FIN 301
Introduction to Business Finance
Resources:
 CFI: Corporate Finance Institute (YouTube)
 Finance and capital markets (Khan Academy)
 AppliedFinanceModels
Class Textbook: Fundamentals of Corporate Finance, 13th edition
Topics:
 (CH 1) Introduction to Corporate Finance
 (CH 2) Financial statements, taxes and cash flow
 (CH 3) Working with financial statements
 (CH 5) Time value of money
 (CH 6) Discounted cash flow valuation
 (CH 7) Interest rates and bond valuation
 (CH 8) Stock valuation
 (CH 12) Capital market history
 (CH 12) Risk and return
 (CH 9) NPV and IRR
 (CH 10) Capital budgeting
 (CH 14) Cost of capital
 (CH 16) Capital structure
Multiple Cash Flows
Variables:
 L1: investments
 L2: accrued years
 R: rate (potentially L3, if it changes between years)
Equations:
 FV = PV * (1+rate)^year
sum( seq( L1(I) * (1+R) ^ (I1), I, 1, dim(L1) ) )
 PV = FV / (1+rate)^year
sum( seq( L1(I) / (1+R) ^ (I1), I, 1, dim(L1) ) )
Annuities
Variables:
 Present Value Annuity (PV Annuity)
 Dollars in Period
 Rate
 Periods
Equations:
 presentValueAnnuity = dollarsInPeriod * ((1 (1 / ((1+ rate) ^ periods))) / rate)
 dollarsInPeriod = presentValueAnnuity / ((1 (1 / ((1+ rate) ^ periods))) / rate)
Chapter 1: Introduction to Corporate Finance
Finance is making decisions to maximize value to owners.
Value is driven by cash flows, not income.
Key equations
Chapter 2: Financial statements, taxes and cash flow
 \[assets = {liabilities + shareholdersEquity}\]
 \[income = revenues  expenses\]

\[cashFlowFromAssets = cashFlowToCreditors + cashFlowToStockholders\]
 \[cashFlowFromAssets = operatingCashFlow (OCF)  netCapitalSpending  changeInNetWorkingCapital (NWC)\]
 \[operatingCashFlow = EBIT + depreciation  taxes\]
 \[netCapitalSpending = endingNetFixedAssets  beginningNetFixedAssets + depreciation\]
 \[changeInNetWorkingCapital = endingNWC  beginningNWC\]
 \[cashFlowToCreditors = interestPaid  netNewBorrowing\]
 \[cashFlowToStockholders = dividendsPaid  netNewEquityRaised\]
 The ratio of net working capital to total assets:
 \[netWorkingCapitalToTotalAssets = {netWorkingCapital \over{totalAssets}}\]
 The interval measure:
 \[intervalMeasure = {currentAssets \over{averageDailyOperatingCosts}}\]
 The total debt ratio:
 \[totalDebtRatio = {totalAssets  totalEquity \over{totalAssets}}\]
 The debtequity ratio:
 \[debtEquityRatio = {totalDebt \over{totalEquity}}\]
 The equity multiplier:
 \[equityMultiplier = {totalAssets \over{equityMultiplier}}\]
 The longterm debt ratio:
 \[longTermDebtRatio = {longTermDebt \over{longTermDebt + totalEquity}}\]
 The times interest earned (TIE) ratio:
 \[timesInterestEarnedRatio = {EBIT \over{interest}}\]
 The cash coverage ratio:
 \[cashCoverageRatio = {EBIT + deprecation \over{interest}}\]
Chapter 7: Interest rates and bond valuation
Bond valuation formula
 rate (r) = Coupon Rate (annual or semiannual)
 time (t) = periods to maturity
 C = dollarCouponPayment (annual or semiannual)
Bond valuation
 Equity ≈ ownership
 Bond ≈ loan, with legal claim to future cashflows
Bond features
 Bond: loan, debt
 Coupons: promised interest payments
 Face Value: promised repayment amount (par value, maturity value)
 Coupon Rate: \({annualDollarCoupon \over{faceValue}}\)
 Maturity: years until face value is paid
Bond values and yields
 Cashflows of a bond: coupons and faceValue
 To find market price: discount cashflows at market discount rate
 Cashflows of a bond are fixed
 Market interest rates change ▶ discount rate changes ▶ price of bond changes (PV)
 Two cashflow components to a bond
 Annuity (level coupons)
 Lump sum (faceValue)
 YieldToMaturity (YTM): required market rate on a bond
 Bond “discount rate,” or “yield”
Bond example
A corporation issues a bond with 10 years to maturity. The bond has an annual coupon of $80, and a face value of $1,000. Similar bonds in the market have a YTM of 8%. What is the price of the bond? What is the coupon rate?
Steps:
 Draw cashflow timeline
 Calculate present value (PV) of coupon payments (using annuity formula)
 Calculate present value (PV) of face value payment
 Calculate the total bond price
 Calculate the coupon rate
Step 2: Calculate present value (PV) of coupon payments with annuity formula:
\[PVA = {coupon \times { {1  {1 \over{(1+r)^t}}} \over{r}}}\] \[$536.81 = {80 \times { {1  {1 \over{(1.08)^{10}}}} \over{.08}}}\] PVA =
coupon*( (1  (1/(1+r)^t)) / r)
 536.81 =
80*( (1  (1/(1+.08)^10)) / .08)
Step 3: Calculate present value (PV) of face value payment
\[presentValue = {facevalue \over{(1+r)^t}}\] \[463.19 = {$1000 \over{(1.08)^{10} }}\] presentValue =
facevalue / (1+r)^t
 $463.19 =
1000 / (1+.08)^10
Step 4: Calculate the total bond price
\[price = {$536.81 + $463.19} = $1000\]Step 5: Calculate the coupon rate
\[couponRate = {annualDollarCoupon \over{faceValue}}\] \[.08 = {$80 \over{$1000}} = 8\%\]As couponRate = yieldToMaturity (YTM), this is a “Par Bond,” implying the price of the bond = faceValue—a unique case.
Now, a year passes in our above example and the company now has nine years until maturity and the market interest rate has risen by 10%. What is the new bond price? What is the relationship between couponRate, yieldToMaturity, and price?
 time (t) = 9
 rate (r) = .10
Step 2: Calculate present value (PV) of coupon payments with annuity formula:
\[PVA = {coupon \times { {1  {1 \over{(1+r)^t}}} \over{r}}}\] \[460.72 = {80 \times { {1  {1 \over{(1.10)^9}}} \over{.10}}}\] PVA =
coupon*( (1  (1/(1+r)^t)) / r)
 $460.72 =
80*( (1  (1/(1+.10)^9)) / .10)
Step 3: Calculate present value (PV) of face value payment
\[presentValue = {facevalue \over{(1+r)^t}}\] \[424.10 = {1000 \over{(1.10)^9}}\]Step 4: Calculate the total bond price
\[price = {couponPV + faceValuePV}\] \[price = {$460.72 + $424.10} = $884.85\]The coupon rate has not changed; however, yieldToMaturity has.
As couponRate (8%) < yieldToMaturity (YTM) (10%), this is a “Discount Bond,” a bond that sells for less than its face value.
What if the market interest rate had fallen to 6% instead of rising to 10%?
\[price = {couponPV + faceValuePV}\] \[price = {\left( 80 \times { {1  {1 \over{(1.06)^9}}} \over{.06}}\right)} + {1000 \over{(1.06)^9}}\] \[1136.04 = {544.14 + 591.90}\]As couponRate (8%) > yieldToMaturity (YTM) (6%), this is a “Premium bond,” a bond that sells for more than its face value.
Bond values
 Par Value Bond: a bond that sells for its face value
 Discount Bond: a bond that sells for less than its face value
 Premium Bond: a bond that sells for more than its face value
 Coupon rate: \({annualDollarCoupon \over{faceValue}}\)
YTM vs. Price
As yieldToMaturity (YTM) increases, bond price decreases—an inverse relation.
Interest Rate vs. Bond Value (Interest rate risk)
 Longer maturity time means greater interest rate risk.
 As maturity increases the change in interest rate is magnified.
 Like the effect of lump sum payments as time increases
 Lower the coupon rate means greater interest rate risk.
 A bond with a lower coupon rate has more of its value built into lump sum payment at maturity.
 As the coupon rate and present value (PV) of lump sum changes this effect is larger on the bond value for a lower coupon bond.
Solving for yieldToMaturity (YTM)
Calculate the YTM for a 6 year, 8% coupon bond (annual payments), price = $955.14, faceValue = $1,000.
 couponRate = \({annualDollarCoupon \over{faceValue}}\)
 8% = \({annualDollarCoupon \over{1000}}\) , annualDollarCoupon = 80
Bond values: Semiannual Coupons
 Semiannual bonds make coupon payments twice a year
 Adjustments
 Halve annual coupon payment
 Halve quoted yieldToMaturity (YTM)
 Double period years (t)
For example, what is the price of a bond with an 11% coupon rate that makes semiannual payments for 20 years at a YTM of 13%? How does the price compare to a bond that makes annual coupon payments?
 couponRate = \({annualDollarCoupon \over{faceValue}}\)

11% = \({annualDollarCoupon \over{1000}}\) , annualDollarCoupon = 110
 Semiannual Adjustments
 Halve annual coupon payment, Semiannual coupon payment = \({110 \over{2}}\) = $55
 Halve quoted yieldToMaturity (YTM) (r), \({.13 \over{2}}\) = $.065
 Double period years (t), 20 * 2 = 40
If we had annual payments:
\[price = {\left( $110 \times { {1  {1 \over{(1.13)^{20}}}} \over{.13}}\right)} + {1000 \over{(1.13)^{20}}} = $859.51\]effectiveAnnualRate (EAR) = \({1 + {yieldToMaturity \over{2}}^{2}}\)
effectiveAnnualRate (EAR) = \({1 + {.13 \over{2}}^{2}}\) = .1342 (13.42%)
effectiveAnnualRate (EAR) 13.42% > yieldToMaturity (YTM) 13%, anytime we have compounding frequency greater than once a year, the EAR > YTM.
Bond features
 Characteristics of debt securities
 Creditors (lenders) generally have no voting rights
 Payment of interest on debt is a tax deductible business expense
 Unpaid debt is a liability, subjects the firm to legal action if they default.
As equity is ownership, equity holders are paid after debt holders.
 Shortterm debt: under 1 year to maturity
 Longterm debt: more than 1 year to maturity
The legal written agreement between borrower and lender is a bond indenture.
 Registered form: ownership is recorded, payment direction to owner of record.
 Bearer form: payment made to holder, the bearer, of bond—no record of owner.
Security, debt classified by collateral:
 Mortgage securities: bonds are backs by mortgage on real property.
 Debenture: unsecured debt with no collateral with greater than 10 years maturity.
 Note: unsecured debt with less than 10 years maturity.
Seniority, order of precedence claims:  Senior vs subordinated debt
Repayment, early repayment is typical:  Sinking fund, early redemption
Call provision, allows issue to call or repurchase part or all of the issue:  Callable in 10 years (10 years of call protection)
Protective covenants, limiting actions of the firm  Negative covenant — “shall not…” (e.g., merger)  Positive covenant — “shall…” (e.g., maintain collateral)
Government bonds:
 Issued by U.S. Treasury
 No default risk
 Treasury bills are bonds less than one year to maturity, “risk free rate”
 Treasury notes are bonds with 210 year maturities
 Treasury bonds are bonds with up to 30 year maturities
 Interest exempt from state taxes
Corporate bonds:
 Issued by corporations
 Have default risk, or credit risk
 Callable and convertible
 Secure and unsecured (“debentures”)
Municipal bonds — “Munis”:
 Issued by state and local governments
 School districts, hospitals, universities
 Varying degrees of default risk
 Interest income (coupons) are exempt from federal taxes, and often state and local taxes
 Attractive to investors in high tax brackets
 Because of tax exemption, yields are lower than comparable taxable bonds
 Need to make a conversion to compare taxEquivalentYield (TEY)
 \[muniTaxEquivalentYield = {taxFreeYieldToMaturity \over{1  taxRate}}\]
Example:
 AA corporate bond yield = 6.6%
 AA municipal bond yield = 4.8%
 In a 35% tax bracket, the AA municipal bond is a better investment as the muniTaxEquivalentYield = \({4.8 \over{1  .35}}\) = 7.38%
 In a 22% tax bracket, the AA corporate bond is a better investment as the municipal bond muniTaxEquivalentYield = \({4.8 \over{1  .22}}\) = 6.15%
Zero coupon bonds:  No coupon payments  Deep discount bonds are bonds with a price much lower than faceValue  Longterm zero coupon bonds have very high interest rate risk  Generally, lower coupons and high maturity = more interest rate risk
Example:
 Zero coupon bond with $1,000 faceValue, 12% YTM, and 5 years to maturity. What is the price of this bond?
A deep discount bond.
Bond ratings
Bond ratings assess the creditworthiness of the issuer
 Assesses the issuers ability to make principal and interest payments
 Bond ratings, or credit ratings, are only concerned with default risk
 Not a measure of interest rate risk
 One could have a highly rated bond with significant interest rate risk
Bond markets
Bond trading volume is significantly higher than stock trading volume; U.S. Treasury bonds are one of the highest volume security markets.
 Many more bond issues than stock issues
 A company can have multiple bond issues
 Federal, state, and local borrowing is significant
Most bonds are traded overthecounter:
 Not traded on listed exchange
 Bonds offer less price transparency compared to stocks
 Bond prices are not easily observable
Treasury bond pricing
Treasury bond prices are quoted as percentage of faceValue
 Example: a quote is 103.22 and faceValue is $1,000, the bond price is $1,032.20

Treasury bonds pay semiannual coupons
 Bid price: price paid for a security
 Ask price: selling price for a security
 Bigask spread: difference between bid and ask; represents dealer’s profit on a roundtrip trade
Inflation vs. Interest Rates
 Nominal rates are rates that have not been adjusted for inflation
 Real rates have been
Example, an investment is bought at $100 and sold a year later for $115.50. Inflation over the year was 5% what was the nominal rate of return? What was the real rate of return?
Nominal rate of return = \({endingValue \over{initialValue}}  1\)
Nominal rate of return = \({115.50 \over{100}}  1 = .155\) (15.5%)
Fisher effect:
\[{(1 + nominalRateOfReturn) = (1 + realRateOfReturn) \times{(1 + inflationRate)}}\] \[{(1 + .115) = (1 + realRateOfReturn) \times{(1 + .05)}}\]realRateOfReturn = .10 (10%)
The Fisher effect describes the relationship between nominal return, real returns, and inflation.
Term structure of interest rates
Term structure is the relationship between shortterm and longterm interest rates.
 Represents the pure time value of money
 Nominal rates on default fee securities
 Upward sloping term structure: longterm rates > shortterm rates
 Downward sloping term structure: longterm rates < shortterm rates
Components of term structure:
 Real rate of interest
 Inflation premium; compensation for expected future inflation
 Interest rate risk premium; compensation for bearing interest rate risk; increases with maturity
Determinants of bond yields
Bond yields reflect several effects:
 Real rate of interest
 Premiums reflect compensation for:
 Expected future inflation
 Interest rate risk
 Default risk
 Taxability
 Liquidity
Chapter 8: Stock valuation
A stock is an equity instrument; a stock issuer is issuing shares of ownership, who purchase for an expected rate of return—stock owners have a claim on the future residual cash flows of the issuer.
Valuing a stock is more difficult relative to a bond as:
 Cash flows are uncertain
 Stocks have infinite life (no maturity—no end date of final cashflow)
 No easily observable required rate of return
Fundamental theory of evaluation: the market value of a financial asset equals the presentValue (PV) of its future cash flows.
Cash flows to a common stock share holder:
 Dividends
 Future sale price
Example: You want to purchase a stock, the firm will pay $2 dividend in one year and you believe you can sell the stock in one year for $88. If you require a 10% rate of return, what price should you pay today?
 presentValue = \({futureValue \over{ (1+rate)^{year}}}\)
 presentValue = \({ ($2 + $88) \over{ (1+.10)^{1}}}\) = $81.82
If the market price of the stock today is $85, would you buy the stock? Based on our evaluation, the stock is overpriced and we will not earn the 10% required rate of return.
If we were to sell it in 2 years and receive 2 cash dividends:
 presentValue = \({cashDividend \over{ (1+rate)^{year}}} + {cashDividend \over{ (1+rate)^{year}}} + {priceInSellPeriod \over{ (1+rate)^{year}} }\)
 presentValue = \({$2 \over{ (1+.10)^{1}}} + {$2 \over{ (1+.10)^{2}}} + {$88 \over{ (1+.10)^{2}} }\)
Side note: Many stocks do not pay dividends. A stock that currently pays no dividends can be valued due to the expectation of future dividends.
Stock valuation: 3 special cases
To assign value to a stock we make assumptions to simplify future dividends:
 Zero growth; dividends do not change
 Constant growth; dividends grow at constant rate
 Nonconstant growth; shortterm fast growth then subsides to constant growth
Zero growth
 presentValue = \({dividend \over{rateOfReturn}}\)
Example: A corporation common stock pays a $1 constant dividend with a required rateOfReturn of 10%. What is the price of the stock today? What is the price of the stock in one year?
 presentValue = \({dividend \over{rateOfReturn}}\)
 presentValue = \({1 \over{.10}}\) = $10
As all dividends have the same value the price of a zerogrowth stock will never change given a constant discount rate and expected cash flow.
Constant growth
Dividend growth model only works if growthRate is less than requiredRateOfReturn.
 \[dividend_{periods} = {dividend_0 \times{(1+growthRate)}^{periods}}\]
 \[period_0 = {dividend_0 \times{(1+growthRate)}^{periods} \over{requiredRateOfReturn  growthRate} }\]
 \[period_t = {dividend_0 \times{(1+growthRate)}^{t+1} \over{requiredRateOfReturn  growthRate} }\]
Example: A common stock paid a $2 dividend. If the dividends grow at a constant rate of 7% and the required rate of return is 10% what is the price of the stock today? What is the price in 4 years? Why can’t growth rate exceed the required return?
Today:
 \[period_0 = {dividend_0 \times{(1+growthRate)}^{periods} \over{requiredRateOfReturn  growthRate} }\]
 \[period_0 = {$2 \times{(1+.07)}^{1} \over{.10  .07} } = $71.33\]
In 4 years:
 \[period_t = {dividend_0 \times{(1+growthRate)}^{t+1} \over{requiredRateOfReturn  growthRate} }\]
 \[period_{4} = {$2 \times{(1+.10)}^{4+1} \over{.10  .07} } = $93.50\]
If growth rate is larger than required rate of return the stock price becomes infinite.
Nonconstant growth
If growthRate exceeds requiredRateOfReturn then we can discount high growth dividends separately, discount the constant growth dividends and sum.
Example: A common stock pays a $0.50 dividend in one year, a $1 dividend in 2 years, and a $1.50 dividend in 3 years. After year 3, dividends will grow at a constant 5% rate. Assume a required return of 10%. What is the price of the stock today?
Step 1: Calculate presentValue of constant growth period (3infinity):
 \[period_0 = {dividend_0 \times{(1+growthRate)}^{periods} \over{requiredRateOfReturn  growthRate} }\]
 \[period_{3} = {$1.50 \times{(1+.10)}^{1} \over{.10  .05}} = $31.50\]
Step 2: Calculate presentValue of nonconstant growth period (t = 1, 2, 3):
 \[period_{t} = {dividend_t \times{(1+growthRate)}^{t} \over{(1 + requiredRateOfReturn)^{t}}}\]
 \[period_{0} = {$0.50 \times{(1+.10)}^{1} \over{(1 + .10)^{1}}} + {$1 \times{(1+.10)}^{2} \over{(1 + .10)^{2}}} + {$1.50 \times{(1+.10)}^{3} \over{(1 + .10)^{3}}} + {$31.50 \over{(1 + .10)^{3}}} = $26.07\]
Required rate of return
\(totalReturn = {dividedYield + capitalGainsYield}\)
dividedYield: (expected cash divided by currentPrice)
capitalGainsYield: rate of investment value growth
 \[price_0 = {dividend_0 \times{(1+growthRate)}^{periods} \over{requiredRateOfReturn  growthRate}}\]
Rearrange dividend growth model for requiredRateOfReturn:
 \[requiredRateOfReturn = {dividend_0 \times{(1+growthRate)}^{periods} \over{price_0}} + growthRate\]
Example: A stock has paid a $5 per share dividend and is projected to grow at 5% per year. What is the required return if the stock sells today for $65.63?
 \(requiredRateOfReturn = {$5 \times{(1+.05)}^{1} \over{$65.63}} + .05 = .13\) (13%)
Common Stock features
 Common stock is equity without priority for dividends—or in bankruptcy.
 Common stock shareholder rights:
 Elect directors—elected at annual stockholders meeting—one share, one vote
 Cumulative voting: all directors elected at once—easier to get minority shareholder representation
 Straight voting: directors elected once at a time—freezes out minority shareholders
 Stagger voting makes takeover less likely to succeed as it’s hard to replace a majority of directors
 Proxy voting grants authority to vote shares on their behalf—common for large corporations.
 Shareholders can vote by attending annual meeting or vote by proxy
 Proxy fights attempt to replace management by electing enough directors
Other rights of common stock ownership:
 Dividends shared proportionally
 Liquidation value shared proportionally
 Voting in important matters (e.g., mergers)
Dividends are a return on shareholder capital paid in the form of cash or stock:
 Not a tax deductible business expense for firm
 Individuals pay tax on dividends (1520% on qualified dividends
 Subject to doubletaxation—taxed when paid to shareholders
Preferred Stock features
Preferred stock represents equity in the first but has many features of debt:
 Stated yield (stated dividend)
 Preference for cash flows in liquidation
 Credit ratings (like bonds)
 Convertible preferred; can be converted to common stock
Features:
 Preferred stock has priority over common stock
 Often no voting rights
 Directors may choose not to pay preferred dividends
Stock markets
 Primary market: original sale of new securities

Secondary market: trading of previously issued securities
 Dealer: maintains inventory, ready to buy/sell at any time
 Broker: arranges for transactions between investors
 Bid price: dealer buying price
 Ask price: dealer selling price
 Bidask spread: dealers compensation for the risk of holding inventory
Stock market trading venues
 Organized exchange: trading done facetoface
 Overthecounter market: trading done through dealers over digital networks
 Electronic communication networks (ECNs): networks that allow investors to trade directly with each other.
Chapter 12: Capital market history
Required return
How do you determine an appropriate requiredRateOfReturn? Based on the associated investment risk—greater the risk, greater the required return.
Example: If you’re considering two different stocks: stockA is a wellestablished company, stockB is a new tech company that just went public.
The startup has more uncertainty and thus a greater requiredRateOfReturn.
Dollar returns
Dollar return has two components:
 Income (dividend payment, coupon payment)
 Capital gain, or loss (price appreciation, or depreciation)
Example: If you purchased 100 shares of a stock at the start of the year at $37 per share and sell all 100 shares at the end of the year for $40.33 per share and the firm paid a dividend of $1.85 per share during the year what is your total dollar return? Is this a good return? What if another investment offered a total dollar return of $700—is this better?
 \[totalDollarReturn = {(shares \times{dividendPerShare}) + (shares \times{sharePurchasePrice  shareSellPrice})}\]
 \[totalDollarReturn = {(100 \times{$1.85}) + (100 \times{($40.33  $37)})} = $518\]
totalDollarReturn needs to be weighted by the size of the investment.
 \[rateOfReturn = {totalDollarReturn \over{totalDollarInvestment}}\]
The same return with a lower totalDollarInvestment has a better rateOfReturn.
Percentage returns
How much do we get in return for each dollar invested?
 \[percentageReturn = {endingDollarAmount  initialDollarInvestment \over{initialDollarInvestment}}\]
 \[percentageReturn = {dividendYield + capitalGainsYield}\]
 \[percentageReturn = {dividendPaid_{t+1} \over{stockPrice_{start}}} + {stockPrice_{end}  stockPrice_{start} \over{stockPrice_{start}}}\]
Example: If you purchased 100 shares of a stock at the start of the year at $37 per share and sell all 100 shares at the end of the year for $40.33 per share and the firm paid a dividend of $1.85 per share during the year.
 Step 1: Calculate the dividendYield
 Step 2: Calculate the capitalGainsYield
 Step 3: Calculate the totalPercentageReturn
Step 1: Calculate the dividendYield
 \[dividendYield = {dividendPaid_{t+1} \over{stockPrice_{start}}}\]
 \(dividendYield = {$1.85 \over{$37}} = 0.05\) (5%)
Step 2: Calculate the capitalGainsYield
 \[capitalGainsYield = {stockPrice_{end}  stockPrice_{start} \over{stockPrice_{start}}}\]
 \(capitalGainsYield = {$40.33  $37 \over{$37}} = 0.09\) (9%)
Step 3: Calculate the totalPercentageReturn
 \[percentageReturn = {dividendYield + capitalGainsYield}\]
 \(percentageReturn = {0.05 + 0.09} = 0.14\) (14%)
Example: Calculate the percentage return for the following investments if you invested $1,000 in each:
 StockA: initialPrice of $50.25, endingPrice of $54.88, dividend of $0.50 per share.

StockB: initialPrice of $17.35, endingPrice of $16.05, dividend of $0.15 per share.
 \[percentageReturn = {dividendPaid_{t+1} \over{stockPrice_{start}}} + {stockPrice_{end}  stockPrice_{start} \over{stockPrice_{start}}}\]
 \[totalDollarReturn = initialInvestment + (initialInvestment \times{percentageReturn})\]
StockA:
 \(percentageReturn = {$0.50 \over{$50.25}} + {$54.88  $50.25 \over{$50.25}}\) = 0.102 (10.2%)
 \(totalDollarReturn = $1000 + ($1000 \times{0.102})\) = $1102
StockB:
 \(percentageReturn = {$0.15 \over{$17.35}} + {$16.05  $17.35 \over{$17.35}}\) = 0.066 (6.6%)
 \(totalDollarReturn = $1000 + ($1000 \times{0.066})\) = $934
Capital Market History
How do we know if a return is normal or not? Compare returns with the historical average portfolio return for five different investment classes:
 Largecap stocks: 500 of largest U.S. companies (S&P 500 Index) (risk of large enterprise highrisk, highreward)
 Smallcap stocks: smallest 20% of stocks listed on NYSE (risk of small enterprise)
 Longterm Corporate bonds: highcredit quality bonds, 20 years to maturity (interest rate risk, default risk)
 Longterm U.S. Treasury bonds: Treasury bonds with 20 years to maturity (interest rate risk)
 U.S. Treasury bills: Tbills with onemonth to maturity (riskfree)
Average returns
averageReturn = sum(annualReturns) / years
Example: calculate the average return over 3 years given the individual annual returns: 8%, 12%, 4%.
\[averageReturn = {(.08 + .12 + .4) \over{3}} = .0533\]Risk premiums
A risk premium is additional required return due to risk:
 Tbill is considered the benchmark riskfree rate of return
 Risky investments earn a risk premium over the riskfree rate
 Investors are riskaverse and demand extra return for taking on risk
riskPremium = averageReturn  usTreasuryBillsReturn
Investment  Average Return (%)  Risk Premium (%) 

Largecompany stocks  12.1  8.7 
Smallcompany stocks  16.3  12.9 
Longterm corporate bonds  6.4  12.9 
Longterm government bonds  6.0  3.0 
U.S. Treasury Bills  3.4  0.0 
Return variability
 Variance: the average of the squared deviations from the mean (\(\sigma^{2}\))
 Standard deviation: squareroot of variance (\(\sqrt{\sigma^{2}} = \sigma\))
Example: calculate the average return and standard deviation for the following stock over three years:
 R1 = 5.77%
 R2 = 54.25%
 R3 = 0.39%
averageReturn = sum(annualReturns) / years
 \[averageReturn = {(.0577 + .5425 + .0039) \over{3}} = .2014\]
\(variance = {\sigma^{2}}\)
 \[variance ={ { {(.0577  .2014)^{2}} + {(.5425  .2014)^{2}} + {(.0039  .2014)^{2}} } \over{3  1} } = .0880\]
\(standardDeviation = \sqrt{\sigma^{2}} = \sigma\)
 \[standardDeviation = \sqrt{.0880} = .2967\]
Investment  Average Return (%)  stdDeviation (%) 

Largecompany stocks  12.1  19.8 
Smallcompany stocks  16.3  31.5 
Longterm corporate bonds  6.4  8.5 
Longterm government bonds  6.0  9.8 
U.S. Treasury Bills  3.4  3.1 
Return distributions
Normal distribution:
 1σ (68%): [7.7%, 31.9%]
 2σ (95%): [27.5%, 51.7%]
 3σ (99%): [47.3%, 71.5%]
Example: If the TBill rate is 5% and an investment with an average relative risk offers a 12% return, is this a good investment?
riskPremium = averageReturn  usTreasuryBillsReturn
 \[riskPremium = 12\%  5\% = 7\%\]
Looking at our table for risk premiums, the riskPremium for largecap stocks is 8.7%. The considered investment (7%) does not meet the riskPremium bar and so according to our calculations it is not a good investment.
Example: If the TBill rate is 5% what return would you demand for an investment with similar risk to smallcap stocks?
Looking at our table for risk premiums, the riskPremium for smallcap stocks is 12.9%. So 12.9% + 5% = 17.9%
Geometric average return
Arithmetic average return: return earned in an average year over a multiyear period.
Geometric average return: average compounded return earned per year over a multiyear period.
 \[gAverageReturn = {[(1+return_t)^{1\over{t}}}]  1\]
prod(1 + L1)^(1/dim(L1))  1
Example: consider the following annual returns on the S&P 500 Index over a 5 year period:
 [11.62%, 37.49%, 43.61%, 8.42%, 24.90%]
What is the arithmetic average return? What is the geometric average return? How much would $10,000 invested at the beginning of the period be worth after the 5 years?
averageReturn = sum(annualReturns) / years
 \[averageReturn = (11.62\% + 37.49\% + 43.61\% + 8.42\% + 24.90\%) / 5 = 11.88\%\]
\(gAverageReturn = {[(return_t)]^{1\over{totalYears}}}  1\)
 \[gAverageReturn = {[11.62\% \times 37.49\% \times 43.61\% \times 8.42\% \times 24.90\%]^{1\over{5}}  1}\]
prod(1 + L1)^(1/dim(L1))  1
= .0867 (8.67%)
How much would $10,000 invested at the beginning of the period be worth after the 5 years?
prod(1 + L1) * 10000
= $15,158
Capital market efficiency
Efficient capital market is a market in which security prices reflect available information. Efficient Market Hypothesis (EMH) asserts that modern stock markets are practicallt efficient.
Three forms of market efficiency:
 Weak form efficiency: current price reflects historical stock prices
 Semistrong form efficiency: current price reflects all public information
 Strong form efficiency: current price reflects public and private information
Chapter 12: Risk and return
Prefer more money and less risk.
Expected returns
Expected return E(R) a mathematical expectation a weighted average of the distribution of possible future returns
Simple example: one stock with two state:
 Economic boom: 50% probability, return 45%
 Economic recession: 50% probability, return 15%
 E(R) = (0.50)(45%) + (0.50)(15%) = 15%
General relation:
 \[expectedReturn = {[(prob_{state})(return_{state})]}\]
sum(L1 * L2)
Example: consider the following and calculate the risk premium for this stock if Tbills are offering a 2.5% return:
State of economy  Probability  Return 

+1% GDP  .25  0.05 
+2% GDP  .50  0.15 
+3% GDP  .25  0.35 
We can calculate the expected risk premium:
 \[riskPremium = {expectedReturn  tBillRate}\]
 \[expectedReturn = {[(prob_{state})(return_{state})]}\]
 expectedReturn =
sum(L1 * L2)
= .15 (15%)  \[riskPremium = {.15  .025} = .125\]
Variance
\(variance = {\sigma^{2}} = {[(prob_{state})(return_{state}  expectedReturn)^2]}\)
 variance =
sum(L1 * (L2  sum(L1*L2))^2 )
example: continuing the example above:
 variance =
sum(L1 * (L2  sum(L1*L2))^2 )
= .02  standardDeviation = \(\sqrt{variance}\) = .1414
Portfolios
A portfolio is a collection of securities like stocks and bonds.
Portfolio expected return
\[portfolioExpectedReturn = {[assetWeight \times{assetExpectedReturn} + …]}\] portfolioExpectedReturn =
sum((assetWeight)(L1 * L2) + (assetWeight)(L1 * …)) =
Example: consider a portfolio invested equally in 3 stocks and calculate the expected return to the portfolio:
State of economy  Probability (L1)  stockA return  stockB return  stockC return 

+1% GDP  .25  0.05  0.00  0.20 
+2% GDP  .50  0.15  0.10  0.10 
+3% GDP  .25  0.35  0.20  0.00 
To do down (?)
 portfolioExpectedReturn =
sum((1/3)( (L1 * L2) + (L1 * L3) + (L1 * L4) ))
 portfolioExpectedReturn = .117
To go across:
 portfolioStateReturn =
seq( (1/dim(L1)) * (L2(I) + L3(I) + L4(I)), I, 1, dim(L1) )
> L5  portfolioExpectedReturn =
seq( L1(I) * (1/dim(L1)) * (L1(I) + L2(I) + L3(I) + L4(I)), I, 1, dim(L1) ) )
 portfolioExpectedReturn = .117
Portfolio variance
\(variance = {\sigma^{2}} = {[(prob_{state})(return_{state}  expectedReturn)^2]}\)
 variance =
sum(L1 * (portfolioStateReturn  portfolioExpectedReturn)^2 )
 variance =
sum(L1 * (L5  .117)^2 )
= .002  stdDev = \(\sqrt{0.0022} = 0.047\)
The variance of a portfolio is NOT the weighted sum of the individual security variances.
Expected vs. Unexpected return
Two components to the actual return on a stock:
 Expected return: normal return expected by market participants
 Unexpected return: the uncertain part of return
Surprises of new information not previously available make stock prices difficult to predict.
Systematic risk: risk that affects a large number of assets
 Also known as market risk
 Examples: unexpected changes in interest rates, GDP inflation
Unsystematic risk: risk that affects a small number of assets
 Also known as unique risk, assetspecific risk, and idiosyncratic risk
 Examples: labor strikes, CEO resignation, corporate takeover
Diversification reduces risk by spreading an investment across a number of assets—thereby reducing variation and standard deviation of returns.
 Diversification can eliminate unsystematic risk.
 Diversification cannot eliminate systematic risk.
Systematic risk and beta
Systematic risk principle: the expected return on an assets depends only on systematic risk.
Beta coefficient (β) is a measure of how much systematic risk an asset has relative to an average risk asset.
 Average asset has a β = 1.0, relative to itself
 Asset with β = 0.5 has half as much systematic risk
 Assets with larger β have greater expected returns.
Total risk vs. Systematic risk
Consider the following standard deviation and beta for two stocks:
stdDev (%)  Beta  

StockA  .35  0.8 
StockB  .20  1.2 
Which stock has greater total risk? StockA; it has a larger standard deviation.
Which stock has greater systematic risk? StockB; it has a larger beta.
Which stock has the larger expected risk premium? StockB; it has a larger beta.
Portfolio beta
 portfolioExpectedReturn =
sum( (invested/totalInvested) * expectedReturn )
 portfolioExpectedReturn =
sum( (L1/sum(L1)) * L2 )
 portfolioBeta =
sum( (invested/totalInvested) * beta )
 portfolioBeta =
sum( (L1/sum(L1)) * L3 )
Consider a portfolio with the following 4 stocks and calculate the portfolio expected return:
Security  Invested  expectedReturn  Beta 

StockA  1000  0.7  0.75 
StockB  2000  0.11  0.90 
StockC  3000  0.14  1.15 
StockD  4000  0.20  1.50 
 portfolioExpectedReturn =
sum( (L1/sum(L1)) * L2 )
= .151  portfolioBeta =
sum( (L1/sum(L1)) * L3 )
= 1.20
Rewardtorisk ratio
Rewardtorisk ratio: the expected return per unit of systematic risk.
A riskfree asset, by definition, has no systematic risk (β = 0).
When a security is combined with a riskfree asset:
 the expected return is the weighted sum of expected returns
 the beta is the weighted sum of beta
Example: consider a portfolio with a riskfree asset and a risky asset:
 assetA expectedReturn = .20
 assetA beta = 1.6
 assetA riskFreeRate = 0.08
What is the expected return and beta of a portfolio that invests 25% in Asset A?
Security  Invested  expectedReturn  Beta 

StockA  .25  0.20  1.6 
StockB  .75  0.08  0 
 portfolioExpectedReturn =
sum( (L1/sum(L1)) * L2 )
= .11  portfolioBeta =
sum( (L1/sum(L1)) * L3 )
= 0.40
\(rewardToRiskRatio = {expectedReturn_{stockA}  riskFreeRate \over{beta_{stockA}  0}}\)
 \[0.075 = {0.20  0.08 \over{1.6  0}}\]
A portfolio weight of greater than 100% means borrowing for leverage.
Security market line (SML)
Security market line displays the relationship between expectedReturn and beta.
Market portfolio: a portfolio of all assets in the market
 Has average systemic risk (B = 1.0)
 As all assets must lie on the securityMarketLine (SML), the market portfolio must as well.
Slope of securityMarketLine (SML) = marketExpectedReturn  riskFreeRate
Market risk premium: SML slope
Capital asset pricing model (CAPM)
Capital asset pricing model (CAPM) displays the relationship between any asset’s expected return and its beta
CAPM states an asset expected return depends on:
 The riskfree rate
 Reward per unit of systematic risk (market risk premium)
 Level of systematic risk (beta)
Since all assets have the same rewardtorisk ratio:
\[{expectedReturn_{asset}  riskFreeRate \over{\beta_{asset}}} = {expectedReturn_{market}  riskFreeRate \over{\beta_{market}}}\]Which we can rearrange to get the following:
 \[expectedReturn_{asset} = {riskFreeRate + \beta_{asset} * [expectedReturn_{market}  riskFreeRate]}\]
 \[expectedReturn_{asset} = {riskFreeRate + \beta_{asset} * [marketRiskPremium]}\]
Example: Assume historic market risk premium has been 8.5%. The riskfree rate is current 5%. A stock has a beta of 0.85.
What return should you expect from an investment in this stock?
 \[expectedReturn_{asset} = {riskFreeRate + \beta_{asset} * [marketRiskPremium]}\]
 \[.12225 = {0.05 + 0.85 * .085}\]
What is the expected return on the market portfolio?
 \[expectedReturn_{market} = {marketRiskPremium + riskFreeRate}\]
 \[.135 = {.085 + .05}\]
Chapter 9: NPV and IRR
Making investment decisions:
 Capital firm structure (how to finance operations)
 Working capital (managing shortterm operations)
 Capital budgeting decisions (what fixed assets to buy)
Net Present Value (NPV)
Net Present Value (NPV) is the difference between the market value of and investment and its costs.
valueAdded = proposedInvestmentValue > proposedInvestmentCosts
Capital budgeting is the process of searching for projects and investments with positive NPV.
Example: You buy a house at $50,000 and sell it for $60,000. Value added—NPV—is $10,000.
Example: You create a house renovation company and sell 50,000 shares at $1 each to raise the $50,000 to buy a house.. Investors buy [15K, 15K, 20K]
(L1) shares. You sell the house at $60,000, and the profit is distributed proportionally to shareholders: [L1/dim(L1)]
:
 Investor 1 receives: 60K * (15K/50K) = $18,000
 Investor 2 receives: 60K * (15K/50K) = $18,000
 Investor 3 receives: 60K * (20K/50K) = $24,000
Value accrues to the owner of investment.
Estimating NPV
NPV Rule: NPV > 0 ? accept : reject.
Discounted Cash Flow (DCF) valuation is finding the market value of assets by taking the presentValue of future cash flows.
Example: An investment costs $1,100 today, will deliver yearlyRevenue[$1,000, $2,000], yearlyExpenses[$500, $1,000]. Assuming a requiredRateOfReturn of 10%, what is the NPV of this investment? Is this a good investment?
\[NPV = cashOutlay + …\left[{cashFlow_{year} \over{(1+requiredRateOfReturn)^{year}}}\right]\]NPV = sum(L_cashflows/(1+R)^L_years)
but, if the cash flows are level, use an annuity formula:
\[NPV = cashOutlay + cashFlow * …\left[{ {1{1\over{(1+requiredRateOfReturn)^{t}} } }\over{requiredRateOfReturn}}\right]\] \[NPV = cashOutlay + {cashFlow_1 \over{(1+requiredRateOfReturn)^1}} + {cashFlow_2 \over{(1+requiredRateOfReturn)^2}}\] yearlyRevenue = [$1,000, $2,000] (L1)
 yearlyExpenses = [$500, $1,000] (L2)
 yearlyCashFlows = L1  L2 > (L3)
NPV = C + sum( seq(L3(I)/(1+R)^I, I, 1, dim(L3) ) )
180.99 = $1,100 + sum( seq(L3(I)/(1+R)^I, I, 1, dim(L3) ) )
As the NPV is positive, this is a good investment.
Example: If you can buy the required equipment to start a cafe for $13,000 and expect to spend $4,000 per year on goods while collecting $5,800 per year in revenue over 12 years—only if you can make 11% return on your investment: What is the NPV? Should you make the investment? At what required return on investment would you be indifferent to take on the project?
 Step 1: Calculate cash flows from investment
 Step 2: Calculate NPV
Because the cash flows are level, we can use an annuity formula:
\[NPV = 13000 + cashFlow * …\left[{ {1{1\over{(1+requiredRateOfReturn)^{t}} } }\over{requiredRateOfReturn}}\right]\] \[NPV = 13000 + 1800 * …\left[{ {1{1\over{(1+.11)^{t}} } }\over{.11}}\right]\] NPV =
13000 + 1800 * ( (1 (1/(1+R)^t ) ) / R )
 NPV =
13000 + 1800 * ( (1 (1/(1+.11)^12 ) ) / .11 )
 NPV = $1,313.76
Negative NPV => Do not invest. To be indifferent (NPV = 0), solve for R and R = .0883, or 8.83%
Level and nonlevel NPVs
Example: To set up a business producing an annual revenue of $20,000, annual costs of $14,000, upfront costs of $30,000, dissolve in 8 years with PPE worth $2,000—if similar projects require a 15% rate of return, what is the NPV?
Cashflows: [ 6000, 6000, 6000, 6000, 6000, 6000, 6000, 8000 ] (L1)
\[NPV = cashOutlay + …\left[{cashFlow_{year} \over{(1+requiredRateOfReturn)^{year}}}\right]\]$2422.26 = 30000 + sum( seq(L1(I) / (1+R)^I), I, 1, dim(L1)) )
Payback rule vs NPV
Example: 2year payback on investments and a 15% return on similar investments:
L1, L2, L3
Year  Long ($)  Short ($) 

0  250  250 
1  100  100 
2  100  200 
3  100  0 
4  100  0 
 \[payback_{long} = 2 + (50/100) = 2.5 years\]
 \[payback_{short} = 1 + (150/200) = 1.75 years\]
 NPV_long =
sum(L2/(1+R)^L1)
= 35.5  NPV_short =
sum(L3/(1+R)^L1)
= 11.82
Internal Rate of Return (IRR)
IRR is the discount rate where NPV = 0.
IRR Rule: IRR > requiredRateOfReturn ? accept : reject.
Example (singleperiod): A project costs $100 today and pays $110 in one year, what is the IRR?
NPV (0) = 100 + (110 / (1+R)^1), solve for r, r = .10
Example (multiperiod): A project with a $200 initial investment, and cash flows of [50, 100, 150], what is the IRR?
 L1: [0, 1, 2, 3]
 L2: [200, 50, 100, 150]
 NPV (0) =
solve( sum(L2/(1+R)^L1), R )
= .1944
Example: Calculate IRR for an investment with an upfront cost of $275 and produces $100 cash flows for the next four years.
 L1: [0, 1, 2, 3, 4]
 L2: [275, 100, 100, 100, 100]
 NPV (0) =
solve( sum(L2/(1+R)^L1), R )
= .1688
Problems with IRR: Nonconventional cash flows
Example: calculate the NPVs of the following project at [.25, .3333, .4286, .6667] (L3):
Year  Cash Flows 

0  252 
1  1431 
2  3035 
3  2850 
4  1000 
 NPV =
sum(L2/(1+R)^L1)
 NPV =
seq(sum(L2/(1+L3(I))^L1), I, 1, dim(L3))
= [0, 0, 0, 0]
When cash flows are not conventional strange things happen to IRR, but NPV rule always works.
Problems with IRR: Mutually exclusive projects
Mutually exclusive investments is when taking one investment prevents us from taking another.
Year  ProjectA  ProjectB 

0  350  250 
1  50  125 
2  100  100 
3  150  75 
4  250  50 
Which project should we take?
 NPV_A (0, IRR) =
solve( sum(L2/(1+R)^L1), R )
= .1618  NPV_B (0, IRR) =
solve( sum(L3/(1+R)^L1), R )
= .19441780  NPV_A =
sum(L2/(1+.10)^L1)
= 61.55  NPV_B =
sum(L3/(1+.10)^L1)
= 36.78
Crossover rate: discount rate where NPV_A = NPV_B
 crossoverRate:
solve( sum((L2L3)/(1+R)^L1) , R )
= .1467
Profitability Index (PI)
Profitability index gives the cots/benefit ratio for an investment.
Drawback: does not show magnitude of investment.
\[PI = {cashFlowsPresentValue \over{initialCost} } = {NPV + C_0 \over{C_0}}\]Want PI > 1, want present value of cash flows to be greater than initial cost.
Example: calculate the PI from ProjectA at 10%.
Year  ProjectA  ProjectB 

0  350  250 
1  50  125 
2  100  100 
3  150  75 
4  250  50 
 PI = (NPV + initialCost) / initialCost
 PI =
(sum(L2/(1+.10)^L1) + abs(L2(1))) / abs(L2(1))
= 1.18
Chapter 14: Cost of capital
Cost of capital: the minimum required return on a new investment.
A discount rate is required to do NPV analysis on projects.
Cost of capital vs. required return
The following mean the same things:
 Required return: investor POV
 Cost of capital: firm POV
 Appropriate discount rate
Cost of capital depends on the risk of the investment—the use of funds, not the source of funds—where the money goes, not where it came from.
Cost of capital and financial policy
Capital structure: the mix of debt and equity financing.
Target capital structure is the debtequity mix that minimizes the cost of capital and maximizes firm value.
Cost of equity
Cost of equity: the return that equity investors require on their investment in the firm.
Cost of equity is more difficult to determine than cost of debt.
Two approaches of estimating cost of equity:
 Dividend growth model
 Security Market Link (SML)
Dividend Growth Model
costOfEquity = dividendYield + growthRate
To calculate the cost of equity we need three variables:
 \(P_0\) — the current stock price
 \(D_0\) — last dividend payment

g — dividend growth rate
 \[price_0 = {D_1 * (1+G)}\over{R_e  G}\]
 \[costOfEquity = { {D_1\over{P_0} } +G}\]
 \[costOfEquity = { {D_1\over{ {D_1 * (1+G)}\over{R_e  G} } } +G}\]
 costOfEquity =
This is the return that shareholders require on the stock, therefore it is the firm’s cost of equity capital.
Example: The stock below is priced at $92 per share, calculate the cost of equity for this firm:
Year  Dividend  $ Change  % Change 

1  4.00  0  0 
2  4.40  0.40  .10 
3  4.75  0.35  .0795 
4  5.25  0.50  .1053 
5  .0565  .040  .0762 
 averageGrowthRate = sum(L4) = .0925
 g = .0925
 P_0 = $92
 D_0 = $5.65
 R_E = ( ( $D_0 * (1 + G) ) / (P_0) ) + G
 R_E = ( ( $5.65 * (1 +.09025) ) / $92 + .09025
 \[costOfEquity = { {D_0\over{ {D_1 * (1+G)}\over{R_e  G} } } +G}\]
Cost of Equity: SML Approach
Depends on:
 Riskfree rate (R_f)
 Expected market risk premium (E(R_m)  R_f)
 Systematic asset risk (Beta)
Using SML and CAPM we can denote a firm’s required return on equity:
\[costOfEquity = riskFreeRate + estimatedBeta (marketRiskPremium  riskFreeRate)\]Example: To estimate the cost of equity for Netflix, which has a beta of 0.85, the current Tbill rate is 0.30% and market risk premium is 8.7%.
\(costOfEquity = riskFreeRate + estimatedBeta (marketRiskPremium)\)
 \[.07695 = 0.003 + 0.85 (.087)\]
Example: A company hasa beta of 1.3, the market risk premium is 7.5% and the current riskfree rate is 1.2%. If the last dividend was $1.75 per share, dividends are expected to grow at 4%, and the current stock price is $24.
 Estimate costOfEquity using SML approach
 \[costOfEquity = riskFreeRate + estimatedBeta (marketRiskPremium)\]
 10.95% = .012 + 1.3(.075)
 Estimate costOfEquity using Dividend growth model
 costOfEquity =
( ( $D_0 * (1 + G) ) / currentStockPrice ) + G
 11.58% =
( ( $1.75 * (1 + .04) ) / $24 ) +.04
 costOfEquity =
Cost of Debt
Cost of debt is the return that lenders require on the firms debt.
Example: A corporation issues a 10year bond 5 years ago with a coupon rate of 8% that currently sells for $1,075. What is the current cost of debt, assuming annual interest payments?
 \[price = {\left( C \times { {1  {1 \over{(1+r)^t}}} \over{r}}\right)} + {faceValue \over{(1+r)^t}}\]
 \[1075 = {\left( 80 \times { {1  {1 \over{(1+r)^5}}} \over{r}}\right)} + {1000 \over{(1+r)^5}}\]
 P =
C * ( (1  (1 / (1+R)^T)) / R ) + (F / (1+R)^T)
 1075 =
80 * ( (1  (1 / (1+R)^5)) / R ) + (1000 / (1+R)^5)
 solve( C * ( (1  (1 / (1+R)^T)) / R ) + (F / (1+R)^T)  1075, R)
 solve for r, r = .0621
 1075 = ( 80 * ( (1  (1 / ((1+R)^5) / R ) + (1000 / (1+r)^5)
costOfDebt = 6.21% = YTM
Cost of preferred stock
Cost of preferred stock is the return that investors demand to buy shares of preferred stock.
\[price_{preferredStock} = dividend \over{price}\]Weighted Average Cost of Capital (WACC)
Weighted average cost of capital (WACC) is the weighted average of the cost of equity and aftertax cost of debt.
\[WACC = (E/V) * R_e + (D/V) * R_d * (1  T_c)\]Example: A firm has 50,000 shares of stock at a price of $80 per share, a beta of 1.15, the current riskfree rate is 2%, the market risk premium is 7%, the firm has a $1M face value of bonds currently priced at 110% of par value at a YTM of 5.5% and the corporate tax rate is 21%.
 costOfEquity (C) = riskFreeRate + estimatedBeta (marketRiskPremium) (SML approach)
 .1005 = .02 + 1.15(.07)
 R_d (R) = 5.5% YTM
 marketDebt (D) = numberOfBonds * marketBondPrice
 1,100,000 = 1,000,000 * 1.10
 marketEquity (E) = numberOfShares * marketSharePrice
 $4,000,000 = 50,000 * $80
 firmTotalMarketValue (V) = marketEquity * marketDebt
 5,000,000 = 4,000,000 * 1,000,000
 WACC = (E/V) * C + (D/V) * R * (1  T)
 WACC = (marketEquity/firmTotalMarketValue) * costOfEquity + (marketDebt/firmTotalMarketValue) * R_d * (1  T_c)
 .0899 = (4,000,000/5,000,000) * .1005 + (1,100,000/5,000,000) * .055 * (1  .21)
Capital structure weights:
 E = market value of equity = numberOfShares * marketSharePrice
 D = market value of debt = numberOfBonds * marketBondPrice
 V = total market value of the firm = E * D
100% = (E/V) + (D/V)
E/V and D/V are the capital structure weights, similar to portfolio weights.
Example: A firm has a market value equity of $200M and debt of $50M.
 V = $200M + $50M = $250M
 E/V = 200 / 250 = .8
 D/V = 50 / 250 = .2
WACC and Taxes
We need to consider tax in aftertax cash flows. Interest payments on debt are taxdeductible (dividend are not).
Aftertax cost of debt = \(preTaxCostOfDebt * (1corpTaxRate)\)
Example: A pre tax cost of debt at 9% and a corporate tax rate of 21%.
 Aftertax cost of debt = \(preTaxCostOfDebt * (1corpTaxRate)\)
 \[.0711 = .09 * (1.21)\]
 Total interest payments = .09 * $1,000,000
 Reduction in taxes = $90,000 * .21 = $18,900
 After tax interest payments = $90,000  $18,900 = $71,100
 After tax interest rate = $71,000 / $1,000,000 = .0711 (7.11%)
Using WACC to Discount Cash Flows
WACC is the appropriate discount rate only if:
 The risk of the proposed investment is similar in risk to the overall business operations
 The proposed investment in financed with the same capital structure weights.
Example: A firm is considering the construction of another production plant to manufacture the same product.
(Another) As the project is similar risk to existing construction, WACC is appropriate.
Example: A firm decides to open a restaurant to experiment with new food offerings.
(New) This project is much riskier than existing operations, WACC is inappropriate.
Example: A firm has 50,000 shares of stock at a price of $80 per share, a beta of 1.15, the current riskfree rate is 2%, the market risk premium is 7%, the firm has a $1M face value of bonds currently priced at 110% of par value at a YTM of 5.5% and the corporate tax rate is 21%. Calculate the WACC.
 costOfEquity (C) = riskFreeRate + estimatedBeta (marketRiskPremium) (SML approach)
 .1005 = .02 + 1.15(.07)
 R_d (R) = 5.5% YTM
 marketDebt (D) = numberOfBonds * marketBondPrice
 1,100,000 = 1,000,000 * 1.10
 marketEquity (E) = numberOfShares * marketSharePrice
 $4,000,000 = 50,000 * $80
 firmTotalMarketValue (V) = marketEquity * marketDebt
 5,000,000 = 4,000,000 * 1,000,000
 WACC = (E/V) * C + (D/V) * R * (1  T)
 WACC = (marketEquity/firmTotalMarketValue) * costOfEquity + (marketDebt/firmTotalMarketValue) * R_d * (1  T_c)
 .0899 = (4,000,000/5,000,000) * .1005 + (1,100,000/5,000,000) * .055 * (1  .21)
The firm is considering the construction of a new chemical processing facility. Initial cost is $79M and generates aftertax cash flows of $14M per year over 10 years. Should the project be accepted?
 \[NPV = cashOutlay + …\left[{cashFlow_{year} \over{(1+requiredRateOfReturn)^{year}}}\right]\]
 Using WACC for requiredRateOfReturn
 Cashflows: [79000000, 14000000, 14000000, 14000000, 14000000, 14000000, 14000000, 14000000, 14000000, 14000000, 14000000]
 NPV =
sum(L_cashflows/(1+R)^L_years)
 NPV =
sum(L2/(1+.0899)^L1)
= 10886768.69  isNpvPositive ? accept : reject => accept
Note on debttoequity ratios
For WACC problems we will be given the debtequity ratio.
Example: if a firm has a debtequity ratio of 1/3, there are four total units of capital:
 D = 1
 E = 3
 V = (D+E) = 4
 D/V = .25
 E/V = .75
WACC and the SML
WACC is only the appropriate discount rate when the proposed investment is similar to overall firm. For example, a division within a firm might have a different WACC than the overall firm.
Example: a firm that always uses a WACC of 15% to make investment decisions:
 investmentReturn > 15% ? accept : reject
We can use the SML to guide:
 isAboveSmlLine ? goodInvestment : badInvestment
Example: An allequity firm with a beta of 1, a WACC of 15%, the riskfree rate is 7% and the marketrisk premium is 8%. The firm is considering a project with a beta of .6 (the project is less risky than the overall firm), you expect this project will return 14%—should it be accepted?
Using WACC as a benchmark—is an inappropriate benchmark:
 WACC = .07 + 1.0 * .08 = 15% > 14 => reject
Taking into account the project beta, the required return for the project:
 return_project = 7% + 0.6 * 0.8 = 11.8% < 14 => accept
A firm that uses WACC as a cutoff will:
 Tend to reject profitable projects with less risk than the overall firm
 Tend to accept unprofitable projects with more risk than the overall firm
Flotation Costs and WACC
Weighted average flotation cost: sum of all flotation costs as a percent of the amount of security issued multiplied by the target capital structure weights.
\[overallFlotation = (E/V) * equityFlotation + (D/V) * debtFlotation\]To determine gross proceeds from issuance to ensure sufficient funds for investment after paying flotation costs, use mutliplier:
1 / (1  overallFlotation)
Example:
 overallFlotation = .10
 project cost = $500,000
 total funds to raise = $500,000 / (1  0.10) = $555,555.5
 flotation costs = $500,000 = $55,555.5
Example: A company is considering opening another office at the cost of $50,000 and is expected to generate aftertax cash flows of $10,000 in perpetuity. The firm has a target debtequity ratio of 1/2. New equity has a flotation cost of 10% and a required return of 15%. New debt has flotation costs of 15% and a pretax required return of 10%. The corporate tax rate is 21%. Using NPV, should the firm accept the project?
 cashOutlay = $50,000
 returnOnEquity (C) = 15%
 returnOnDebt (R) = 10%
 equityFlotation = 10%
 debtFlotation = 5%
 marketDebt/marketEquity (D/E) = 1/2
 marketDebt (D) = 1
 marketEquity (E) = 2
 V = D + E = 3
 firmTotalMarketValue (V) = marketEquity * marketDebt
 5,000,000 = 4,000,000 * 1,000,000
 WACC = (E/V) * C + (D/V) * R * (1  T)
 .1263 = (2/3) * 15% + (1/3) * 10% * (1  .21)

\[overallFlotation = (E/V) * equityFlotation + (D/V) * debtFlotation\]
 .0833 = (2/3) * 10% + (1/3) * 5%
 totalInvestment = 50000 / (1 .0833) = $54,526
 \[NPV = cashOutlay + …\left[{cashFlow_{year} \over{(1+requiredRateOfReturn)^{year}}}\right]\]
 Using WACC for requiredRateOfReturn
 NPV = $54,526 + ($10,000 / 0.1263) = $24,650.56
 Cashflows: [54526, 10000]
 NPV =
sum(L_cashflows/(1+R)^L_years)
 NPV =
sum(L2/(0.1263)^L1)
= $24,650.56  isNpvPositive ? accept : reject => accept
Chapter 16: Capital structure
What debt/equity ratio should a firm choose?
The goal of managing capital structure is to maximize the value of the firm.
Optimal capital structure is the debt/equity ratio that minimizes the WACC.
 WACC = (E/V) * C + (D/V) * R * (1  T)
Effect of financial leverage
Financial leverage is the extent a firm relies on debt.
Example: A corporation is considering restructuring: issuing new debt to buy back some of its equity. It currently has assets with a market value of $8M, no debt, and 400,000 shares. (since no debt, share price = $20). What is the effect of financial leverage on gains and losses to shareholders?
Proposal: issue $4M debt at 10% to buy back shares.
 At $20 share price, $4M debt buys back 200,000 shares
 After debt issue, new D/E ratio is 1.0
 Assumption: restructuring has no influence on share price
Expansion: No leverage (no debt, D/E = 0)
 returnOnEquity (ROE) = netIncome / totalEquity = $1.5M / $8M = 18.7%
 earningsPerShare (EPS) = netIncome / shares = $1.5M / 400,000 = $3.75
Expansion: Leverage (D/E = 1.0)
 returnOnEquity (ROE) = netIncome / totalEquity = $1.1M / $4M = 27.5%

earningsPerShare (EPS) = netIncome / shares = $1.1M / 200,000 = $5.50
 Breakeven EBIT => Set 2 EPS equal
 noLeverageEPS = leverageEPS
 noLeverageEPS = NI = EBIT
 leverageEPS = NI = EBIT  interest
 earningsPerShare (EPS) = netIncome / shares
 earningsPerShare (EPS) = EBIT / 400,000
 So the equation to solve for becomes: EBIT / 400,000 = (EBIT  400,000) / 200,000
 EBIT = 2(EBIT  400,000)
 EBIT = $800,000 (breakeven)
Leverage magnifies returns.
Homemade leverage
Homemade leverage is the use of personal borrow to change the overall amount of exposed leverage.
 Stockholder can lend and borrow on their own.
 Investors can undo capital structure effects taken at the firm level.
To create leverage an investor can borrow on their own. To undo leverage and investor can lend money.
Example: A company adopts the proposed capital structure (e.g., increases leverage), an investor can undo this effect on their own by “unlevering” the stock. The leverage leg to borrowing an amount equal to half of the firms overall value (D/E = 1).
 If an investor has 100 shares and is unhappy with leverage decision
 Net investment of the firm of $2,000 = (20)(100 shares)
 Investor sells 50 shares for $1,000.
 Lends the sale proceeds of $1,000 at 10% interest.
 Investor now has “homemade” D/E = 0
Recession($)  Expected($)  Expansion($)  

EPS (levered structure)  0.50  3.00  5.50 
Earnings on 50 shares  25  150  275 
Interest on lending  100  100  100 
Total payoff  $125.00  $250.00  $375.00 
ROI  6.25%  12.50%  18.75% 
Result of unlevering the stock: the investor has replicated the payoffs and return on investment of the unlevered firm under the original capital structure.
Conclusion: Homemade leverage allows investors to undo firms capital structure decisions. Investors won’t pay firms to do what they can do on their own.
M&M Porposition 1 (No Taxes)
M&M Proposition I is the value of a firm is independent of its capital structure.
 Assuming no taxes and individuals can borrow at same rate of firms.
 Also known as the pie model “the size of the pie doesn’t depend on how its sliced”
Consider two firms with identical asset composition (lefthand side of balance sheet) but different financing of those assets (righthand side of balance sheet). M&M Prop I states that these two firms will have the same value.
M&M Proposition 2 (No Taxes)
M&M Proposition 2 is a firm’s cost of equity capital is a positive linear function of its capital structure. While change in capital structure doesn’t affect overall firm value (M&M Prop 1), it does affect the debt/equity ratio.
Let WACC represent the required return on firm’s total assets:
 \[WACC = totalAssetRequiredReturn = (E/V)*returnEquity + (D/V)*returnDebt\]
Rearrange to solve for cost of equity:
 \[returnEquity = totalAssetRequiredReturn + (totalAssetRequiredReturn  returnDebt) * (D/E)\]
returnEquity (cost of equity) depends on three things:
 Required return on firms assets (totalAssetRequiredReturn)
 Firm’s cost of debt (returnDebt)
 Firm’s debt/equity ratio (D/E)
returnEquity (cost of equity) is a straight line with a slope (totalAssetRequiredReturn  returnDebt)
Example: a firm with a WACC of 12% that can borrow at 8% (ignore taxes).
 If target capital is 80% equity and 20% debt what is the cost of equity?
 D/E = 20/80 = .25
 \[returnEquity = totalAssetRequiredReturn + (totalAssetRequiredReturn  returnDebt) * (D/E)\]
 \[.13 = .12 + (.12  .08) * (20/80)\]
 If target capital is 50% equity and 50% debt what is the cost of equity and WACC?
 D/E = 50/50 = 1
 \[returnEquity = totalAssetRequiredReturn + (totalAssetRequiredReturn  returnDebt) * (D/E)\]
 \[.16 = .12 + (.12  .08) * (50/50)\]
The firms cost of equity rises when the firm increases its use of financial leverage as the financial risk of equity increases while the business risk is unchanged.
M&M With Corporate Taxes
Two important features of debt that need to be considered:
 Interest paid on debt is tax deductible (benefit)
 Debt introduces risk of bankruptcy (cost)
Example: Consider two otherwise identical firms: FirmA is unlevered and FirmB is levered. Both firms have EBIT of $1,000 every year and pay tax rate of 21%. FirmB issues $1,000 in bonds (in perpetuity) at 8% interest rate.
FirmA  FirmB  

EBIT  1,000  1,000 
Interest  0  80 
Taxable Income  1000  920 
Taxes (21%)  210  193.20 
Net Income  790.00  726.80 
Calculate cash flow from assets (assume zero depreciation and change in net working capital)
 FirmA Cash flow: EBIT  Taxes = 1000  210 = $790
 FirmB Cash flow: EBIT  Taxes = 1000  193.2 = $806.80
Calculate cash flow to stockholders and bondholders:
 FirmA cash flow: $790
 to stock holders: $790
 to bond holders: $0
 FirmB cash flow: $806.80
 to stock holders: $727
 to bond holders: $80
FirmB cash flow  FirmA cash flow = $16.90
Interest tax shield is the tax saving arising from the deductibility of interest. At 21% tax rate, tax savings on interest = ($80)(0.21) = $16.80
Because FirmB pays interest on debt every year (perpetual), FirmB will have larger aftertax cash flows than FirmA as a result of the tax shield. Therefore, FirmB has greater value by the present value of the perpetuity of the interest tax shield.
Interest tax shield valued using cost of debt:
 \[interestTaxShieldPV ={ {T_C * (D * R_D)} \over{R_D} } = corporateTaxRate * Debt\]
 interestTaxShieldPV = corporateTaxRate * Debt
Calculate the PV of the interest tax shield on FirmB
 perpetuityPV = cashFlow / discountRate
 taxShieldPV = $16.80 / .08 = ()
M&M Prop 1 (With Taxes)
A levered firm will exceed the value of an unlevered firm by the interestTaxShieldPV (interestTaxShieldPV = corporateTaxRate * Debt).
\[leveredFirmValue = unleveredFirmValue + interestTaxShieldPV\]The value of the unlevered firm is given by: \(unleveredFirmValue = { {EBIT * (1  corporateTaxRate)} \over{unleveredDiscountRate} }\)
Example: Calculate the value of the levered and unlevered firm if the unlevered cost of capital is 10% and the firm has a $790 cash flow in perpetuity.
 unleveredDiscountRate = 10%

\[unleveredFirmValue = { {EBIT * (1  corporateTaxRate)} \over{unleveredDiscountRate} }\]
 $7,900 = ( 1000(1.21) ) / .10

\[leveredFirmValue = unleveredFirmValue + interestTaxShieldPV\]
 $8,110 = 7900 + (.21)($1,000)
M&M Prop 2 (With Taxes)
Recall WACC when there are taxes:
 \[WACC = {(E/V) * returnEquity + (D/V) * returnDebt * (1corporateTaxRate)}\]
Rearrange for the cost of equity:
 \[returnEquity = {unleveredCostOfCapital + (unleveredCostOfCapital  returnDebt) * (D/E) * (1corporateTaxRate)}\]
Example: Calculate the cost of equity and the WACC for the levered firm assuming an unlevered cost of capital of 10%/
 V = D + E = $8,110
 E = V  D = 8,110  1,000 = $7,110
 returnEquity = (unleveredCostOfCapital + (unleveredCostOfCapital  returnDebt) * (D/E) * (1corporateTaxRate) )
 .1022 = (.10 + (.10  .08) * (1000/7110) * (1.21) )
 WACC = (E/V) * returnEquity + (D/V) * returnDebt * (1corporateTaxRate)
 .0974 = (7110/8110) * .1022 + (1000/8110) * .08 * (1.21)
A lower WACC implies a lower discount rate which implies higher present value of future cash flows which implies a higher NPV.
M&M Summary (No Taxes)
Prop 1:
 Value of levered firm = value of unlevered firm
 A firms capital structure is irrelevant (homemade leverage)
 A firms WACC is the same no matter what D/E is used
Prop 2:
 Cost of equity rises as firm increases use of debt financing
 Risk of equity depends on business risk and financial risk
 returnEquity = returnAssets + (returnAssets  returnDebt) * (D/E)
Bankruptcy costs
 Second implication of debt financing is possibility of bankruptcy
Under bankruptcy: value of assets = value of debt; a transfer of ownership from stockholders to bondholders. Result: costs associated with bankruptcy or financial distress act as a disincentive to debt financing
Optimal cost structure
Two effects of debt on cost of capital:
 Firms borrow because tax shields are valuable
 Borrowing is constrained by costs of bankruptcy or financial distress
At low levels of debt:
 Probability of financial distress is small.
 Benefits of tax shield outweigh costs of financial distress
At high levels of debt:
 As D/E increases probability of bankruptcy increases.
 Costs of financial distress outweigh the benefits of tax shield.
Static theory of capital structure
A firm borrows up to the point where the tax benefit from an extra dollar in debt equals the cost from increased probability of financial distress. Optimal capital structure: marginal benefit = marginal cost.