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**Unformatted text preview: **Equation T—l. The Expected Value, or Mean (u), of a Probability Distribution: Ll = Z (V P)
where: p = the expected value, or mean
2 = the sum of V = the possible value for some variable P = the probability of the value V occurring Equation T—Z. The Standard Deviation: = it: PM! — pf
where: o = the standard deviation
2 = the sum of P = the probability of the value V occurring
V = the possible value for a variable [1 = the expected value Equation 7-3. The Coefﬁcient of Variation of a Probability Distribution: CV = Standard Tiles-'ialion
Mean Equation 14. The Expected Rate of Return, Epr), of a portfolio comprised of
Two Assets, A and B: HIE.) = (Wa HRH?!) + (Wb E0213} where: BER”) = the expected rate of return of the portfolio composed of Asset A
and Asset B wa = the weight of Asset A in the portfolio
EfRa] = the expected rate of return of As set A
wb = the weight of Asset B in the portfolio
Ebe] = the expected rate of return of As set E Equation 7-5. The Standard Deviation of a Two—Asset Portfolio: (Ff! = WIIWJGJ + + 2"V:I“rhr:l.h”:iﬁ'n where: on = the standard deﬂation of the returns of the combined portfolio
containing Asset A and Asset B wa = the weight of Asset A in the two—as set portfolio
c1 = the stande deviation of the returns of AssetA
wk = the weight of Asset B in the two—as set portfoh'o
ob = the standard deviation of the returns of Asset B
rib = the con-elation coeﬂicient of the returns of Asset A and Asset B Equation 7-6. The Capital Asset Pricing Model (CAPM): kp=ltd+ (km—kﬂ] |3
where: kg = the required rate of return appropriate for the investment
project
led: the risk—free rate of return
1c}m = the required rate of return on the overall market
[3 = the proj ect’s beta Equation 9-]. Formula for the After-Tax Cost of Debt (AT 1: :1l: ATkﬁ=kd(l—Tj where: ltd = The before—tax cost of debt
T = The ﬁrm’s marginal tax rate Equation 9-2. Formula for the Cost of Preferred Stock (kn): 13:. k“ 2 (P: 4P) where: kg = The cost of the preferred stock issue; the expected return DU 2 The amount of the expected preferred stock dividend Pa = The current price of the preferred stock F = The ﬂotation cost per share Equation 9-3. Formula for the Cost of Common Stock Equity (k5)
(Dividend Growth Model Approach): k<=ﬂ+g
Pu where: Pu = The current price of the common stock II}1 = The amount of the common stock dividend expected one period
from now g = The expected constant growth rate of the company’s common stock
dividends Equation 9-4. CﬁPM Formula for the Cost of Common Equityr (k5):
k: = k: + ac: — 1e) B
where:l:s = The required rate of return from the company's common stock equit}r 1cIf = The risk—free rate of return kn1 = The expected rate of return on the overall stock market B = The beta of the company’s common stoclc= a measure of the amount
of nondiversiﬁahle risk Equation 9-5. Formula for the Cost of New Common Equityr (kn): k“ = D1
Pu—F +E where: kn = The cost of new common stock equityr
PEl = The price of one share of the common stock D1 = The amount of the common stock dividend expected to he paid in
one year F = The ﬂotation cost per share
g = The expected constant growth rate of the company’s common stock
dividends
Equation 9-6. Formula for the Weighted Average Cost of Capital (WACC):
l»:a = {WTc1 AT ltd) + (WTD lip} + (“EFTs k5) where: 1ta = The weighted average cost of capital (“ECCJ
"WTi = The weight, or proportion, of debt used to ﬁnance the ﬁrm’s assets
AT k:1 = The after-tax cost of debt WTD = The weight, or proportion, of preferred stock being used to ﬁnance
the ﬁrm’s assets kg = The cost of preferred stock WT5 = The weight, or proportion, of common equity being used to ﬁnance
the ﬁrm’s assets k5 = The cost of common equitj,r Equation 9-1 Formula for the Marginal Cost of Capital (MCC) Break Point: BF 2 himil
Proporllon of Total where:BP = The capital budget size at which the MCC changes [break point) Limit = The point at which the cost of the source of capital changes Proportion
of Total = The percentage of this source of capital in the ﬁrrn's capital oh-i 11"1'I'I'I‘t1 Equation 10-1a. NPV Formula, Algebraic Method: NPV= + + + — Initial Investment
{I + k] (1 + k)‘ [I + k] where: CF = ICash ﬂow at the indicated times
k 2 Discount rate= or required rate of return for the project
11 = Life of the project measured in the number of Time periods Equation 10-1h. NPV Formula, Table Method: NPV = crimson]; 1) + CHEW“) + . . . + [Intimate1L n) — Initial Investment where: PW = Present Value Interest Factor
k = Discount rate, or required rate of return for the project
11 = Life of the project measured in the number of time periods Equation 10-2. Formula for IRR Using Equation 10—] a: NFV = [II = + + — — Initial Inveslmcm
+ ' + ‘ + ‘ where: k = the IRR value Fill in cash ﬂows {[3st and periods (11}. Then choose values for k by trial and error
until the NPV equals zero. The value of k that causes the NPV to equal zero is the IRR. Equation 3-15. Future Value of a .5er Mount—Algebraic Method: FV = PV {1 + k)“
where: FV = Future Value: the ending amount
PV = PresentValue, the starting amount, or 1: = Rate of interest per period [expressed as a decimal)
n = Number of time periods Equation 3-111. Future Value of a Single Arnount—Table hietbod: F: = PT {F‘u’lE n} where: FV = Future Value: the ending amount
PV = PresentValue, the starting amount F‘i.-"l'_|:1'll = Future Value Interest Factor given interest rate= k: and number of
periods, 11, from Table I Equation 3-25. Present Value of a Single Aniount—Algebraie Metbnd: l psi—va—n
[l+kJ where: P"-.-' = PresentValue, the starting amount
F? = Future Value, the ending amount
1: = Discount rate of interest per period [expressed as a decimal} n = Number of time periods Equation 3-211. Present Value of a Single Aninunt—Table Method:
P1: = F‘.‘ {PVIF‘ n} where: P‘V = Present Value
FV = Future Value P‘s-1ft.” = Present Value Interest Factor given discount rate, k, and numb-er"
of periods, :1, from Table II Equaﬁon 3-3a. Future Value of an Pumuity—Algebraic Method: Fan-1. = pM'r 32-: where: File. = Future Value of an Annuity
Pl'wlT = Arnonnt of each annuity payment
l-r = Interest rate per lime period expressed as a decimal n = Number of annuity payments
Equation 3-311. Future Value of an Armuity—Table Method: PW. = M Frans“ where: F‘ﬂﬁr = Future Value of an Annuity
EMT = mount of each annuity payment
F'r'lFAkln = Future Value Interest Factor for an Annuity from Table II]
1: = Interest rate per period n = Number of armuity payments Equaﬁon 34a. Present Value of anﬂnnuity—Algebraic Method: I
II— II
{1+k}
k PVA = PMT x. where: "Pl-.955. = Present 1falue of an .i'umuity
EMT = mount of each annuity payment
l-r = Discount rate per period expressed as a deminal n = Number of annuity payments
Equation 3-411. Present 1‘r’alne of anAnnnity—Table Method: Pwt = Psrr PTIFAU where: P1955. = Present Value of an .i'umuity
P'MT = Amount of each annuity payment
PVlFAkIn = Present Value Interest Factor for an .i'uuruity ﬁ'om Table IV
1: = Discount rate per period n = Number of annuity payments Equation 3-5. Present Value of a PEI'pEtIJlt‘j-‘i FVF = FMT K where: Pit-T = Present‘t’alue of a Perpeluitjr' 1: = Discount rate expressed as a decimal Equation 3-6. Rate of Return: I
k = — 1
PF
where: 1-: = Rate of return expressed as a decimal
F‘J = Future Value P? = Present‘r'alue
n = Number of compounding periods Equation 3-7. Future Value nit]: Continuous Compounding:
FT = PT e353] where: F‘J = Future Value
PT = Present‘t-‘alue
e =Natural antilog oil
1-: = Stated annual interest rate expressed as a decimal n = Number of years Summary.F 1. Explain the capital budgeting process. Capital budgeting is die process of evaluating proposed investment projects. Capital
budgeting projects may be independent—the projects do not compete with each other—
or mutually exclusive—accepting. one project precludes the acceptance of the othertis}
in that group. Financial managers applj.r two decision practices when selecting capital budgeting
projects: accepti‘reject decisions and ranking decisions. The acoeptt'reject decision is
the determination of which independent projects are acceptable in light of the ﬁlm‘s
ﬁnancial objectives. Ranking is a process of comparing projects to a standard measure and ordering the mutually exclusive projects based on how well they meet the measure.
The capital budgeting process is concerned only with incremental cash ﬂows; that is,
those cash ﬂows that will occur ifaninvestment is undertaken: btit won’t occur ifit isn’t. 2. Calculate the payback period, net present value, internal rate of return, and
modiﬁed internal rate ofreturn for a proposed budgeting project The payback period is deﬁned as the number of time periods it will take before the
cash inﬂows from a proposed capital budgeting project will equal the amount of the initial invesunent. To ﬁnd a proj ect’s payback period= add all the project’s positive cash
ﬂows, one at a time, until the sum equals the amount of the initial cash outlay
for the project The number of time periods it takes to recoup the irutial cash outlay is
the payback period. A project is acceptable if it pays back the initial investment within
a time frame set by ﬁrm management. The net present value (NPR? of a proposed capital budgeting project is The dollar
amount ofthe change in thes'alue of the ﬁrm that will occur iftheprojectis undertake-n. To calculate NPV, total the present values of all the projected mere-mental cash ﬂows associated uith a project and subtract the amount of the initial cash outlay. A project
with an HP? greater than or equal to zero is acceptable. An Wit" proﬁle—a graph that shows a project=s I‘x—P‘J at many different discount rates {required rates of remrn}— shows
how sensitive a proj ect’s Iii—FY is to changes in the discount rate. The internal rate of return (ERR) is the projected percentage rate of return that a
proposed project will earn: given its incremental cash flows and required initial cash
outlay. To calculate IRE, ﬁnd the discount rate that makes the proj ect’s I‘CP‘T equal to
zero. That rate ofreturn is the [RR. A project is acceptable if its [ER is greater than or
equal to the ﬁrm’s required rate of return {the hurdle rate}. The IRE method assumes that intervening cash ﬂows in a project are reiiu'ested
at a rate ofreturn equal to the ERR. This can result in overly optimistic expectations
when the [ER is high and the interverung cash flows can’t be reinvested at that rate. To
produce more realistic results, the Modiﬁed Internal Rate of Return @1133.) method
was developed. The MERE method calls for assuming that the intervening cash ﬂows
from a project are reinvested at a rate of return equal to the cost of capital. To ﬁnd the
lit-1M, ﬁrst calculate how much you would end up 1t'tith at the end ofa project, assuming
the intervening cash ﬂows were invested at the cost of capital. The result is called the
project's terminal value. Next calculate the annual rate of return it wouldtalre to produce
that end amount from the beginning invesu'nent. That rate is the MIRR. 3. Describe capital rationing and how ﬁrms decide which projects to select. The practice ofplacing dollar limits on the total size ofthe capital budget is called capital
rationing. Under capital rationing, the ﬁrm still select the combination of projects that
yields the highest N'P'V 1without exceeding the budget limit. 4. Iv-Ieasure the 1isk of a capital budgeting project To measure the risk of a capital budgeting project; we determine how the project 1t'nsould
aﬁect the risk of the ﬁrm's existing asset portfolio. 1We compare the coefﬁcient of
variation of possible returns of the ﬁrm‘s asset portfolio with the proposed project and
without it The diﬂ'erence between the two coeﬁicients of variation is the measure of
the risk of the budgeﬁng project. a. Emlaiu risk—adjusted discount rates.
To compausatc for thc dogma of riskin capital budgeting: ﬁrms may adjust the discount
rate. for each project according to risk. The: more riskis increased, thcliighcrthc discount rate. The: more risk is dvscrnz'asnsd= tho lower L116 discount rate. Rates adjusted for the risk
of projects. are called risk-adjusted discount rates ['RADRs). 1. Deﬁne risk, risk aversion, and the risk—return relationship. In everything you do, or don’t do, there is a chance that something still happen that you
didn’t count on. Risk is the potential for unexpected events to occur. Given two ﬁnancial alternatives that are equal except for their degree of risk, most
people will choose the less risk}r alternative because they are 1isk averse. Risk aversion
is a common trait among almost all investors. Most investors avoid risk if they can,
unless they are compensated for accepting risk. In an imestment context, the additional
compensation is a higher expected rate of return. The 1isk—return relationship refers to the positive relationship between risk and
the required rate of return. Due to risk aversion, the higher the risk, the more return
investors expect 2. Measure risk using the standard deviation and tire coefﬁcient of variation. Risk is the chance, or probability, that outcomes other man what is expected will occur. This probability is reﬂectedin the narrctsness or width ofthe distribution ofthepossible
values of the ﬁnancial variable. In a distribution of variable values the standard deviation is a number that indicates how rtidely dispersed tire possible values are around the
expected value. The more 1widely dispersed a distribution is, the larger the standard
deviation, and the greater the probability that an actual value still be different than the expected value. The standard deviation, then, canbe used to measure the likelihood that
some outcome substantially diﬂerent than 1.vhat is expected will occur. ‘Wheu the degrees of risk in distributions of different sizes are compared, the
coefﬁcient of variation is a statistic used to measure relative rislciness. The coefﬁcient
of variation measures the standard deviation's percentage of the expected value. It relates the standard deviation to its mean to give a risk measure that is independent of
the magnitude of the possible returns. 3. Identify the types ofiisk that business ﬁrms encounter. Business risk is the risk that a company’s operating income still differ from what is The more volatile a company’s operating income, the more business risk the
ﬁrrncontains. Business risk is a result of sales volatility, which translates into operating
income volatility. Business riskis increasedbythepresence of ﬁxed costs, which magnify
the effect on operating income of changes in sales. Finmcirﬂ risk occurs when companies borrow money and incur interest charges
that show up as ﬁxedexpenses on their income statements. Fixed interest charges act on
a ﬁrm's net income the same way ﬁxed operating expenses act on operating income—
they increase volatility. The additional volatility of a ﬁrm‘s net income caused by the
presence of ﬁxed interest expense is called ﬁnancial Porg‘ot‘io risk is the chance that investors 1won’t get the return they expect from a
portfolio. Portfolio risk can be measured by the standard deviation of possible returns of a portfolio. It is affected by the correlation of returns of the assets making up the
portfolio. The less correlated these returns are, the more gains on someassets offset losses on others, resulting in a reduction of the portfolio=s This phenomenon is known as
the diversiﬁcation effect. Nondiversiﬁable risk is risk that remains in a portfolio after
all diversiﬁcation beneﬁts have been achieved. Nondiversiﬁable 1isk is measured by a term called beta {B}. The market has a beta of 1.1]. Portfolios with betas greater than ELI] contain more nondiversiﬁable risk than the market, and portfolios with betas less
Than LEI contain less nondiversiﬁable risk than the market. cl. Explain methods of risk reduction. Firms can reduce the degree of risk by taking steps to reduce the volatility of sales or
Their ﬁxed costs. Firms also obtain insurance policies to protect againstmanj.r risks, and
The}: d.i'.'s:;rsi.‘fj.-r Their asset portfolios to reduce the risk of income loss. 3. Describe how ﬁrms compensate for assuming risk. Firms almost always demand a higher rate of return to compensate for assuming risk.
The more risky a project, the higher the return ﬁrms demand. 6. Explain how The capital asset pricing model [CAPMJ relates risk and return. 1W'hesi investors adjust theirrequired rates of return to compensate for risk, the question
arises as to how much return is appropriate for a given degree oftisk. The capital asset pricing model {CAPIVU is a model that measures the required rate of return for an
investment or project, given its degree of nondis'ersciﬁable risk as measured by beta {B}; 1. Explain the time value of money and its inip-ortance in the business world. haloney grows mm time when it earns interest. Money expected or promised in the
future is worth less than the same amount of money in hand today. This is because we
lose the opportunity to earn interest when we have to wait to receive money. Similarly,
money we one is less burdensome if itis to be paid in the future rather than now. These
concepts are at the heart of investment and valuation decisions of a ﬁrm. 2. Calculate the future value and present value of a single amount To calculate the future value and the present value of a single do]lar amount; we may
use the algebraic, table, or calculator methods. Future value andpresent value are mirror
images of each other. They are compounding and discounting, respectively. ‘With future
value, increases in l: and 11 result in an exponential increase in future value. Increases
in k and n result in an exponential decrease in present value. 3. Find the future and present values of an annuity. Annuities are a series of equal cash flows. An annuity that has payments that occur at
the end of each is an annuity. An annuity that has payments that occur
at the beginning of each period is an armuity due. .5; perpetuity is a perpetual annuity.
To ﬁnd the future and present values of an ordinary annuity, we may use the
algebraic, table, or ﬁnancial calculator method. To ﬁnd the future and present values of an annuity due, multiplythe app1icab1e formula by if 1. + k) to reﬂect the earlier payment. 4. Solve time value of money problems with uneven cash ﬂows. To solve time value of money problems with uneven cash ﬂows, we ﬁnd the value of each payment l[each single amount] in the cash ﬂow series and total each single amount.
Sometimes the series has several cash flows of the same amount. If so, calculate the present value of those cash ﬂows as an armuity and addthe total to the sum ofthe present
values of the amounts to ﬁnd the total present value of the uneven cash ﬂow series. 5. Solve special time value of money problems, such as ﬁnding the interest rate, number
or amount of payments, or number of periods in a future or present value problem. To solve special time value of money problems, we use the present value and future
value equations and solve for the missing variable, such as the loan payment, k, or n.
"e may also solve for the present and future values of single amounts or annuities in
which the interest rate, payments, and number of time periods are expressed in terms
other than a year. The more often interest is compounded, the larger the future value. 1. Desciibe the sources of capital and how ﬁrms raise Firms raise debt capital from lenders or bondholders. They also raise funds from preferred
stockholders, from current stockholders, and from investors who bu}r newly issued shares
of common stock. J51” suppliers of capital expect a rate of return proportionate to the risk
they take. To ensure a supply of capital to meet their capital budgeting needs, ﬁrms must pay that return to suppliers. To compensate creditors, ﬁrms must pay the interest
andplincipal on loans. For bondholders, ﬁrms mu...

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