Risk and return are crucial concepts in finance. This section dives into how we measure them for individual assets and portfolios. It explains expected returns, standard deviations, and the Security Market Line, which shows the relationship between risk and return.

The Capital Asset Pricing Model (CAPM) is introduced as a key tool for estimating required returns. We learn about systematic and unsystematic risks, and how diversification can reduce overall portfolio risk without sacrificing expected returns.

Expected Returns and Standard Deviations

Calculating Returns for Individual Assets

Expected return represents the weighted average of all possible returns
- Weights correspond to probabilities of each return occurring
Calculate expected return using the formula $E(R) = \sum (P_i \times R_i)$ $E (R) = \sum (P_{i} \times R_{i})$
- $P_i$ denotes probability of return i occurring
- $R_i$ signifies return for scenario i
Standard deviation measures dispersion of returns around expected return
- Serves as a quantitative measure of risk
Compute standard deviation using $\sigma = \sqrt{\sum (P_i \times (R_i - E(R))^2)}$ $σ = \sum (P_{i} \times (R_{i} - E (R))^{2})$
- $E(R)$ represents the expected return

Portfolio Returns and Risk

Portfolio expected return equals weighted average of individual asset expected returns
- Weights reflect proportion invested in each asset
Portfolio standard deviation incorporates individual asset risks and correlations between assets
- Reflects benefits of diversification
Covariance and correlation concepts explain asset interactions within portfolio
- Affect overall portfolio risk
Diversification reduces portfolio risk without sacrificing expected return
- Combining assets with low or negative correlations (stocks and bonds)

Security Market Line and Asset Pricing

Calculating Returns for Individual Assets, Compute the Sample Mean, Variance and Standard Deviation Using a Spreadsheet – Math FAQ

Fundamentals of the Security Market Line

Security Market Line (SML) graphically depicts relationship between systematic risk and expected return
- Derived from Capital Asset Pricing Model (CAPM)
SML plots expected return against beta on a graph
- Beta measures asset's sensitivity to market movements
Slope of SML represents market risk premium
- Additional return investors demand for taking on market risk
Y-intercept of SML equals risk-free rate
- Typically yield on short-term government securities (Treasury bills)

Applications and Implications of the SML

SML serves as benchmark for evaluating asset performance
- Assets above SML considered undervalued (stocks with high dividend yields)
- Assets below SML deemed overvalued (overpriced growth stocks)
SML shifts due to changes in risk-free rate or market risk premium
- Affects required returns for all assets in market
Investors use SML to assess risk-return tradeoffs
- Helps in portfolio construction and asset allocation decisions

Systematic vs Unsystematic Risk

Understanding Risk Components

Systematic risk affects all securities in market to varying degrees
- Cannot be diversified away (economic recessions, interest rate changes)
Unsystematic risk unique to individual securities or sectors
- Can be reduced through diversification (company-specific events, industry regulations)
Beta (β) measures systematic risk
- Indicates sensitivity of asset's returns to market movements
Total risk equals sum of systematic and unsystematic risk
- Measured by security's standard deviation

Risk and Portfolio Diversification

Beta of 1 indicates asset moves in line with market
- Beta greater than 1 suggests higher volatility than market (technology stocks)
As portfolio becomes more diversified, unsystematic risk decreases
- Portfolio's total risk approaches its systematic risk
Investors compensated only for bearing systematic risk
- Unsystematic risk can be eliminated through proper diversification
Well-diversified portfolio contains 20-30 stocks from different sectors
- Further diversification provides minimal additional risk reduction

Capital Asset Pricing Model for Required Returns

CAPM Formula and Components

Capital Asset Pricing Model (CAPM) expressed as $E(R_i) = R_f + \beta_i(E(R_m) - R_f)$ $E (R_{i}) = R_{f} + β_{i} (E (R_{m}) - R_{f})$
- $E(R_i)$ represents expected return on asset i
- $R_f$ denotes risk-free rate
- $\beta_i$ signifies beta of asset i
- $E(R_m)$ equals expected market return
Market risk premium calculated as $(E(R_m) - R_f)$
- Additional return investors expect for taking on market risk
Calculate beta as covariance between asset's returns and market returns divided by variance of market returns
- Measures asset's sensitivity to market movements

CAPM Applications and Limitations

CAPM assumes investors are rational, risk-averse, and hold well-diversified portfolios
- Eliminates unsystematic risk from consideration
Model implies linear relationship between expected return and systematic risk
- Represented by beta in the equation
Use CAPM to estimate cost of equity and evaluate investment performance
- Widely applied in corporate finance and portfolio management
Limitations of CAPM include reliance on historical data and single-period investment horizon
- Difficulty in identifying true market portfolio (S&P 500 often used as proxy)
Alternative models developed to address CAPM limitations
- Fama-French Three-Factor Model incorporates size and value factors

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