A curve with the leftmost point as the minimum variance portfolio; moving up the curve increases return and risk.

Question 37

How can investors utilize the Tangent Portfolio based on their risk preferences?

Answer 37

Low Risk: Mix risk-free assets with the Tangent Portfolio.

High Risk: Borrow at the risk-free rate to invest more in the Tangent Portfolio.

The CML reduces risk without sacrificing returns, maximizing the Sharpe ratio.

*move up or down the tangent line depending on your risk preferences

Question 38

How do investor preferences affect investment in the Tangent Portfolio?

Answer 38

Every investor should invest in the Tangent Portfolio, regardless of risk aversion. Preferences determine the allocation between risk-free investments and the Tangent Portfolio.

Question 39

What are the assumptions of the Capital Asset Pricing Model (CAPM)?

Answer 39

Competitive Markets: Investors can trade all securities at competitive market prices without taxes or transaction costs and can borrow/lend at the risk-free rate.

Efficient Portfolios: Investors hold only efficient portfolios and seek to maximize risk-adjusted returns.

Homogeneous Expectations: Investors have identical expectations about volatilities, correlations, and expected returns of securities.

Question 40

What is the result of the CAPM assumptions regarding the efficient portfolio?

Answer 40

With these assumptions, all investors will identify and demand the same efficient portfolio of risky assets, making the market portfolio the efficient portfolio.

Question 41

How should investors choose a portfolio according to the CAPM?

Answer 41

According to CAPM, all investors should choose a portfolio on the CML by holding a combination of the risk-free security and the market portfolio. This provides the optimal risk-return tradeoff.

Question 42

How does the Capital Market Line dominate as an investment strategy?

Answer 42

Rational investors will hold a combination of risk-free assets (e.g., treasury bills) and the market portfolio, maximizing the Sharpe ratio.

They can reduce risk by combining with the risk-free asset or increase risk by leveraging (borrowing at the risk-free rate) to invest more in the tangent portfolio.

Question 43

How does CAPM determine the risk premium for an investment?

Answer 43

CAPM rescale the market risk premium to determine the risk premium for any investment, ensuring investments with similar risk have the same expected return.

Question 44

Why is Beta the right risk measure in CAPM, and how should it be used?

Answer 44

Beta is the right risk measure because it accounts for market risk, excluding firm-specific risk.

An investment’s expected return should match the CML’s return at the same market risk level, calculated as 𝛽×marketriskpremium.

Question 45

What are some criticisms and implications of the CAPM?

Answer 45

Strong Assumptions: CAPM’s assumptions are considered too strong and unrealistic.

Inaccurate Implications: Not all investors hold the market portfolio or a well-diversified portfolio, which is an implication of CAPM.

Practical Use: Despite its limitations, CAPM is seen as intuitive and a useful approximation for understanding risk and return.

Question 46

What defines the efficient and inefficient frontiers?

Answer 46

Efficient Frontier: The upper part of the curve, starting from the minimum variance point and moving upwards.

Inefficient Frontier: The lower part of the curve, representing the worst risk-return tradeoff.

Question 47

How is CAPM used by investors?

Answer 47

Compare Investments: Evaluate different investments.

Pricing Securities: Determine if an investment is under- or over-valued based on its risk.

Risk-Return Tradeoff: Shows the return an investor should demand for the risk they’re taking.