Lesson 11.1: Return, Risk, and the Capital Market

Introduction

In this lesson, we aim to explore the central concepts of return, risk, and the capital market in the context of portfolio management. The primary objectives are to understand how to measure portfolio return and risk, to apply the Capital Asset Pricing Model (CAPM), and to analyze the relationships between risk and return utilizing the Capital Market Line (CML).

Objectives:

Understand portfolio return and risk measures and the capital allocation line.
Learn about the Capital Asset Pricing Model and the Security Market Line.
Compute portfolio return, risk, and risk-adjusted measures.
Apply the Capital Asset Pricing Model in valuation.
Explain the main ideas and terminology behind Return, Risk, and the Capital Market.

Understanding Return

Return measures the gain or loss made on an investment over a specific period. It is often expressed as a percentage of the initial investment. Returns can be categorized into various types: holding period returns, excess returns, and risk-adjusted returns.

Holding Period Return

The holding period return (HPR) is the total return received from holding an asset or portfolio over a specified period. It is calculated as follows:

HPR = $\frac{(P_f - P_i) + D}{P_i}$ $\times 100$\%

Where:

$P_f$ = Price at the end of the period
$P_i$ = Price at the beginning of the period
$D$ = Dividends received during the period

Example:

Suppose you purchase a stock for $50, hold it for one year, and it pays a dividend of $2. If the stock price rises to $55 by the end of the year, the holding period return would be calculated as follows:

HPR = $\frac{(55 - 50) + 2}{50}$ $\times 100$\% = $\frac{7}{50}$ $\times 100$\% = 14%

Excess Return

Excess return refers to the return of an asset above the risk-free rate. The risk-free rate is the return on an investment with zero risk, typically represented by government bonds. Excess return is important in evaluating the performance of an investment against a benchmark.

Risk-Adjusted Return

Risk-adjusted return considers the amount of risk taken to achieve a return. Popular measures for risk-adjusted returns include the Sharpe Ratio and the Treynor Ratio.

Understanding Risk

Risk is the possibility of experiencing a loss or an unwanted outcome in an investment. Understanding risk is crucial for investors to make informed decisions. Two major types of risk in finance are systematic risk and unsystematic risk.

Systematic Risk

Systematic risk, also known as market risk, is the risk inherent to the entire market or market segment. This type of risk is unavoidable, and factors contributing to systematic risk include economic changes, political events, and overall market volatility.

Unsystematic Risk

Unsystematic risk, or specific risk, refers to the risk associated with a particular asset or industry. Unlike systematic risk, this type can be mitigated through diversification within a portfolio.

Measuring Risk

Risk can be quantified using various metrics, including standard deviation and beta. Standard deviation measures the total variability of returns from the mean return, while beta measures an asset's sensitivity to market movements.

\text{Standard Deviation} = $\sqrt{\frac{1}{N} \sum_{i=1}^{N} (R_i - \bar{R})^2}$

The Capital Market Line (CML)

The Capital Market Line represents the relationship between expected return and portfolio risk in a market characterized by optimal portfolios. It is derived from the efficient frontier and illustrates the best possible risk-return combinations available to investors.

CML Equation

The equation for the Capital Market Line is:

E(R_p) = R_f + $\frac{E(R_m) - R_f}{\sigma_m}$ $\cdot$ $\sigma$_p

Where:

$E(R_p)$ = Expected return on the portfolio
$R_f$ = Risk-free rate
$E(R_m)$ = Expected return on the market
$\sigma_m$ = Standard deviation of the market portfolio
$\sigma_p$ = Standard deviation of the portfolio

Example of CML Calculation

Let’s assume the risk-free rate is 3%, the expected return on the market is 8%, and the standard deviation of the market portfolio is 10%. If you hold a portfolio that has a standard deviation of 5%, the expected return on your portfolio would be:

E(R_p) = 3\% + $\frac{8\% - 3\%}{10\%}$ $\cdot 5$\% \quad = 3\% + $0.25 \cdot 5$\% \quad = 3\% + 1.25\% \quad = 4.25\%

This indicates that your portfolio, given the risk level, is expected to yield a return of 4.25%.

The Capital Asset Pricing Model (CAPM)

The Capital Asset Pricing Model is a fundamental finance theory that establishes a linear relationship between the expected return of an asset and its systematic risk, represented by beta. CAPM helps investors understand the risk-return trade-off.

CAPM Formula

The formula for CAPM is:

E(R_i) = R_f + $\beta$_i (E(R_m) - R_f)

Where:

$E(R_i)$ = Expected return of the asset
$R_f$ = Risk-free rate
$\beta_i$ = Beta of the asset
$E(R_m)$ = Expected return of the market

Example of CAPM Calculation

Assume the risk-free rate is 4%, the expected return on the market is 10%, and a stock has a beta of 1.5. The expected return on the stock can be calculated as:

E(R_i) = 4\% + $1.5 \cdot$ (10\% - 4\%) = 4\% + $1.5 \cdot 6$\% = 4\% + 9\% = 13\%

This means that, according to CAPM, the expected return on the stock is 13% given its level of risk relative to the market.

Conclusion

Understanding the interplay between return and risk is essential in portfolio management. The Capital Market Line and the Capital Asset Pricing Model provide frameworks that help in measuring and evaluating investments. By applying these concepts, investors can make informed choices about their portfolios, balancing risk and expected return in alignment with their financial goals.

Study Notes

Return measures the gain or loss from an investment.
Holding Period Return (HPR) formula: $HPR = \frac{(P_f - P_i) + D}{P_i} \times 100\%$.
Systematic risk is market-wide risk; unsystematic risk is specific to an asset.
CML shows the optimal portfolios and is represented by $E(R_p) = R_f + \frac{E(R_m) - R_f}{\sigma_m} \cdot \sigma_p$.
The CAPM formula is $E(R_i) = R_f + \beta_i (E(R_m) - R_f)$.