Risk and Return in Capital Markets Study Guide

Overview of Risk and Return in Capital Markets

• Course Context: Within the broader financial curriculum, this topic connects the foundational "Time Value of Money," "Financial Statement Analysis," and "Valuation" modules (Bond and Stock valuation) to "Capital Budgeting" and the "Cost of Capital."

• Central Objective: The goal is to quantify the relationship between risk and required return to produce a discount rate for valuation. A discount rate must accurately reflect the risk involved in the future benefits of an investment.

• Key Challenges:

  • Determining how to measure risk accurately.

  • Determining how to compare risks across different investments.

The NPV Rule and Forward-Looking Valuation

• NPV Rule Revisited: Investment decisions by corporations and investors depend on the valuation of future cash flows.

  • Projects with a positive Net Present Value (NPV) should be chosen.

  • The market price of any asset is equal to the Present Value (PV) of its expected future cash flows.

• Valuation Characteristics: Valuation is inherently forward-looking, requiring estimates of:

  • Cash flow amounts: Based on revenues, costs, growth rates, etc.

  • Timing of cash flows: Impatient investors require compensation for late payments (Time Value of Money).

  • Opportunity cost: This is the combination of the Time Value of Money and the riskiness of the cash flows.

• Risk Compensation: Risk-averse investors require compensation for accepting risky payments.

Estimating the Discount Rate

• Conceptual Framework: Theory dictates that the discount rate ($r_E$) should be the expected return from holding the stock, which is a forward-looking metric.

• Practical Limitations: Because future returns are unknown, estimates usually rely on historical data, which is backward-looking.

• Estimating Methods:

  • Average Historical Returns of Matching Companies: This involves looking at the performance of companies in the same industry with similar size and risk. However, these measures are often "noisy" and less precise.

  • Advanced Models: Better alternatives include the Capital Asset Pricing Model (CAPM) and various factor models.

Risk Aversion and the Lottery Example

• Defining Risk Aversion through Lotteries:

  • Lottery 1: A guaranteed payment of $\$1,000$.

  • Lottery 2: A $50\%$ probability of receiving $\$2,000$ and a $50\%$ probability of receiving $\$0$.

  • Expected Payoff calculation: For Lottery 2, the expected payoff is (0.5 \times \2,000) + (0.5 \times \$0) = \$1,000$.

  • Result: If an individual prefers the sure $\$1,000$ in Lottery 1, they are considered risk averse.

• Value of Uncertainty: A risk-averse investor will value Lottery 2 at less than its expected payoff (e.g., valuing it at $\$800$).

  • The difference (e.g., $\$200$) is the compensation required for the investor to hold uncertain outcomes.

• Logic in Financial Markets:

  • Given a constant level of return, investors prefer the asset with lower risk.

  • Taking on more risk necessitates a higher required return.

  • Discount rates must reflect both the time value of money and the riskiness of the security.

Historical Performance of Asset Classes (1925-2015)

• Cumulative Growth of a $100 Investment:

  • Small Stocks: Grew to $\$4,723,025$.

  • S&P 500 (Large Stocks): Grew to $\$635,430$.

  • World Portfolio: Grew to $\$227,975$.

  • Corporate Bonds: Grew to $\$22,253$.

  • Treasury Bills: Grew to $\$2,098$.

  • Consumer Price Index (CPI/Inflation): Grew to $\$1,404$.

• Notable Periods of Volatility and Market Stress:

  • 1928-1932: Small stocks dropped $-92\%$, S&P 500 dropped $-84\%$.

  • 1937-1938: Market decline of $-72\%$ to $-50\%$.

  • 1968-1974: Range of $+63\%$ to $-37\%$.

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

  • 1987: Single period drop of $-34\%$ to $-30\%$.

  • 2000-2002: Tech bubble crash, $-26\%$ to $-45\%$

  • 2007-2009: Financial crisis, series of $+67\%$ and $-51\%$ movements.

Calculating Realized Returns

• Realized Return (Non-Annual Holding Period): This is the total return that actually occurs over a specific period, assuming no interim cash flows except the final dividend and sale.

  • Formula: $R_{t+1} = \frac{Div_{t+1} + (P_{t+1} - P_t)}{P_t} = \frac{Div_{t+1} + P_{t+1}}{P_t} - 1$

  • Direct Components:

    • Dividend Yield: $\frac{Div_{t+1}}{P_t}$

    • Capital Gain Yield: $\frac{P_{t+1} - P_t}{P_t}$

• Microsoft (MSFT) Example (Nov 1 to Nov 15, 2004):

  • Purchase Price ($P_t$): $\$28.08$

  • Sale Price ($P_{t+1}$): $\$27.39$

  • Dividend ($Div_{t+1}$): $\$3.08$

  • Dividend Yield: $\frac{3.08}{28.08} = 10.97\%$

  • Capital Gain Yield: $\frac{27.39 - 28.08}{28.08} = -2.46\%$

  • Total Realized Return: $10.97\% - 2.46\% = 8.51\%$

Annual Realized Returns and Compounding

• Annualization Logic: To compute return over a year with multiple dividend periods, realize the returns per period and assume immediate reinvestment.

• Compounding Formula:

  • $1 + R_{annual} = (1 + R_1) \times (1 + R_2) \times (1 + R_3) \times (1 + R_4)$

• Microsoft Annual Example (Nov 1, 2004 – Oct 31, 2005):

  • Quarterly realized returns including dividends: $8.51\%$, $-5.04\%$, $-1.39\%$, $6.75\%$, and a fifth period return of $-5.27\%$.

  • Calculation: $1 + R_{annual} = (1.0851) \times (0.9496) \times (0.9861) \times (1.0675) \times (0.9473) = 1.0275$

  • Final Return: $2.75\%$

  • Observation: Even though the selling price ended lower than the purchase price, the total return was positive due to dividends.

Arithmetic vs. Geometric Average Returns

• Arithmetic Average ($R_A$):

  • Definition: The simple sum of returns divided by the number of periods.

  • Calculation: $R_A = \frac{R_1 + R_2 + \dots + R_T}{T}$

  • Purpose: Answers "What was your return in an average year?" and is used to estimate the expected return for a future year.

• Geometric Average ($R_G$):

  • Definition: Also known as the Compound Annual Growth Rate (CAGR).

  • Calculation: $R_G = \left[ (1+R_1) \times (1+R_2) \times \dots \times (1+R_T) \right]^{1/T} - 1$

  • Purpose: Answers "What was the average compound return earned per year?" and is used to measure historical performance.

• Relationship and Volatility:

  • The larger the volatility, the larger the gap between the arithmetic and geometric averages.

  • Example: Returns of $+20\%$ and $-20\%$

    • Arithmetic Average: $\frac{20\% - 20\%}{2} = 0\%$

    • Geometric Average: $\sqrt{(1.20) \times (0.80)} - 1 = \sqrt{0.96} - 1 = -2\%$

    • A $\$1$ investment would be worth $\$0.96$ after two years.

Calculating Variance and Volatility

• Goal: Measure the spread in returns and how far actual returns deviate from the mean.

• Variance ($Var(R)$): The average of the squared deviations from the mean.

  • Formula: $Var(R) = \frac{1}{T-1} \sum_{t=1}^T (R_t - \bar{R})^2$

• Standard Deviation ($SD(R)$): Known as volatility. It is the square root of the variance.

  • Formula: $SD(R) = \sqrt{Var(R)}$

  • Significance: A larger SD indicates higher uncertainty and risk.

Sample Problem: Computing S&P 500 Volatility (2005-2009)

• Data Points (Annual Returns):

  • 2005: $4.9\%$

  • 2006: $15.8\%$

  • 2007: $5.5\%$

  • 2008: $-37.0\%$

  • 2009: $26.5\%$

• Step 1: Compute Mean ($R_A$):

  • $(4.9 + 15.8 + 5.5 - 37 + 26.5) / 5 = 3.14\%$

• Step 2: Compute Squared Differences:

  • $(4.9 - 3.14)^2 = 3.10$

  • $(15.8 - 3.14)^2 = 160.00$

  • $(5.5 - 3.14)^2 = 5.57$

  • $(-37.0 - 3.14)^2 = 1611.00$

  • $(26.5 - 3.14)^2 = 545.70$

• Step 3: Variance Calculation:

  • $Var(R) = \frac{3.10 + 160 + 5.57 + 1611 + 545.7}{5 - 1} = \frac{2325.37}{4} = 581.46$

• Step 4: Standard Deviation:

  • $SD(R) = \sqrt{581.46} = 24.11\%$

Common Calculation Pitfalls

1. Unit Consistency: Use percent as the unit to minimize confusing decimal places.

2. Order of Operations: Ensure you square the deviation from the mean before summing them up.

3. Degree of Freedom: Remember to divide by $T-1$ (for a sample) rather than $T$.

4. Final Root: Do not stop at the variance; remember to take the square root for standard deviation.

The Normal Distribution and Future Predictions

• Definition: A symmetric, bell-shaped curve completely defined by its mean and standard deviation.

• Historical Average as Predictor: We assume the return distribution is identical each year and independent of prior years' returns.

• Confidence Intervals:

  • $68\%$ Confidence Interval: Mean $\pm$ (1 x Standard Deviation)

  • $95\%$ Confidence Interval: Mean $\pm$ (2 x Standard Deviation)

• Example Case (Mean = $7\%$, Variance = $129\%^2$):

  • Standard Deviation = $\sqrt{129} \approx 11.36\%$

  • $95\%$ range: $7 - 2(11.36)$ to $7 + 2(11.36) = -15.72\%$ to $29.72\%$

Questions & Discussion

• Q: If a stock does not pay dividends, can the annual return be computed as just price appreciation?

  • A: Yes. Without dividends, the dividend yield component of the return formula is zero.

• Q: Which average should be used to predict next year's return?

  • A: The Arithmetic Average. It is the best prediction for a single future period's return based on the assumption that past years represent equally likely outcomes.

• Scenario: If mean is $7\%$ and range is $-15.72\%$ to $29.72\%$ for a $95\%$ confidence interval, can we be $95\%$ confident the portfolio won't lose more than $20\%$?

  • A: Yes. Since the lower bound of the $95\%$ confidence interval is $-15.72\%$, any loss greater than $-15.72\%$ (like $-20\%$) falls outside that range of likely outcomes.

• Q: Assuming a mean of $7\%$ and variance of $129\%^2$, on average, how often would you expect to lose more than $15.72\%$?

  • A: By the Normal Distribution, outcomes outside the $2$ SD range ($\pm 15.72\%$ from the mean of $7\%$) happen $5\%$ of the time ($2.5\%$ in the upper tail, $2.5\%$ in the lower tail). A loss worse than $-15.72\%$ is the lower tail ($2.5\%$).

  • Frequency: $1 / 0.025 = 40$ years. Therefore, on average, once every $40$ years.