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Ch 12 Capital market history

- Returns
- The Historical Record
- Average Returns: The First Lesson
- The Variability of Returns: The Second Lesson
- More on Average Returns
- Capital Market Efficiency

Percentage Returns

- It is generally more intuitive to think in terms of percentages than in dollar returns
- Dividend yield = income / beginning price
- Capital gains yield = (ending price – beginning price) / beginning price
- Total percentage return = dividend yield + capital gains yield

Example – Calculating Returns

- You bought a stock for $35 and you received dividends of $1.25. The stock is now selling for $40.
- What is your dollar return?
- Dollar return = 1.25 + (40 – 35) = $6.25
- What is your percentage return?
- Dividend yield = 1.25 / 35 = 3.57%
- Capital gains yield = (40 – 35) / 35 = 14.29%
- Total percentage return = 3.57 + 14.29 = 17.86%

Risk Premiums

- The “extra” return earned for taking on risk
- Treasury bills are considered to be risk-free
- The risk premium is the return over and above the risk-free rate

Variance and Standard Deviation

- Variance and standard deviation measure the volatility of asset returns
- The greater the volatility, the greater the uncertainty
- Historical variance = sum of squared deviations from the mean / (number of observations – 1)
- Standard deviation = square root of the variance

Example – Variance and Standard Deviation

Variance = .0045 / (4-1) = .0015 Standard Deviation = .03873

Arithmetic vs. Geometric Mean

- Arithmetic average – return earned in an average period over multiple periods
- Geometric average – average compound return per period over multiple periods
- The geometric average will be less than the arithmetic average unless all the returns are equal
- Which is better?
- The arithmetic average is overly optimistic for long horizons
- The geometric average is overly pessimistic for short horizons
- So the answer depends on the planning period under consideration
- 15 – 20 years or less: use arithmetic
- 20 – 40 years or so: split the difference between them
- 40 + years: use the geometric

Example: Computing Averages

- What is the arithmetic and geometric average for the following returns?
- Year 1 5%
- Year 2 -3%
- Year 3 12%
- Arithmetic average = (5 + (–3) + 12)/3 = 4.67%
- Geometric average = [(1+.05)*(1-.03)*(1+.12)]1/3 – 1 = .0449 = 4.49%

Efficient Capital Markets

- Stock prices are in equilibrium or are “fairly” priced
- If this is true, then you should not be able to earn “abnormal” or “excess” returns
- Efficient markets DO NOT imply that investors cannot earn a positive return in the stock market

What Makes Markets Efficient?

- There are many investors out there doing research
- As new information comes to market, this information is analyzed and trades are made based on this information
- Therefore, prices should reflect all available public information
- If investors stop researching stocks, then the market will not be efficient

Strong Form Efficiency

- Prices reflect all information, including public and private
- If the market is strong form efficient, then investors could not earn abnormal returns regardless of the information they possessed
- Empirical evidence indicates that markets are NOT strong form efficient and that insiders could earn abnormal returns

Semistrong Form Efficiency

- Prices reflect all publicly available information including trading information, annual reports, press releases, etc.
- If the market is semistrong form efficient, then investors cannot earn abnormal returns by trading on public information
- Implies that fundamental analysis will not lead to abnormal returns

Weak Form Efficiency

- Prices reflect all past market information such as price and volume
- If the market is weak form efficient, then investors cannot earn abnormal returns by trading on market information
- Implies that technical analysis will not lead to abnormal returns
- Empirical evidence indicates that markets are generally weak form efficient

Ch 13 Risk, return and the security market line

- Expected Returns and Variances
- Portfolios
- Announcements, Surprises, and Expected Returns
- Risk: Systematic and Unsystematic
- Diversification and Portfolio Risk
- Systematic Risk and Beta
- The Security Market Line
- The SML and the Cost of Capital: A Preview

Expected Returns

- Expected returns are based on the probabilities of possible outcomes
- In this context, “expected” means average if the process is repeated many times
- The “expected” return does not even have to be a possible return

Example: Expected Returns

- Suppose you have predicted the following returns for stocks C and T in three possible states of nature. What are the expected returns?
- State Probability C T
- Boom 0.3 15 25
- Normal 0.5 10 20
- Recession ??? 2 1
- RC = .3(15) + .5(10) + .2(2) = 9.9%
- RT = .3(25) + .5(20) + .2(1) = 17.7%

Variance and Standard Deviation

- Variance and standard deviation still measure the volatility of returns
- Using unequal probabilities for the entire range of possibilities
- Weighted average of squared deviations

Example: Variance and Standard Deviation

- Consider the previous example. What are the variance and standard deviation for each stock?
- Stock C
- 2 = .3(15-9.9)2 + .5(10-9.9)2 + .2(2-9.9)2 = 20.29
- = 4.5
- Stock T
- 2 = .3(25-17.7)2 + .5(20-17.7)2 + .2(1-17.7)2 = 74.41
- = 8.63

Exercise in class

- Consider the following information:
- State Probability ABC, Inc. (%)
- Boom .25 15
- Normal .50 8
- Slowdown .15 4
- Recession .10 -3
- What is the expected return?
- What is the variance?
- What is the standard deviation?

Portfolios

- A portfolio is a collection of assets
- An asset’s risk and return are important in how they affect the risk and return of the portfolio
- The risk-return trade-off for a portfolio is measured by the portfolio expected return and standard deviation, just as with individual assets

Example: Portfolio Weights

- Suppose you have $15,000 to invest and you have purchased securities in the following amounts. What are your portfolio weights in each security?
- $2000 of DCLK
- $3000 of KO
- $4000 of INTC
- $6000 of KEI

- DCLK: 2/15 = .133
- KO: 3/15 = .2
- INTC: 4/15 = .267
- KEI: 6/15 = .4

Portfolio Expected Returns

- The expected return of a portfolio is the weighted average of the expected returns of the respective assets in the portfolio
- You can also find the expected return by finding the portfolio return in each possible state and computing the expected value as we did with individual securities

Example: Expected Portfolio Returns

- Consider the portfolio weights computed previously. If the individual stocks have the following expected returns, what is the expected return for the portfolio?
- DCLK: 19.69%
- KO: 5.25%
- INTC: 16.65%
- KEI: 18.24%
- E(RP) = .133(19.69) + .2(5.25) + .267(16.65) + .4(18.24) = 15.41%

Portfolio Variance

- Compute the portfolio return for each state:RP = w1R1 + w2R2 + … + wmRm
- Compute the expected portfolio return using the same formula as for an individual asset
- Compute the portfolio variance and standard deviation using the same formulas as for an individual asset

Example: Portfolio Variance

- Consider the following information
- Invest 50% of your money in Asset A
- State Probability A B
- Boom .4 30% -5%
- Bust .6 -10% 25%
- What are the expected return and standard deviation for each asset?
- What are the expected return and standard deviation for the portfolio?

Portfolio

12.5%

7.5%

Expected versus Unexpected Returns

- Realized returns are generally not equal to expected returns
- There is the expected component and the unexpected component
- At any point in time, the unexpected return can be either positive or negative
- Over time, the average of the unexpected component is zero

Systematic Risk

- Risk factors that affect a large number of assets
- Also known as non-diversifiable risk or market risk
- Includes such things as changes in GDP, inflation, interest rates, etc.

Unsystematic Risk

- Risk factors that affect a limited number of assets
- Also known as unique risk and asset-specific risk
- Includes such things as labor strikes, part shortages, etc.

Returns

- Total Return = expected return + unexpected return
- Unexpected return = systematic portion + unsystematic portion
- Therefore, total return can be expressed as follows:
- Total Return = expected return + systematic portion + unsystematic portion

Diversification

- Portfolio diversification is the investment in several different asset classes or sectors
- Diversification is not just holding a lot of assets
- For example, if you own 50 Internet stocks, you are not diversified
- However, if you own 50 stocks that span 20 different industries, then you are diversified

The Principle of Diversification

- Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns
- This reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another
- However, there is a minimum level of risk that cannot be diversified away and that is the systematic portion

Diversifiable Risk

- The risk that can be eliminated by combining assets into a portfolio
- Often considered the same as unsystematic, unique or asset-specific risk
- If we hold only one asset, or assets in the same industry, then we are exposing ourselves to risk that we could diversify away

Total Risk

- Total risk = systematic risk + unsystematic risk
- The standard deviation of returns is a measure of total risk
- For well-diversified portfolios, unsystematic risk is very small
- Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk

Systematic Risk Principle

- There is a reward for bearing risk
- There is not a reward for bearing risk unnecessarily
- The expected return on a risky asset depends only on that asset’s systematic risk since unsystematic risk can be diversified away

Measuring Systematic Risk

- How do we measure systematic risk?
- We use the beta coefficient to measure systematic risk
- What does beta tell us?
- A beta of 1 implies the asset has the same systematic risk as the overall market
- A beta < 1 implies the asset has less systematic risk than the overall market
- A beta > 1 implies the asset has more systematic risk than the overall market

Total versus Systematic Risk

- Consider the following information:

Standard Deviation Beta

- Security C 20% 1.25
- Security K 30% 0.95
- Which security has more total risk?
- Which security has more systematic risk?
- Which security should have the higher expected return?

Beta and the Risk Premium

- Remember that the risk premium = expected return – risk-free rate
- The higher the beta, the greater the risk premium should be
- Can we define the relationship between the risk premium and beta so that we can estimate the expected return?
- YES!

Reward-to-Risk Ratio: Definition and Example

- The reward-to-risk ratio is the slope of the line illustrated in the previous example
- Slope = (E(RA) – Rf) / (A – 0)
- Reward-to-risk ratio for previous example = (20 – 8) / (1.6 – 0) = 7.5
- What if an asset has a reward-to-risk ratio of 8 (implying that the asset plots above the line)?
- What if an asset has a reward-to-risk ratio of 7 (implying that the asset plots below the line)?

Market Equilibrium

- In equilibrium, all assets and portfolios must have the same reward-to-risk ratio and they all must equal the reward-to-risk ratio for the market

Security Market Line

- The security market line (SML) is the representation of market equilibrium
- The slope of the SML is the reward-to-risk ratio: (E(RM) – Rf) / M
- But since the beta for the market is ALWAYS equal to one, the slope can be rewritten
- Slope = E(RM) – Rf = market risk premium

The Capital Asset Pricing Model (CAPM)

- The capital asset pricing model defines the relationship between risk and return
- E(RA) = Rf + A(E(RM) – Rf)
- If we know an asset’s systematic risk, we can use the CAPM to determine its expected return
- This is true whether we are talking about financial assets or physical assets

Example - CAPM

- Consider the betas for each of the assets given earlier. If the risk-free rate is 2.13% and the market risk premium is 8.6%, what is the expected return for each?

Ch 15 The cost of capital

- The Cost of Capital: Some Preliminaries
- The Cost of Equity
- The Costs of Debt and Preferred Stock
- The Weighted Average Cost of Capital
- Divisional and Project Costs of Capital
- Flotation Costs and the Weighted Average Cost of Capital

Cost of Equity

- The cost of equity is the return required by equity investors given the risk of the cash flows from the firm
- Business risk
- Financial risk
- There are two major methods for determining the cost of equity
- Dividend growth model
- SML or CAPM

The Dividend Growth Model Approach

- Start with the dividend growth model formula and rearrange to solve for RE

Dividend Growth Model Example

- Suppose that your company is expected to pay a dividend of $1.50 per share next year. There has been a steady growth in dividends of 5.1% per year and the market expects that to continue. The current price is $25. What is the cost of equity?

Example: Estimating the Dividend Growth Rate

- One method for estimating the growth rate is to use the historical average
- Year Dividend Percent Change
- 2002 1.23 -
- 2003 1.30
- 2004 1.36
- 2005 1.43
- 2006 1.50

(1.30 – 1.23) / 1.23 = 5.7%

(1.36 – 1.30) / 1.30 = 4.6%

(1.43 – 1.36) / 1.36 = 5.1%

(1.50 – 1.43) / 1.43 = 4.9%

Average = (5.7 + 4.6 + 5.1 + 4.9) / 4 = 5.1%

The SML Approach

- Use the following information to compute our cost of equity
- Risk-free rate, Rf
- Market risk premium, E(RM) – Rf
- Systematic risk of asset,

Example - SML

- Suppose your company has an equity beta of .58 and the current risk-free rate is 6.1%. If the expected market risk premium is 8.6%, what is your cost of equity capital?
- RE = 6.1 + .58(8.6) = 11.1%
- Since we came up with similar numbers using both the dividend growth model and the SML approach, we should feel pretty good about our estimate

Example – Cost of Equity

- Suppose our company has a beta of 1.5. The market risk premium is expected to be 9% and the current risk-free rate is 6%. We have used analysts’ estimates to determine that the market believes our dividends will grow at 6% per year and our last dividend was $2. Our stock is currently selling for $15.65. What is our cost of equity?
- Using SML: RE = 6% + 1.5(9%) = 19.5%
- Using DGM: RE = [2(1.06) / 15.65] + .06 = 19.55%

Cost of Debt

- The cost of debt is the required return on our company’s debt
- We usually focus on the cost of long-term debt or bonds
- The required return is best estimated by computing the yield-to-maturity on the existing debt
- We may also use estimates of current rates based on the bond rating we expect when we issue new debt
- The cost of debt is NOT the coupon rate

Example: Cost of Debt

- Suppose we have a bond issue currently outstanding that has 25 years left to maturity. The coupon rate is 9% and coupons are paid semiannually. The bond is currently selling for $908.72 per $1,000 bond. What is the cost of debt?
- N = 50; PMT = 45; FV = 1000; PV = -908.72; CPT I/Y = 5%; YTM = 5(2) = 10%

The Weighted Average Cost of Capital

- We can use the individual costs of capital that we have computed to get our “average” cost of capital for the firm.
- This “average” is the required return on the firm’s assets, based on the market’s perception of the risk of those assets
- The weights are determined by how much of each type of financing is used

Capital Structure Weights

- Notation
- E = market value of equity = # of outstanding shares times price per share
- D = market value of debt = # of outstanding bonds times bond price
- V = market value of the firm = D + E
- Weights
- wE = E/V = percent financed with equity
- wD = D/V = percent financed with debt

Example: Capital Structure Weights

- Suppose you have a market value of equity equal to $500 million and a market value of debt = $475 million.
- What are the capital structure weights?
- V = 500 million + 475 million = 975 million
- wE = E/V = 500 / 975 = .5128 = 51.28%
- wD = D/V = 475 / 975 = .4872 = 48.72%

Taxes and the WACC

- We are concerned with after-tax cash flows, so we also need to consider the effect of taxes on the various costs of capital
- Interest expense reduces our tax liability
- This reduction in taxes reduces our cost of debt
- After-tax cost of debt = RD(1-TC)
- Dividends are not tax deductible, so there is no tax impact on the cost of equity
- WACC = wERE + wDRD(1-TC)

Equity Information

50 million shares

$80 per share

Beta = 1.15

Market risk premium = 9%

Risk-free rate = 5%

Debt Information

$1 billion in outstanding debt (face value)

Current quote = 110

Coupon rate = 9%, semiannual coupons

15 years to maturity

Tax rate = 40%

Extended Example – WACC - IExtended Example – WACC - II

- What is the cost of equity?
- RE = 5 + 1.15(9) = 15.35%
- What is the cost of debt?
- N = 30; PV = -1,100; PMT = 45; FV = 1,000; CPT I/Y = 3.9268
- RD = 3.927(2) = 7.854%
- What is the after-tax cost of debt?
- RD(1-TC) = 7.854(1-.4) = 4.712%

Extended Example – WACC - III

- What are the capital structure weights?
- E = 50 million (80) = 4 billion
- D = 1 billion (1.10) = 1.1 billion
- V = 4 + 1.1 = 5.1 billion
- wE = E/V = 4 / 5.1 = .7843
- wD = D/V = 1.1 / 5.1 = .2157
- What is the WACC?
- WACC = .7843(15.35%) + .2157(4.712%) = 13.06%

Ch 17 Capital structure

- The Capital Structure Question
- The Effect of Financial Leverage
- Capital Structure and the Cost of Equity Capital
- M&M Propositions I and II with Corporate Taxes
- Bankruptcy Costs
- Optimal Capital Structure
- The Pie Again
- Observed Capital Structures
- A Quick Look at the Bankruptcy Process

Capital Restructuring

- We are going to look at how changes in capital structure affect the value of the firm, all else equal
- Capital restructuring involves changing the amount of leverage a firm has without changing the firm’s assets
- The firm can increase leverage by issuing debt and repurchasing outstanding shares
- The firm can decrease leverage by issuing new shares and retiring outstanding debt

Capital Structure Theory

- Modigliani and Miller Theory of Capital Structure
- Proposition I – firm value
- Proposition II – WACC
- The value of the firm is determined by the cash flows to the firm and the risk of the assets
- Changing firm value
- Change the risk of the cash flows
- Change the cash flows

Capital Structure Theory Under Three Special Cases

- Case I – Assumptions
- No corporate or personal taxes
- No bankruptcy costs
- Case II – Assumptions
- Corporate taxes, but no personal taxes
- No bankruptcy costs
- Case III – Assumptions
- Corporate taxes, but no personal taxes
- Bankruptcy costs

Case I – Propositions I and II

- Proposition I
- The value of the firm is NOT affected by changes in the capital structure
- The cash flows of the firm do not change; therefore, value doesn’t change
- Proposition II
- The WACC of the firm is NOT affected by capital structure

Case I - Equations

- WACC = RA = (E/V)RE + (D/V)RD
- RE = RA + (RA – RD)(D/E)
- RA is the “cost” of the firm’s business risk, i.e., the risk of the firm’s assets
- (RA – RD)(D/E) is the “cost” of the firm’s financial risk, i.e., the additional return required by stockholders to compensate for the risk of leverage

Case I - Example

- Data
- Required return on assets = 16%, cost of debt = 10%; percent of debt = 45%
- What is the cost of equity?
- RE = 16 + (16 - 10)(.45/.55) = 20.91%
- Suppose instead that the cost of equity is 25%, what is the debt-to-equity ratio?
- 25 = 16 + (16 - 10)(D/E)
- D/E = (25 - 16) / (16 - 10) = 1.5
- Based on this information, what is the percent of equity in the firm?
- E/V = 1 / 2.5 = 40%

The CAPM, the SML and Proposition II

- How does financial leverage affect systematic risk?
- CAPM: RA = Rf + A(RM – Rf)
- Where A is the firm’s asset beta and measures the systematic risk of the firm’s assets
- Proposition II
- Replace RA with the CAPM and assume that the debt is riskless (RD = Rf)
- RE = Rf + A(1+D/E)(RM – Rf)

Business Risk and Financial Risk

- RE = Rf + A(1+D/E)(RM – Rf)
- CAPM: RE = Rf + E(RM – Rf)
- E = A(1 + D/E)
- Therefore, the systematic risk of the stock depends on:
- Systematic risk of the assets, A, (Business risk)
- Level of leverage, D/E, (Financial risk)

Case II – Cash Flow

- Interest is tax deductible
- Therefore, when a firm adds debt, it reduces taxes, all else equal
- The reduction in taxes increases the cash flow of the firm
- How should an increase in cash flows affect the value of the firm?

Interest Tax Shield

- Annual interest tax shield
- Tax rate times interest payment
- 6,250 in 8% debt = 500 in interest expense
- Annual tax shield = .34(500) = 170
- Present value of annual interest tax shield
- Assume perpetual debt for simplicity
- PV = 170 / .08 = 2,125
- PV = D(RD)(TC) / RD = DTC = 6,250(.34) = 2,125

Case II – Proposition I

- The value of the firm increases by the present value of the annual interest tax shield
- Value of a levered firm = value of an unlevered firm + PV of interest tax shield
- Value of equity = Value of the firm – Value of debt
- Assuming perpetual cash flows
- VU = EBIT(1-T) / RU
- VL = VU + DTC

Example: Case II – Proposition I

- Data
- EBIT = 25 million; Tax rate = 35%; Debt = $75 million; Cost of debt = 9%; Unlevered cost of capital = 12%
- VU = 25(1-.35) / .12 = $135.42 million
- VL = 135.42 + 75(.35) = $161.67 million
- E = 161.67 – 75 = $86.67 million

Case II – Proposition II

- The WACC decreases as D/E increases because of the government subsidy on interest payments
- RA = (E/V)RE + (D/V)(RD)(1-TC)
- RE = RU + (RU – RD)(D/E)(1-TC)
- Example
- RE = 12 + (12-9)(75/86.67)(1-.35) = 13.69%
- RA = (86.67/161.67)(13.69) + (75/161.67)(9)(1-.35)RA = 10.05%

Example: Case II – Proposition II

- Suppose that the firm changes its capital structure so that the debt-to-equity ratio becomes 1.
- What will happen to the cost of equity under the new capital structure?
- RE = 12 + (12 - 9)(1)(1-.35) = 13.95%
- What will happen to the weighted average cost of capital?
- RA = .5(13.95) + .5(9)(1-.35) = 9.9%

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