Risk and Return in Finance

FIN 3060 Corporation Finance - Chapter 11: Risk and Return

The Journey So Far

  • Overview of prior chapters leading up to Chapter 11:

    • Ch 1: Introduction to Financial Management

    • Ch 2: Financial Statements, Taxes, and Cash Flow

    • Ch 3: Working with Financial Statements

    • Ch 4: Introduction to Valuation: The Time Value of Money

    • Ch 5: Discounted Cash Flow Valuation

    • Ch 6: Interest Rates and Bond Valuation

    • Ch 7: Equity Markets and Stock Valuation

    • Ch 8: Net Present Value and Other Investment Criteria

    • Ch 9: Making Capital Investment Decisions

    • Ch 10: Some Lessons from Capital Market History

    • Ch 11: Risk and Return

    • Ch 12: Cost of Capital

Key Concepts

After studying this chapter, one should be able to:

  • Calculate expected returns.

  • Explain the impact of diversification on risk and return.

  • Define the systematic risk principle.

  • Discuss the security market line (SML) and the risk-return trade-off.

Expected Returns

  • Definition: Expected returns are based on the probabilities of various possible outcomes.

  • Formula to calculate expected returns: E(R) = ext{Probability of state } i imes ext{Return in state } i E(R) = ext{sum of all } piRi Where:

    • $p_i$ = Probability of state $i$ occurring.

    • $E(R_i)$ = Expected return on an asset in state $i$.

Example: Expected Returns

  • Stock A:

    • State (i): Recession, Neutral, Boom

    • Probability (P_i): 0.25, 0.50, 0.25

    • Expected Return (E(R_A)): -20%, 15%, 35%

  • Stock B:

    • Expected Return (E(R_B)): 30%, 15%, -10%

  • Overall calculation:
    E(R) = ext{sum of all } p_i E(R)

Variance and Standard Deviation

  • Variance and standard deviation measure the volatility of returns.

  • Variance: Calculated as the weighted average of squared deviations from the mean.
    ext{Variance} = rac{1}{n} imes ext{sum of } (E(R_i) - ext{mean})^2

  • Standard Deviation (σ): The square root of variance.
    ext{Standard Deviation} = ext{sqrt(Variance)}

Variance and Standard Deviation Calculation Example

  • For Stock B:

    • States (Recession, Neutral, Boom)

    • Weights (P_i): 0.25, 0.50, 0.25

    • E(R): 30%, 15%, -10%

    • Variance: 0.0207

    • Standard Deviation: 14.4%

Portfolios

  • Definition: A portfolio is a collection of assets.

  • The risk and return of individual assets affect the overall portfolio's risk and return.

  • Portfolio expected return formula:
    E(RP) = ext{sum of } wj E(Rj) Where $wj$ = Percentage of portfolio invested in each asset.

Example: Portfolio Weights

  • Assets with invested dollars and expected returns:

    • Asset A: $15,000 (30%, 12.5%); Contribution: 3.735%

    • Asset B: $8,600 (17%, 9.5%); Contribution: 1.627%

    • Asset C: $11,000 (22%, 10.0%); Contribution: 2.191%

    • Overall Expected Return = 10%.

Expected Portfolio Returns Calculation

  • For each state (Recession, Neutral, Boom):

    • Calculate expected portfolio return by applying weights to expected returns from individual stocks.

Portfolio Variance

  • Method to compute portfolio return for each state:
    R{P, i} = w1R{1, i} + w2R{2, i} + … + wmR_{m, i}

  • Compute expected portfolio return using previous formulas.

  • Calculate portfolio variance and standard deviation similar to individual asset calculations.

Portfolio Risk

  • Definition: Total risk is the sum of systematic risk and unsystematic risk.

  • Important considerations include:

    • Calculate deviations of expected returns from overall expected portfolio return.

    • Square these deviations, multiply each by probability of state, and sum to get portfolio variance:
      ext{Portfolio Variance} = ext{sum of } Pi imes (DEVi)^2

Announcements and Returns

  • Announcement news often contains both expected and unexpected components, impacting stock prices.

  • Total Return formula: R = E(R) + U Where:

    • $U$ = Unexpected return, which can be further detailed as the sum of systematic ($m$) and unsystematic ($$) components.

  • Formula becomes:
    R = E(R) + m + 

Systematic vs. Unsystematic Risk

  • Systematic Risk: Factors affecting many assets, known as non-diversifiable risk or market risk.

    • Examples include changes in GDP, inflation, interest rates, etc.

  • Unsystematic Risk: Diversifiable risk, affecting limited assets, which can be mitigated through portfolio diversification.

    • Known as unique risk (e.g., strikes, shortages).

The Principle of Diversification

  • Diversification reduces risk without significantly lowering expected returns by offsetting poor-performing assets with better-performing ones.

  • Minimum risk that cannot be diversified away is the systematic component.

Portfolio Diversification Insights

  • Adds stocks to portfolios, diminishing marginal returns on diversification after around 10 initial stocks are added. After 30 stocks, minimal additional benefits realized.

Total Risk

  • Total risk is expressed as:
    ext{Total Risk} = ext{Systematic Risk} + ext{Unsystematic Risk}

  • Standard deviation is a common measure. For diversified portfolios, primarily systematic risk remains.

Market Risk for Individual Securities

  • Measured by the stock’s beta (β), reflecting volatility in relation to market movements.

Interpretation of Beta

  • β = 1.0: Stock has average risk.

  • β > 1.0: Stock is riskier than average.

  • β < 1.0: Stock is less risky than average.

  • Common beta ranges for stocks: 0.5 to 1.5.

  • Market’s beta = 1.0; treasury bills = 0.

Beta Coefficients for Selected Companies

Company

Beta Coefficient (β_i)

Coca-Cola

0.59

Pfizer

0.64

McDonald's

0.71

Cisco

0.85

Home Depot

1.00

Meta

1.21

Apple

1.25

Amazon

1.41

Tesla

2.32

Quick Quiz: Total vs. Systematic Risk

  • Total Risk (Standard Deviation):

    • Security C: 20%, β = 1.25

    • Security K: 30%, β = 0.95

  • Questions:

    • Which has more total risk?

    • Which has more systematic risk?

    • Which should have a higher expected return?

Portfolio Beta

  • The formula for portfolio beta (βP): etaP = ext{sum of } wj etaj
    Where $w_j$ = % of portfolio invested in Asset j.

Beta and the Risk Premium

  • Risk Premium Formula: RP = E(R) - R_f

    • Reward-to-Risk Ratio:
      rac{E(Rj) - Rf}{eta_j}

    • Higher beta results in a larger expected risk premium.

Market Equilibrium and Security Market Line (SML)

  • In equilibrium, all investments must yield the same reward-to-risk ratio.

  • Expressed mathematically:
    rac{E(RA) - E(Rf)}{etaA} = rac{E(RM) - Rf}{etaM}

Security Market Line (SML) Equation

  • Describes the relationship between expected return and systematic risk:
    E(R) = Rf + [E(RM) - R_f] imes eta

Required Return and Factors Affecting It

  • The expected return based on CAPM: E(Rj) = Rf + (E(RM) - Rf) imes eta_j Where:

    • $R_f$: Risk-free rate (e.g., T-Bill rate).

    • $E(R_M)$: Expected market return (e.g., S&P 500).

Quick Quiz: Expected Return Calculation

Example scenario:

  • Beta: 1.2

  • Risk-free rate: 5%

  • Market return: 13%

  • Expected return calculation:
    E(R) = 5 ext{%} + (13 ext{%} - 5 ext{%}) imes 1.2 = 14.6 ext{%}

Next Time

  • Review of previous chapters leading through the understanding of financial management basics and continued exploration into risk and return dynamics, concluding with Cost of Capital in Chapter 12.