Quick MBA Finance Primer

8/14/2019 Quick MBA Finance Primer

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Corporate Finance

IntroductionArguably, the role of a corporation's management is to increase the value of the firm to itsshareholders while observing applicable laws and responsibilities. Corporate finance deals withthe strategic financial issues associated with achieving this goal, such as how the corporationshould raise and manage its capital, what investments the firm should make, what portion ofprofits should be returned to shareholders in the form of dividends, and whether it makes senseto merge with or acquire another firm.

Balance Sheet Approach to ValuationIf the role of management is to increase the shareholder value, then managers can make betterdecisions if they can predict the impact of those decisions on the firm's value. y observing thedifference in the firm's equity value at different points in time, one can better evaluate theeffectiveness of financial decisions. A rudimentary way of valuing the equity of a company issimply to take its balance sheet and subtract liabilities from assets to arrive at the equity value.!owever, this book value has little resemblance to the real value of the company. First, theassets are recorded at historical costs, which may be much greater than or much less their

present market values. "econd, assets such as patents, trademarks, loyal customers, andtalented managers do not appear on the balance sheet but may have a significant impact on thefirm's ability to generate future profits. "o while the balance sheet method is simple, it is notaccurate# there are better ways of accomplishing the task of valuation.

Cash vs. ProfitsAnother way to value the firm is to consider the future flow of cash. "ince cash today is worthmore than the same amount of cash tomorrow, a valuation model based on cash flow candiscount the value of cash received in future years, thus providing a more accurate picture of thetrue impact of financial decisions.$ecisions about finances affect operations and vice versa# a company's finances and operationsare interrelated. %he firm's working capital flows in a cycle, beginning with cash that may be

converted into equipment and raw materials. Additional cash is used to convert the raw materialsinto inventory, which then is converted into accounts receivable and eventually back to cash,completing the cycle. %he goal is to have more cash at the end of the cycle than at the beginning.

%he change in cash is different from accounting profits. A company can report consistent profitsbut still become insolvent. For e&ample, if the firm e&tends customers increasingly longer periodsof time to settle their accounts, even though the reported earnings do not change, the cash flowwill decrease. As another e&ample, take the case of a firm that produces more product than itsells, a situation that results in the accumulation of inventory. In such a situation, the inventory willappear as an asset on the balance sheet, but does not result in profit or loss. ven though theinventory was not sold, cash nonetheless was consumed in producing it.

(ote also the distinction between cash and equity. "hareholders' equity is the sum of commonstock at par value, additional paid)in capital, and retained earnings. "ome people have beenknown to picture retained earnings as money sitting in a shoe bo& or bank account. utshareholders' equity is on the opposite side of the balance sheet from cash. In fact, retainedearnings represent shareholders' claims on the assets of the firm, and do not represent cash thatcan be used if the cash balance gets too low. In this regard, one can say that retained earningsrepresent cash that already has been spent.

"hareholder equity changes due to three things* net income or losses


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payment of dividends share issuance or repurchase.

Changes in cash are reported by the cash flow statement, which organi+es the sources and usesof cash into three categories* operating activities, investing activities, and financing activities.

Cash Cycle%he duration of the cash cycle is the time between the date the inventory or raw materials- ispaid for and the date the cash is collected from the sale of the inventory. A company's cash cycleis important because it affects the need for financing. %he cash cycle is calculated as*

days in inventory days in receivables ) days in payables

Financing requirements will increase if either of the following occurs* "ales increase while the cash cycle remains fi&ed in duration. Increased sales increase

the value of assets in the cycle. "ales remain flat but the cash cycle increases in duration.

/hile financially it makes sense to reduce the length of the cash cycle, such a reduction shouldnot be done without considering the impact on operations. For e&ample, one must consider the

impact on customer and supplier relations as well as the impact on order fill rates.

Revenue, Expenses, and InventoryA firm's income is calculated by subtracting its e&penses from its revenue. !owever, not all costsare considered e&penses# accounting standards and ta& laws prohibit the e&pensing of costsincurred in the production of inventory. 0ather, these costs must be allocated to inventoryaccounts and appear as assets on the balance sheet. 1nce the finished goods are drawn frominventory and sold, these costs are reported on the income statement as the cost of goods soldC12"-. If one wishes to know how much product the firm actually produced, the cost of goodsproduced in an accounting period is determined by adding the change in inventory to the C12".

AssetsAssets can be classified as current assets and long)term assets. It is useful to know the numberof days of certain assets and liabilities that a firm has on hand. %hese numbers are easilycalculated from the financial statements as follows*

(umber of days of accounts receivable 3 accounts receivable 4 annual credit sales - 567 -.%his also is known as the collection period.

(umber of days of inventory 3 inventory 4 annual C12" - 567 -. %his also is known as theinventory period.

(umber of days of accounts payable 3 accounts payable 4 C12" - 567 -, assuming that allaccounts payable are for the production of goods. %his also is known as the payables period.

Profitability RatiosA firm's profitability can be evaluated using financial ratios. 0eferencing these ratios to those ofother firms allows a comparison to be made. %he following is a listing of some useful profitabilityratios.8everage* Assets 4 "hareholder's quity

2ross 9argin* defined as 2ross :rofit 4 "ales. 2ross margin measure the profitability consideringonly variable costs and is a measure of the percentage of revenue that goes to fi&ed costs and


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profit.

(et :rofit 9argin* defined as (et Income 4 "ales

%otal Asset %urnover* defined as "ales 4 %otal Assets

0eturn on Assets 01A-* defined as (et Income 4 Assets. 01A is a measure of the return onmoney provided by both owners and creditors, and is a measure of how efficiently all resourcesare managed.

0eturn on quity 01-* defined as (et Income 4 quity, where the equity value is theshareholder's equity at the endof the period in which the income was earned. 01 is a measureof the return on money provided by the firm's owners.

01 can be calculated indirectly as* 01 3 (et Income 4 %otal Assets - %otal Assets 4 quity -

01 also can be calculated using $u:ont analysis*$u:ont Analysis* 01 3 (et Income 4 "ales-"ales 4 %otal Assets-%otal Assets 4 quity-

%his states that 01 is determined by multiplication of three levers*

01 3 net profit margin-total asset turnover-leverage-

%hese levers are readily viewed on the company's financial statements. /hile 01's may besimilar among firms, the levers may differ significantly.

i!uidity%he term working capitalis used to describe the current items of the balance sheet. /orkingcapital includes current assets such as cash, accounts receivable, and inventory, and currentliabilities such as accounts payable and other short term liabilities. Net working capitalis definedas non)cash current operating assets minus non)debt current operating liabilities. Cash, short)term debt, and current portion of long)term debt are e&cluded from the net working capitalcalculation because they are related to financing and not to operations.

%wo measures of liquidity are the current ratioand the quick ratio.

Current 0atio* defined as Current Assets 4 Current 8iabilities. %he current ratio is a measure ofthe firm's ability to pay off current liabilities as they become due.

;uick 0atio* defined as ;uick Assets 4 Current 8iabilities. Also known as the acid test. ;uickassets are defined as cash, accounts receivable, and notes receivable ) essentially current assetsminus inventory.

Ban" oansank loans can be classified according to their durations. %here are short)term loans one year or

less-, long)term loans also known as term loans-, and revolving loans that allow one to borrowup to a specified credit level at any time over the duration of the loan. "ome revolving loansautomatically renew at maturity# these loans are said to be


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Sustainable $ro%thA company's sustainable growth rate is calculated by multiplying the 01 by the earningsretention rate.

&ir' Value, E!uity Value, and (ebt Value%he value of the firm is the value of its assets, or rather, the present value of the unlevered freecash flow resulting from the use of those assets. In the case of an all)equity financed firm, theequity value is equal to the firm value. /hen the firm has issued debt, the debt holders have apriority claim on their interest and principal, and the equity holders have a residual claim on whatremains after the debt obligations are met. %he sum of the value of the debt and the value of theequity then is equal to the value of the firm, ignoring the ta& benefits from the interest paid on thedebt. Considering ta&es, the effective value of the firm will be higher since a levered firm has ata& benefit from the interest paid on the debt. If there is outstanding preferred stock, the firm valueis the sum of the equity value, debt value, and preferred stock value, plus the value of the interestta& shield.

%he debt holders and stock holders each have a claim on the cash flows of the firm. In a giventime period, the debt holders have a claim equal to the interest payments during that period plus

any principal payments that are due. %he stock holders then have a claim equal to the unleveredfree cash flow in that period plus the cash generated by the interest ta& shield, minus the claimsof the debt holders.

Capital Structure%he proportion of a firm's capital structure supplied by debt and by equity is reported as either thedebt to equity ratio $4- or as the debt to value ratio $4=-, the latter of which is equal to the debtdivided by the sum of the debt and the equity.

1ne can quickly convert between the $4 ratio and the $4= ratio by using the followingrelationships*

$ 4 = 3 $ 4 - 4 > $ 4 -$ 4 3 $ 4 = - 4 > ) $ 4 = -

Ris" Pre'iu's Business riskis the risk associated with a firm's operations. It is the undiversifiable

volatility in the operating earnings I%-. usiness risk is affected by the firm'sinvestment decisions. A measure for the business risk is the asset beta, also known thelevered beta. In terms of the discount rate, the return on assets of a firm can bee&pressed as a function of the risk)free rate and the business risk premium 0:-*

rA3 rF 0: Financial riskis associated with the firm's capital structure. Financial risk magnifies the

business risk of a firm. Financial risk is affected by the firm's financing decision.

%he total corporate riskis the sum of the business and financial risks and is measured bythe equity beta, also known as the levered beta. %he business risk premium 0:- andfinancial risk premium F0:- are reflected in the levered equity- beta, and the return onlevered equity can be written as*

r3 rF 0: F0:

$ebt beta is a measure of the risk of a firm's defaulting on its debt. %he return on debt can be


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written as*

r$3 rF default risk premium

Cost of Capital%he cost of capital is the rate of return that must be reali+ed in order to satisfy investors. %he costof debt capital is the return demanded by investors in the firm's debt# this return largely is relatedto the interest the firm pays on its debt. In the past some managers believed that equity capitalhad no cost if no dividends were paid# however, equity investors incur an opportunity cost inowning the equity of the firm and they therefore demand a rate of return comparable to what theycould earn by investing in securities of comparable risk.

%he return required by debt holders is found by applying the CA:9*

r$3 rF betadebt r9) rF-

%he required rate of return on assets that is, on unlevered equity- can be found using the CA:9*

rA3 rF betaunlevered r9) rF-

?sing the CA:9, a firm's required return on equity is calculated as*

r3 rF betalevered r9) rF-

?nder the 9odigliani)9iller assumptions of constant cash flows and constant debt level, therequired return on equity is*

r3 rA >)t- rA) r$-$ 4 -

where t is the corporate ta& rate.

%he overall cost of capital is a weighted)average of the cost of its equity capital and the after)ta&

cost of its debt capital. %he weighted average cost of capital /ACC- then is given by*

/ACC 3 r 4 =8- r$>)t- $ 4 =8-

Assuming perpetuities for the cash flows, the weighted average cost of capital can be calculatedas*

/ACC 3 rA@ > ) t$ 4 =8-

(eglecting ta&es, the /ACC would be equal to the e&pected return on assets because the/ACC is the return on a portfolio of all the firm's equity and all of its debt, and such a portfolioessentially has claim to all of the firm's assets.

For arbitrary cash flows, and under the assumption that the debt to value ratio is held constant,the following relationship derived by Bames A. 9iles and Bohn 0. ++ell is applicable*

/ACC 3 rA) t r$$ 4 =8->rA- 4 >r$-

?nder the same assumptions, the cost of equity capital can be calculated from rAand r$using thefollowing relationship from 9iles and ++ell*

r3 rA @ > ) t r$4 >r$- @ rA) r$ $4


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For low values of r$, @ > ) t r$4 >r$- is appro&imately equal to one, and the e&pression can besimplified if high precision is not required.

If one cannot assume a constant debt to value ratio, then the A:= method should be used.

Esti'atin) BetaIn order to use the CA:9 to calculate the return on assets or the return on equity, one needs toestimate the asset unlevered- beta or the equity levered- beta of the firm. %he beta that often isreported for a stock is the levered beta for the firm. /hen estimating a beta for a particular line ofbusiness, it is better to use the beta of an e&isting firm in that e&act line of business a pure play-rather than an average beta of several firms in similar lines of business that are not e&actly thesame.

&pressing the levered beta, unlevered beta, and debt beta in terms of the covariance of theircorresponding returns with that of the market, one can derive an e&pression relating the threebetas. %his relationship between the betas is*

betalevered3 betaunlevered@ > > ) t- $4 ) betadebt>) t - $4

betaunlevered3 @ betalevered betadebt>) t- $4 4 @ > > ) t- $4

%he debt beta can be estimated using CA:9 given the risk)free rate, bond yield, and market riskpremium.

#nlevered &ree Cash &lo%s/hen valuing the operations of a firm using a discounted cash flow model, the operating cashflow is needed. %his operating cash flow also is called the unlevered free cash flow ?FCF-. %heterm


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have a positive impact on cash flow.

*er'inal Value%he terminal value of a firm is the value at the end of the last year that the unique cash flows areproEected. %his value is the discounted value of all subsequent cash flows. /hile the terminalvalue associated with a piece of manufacturing equipment typically is less than >G of thepresent value, the terminal value associated with a business often is more than 7G of the totalpresent value. For this reason, the terminal value calculation often is critical in performing avaluation. %he terminal value can be calculated either based on the value if liquidated or basedon the value of the firm as an ongoing concern.

If the firm is to be liquidated, the liquidation value can be based on book value, salvage value, orbreak)up value, but liquidation value usually understates the terminal value of a healthy business.1ne must make assumptions about the salvage value of the assets and net working capital. %henet working capital may have a certain recovery rate since it might not be readily liquidated atbalance sheet values. In the pro forma proEections, one often may assume that net workingcapital will grow at the same rate as cash flow. %he terminal value if the firm is liquidated then isthe sum of the discounted value of the cash flow, the recovered net working capital, and thesalvage value of the long)term assets, including any ta& benefits.

For an on)going firm, the terminal value may be determined by either using discounted cash flowestimates or by using multiples from comparable firms. For the $CF method, if the ?FCF isgrowing at a rate of gper year for a set number of years, the terminal value can be calculated bymodeling the cash flow as a %)year growing perpetuity. At the end of % years, one can assume adifferent growth rate possibly +ero- or liquidation. If multiples from comparable firms are used,the price4earnings ratio, market4book values, or cash flow multiples are commonly used.

%he unlevered terminal value is calculated using rA as the discount rate, and the levered terminalvalue is calculated using the /ACC as the discount rate.

%he terminal value of debt or preferred stock is simply the proEected book value of the debt orpreferred stock in the year that the terminal value is being calculated.

%he terminal value of the common stock is the total levered terminal value less the terminal valueof the debt, less the terminal value of the preferred stock adding in the amount from any warrantsthat are e&ercised at their e&ercise price-, plus the cash gained from the e&ercise of any commonand preferred warrants.

*hree (iscounted Cash &lo% +ethods for Valuin) evered Assets

A:= AdEusted :resent =alue- 9ethod%he A:= approach first performs the valuation under an unlevered all)equity assumption, thenadEusts this value for the effect of the interest ta& shield. ?sing this approach,

=83 =? :=I%"where =83 value if levered=?3 value if financed >G with equity:=I%" 3 present value of interest ta& shield

%he unlevered value is found by discounting the unlevered free cash flow at the required returnon assets. %he present value of the interest ta& shield is found by discounting the interest ta&shield savings at the required return on debt, r$.


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%he A:= method is useful for valuing firms with a changing capital structure since the return onassets is independent of capital structure. For e&ample, in a leveraged buyout, the debt to equityratio gradually declines, so the required return on equity and the weighted average cost of capitalchange as the lenders are repaid. !owever, when calculating the terminal value it may beappropriate to assume a stable capital structure, so in calculating the terminal value in aleveraged buyout situation the /ACC method may be a better approach.

Flows to quity 9ethod%he flows to equity method sums the (:= of the cash flows to equity and to debt.

%hen, =83 $

/ACC 9ethod%he /ACC method discounts the unlevered free cash flow at the weighted average cost ofcapital to arrive at the levered value of the firm.

Cash &lo%s to (ebt and E!uity

/hen calculating the amount of cash flowing to debt and equity holders, it is not appropriate touse the unlevered free cash flows because these cash flows do not reflect the ta& savings fromthe interest paid. "tarting with the ?FCF, add back the ta&es saved to obtain the total amount ofcash available to suppliers of capital.

urdle Price

At times a firm may wish to know at what price it would have to sell its product for a particularinvestment to have a positive net present value. A procedure for determining this price is asfollows*

&press the operating cash flow in terms of price. %here may be multiple phases such as

a short start)up period, a long operating period, and a final year in which the terminalvalue is calculated.

/rite out the e&pression for the (:= using the appropriate discount rate. For the longeroperating period, one can calculate an annuity factor to multiply by the operating cashflow e&pression. "olve the e&pression for the cash flow that would result in an (:= of+ero.

"ince the operating cash flow was written in terms of price, the price now can be found.

(ebt Valuation

/hile debt may be issued at a particular face value and coupon rate, as the debt value changesas market interest rates change. %he debt can be valued by determining the present value of the

cash flows by discounting the coupon payments at the market rate of interest for debt of the sameduration and rating. %he final period's cash flow will include the final coupon payment and theface value of the bond.

Invest'ent (ecisionIf the unlevered (:= of a proEect is negative, aside from potential strategic benefits, the proEect isdestroying value, even if the levered (:= is positive. %he firm always could benefit from the ta&shield of debt by borrowing money and putting it to other uses such as stock buybacks.


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-pti'al Capital Structure%he total value of a firm is the sum of the value of its equity and the value of its debt. %he optimalcapital structure is the amount of debt and equity that ma&imi+es the value of the firm.

Share Buybac"If a firm has e&tra cash on hand it may choose to buy back some of its outstanding shares. 1neinteresting aspect of such transactions is that they can be based on information that the firm hasthat the market does not have. %herefore, a share buyback could serve as a signal that the shareprice has potential to rise at above average rates.

+er)ers and Ac!uisitionsy some definitions, a merger is a type of an acquisition in which one firm acquires all of theassets and liabilities of another to form a combined business entity. In a merger of firms that areappro&imate equals, there often is an e&change of stock in which one firm issues new shares tothe shareholders of the other firm at a certain ratio. For the sake of this discussion, the firm'swhose shares continue to e&ist possibly under a different company name- will be referred to as

the acquiring firm and the firm's whose shares are being replaced by the acquiring firm will bereferred to as the target firm. &cluding any synergies resulting from the merger, the total post)merger value of the two firms is equal to the pre)merger value. !owever, the individual post)merger values of the firms likely will be different from their pre)merger values because thee&change ratio of the shares probably will not e&actly reflect the firms' values with respect to oneanother. %he e&change ratio is skewed because the target's shareholders are paid a premium fortheir shares.

"ynergies take the form of revenue enhancements and cost savings. /hen two companies in thesame industry merge, such as two banks, the revenue

For the merger to make sense for the acquiring firm's shareholders, the synergies resulting fromthe merger must be more than the initial lost value. %he minimum required synergies are found

solving for the synergies in the following equation*

pre)merger value of both firms synergies-)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 3 pre)merger stock pricepost)merger number of shares

where the pre)merger stock price refers to the price of the acquiring firm.

Appendi&

Co'poundin) and (iscountin)

Continuous compounding* F=t 3 C e r t

:erpetuity* := 3 C 4 r

2rowing perpetuity* := 3 C 4 r )g -

%)year annuity % equally spaced payments-* := 3 C 4 r - @ > ) >4>r-%


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%)year growing := 3 @C 4 r)g- H>) @>g- 4 >r-%


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Investment 9anagement

Investment management is about attaining investment obEectives under specified constraints# fore&ample, achieving the best possible return for a given level of risk. %o meet these obEectives, the

investor may buy equity in an asset such a stock, a fund, or real estate, or buy debt issued bygovernments and corporations. y effectively managing such investments the investmentmanager can achieve a higher return for a specified acceptable level of risk. %here are manytools for reaching this goal.

Expected Return and Portfolio Variance%he two basic metrics for an investment portfolio are the return and the variance.In the case of an individual dividend)paying stock, the return is given by*0i3 @:> $>- 4 : ) >,where $> is the dividend paid at time t 3 >.

%he future return of a stock or a portfolio is not known with certainty# there are different

probabilites for different return scenarios, one of which actually will unfold.2iven npossible return scenarios, each with its own probabilitypi, the e&pected return is*0- 3 i3>,n pi0i%he variance of such a stock or portfolio is given by*J= i3>,n pi@0i) 0-J

PortfoliosA portfolio has certain advantages over a single security. %he return of one security may tend tomove in the same direction as the return of another security, but in the opposite direction of thereturn of a third security. ecause of these tendencies, when securities are grouped into aportfolio, for a given e&pected return the variance of that return can be reduced. %he Eointtendencies between the returns can be measured by covariances.%he covariance in two securities' returns is given by*

Cov0>, 0J) = >J3 >J>J%he correlation coefficient between security iand the market is given by*im= im/ imFor two securities,

Jp = i3>,nE3>,n&i&EiEJp 3 &J>J> &JJJJ J &> &J >J3 &J>J> &JJJJ J &> &J >J>Jwhere &J3 > ) &>(ote that if %)bills that earn the risk)free rate are included, for RF3 .2iven two securities, many different portfolios can be constructed by varying the weighting ofeach security in the portfolio. %o find the minimum variance portfolio,set d>4 d&>3 3K &>= (JJ- >J>J) / (J>+ JJ- 2 >J>J-For an equally weighted portfolio with all standard deviations equal and all covariances equal to+ero*=ar0p- 3 >4(J) i3>,n=ar0i-3 >4(- =ar0i-= (1/N) Jiandp 3 >4(>4J) i


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Ris" Adusted Return$ifferent investors have different aversions to risk. /hen managing a portfolio for a particularinvestor, the goal is to ma&imi+e the portfolio return for the level of risk that the investor is willingto take. %he following model can be used*9a&imi+e L 3 0p- ) A =ar0p-

where A 3 investor's aversion to risk as measured by the variance of the portfolio return.%o ma&imi+e the function assuming the investor's assets are only in the market portfolio and theriskfree asset,first let wm3 the fraction of assets in the market portfolio. %hen0p- 3 rF wm0m) rF-and=ar0p- 3 wJmJm.%henL 3 rF wm@0m-) rF ) .7 A wJmJmanddL4dwm3 0m- ) rF) A wmJm3 "olving forA,

A 3 @0m- ) rF 4 wmJ

m-Beta%he risk of an individual security in a well diversified portfolio can be measured by its beta. "uchrisk is nondiversifiable.eta of an individual security with respect to the market is*

im= im/ Jm 3 Cov0i, 0":7- 4 =ar0":7-

eta of a risk)free asset with respect to the market 3 .etas determined using historical data are subEect to estimation error. 9errill 8ynch and someother firms adEust this value back towards the mean beta of the market 3>- or industry usingadEusted3 whistorical >)w) 0>- >)w>- 0J-

%he covariance between any two portfolios on the efficient frontier can be found by finding theweights needed to emulate the market portfolio and then solving for N >Jin the following equation*Jp 3 wJ>J> wJJJJ J w> wJ >JCAP+%he "harpe)8intner version of the capital asset pricing model implies that as a result of allinvestors holding the market portfolio, there is a linear relation between the e&pected return on asecurity and its M.%he following is the security market line ) any security's e&pected return will lie on this line. %hisline applies to all securities, not Eust efficient portfolios.


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0i- 3 0F @ 0m- ) 0F] im"harpe)8intner0i- 3 0+ @ 0m- ) 0+] imlack&pected return of a portfolio using CA:9*0p- 3 0F @ 0m- ) 0F] pmIf the assumption of equal borrowing and lending rates is rela&ed, investors no longer arerequired to hold the market portfolio# instead, they can hold a range of portfolios along theefficient frontier between the point of tangency of the lending line and the point of tangency of theborrowing line.CA:9 requires the measure of two unknown quantities ) market risk premium and beta.!owever, attempts to estimate e&pected returns by using historical stock return data haveresulted in std errors about double those of CA:9, because for CA:9 the better precision in theestimate of the market risk premium more than offsets the additional estimation error in beta.%here have been many difficulties in testing CA:9. 0oll argued that the CA:9 must always holdfor ex postdata if the pro&y chosen for the market is efficient. !e also argued that it is impossibleto measure the true market, so the CA:9 cannot be tested. !owever, in >OPJ "tambaugh foundthat adding other risky assets such as corporate bonds, real estate, and consumer durables tothe market portfolio did not materially affect the tests.

Sin)le &actor +odel /+ar"et +odel00it3 ai+ i0mt eitt = 1, ..., where eit is the distance from the regression line at time t. %he mean value of e it3 and thecovariance between 0mand ei3 . %his is a regression model that characteri+es the risk of asecurity over time by measuring its beta over a time interval. M iis different from the Mimused in theCA:9 in that Mimis more of a present)day beta rather than one taken over time. In the traditionalapproach of testing the CA:9, in the first step one uses this model to measure the beta of allsecurities or portfolios-. In the second step one estimates the CA:9 itself by regressing thesecurity returns on the estimated betas. /hen testing CA:9 in this manner, one must questionthe validity of tests using ex postdata to test the ex anteCA:9. Also, there is measurement errorin individual security betas. ?sing portfolios instead in the first)pass regression helps.=ariance using the single)factor model*=ar0i) = Ji=ar0m- =arei-where 0i, 0m, and ei are random variables. %he variance of the mean return a i is +ero bydefinition, so this term falls out. In a well)diversified portfolio, =are i- 3 . In this equation, M Ji=ar0m- is the variance e&plained by the market. %he percent of variance e&plained by the marketthen is given byJi=ar0m) / Ji3 0J

(ote that >)0J- is the idiosyncratic variance.%hese e&pressions apply to portfolios as well by replacing i with p.For two portfolios or securities in which their e i's are uncorrelated, the covariance between themis given by*iE= i E

Jm.

%his is derived by finding the covariance between*0it3 ai+ i0mt eitand 0Et3 ai+ E0mt eEtCross Section of Co''on Stoc" Returns

Fama and French used a multi)factor model using additional risk factors related to si+e,price4book, etc.%hey concluded that three


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+ar"et12eutral Strate)ies9arket)neutral strategies balance the market risk by going long on some securities and short onothers. "ome people propose using the %)bill rate as a benchmark against which to compare themarket return of a such a strategy. 1ne can argue that even though the market)neutral strategy isrisky, since it has +ero beta it does not contribute to the risk of the market portfolio and thereforeshould not command a premium over the risk)free rate. 1n the other hand, if the e&pected returnson the long side are higher than those on the short, the benchmark return should e&ceed the risk)free rate.

*radin) CostsAn important factor in the performance of high)turnover portfolios is the amount of the tradingcosts,including e&plicit costs such as commissions, fees, and ta&es, the market maker spread,the impact of trading on market price, and the opportunity cost incurred during the delay betweenthe time the decision is made and the time the trade is e&ecuted.%rading costs can be reduced through passive fund management and electronic trading.

on)1*er' Investin)%he conventional wisdom is that over the long run, stock will generate returns superior to those ofbonds. ut while the variance of the geometric means of the returns declines as the time hori+onincreases, the variance of the terminal wealth increases. If a put option were purchased to insurea certain terminal wealth, the cost of that option would increase as the time hori+on increases. %othe e&tent that option prices are a measure of risk, the risk of stock investments then increases asthe time hori+on lengthens.%he optimal asset allocation is a function of the present wealth, target future wealth, risktolerance, and time hori+on. 8ong)term returns are difficult to analy+e statistically because as thehistorical time hori+on increases, the number of possible independent samples of returnsdecreases. In >OO>, utler and $omian illustrated a procedure that attempts to overcome thisdifficulty by first listing the monthly returns for the "R: 7 and the long)term bonds over a longhistorical time hori+on. y randomly selecting data, returns over various long)term holding periods

can be emulated by multiplying the appropriate number of random samples. An almost limitlessnumber of samples for each holding period can be generated using this method. :erforming suchan analysis with data taken from the SOJ months from >OJ6)>OO> indicates that over a >)yeartime period, there is an >>G chance that stocks will underperform bonds# over a J)year timeperiod this probability reduces to 7G.

(efined1Benefit Pension PlansIn a defined)benefit plan, the plan sponsor usually an employer- guarantees a level of futurebenefits to the plan participants, taking responsibility for any shortfall in the investmentperformance of the plan. FA" PS requires that any unfunded liability in the present value of thebenefits appear on the balance sheet of the employer. 1ne alternative for the plan sponsor is toplace the present value of the plan liability into government bonds of the same duration as the

liability, in which case there is no chance of shortfall and the liability is fully immuni+ed.Furthermore, because pension plans are not ta&ed, the incentive to hold equity in order to takeadvantage of lower ta&es on capital gains is diminished. For a given level of risk, ta& implicationsincrease the return most for investments such as bonds, which have a large spread between pre)ta& and after)ta& return. An alternative to bonds is for the pension fund to place its money intoriskier assets such as common stocks. ?nder this latter alternative, there e&ists both the chanceof a shortfall and the chance of a surplus. !owever, FA" PS does not permit a surplus to bereported as an asset on the sponsor's balance sheet, and the surplus often gets allocated to theplan participants. (onetheless, for many reasons it is common for firms to hold equity in theirpension funds. %he :ension enefit 2uarantee Corporation a federal agency- guarantees the
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benefits, and the sponsor's premiums are independent of the risk level of the pension fund'sinvestments. For employers in financial distress, the pension guarantee from :2C effectively isa put option. 2iven that put options increase in value as risk increases, there is an incentive forsome firms to invest the pension fund in risky assets.In defined)benefit plans there e&ists the opportunity for ta& arbitrage. %he plan sponsor can issuedebt in order to buy equity in the pension plan. %he pension plan then can invest the funds inbonds. ecause of the ta& status of the pension fund, the ta&es on the pension plan's bondinterest will be deferred, and the sponsor will enEoy the interest ta& shield from its debt issuance.%he sponsor then reali+es an arbitrage profit equal to the interest rate multiplied by the corporateta& rate, with no increase in the firm's overall risk.

Arbitra)e Pricin) *heoryIn >OS6, "teve 0oss presented the arbitrage pricing theory A:%- as an alternative to the CA:9that requires fewer assumptions. %he A:% is an equilibrium theory, which differs from a factormodel in that it specifies relationships between e&pected returns across securities and attributesthat influence those securities. A factor model allows the first term in the model, the e&pectedreturn, to differ across securities and therefore can represent either and efficient or an inefficientmarket. 0oss assumed that returns have the first term in common, and the other terms dependon several different systematic factors, as opposed to the single market risk premium factor of the

CA:9. %he model takes the form*Rpt= !"Rp# $ %p1&1t$ %p'&'t$ ... $ %p(&(t$ eptwhere &i3 value of the it)factor,%pi3 sensitivity of the return to the it)factor, k3 number of factors,and eptequals the idiosyncratic variation in the return. Assuming an efficient market in equilibrium,the first term to the right of the equal sign is the same for all securities and is appro&imately equalto the risk)free rate. &amples of factors that could be included in the model are monthlyindustrial production, changes in e&pected inflation, une&pected inflation, une&pected changes inthe risk premium, and une&pected shifts in the term structure of interest rates. "uch variableslikely affect most or all stocks.

+ar"et12eutral Strate)ies Revisited2iven that


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and reports of portfolio holdings may not be very representative since they are only snapshotstaken at one point in time. %his limitation makes the public's view of the holdings subEect todistortions such as . $efine the benchmark return for period tto be*RBenc)*ark,t= w1R1,t$ w'R',t$ w+R+,t$ wR,t$ w-R-,t

J. $efine the tracking error to be*et= RFund,t RBenc)*ark,t

5. "olve for the weights by minimi+ing the standard deviation of the mean tracking errorover the entire time period being analy+ed under the constraint that the weights sum toone and are each greater than or equal to +ero unless net short positions are permitted

in the fund-. %he standard deviation of the tracking error is given by*(e) = [ ( 1 / T-1 ) t=1,"et e*ean#

'>4J

Evaluatin) &und Perfor'ance/hen the popular press publishes mutual fund performance rankings, it usually does not considerthe risk that the portfolio manager took to achieve that return. "uch rankings do not necessarilyreflect the skill of the manager. %o adEust for risk, one should consider the ratio of e&cess returnsto risk, or consider risk)adEusted differential returns. For the risk, one can use standard deviationsor betas.%he


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Coupon bearing notes and bonds typically make fi&ed interest payments two times per year. Lerocoupon bonds are sold at a discount and pay off their face values at maturity. Lero coupontreasury securities are issued by commercial institutions who separate the interest and principalpayments. %hese +ero coupon bonds are known as CA%'s, %I20's, and "%0I:'s.ond prices often are quoted in the format &*y, where & is the integer dollar amount and y is thefractional amount in 5Jnd's of a dollar.%he spot rate is the rate that would correspond to a single cash flow at maturity for a bondpurchased today, as is the case with a +ero coupon bond. A notation used for spot rates is rn,where n is the number of periods e.g. years- into the future when a loan made today is tomature. %he forward rate is the rate at which a future loan is made today. A notation used forforward rates is 3*,n, where * is the number of periods from the present when the loan is tocommence, and nis the number of periods into the future when the loan is to end. Forward ratescan be e&pressed in terms of spot rates*> 3*,n= " > rn# " > r*#%he ask price of a ?.". %reasury bill is calculated from the , @ > ) asked rate ( 4 56 - where ( 3 the number of days until maturity.%he implied rate spot rate- is >, 4 Ask :rice ) > -. %his implied rate does not represent anannuali+ed basis. %he annuali+ed rate is found by raising the implied rate to the 5674( power*

Annuali+ed 0ate 3 Implied 0ate -567 4 (%he bond equivalent yield is the yield to maturity ythat satisfies the following equation*: 3 Tn3>,(Cn4 > y4J -n

where : 3 price, Cn3 cash flow at the end of each period, ( 3 number of periods.For a +ero coupon bond there is only one cash flow at maturity.%he value of a coupon bond can be modeled as a portfolio of +ero)coupon bonds having facevalues and maturity dates that correspond to the coupon payments and dates. "umming theprices of the +ero)coupon bonds then would give the value of the coupon bond, and anydifference would represent an arbitrage opportunity.Forward rates can be calculated using the prices and returns of bills, notes, or bonds providedthey cover the proper time periods. For e&ample, given the si& month spot rate r4.-, one cancalculate the one year spot rate r1.4by using the data for a one year note the following equation*5rice = coupon1 "1$r4.-# $ "coupon' $ 3ace value# "1$r1.4#

1nce the spot rates are known, the forward rate can be calculated as already illustrated.%he spot rate is not quoted on an annuali+ed basis. %o annuali+e it*Annuali6ed 7ield = "8pot Rate#xy

wherexis the number of periods in one year, and yis the number of periods included in the spotrate.%he duration of a bond often is thought of in terms of time until maturity. !owever, in addition tothe payoff of the face value at maturity, there are the coupon payment cash flows that influenceeffective duration. %wo bond with equal yield)to)maturities and maturity dates will have differenteffective durations if their coupon rates are different. Frederick 9acaulay suggested the followingmethod of determining duration*ffective $uration 3 Tt3>,%t H @ Ct4 >y4J -t 4 @Tt3>,%Ct4 >y4J -t where % 3 life of the bond in semiannual periods,Ct3 cash flow at end of t thsemiannual period,

y 3 yield to maturity, e&pressed as a bond)equivalent yield.A +ero)coupon bond has no coupon payments and therefore its effective duration always is equalto the time until maturity and does not change as yield)to)maturity changes. $uration essentiallymeasures the sensitivity of a bond's price to movements in interest rates. y this definition,duration is defined as9 : " 9 5 5 # / 9 " 1 $ r # " 1 $ r # 0If one plots the price of a non)callable bond as a function of its yield, the plot will be concave upconve& down- rather than linear. %his curvature is called conve&ity, and in this case, positiveconve&ity. Conve&ity is due to the fact that effective duration increases as interest rates decrease.


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the Customer's Afternoon 8etter, which later became the /all "treet Bournal. At that time, therewere O railroad stocks and J industrial stocks in the inde&. In >PPV, the railroads were the largestand most stable companies. %he stocks of industrial companies were considered speculativeinvestments. In >PO6, Charles $ow introduced an inde& for industrial stocks and the original $owaverage became a railroad stock inde&. 9ore companies were added to the industrial inde& until>OJP, when the number was increased to 5.

%he $ow Bones Industrial Average uses a divisor to adEust for events that result in no change in acompany's value but that would otherwise influence the inde&. 1ne such event is a stock split#another is the replacement of one company in the inde& by another. /hile this adustment doesnot result in a change in the inde& value when a stock splits, because the inde& is price)weightedthe newly split stock will have a lower price and therefore less influence on the inde&.

$ow Bones %ransportation Average

%he $ow Bones %ransportation Average is a price)weighted inde&. It originated from the inde& ofO railroad stocks and J industrial stocks that $ow Bones R Co. introduced in >PPV. In >PO6 whenthe original inde& became the $ow Bones 0ailroad Average the industrial stocks were removedfrom it. 8ater, the 0ailroad Average was renamed to the %ransportation Average. In addition to

railroads, today the average includes other transportation stocks such as airlines and truckingcompanies.

$ow Bones ?tility Average

%he $ow Bones ?tility Average is a price)weighted inde& of >7 utility stocks, especially electricutilities and gas utilities. It was created in >OJO with >P stocks, was increased to J stocks si&months later, then reduced to >7 stocks in >O5P.

(asdaq Composite Inde&

%he (asdaq Composite Inde& is a market capitali+ation weighted inde& of more than 7 stocks.Comprising all (asdaq)listed common stocks, it is the most commonly used inde& for tracking the(asdaq.

0ussell J

%he 0ussell J is a market capitali+ation weighted inde&. It was created in >OPV by the Frank0ussell Company. %he 0ussell universe of stocks covers 5 companies, and the 0ussell Jrepresents the smallest two)thirds of those companies. As such, it is a small)cap inde&.

"R: >

%he "R: > is a market capitali+ation weighted inde& of large)cap companies. %his inde& also isknown by its ticker symbol, 1. It comprises > large blue)chip companies across a widerange of industries.

"R: 7


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%he "R: 7 is a market cap weighted inde& of large)cap companies from a variety of industries.It includes industrial, utility, transportation, and financial stocks. %he "R: 7 is widely used as abenchmark by institutional investors.

=alue 8ine Composite Inde&

%he =alue 8ine Composite Inde& is a broad, unweighted inde& of appro&imately >S companiescovered in the =alue 8ine Investment "urvey.

/ilshire 7

%he /ilshire 7 is a market capitali+ation weighted inde&. It was created by /ilshireAssociates in >OSV. It is the broadest inde&, including virtually every actively traded ?.". stock.


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%rading Costs

%he cost associated with trading securities can have a non)negligible impact on portfolio return.%rading costs include the following*&plicit costs ) commissions, fees, and ta&es.

9arket maker spread ) difference between the bid and ask prices that the specialist sets for astock# the specialist keeps the difference as compensation for providing immediacy. For lessliquid stocks, the specialist has greater e&posure to adverse price movements and likely will makethe spread larger.

9arket impact ) results when high volume trades influence the market price. 9arket impact canbe broken into two components ) a temporary one and a permanent one. %he temporarycomponent is due to the need for liquidity to fill the order. %he permanent impact is due to thechange in the market's perception of the security as a result of the block trade.

1pportunity cost ) the effective cost of price movements that occur before the trade e&ecutes.(D" specialists sometimes may appear to have a monopoly on trading their respective

securities, creating a larger than necessary spread between bid and ask. !owever, there is morecompetition than is initially obvious. First, there is competition for the specialist positions,providing the specialist incentive to price fairly. Furthermore, there are other specialists on thefloor who may be willing to trade within the spread if it is too wide.

%he total trading cost of a buy transaction is calculated by taking the percentage increase of theaverage purchase price as compared to the price when the buy decision was made, and addingthe commissions, fees, and ta&es as a percentage of the price when the buy decision was made.

Active portfolio managers attempt to outperform passive benchmarks, but trading costs reduceany reali+ed advantages. %ypical trading commissions run .JG of the transaction amount, andthe typical cost due to bid)ask spread and market impact is .77G. %he total cost of a trade thenis .S7G of the trade amount. If a fund has a portfolio turnover rate of PG, and for every sell

transaction the stock is replaced via a buy transaction, a total of >6G of the portfolio value willbe transacted each year. For trading costs of .S7G per transaction, the annual trading costsamount to >.6-.S7G- 3 >.JG of the portfolio value. If one adds a .5G management fee to thisamount, the total becomes >.7G.

0educing %rading Costs* :assively %raded Funds

:assive portfolios have lower transaction costs and overall trading costs. %he transaction cost istypically .J7G of the transaction value, since a passive portfolio does not have to trade asquickly and can be more patient with each transaction. A typical turnover rate for a passiveportfolio is about VG per year, and assuming replacement PG of the portfolio value will betransacted each year for annual trading costs of only .P-.J7G- 3 .JG of the portfolio value.

:assive portfolios have lower management fees, for e&ample, .>G, so the total of trading costsand management fees is only .>JG, compared to >.7G for a typical actively managed fund.

:assively managed funds that track an inde& often have returns less than that of the inde&because of trading costs, especially for small)cap indices in which the securities are less liquid.%hese trading costs can be reduced if the weights of the securities in the fund are allowed todeviate somewhat from the inde&, since both trading volume and the need for immediacy arereduced. %he correlation with the inde& still can remain quite high under the rela&ed weights.


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In >OPJ $imensional Fund Advisors $FA- introduced a passive small)cap < fund composedof the lower two deciles of (D" market capitali+ation. %he fund sacrificed tracking accuracy byallowing the weights to deviate in order to minimi+e trading costs. %he result was higherperformance than other small)cap funds. %he O)> fund even outperformed the stocks in thelower two market capitali+ation deciles of the (D", partly due to the following strategies*

%he Pth decile is treated as a hold range, not a sell range,

%he $FA waits a minimum of one year before buying I:1's,

%he fund does not buy stocks selling for less than UJ or having less than U> million in marketcapitali+ation,

%he fund does not buy (A"$A; stocks having fewer than four market makers,

%he fund does not buy bankrupt stocks, and

%he fund is passive, not rigidly inde&ed.(ote that using the Pth decile as a hold range effectively increases the average market cap of theportfolio and increases returns in periods in which large caps outperform small caps, such as in

the >OP's.

0educing %rading Costs* lectronic %radinglectronic crossing networks have lower trading costs than do e&changes because of lowercommissions, no bid)ask spread, and elimination of market impact. y matching the naturalbuyers and sellers of a security at some predetermined price, for e&ample, the (D" closingprice, electronic crossing networks eliminate the need for a market maker to provide liquidity.!owever, crossing networks require buyers and sellers to participate in order for there to beliquidity. Furthermore, there are the disadvantages of potentially limited liquidity and no inherentprice discovery mechanism.

lectronic communications networks are computeri+ed bulletin boards for matching trades.ecause the traders can remain anonymous, price impact is diminished.

Another electronic trading mechanism is the single)price call auction in which buyers and sellerssimply place limit orders. %he market clearing price is set at the intersection of the supply anddemand curves.


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"ecurity Analysis

Introduction

"ecurity analysis is about valuing the assets, debt, warrants, and equity of companies from theperspective of outside investors using publicly available information. %he security analyst musthave a thorough understanding of financial statements, which are an important source of thisinformation. As such, the ability to value equity securities requires cross)disciplinary knowledge inboth finance and financial accounting.

/hile there is much overlap between the analytical tools used in security analysis and those usedin corporate finance, security analysis tends to take the perspective of potential investors,whereas corporate finance tends to take an inside perspective such as that of a corporatefinancial manager.

E!uity Value and Enterprise Value

%he equity value of a firm is simply its market capitali+ation# that is, the market price per sharemultiplied by the number of outstanding shares. %he enterprise value, also referred to as the firmvalue, is the equity value plus the net liabilities. %he enterprise value is the value of the productiveassets of the firm, not Eust its equity value, based on the accounting identity*

Assets 3 (et 8iabilities quity

(ote that net values of the assets and liabilities are used. Any cash and cash)equivalents wouldbe used to offset the liabilities and therefore are not included in the enterprise value.

As an analogy, imagine purchasing a house with a market value of U>,, for which the ownerhas U7, in equity and a U7, assumable mortgage. %o purchase the house, the newowner would pay U7, in cash and assume the U7, mortgage, for a total capital structure

of U>,. If UJ, of that market value were due to UJ, in cash locked in a safe in thebasement, and the owner pledged to leave the money in the house, the cash could be used topay down the U7, mortgage and the net assets would become UP, and the net liabilitieswould become U5,. %he


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significant liabilities that are part of operations.

%he cash flow to the enterprise approach values the equity of the firm as the value of theoperations less the value of the debt. %he value of the operations is the present value of thefuture free cash flows e&pected to be generated. %he free cash flow is calculated by taking theoperating earnings earnings e&cluding interest e&penses-, subtracting items that required cashbut that did not reduce reported earnings, and adding non)cash items that did reduce reportedearnings but that did not result in cash e&penditures. Interest and dividend payments are notsubtracted since we are calculating the free cash flow available to all capital providers, bothequity and debt, before financing. %he result is the cash generated by operations. %he free cashflow basically is the cash that would be available to shareholders if the firm had no debt ) the cashproduced by the business regardless of the way it is financed. %he e&pected future cash flow thenis discounted by the weighted average cost of capital to determine the enterprise value. %hevalue of the equity then is the enterprise value less the value of the debt.

/hen valuing cash flows, pro forma proEections are made a certain number of years into thefuture, then a terminal value is calculated for years thereafter and discounted back to the present.

&ree Cash &lo% Calculation

%he free cash flow FCF- is calculated by starting with the profits after ta&es, then adding backdepreciation that reduced earnings even though it was not a cash outflow, then adding back after)ta& interest since we are interested in the cash flow from operations-, and adding back any non)cash decrease in net working capital (/C-. For e&ample, if accounts receivable decreased, thisdecrease had a positive effect on cash flow.

If the accounting earnings are negative and the free cash flow is positive, the carry)forward ta&benefit will is in effect reali+ed in the current year and must be added to the FCF calculation.

evera)e

In >O7P, economists and now (obel laureates Franco 9odigliani and 9erton !. 9iller proposedthat the capital structure of a firm did not affect its value, assuming no ta&es, no bankruptcy costs,no transaction costs, that the firm's investment decisions are independent of capital structure, andthat managers, shareholders, and bondholders have the same information. %he mi& of debt andequity simply reallocates the cash flow between stockholders and bondholders, but the totalamount of the cash flow is independent of the capital structure. According to 9odigliani and9iller's first proposition, the value of the firm if levered equals the value if unlevered*

=83 =?

!owever, the assumptions behind :roposition I do not all hold. 1ne of the more unrealisticassumptions is that of no ta&es. "ince the firm benefits from the ta& deduction associated withinterest paid on the debt, the value of the levered firm becomes*

=83 =? tc$

where tc3 marginal corporate ta& rate.

/hen considering the effect of ta&es on firm value, it is worthwhile to consider ta&es from apotential investors point of view. For equity investors, the firm first must pay ta&es at thecorporate ta& rate, tc, then the investor must pay ta&es at the individual equity holder ta& rate, t e.%hen for debt holders,


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re3 rf > ) tc- be@ rm) rf > ) tc-

where tc3 corporate ta& rate.

1nce the e&pected return on equity and on debt are known, the weighted average cost of capitalcan be calculated using 9odigliani and 9iller's second proposition*

/ACC 3 re 4 $ - rd$ 4 $ -

%aking into account the ta& shield*

/ACC 3 re 4 $ - rd > ) tc- $ 4 $ -

For % 3 no ta& advantage for debt- the /ACC is equivalent to the return on assets, ra.

rdis calculated using the CA:9*

rd3 rf bd@ rm) rf > ) tc-

For a levered firm in an environment in which there are both corporate and personal income ta&es

and in which there is no ta& advantage to debt %3-, /ACC is equal to ra, and the above /ACCequation can be rearranged to solve for re*

re3 ra $4-@ra) rd>)tc-

From this equation it is evident that if a firm with a constant future free cash flow increases itsdebt)to)equity ratio, for e&ample by issuing debt and repurchasing some of its shares, its cost ofequity will increase.

raalso can be calculated directly by first obtaining a value for the asset beta, b a, and thenapplying the CA:9. %he asset beta is*

ba3 be 4 = - bd $ 4 = - > ) tc-

%hen return on assets is calculated as

ra3 rf > ) tc- ba@ rm) rf > ) tc-

In summary, for the case in which there is personal ta&ation and in which 9iller's quilibriumholds % 3 -, the following equations describe the e&pected returns on equity, debt, and assets*

re3 rf > ) tc- be@ rm) rf > ) tc- ra3 rf > ) tc- ba@ rm) rf > ) tc-

rd3 rf bd@ rm) rf > ) tc-

%he cost of capital also can be calculated using historical averages. %he arithmetic meangenerally is used for this calculation, though some argue that the geometric mean should beused.

Finally, the cost of equity can be determined from financial ratios. For e&ample, the cost ofunleveraged equity is*

re,?3 @ re,8 rf,debt > ) tc- $4 4 > $4 -


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re,83 b>g- 4 :4- g

where b 3 dividend payout ratio

g 3 > ) b - 01-

where >)b- 3 plowback ratio.

%he payout ratio can be calculated using dividend and earnings ratios*

b 3 $ividend 4 :rice - :rice 4 arnings-

Share Buy1Bac"

%ake a firm that is >G equity financed in an environment in which % is not equal to +ero# i.e.,there is a net ta& advantage to debt. If the firm decides to issue debt and buyback shares, thelevered value of the firm then is*

=83 =? % debt-

%he number of shares that could be repurchased then is*

n 3 debt- 4 price per share after relevering-

where the price per share after relevering is*

=84 original number of outstanding shares-

%he buyback will lower the firm's /ACC.

Proect Valuation

%he (:= of a capital investment made by a firm, assuming that the investment results in anannual free cash flow : received at the end of each year beginning with the first year, andassuming that the asset is financed using current debt4equity ratios, is equal to*

(:= 3 ): : 4 /ACC

3arrants and -ptions

/arrants are call options issued by a corporation. %hey tend to have longer durations than doe&change)traded call options.

In >OS5, Fischer lack and 9yron "choles published an option valuation model that today is

known as the lack)"choles model. %he formula calculates the price of a call option to be*C 3 " (d>- ) e)r%(dJ-

whereC 3 price of the call option" 3 price of the underlying stock 3 option e&ercise pricer 3 interest rate% 3 current time until e&piration


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( 3 area under the normal curved>3 @ ln"4- r sJ4J- % 4 s %>4J

dJ3 d>) s %>4J

:ut)call parity requires that*: 3 C ) " e)r%

%hen the price of a put option is*

: 3 e)r%()dJ- ) " ()d>-

/arrants require some modifications to the lack)"choles parameters. /hen warrants aree&ercised, the company typically issues new shares at the e&ercise price to fill the order. %heresulting increase in shares outstanding dilutes the share value. If there were nsharesoutstanding, and *warrants are e&ercised, a represents the percentage of the value of the firmthat is represented by the warrants, where

a 3 *4 * n-

/hen using the lack)"choles model to value the warrants, it is worthwhile to use total amountsinstead of per share amounts in order to better account for the dilution. %he current share price 8becomes the enterprise value less debt- to be acquired by the warrant holders. %he e&erciseprice is the total warrant e&ercise amount, adEusted for the fact that in paying cash to the firm toe&ercise the warrants, the warrant holders in effect are paying a portion of the cash, a, tothemselves. %he inputs to the model are outlined in the following table.

Blac"1Scholes Para'eters for Pricin) -ptions and 3arrants

InputPara'eter

-ption Pricin) 3arrant Pricin)

" current share price a =, where = is enterprise value minus debt.

e&ercise price per share total warrant e&ercise amount multiplied by > ) a.% current time to e&piration average % for warrantsr interest rate interest rate

sstandard deviation of stockreturn

standard deviation for returns on enterprise value,including warrants

Valuation Calculation

1nce the free cash flow and /ACC are known, the valuation calculation can be made. If the freecash flow is equally distributed across the year, an adEustment is necessary to shift the year)end

cash flows to mid)year. %his adEustment is performed by shifting the cash flow by one)half of ayear by multiplying the valuation by > /ACC ->4J.

%he enterprise value includes the value of any outstanding warrants. %he value of the warrantsmust be subtracted from the enterprise value to calculate the equity value. %his result is divided

by the current number of outstanding shares to yield the per share equity value.

PE$ Ratio


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As a rule of thumb, the :4 ratio of a stock should be equal to the earnings growth rate.9athematically, this can be shown as follows*

: 3 $ 4 re :=21

where : 3 price, $ 3 annual dividend, re3 return on equity, and :=21 3 present value of growthopportunities. For high growth firms, :=21 usually dominates $ 4 re. :=21 is equal to theearnings divided by the earnings growth rate

(ebt Valuation

%o valuing bonds, one calculates the e&pected payoffs considering the probability of default andthe amount that is recovered in case of default. It is assumed that the cash flow from therecovered amount is reali+ed at the end of the year of default.

%wo approaches to valuing bonds are*>. $iscount the e&pected cash flow at the e&pected bond return# orJ. $iscount the scheduled bond payments at the rating)adEusted yield)to)maturity.

9ethod >* $iscount the e&pected cash flow at the e&pected bond return

:rice 3 the sum for each year tof CF-t4 > rdebt-t

where CF-t3 e&pected cash flow in year t.

For a one year bond* : 3 CF- 4 @> rd-

%he e&pected bond return is the risk)adEusted discount rate, rdebt.%he e&pected cash flow is the cash flow considering the probability of default*

CFdebt- 3 p > C- F > ) p- l Fwhere p 3 probability of no default

l 3 recovery rate in case of default, percentage of face value-C 3 annual coupon rate of the bondF 3 face value of the bond

rdebtcan be calculated using the CA:9*

rdebt3 rf bdebtX"R:7

whereX"R:73 risk premium for the market portfoliobdebt3 covariance between rdebtand the market return#rf3 D%9 on a risk)free bond having the same maturity.

If bdebtis not known, it can be found using ordinary least squares regression.If p3 > no default risk-, then rdebt3 D%9.%he difference in rdebtand D%9 is the default risk.

9ethod J* $iscount the scheduled bond payments at the rating)adEusted yield)to)maturity

For this method, estimate the 0AD%9 by averaging the market D%9's of bonds in the samegroup. %he promised cash flows then are discounted at this rate.


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9arkov Chain 0epresentation

ond valuation can be modeled as a 9arkov Chain problem in which a transition matri& isconstructed for the probabilities of moving from one rating to another. For e&ample, if there arefive possible ratings* A,,C,$,, and F# and p&yrepresents the probability of moving from statexto state y, then the transition matri& would look like the following*

pAA pA pAC pA$ pApA p pC p$ ppCA pC pCC pC$ pCp$A p$ p$C p$$ p$pA p pC p$ p

For multiple periods, the transition matrices for each period must be multiplied in order tocalculate the multi)period probabilities. %his multiplication easily can be performed by

spreadsheet software.

Accountin) Issues

$epreciation of goodwill is an e&pense on the income statement, but unlike depreciation of fi&edassets, depreciation of goodwill is not ta& deductible.

$lossary

A:=* AdEusted :resent =alue

CA:9* Capital Asset :ricing 9odel

nterprise =alue* 9arket value of a firm's equity plus the net market value of its debt. nterprise value 3 market cap 8%$ ) net cash R investments

FCF* Free Cash Flow

8%$* 8ong)%erm $ebt

90:* 9arket risk premium, defined as rm) rf, unless it specifically is referred to as ta&)adEustedmarket risk premium, in which case there would be a factor to adEust rffor ta&es.

(1:8A%* (et 1perating :rofits 8ess AdEusted %a&es

18"* 1rdinary 8east "quares method of regression-

:2* %he ratio of :4 to growth rate in earnings.

0A$0* 0isk AdEusted $iscount 0ate

0AD%9* 0ating)AdEusted Dield)%o)9aturity

01* 0eturn 1n quity# equivalent to the e&pected return on retained earnings


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D%9* Dield %o 9aturity

Quick MBA Finance Primer

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