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FIN 301

Introduction to Business Finance

Resources:

Class Textbook: Fundamentals of Corporate Finance, 13th edition

Topics:

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) ^ (I-1), I, 1, dim(L1) ) )
  • PV = FV / (1+rate)^year
    • sum( seq( L1(I) / (1+R) ^ (I-1), 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 debt-equity ratio:
    • \[debtEquityRatio = {totalDebt \over{totalEquity}}\]
  • The equity multiplier:
    • \[equityMultiplier = {totalAssets \over{equityMultiplier}}\]
  • The long-term 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)
\[price = {couponPV + faceValuePV}\] \[price = {\left( C \times { {1 - {1 \over{(1+r)^t}}} \over{r}}\right)} + {faceValue \over{(1+r)^t}}\]

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:

  1. Draw cashflow timeline
  2. Calculate present value (PV) of coupon payments (using annuity formula)
  3. Calculate present value (PV) of face value payment
  4. Calculate the total bond price
  5. 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
\[price = {\left( C \times { {1 - {1 \over{(1+r)^t}}} \over{r}}\right)} + {faceValue \over{(1+r)^t}}\] \[955.14 = {\left( $80 \times { {1 - {1 \over{(1+r)^6}}} \over{r}}\right)} + {1000 \over{(1+r)^6}}, r = .09\]

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
\[price = {\left( C \times { {1 - {1 \over{(1+r)^t}}} \over{r}}\right)} + {faceValue \over{(1+r)^t}}\] \[price = {\left( $55 \times { {1 - {1 \over{(1.065)^{40}}}} \over{.065}}\right)} + {1000 \over{(1.065)^{40}}} = $858.54\]

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.

  • Short-term debt: under 1 year to maturity
  • Long-term 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 2-10 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 - Long-term 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?
\[price = {couponPV + faceValuePV}\] \[price = { 1000 \over{(1+.12)^{5}}} = $567.43\]

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 over-the-counter:

  • 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
  • Big-ask spread: difference between bid and ask; represents dealer’s profit on a round-trip 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 short-term and long-term interest rates.

  • Represents the pure time value of money
  • Nominal rates on default fee securities
  • Upward sloping term structure: long-term rates > short-term rates
  • Downward sloping term structure: long-term rates < short-term 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:

  1. Cash flows are uncertain
  2. Stocks have infinite life (no maturity—no end date of final cashflow)
  3. 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:

  1. Dividends
  2. 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:

  1. Zero growth; dividends do not change
  2. Constant growth; dividends grow at constant rate
  3. Non-constant growth; short-term 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 zero-growth 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.

Non-constant 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 (3-infinity):

  • \[period_0 = {dividend_0 \times{(1+growthRate)}^{periods} \over{requiredRateOfReturn - growthRate} }\]
  • \[period_{3} = {$1.50 \times{(1+.10)}^{1} \over{.10 - .05}} = $31.50\]

Step 2: Calculate presentValue of non-constant 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 (15-20% on qualified dividends
  • Subject to double-taxation—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
  • Bid-ask spread: dealers compensation for the risk of holding inventory

Stock market trading venues

  • Organized exchange: trading done face-to-face
  • Over-the-counter 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 well-established company, stockB is a new tech company that just went public.

The start-up 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:

  1. Large-cap stocks: 500 of largest U.S. companies (S&P 500 Index) (risk of large enterprise high-risk, high-reward)
  2. Small-cap stocks: smallest 20% of stocks listed on NYSE (risk of small enterprise)
  3. Long-term Corporate bonds: high-credit quality bonds, 20 years to maturity (interest rate risk, default risk)
  4. Long-term U.S. Treasury bonds: Treasury bonds with 20 years to maturity (interest rate risk)
  5. U.S. Treasury bills: T-bills with one-month to maturity (risk-free)

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:

  • T-bill is considered the benchmark risk-free rate of return
  • Risky investments earn a risk premium over the risk-free rate
  • Investors are risk-averse and demand extra return for taking on risk

riskPremium = averageReturn - usTreasuryBillsReturn

Investment Average Return (%) Risk Premium (%)
Large-company stocks 12.1 8.7
Small-company stocks 16.3 12.9
Long-term corporate bonds 6.4 12.9
Long-term 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: square-root 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 (%)
Large-company stocks 12.1 19.8
Small-company stocks 16.3 31.5
Long-term corporate bonds 6.4 8.5
Long-term 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 T-Bill 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 large-cap 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 T-Bill rate is 5% what return would you demand for an investment with similar risk to small-cap stocks?

Looking at our table for risk premiums, the riskPremium for small-cap stocks is 12.9%. So 12.9% + 5% = 17.9%

Geometric average return

Arithmetic average return: return earned in an average year over a multi-year period.

Geometric average return: average compounded return earned per year over a multi-year 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:

  1. Weak form efficiency: current price reflects historical stock prices
  2. Semi-strong form efficiency: current price reflects all public information
  3. 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 T-bills 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 )
\[standardDeviation = \sqrt{\sigma^{2}} = \sigma\]

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
\[totalReturn = {expectedReturn + unexpectedReturn}\]

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, asset-specific risk, and idiosyncratic risk
  • Examples: labor strikes, CEO resignation, corporate takeover
\[return = {expectedReturn + marketRisk + unsystematicRisk}\]

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

Reward-to-risk ratio

Reward-to-risk ratio: the expected return per unit of systematic risk.

A risk-free asset, by definition, has no systematic risk (β = 0).

When a security is combined with a risk-free 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 risk-free 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 risk-free rate
  • Reward per unit of systematic risk (market risk premium)
  • Level of systematic risk (beta)

Since all assets have the same reward-to-risk 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 risk-free 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:

  1. Capital firm structure (how to finance operations)
  2. Working capital (managing short-term operations)
  3. 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)
\[NPV = -1100 + {500\over{(1+.10)^1}} + {1000\over{(1+.10)^2}}\]

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
\[annualCashFlow = 5800 - 4000 = 1800\]

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 non-level 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: 2-year 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 (single-period): 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 (multi-period): 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((L2-L3)/(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 debt-equity 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:

  1. Dividend growth model
  2. 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:

  • Risk-free 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 T-bill 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 risk-free 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

Cost of Debt

Cost of debt is the return that lenders require on the firms debt.

Example: A corporation issues a 10-year 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 after-tax 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 risk-free 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 after-tax cash flows. Interest payments on debt are tax-deductible (dividend are not).

After-tax cost of debt = \(preTaxCostOfDebt * (1-corpTaxRate)\)

Example: A pre tax cost of debt at 9% and a corporate tax rate of 21%.

  • After-tax cost of debt = \(preTaxCostOfDebt * (1-corpTaxRate)\)
  • \[.0711 = .09 * (1-.21)\]
  • 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 risk-free 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 after-tax 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 debt-to-equity ratios

For WACC problems we will be given the debt-equity ratio.

Example: if a firm has a debt-equity 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 all-equity firm with a beta of 1, a WACC of 15%, the risk-free rate is 7% and the market-risk 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 after-tax cash flows of $10,000 in perpetuity. The firm has a target debt-equity 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 pre-tax 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 (left-hand side of balance sheet) but different financing of those assets (right-hand 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 after-tax 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:

  • \[WACC = {(E/V) * returnEquity + (D/V) * returnDebt * (1-corporateTaxRate)}\]

Rearrange for the cost of equity:

  • \[returnEquity = {unleveredCostOfCapital + (unleveredCostOfCapital - returnDebt) * (D/E) * (1-corporateTaxRate)}\]

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) * (1-corporateTaxRate) )
  • .1022 = (.10 + (.10 - .08) * (1000/7110) * (1-.21) )
  • WACC = (E/V) * returnEquity + (D/V) * returnDebt * (1-corporateTaxRate)
  • .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.