Module 3 Risk Aversion, Premia and Sharpe Ratios

Risk Aversion, Risk Premia, and Sharpe Ratios

Lecture Details

• Course: F303 – Intermediate Investments

• 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

Review Questions

Holding Period Returns (HPR)

• Example: You invested $435 in Bitcoin in January of 2016. By January 2023, you sold your investment for $20,750.

  • Holding Period Return (HPR): HPR is calculated as:
    $HPR = \frac{Final\ Value - Initial\ Value}{Initial\ Value} = \frac{20750 - 435}{435}$

  • Simple Annualized HPR:
    Formula: $Simple\ Annualized\ HPR = \frac{HPR}{Holding\ Period\ In\ Years}$

  • Compounded Annualized HPR:
    Formula: $Compounded\ Annualized\ HPR = (1 + HPR)^{\frac{1}{Holding\ Period}} - 1$

Return Averages

• Example: You bought 50 shares of Amazon at $30/share. After 1 year, the share price was $25. After 2 years, it increased to $40.

  • HPR for Year 1:
    $HPR = \frac{25 - 30}{30}$

  • HPR for Year 2:
    $HPR = \frac{40 - 25}{25}$

  • Arithmetic Average of Returns:
    Formula: $Arithmetic\ Average = \frac{HPR<em>1 + HPR</em>2}{2}$

  • Geometric Average of Returns:
    Formula: $Geometric\ Average = (1 + HPR<em>1)(1 + HPR</em>2)^{\frac{1}{2}} - 1$

Walmart Stock Returns

• Historical returns: -5%, 14%, 7%, -3%, 6%.

  • Expected Return:
    Formula: $Expected\ Return = \frac{(-5 + 14 + 7 - 3 + 6)}{5}$

  • Variance Calculation:
    Formula: $Variance = \frac{(R<em>1 - E(R))^2 + (R</em>2 - E(R))^2 + … + (R_n - E(R))^2}{n}$

Covariance

• Definition: Covariance measures the degree to which two random variables move together.

• Importance: This metric is crucial for analyzing portfolios of assets.

• Example graphical analysis: Co-movement of stocks (Show pricing charts for Home Depot vs Hewlett Packard and Home Depot vs Lowe's).

Covariance Formula

• Given X and Y are random variables, the covariance between X and Y is defined as:
  $Cov(X,Y) = E\left[{X - E(X)}{Y - E(Y)}\right] = E(XY) - E(X)E(Y)$

• Interpretation: Covariance indicates how much two variables co-move together.

Covariance Identities

• Independence: If X and Y are independent, $Cov(X,Y) = 0$, but the reverse is not always true.

• Variance properties:

  • For any constants a and b:
    $Var(aX + bY) = a^2Var(X) + b^2Var(Y) + 2abCov(X,Y)$

• Example: Variance calculation for two stocks, AMZN and GOOG with respective portfolio weights 0.3 and 0.7.

Correlation

• Denotation: Often denoted by ρ (rho)

• Range: Value between -1 and 1:

  • If ρ = -1: Perfect negative correlation

  • If ρ = 1: Perfect positive correlation

  • If ρ = 0: No correlation

Risk Aversion and Risk Premia

• Risk Aversion Definition: Investor’s reluctance to accept risk.

• Risk Premium Definition: Compensation above what an investor receives for no risk.

• Fair Game: A risky investment with a risk premium of zero that risk-averse investors would reject.

Examples of Risk Decisions

1. Gamble vs No Gamble:

   • Gamble: Toss a coin; heads wins $40,000; tails pays $20,000.

   • No Gamble: $10,000 for sure.

2. Repeated trials of similar gambles show differing preferences based on risk premiums.

   • More students chose the security of certain amounts over the gamble for the larger guaranteed amounts ($10,000) than smaller amounts.

Risk Premium and Risk-Free Rate

• Risk-Free Rate: The return investors expect from a riskless investment, often linked to government bonds.

• Risk Premium Formula:
  $Excess\ Return<em>a = E(R</em>a) - r_f$

• Example: General Motors expected return of 15% against a 2% T-bill.

Portfolio Problem

• Wealth Allocation: Investors distribute their wealth between a risky asset and a risk-free asset. The proportion allocated to each must be managed.

• Weights: Portfolio weight (w) for risky asset must be between 0 and 1.

• Expected Portfolio Return and Variance:

  • $E(R) = wE(R<em>a) + (1 - w)r</em>f$

  • Variances also apply given the configuration of the assets.

Utility Function

• Utility is defined mathematically as:
  $U = E(R) - \frac{1}{2}A\sigma^2$

• A coefficient reflects individual risk aversion, where higher A denotes lower tolerance for risk.

• To maximize utility, the optimal weight in risky assets can be calculated as:
  $w^* = \frac{E(R<em>a) - r</em>f}{A\sigma_a^2}$

Final Thoughts

• Next Topics:

1. Indifference curves and the Capital Market Line (CML)

2. Portfolio Theory
   The exploration of risk management via more advanced models such as the CML and understanding how portfolios behave in conjunction with market forces and personal risk profiles will be discussed in subsequent sessions.