Business Finance - Lecture Notes

## Lecture 1: Time Value of Money

### Compounding

- $F$: Future value
- $P$: Present value
- $I$: Interest rate per period
- $N$: Total number of periods

### Annuity Due

- $F$: Future value
- $P$: Present value
- $I$: Interest rate per period
- $N$: Total number of periods
- $M$: Payment that occurat the end of each period

### Annuity Due

- $F$: Future value
- $P$: Present value
- $F_{ord}$: Future value of ordinary annuity given the same conditions
- $P_{ord}$: Present value of ordinary annuity given the same conditions

### Perpetuity

- $M$: Payment that occurat the end of each period
- $I$: Interest rate per period

### Uneven Cash Flow

- $P$: Present value
- $i$: Period number, from $0$ to $N$
- $C_{i}$: Cash flow at the end of period $i$
- $I_{i}$: Interest rate during period $i$

### Interest Rate

- $I_{n}$: Nominal interest rate
- $I_{p}$: Periodic interest rate
- $I_{e}$: Effective anual rate
- $K$: Number of periods per year

### Periodic Compounding

- $F$: Future value
- $P$: Present value
- $K$: Number of periods per year
- $N$: Number of years

### Loan Amortization

- $A$: Amount of payment per period
- $P$: Initial principle (loan amount)
- $I$: Interest rate per period
- $N$: Total number of periods

## Lecture 2: Bond Evaluation

### Formula

### Types of Bond

Type | Coupon Rate $?$ Yield | Bond Price $?$ Par Value |
---|---|---|

Par Bond | $=$ | $=$ |

Premium Bond | $>$ | $<$ |

Discount Bond | $<$ | $<$ |

### The Total Return Identity

### Risks

#### Investment Risk

Definition | Short Term | Long Term | |
---|---|---|---|

Interest Rate Risk | The concern that the rising interest rate cause the value of a bond to fall | Low | High |

Reinvestment Risk | The concern that the interest rate will fall, and future coupons will have to be reinvested at lower rates | High | Low |

#### Default Risk

- If an issuer defaults, investors receive less than the promised return.
- Affected by the issuer’s financial strength and the terms of the bond contract.
- Bond ratings reflect the probability of a bond issue going into default.

## Lecture 3: Risks & Rate of Return

### Risks

#### Single Investment

For example, given the probability distribution of expected returns for a certain investment:

Scenario | Probability ($p_{i}$) | Rate of Return ($r_{i}$) |
---|---|---|

Recession | 0.1 | -27% |

Below Average | 0.2 | -7% |

Average | 0.4 | 15% |

Above Average | 0.2 | 30% |

Boom | 0.1 | 45% |

The **Expected Return** ($r$) of this investment:

The **Risk** ($\sigma$) is defined as the standard deviation of the distribution of expected returns for a specific investment:

The **Coefficient of Variation** ($c$) is defined as the risk per unit return:

#### Investments in a Portfolio

For example, given the expected return, risk and weight of each investment in the portfolio:

Investment | Rate of Return ($r_{i}$) | Weight ($w_{i}$) |
---|---|---|

1 | $r_1$ | $w_1$ |

2 | $r_2$ | $w_2$ |

$\vdots$ | $\vdots$ | $\vdots$ |

$n$ | $r_n$ | $w_n$ |

The **Expected Return** ($r$) of the portfolio:

The portfolio’s **Risk** ($\sigma$) and **Coefficient of Variation** ($c$) can be calculated by using the same method as above.

#### Diversification

- Diversification benefits exist if stocks are not perfectly positively correlated (i.e. )
- Most stocks are positively (but not perfectly) correlated with the market
- Combining stocks in a portfolio generally lowers risk
- From about 10 stocks and more, converges to 20%
- Standalone risk = diversifiable risk + market risk

### Capital Asset Pricing Model (CAPM)

#### Formula

- $r$: Actual
- $r_f$: Risk free return (typically a 10-year government bond yield)
- $r_m$: Expected market return
- $\beta$: Beta of security
- $(r_m - r_f)$: Equity market premium

#### Conclusions

- The return on an individual stock, or a portfolio of stocks, should equal its cost of capital.
- Equity market premium: the amount that equity investors demand to compensate them for the extra risk they accept.

#### Beta ($\beta$)

##### Definition

A stock’s relative volatility, which shows how much the price of a specific stock jumps up and down compared with how much the stock market as a whole jumps up and down.

##### Calculation

The slope of the regression line of the security’s past returns and the market’s past returns.

E.g.

##### Analysis

$\beta \ ? \ 1$ | Risk of Security $?$ Risk of Average Stock |
---|---|

$=$ | $=$ |

$>$ | $>$ |

$<$ | $<$ |

#### Security Market Line

Given

- $r_f$
- $r_m$
- $\beta$

, calculate Required Rate of Return ($r$) from

Graph:

#### Other Factors

Factor | Example | Changed | Illustration |
---|---|---|---|

Inflation | The investors raised inflation expectations by 3% | Intersection | |

Risk Aversion | The investors’ aversion to risk increased, causing the market risk premium to increase by 3% | Slope |

## Lecture 4: Stocks and Their Valuation

### Discounted Dividend with Constant Growth

Right side:

- $P_0$: Stock price now (or, intrinsic value)
- $D_0$: Divident that has just been announced now
- $g$: Growth rate
- $r_s$: Required rate of return

Left side:

- $P_0$: Current stock price (intrinsic value)
- $P_i$: Stock price (intrinsic value) $i$ years from now
- $y_cd$: Expected divident yield
- $y_c$: Expected capital gains yield
- $r$: Expected total return

### Growth

### Preferred Stock

- Preferred stockholders receive a fixed dividend that must be paid before dividends are paid to common stockholders
- Companies can omit preferred dividend payments without fear of pushing the firm into bankruptcy

### Market Equilibrium

Since

and

The market is in equilibrium when $\widehat{r} = r_s$.

## Lecture 5: The Cost of Capital

### Capital Structure

### Weighted Average Cost of Capital

- $w_i$: The weight of each component
- $r_i$: Cost of each component

### Basic Concepts

- Use the target capital structure (desired optimal mix of equity and debt financing that most firms attempt to maintain) instead of actual financing
- Use weights calculated according to market value instead of book values
- WACC is calculated at a point of time, reflects the marginal cost of raising additional money; the historical cost of existing financing is irrelevant
- Use after-tax capital cost (only $r_d$ needs adjustment)

### Components

#### Long-Term Debt

- $r_d$: Yield-to-maturity on outstanding long-term debt
- $T$: Tax rate

#### Preferred Stock

and

- $D$: Divident
- $P$: Price of stock

#### Common Equity

Average of the following methods:

Method | Formula |
---|---|

Capital Asset Pricing Model | $r_s = r_f + (r_m - r_f) \beta$ |

Discounted Cash Flow | $r_s = \frac{D_0 (1+g)}{P} + g$ |

Bond Yield + Risk Premium | $r_s = r_d + (r_m - r_f)$ |

### Hurdle Rate

- Firms with risker projects generally have a higher WACC
- Projects should be accepted only if their estimated returns exceed their cost of capital
- WACC only represents the hurdle rate for a typical project whose risk is similar to the firm’s average risk

## Lecture 6: The Basics of Capital Budgeting

### Problem Definition

- Given the cost of capital
- Given projects’ cash flows
- Select one or more projects that are worth doing

### Example

### Discounted Payback Period

### Net Present Value

### Internal Rate of Return

Force $NPV = 0$, and solve the equation for $IRR$:

### Modified Internal Rate of Return

- Calculate the future value of all income using cost of capital
- MIRR is the discount between the future value to the current outflow (present value)

### Comments

- IRR method assumes intermediate cash flows are reinvested at IRR
- MIRR method assumes intermediate cash flows are reinvested at cost of capital (more realistic than IRR method)
- NPV is superior to MIRR when evaluating mutually exclusive projects

## Lecture 7: Distribution to Shareholders

### Dividend Preference Theories

#### Dividend Irrelevance

- Investors are indifferent between dividends and retention-generated capital gains.
- Investors can create their own dividend policy
- If they want cash, they can sell stock.
- If they do not want cash, they can use dividends to buy stock.

- Any payout is OK

#### Bird-in-Hand Theory

- Investors may think dividends obtained today are less risky than potential future capital gains, hence investors prefer dividends.
- Set a high payout

#### Tax Issues

- To the extent that dividends have a tax disadvantage relative to capital gains, shareholders prefer capital gains.
- Set a low payout

#### Signalling Hypothesis

#### Clientele Effect

- Certain types of investors are attracted to companies with specific dividend policies
- Changing dividend policy $\rightarrow$ taxes and brokerage costs $\rightarrow$ hurts investors who have to switch companies

### Dividend Reinvestment Plan

#### Overview

#### Open Market Purchase

- Brokerage costs are reduced by volume purchases.
- Convenient, easy way to invest thus useful for investors.

#### New Stock Plan

- Helps conserve cash.
- Companies that need capital use new stock plans.

### Stock Repurchase

#### Overview

- When companies decide to pay out cash instead of retaining it, they can choose to pay cash dividends or buy back their own stock from stockholders.
- Shares outstanding is reduced. The shares bought back are held as treasury stock and can be resold in the future to raise capital.

#### Advantages

- May be viewed as a positive signal that the management thinks stock is undervalued.
- Shareholders can choose to sell or hold.
- Repurchases can be used to dispose off temporary excess cash flows and avoid setting high dividend payout.
- Can be used to make large capital changes.

#### Disadvantages

- May be viewed as a negative signal that the firm has poor investment opportunities.
- Cash dividends are dependable but repurchases are not.
- Firm may have to bid up price to complete purchase, thus paying too much for its own stock.

### Stock Dividends/Splits

#### Examples

- Stock dividends: If a 10% stock dividend is announced, shareholders get 10 shares for each 100 shares owned.
- Stock splits: Assume a company has 100 shares outstanding and each share is trading at 10 dollars dollars. The company announces a 2-for-1 stock split. After the split, the company would have 200 shares outstanding and each share should be worth 5 dollars.

#### Effects on Stock Price

- Increase the number of shares outstanding
- The stock price falls and keeps each investor’s wealth unchanged
- May get us to an “optimal price range” (e.g. 20 - 80 dollars)

#### Effects on Firm’s Value

- Stock splits and stock dividends are viewed as positive signals that the management is confident about future earnings $\rightarrow$ Stock price increases
- By creating more shares and lowering the stock price, stock splits may increase the stock’s liquidity $\rightarrow$ Firm’s value increases