Module 3 Risk Aversion, Premia, and Sharpe Ratios

Risk Aversion, Risk Premia, and Sharpe Ratios

Course Details

  • Course Code: F303

  • Course Title: Intermediate Investments

  • Instructor: Professor Mathias S. Kruttli

  • Semester: Spring 2026

Agenda

  1. Previous lecture review questions

  2. Review of covariance and correlation

  3. Risk aversion and risk premia

  4. The portfolio problem

  5. Sharpe Ratios

Holding Period Return (HPR) Calculations

  • Example: Investment in Bitcoin

    • Investment: $435 in January 2016

    • Sale Price: $20,750 in January 2023

    • Calculations:

    • Holding Period Return (HPR)

    • Simple Annualized HPR

    • Compounded Annualized HPR

  • Example: Amazon Stocks

    • Shares Purchased: 50 shares at $30/share

    • Year 1 Price: $25/share

    • Year 2 Price: $40/share

    • Calculations:

    • Compute HPR for each year

    • Arithmetic average of returns

    • Geometric average of returns

  • Example: Walmart Inc. Stock Returns

    • Return Data: -5%, 14%, 7%, -3%, and 6% over 5 years

    • Calculations:

    • Compute expected return

    • Compute variance

Covariance

  • Definition: Covariance measures how two random variables move together and is crucial for understanding asset portfolios.

  • Importance: It influences portfolio construction by showing how asset returns interact.

Co-Movement of Stocks

  • Examples:

    • Price comparison between Home Depot (HD) and Hewlett Packard (HPQ)

    • Price comparison between Home Depot (HD) and Lowes (LOW)

Covariance Formula

  • Covariance between two random variables X and Y:
    Cov(X,Y)=E[(X−E(X))(Y−E(Y))]=E(XY)−E(X)E(Y)Cov(X,Y) = E\bigg[\bigg(X - E(X)\bigg)\bigg(Y - E(Y)\bigg)\bigg] = E(XY) - E(X)E(Y)

Covariance Identities

  • Key points:

    • If variables X and Y are independent, then:
      Cov(X,Y)=0Cov(X,Y) = 0

    • Considerations around Cov(X,X) which equals the variance of X.

  • If Cov(X,Y) = 0: This does not imply X and Y are independent.

More on Covariance

  • Positive Covariance: Variables move in the same direction.

  • Negative Covariance: Variables move in opposite directions.

  • Example consideration of assets with positive or negative correlation.

Additional Covariance Identities

  • Overview of portfolio variance:
    Var(aX+bY)=a2Var(X)+b2Var(Y)+2abCov(X,Y)Var(aX + bY) = a^2Var(X) + b^2Var(Y) + 2abCov(X,Y)

  • Independence leads to separate variances:
    Var(Xext±Y)=Var(X)+Var(Y)Var(X ext{±} Y) = Var(X) + Var(Y)

  • Example Calculation: Variance of a portfolio of stocks - AMZN and GOOG with weights 0.3 and 0.7 respectively.

Correlation

  • Denoted by ρ: Range between -1 and 1.

  • Interpretation of extreme values:

    • ρ = -1: Perfect negative correlation

    • ρ = 1: Perfect positive correlation

    • ρ = 0: No correlation

Risk Aversion and Risk Premia

  • Concept of Risk Aversion: Reluctance to take on risk.

  • Solution for Risk Aversion: Overcome through a risk premium.

  • Definition of Risk Premium: Compensation received over what would be earned from a riskless investment.

    • Risk-averse investors reject fair games (those with zero risk premiums).

Examples of Risk Gambling Choices

  • For each scenario (questions 1–4):

    • Gamble description vs No Gamble choice

    • Question 1: $40,000 risk with a potential $20,000 loss vs $10,000 certain.

    • Question 2: Similar setup but $5,000 certain instead.

    • Question 3: $1,000 for certain with equivalent gamble.

    • Question 4: $20,000 risk with $0 loss vs $10,000 certain.

Risk-Free Rate

  • Definition: The return on a riskless investment (e.g., government securities).

  • Risk Premium Formula:
    ExcessReturn<em>a=E(R</em>a)−rfExcess Return<em>a = E(R</em>a) - r_f

  • Example: General Motors expected return of 15% against a risk-free rate of 2%, yields a risk premium of 13%.

Relationship of Risk Premia and Risk Aversion

  • Proportional Relationship: The required risk premium increases with both risk aversion (A) and portfolio risk ($ ext{σ}^2$):
    E(R<em>p)−r</em>f∝Aσ2E(R<em>p) - r</em>f ∝ Aσ^2

The Portfolio Problem

  • Investors need to decide allocation between risky assets and risk-free assets.

  • Portfolio Weight (w) can range from 0 to 1.

  • Expected Return of risk assets (E(R_a)): Critical in deciding on asset allocation.

Example Calculation with Google and Risk-Free Asset

  • Excel model: Weight on Google (w = 0.6) for portfolio returns comparison.

Variance of Portfolio Calculations

  • Understand variance of the overall portfolio and covariance components.

Investor Decision-Making

  • Determining the optimal weight (w) for the risky asset in portfolio.

    • Remaining weight is allocated to the risk-free asset.

  • Utility Function Relation:
    U=E(R)−rac12Aσ2U = E(R) - rac{1}{2} Aσ^2

Maximizing Utility

  • From the utility function, derive optimal weight: w∗=racE(R<em>a)−r</em>fAσa2w^* = rac{E(R<em>a) - r</em>f}{Aσ_a^2}

    • Explore investor types (high vs low risk aversion coefficient A) and impact of asset risk.

Risk Aversion Coefficient Calculation

  • Use an Excel model to adjust weights and calculate risk aversion coefficient (A) using the utility equation.

Future Topics

  1. Indifference Curves and Capital Market Line (CML)

  2. Advanced Portfolio Theory