Finance Chapters 10 and 11 Quiz

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Chapters 10 and 11

• Topic: Capital Markets and the Pricing of Risk
• Topic: Optimal Portfolio Choice and the Capital Asset Pricing Model

Learning Objectives

• Compute the realized return and variance for an investment.
• Compute the expected return, variance, and standard deviation (or volatility) of returns.
• Compute the expected return on a portfolio given a portfolio of stocks and the variance of the portfolio.
• Use the Capital Asset Pricing Model (CAPM) to calculate the expected return for a risky security.

Historical Returns of Stocks and Bonds

• Realized Return: The gain or loss from an investment over a specific time period.

• Components of Return:

  • Income component (cash received directly)
  • Capital gain/loss (change in asset value)

• Formula for Realized Return (Rt+1): Rt+1= (Divt+1+Pt+1−Pt)/ Pt

• Where:

  • Pt = share price at beginning of period
  • Pt+1 = share price at end of period
  • Divt+1 = dividend received at end of period

Example 1

• A stock purchased for $10 is sold for $11 with a dividend of $0.50.
• Realized return = 5% + 10% = 15%

Alternative Example 10.2

• This example shows how to calculate annualized returns for a stock (NRG) in 2012 and 2014, using price and dividend data.
• Formulae for calculating the return is provided in this example.

Table 10.2

• Table of S&P 500, Microsoft, and Treasury Bill realized returns annually.
• Data spans 2002-2014.
• Includes percentage returns for each year.

Average Annual Return

• The average annual return is calculated by summing the realized returns for each year (t = 1 to T) and dividing by the number of years (T).
• Example values for different investments are provided in a table (Small stocks, S&P 500, Corporate bonds, Treasury bills)

Variance and Volatility of Returns

• Variance: Measures the average squared difference between actual returns and the average return.
• Larger variance indicates greater returns that differ from average returns.
• Standard Deviation (or Volatility): The square root of the variance, describing the range of returns.
• Formula for Variance (Var(R)): Var(R) = (1/T) Σ(Rt-R)^2; where R is average return and Rt is actual return in year t
• Formula for Standard Deviation (SD(R)) or volatility (σ): SD(R) = √Var(R)

Variance and Volatility Example

• An example table demonstrates how to calculate variance and standard deviation.
• Data on realised returns and average return in each year is included.
• Results include the standard deviation.

Table 10.4

• Return Volatility (Standard Deviation) is displayed for various investments (small stocks, S&P 500, Corporate Bonds, Treasury Bills).
• Data is for 1926-2014.
• Standard deviation is compared to quantify risk.

The Historical Trade-Off Between Risk and Return

• Excess Return/Risk Premium: The difference between the average return of an investment and the average return on T-Bills (a risk-free investment).
• Risk-Free Return (T-Bills): Government bills that are virtually free from default risk.
• Data from the table on various investments with their return volatilities (Standard Deviations) are provided.

Common Measures of Risk and Return – Expected Return

• Expected Return (E[R]): The weighted average of possible returns, where weights correspond to their probabilities, representing the anticipated return of a risky asset in the future.
• Formula for Expected Return (E[R]): E[R] = Σ (PR × R); where PR is the probability of return R

Expected Return Example

• An example table presents probabilities of different economic states (Recession, Boom) and their associated returns for JO's Secretarial service and Tracey's Educational Supplies for each state.

Variance and Standard Deviation

• Formula for Variance: σ2 = Var(R) = E[(R − E[R])^2] = Σ PR × (R − E[R])^2
• Formula for Standard Deviation: σ = √Var(R)

Textbook Example 10.1

• Example calculation of expected return and volatility of AMC stock, given equal likelihood of 45% return or -25% return.

The Expected Return of a Portfolio

• Portfolio Weights: Fraction of total investment in each asset in a portfolio. Portfolio weights add to 1.
• Formula for Portfolio Weights: xi = Value of investment i / Total value of portfolio
• Expected Portfolio Return (E[Rp]): Weighted average of the expected return of each asset in the portfolio.

Textbook Example 11.2

• Portfolio expected return calculation example using $10,000 in Ford stock and $30,000 in Tyco International stock.

Determining Covariance and Correlation

• Covariance: Expected product of deviations of two returns from their means. Measures the degree to which returns move together.

• Formula for Covariance: Cov(Ri, Rj) = E[(Ri − E[Ri])(Rj − E[Rj])]

• Correlation: Measure of common risk shared by stocks without depending on volatility.

• Formula for Correlation: Corr(Rᵢ, Rⱼ) = Cov(Rᵢ, Rⱼ) / (SD(Rᵢ) * SD(Rⱼ))

• Positive covariance/correlation: Two returns tend to move together.

• Negative covariance/correlation: Two returns tend to move in opposite directions

Table 11.2

• Calculations for covariance and correlation of returns between pairs of stocks (North Air, West Air and Tex Oil).

Computing a Portfolio's Variance and Volatility

• Portfolio Variance (Var(Rp)): The variance of a two-stock portfolio is calculated by considering individual variances, covariances and weights of the stocks in the portfolio.
• Formula for Portfolio Variance: σ²p=x²Var(R₁) + x₂Var(R₂) + 2x₁x₂Cov(R₁,R₂)
• Explains how correlation affects portfolio risk. Lower correlation leads to a lower portfolio standard deviation.

Textbook Example 11.6

• Calculation of portfolio volatility for various combinations of stocks (Microsoft, Hewlett-Packard, and Alaska Air). Includes correlation coefficients of stock pairs, demonstrating calculations based on correlation data.

Diversification in Stock Portfolios

• Total Risk: Composed of firm-specific risk and systematic risk. Total risk is measured by variance or standard deviation
• Firm-Specific Risk (Non-Systematic): Fluctuations due to company- or industry-specific news (e.g., introduction of competition, strikes)
• Systematic Risk: Fluctuations related to market-wide news, like economic trends, affecting all firms (e.g., interest rates, inflation)
• Diversification effect: Firm-specific risk is reduced in larger, diversified portfolios. Systematic risk, however, remains undiversifiable.

Figure 11.2

• Shows how the volatility of an equally weighted portfolio decreases as the number of stocks in the portfolio increases. Illustrates the effect of diversification.

Measuring Systematic Risk

• Efficient Portfolio: A portfolio containing only systematic risk. Reducing volatility requires lowering potential/expected return.
• Market Portfolio: Portfolio holding all shares and securities in the market. The S&P 500 is a frequent proxy.
• Beta (β): Measures systematic risk of a security relative to the overall market. Beta, unlike volatility, measures only systematic risk.
• Formula for Beta: β₁ = Cov(Rᵢ, Rмkt)/ Var(Rмkt)

Table 10.6

• Table showing beta values for individual stocks relative to the S&P 500 index, based on monthly stock data (2010-2015).

Capital Asset Pricing Model (CAPM)

• CAPM: Equilibrium model describing the relationship between the risk and return. Expected return is based on Beta.
• Formula for Expected Return (using CAPM): E[R] = rf + β × (E[RMkt] − rf). Where: rf = risk-free return ; β=beta, E[RMkt] is expected return on the market portfolio.

Example - CAPM

• Example application of the Capital Asset Pricing Model (CAPM) to find expected return given risk-free rate, market risk premium, and beta value.

Test your knowledge on Capital Markets and the Pricing of Risk, along with Optimal Portfolio Choice and the Capital Asset Pricing Model. This quiz covers key concepts such as realized return, variance, and expected returns for investments and portfolios. Ideal for students studying finance and investment strategies!