Unit 4 - Notes

FIN212

Unit 4: Risk & Return and Leverage

Part A: Risk and Return

1. Introduction to Return

Return is the primary motivation for any investment. It represents the reward for undertaking an investment and the compensation for the time value of money and the risk involved.

Components of Total Return

Total return generally consists of two distinct components:

Current Income (Periodic Cash Flow):
- Cash received periodically during the holding period of the asset.
- Examples: Dividends from stocks, interest payments from bonds, rent from real estate.
Capital Appreciation (Capital Gain/Loss):
- The change in the market value of the asset over time.
- Formula: $Selling\ Price - Purchase\ Price$ .
- If the selling price is lower than the purchase price, it is a Capital Loss.

Real vs. Nominal Return

Nominal Return: The return calculated in current monetary terms without adjusting for inflation.
Real Return: The return adjusted for changes in purchasing power (inflation).
- Fisher Effect Approximation: $Real\ Rate \approx Nominal\ Rate - Inflation\ Rate$ .

2. Single Asset Return Calculation

The most common method to calculate the return on a single asset over a specific period is the Holding Period Return (HPR).

Formula

HPR = \frac{(P_1 - P_0) + I}{P_0}

Where:

$P_1$ = Price at the end of the period
$P_0$ = Price at the beginning of the period (Purchase Price)
$I$ = Income received during the period (Dividends or Interest)

Example Calculation

Scenario: An investor buys a share for $100 ( $P_0$ ). After one year, the share price rises to $110 ( $P_1$ ), and the investor receives a dividend of $5 ( $I$ ).
Calculation:
$Return = \frac{(110 - 100) + 5}{100} = \frac{15}{100} = 0.15$
Result: The return is 15%.

3. Introduction to Risk

Risk in financial management is defined as the variability of actual returns from expected returns. It represents the uncertainty associated with the future outcome of an investment.

The Risk-Return Trade-off

There is a direct positive relationship between risk and return:

Low Risk: Associated with lower expected returns (e.g., Government Bonds).
High Risk: Associated with higher expected returns to compensate the investor for the uncertainty (e.g., Equity Shares).

Types of Risk

Systematic Risk (Non-diversifiable / Market Risk):
- Risks that affect the entire market or economy.
- Cannot be eliminated through diversification.
- Examples: Inflation, interest rate changes, war, political instability.
Unsystematic Risk (Diversifiable / Specific Risk):
- Risks unique to a specific company or industry.
- Can be reduced or eliminated through diversification (holding a portfolio of assets).
- Examples: Strikes, management failure, lawsuits, raw material shortage.

4. Single Asset Risk Calculation

When analyzing a single asset, risk is typically measured by the Variance and Standard Deviation of the asset's returns. These statistical measures quantify how spread out the returns are from the mean (average).

Key Concepts

Expected Return ( $\bar{R}$ or $E(R)$ ): The weighted average of possible returns, where weights are the probabilities of occurrence.
Standard Deviation ( $\sigma$ ): The absolute measure of risk. A higher $\sigma$ indicates higher risk (greater volatility).

Step-by-Step Calculation (Probability Method)

Step 1: Calculate Expected Return

\bar{R} = \sum (R_i \times P_i)

Where

R_i

is the return in state

i

and

P_i

is the probability of state

i

Step 2: Calculate Variance ( $\sigma^2$ )
Variance measures the average squared deviation from the mean.

\sigma^2 = \sum [ (R_i - \bar{R})^2 \times P_i ]

Step 3: Calculate Standard Deviation ( $\sigma$ )

\sigma = \sqrt{\text{Variance}}

Coefficient of Variation (CV)

Used to compare risk between assets with different expected returns. It measures risk per unit of return.

CV = \frac{\sigma}{\bar{R}}

Rule: Lower CV is better.

Part B: Leverage

5. Introduction to Leverage

Leverage refers to the use of fixed-cost assets or funds to magnify the returns to the firm's owners. It acts like a physical lever: a small change in one variable (like Sales) results in a large change in another variable (like Profit).

The Income Statement Structure for Leverage

To understand leverage, one must view the income statement in the "Contribution" format:

Sales
Less: Variable Costs
Equals: Contribution
Less: Fixed Operating Costs (Rent, Salaries)
Equals: Operating Profit (EBIT)
Less: Fixed Financial Costs (Interest)
Equals: Earnings Before Tax (EBT)
Less: Tax
Equals: Earnings After Tax (EAT) / Net Income
Divide by # of shares: Earnings Per Share (EPS)

6. Operating Leverage

Operating leverage results from the existence of fixed operating costs in the firm’s income stream. It measures the effect of a change in Sales on Operating Profit (EBIT).

High Operating Leverage: A firm has high fixed costs relative to variable costs. Once the break-even point is reached, profit rises rapidly.
Risk Implication: High operating leverage indicates high Business Risk.

Degree of Operating Leverage (DOL)

The DOL quantifies the responsiveness of EBIT to changes in Sales.

Formula 1 (At a specific level of Sales):

DOL = \frac{\text{Contribution}}{\text{EBIT}}

Note: Contribution = Sales - Variable Costs

Formula 2 (Based on changes):

DOL = \frac{\%\ \text{Change in EBIT}}{\%\ \text{Change in Sales}}

Interpretation:
If DOL = 3, a 1% increase in Sales results in a 3% increase in EBIT.

7. Financial Leverage

Financial leverage results from the presence of fixed financial charges (specifically Interest on Debt and Preference Dividends). It measures the effect of a change in Operating Profit (EBIT) on Earnings Per Share (EPS).

Concept: Trading on Equity. If the firm earns more on its assets (ROI) than the cost of debt, the surplus increases the return to equity shareholders.
Risk Implication: High financial leverage indicates high Financial Risk (risk of default/insolvency).

Degree of Financial Leverage (DFL)

The DFL quantifies the responsiveness of EPS to changes in EBIT.

Formula 1 (At a specific level of EBIT):

DFL = \frac{\text{EBIT}}{\text{EBT}}

(Note: If Preference Dividend ( $PD$ ) exists, the denominator becomes $EBIT - Interest - \frac{PD}{1-TaxRate}$ )

Formula 2 (Based on changes):

DFL = \frac{\%\ \text{Change in EPS}}{\%\ \text{Change in EBIT}}

Interpretation:
If DFL = 2, a 1% increase in EBIT results in a 2% increase in EPS.

8. Combined Leverage

Combined leverage analyzes the total effect of fixed costs (both operating and financial) on the firm. It examines the relationship between the top of the income statement (Sales) and the bottom (EPS).

Concept: It represents the total magnification of shareholder returns due to sales volume changes.
Risk Implication: High combined leverage indicates high Total Risk.

Degree of Combined Leverage (DCL)

The DCL quantifies the responsiveness of EPS to changes in Sales.

Formula 1 (Relationship method):

DCL = DOL \times DFL

Formula 2 (Direct calculation):

DCL = \frac{\text{Contribution}}{\text{EBT}}

Formula 3 (Based on changes):

DCL = \frac{\%\ \text{Change in EPS}}{\%\ \text{Change in Sales}}

Interpretation:
If DCL = 6, a 1% increase in Sales results in a 6% increase in Earnings Per Share (EPS).

Summary of Formulas

Leverage Type	Focus Costs	Relationship Measured	Formula (Ratio)	Risk Type
Operating	Fixed Operating Costs	Sales $\to$ EBIT	$\frac{\text{Contribution}}{\text{EBIT}}$	Business Risk
Financial	Fixed Interest Costs	EBIT $\to$ EPS	$\frac{\text{EBIT}}{\text{EBT}}$	Financial Risk
Combined	Total Fixed Costs	Sales $\to$ EPS	$\frac{\text{Contribution}}{\text{EBT}}$	Total Risk

Unit 4 - Notes

Table of Contents

Unit 4: Risk & Return and Leverage

Part A: Risk and Return

1. Introduction to Return

Components of Total Return

Real vs. Nominal Return

2. Single Asset Return Calculation

Formula

Example Calculation

3. Introduction to Risk

The Risk-Return Trade-off

Types of Risk

4. Single Asset Risk Calculation

Key Concepts

Step-by-Step Calculation (Probability Method)

Coefficient of Variation (CV)

Part B: Leverage

5. Introduction to Leverage

The Income Statement Structure for Leverage

6. Operating Leverage

Degree of Operating Leverage (DOL)

7. Financial Leverage

Degree of Financial Leverage (DFL)

8. Combined Leverage

Degree of Combined Leverage (DCL)

Summary of Formulas