Corporate Finance Fundamentals
Module 5: Risk and Return
Module Overview
Welcome to Module 5! Until now, we've assumed we know the discount rate to use in NPV calculations. But where does that rate come from? Why is it 10% for one project and 15% for another?
The answer lies in understanding risk and return—one of the most fundamental concepts in all of finance. This module explores how risk is measured, why risk matters, and how it determines the returns investors require.
Learning Objectives:
By the end of this module, you will be able to:
- Define and measure investment risk using statistical tools
- Calculate expected returns and standard deviation
- Understand the relationship between risk and return
- Explain how diversification reduces risk
- Distinguish between systematic and unsystematic risk
- Understand portfolio theory basics
- Calculate and interpret beta
- Introduce the Capital Asset Pricing Model (CAPM)
- Apply these concepts to determine appropriate discount rates
Estimated Time: 5-6 hours
Prerequisites:
- Basic statistics (mean, variance, standard deviation)
- Understanding of probability concepts
- Comfortable with Module 3's time value of money
5.1 Introduction to Risk and Return
What is Return?
Return is what you earn from an investment, expressed as a percentage of the amount invested.
Total Return Formula:
Return = (Ending Value - Beginning Value + Income) / Beginning Value
Or:
Return = Capital Gain + Dividend (or Interest)
─────────────────────────────────────
Beginning Value
Example 1: Stock Return
You buy a stock for $100. One year later:
- Stock price: $110
- Dividend received: $3
Return = ($110 - $100 + $3) / $100
Return = $13 / $100
Return = 0.13 = 13%
Example 2: Bond Return
You buy a bond for $1,000 paying 5% interest ($50/year). After one year, the bond trades at $1,020.
Return = ($1,020 - $1,000 + $50) / $1,000
Return = $70 / $1,000
Return = 0.07 = 7%
Example 3: Negative Return
You buy a stock for $50. After one year:
- Stock price: $42
- Dividend: $2
Return = ($42 - $50 + $2) / $50
Return = -$6 / $50
Return = -0.12 = -12%
You lost 12%.
What is Risk?
Risk is uncertainty about future returns. The more uncertain the return, the riskier the investment.
Common Definition: Risk is the variability or volatility of returns.
Intuitive Understanding:
Low Risk Investment:
- Returns: 5%, 6%, 5%, 6%, 5%
- Predictable, stable
- Example: U.S. Treasury bonds
High Risk Investment:
- Returns: -20%, 35%, -10%, 40%, 15%
- Unpredictable, volatile
- Example: Cryptocurrency or startup stock
Key Insight: Risk isn't just about losing money—it's about uncertainty. An investment could be risky even if it ultimately does well, if the outcome was uncertain.
Historical Risk and Return: The Evidence
U.S. Market Data (1926-2023 approximately):
| Asset Class | Average Annual Return | Standard Deviation (Risk) |
|---|---|---|
| U.S. Treasury Bills | 3.3% | 3.1% |
| Long-term Gov Bonds | 5.7% | 9.8% |
| Corporate Bonds | 6.3% | 8.4% |
| Large Company Stocks | 10.3% | 19.8% |
| Small Company Stocks | 12.1% | 31.9% |
Key Observations:
1. Risk-Return Tradeoff:
- Higher risk → Higher average return
- Lower risk → Lower average return
2. The Spread:
- T-bills (safe): 3.3% with little volatility
- Small stocks (risky): 12.1% with huge volatility
3. Historical Pattern: Stocks have outperformed bonds over long periods, but with much more volatility.
The Fundamental Question: Why do riskier investments offer higher returns?
Answer: Investors are risk-averse. They demand higher returns to compensate for taking more risk. This is the risk premium.
Risk Premium
Risk Premium = Expected Return on Risky Asset - Risk-Free Rate
Example:
Risk-free rate (T-bills): 3% Expected return on stocks: 11%
Risk Premium = 11% - 3% = 8%
This 8% is the extra return investors require for bearing stock market risk.
Different investments have different risk premiums:
- Investment-grade corporate bonds: 2-3% over T-bills
- High-yield bonds: 5-6% over T-bills
- Large-cap stocks: 7-8% over T-bills
- Small-cap stocks: 9-10% over T-bills
- Emerging market stocks: 10-12% over T-bills
Higher risk → Higher required premium
5.2 Measuring Returns
Expected Return
When we don't know what return we'll get, we calculate the expected return—the probability-weighted average of possible returns.
Formula:
E(R) = Σ [Probability × Return]
E(R) = p₁R₁ + p₂R₂ + ... + pₙRₙ
Example 1: Simple Expected Return
An investment has three possible outcomes:
| Outcome | Probability | Return |
|---|---|---|
| Good | 30% | 25% |
| Normal | 50% | 12% |
| Bad | 20% | -8% |
Expected Return:
E(R) = (0.30 × 25%) + (0.50 × 12%) + (0.20 × -8%)
E(R) = 7.5% + 6.0% - 1.6%
E(R) = 11.9%
Example 2: Investment Comparison
Investment A:
| Outcome | Probability | Return |
|---|---|---|
| Boom | 25% | 40% |
| Normal | 50% | 15% |
| Recession | 25% | -10% |
E(R_A) = (0.25 × 40%) + (0.50 × 15%) + (0.25 × -10%)
E(R_A) = 10% + 7.5% - 2.5%
E(R_A) = 15%
Investment B:
| Outcome | Probability | Return |
|---|---|---|
| Boom | 25% | 20% |
| Normal | 50% | 15% |
| Recession | 25% | 10% |
E(R_B) = (0.25 × 20%) + (0.50 × 15%) + (0.25 × 10%)
E(R_B) = 5% + 7.5% + 2.5%
E(R_B) = 15%
Both have 15% expected return—but which is riskier?
Look at the range:
- Investment A: -10% to +40% (50% range)
- Investment B: +10% to +20% (10% range)
Investment A is much riskier! We need a better measure of risk.
Historical Average Return
When we have past data, we can calculate the average historical return:
Average Return = Σ Returns / Number of Periods
Example:
Stock returns over 5 years: 12%, -5%, 18%, 8%, 22%
Average = (12% - 5% + 18% + 8% + 22%) / 5
Average = 55% / 5
Average = 11%
Arithmetic vs. Geometric Average:
Arithmetic Average: Simple average (as above)
- Use for expected single-period returns
Geometric Average: Compound average return
Geometric Average = [(1+R₁)(1+R₂)...(1+Rₙ)]^(1/n) - 1
Example:
Returns: 20%, -10%, 15%
Arithmetic:
(20% - 10% + 15%) / 3 = 8.33%
Geometric:
[(1.20)(0.90)(1.15)]^(1/3) - 1
= [1.242]^(1/3) - 1
= 1.075 - 1
= 7.5%
Which to use?
- Arithmetic: Expected return for one period
- Geometric: Actual compound return achieved over multiple periods
For expected returns in finance, we typically use arithmetic average.
5.3 Measuring Risk
Variance and Standard Deviation
Variance (σ²) measures the average squared deviation from the expected return.
Standard Deviation (σ) is the square root of variance.
Why Standard Deviation?
- Measured in same units as return (percentages)
- Most commonly used risk measure
- Higher standard deviation = more volatile = riskier
Calculating Standard Deviation: Expected Return Approach
Formula:
Variance = σ² = Σ [Probability × (Return - Expected Return)²]
Standard Deviation = σ = √Variance
Example: Investment A from Earlier
Expected Return: 15%
| Outcome | Prob | Return | Deviation | Squared Dev | Weighted |
|---|---|---|---|---|---|
| Boom | 0.25 | 40% | 25% | 625 | 156.25 |
| Normal | 0.50 | 15% | 0% | 0 | 0 |
| Recession | 0.25 | -10% | -25% | 625 | 156.25 |
Variance = 156.25 + 0 + 156.25 = 312.5
Standard Deviation = √312.5 = 17.68%
Investment B:
Expected Return: 15%
| Outcome | Prob | Return | Deviation | Squared Dev | Weighted |
|---|---|---|---|---|---|
| Boom | 0.25 | 20% | 5% | 25 | 6.25 |
| Normal | 0.50 | 15% | 0% | 0 | 0 |
| Recession | 0.25 | 10% | -5% | 25 | 6.25 |
Variance = 6.25 + 0 + 6.25 = 12.5
Standard Deviation = √12.5 = 3.54%
Comparison:
- Both have 15% expected return
- Investment A: σ = 17.68% (high risk)
- Investment B: σ = 3.54% (low risk)
Investment B is much less risky!
Calculating Standard Deviation: Historical Data
When using historical returns:
Formula:
Variance = Σ(Return - Average Return)² / (n - 1)
Standard Deviation = √Variance
(Note: We divide by n-1 for sample variance)
Example:
Stock returns: 12%, -5%, 18%, 8%, 22% Average: 11%
| Year | Return | Deviation | Squared |
|---|---|---|---|
| 1 | 12% | 1% | 1 |
| 2 | -5% | -16% | 256 |
| 3 | 18% | 7% | 49 |
| 4 | 8% | -3% | 9 |
| 5 | 22% | 11% | 121 |
Variance = (1 + 256 + 49 + 9 + 121) / (5 - 1)
Variance = 436 / 4
Variance = 109
Standard Deviation = √109 = 10.44%
In Excel:
Returns in cells A1:A5
Average: =AVERAGE(A1:A5)
Variance: =VAR.S(A1:A5)
Std Dev: =STDEV.S(A1:A5)
Much easier!
Coefficient of Variation
Coefficient of Variation (CV) adjusts risk for the level of return:
CV = Standard Deviation / Expected Return
Use: Comparing investments with very different expected returns.
Example:
Investment A:
- E(R) = 15%
- σ = 17.68%
- CV = 17.68% / 15% = 1.18
Investment B:
- E(R) = 15%
- σ = 3.54%
- CV = 3.54% / 15% = 0.24
Investment B has much lower risk per unit of return.
Another Example:
Investment C:
- E(R) = 30%
- σ = 25%
- CV = 25% / 30% = 0.83
Investment D:
- E(R) = 10%
- σ = 5%
- CV = 5% / 10% = 0.50
Despite Investment C having higher absolute risk (25% vs. 5%), Investment D actually has lower risk per unit of return (0.50 vs. 0.83).
5.4 Portfolio Theory and Diversification
The Power of Diversification
Key Insight: By holding multiple investments, you can reduce risk without reducing expected return!
This is one of the most important concepts in finance.
The Saying: "Don't put all your eggs in one basket."
Portfolio Return
A portfolio is a collection of investments.
Portfolio Return Formula:
E(R_p) = w₁E(R₁) + w₂E(R₂) + ... + wₙE(Rₙ)
Where:
- w = weight (proportion) of each asset
- E(R) = expected return of each asset
Example: Two-Asset Portfolio
Stock A:
- Weight: 60%
- Expected return: 12%
Stock B:
- Weight: 40%
- Expected return: 18%
Portfolio Return:
E(R_p) = (0.60 × 12%) + (0.40 × 18%)
E(R_p) = 7.2% + 7.2%
E(R_p) = 14.4%
Simple rule: Portfolio return is the weighted average of individual returns.
Portfolio Risk: Not So Simple!
Portfolio risk is NOT simply the weighted average of individual risks.
Why? Because of correlation—how assets move together.
Correlation
Correlation measures how two variables move together.
Correlation Coefficient (ρ or r):
- Ranges from -1 to +1
- +1: Perfect positive correlation (move exactly together)
- 0: No correlation (independent)
- -1: Perfect negative correlation (move exactly opposite)
Examples:
High Positive Correlation (+0.8 to +1):
- Ford stock and GM stock
- Coca-Cola and PepsiCo
- Most stocks in same industry
Low/Zero Correlation (-0.2 to +0.2):
- Technology stocks and utility stocks
- U.S. stocks and bonds
- Oil prices and food prices
Negative Correlation (-0.5 to -1):
- Umbrella sales and sunscreen sales
- Gold and stock market (sometimes)
- Insurance company and natural disasters
Key for Diversification: Combining assets with low or negative correlation reduces portfolio risk!
Portfolio Risk Formula
For a two-asset portfolio:
σ_p² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρ₁₂σ₁σ₂
Where:
- σ_p = portfolio standard deviation
- w = weights
- σ = standard deviations
- ρ₁₂ = correlation between assets 1 and 2
Then:
σ_p = √(σ_p²)
Example: Perfect Positive Correlation (ρ = +1)
Stock A:
- Weight: 50%
- σ = 20%
Stock B:
- Weight: 50%
- σ = 30%
Correlation: +1
σ_p² = (0.5)²(20)² + (0.5)²(30)² + 2(0.5)(0.5)(1)(20)(30)
σ_p² = 0.25(400) + 0.25(900) + 0.5(1)(600)
σ_p² = 100 + 225 + 300
σ_p² = 625
σ_p = √625 = 25%
Note: With perfect positive correlation, portfolio risk (25%) is simply the weighted average of individual risks: (0.5 × 20%) + (0.5 × 30%) = 25%
No diversification benefit!
Example: Zero Correlation (ρ = 0)
Same stocks, but correlation = 0
σ_p² = (0.5)²(20)² + (0.5)²(30)² + 2(0.5)(0.5)(0)(20)(30)
σ_p² = 100 + 225 + 0
σ_p² = 325
σ_p = √325 = 18.0%
Portfolio risk (18%) is less than the weighted average (25%)!
Diversification works!
Example: Perfect Negative Correlation (ρ = -1)
Same stocks, but correlation = -1
σ_p² = (0.5)²(20)² + (0.5)²(30)² + 2(0.5)(0.5)(-1)(20)(30)
σ_p² = 100 + 225 - 300
σ_p² = 25
σ_p = √25 = 5%
Portfolio risk is only 5%!
With perfect negative correlation, massive risk reduction.
Key Insight:
- ρ = +1: No diversification benefit
- ρ = 0: Significant diversification benefit
- ρ = -1: Maximum diversification benefit
Most real assets have positive correlation (0.3 to 0.7), but still provide diversification benefits.
Practical Diversification
How many stocks do you need?
Research shows:
- 1 stock: Very risky
- 10 stocks: Much less risk
- 20-30 stocks: Most diversification achieved
- 100+ stocks: Little additional benefit
The Law of Diminishing Returns:
- First few stocks add lots of diversification
- Additional stocks add progressively less
Practical approach:
- 15-20 different stocks across different industries
- Or: Buy an index fund (instant diversification)
5.5 Systematic vs. Unsystematic Risk
Two Types of Risk
Total Risk = Systematic Risk + Unsystematic Risk
Unsystematic Risk (Firm-Specific Risk)
Also called:
- Firm-specific risk
- Idiosyncratic risk
- Diversifiable risk
- Unique risk
Definition: Risk that affects only one company or small group of companies.
Examples:
- CEO gets caught in scandal
- Factory burns down
- New competitor enters market
- Product liability lawsuit
- Labor strike
- Patent expires
Key Property: Can be eliminated through diversification!
Why? When you hold many stocks:
- One company's bad news is offset by another's good news
- Firm-specific events average out
- Only the market-wide risk remains
Systematic Risk (Market Risk)
Also called:
- Market risk
- Non-diversifiable risk
- Systematic risk
Definition: Risk that affects all (or most) companies simultaneously.
Examples:
- Economic recession
- Interest rate changes
- Inflation
- Political instability
- Natural disasters
- Pandemic
- War
Key Property: Cannot be eliminated through diversification!
Why? These factors affect the entire market. No matter how many stocks you hold, you can't escape market risk.
The Diversification Effect
Graphical Representation:
Total Risk
↑
| _____________ Unsystematic Risk
| / (can be eliminated)
| /
| /
| /_____________________ Systematic Risk
| (cannot be eliminated)
|
└────────────────────→
Number of Stocks
Key Points:
- With 1 stock: High total risk (both types)
- With 20-30 stocks: Unsystematic risk nearly eliminated
- With 100+ stocks: Only systematic risk remains
- Cannot reduce risk below systematic risk level
Implications for Investors
Rational investors:
- Hold diversified portfolios
- Eliminate unsystematic risk (it's free!)
- Only bear systematic risk
Therefore: Investors should only be compensated for systematic risk!
Unsystematic risk:
- Can be eliminated → Investors aren't compensated for it
- Taking unsystematic risk is unwise (taking risk without reward)
Systematic risk:
- Cannot be eliminated → Investors must be compensated
- Risk premium depends on amount of systematic risk
This leads us to beta...
5.6 Beta: Measuring Systematic Risk
What is Beta?
Beta (β) measures an investment's systematic risk relative to the market.
Formula:
β = Covariance(Stock, Market) / Variance(Market)
Or equivalently:
β = ρ(Stock,Market) × σ(Stock) / σ(Market)
But what does it mean?
Interpreting Beta
β = 1.0: Stock moves with the market
- If market goes up 10%, stock goes up 10% (on average)
- Average systematic risk
β > 1.0: Stock is more volatile than market
- If market goes up 10%, stock goes up more than 10%
- Higher systematic risk (aggressive stock)
- Example: Tech stocks, growth stocks
β < 1.0: Stock is less volatile than market
- If market goes up 10%, stock goes up less than 10%
- Lower systematic risk (defensive stock)
- Example: Utility stocks, consumer staples
β = 0: No systematic risk
- Stock doesn't move with market
- Example: Risk-free asset (T-bills)
β < 0: Negative correlation with market (rare)
- When market goes up, stock goes down
- Very unusual
Beta Examples
Typical Betas:
| Company/Asset | Beta | Interpretation |
|---|---|---|
| U.S. Treasury Bills | 0 | No systematic risk |
| Procter & Gamble | 0.5 | Defensive (consumer staples) |
| Walmart | 0.6 | Defensive (retail) |
| Johnson & Johnson | 0.7 | Below-average risk |
| S&P 500 Index | 1.0 | Market average |
| Apple | 1.2 | Above-average risk |
| Tesla | 1.8 | High risk (tech/auto) |
| Small-cap growth funds | 1.3 | More volatile |
| Crypto-related stocks | 2.0+ | Very high risk |
Calculating Beta
Method 1: Regression Analysis
Run a regression of stock returns against market returns:
R_stock = α + β × R_market + ε
The slope coefficient is beta.
Example Data:
| Month | Market Return | Stock Return |
|---|---|---|
| 1 | 2% | 3% |
| 2 | -1% | -2% |
| 3 | 3% | 4.5% |
| 4 | 1% | 1% |
| 5 | -2% | -3% |
In Excel:
=SLOPE(stock_returns, market_returns)
Result: β ≈ 1.5
Interpretation: This stock is 50% more volatile than the market.
Method 2: Using Formula
β = Correlation × (σ_stock / σ_market)
Example:
- Correlation between stock and market: 0.80
- Stock standard deviation: 30%
- Market standard deviation: 20%
β = 0.80 × (30% / 20%)
β = 0.80 × 1.5
β = 1.2
Portfolio Beta
Portfolio beta is the weighted average of individual betas:
β_p = w₁β₁ + w₂β₂ + ... + wₙβₙ
Example:
Portfolio consists of:
- 40% Stock A (β = 1.2)
- 35% Stock B (β = 0.8)
- 25% Stock C (β = 1.5)
β_p = (0.40 × 1.2) + (0.35 × 0.8) + (0.25 × 1.5)
β_p = 0.48 + 0.28 + 0.375
β_p = 1.135
Portfolio beta ≈ 1.14
This portfolio is about 14% more volatile than the market.
Using Beta to Predict Risk
Expected volatility relative to market:
If market moves by X%, stock expected to move by β × X%
Examples:
Stock with β = 1.5:
- Market up 10% → Stock expected up 15%
- Market down 8% → Stock expected down 12%
Stock with β = 0.6:
- Market up 10% → Stock expected up 6%
- Market down 8% → Stock expected down 4.8%
Important: Beta predicts average behavior, not exact movements. Individual stocks can deviate significantly from predictions.
5.7 The Capital Asset Pricing Model (CAPM)
Introduction to CAPM
The Capital Asset Pricing Model is one of the most important models in finance. It tells us what return we should expect on an investment based on its systematic risk (beta).
Developed by: William Sharpe, John Lintner, and Jan Mossin (1960s) Sharpe won Nobel Prize in Economics (1990) for this work
The CAPM Formula
E(R_i) = R_f + β_i[E(R_m) - R_f]
Where:
- E(R_i) = Expected return on investment i
- R_f = Risk-free rate (T-bill rate)
- β_i = Beta of investment i
- E(R_m) = Expected return on the market
- [E(R_m) - R_f] = Market risk premium
In words:
Expected Return = Risk-Free Rate + Beta × Market Risk Premium
CAPM Components
1. Risk-Free Rate (R_f)
- Return on U.S. Treasury bills
- Usually 3-month or 10-year T-bills
- Currently around 4-5% (but varies)
- Represents time value of money with zero risk
2. Market Risk Premium [E(R_m) - R_f]
- Extra return for bearing market risk
- Historically about 7-8% in U.S.
- "What do investors require above risk-free rate?"
3. Beta (β_i)
- Measures stock's systematic risk
- Adjusts market premium for this specific stock
Using CAPM: Examples
Example 1: Average Stock
- Risk-free rate: 4%
- Market return: 12%
- Beta: 1.0
E(R) = 4% + 1.0(12% - 4%)
E(R) = 4% + 1.0(8%)
E(R) = 4% + 8%
E(R) = 12%
Stock with beta of 1.0 should earn the market return (12%).
Example 2: Conservative Stock
- Risk-free rate: 4%
- Market return: 12%
- Beta: 0.6
E(R) = 4% + 0.6(12% - 4%)
E(R) = 4% + 0.6(8%)
E(R) = 4% + 4.8%
E(R) = 8.8%
Less risky stock requires lower return.
Example 3: Aggressive Stock
- Risk-free rate: 4%
- Market return: 12%
- Beta: 1.5
E(R) = 4% + 1.5(12% - 4%)
E(R) = 4% + 1.5(8%)
E(R) = 4% + 12%
E(R) = 16%
Riskier stock requires higher return.
Example 4: Risk-Free Asset
- Beta: 0
E(R) = 4% + 0(8%)
E(R) = 4%
Risk-free asset earns risk-free rate (by definition).
The Security Market Line (SML)
The SML is a graphical representation of CAPM.
Graph:
- X-axis: Beta (systematic risk)
- Y-axis: Expected Return
- SML: Straight line from R_f through market portfolio
Key Points on SML:
- At β = 0: E(R) = R_f (4%)
- At β = 1.0: E(R) = R_m (12%)
- Slope = Market risk premium (8%)
Interpretation:
On the line: Fair expected return for the risk Above the line: Undervalued (earn more than required) Below the line: Overvalued (earn less than required)
Example:
Stock X has β = 1.2 and expected return = 18%
CAPM says it should earn:
E(R) = 4% + 1.2(8%) = 13.6%
But it's expected to earn 18%!
Conclusion: Stock X is undervalued. It provides 18% return when only 13.6% is required.
Investment decision: Buy Stock X!
Using CAPM for Discount Rates
This is where CAPM becomes practical for capital budgeting.
Question: What discount rate should we use for NPV?
Answer: The required return from CAPM!
Example: Project Evaluation
Company considering a project:
- Project beta: 1.3
- Risk-free rate: 4%
- Market return: 11%
Calculate required return:
Required return = 4% + 1.3(11% - 4%)
Required return = 4% + 1.3(7%)
Required return = 4% + 9.1%
Required return = 13.1%
Use 13.1% as discount rate in NPV calculation.
If the project has:
- Positive NPV at 13.1% → Accept
- Negative NPV at 13.1% → Reject
Assumptions of CAPM
CAPM is based on several assumptions:
1. Investors are rational and risk-averse 2. All investors have same holding period 3. Investors can borrow/lend at risk-free rate 4. No taxes or transaction costs 5. All information is free and available to everyone 6. Investors hold diversified portfolios (only care about systematic risk)
Are these realistic? Not entirely. But CAPM still provides useful framework.
Limitations of CAPM
1. Difficult to Measure Market Return
- What is "the market"? S&P 500? Total stock market? World stocks?
- Historical average or forward-looking estimate?
2. Beta is Unstable
- Company beta changes over time
- Varies with time period and frequency of data
- Past beta may not predict future beta
3. Risk-Free Rate Varies
- Which maturity? 3-month? 10-year?
- Risk-free rate changes constantly
4. Assumptions Are Unrealistic
- Investors aren't perfectly rational
- Taxes and transaction costs do exist
- Information isn't free or equally available
5. Other Factors Matter
- Company size
- Value vs. growth
- Momentum
- Other factors beyond beta explain returns
Despite limitations, CAPM is:
- Widely used in practice
- Conceptually sound
- Better than alternatives for many applications
5.8 Practical Applications
Application 1: Cost of Equity
Use CAPM to estimate company's cost of equity:
Example: ABC Corporation
- Beta: 1.15
- Risk-free rate: 4%
- Market return: 11%
Cost of Equity = 4% + 1.15(11% - 4%)
Cost of Equity = 4% + 1.15(7%)
Cost of Equity = 4% + 8.05%
Cost of Equity = 12.05%
ABC's shareholders require 12.05% return.
Use in capital budgeting: Discount equity cash flows at 12.05%.
Use in valuation: Equity is worth the PV of dividends discounted at 12.05%.
Application 2: Project Risk Assessment
Not all projects have same risk as company overall.
Risk Adjustment:
Low-risk projects:
- More predictable cash flows
- Established markets
- Use lower discount rate (β < company β)
High-risk projects:
- Uncertain cash flows
- New markets
- Use higher discount rate (β > company β)
Example:
Company beta: 1.2 Risk-free rate: 4% Market return: 11% Company's cost of capital: 12.4%
Project A (Low-risk maintenance):
- Estimated beta: 0.8
- Required return: 4% + 0.8(7%) = 9.6%
- Discount at 9.6%, not 12.4%
Project B (High-risk new product):
- Estimated beta: 1.6
- Required return: 4% + 1.6(7%) = 15.2%
- Discount at 15.2%, not 12.4%
Using company's overall rate for all projects can lead to:
- Rejecting good low-risk projects (discount rate too high)
- Accepting bad high-risk projects (discount rate too low)
Application 3: Performance Evaluation
Evaluate whether investment manager added value.
Jensen's Alpha:
α = Actual Return - CAPM Expected Return
α = R_actual - [R_f + β(R_m - R_f)]
Positive alpha (α > 0): Manager beat expectations (added value) Negative alpha (α < 0): Manager underperformed (destroyed value)
Example:
Portfolio had:
- Actual return: 14%
- Beta: 1.1
- Risk-free rate: 3%
- Market return: 11%
Expected return:
E(R) = 3% + 1.1(11% - 3%)
E(R) = 3% + 1.1(8%)
E(R) = 11.8%
Alpha:
α = 14% - 11.8% = 2.2%
Conclusion: Portfolio manager added 2.2% value above what was expected given the risk taken.
Application 4: Capital Budgeting Decision
Complete Example:
Company evaluating new product line:
- Initial investment: $5 million
- Expected cash flows: $1.2M/year for 7 years
- Project beta: 1.4
- Risk-free rate: 4%
- Market return: 12%
Step 1: Calculate required return
r = 4% + 1.4(12% - 4%)
r = 4% + 11.2%
r = 15.2%
Step 2: Calculate NPV
NPV = -$5M + $1.2M × [(1 - 1.152^-7) / 0.152]
NPV = -$5M + $1.2M × 4.160
NPV = -$5M + $4.99M
NPV = -$10,000
Decision: Reject. NPV is slightly negative at the risk-adjusted rate.
What if we used 12% (company's average rate)?
NPV = -$5M + $1.2M × 4.564
NPV = +$477,000
We'd incorrectly accept! Risk adjustment matters.
Module 5 Practice Problems
Problem Set 1: Returns and Risk Measures
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Calculate Return: You bought stock for $45. One year later, it trades at $52 and paid a $2 dividend. a. What was your total return? b. What if the stock dropped to $40? What's your return?
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Expected Return: Investment has these possible outcomes:
State Probability Return Boom 20% 30% Growth 50% 15% Recession 30% -5% Calculate expected return.
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Standard Deviation (Historical): Stock returns over 6 years: 12%, -3%, 18%, 9%, -6%, 15%
a. Calculate average return b. Calculate standard deviation
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Standard Deviation (Expected Returns): Using the investment from Problem 2: a. Calculate variance b. Calculate standard deviation
Problem Set 2: Portfolio Analysis
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Portfolio Return: Portfolio consists of:
- 60% Stock A (expected return 14%)
- 40% Stock B (expected return 10%)
Calculate portfolio expected return.
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Portfolio Risk: Two-stock portfolio:
- Stock A: 50% weight, σ = 20%
- Stock B: 50% weight, σ = 30%
- Correlation: 0.4
Calculate portfolio standard deviation.
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Diversification Effect: Same stocks as Problem 6. Calculate portfolio risk if: a. Correlation = 1.0 b. Correlation = 0 c. Correlation = -1.0
What do you observe?
Problem Set 3: Beta
-
Calculate Beta: Stock has correlation of 0.75 with market. Stock standard deviation: 40% Market standard deviation: 20%
Calculate beta.
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Portfolio Beta: Portfolio:
- 30% Stock X (β = 1.5)
- 50% Stock Y (β = 1.0)
- 20% Stock Z (β = 0.6)
Calculate portfolio beta.
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Interpret Beta: Stock has β = 1.8
a. Is this stock more or less risky than market? b. If market goes up 10%, what's expected stock return? c. If market goes down 12%, what's expected stock return?
Problem Set 4: CAPM
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Basic CAPM: Risk-free rate: 3% Market return: 11% Stock beta: 1.3
Calculate expected return.
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Compare Required Returns: Risk-free rate: 4% Market return: 12%
Calculate required return for: a. Stock with β = 0.5 b. Stock with β = 1.0 c. Stock with β = 1.5 d. Stock with β = 2.0
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Over/Under Valued: Risk-free rate: 3% Market return: 10%
Stock A: β = 1.2, expected return = 14% Stock B: β = 0.8, expected return = 8% Stock C: β = 1.5, expected return = 12%
Which stocks are undervalued, fairly valued, or overvalued?
Problem Set 5: Applications
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Cost of Equity: Company beta: 1.25 Risk-free rate: 4% Market risk premium: 7%
Calculate cost of equity.
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Project Evaluation: Project details:
- Investment: $500,000
- Cash flows: $150,000/year for 5 years
- Project beta: 1.4
- Risk-free rate: 3.5%
- Market return: 11%
a. Calculate required return using CAPM b. Calculate NPV c. Accept or reject?
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Risk-Adjusted Discount Rates: Company's overall beta: 1.0 Company's cost of capital: 12% Risk-free rate: 4% Market return: 12%
Evaluate three projects:
Project A (Low risk, β = 0.7):
- Investment: $100,000
- Annual cash flows: $25,000 for 6 years
Project B (Average risk, β = 1.0):
- Investment: $100,000
- Annual cash flows: $30,000 for 5 years
Project C (High risk, β = 1.5):
- Investment: $100,000
- Annual cash flows: $35,000 for 5 years
a. Calculate appropriate discount rate for each b. Calculate NPV for each c. Which projects should be accepted?
Additional Resources
Excel Practice
Download templates for:
- Return and risk calculations
- Portfolio analysis
- Beta estimation
- CAPM calculator
Further Reading
Books:
- "A Random Walk Down Wall Street" by Burton Malkiel
- "The Intelligent Investor" by Benjamin Graham
- "Common Stocks and Uncommon Profits" by Philip Fisher
Online:
- Yahoo Finance (get betas for any stock)
- Morningstar (portfolio analysis tools)
- Damodaran Online (comprehensive data on returns and risk)
Looking Ahead to Module 6
You now understand risk, return, and how they relate. You can measure both and use CAPM to determine required returns.
In Module 6, we'll explore the Cost of Capital—bringing together everything you've learned:
- Cost of debt
- Cost of equity (using CAPM!)
- Weighted Average Cost of Capital (WACC)
- How to use WACC in capital budgeting
- The relationship between capital structure and cost of capital
This is where risk and return meet financing decisions!
Prepare for Module 6 by:
- Reviewing CAPM thoroughly
- Understanding that different sources of capital have different costs
- Recognizing that WACC will be your discount rate for many projects
Summary
Congratulations on completing Module 5! You now understand:
✓ How to measure investment returns ✓ How to measure risk using variance and standard deviation ✓ Expected returns and probability-weighted outcomes ✓ Portfolio theory and the power of diversification ✓ Correlation and its effect on portfolio risk ✓ Systematic vs. unsystematic risk ✓ Beta as a measure of systematic risk ✓ The Capital Asset Pricing Model (CAPM) ✓ The Security Market Line ✓ How to use CAPM to determine required returns
Risk and return are at the heart of finance. Every investment decision involves trading off these two factors. You now have the tools to quantify both and make informed decisions.
The concepts in this module—especially CAPM—will be used repeatedly in the remaining modules. They're the bridge between theory and practice.
Ready for the next step? Proceed to Module 6: Cost of Capital to learn how companies determine their overall cost of financing.
"Risk comes from not knowing what you're doing." — Warren Buffett
"The essence of investment management is the management of risks, not the management of returns." — Benjamin Graham
You now know how to measure and manage risk. That's what separates good investors from lucky ones.
See you in Module 6!