Comprehensive Study Notes: Risk and Return in Capital Markets

Overview of Risk and Return in Capital Markets

• Historical Perspective and Growth of Investments

  • In the period from $1925$ to $2015$, the value of a $\$100$ investment varied wildly depending on the asset class:

    • Small Stocks: Provided the highest growth, reaching an end value of approximately $\$4,723,025$.

    • S&P 500: Reached an end value of approximately $\$635,430$.

    • World Portfolio: Reached an end value of approximately $\$227,975$.

    • Corporate Bonds: Reached an end value of approximately $\$22,253$.

    • Treasury Bills: Reached an end value of approximately $\$2,098$.

    • CPI (Consumer Price Index): Reached an end value of approximately $\$1,404$.

  • Significant Market Events (Historical Drawdowns):

    • 1928-1932: Small stocks fell by $92\%$, while the S&P 500 fell by $84\%$.

    • 1937-1938: Declines of $72\%$ and $50\%$.

    • 1968-1974: Growth of $63\%$ followed by a decline of $37\%$.

    • 1979-1981: High inflation period; Treasury Bill yields peaked at $15\%$.

    • 1987: One-year fluctuations of $-34\%$ and $-30\%$.

    • 2000-2002: Declines of $-26\%$ and $-45\%$

    • 2007-2009: Growth of $67\%$ followed by a decline of $-51\%$.

Historical Trade-off Between Risk and Return (1926-2018)

• Large Portfolio Performance Data:

  • Small Stocks:

    • Historical Average Annual Return: $18.04\%$

    • Standard Deviation of Returns (Volatility): $39.25\%$

  • S&P 500:

    • Historical Average Annual Return: $11.71\%$

    • Standard Deviation of Returns (Volatility): $19.76\%$

  • Corporate Bonds:

    • Historical Average Annual Return: $6.16\%$

    • Standard Deviation of Returns (Volatility): $6.36\%$

  • Treasury Bills (T-Bills):

    • Historical Average Annual Return: $3.39\%$

    • Standard Deviation of Returns (Volatility): $3.12\%$

• Historical Statistics Summary (1928-2010):

  • Stocks:

    • Simple Mean: $11.31\%$

    • Geometric Mean: $9.32\%$

    • Standard Deviation: $20.21\%$

    • Minimum: $-43.84\%$

    • Maximum: $52.56\%$

  • Treasury Bonds:

    • Simple Mean: $5.28\%$

    • Geometric Mean: $5.01\%$

    • Standard Deviation: $7.74\%$

    • Minimum: $-11.12\%$

    • Maximum: $32.81\%$

  • Treasury Bills:

    • Simple Mean: $3.70\%$

    • Geometric Mean: $3.66\%$

    • Standard Deviation: $3.04\%$

    • Minimum: $0.03\%$

    • Maximum: $14.30\%$

• Key Observations on Portfolio Risk:

  • For large portfolios, there is a general positive increasing relationship between historical volatility and average return.

  • Volatility serves as a reasonable measure of risk to evaluate a large portfolio.

  • The S&P 500 has a historical excess return of approximately $8.5\%$ over Treasury Bills.

Risk and Return in Individual Stocks

• Lack of Correlation: Unlike large portfolios, there is no clear relationship between volatility and returns for individual stocks.

• Individual Stock Characteristics:

  • Larger stocks generally exhibit lower volatility overall compared to smaller stocks.

  • Even the largest individual stocks are typically more volatile than a diversified portfolio of large stocks.

  • All individual stocks typically possess lower returns and/or higher risk (volatility) than diversified portfolios.

Common vs. Independent Risk

• Independent Risk:

  • Definition: Risks that bear no relation to each other. Knowing the outcome of one event provides no information about the other.

  • Example: Theft Insurance. If an insurance company writes $100,000$ policies in San Francisco with a $1\%$ chance of theft, on average, $1,000$ claims will be filed annually. There is very small variation in payouts year-to-year, making the risk almost zero for the company.

  • Diversification: Independent risks can be averaged out in a large portfolio.

• Common Risk:

  • Definition: Risk that is perfectly correlated and linked across outcomes.

  • Example: Earthquake Insurance. If there is a $1\%$ chance of an earthquake damaging San Francisco, there is a $99\%$ chance of zero claims and a $1\%$ chance that all $100,000$ claims are filed simultaneously. This creates massive variation in outcomes.

  • Implications: The insurance company must hold massive cash reserves to cover the $1\%$ catastrophic event; the risk of the portfolio is identical to the risk of a single policy.

Systematic vs. Unsystematic Risk

• Types of News Driving Stock Fluctuations:

  1. Market-wide News: Information affecting the entire economy and all stocks (e.g., changes in GDP, inflation, interest rates).

  2. Firm-specific News: Good or bad news about an individual company or a specific industry (e.g., product recalls, discovery of a new drug).

• Categorization of Risk:

  • Systematic Risk:

    • Also known as: Market Risk, Non-diversifiable Risk, Common Risk.

    • Source: Market-wide news.

    • Diversification: Cannot be eliminated by adding more stocks to a portfolio.

  • Unsystematic Risk:

    • Also known as: Firm-specific Risk, Diversifiable Risk, Independent Risk.

    • Source: Firm-specific news.

    • Diversification: Can be eliminated (diversified) in a large portfolio as firm-specific events average out.

• Diversification Statistics:

  • As the number of stocks in a portfolio increases (from $1$ to $1,000$), the total volatility declines until it plateaus. This plateau represents the remaining systematic risk.

Risk Premium and Compensation

• Definition of Risk Premium:

  • The additional return above the risk-free rate resulting from bearing risk.

  • $\text{Risk Premium} = \text{Risky Investment Return} - \text{Risk-free Rate}$.

  • The benchmark for the risk-free asset is typically Treasury Bills (T-Bills).

• Historical Risk Premium (1928-2010):

  • U.S. Common Stocks: $11.31\% - 3.70\% = 7.61\%$

  • Treasury Bonds: $5.28\% - 3.70\% = 1.58\%$

• Compensation Rule:

  • Unsystematic Risk: The risk premium for diversifiable risk is zero. Investors are not compensated for holding firm-specific risk because they can eliminate it for free via diversification.

  • Systematic Risk: The risk premium of a stock depends solely on its systematic risk.

  • No Arbitrage Principle: If diversifiable risk earned a premium, investors could buy the stock, earn the extra return, and diversify away the risk, creating an arbitrage opportunity. Markets eliminate this through trading.

Comparative Portfolio Analysis Examples

• Portfolio Comparison (S Firm vs. U Firm):

  • S Firm (Systematic Risk):

    • Volatility (Standard Deviation): $30\%$

    • Risk-Free Rate: $5\%$

    • Expected Return: $10\%$

    • Calculated Risk Premium: $10\% - 5\% = 5\%$

  • U Firm (Unsystematic Risk Only):

    • Volatility (Standard Deviation): $30\%$

    • Risk-Free Rate: $5\%$

    • Expected Return: $5\%$

    • Calculated Risk Premium: $5\% - 5\% = 0\%$

• Lesson: Even if two firms have the same total volatility ($30\%$), the firm with only unsystematic risk generates no risk premium and offers only the risk-free rate.

Questions & Discussion

• Q: What is the relationship between the volatility and returns for large portfolios of assets?

  • A: Positive. Higher volatility in large portfolios generally correlates with higher average returns.

• Q: Risk of theft is what type of risk?

  • A: Independent risk. It can be diversified away across many policies.

• Q: Which risks are likely firm-specific and diversifiable?

  • A: A product design flaw leading to a recall is a firm-specific, diversifiable risk. (By contrast, oil price spikes or economic slowdowns are systematic risks affecting most firms).

• Q: Does a firm with an expected return equal to the risk-free rate have systematic risk?

  • A: No. If the expected return equals the risk-free rate, the risk premium is zero, implying there is no systematic risk.

• Q: If an investor holds firm-specific risk, do they earn a risk premium?

  • A: No. The risk premium for diversifiable risk is zero; they will earn only the risk-free rate for that portion of risk.