Business Finance - Lecture Notes | Nathaniel's Blog

Lecture 1: Time Value of Money

Compounding

$$F = P ( 1 + I )^{N}$$

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

Annuity Due

$$F = \frac{M}{I} \big[ ( 1+I )^{N} - 1 \big]$$ $$P = \frac{M}{I} \bigg[1 - \frac{1}{ ( 1 + I )^{N}} \bigg]$$

• $F$: 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

$$F = F_{ord} (1+I)$$ $$P = P_{ord} (1+I)$$

• $F$: 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

$$P = \frac{M}{I}$$

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

Uneven Cash Flow

$$P = \sum_{i=0}^N \frac{C_{i}}{(1 + I_{i})^{N}}$$

• $P$: 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

$$I_{p} = \frac{I_{n}}{K}$$ $$I_e = \Big(1 + \frac{I_{n}}{K} \Big) ^{K} - 1$$

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

Periodic Compounding

$$F = P \Big( 1 + \frac{I_{n}}{K} \Big)^{N \cdot K}$$

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

Loan Amortization

$$A = P \frac{I (1+I)^{N}}{(1+I)^{N}-1}$$

• $A$: 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

$$V = \frac{p}{(1 + r)^{n}} + \sum_{i = 1}^{n} \frac{c}{(1+r)^{i}}$$

Types of Bond

Type	Coupon Rate $?$ Yield	Bond Price $?$ Par Value
Par Bond	$=$	$=$
Premium Bond	$>$	$<$
Discount Bond	$<$	$<$

The Total Return Identity

$$ExpectedTotalReturn = YieldToMaturity = CurrentYield + CapitalGainsYield$$ $$CurrentYield = \frac {Coupon} {BondPrice}$$ $$CapitalGainsYield = \frac {BondPriceNextYear - Bond PriceThisYear} {BondPriceThisYear}$$

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

1. If an issuer defaults, investors receive less than the promised return.
2. Affected by the issuer’s financial strength and the terms of the bond contract.
3. 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:

$$r = \sum_{i} r_{i} \cdot p_{i}$$

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

$$\sigma = \sqrt{\sigma_{i} (r_{i} - r)^{2} \cdot p_{i}}$$

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

$$c = \frac{\sigma}{r}$$

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:

$$r = \sum_{i = 1}^{n} w_i \cdot r_i$$

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

Diversification

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

Capital Asset Pricing Model (CAPM)

Formula

$$r = r_f + (r_m - r_f) \beta$$

• $r$: 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

1. The return on an individual stock, or a portfolio of stocks, should equal its cost of capital.
2. 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

$$r = r_f + (r_m - r_f) \beta$$

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

$$P_0 = \frac{D_0 (1+g)}{r_s - g}$$ $$P_i = \frac{D_0 (1+g)^{i+1}}{r_s - g}$$ $$y_d = \frac{D_0 (1+g)}{P}$$ $$y_c = \frac{P_1 - P_0}{P}$$ $$r = y_d + y_c$$

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

$$PayoutRatio = \frac{Dividents}{NetIncome}$$ $$ReturnOnEquity = \frac{NetIncome}{TotalCommonEquity}$$ $$RententionRatio = 1 - PayoutRatio$$ $$Growth = RetentionRatio \ \times \ ReturnOnEquity = \frac{NetIncome - Dividents}{TotalCommonEquity}$$

Preferred Stock

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

Market Equilibrium

Since

$$\widehat{r} = \frac{D_1}{P}$$

and

$$r_s = r_f + (r_m - r_f) \beta$$

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

Lecture 5: The Cost of Capital

Capital Structure

Weighted Average Cost of Capital

$$C = w_d r_d (1-T) + w_p r_p + w_c r_s$$

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

Basic Concepts

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

Components

Long-Term Debt

$$w_d r_d (1-T)$$

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

Preferred Stock

$$w_p r_p$$

and

$$r_p = \frac{D}{P}$$

• $D$: 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

1. Firms with risker projects generally have a higher WACC
2. Projects should be accepted only if their estimated returns exceed their cost of capital
3. 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

1. Given the cost of capital
2. Given projects’ cash flows
3. 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

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

Comments

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

Lecture 7: Distribution to Shareholders

Dividend Preference Theories

Dividend Irrelevance

1. Investors are indifferent between dividends and retention-generated capital gains.
2. 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.
3. Any payout is OK

Bird-in-Hand Theory

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

Tax Issues

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

Signalling Hypothesis

Clientele Effect

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

Dividend Reinvestment Plan

Overview

Open Market Purchase

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

New Stock Plan

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

Stock Repurchase

Overview

1. 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.
2. Shares outstanding is reduced. The shares bought back are held as treasury stock and can be resold in the future to raise capital.

Advantages

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

Disadvantages

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

Stock Dividends/Splits

Examples

1. Stock dividends: If a 10% stock dividend is announced, shareholders get 10 shares for each 100 shares owned.
2. 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

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

Effects on Firm’s Value

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