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FNCE 604 Accelerated Corporate Finance Alex Edmans The Wharton School Summer 2013 Alex Edmans FNCE 604 Summer 2013 1

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Page 1: 604 linearity

FNCE 604 —Accelerated Corporate Finance

Alex Edmans

The Wharton School

Summer 2013

Alex Edmans FNCE 604 Summer 2013 1

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Introduction

Acknowledgements

A significant amount of this material is taken from lecture notes byProfessor Simon Gervais, who taught this course for many years atWharton. I am extremely grateful to Simon for allowing me to use hisnotes.

I also thank present and past teaching assistants Lucia Bonilla, NacerBouhitem, Michael Graham, Adeel Ikram, Michelle Khundakar, JonMensing, Ryan Peters, Jonathan Vogan, and Austin Zalkin for valuedinput, and Indraneel Chakraborty and James Park for their help intypesetting this packet.

Alex Edmans FNCE 604 Summer 2013 2

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Introduction

0. Introduction

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Introduction

0. Introduction

Readings: Brealey, Myers and Allen, Chapter 1

This section will be especially relevant for:

— FNCE 726: Advanced Corporate Finance.— FNCE 731: International Corporate Finance

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Introduction

The Finance Function and the Financial Manager

Broadly stated, the finance function is concerned with the flow offunds between the capital markets and the firm’s operations

These flows include: (1) issues of securities to raise cash; (2)purchases of real assets used in the firm’s operations; (3) cash inflowsgenerated by the real assets; this cash is either (4a) reinvested in thefirm or (4b) returned to the firm’s investors.The financial manager therefore faces two main tasks:— Investment decisions (allocating funds to investments)— Financing decisions (choosing what instruments to issue to raise funds)Alex Edmans FNCE 604 Summer 2013 5

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Introduction

The Objective of the Financial Manager

Although many claimholders have a stake in the firm’s income, theshareholders are the owners, and managers should act in their interest

We will see that shareholders are made better off by any decisionwhich increases the value of their stake in the firm. Therefore,managers should act to maximize the value of the firm’s shares.

— This is more complex than profit maximization and requires anunderstanding of how financial assets are valued.

Several institutional arrangements exist to ensure that managers willindeed follow this objective:

— Stock and option compensation— Reputation in managerial labor markets— Hostile takeovers— Boards

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Introduction

Objective of this Course

The course is intended to provide a framework for analyzing theinvestment and financing decisions made by corporations.

Since such a framework requires an understanding of thedeterminants of value, the course provides an introduction to theconcepts underlying both corporate finance and asset pricing.

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Introduction

Checkpoint: BMA - Sections 1-3 and 1-4

Material relevant to this section:

BMA: chapter 1

— Problem set: 1, 2, 4

Bulk pack problem set: none

What is next?

BMA: chapter 2 and section 3-5

Look at the “Math/Stat reminder” in Additional Materials; this willbe useful for the problem sets

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Investment Decisions

I. Investment Decisions

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Investment Decisions

I. Investment Decisions

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Investment Decisions Compounding and Discounting

I.1 Compounding and Discounting

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Investment Decisions Compounding and Discounting

I.1.1 Constant Interest Rate

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Investment Decisions Compounding and Discounting

I.1.1 Constant Interest Rate

Readings:

— BMA chapter 2 and section 3-5

This section will be especially relevant for:

— FNCE 725: Fixed Income Securities.— FNCE 728: Corporate Valuation.

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Investment Decisions Compounding and Discounting

Motivation

At the most general level, an investment is a claim to a stream ofcash flows.

— This definition encompasses both real and financial investments.

In order to choose between alternative investments, we must thereforefind a way to compare cash flows differing in size, timing, and risk.

We will at first ignore risk and compare certain cash flows.

The techniques of compounding and discounting allow us to comparecash flows differing in size and timing.

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Investment Decisions Compounding and Discounting

Compounding and Future Value

Suppose that you invest $C in a bank account paying r%, and thatinterest is credited once a year.Money in the account after one year:Investment: $CInterest (C × r%): $CrTotal: $C (1+ r)Money in the account after two years:Investment: $C (1+ r)Interest (C × r%): $C (1+ r)r (> $Cr)Total: $C (1+ r)2 = $C (1+ r)(1+ r)Continuing this reasoning, you have $C (1+ r)T after T years$C (1+ r)T is the future value in T years of $C at r% compoundedannually. The quantity

CFT = (1+ r)T

is called the T -period compounding factor.Alex Edmans FNCE 604 Summer 2013 15

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Investment Decisions Compounding and Discounting

More Frequent Compounding (Numerical Example)

Suppose that you invest $100 in a bank account paying an interestrate of 10%, and that interest is credited twice a year.

— Every six months, your account will generate 10%2 = 5% interest.

Money in the account after 6 months:Investment: $100.00Interest (100× 10%

2 ): $5.00Total: $105.00 = $100(1.05)

Money in the account after one year (12 months):Investment: $105.00Interest (105× 10%

2 ): $5.25 (> $5.00)Total: $110.25 = $100(1.05)2

Does it make sense that you get more than $110.00?

You can continue this reasoning to show that you’ll have$100(1.05)2T after T years (i.e. 2T periods of six months)

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Investment Decisions Compounding and Discounting

More Frequent Compounding

More generally, suppose that you invest $C in a bank account payingan interest rate of r , and that interest is credited to your accounttwice a year.— You can then show that you will have $C (1+ r

2 ) after 6 months,$C (1+ r

2 )2 after a year, ..., $C (1+ r

2 )2T after T years.

— $C (1+ r2 )2T is the future value in T years of $C at the annual rate r

compounded semiannually (i.e. two times a year)

Even more generally, the future value in T years of $C at the annualrate r compounded m times a year is

FVT = C(1+

rm

)mTand the T -year compounding factor is

CFT =(1+

rm

)mTAlex Edmans FNCE 604 Summer 2013 17

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Investment Decisions Compounding and Discounting

The Effect of More Frequent Compounding

The following table shows the value after T years of $100 invested atthe rate of 10%, compounded m times a year.

Tm 1 5 10 301 110.00 161.05 259.37 1,744.942 110.25 162.89 265.33 1,867.924 110.38 163.83 268.51 1,935.8112 110.47 164.53 270.70 1,983.74365 110.51 164.86 271.79 2,007.73

Notice that more frequent compounding (as you go down eachcolumn) means a higher effective annual rate. How far can we pushthe benefits of more and more frequent compounding?

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Investment Decisions Compounding and Discounting

The Effect of More Frequent Compounding (cont’d)

A standard result from algebra states that

limm→∞

C(1+

rm

)mT= CerT ,

where e ' 2.718 is the base for natural logarithms.We express this result by saying that the future value in T years of$C invested at an interest rate of r continuously compounded is

FV = CerT

Important note: In this course (and all finance courses that you willtake here), “log”always means the natural logarithm “ln”.

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Investment Decisions Compounding and Discounting

Effective Annual Rate (Numerical Example)

Suppose that you invest $100 at an annual rate of 10% compoundedsemi-annually.From slide 16, you get $100(1+ 10%

2 )2 = $110.25 after one year.

Question: What is the interest rate r (compounded annually) thatwould generate the same amount after one year?— We need to solve

100(1+ r) = 110.25⇒ r = 10.25%.

— So the future value of $100 invested at an annual rate of 10%compounded semi-annually is the same as the future value of $100invested at an annual rate of 10.25% compounded annually.

This 10.25% is called the effective annual rate or the equivalentannual rate (EAR).The 10% is called the annual percentage rate (APR) or the annualrate. It tells you how much interest is paid each year —but not howfrequently the interest is paid. Therefore, it is an incomplete picture

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Investment Decisions Compounding and Discounting

Effective Annual Rate

More generally, suppose that you invest $C at an APR of rcompounded m times a year.

From slide 17, you will have $C (1+ rm )

m after one year.

The EAR r corresponding to an APR r compounded m times a yearsatisfies

1+ r =(1+

rm

)mSimilarly, the EAR r corresponding to an APR r continuouslycompounded satisfies

1+ r = er

Notice that r = r if r is compounded annually (m = 1).

In what follows, unless otherwise specified, we will always assume thatAPRs are compounded annually, i.e. are also EARs.

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Investment Decisions Compounding and Discounting

Effective Annual Rate: Example

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Investment Decisions Compounding and Discounting

Effective Annual Rate: Example (cont’d)

Why do these certificates of deposits (CD’s) seem to offer twodifferent interest rates?

The small print says that the interest rate (APR) is a dailycompounded rate. This means that $1 invested in these CD’s willgrow to

CF1 =(1+

0.065365

)365= 1.0672

after one year

What they call the “annual percentage yield” is simply the effectiveannual rate r , which can be found as follows:

1+ r = 1.0672⇒ r = 6.72%.

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Investment Decisions Compounding and Discounting

Effective Monthly (Weekly, Daily, etc.) Rate

Just as we can compute the effective annual rate, we can computethe effective rate over a month (or a week, or a day).The effective monthly rate rM corresponding to an annual interestrate r compounded m times a year satisfies

(1+ rM )12 =

(1+

rm

)m⇒ rM =

(1+

rm

) m12 − 1

In particular, if r is compounded annually (m = 1), then

rM = (1+ r)1/12 − 1 = 12

√1+ r − 1.

— If r is compounded monthly (m = 12), then rM = r12 .

Similarly, the effective weekly rate rW satisfies

(1+ rW )52 =

(1+

rm

)m⇒ rW =

(1+

rm

) m52 − 1

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Investment Decisions Compounding and Discounting

A Source of Confusion

People can get confused about the link between the following rates:— Annual percentage rate r compounded monthly vs. monthly rate rM :

CF1 =(1+

r12

)12= (1+ rM )

12 .

— Annual percentage rate r compounded weekly vs. weekly rate rW :

CF1 =(1+

r52

)52= (1+ rW )

52 .

— Annual percentage rate r compounded daily vs. daily rate rD :

CF1 =(1+

r365

)365= (1+ rD )

365 .

To practice, check the following are equivalent to a 10% EAR:— 9.5690% compounded monthly, or a monthly rate of 0.7974%;— 9.5398% compounded weekly, or a weekly rate of 0.1835%;— 9.5323% compounded daily, or a daily rate of 0.0261%.

On Canvas I have posted a rate calculator, RateEquiv.xls, to helpconvert between rates

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Investment Decisions Compounding and Discounting

Compounding and Effective Rates: Example

You want to invest $1,000 in a savings account for two years.

— After visiting three different banks, you discover that you have threepossible options:

1 an annual percentage rate of 12.5%, compounded annually;2 an annual percentage rate of 12%, compounded quarterly;3 a continuously compounded rate of 11.75%.

Which savings account should you choose?

What is the effective monthly rate that you will then be getting?

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Investment Decisions Compounding and Discounting

Compounding and Effective Rates: Example (cont’d)

In two years, the three different accounts would respectively accrue to1 1, 000(1+ 0.125)2 = 1, 265.63;

2 1, 000(1+ 0.12

4

)4×2= 1, 266.77;

3 1, 000e0.1175×2 = 1, 264.91.

You should therefore invest in the quarterly compounded account

You could also solve this problem by calculating CF1 for all threerates, or by comparing EARs (denoted by r below):

1 Obviously, r = 12.50%;

2 1+ r =(1+ 0.12

4

)4⇒ r = 12.55%;

3 1+ r = e0.1175×1 ⇒ r = 12.47%;

Again, the second bank offers the best deal.

As shown on Slide 24, the effective monthly rate rM must satisfy

(1+ rM )12 =

(1+

0.124

)4⇒ rM =

(1+

0.124

)1/3

− 1 = 0.9902%.Alex Edmans FNCE 604 Summer 2013 27

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Investment Decisions Compounding and Discounting

Discounting and Present Value

Suppose you want $C in your account in T years, and that thecurrent EAR is r . How much must you invest today?Again, let PV denote this amount.

0 1 2 3 ... T

PV C

So PV must satisfy: PV (1+ r)T = C , or PV = C/(1+ r)T .PV is the present value of $C delivered in T years from now.You are indifferent between $PV now and $C in T years. (Why?)We can also write PV = C ×DFT , where

DFT =1

(1+ r)T

is called the T-period discount factor.Alex Edmans FNCE 604 Summer 2013 28

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Investment Decisions Compounding and Discounting

Discounting and Present Value (cont’d)

More generally, suppose you want to receive $C1 in one year, $C2 intwo years, ..., $CT in T years. How much must you invest today?We wish to find the PV of an investment paying $C1 in one year, $C2in two years, ..., $CT in T years.

0 1 2 3 ... T

PV C1 C2 C3 ... CTUsing arguments similar to the previous two slides, we should find that

PV =C11+ r

+C2

(1+ r)2+ ...+

CT(1+ r)T

=T

∑t=1

Ct(1+ r)t

The present value of a sequence of cash flows is the sum of thepresent values of each individual cash flow.— Value additivity: you value an investment by valuing each constituentcash flow.

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Investment Decisions Compounding and Discounting

Discount Factors

The following figure shows discount factors DFT as functions of timeand EAR.

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Investment Decisions Compounding and Discounting

Discounting: Example

Two years ago, you put $10,000 in a savings account earning an APR of8% compounded semiannually. At the time, you thought that thesesavings would grow enough for you to buy a new car five years later (i.e. inthree years from now). However, you just reestimated the price that youwill have to pay for the new car in three years at $18,000.

1 How much more money do you need to put in your savings accountnow for it to grow to this new estimate in three years?

2 Now suppose that you know that the car company will offer you topay for the car over some time. In particular, you will have theopportunity to make a downpayment of $6,000 at the time you getthe car (three years from now) and to make additional payments of$6,500 at the end of each of the following two years. With this offer,how much money do you need to add to your account now?

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Investment Decisions Compounding and Discounting

Discounting: Example (cont’d)

1 Let us first figure out how much money FV is now in the account.

-2 0 ... 3

10, 000 ×(1.04)4−−−−−→

FV ...

FV = 10, 000(1+

0.082

)2×2= 10, 000(1.04)4 = 11, 698.59.

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Investment Decisions Compounding and Discounting

Discounting: Example (cont’d)

Now, the account should have an amount PV in it for it to grow to$18,000 in three years.

-2 0 ... 3

PV (1.04)−6←−−−−−

PV =18, 000(

1+ 0.082

)2×3 = 18, 000

(1.04)6= 14, 225.66.

So, you need to put $14, 225.66− $11, 698.59 = $2, 527.07 in theaccount.

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Investment Decisions Compounding and Discounting

Discounting: Example (cont’d)

2. The PV (at time 0, or two years after the initial investment) of thesethree payments is

PV =6, 000(

1+ 0.082

)2×3 + 6, 500(1+ 0.08

2

)2×4 + 6, 500(1+ 0.08

2

)2×5 = 13, 882.54.So, you need to add $13, 882.54− $11, 698.59 = $2, 183.95 to theaccount

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Investment Decisions Compounding and Discounting

Shortcuts to Calculating PVs: Perpetuities

A perpetuity is an investment paying a fixed sum C1 at the end ofevery period forever. (We will typically consider a period being oneyear, but other period lengths are possible).

0 1 2 3 4 ...

PV C1 C1 C1 C1 ...From our general formula, the PV of the perpetuity is given by:

PV0 =C11+ r

+C1

(1+ r)2+

C1(1+ r)3

+ ...

How do we find the value of this infinite sum?

Dividing both sides of the above equation by 1+ r givesPV01+ r

=C1

(1+ r)2+

C1(1+ r)3

+C1

(1+ r)4+ ...

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Investment Decisions Compounding and Discounting

Shortcuts to Calculating PVs: Perpetuities (cont’d)

We now subtract the second equation from the first to obtain

PV0 −PV01+ r

=C11+ r

+C1

(1+ r)2+

C1(1+ r)3

+ ...

− C1(1+ r)2

− C1(1+ r)3

− ...

Notice that all but one term on the right can be canceled out.

We now solve for PV as follows (provided r > 0):

PV0 −PV01+ r

=C11+ r

⇔ PV0(1+ r)− PV0 = C1⇔ PV0 · r = C1⇔ PV0 =

C1r

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Investment Decisions Compounding and Discounting

Example: UK Consols

In the 1800’s, the British government decided to consolidate the hugedebt accumulated during the Napoleonic wars and to replace it with asingle issue of bonds with no termination date and a coupon rate of2 12%. These bonds, called consols, are still traded today.

Suppose that the current interest rate in the U.K. is 9%. What is thePV of a consol with a £ 1, 000 face value?

This is just a perpetuity promising to pay £ 25 each year. Its PV is

PV0 =250.09

= £ 277.78.

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Investment Decisions Compounding and Discounting

A More Challenging Example: Deferred Perpetuities

A rich entrepreneur would like to set up a foundation that, every year,will pay $5,000 in the form of a scholarship to one deserving student

The first such scholarship is to be awarded in three years, and ascholarship will be awarded in perpetuity every year after that (evenafter the entrepreneur’s death).

How much money should the entrepreneur put in the foundation’saccount, if that account earns 8% compounded annually?

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Investment Decisions Compounding and Discounting

A More Challenging Example: Deferred Perp’s (cont’d)

First, let us calculate how much money will need to be in the accountat the end of the second year; let us denote that amount by PV2

0 1 2 3 4 5 ...

PV ∗ = 50000.08 5000 5000 5000 ...

For the account to be worth this much in two years, the amount thatthe entrepreneur needs to contribute initially is

PV0 = PV2 ×1

1.082=5, 0000.08

× 11.082

= 53, 583.68.

The perpetuity starts in three years, but the exponent on the discountfactor is a two.— The perpetuity formula used in the first step calculates the value of astream of cash flows starting a year later, i.e. the perpetuity formulafrom slide 36 gives us the value of the stream at time 2.

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Investment Decisions Compounding and Discounting

Shortcuts to Calculating PVs: Growing Perpetuities

A growing perpetuity is an investment paying, at the end of eachperiod, an amount C1 that grows at an annual rate g forever.

0 1 2 3 4 ...

PV C1 C1(1+ g) C1(1+ g)2 C1(1+ g)3 ...From our general formula, the PV of the growing perpetuity is:

PV0 =C11+ r

+C1(1+ g)

(1+ r)2+C1(1+ g)2

(1+ r)3+ ...

We now use a “trick” similar to that on slide 35. We multiply bothsides of the above equation by 1+g

1+r :

PV0

(1+ g1+ r

)=C1(1+ g)

(1+ r)2+C1(1+ g)2

(1+ r)3+ ...

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Investment Decisions Compounding and Discounting

Shortcuts to Calculating PVs: Growing Perp’s (cont’d)

Again, we subtract the second equation from the first:

PV0 − PV0(1+ g1+ r

)=

C11+ r

+C1(1+ g)

(1+ r)2+C1(1+ g)2

(1+ r)3+ ...

−C1(1+ g)(1+ r)2

− C1(1+ g)2

(1+ r)3+ ...

As before, all but one term on the right can be canceled outWe can now solve for PV as follows (provided that g < r):

PV0 − PV0(1+ g1+ r

)=

C11+ r

⇔ PV0(1+ r)− PV0(1+ g) = C1

⇔ PV0(r − g) = C1

⇔ PV0 =C1r − g

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Investment Decisions Compounding and Discounting

Example: A Stock

Morgan Stanley has just paid a dividend of $1. It will grow itsdividend at 5% forever. With an interest rate of 10%, what is itsshare price?

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Investment Decisions Compounding and Discounting

Example: A Stock (cont’d)

We have r = 10%, g = 5%. Note that $1 is C0, not C1.C1 = $1× 1.05 = $1.05. Therefore,

PV0 =C1r − g =

1.050.10− 0.05 = $21.

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Investment Decisions Compounding and Discounting

Another Shortcut to Calculating PVs: Annuities

An annuity is an investment paying a fixed sum C1 at the end ofevery period for a given number T periods.

0 1 2 ... T − 1 T T + 1 T + 2 ...

PV C1 C1 C1 C1 C1From our general formula, we can write the PV of the annuity as:

PV0 =C11+ r

+C1

(1+ r)2+ ...+

C1(1+ r)T

.

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Investment Decisions Compounding and Discounting

Another Shortcut to Calculating PVs: Annuities (cont’d)

A more convenient expression can be obtained by observing that thecash flows from the annuity equal the difference between the cashflows of two perpetuities, one starting at time 1 and the other startingat time T + 1:

0 1 2 ... T − 1 T T + 1 T + 2 ...

PV1 C1 C1 C1 C1 C1 C1 C1 ...−PV2 −C1 −C1 ...

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Investment Decisions Compounding and Discounting

Another Shortcut to Calculating PVs: Annuities (cont’d)

The PV of the first perpetuity is PV 10 =C1r , as derived on slide 36

What about the second perpetuity, which is deferred for T periods?— Let us first calculate the value of that perpetuity after T periods. Wecall this value PV 2T .0 1 2 ... T − 1 T T + 1 T + 2 ...

PV ∗2 =C1r C1 C1 ...

— Now, since PV 2T is the value in T periods from now, we need todiscount this value to time 0 to get the value of the perpetuity:0 1 2 ... T − 1 T T + 1 T + 2 ...

PV2 (1+ r)−T←−−−−−−

PV ∗2

PV 20 =PV 2T

(1+ r)T=

C1/r(1+ r)T

.

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Investment Decisions Compounding and Discounting

Another Shortcut to Calculating PVs: Annuities (cont’d)

The calculation for the PV of the annuity then simply involves adifference of two perpetuities:

Perpetuity Cash Flow Presentstarting at 1 2 ... T T + 1 T + 2 ... Value1 C1 C1 ... C1 C1 C1 ... PV 10 =

C1r

T + 1 0 0 ... 0 C1 C1 ... PV 20 =C1r

1(1+r )T

Annuity C1 C1 ... C1 0 0 ... PV0 = PV 10 − PV 20

The PV of the annuity is therefore given by:

PV0 =C1r

(1− 1

(1+ r)T

).

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Investment Decisions Compounding and Discounting

Example: DiMaggio’s Vow

When Marilyn Monroe died, her ex-husband Joe DiMaggio vowed toplace fresh flowers on her grave every Sunday (starting the week afterher death) as long as he lived.A bouquet of fresh flowers cost $4.00 a week after she diedBased upon actuarial tables, Joe could expect to live for 30 moreyears when Monroe died.Assuming that the cost of flowers would grow every week at a rateequivalent to 2% per year and that 6% was an appropriate annuallycompounded discount rate, what was the PV of this commitment?Hint: First derive the PV formula for an annuity that grows at rate g :

PV0 =C1r − g

[1−

(1+ g1+ r

)T ].

Note: the detailed solution is included in Additional Materials.Alex Edmans FNCE 604 Summer 2013 48

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Investment Decisions Compounding and Discounting

Example: DiMaggio’s Vow (cont’d)

To use the formula, we must calculate the equivalent weekly interestrate rW , the equivalent weekly growth rate gW , as well as the numberof periods T . The first cash flow is simply C1 = 4.— Using slide 24, we have

(1+ rW )52 = 1.06⇒ rW = (1.06)1/52 − 1 = 0.1121%.

— Using slide 24 again, we have

(1+ gW )52 = 1.02⇒ gW = (1.02)1/52 − 1 = 0.0381%.

— The number of weeks over which flowers will have to be bought is:

T = 52× 30 = 1, 560.

Therefore, the PV of DiMaggio’s commitment is

PV =4

0.1121%− 0.0381%

[1−

(1.0003811.001121

)1,560]= 3, 699.21.

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Investment Decisions Compounding and Discounting

Other Useful Formulas

The formulas for perpetuities and annuities that we derived on slides36, 40, 44 and 47 all assume that payments are always made at theend of the year.

If the payments are made at the beginning of the year (i.e. the firstpayment is C0, paid immediately), these PV formulas become:

perpetuity: PV0 =C0(1+ r)

r;

growing perpetuity: PV0 =C0(1+ r)r − g ;

annuity: PV0 =C0(1+ r)

r

[1− 1

(1+ r)T

];

growing annuity : PV0 =C0(1+ r)r − g

[1−

(1+ g1+ r

)T ].

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Some Notation

The following notation for annuities will sometimes be useful. rrepresents the EAR.

— The PV of a T -year annuity of $1, payable at the end of each year:

aT |r =1r

[1− 1

(1+ r)T

].

— The PV of a T -year annuity of $1, payable at the start of each year:

aT |r =1+ rr

[1− 1

(1+ r)T

].

These are called annuity factors, as any (constant) annuity can becalculated using these factors.

— For example, the PV of a T -year annuity of $C payable at the end ofeach year is equal to PV = CaT |r .

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Nominal versus Real Interest Rates: Numerical Example

Suppose that all you buy are apples, which cost $1 each today, andthat r = 26%. Can you buy 26% more apples next year?

Answer: It will depend on the price of apples in one year.

Example:

— Suppose that you have $100 today, and that the price of apples goesup by 5% during the year.

— Let us compare how many apples you could buy today vs. in one year.today in one year

money available $100 $126price of apples $1.00 $1.05can buy 100

1.00 = 100 apples1261.05 = 120 apples

— Since you can only buy 20% more apples, you are only 20% better off(not 26%). In other words, your real rate of return is 20%.

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Nominal versus Real Interest Rates

More generally, suppose that you invest for one year in the bondmarket. Your investment next year is worth $ (1+ r) for each dollarinvested. Does this mean you are better off by saving?

The answer depends on what happens to inflation. Suppose that theone-year rate of inflation is i . Then, in order to buy the same amountof goods you could have purchased with $1 today, you will need$ (1+ i) a year from now. This means that your actual return,measured in today’s dollars is given by

1+ R =1+ r1+ i

=⇒ R =1+ r1+ i

− 1.

R is known as the real rate of return, as opposed to r , which is anominal rate.

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Nominal versus Real Interest Rates (cont’d)

People often calculate the real rate of return R to be the differencebetween the nominal rate of return r and the inflation rate i :

R = r − i .

This is only approximately true.

Indeed, notice that

1+ r = (1+ R)(1+ i) = 1+ R + i + (R × i)︸ ︷︷ ︸small

' 1+ R + i ,

so that

R ' r − i .

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Why Real Rates Are Important

Suppose you are in Germany at the beginning of 1923. Someoneoffers you a one-year German Treasury bill denominated in marks witha face value of DM10,000,000 (' $700) at the “bargain”price ofDM5,000,000. Since this implies a rate of return of

r =10, 000, 0005, 000, 000

− 1 = 100%,

you accept. Did this turn out to be a good deal?

The inflation rate in Germany in 1923 turned out to be about4,530,000,000%, so that the real return on your investment was

R =1+ 100%

1+ 45, 300, 000− 1 ' −99.9999956%.

In other words, your investment was practically worthless at the endof 1923!

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Key Takeaways: Constant Interest Rate

Compounding factor gives future value of $1

— CFT =(1+ r

m

)mT= erT if m→ ∞

Effective annual rate gives equivalent rate under once-a-yearcompounding

— r =(1+ r

m

)m − 1Discount factor gives present value of $1

— DFT = 1/CFT

Growing perpetuity: PV0 = C1/ (r − g)Annuity factor gives PV of $1 for T years. aT |r =

1r

[1− 1

(1+r )T

]Real rate of return R = 1+r

1+i − 1

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Checkpoint: Constant Interest Rate

Material relevant to this section:

BMA: chapter 2

— Problem set: 1, 2, 5, 9, 13, 22, 24, 29, 32

BMA: chapter 3

— Problem set: none

Bulk pack problem set #1

What is next?

— BMA: sections 3-3 and 3-4

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I.1.2 Term Structure

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I.1.2 Term Structure

Readings: BMA sections 3-3 and 3-4

This section will be especially relevant for:

— FNCE 717: Financial Derivatives.— FNCE 725: Fixed Income Securities.— FNCE 731: International Corporate Finance.— FNCE 738: Funding Investments.

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Accounting for the Term Structure of Interest Rates

Our PV formulas have assumed that the interest rate is the same forall maturities. In practice, the interest rate at which you canborrow/invest for, say, 1 year is typically different from the rate atwhich you can borrow/invest for, say, 5 years.The relationship among interest rates for different maturities is knownas the term structure of interest rates— In a flat term structure, the interest rates are the same for all maturities

With a non-flat term structure,

the cash flow at time: 1 2 3 ...should be discounted at: r1 r2 r3 ...

where rt is known as the t-year spot rate. The PV formula of slide 29 hasto be modified to

PV =C11+ r1

+C2

(1+ r2)2+ ...+

CT(1+ rT )T

=T

∑t=1

Ct(1+ rt )t

.

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Term Structure: Example

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More on Discount Factors

Letting

DFt =1

(1+ rt )t

we can write the PV formula of slide 60 as

PV = (C1 ×DF1) + ...+ (CT ×DFT ) =T

∑t=1(Ct ×DFt ) .

DFt is known as the t-period discount factor, since multiplication byDFt converts a cash flow Ct in t periods into its PV.

Since rt ≥ 0, DFt < 1, i.e. $1 is worth less than $1 today. In fact, wemust have

1 > DF1 > DF2 > · · · > DFT ,i.e. $1 the day after tomorrow is worth less than $1 tomorrow, and soon.

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Proof that DFt > DFt+1

Suppose that r1 = 20% and r2 = 7%. This implies

DF1 =11.20

= 0.83 and DF2 =1

(1.07)2= 0.87,

so that DF1 < DF2. Why can such a situation not occur?

The reason is that anyone who could borrow and lend at these ratescould become a billionaire overnight. How would you proceed?

Cash flows at the end of year

Strategy 0 1 2

Borrow $1,000 at 7% for 2 yrs +1,000.00 0 -1,144.90

Invest (lend) $954.08 at 20% for 1 yr -954.08 +1,144.90 0

Total +45.92 +1,144.90 -1,144.90

Since you can store the $1,144.90 that you receive after the first year torepay your loan at the end of the second year, the $45.92 represents anarbitrage opportunity .

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Proof that DFt > DFt+1 (cont’d)

You borrow $1,000 at 7% for two years and invest $954.08 out ofthese $1,000 at the rate of 20% for one year. After one year, yourinvestment will be worth $954.08(1.20) = $1,144.90. You owe thebank $1,000(1.07)2 = $1,144.90 at the end of the second year. If youjust store the proceeds from your investment under your mattress forone year, you can be sure to have enough money to repay the loan.The remaining $(1,000-954.08) = $45.92 is a free lunch.

Of course, there is no reason to limit yourself to borrowing $1,000.Similarly, other investors will rush into borrowing at 7% and lendingat 20%. Eventually, r1 will have to decrease and r2 will have toincrease until DF1 becomes greater than DF2.

Money machines like the one we just discussed are called arbitrageopportunities. Arbitrage opportunities cannot exist for long in a wellfunctioning market.

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Forward Rates: Numerical Example

Your friend tells you that he will have to borrow money for one year inone year from now.You agree (today) to lend him the money at that time (in one year)at a rate f2 specified today, i.e. you enter a forward rate agreementwith your friend. What is the correct rate?For every dollar that you invest for two years, you now have twopossible investment alternatives:— Lend for 2 years at a rate of r2: FV2 = (1+ r2)2.— Lend for 1 year at a rate of r1, and then lend to your friend at thepre-specified rate of f2: FV2 = (1+ r1)(1+ f2).

Since both investment alternatives involve no risk, neither shouldresult in a larger future value in two years, that is

(1+ r2)2 = (1+ r1)(1+ f2)⇒ f2 =(1+ r2)2

1+ r1− 1.

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Forward Rates

More generally, suppose you were offered a forward rate agreement(FRA), structured as follows. Your counterparty wishes to borrow fora year at the end of year t − 1 at a rate ft specified today. Whatshould this rate be?

Again, you could invest $1 today for t years in two different ways:

— Lend for t years at the rate rt , resulting in (1+ rt )t at the end of yeart.

— Lend for t − 1 years at the rate rt−1, and enter into a FRA, resulting in(1+ rt−1)t−1(1+ ft ) at the end of year t.

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Forward Rates (cont’d)

Since both strategies involve no risk, we must have

(1+ rt )t = (1+ rt−1)t−1(1+ ft ),

which implies

ft =(1+ rt )t

(1+ rt−1)t−1− 1 = DFt−1

DFt− 1

The rates ft defined by the above equation are known as forwardrates.

Notice that saying that DFt−1 > DFt is equivalent to saying thatft > 0.

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The Relationship Between Interest Rates and ForwardRates

The following figures show the relationship that exists between spotrates and forward rates.

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Nominal versus Real Interest Rates Revisited

The relationship between nominal and real interest rates derived onslide 53 also applies to the one-year spot rates:

R1 =1+ r11+ i1

− 1.

More generally, the t-period real interest rate Rt satisfies

1+ Rt =1+ rt1+ it

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Key Takeaways: Term Structure

Spot rate rT is the rate per year for T years starting today

The term structure is the graph of r1, r2, ... for different T

With a non-flat term structure, PV = ∑Tt=1

Ct(1+rt )t

Forward rate ft is the rate for 1 year starting at t − 1 and ending at t:— ft =

(1+rt )t

(1+rt−1)t−1− 1

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Checkpoint: Term Structure

Material relevant to this section:

BMA: chapter 3

— Problem set: 14, 18, 20, 25, 32

What is next?

BMA: section 3-1

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Investment Decisions The Valuation of Certain Cash Flows: Pricing Bonds

I.2 The Valuation of Certain Cash Flows:Pricing Bonds

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I.2 The Valuation of Certain Cash Flows: Pricing Bonds

Readings: BMA section 3-1

This section will be especially relevant for:

— FNCE 717: Financial Derivatives.— FNCE 725: Fixed Income Securities.— FNCE 731: International Corporate Finance.— FNCE 738: Funding Investments.

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What Is a Bond?

A bond is essentially a loan: the issuer (borrower) promises to repaythe investor (lender) the amount borrowed plus interest over somespecified period of time.

— A coupon bond promises a periodic interest payment (e.g. every sixmonths) and repayment of the face value (F ) at the maturitydate (T ). The periodic interest payment is known as the coupon (C )and the APR on the face value is called the coupon rate (i.e., CF is thecoupon rate).

0 1 2 3 ... T − 1 T (maturity date)

C (coupon) C C ... C C + F (face value)

For a zero coupon bond there are no periodic coupon payments, andboth principal and interest are paid together at the maturity date.

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Treasury Securities

The U.S. Treasury is the largest single issuer of debt in the world.U.S. Treasury securities are backed by the full faith and credit of theU.S. government, so that they are viewed by market participants ashaving no (or very low) default risk (i.e. no risk that the issuer willdefault on his payments of interest and/or principal).— There are three major types of Treasury securities:— Treasury Bills are issued with maturities of 3, 6, or 12 months.— Treasury Notes are issued with maturities between 2 and 10 years.— Treasury Bonds are issued with maturities greater than 10 years.

Treasury bills are zero-coupon bonds (ZCBs), while Treasury notesand bonds are coupon bonds with interest paid every 6 months.A coupon bond is basically a portfolio of several zero-coupon bonds,one for each coupon or principal payment. The Treasury allows buyersof T-bonds or T-notes to exchange them for the individualcomponent ZCBs. These ZCBs corresponding to unbundled Treasurycoupon bonds are called Treasury Strips.

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How Treasury Securities Are Quoted

Treasury bills are quoted in terms of a discount rate (in percent), notof a price. If d is the quoted discount rate, N the maturity (in days)and F the face value, then the price is computed according to theformula

P = F(1− d N

360

)— For example, if the quote for a 100-day T-bill with a face value of$100,000 is 8.75, then the price is

$100,000(1− 0.0875100

360

)= $97,569.

Treasury coupon securities and Treasury strips are quoted in terms ofprices (in percent of face value), with the decimal part expressed inunits of 1/32. For example, a quote of 92:14 refers to a price of 92and 14/32, or 92.4375 for a security with $100 face value.

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How Treasury Securities Are Quoted: Example (cont’d)

For example, if we look at the Treasury bill with the “Dec 26 ’96”maturity date, the Wall Street Journal tells us:

The bond has 135 days to maturity.The (best) bid discount is 5.05%, which means that the price atwhich you can sell that T-bill with a face value of $100,000 is

100,000(1− 0.0505135

360

)= 98,106.25.

The (best) ask discount is 5.03%, which means that the price atwhich you can buy that T-bill with a face value of $100,000 is

100,000(1− 0.0503135

360

)= 98,113.75.

See Additional Materials for the meaning of “ask yld” (not part ofthis course)

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Present Value and Market Price

Consider a Treasury bill with 1 year to maturity and a face value of$100. If the 1-year interest rate is 10%, we know that the PV of theT-bill is 100/1.10 = 90.91.Why must $90.91 be its market price?— Since the current interest rate is 10%, nobody would buy the bill if P> $90.91, since it would be possible to obtain $100 in a year bylending $90.91 at 10% for one year.

— Conversely, if P < $90.91, nobody would sell it, since it would bepossible to obtain $90.91 today by borrowing this amount from a bankat 10% and using the payoff from the bill a year from now to repay theloan.

— Therefore, the market price of the bill must be $90.91.— These “no-arbitrage”arguments are shown on the following pages:

In a well-functioning (effi cient) market, the price of an investmentequals its present value.Things of equal value trade at equal prices.

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Present Value and Market Price (cont’d)

If the T-bill on page 79 were priced at $90:

Cash flowStrategy 0 1Buy T-bill -90.00 100Borrow $90.91 90.91 -100Total 0.91 0

This $0.91 is an arbitrage opportunity.If the T-bill on page 79 were priced at $92:

Cash flowStrategy 0 1Sell (short) T-bill 92.00 -100Lend $90.91 -90.91 100Total 1.09 0

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Pricing a Coupon Bond

Suppose that the term structure of interest rates is as follows:r0.5 = 4%, r1 = 4.1%, r1.5 = 4.3%, r2 = 4.5%. What is the currentprice of a T-note with 2 years to maturity, a coupon rate of 8%semiannual, and a face value of $100?

Using the general PV formula, we have:

P =4

(1.040)0.5+

4(1.041)

+4

(1.043)1.5+

104(1.045)2

= 106.756

Does it make sense that P > $100?

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Estimating the Term Structure

Since bond prices reflect the current term structure, we can use bondprices to estimate the term structure, provided we have a suffi cientnumber of bonds with differing maturities and/or coupons.

Some find it simpler to first solve for the discount factors, and thenobtain the interest rates from the discount factors.

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Estimating the Term Structure: Example

Suppose you observe the following prices for a bond with $100 facevalue:

Bond Security Coupon Maturity Quote PriceA T-bill — 180 days 4.98 97.51000B T-bill — 360 days 5.44 94.56000C T-note 6% 2 years 99:10 99.31250D T-note 8% 2 years 103:01 103.03125

Determine the 6, 12, 18 and 24 month interest rates (spot rates).

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Estimating the Term Structure: Example (cont’d)

First, let us figure out the timing of the bonds’payments

Cash flow at the end of6 months 12 months 18 months 24 months(0.5 year) (1 year) (1.5 year) (2 years)

Bond A 100 0 0 0Bond B 0 100 0 0Bond C 3 3 3 103Bond D 4 4 4 104

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Estimating the Term Structure: Example (cont’d)

Let us now calculate the discount factors that are consistent with thebonds’prices and the above payments. In particular, each bond mustsatisfy the PV formula on slide 52:

97.51000 = (100×DF0.5)94.56000 = (0×DF0.5) + (100×DF1)99.31250 = (3×DF0.5) + (3×DF1) + (3×DF1.5) + (103×DF2)103.03125 = (4×DF0.5) + (4×DF1) + (4×DF1.5) + (104×DF2)

Solving the above system gives DF0.5 = 0.9751, DF1 = 0.9456,DF1.5 = 0.9165 and DF2 = 0.8816.

We can then use the fact that DFt = 1(1+rt )t

(see slide 62) to findr0.5 = 5.17%, r1 = 5.75%, r1.5 = 5.99% and r2 = 6.51%.

On Canvas I have posted a spreadsheet, TermStructure.xls, whichestimates the term structure from bonds.

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Bond Yields

The yield to maturity or internal rate of return for a bond is the ratey that solves

P =T

∑t=1

C(1+ y)t

+F

(1+ y)T(1)

If you recall (from the PV formula on slide 60) that

P =T

∑t=1

C(1+ rt )t

+F

(1+ rT )T,

you will see that the yield of a bond is a complicated average of thecurrent interest rates. In particular, the yields of two bonds with thesame maturity but with different coupon rates will generally differ.Solving for y in (1) involves solving a polynomial of degree T .For a zero-coupon bond with maturity T , or if the term structure isflat, we have y = rT .

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Computing the Yield of a Bond

We compute yields by trial and error: pick a y and compute the PVat this rate; if PV > bond price, try a higher y and vice-versa.For example, suppose we want to compute the yield of a 3-year bondwith a face value of $100, paying an annual coupon of 10%, andselling for $103.83. That is, we want to solve for y in

103.83 =101+ y

+10

(1+ y)2+

110(1+ y)3

.

The following table and graph show the required computations.

y PV9.00% 102.538.00% 105.158.50% 103.85

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Yields of Treasury Securities

When financial practitioners talk of the yield of a Treasury security,they don’t generally refer to the rate y as defined on page 86, but tothe annual percentage rate under semi-annual compounding y givenby (

1+y2

)2= 1+ y .

where y is the effective annual yield (under annual compounding).Note that the effective semi-annual yield (see slide 25) is yS = y/2y is also referred as a bond-equivalent yield.

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Problems with Bond Yields

You should never use the information on bond yields reported by thefinancial press as a substitute for the term structure of interest rates.

— Knowing the term structure of interest rates allows you to price anyriskless asset.

— Knowing the yield of a bond tells you the price of that particular bondonly.

Two bonds with the same maturity but different coupons will ingeneral have different yields.

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Key Takeaways: Pricing Bonds

Treasury bills pay no coupon, and are quoted in terms of a discountrate: P = F

(1− d N

360

)Treasury notes and bonds pay a semiannual coupon (C2 every 6months), and are quoted in units of 1/32

The price of a bond is P = ∑Tt=1

C(1+rt )t

+ F(1+rT )T

where rt is thet-year spot rate

The yield of a bond is the single constant discount rate y that solvesP = ∑T

t=1C

(1+y )t +F

(1+y )T . It is a (weighted geometric) average ofthe spot rates

— For Treasuries, quoted yields are semiannual APRs: y given by(1+ y

2

)2= 1+ y

The coupon of a bond is fixed. Its yield depends on marketconditions, and is inversely related to its price

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Checkpoint: Pricing Bonds

Material relevant to this section:

BMA: chapter 3

— Problem Set: 4, 8.

Bulk pack problem set #2.

The institutional details of “short-selling”are carefully explained inAdditional Materials.

What is next?

BMA: section 5-1

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Investment Decisions Capital Budgeting Under Certainty

I.3 Capital Budgeting Under Certainty

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Investment Decisions Capital Budgeting Under Certainty

I.3 Capital Budgeting Under Certainty

Three questions:

What is a good rule for selecting projects? (section I.3.1)How do we apply it? (section I.3.2)What are the alternatives, and are they useful at all? (section I.3.3)

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Investment Decisions Capital Budgeting Under Certainty

I.3.1 The NPV Rule: Theoretical Foundation

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I.3.1 The NPV Rule: Theoretical Foundation

Readings: BMA section 5-1

This section will be especially relevant for:

— FNCE 726: Advanced Corporate Finance.— FNCE 728: Corporate Valuation.

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Investment Decisions Capital Budgeting Under Certainty

Motivation

Suppose you are at a Volkswagen shareholders’meeting. Twoshareholders are quite vocal about what the firm should do.

— An old man wants money right now: he wants VW to invest in sportcars which would yield a quick profit.

— A little child’s trust fund representative wants money a long way in thefuture: he wants VW to invest in developing electric cars.

What do you think VW’s managers should do?

To answer this question we will look at how an individual shouldchoose among different investment opportunities. We will show thatthere is a simple rule that managers should follow, regardless ofshareholders’preferences. Hence both VW shareholders will agree onthe same project.

— Until Part III, we will ignore the impact on investment decisions oftaxes and other complications.

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Investment Decisions Capital Budgeting Under Certainty

Mr. Rossi’s Problem

Mr. Rossi has inherited $1M. He grew up in Italy and has developed areal aversion to work, which he completely detests. He therefore plansto use his inheritance to finance himself for the rest of his life.

For simplicity, we will divide his life into two periods, youth and oldage.

We are going to assume that the current interest rate available incapital markets (for borrowing or lending) is 20%, so that for everydollar Mr. Rossi saves in his youth, he gets $1.20 in old age.

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How the Capital Market Helps Smooth Consumption

Once the possibility of borrowing/lending is taken into account, hereare some of the possibilities available to Mr. Rossi:— Go on a fantastic trip around the world, spend the whole $1M and thenlive in poverty in his old age.

— Spend $0.5M in his youth, put $0.5M in the bank, and have $0.6M inhis old age.

— Spend nothing in his youth and take a $1.2M trip in old age.

More generally, the capital markets allow him to choose anycombination in the following figure:

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The Effect of Real Investment Opportunities

While borrowing/lending gives him some choice as to how to allocatehis money, Mr. Rossi also has real investment opportunities. Hefancies himself as an entrepreneur and sits down to work out whatinvestments he can make.

— Mr. Rossi is a wine lover. He reckons that a small vineyard that hasjust come on the market will cost him $50,000 and will yield $200,000for his old age. This is the best project he can think of.

— Mr. Rossi is also a gourmet. His next best project is to run arestaurant. This will cost $100,000 now and give $140,000 in old age.

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The Effect of Real Investment Opportunities (cont’d)

We can represent these and the other projects Mr. Rossi can think ofthrough the following curve:

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Should Mr. Rossi Invest in the Vineyard?

Suppose Mr. Rossi invests in the vineyard. He is left with

◦$1M − $0.05M = $0.95M now;◦$0.20M later

Instead of simply consuming these amounts, Mr. Rossi could useborrowing and lending to achieve other consumption patterns:— Suppose he invests the whole $0.95M. This will generate

$0.95M × 1.2 = $1.14M later, so he is left with

◦$0 now (since the whole $1M was invested);◦$1.14M + $0.2M = $1.34M later

— Or he can borrow the PV of $0.2M, that is $0.2/1.2 = $0.167M, andrepay $0.2M later. So Mr. Rossi is left with

◦$0.95M + $0.167M = $1.117M now;◦$0.2M − $0.2M = $0 later

— In fact, by borrowing or lending, Mr. Rossi could achieve anyconsumption on the following line. (see next slide)

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Should Mr. Rossi Invest in the Vineyard? (cont’d)

Investing in the vineyard expands his consumption possibility frontier

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Should Mr. Rossi Invest in the Vineyard? (cont’d)

Mr. Rossi is better off by investing in the vineyard.

If he chose to consume everything today, he could consume $1.117Mnow. In other words, his current wealth is increased by

$1.117M − $1M = $0.117M.

Notice that this increase in wealth corresponds exactly to theproject’s net present value (the PV minus the initial cost):

NPVVineyard =$0.2M1.2

− $0.05M = $0.117M.

More generally, Mr. Rossi can consume more both in his young andold age.

Optimal decision rule: accept (reject) all projects with positive(negative) NPV

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Should Mr. Rossi Invest in the Restaurant?

Similarly, if Mr. Rossi also invests in the restaurant, in addition toinvesting in the vineyard, he will be left with

◦$1M − $0.05M − $0.1M = $0.85M now◦$0.20M + $0.14M = $0.34M later.

As before, instead of simply consuming these amounts, Mr. Rossi canborrow and lend to achieve other consumption patterns:— Suppose he invests the whole $0.85M. This will generate

$0.85M × (1.2) = $1.02M later, so he is left with

◦$0 now (since the whole $1M was invested);◦$1.02M + $0.34M = $1.36M later

— Or he can borrow the PV of $0.34M, that is $0.34M/1.2 = $0.283M,and repay $0.34M later. So he is left with:

◦$0.85M + $0.283M = $1.133M now◦$0.34M − $0.34M = $0 later

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Should Mr. Rossi Invest in the Restaurant? (cont’d)

In fact, by borrowing or lending the appropriate amounts, Mr. Rossican achieve any consumption on the following line:

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Should Mr. Rossi Invest in the Restaurant? (cont’d)

Mr. Rossi is better off by investing in both the vineyard and therestaurant, than just the vineyard:

If he chose to consume everything today, he could consume $1.133Mnow. In other words, his current wealth is increased by

$1.133M − $1.117M = $0.016M.

Again, this increase in wealth corresponds exactly to the project’sNPV (i.e. the PV minus the initial cost)

NPVrestaurant =$0.14M1.2

− $0.1M = $0.016M.

The consumption possibility frontier is pushed further out, i.e. he canconsume more in both periods.Therefore, Mr. Rossi should invest in the restaurant: this is true nomatter what his preferences are, as long as he prefers more to less.

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How Much Should Mr. Rossi Invest?

Clearly, Mr. Rossi should keep investing as long as he can keeppushing out the consumption possibility frontier, i.e. as long as he canfind positive-NPV investments. Again, this is true independently ofhis preferences.

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How Much Should Mr. Rossi Invest? (cont’d)

By investing I ∗, Mr. Rossi would be investing up to the point at whichthe slope of the investment opportunity line just equals (minus) 1+ r

The slope of the investment opportunity line tells you how many moredollars tomorrow you can have for an additional dollar of investmenttoday, and thus equals (minus) 1 plus the rate of return of themarginal investment. Therefore, we can also say that Mr. Rossishould invest for as long as he can find investments whose rates ofreturn are above the interest rate.

r is also known as the “opportunity cost of capital.”By investing $1in a project, Mr. Rossi forgoes the opportunity to put the dollar in thebank and earn r . Thus, it is effi cient to invest in the project if andonly if it gives a return of at least r

— This links to the IRR rule analyzed in more detail later

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Solution to Mr. Rossi’s Problem

Now that Mr. Rossi has decided how much to invest in realinvestment opportunities, he can choose how much to borrow/lend.This will depend on his preferences for consumption in youth versusconsumption in old age, as represented by the shape of hisindifference curves.

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Implications for the Financial Manager

When facing investment decisions managers should acceptinvestments with positive NPV —regardless of the preferences ofindividual shareholders.

The manager’s sole objective is to maximize shareholder wealth. Oncetheir wealth is maximized, individual shareholders can use the capitalmarket to achieve their preferred profile of consumption.

This powerful result is the Fisher separation theorem. A firm’soptimal choice of investments is separate from its owners’attitudestowards the investments. This allows different shareholders to bewilling to own shares in the same firm and delegate the running of thefirm to a professional manager.

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A Word of Caution

The FST (and optimality of the NPV rule) requires the market to be:— Complete: markets for shares and borrowing/lending exist— Effi cient: market prices reflect all available information— Perfect: no distorting taxes and frictions (such as transaction costs);individuals can borrow and lend at the same rate.

What happens if investors face different interest rates?

Behavioral finance studies financial decisions under market ineffi ciency.

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Key Takeaways: Capital Budgeting Under Certainty

The required rate of return on a project is determined exclusively bythe rate of return available elsewhere in the capital market

— It is independent of shareholders’preferences for consumption today vs.consumption tomorrow

A financial manager can therefore ignore shareholder preferences.His/her goal is simply to maximize shareholder value, by takingpositive-NPV projects and rejecting negative-NPV projects

— Once shareholder value is maximized, shareholders can use borrowingand lending to choose whatever consumption pattern suits theirindividual preferences

The Fisher separation theorem assumes capital markets are complete,effi cient and perfect

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Checkpoint: Capital Budgeting Under Certainty

Material relevant to this section:

BMA: chapter 5

— Problem set: none

Bulk pack problem set #3

What is next?

BMA: chapter 6

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I.3.2 Using the NPV Rule for Capital Budgeting

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I.3.2 Using the NPV Rule for Capital Budgeting

Readings: BMA, chapter 6.

This section will be especially relevant for:

— FNCE 726: Advanced Corporate Finance.— FNCE 728: Corporate Valuation.— FNCE 731: International Corporate Finance.— FNCE 750: Venture Capital and the Finance of Innovation.

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The NPV Formula

We have seen that financial managers act in the best interest of theshareholders by undertaking investments with positive NPV.

For a one-period investment the NPV formula is

NPV = C0 +C11+ r1

where C0 is the initial cash flow (which is generally negative) and C1is the end-of-period cash flow (which is usually positive).

The general formula is

NPV = C0 +T

∑t=1

Ct(1+ rt )t

=T

∑t=0

Ct(1+ rt )t

We have already discussed how to use bond prices to infer the interestrates rt . We now discuss how to compute the cash flows Ct .

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Free Cash Flow Is Not Profit

Net income (profit) is how much the company earns in year t,according to its accounts

Ct is how much cash physically flows into the firm and is available toinvestors, after all other claimants have been paid off

— Time value of money: cash can be invested elsewhere, unlike profit

Key differences:

— Capital expenditure affects cash flow but not profit— Depreciation affects profit but not cash flow, except via its effect ontaxes T

T = tc (Revenues− Expenses−Depreciation+ Extraordinary Gains)where tc is the corporate tax rate

Free cash flow = Cash from operations —Capex

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Cash From Operations

Three equivalent ways to calculate cash from operations:

1 Start from net income and add back depreciation:

CFO = (R − E −D + X )(1− tc ) +D

2 Add up only the cash items

CFO = R − E + X − tc (R − E −D + X )

3 Tax the cash items and add back the depreciation tax shield

CFO = (R − E + X )(1− tc ) + tcD

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Working Capital

Where there is working capital, the Free Cash Flow equation becomes

FCF = CFO − Increase in Working Capital − Capex

Working capital = Current Assets (restricted cash, inventories,accounts receivable, others) − Current Liabilities (accounts payable,others)

An investment in working capital is a cash outflow, just like aninvestment in capital expenditure

Cash is treated as working capital only if it is necessary for thecompany’s operations —e.g. cash in a cash register cannot be put ina bank to earn interest. In most cases, cash is assumed to be earninginterest in a bank and thus not counted as working capital

We will not consider working capital in FNCE604; however, it isfeatured in the case in FNCE612

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H. O. Coy

H. O. Coy Company is considering purchasing a machine costing$26,000 using excess cash generated through other projects.

The machine will generate revenues of $50,000 per year for five years.The cost of materials and labor needed to generate these revenueswill total $35,000 per year.

Even though the machine is expected to sell for $1,500 in 5 years, itwill be depreciated on a straight-line basis over 5 years to a $1,000book value.

The firm’s tax rate is 34% and its opportunity cost of capital is 10%.

Should the company purchase the machine?

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H. O. Coy (cont’d)

The cash flows are shown below:

End of Year

0 1 2 3 4 5

(1) Revenues 50,000 50,000 50,000 50,000 50,000

(2) Material & labor cost -35,000 -35,000 -35,000 -35,000 -35,000

(3) Profit on machine sale 500

(4) Depreciation -5,000 -5,000 -5,000 -5,000 -5,000

(5) Pre-tax profit 10,000 10,000 10,000 10,000 10,500

(6) Tax @ 34% -3,400 -3,400 -3,400 -3,400 -3,570

(7) Net income 6,600 6,600 6,600 6,600 6,930

(8) Add back depreciation 5,000 5,000 5,000 5,000 5,000

(9) Machine purchase/sale -26,000 1,000

(10) Free cash flow -26,000 11,600 11,600 11,600 11,600 12,930

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H. O. Coy (cont’d)

Purchase of machine for $26,000 does not affect profit: merelytransforms one asset (cash) into another (machine). However, itaffects cash flowSale of machine for $1,500 is comprised of 2 elements:— Sale for book value of $1,000 affects cash flow, but not profit— Profit of $500 above book value affects profit and cash flow

Ct can be calculated in three ways:— Net income + depreciation - capex ((7) + (8) + (9))— Add up only the cash items ((1) + (2) + (3) + (6) + (9))— Tax the cash items and add the DTS(((1) + (2) + (3)) (1− tc ) + (4)tc + (9))

NPV is calculated as:

NPV = −26,000+ 11,6001.10

+ ...+11,600(1.10)4

+12,930(1.10)5

= 18,798.95 > 0 so purchase

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Estimating Cash Flows

Only cash flows are relevant

— Cash flows are simply the difference between dollars received anddollars paid out. Do not confuse cash flows with accounting profits:ignore depreciation

— Estimate cash flows on an after-tax basis— Record cash flows at the time they actually occur (e.g. if selling itemson credit)

Estimate cash flows on an incremental basis

— Include all incidental effects, including opportunity costs— Forget sunk costs

Separate the effect of investment and financing decisions

Treat inflation consistently

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How to Treat Inflation

Either discount nominal cash flows at the nominal interest rate ordiscount real cash flows at the real interest rate. If properly applied,both methods should produce the same NPV, that is

NPVreal =T

∑t=0

C realt

(1+ Rt )t=

T

∑t=0

C nominalt

(1+ rt )t= NPVnominal.

To see this, remember (from slide 69) that the real rate Rt is given by

1+ Rt =1+ rt1+ it

,

so thatC nominalt

(1+ rt )t=C realt (1+ it )t

(1+ rt )t=

C realt

(1+ Rt )t.

This makes intuitive sense: real and nominal dollars are the sametoday .

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Example: BICC’s Toad Ranch

The Biological Insect Control Corporation (BICC) is planning toinvest in a ranch to breed toads, which the company plans to sell asecologically desirable insect-control mechanisms.

The company anticipates that the business will continue in perpetuity.

Following negligible start-up costs, BICC will incur the following(nominal) cash flows at the end of the first year:

Revenues $150,000Labor costs 80,000Other costs 40,000

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Example: BICC’s Toad Ranch (cont’d)

The company will lease the ranch and equipment for $20,000 a year.The first lease payment will be due at the end of the year.

The rate of inflation is expected to be 6%. Revenues and other costsare expected to remain constant in real terms. However, labor costswill increase at 1 percent per year in real terms. Lease payments arefixed in nominal terms.

The real discount rate is 5%.

There are no taxes.

What is the NPV of BICC’s toad ranch? (Solution in AdditionalMaterials)

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Example: BICC’s Toad Ranch (cont’d)

We will do the NPV calculations in nominal terms and in real terms.Of course, according to slide 124, the results should be exactly thesame. More precisely, we will calculate

NPV = PV (Rev’s)− PV (Other Costs)− PV (Labor Costs)− PV (Lease).We have i = 6% and R = 5%. From slide 124, we get

1+ R =1+ r1+ i

⇒ r = (1+ R)(1+ i)− 1 = (1.05)(1.06)− 1 = 11.3%.

In nominal terms, the cash flows (below) need to be discounted at r :Cash Flow at End of Year

1 2 3 ...

Revenues 150, 000 150, 000(1+ i) 150, 000(1+ i)2 ...

Other 40, 000 40, 000(1+ i) 40, 000(1+ i)2 ...

Labor 80, 000 80, 000(1.01)(1+ i) 80, 000(1.01)2(1+ i)2 ...

Lease 20, 000 20, 000 20, 000 ...

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Example: BICC’s Toad Ranch (cont’d)

The revenues and other costs are perpetuities growing at rate i :

PV (Revenues) =150,000r − i =

150,0000.113− 0.06 = 2,830,188.68;

PV (Other Costs) =40,000r − i =

40,0000.113− 0.06 = 754,716.98.

The labor costs are a perpetuity growing at rate g , where

1+ g = (1.01)(1+ i) ⇒ g = 7.06%.

Therefore,

PV (Labor Costs) =80,000r − g =

80,0000.113− 0.0706 = 1,886,792.45.

The lease is a regular perpetuity:

PV (Lease) =20,000r

=20,0000.113

= 176,991.15.

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Example: BICC’s Toad Ranch (cont’d)

In real terms, the cash flows need to be discounted at R, and they areas follows:

Cash Flow at End of Year1 2 3 ...

Revenues 150,0001+i

150,0001+i

150,0001+i ...

Other costs 40,0001+i

40,0001+i

40,0001+i ...

Labor costs 80,0001+i

80,000(1.01)1+i

80,000(1.01)2

1+i ...Lease 20,000

1+i20,000(1+i )2

20,000(1+i )3

...

The revenues and the other costs are both regular perpetuities:

PV (Revenues) =150,000/(1+ i)

R=150,000/(1.06)

0.05= 2,830,188.68;

PV (Other Costs) =40,000/(1+ i)

R=40,000/(1.06)

0.05= 754,716.98.

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Example: BICC’s Toad Ranch (cont’d)

The labor costs are a perpetuities growing at 1%:

PV (Labor Costs) =80,000/(1+ i)R − 0.01 =

80,000/1.060.05− 0.01 = 1,886,792.45.

The lease is a decreasing perpetuity. If G is the real growth rate, wehave 1+ g = (1+ G ) (1+ i). Thus,

G =11.06

− 1 = −5.66038%.

We then find:

PV (Lease) =20,000/(1+ i)

R − G =20,000/1.06

0.05− (−0.0566038) = 176,991.15.

Notice that all the PVs are the same as on page 128. So, asexpected, we get the same NPV as before.

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Investment Decisions Capital Budgeting Under Certainty

Example: Project Interaction

Until now we have assumed that the company is free to undertake allthe investments that have positive NPV. In fact, the financialmanager often faces “either-or”decisions.

Instances of such decisions typically arise in the following cases:

— Choosing between mutually exclusive projects.— Deciding when to replace an existing machine.— Deciding how to invest when resources are limited.

We next turn to these issues.

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Mutually Exclusive Projects

The baseline rule for choosing between alternative investments issimple: pick the one with the highest NPV. However, care should betaken when comparing investments with different lives if theinvestment can/must be repeated at the end of its life.

In this case we can use one of three equivalent criteria:

— Repeat each project enough times to make the investment horizonscomparable, and then use the NPV rule to choose between them.

— Compute an equivalent annual cash flow for each investment andchoose the investment with the highest EACF. The EACF is the cashflow of an annuity having the same life and PV as the investment. It isthe rate you would pay to lease the asset in a competitive market

— Repeat each project indefinitely, and compare with the NPV rule.

The implicit assumption in using the above criteria is that once aninvestment is chosen, the company will reinvest in the same project atall subsequent dates.

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Example: Choosing Between Machines with Different Lives

Suppose you are considering two alternative machines, A and B,having identical capacity. The first machine will last three years andthe second two years. The costs associated with operating them areas follows:

End-of-year costs (in $000)Machine C0 C1 C2 C3 PV at 6%A 15 4 4 4 25.69B 10 6 6 - 21.00

Should we choose machine B, which has a lower PV of costs? Notnecessarily, because B will have to be replaced a year earlier.

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Example: Machines with Different Lives (cont’d)

One way to compare the two investments is to compute the PV ofcost over a 6 year period, at the end of which both machines wouldhave to be replaced.

End-of-year costs (in $000)Machine C0 C1 C2 C3 C4 C5 C6 PV at 6%A 15 4 4 19 4 4 4 47.26B 10 6 16 6 16 6 6 56.32

Therefore, we should prefer machine A.

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Example: Machines with Different Lives (cont’d)

An equivalent way is to compute the equivalent annual cost: the cashflow C of an annuity having the same life and PV as the machine.Thus, for machine A, we look for CA that will make the PV of thefollowing two streams of cash flows equal.

0 1 2 3

15 4 4 4CA CA CA

PV = 25.69

↪→ CAa3|6% =CA0.06

(1− 1

(1.06)3

)= 25.69

⇒ CA = 9.61.

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Example: Machines with Different Lives (cont’d)

Similarly, for machine B:0 1 2 3

10 6 6CB CB

PV = 21.00

↪→ CBa2|6% =CB0.06

(1− 1

(1.06)2

)= 21.00

⇒ CB = 11.45.

Again, since CB > CA, we should choose machine A.

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Example: Machines with Different Lives (cont’d)

If we repeat the project (either over a 6-year horizon or to infinity), itis much easier to compare the EACFs than the original cash flows.— Original cash flows:

0 1 2 3 4 5 6 ...

15 4 4 4+ 15 4 4 4+ 15 ...10 6 6+ 10 6 6+ 10 6 6+ 10 ...

EACFs:

0 1 2 3 4 5 6 ...

9.61 9.61 9.61 9.61 9.61 9.61 9.61 ...11.45 11.45 11.45 11.45 11.45 11.45 11.45 ...

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Example: Arnold’s Armory

Arnold is tired of the Commando business, so he starts a gun shop forpeople with annoying neighbors.His shop needs a new delivery van. Arnold has narrowed his choicedown to two different models. Both require $1,000 in annualmaintenance, and have exactly the same cargo space. The differencesare their fuel effi ciency, useful life and initial purchase price.

Miles Purchase Useful ResaleModel per gallon price Life ValueA 20 $12,000 3 years $1,100B 30 $18,000 5 years $2,000

The van will be used approximately 24,000 miles per year, and thecost of gasoline is $1 per gallon. The cost of capital is 10%. Allfigures are in nominal terms.Suppose that the tax rate is 30% and that the vans will bedepreciated straight-line (to zero) over their useful life. Which vanshould Arnold buy?

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Example: Arnold’s Armory (cont’d)

Here are the after-tax cash flows for van A:

End of Year

Cash Flows 0 1 and 2 3 calculations

Purchase van A -12,000

Maintenance -700 -700 [= −1, 000× (1− 30%)]Fuel -840 -840 [= − 24,00020 ×1× (1− 30%)]Depreciation tax shield 1,200 1,200 [= 12,000

3 ×30%]Sale of van A 770 [= 1, 100× (1− 30%)]Original cash flows -12,000 -340 430

Equivalent Cash Flows CFA CFA

↪→ NPVA = −12,000− 340a2|10% +430(1.10)3

= −12, 267.02 = CFAa3|10%

⇒ CFA = −4, 932.75

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Example: Arnold’s Armory (cont’d)

Here are the after-tax cash flows for van B:

End of Year

Cash Flows 0 1 to 4 5 calculations

Purchase van B -18,000

Maintenance -700 -700 [= −1, 000× (1− 30%)]Fuel -560 -560 [= − 24,00030 ×1× (1− 30%)]Depreciation tax shield 1,080 1,080 [= 18,000

5 ×30%]Sale of van B 1,400 [= 2, 000× (1− 30%)]Original cash flows -18,000 -180 1,220

Equivalent Cash Flows CFB CFB

↪→ NPVB = −18,000− 180a4|10% +1, 220(1.10)5

= −17,813.05 = CFBa5|10%

⇒ CFB = −4,699.04 > CFA

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Example: When to Replace Existing Equipment

In the previous example we have taken the life of each machine asgiven. In fact, the time at which to replace a machine should also bethe result of a capital budgeting decision.

Suppose your machine is expected to produce a cash flow of $4,000 inone year and $3,000 in two years. After that it will have to bereplaced. The resale value is $1,000 if you replace now, $500 nextyear, and zero otherwise. The best alternative is a new machine thatwill be optimally replaced every 3 years, giving the following cashflows (inclusive of the resale value in year 3):

Cash FlowsC0 C1 C2 C3 PV at 6%-15,000 8,000 8,000 10,000 8,063.33

When should you replace your existing machine? (Assume no taxes.)

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Example: When to Replace Existing Equipment (cont’d)

First, let us calculate the equivalent annual cash flow EACF for thenew machine:

0 1 2 3

-15, 000 8, 000 8, 000 10, 000PV = 8, 063.33 EACF EACF EACF

↪→ EACFa3|6% =EACF0.06

(1− 1

(1.06)3

)= 8,063.33

⇒ EACF = 3,016.57.

The three options are:

(a) replace now;(b) replace in one year;(c) replace in two years.

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Example: When to Replace Existing Equipment (cont’d)

The cash flows for each of these options can be represented as follows:0 1 2 3 4 ...

(a) 1, 000 3, 016.57 3, 016.57 3, 016.57 3, 016.57 ...(b) 4, 500 3, 016.57 3, 016.57 3, 016.57 ...(c) 4, 000 3, 000 3, 016.57 3, 016.57 ...

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Example: When to Replace Existing Equipment (cont’d)

The PVs of these cash flows are calculated as follows:

PVa = 1,000+3,016.570.06

= 51,276.21;

PVb =4,5001.06

+11.06

(3,016.570.06

)= 51,675.67;

PVc =4,0001.06

+3,000(1.06)2

+1

(1.06)2

(3,016.570.06

)= 51,189.22.

The best option is to replace in one year.

Notice that the cash flows after year 2 are the same for all threeoptions. Thus we could have compared the three options bycomparing their PVs over the first two years only.

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Capital Budgeting Under Resource Constraints

Projects also interact if they require the same limited resources. Weassume that this resource is money, but the same principle applies toother resources.

One possibility is to get the biggest “bang for the buck”by choosingthe projects with the highest ratio of NPV to initial outlay (theprofitability index). PI = NPV

−C0 .

This rule has two key limitations:

— It fails when more than one resource is constrained or there is anyother constraint on project choice (e.g. mutual exclusivity).

— It is not valid if it is not possible to undertake “fractional” investments.If you have $11m in the following example, you will choose B and C

Project Cost (millions) PV (millions) Profitability IndexA 10 31.0 2.1B 6 18.0 2.0C 5 14.5 1.9

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Key Takeaways: Using the NPV Rule

Free Cash Flow = Cash From Operations - Capex

CFO is not profit. It is after tax, ignores sunk costs, and includesopportunity costs. 3 ways to calculate:

1 (R − E −D + X ) (1− tc ) +D2 R − E + X − tc (R − E −D + X )3 (R − E + X ) (1− tc ) + tcD

Discount nominal cash flows at a nominal discount rate; discount realcash flows at a real discount rate

If different lives, calculate Equivalent Annual Cash Flow: the cashflow of an annuity with the same life and PV as the investment.Equals the annual lease rate

If resource constraints, calculate Profitability Index: PI = NPV−C0

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Checkpoint: Using the NPV Rule

Material relevant to this section:

BMA: chapter 6

— Problem set: 1, 6, 8, 9, 14, 26, 29

Bulk pack problem set #4, questions 1-6

Additional Materials provides an overview of Linear Programming,which is sometimes used with resource constraints

What is next?

BMA: chapter 5

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I.3.3 Alternatives to the NPV Investment Rule

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I.3.3 Alternatives to the NPV Investment Rule

Readings: BMA, chapter 5.

This section will be especially relevant for:

— FNCE 728: Corporate Valuation.— FNCE 750: Venture Capital and the Finance of Innovation.

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NPV’s Competitors

Despite the optimality of the NPV rule, alternative investment ruleshave been– and to some extent still are– used by businesses.

We will now look at four common alternatives to NPV. They are:

1. Payback period.2. Accounting rate of return.3. Internal rate of return.4. Profitability index.

We will see that the internal rate of return and the profitability index,when properly used, lead to the same decisions as the NPV rule.

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NPV’s Competitors (cont’d)

As the following survey shows, many of the above investment ruleswere still broadly used in the 1980’s.

U.S. U.S. JapanMethod (1950’s) (1980’s) (1980’s)Payback 34% 12% 40%Acctg rate of return 34% 8% 19%IRR 19% 49% 15%NPV - 19% 9%Other 6% 10% 2%None 6% 2% 15%

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NPV’s Competitors (cont’d)

The most recently available comprehensive survey shows that netpresent value (NPV) and internal rate of return (IRR) are beingincreasingly commonly used in the U.S.:

% of CFOs usingProfitability index 12%Acctg rate of return 20%Payback 57%NPV 75%IRR 76%

Source: Graham and Harvey (2001): “The Theory and Practice ofFinance: Evidence From the Field.” Journal of Financial Economics

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Features of the NPV Rule

When looking at NPV’s competitors, it is important to keep in mindthe main features of the NPV rule:— Time value of money: $1 today > $1 tomorrow.— NPV depends only on all the forecasted cash flows from the projectand the opportunity cost of capital.

* Any rule ignoring some of the project’s cash flows will lead tosuboptimal decisions.

* Any rule affected by the manager’s tastes, the profitability of thecompany’s existing business, or the profitability of other independentprojects will lead to inferior decisions.

— Since PVs are all measured in today’s dollars, you can add them up.

* The NPV rule can thus identify whether joint projects are better thansingle projects.

— Clear benchmark: accept if NPV > 0

The alternatives to the NPV rule often fail to satisfy one or more ofthese critical features.

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The Payback Period Rule

The payback period of a project is the number of years it takes torecover the initial investment. The payback period rule accepts aproject if the payback period is less than some given cutoff.Here are some examples:

Cash Flows Payback NPVProject C0 C1 C2 C3 Period at 10%A −2, 000 2, 000 1 −181.82B −2, 000 1, 000 1, 000 1, 000 2 486.85C −2, 000 1, 000 1, 000 10, 000 2 7, 248.69The basic weaknesses of the payback rule are:— It ignores the time value of money.— It ignores the cash flows beyond the cutoff period.— It gives no indications on what the cutoff rule should be.

Some companies discount the cash flows before computing thepayback rule. This only addresses the first weakness.

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The Accounting Rate of Return

The accounting rate of return (also known as average return on bookvalue)is computed by dividing the average net income from a project(profit after taxes) by the average book value of the investment:(beginning investment - ending investment)/2

This ratio is then compared with the book rate of return for the firmas a whole (or some other equally absurd yardstick).

This criterion suffers from several defects:

— It ignores the relevant cash flow from investment and instead considersthe accounting profits (in particular, it depends critically on theaccountants’choice of a depreciation method).

— It ignores the time value of money (as well as the risk of the project).— The choice of a yardstick is totally arbitrary.

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The Internal Rate of Return Rule

The internal rate of return (IRR) of a project is defined as theconstant discount rate y which makes NPV = 0. In other words, ysolves

NPV =T

∑t=0

Ct(1+ y)t

= 0

Analogy: bond yield is the discount rate that makes PV = P0, i.e.NPV = 0

The IRR rule says that a project should be accepted if and only if yexceeds the yield on financial securities (bonds) with comparablematurity, cash flows and risk (the opportunity cost of capital).

Notice that with a flat term structure (rt = r for all t), the IRR ruleimplies that we should accept a project if and only if y > r .

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The Internal Rate of Return Rule (cont’d)

For example, consider the following project:C0 C1 C2 C3−5, 000 2, 000 2, 000 2, 000

The graph below shows that, with a flat term structure, the IRR ruleis equivalent to the NPV rule.

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Limitations of the IRR Rule: 1. Non-Flat Term Structure

The IRR rule is very diffi cult to apply with a non-flat term-structure,since the opportunity cost of capital is now a complicated average ofthe interest rates r1, r2, . . ., rT .For example, assume the following term structure,

t1 2 3 4 5

rt 4.00% 4.50% 5.00% 5.50% 6.00%and consider the two projects:Project C0 C1 C2 C3 C4 C5 IRR NPVA −1, 000 20 20 20 20 1, 200 5.24% −32.32B −1, 000 50 50 1, 050 5.00% 0.89

Why does project A have higher IRR but lower NPV?

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Limitations of the IRR Rule: 1. Non-Flat Term Structure(cont’d)

Answer: the IRR of A should be compared to a cutoff different fromB. The IRR for project A (B) should be compared to the yield on a5-year (3-year) bond with the same cash flows.

The prices P5 and P3 of such 5-year and 3-year bonds are

P5 =201.04

+20

(1.045)2+ · · ·+ 1,200

(1.06)5= 967.68,

P3 =501.04

+50

(1.045)2+

1,050(1.05)3

= 1,000.89.

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Limitations of the IRR Rule: 1. Non-Flat Term Structure(cont’d)

The yields on these two bonds can be calculated as follows:

967.68 =20

1+ yA+

20(1+ yA)2

+ · · ·+ 1,200(1+ yA)5

⇒ yA = 5.96%.

1,000.89 =50

1+ yB+

50(1+ yB )2

+1,050

(1+ yB )3⇒ yB = 4.97%.

Since IRRA < yA, we should reject project A. However, sinceIRRB > yB , we should accept project B.

Note that, since NPVA = −1, 000+ 967.68 andNPVB = −1,000+ 1,000.89, we have essentially gone back to theNPV rule!

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Limitations of the IRR Rule: 2. Lending or Borrowing?

Another problem with the IRR rule is that it is necessary todistinguish between borrowing and lending opportunities, as thefollowing example shows:

Project C0 C1 C2 IRRA (lending) −5, 000 3, 000 3, 000 13%B (borrowing) 5, 000 −3, 000 −3, 000 13%

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Limitations of the IRR Rule: 2. Lending or Borrowing?(cont’d)

In the above example, it was easy to tell that project A was lending(and so we want IRR > discount rate) and that project B wasborrowing (and so we want IRR < discount rate).

But, how about the following case? Is this borrowing or lending?

Project C0 C1 C2 C3 IRRC 1, 000 −3, 600 4, 320 −1, 728 20%

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Limitations of the IRR Rule: 3. Multiple or no IRRs

A project can have more than one IRR (in general, there can be asmany different IRRs as there are changes in the sign of cash flows). Infact, it is also possible that the IRR does not exist for some projects.As an example, consider the following two projects

Project C0 C1 C2 IRRA −5, 000 20, 000 −18, 000 37% and 163%B −5, 000 15, 000 −12, 500 none

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Limitations of the IRR Rule: 4. Mutually Exclusive Projects

Finally, the IRR rule can be misleading when choosing betweenmutually exclusive projects, as the following example shows (weassume that the term structure is flat at 10%):

Project C0 C1 C2 IRR NPV at 10%A -2,000 2,000 3,000 82.29% 2,298B -5,000 1,000 9,000 44.54% 3,347

The IRR can be salvaged in the case of mutually exclusive projects bycomputing the IRR for the incremental cash flows.

Project C0 C1 C2 IRRB − A −3, 000 −1, 000 6, 000 25.73%

Since the IRR of the incremental cash flows is greater than 10%, weshould choose B over A.

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The Profitability Index Rule

The profitability index is defined as the NPV of future cash flowsdivided by the initial investment:

PI =NPV−C0

=∑Tt=0 Ct/(1+ rt )t

−C0.

The PI index rule entails accepting a project if and only if PI > 0.

Note that since

PI = − NPVC0

,

the PI rule is equivalent to the NPV rule (provided that C0 < 0).

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The Profitability Index Rule (cont’d)

As with the IRR rule, however, the acceptance rule has to be reversedfor borrowing projects (C0 > 0).

Moreover, the PI rule can be misleading when applied to mutuallyexclusive projects, unless we look at the incremental cash flows. Thisis shown in the following example (where once again we assume thatthe term structure is flat at 10%):

Project C0 C1 PI NPV at 10%A -1,000 2,000 0.82 818B -10,000 15,000 0.36 3,636B-A -9,000 13,000 0.31 2,818

The PI rule assumes that you invest only once, in year 0, and that youcannot reinvest the proceeds in the same project or other projects later

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Key Takeaways: Alternatives to the NPV Rule

Investment rule should:

— Consider time value of money— Consider all of a project’s cash flows, and only these cash flows: ignoreexisting projects.

— Have a clear benchmark

Payback period: number of years to recover initial investment

Accounting return: net income / average investment

Internal rate of return: constant discount rate which makes NPV = 0.Like bond yield. Accept if IRR > r . Problems if:

— Non-flat term structure: what r to compare against?— Lending or borrowing: if latter, accept if IRR < r— Multiple IRRs or no IRRs— Mutually exclusive projects

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Checkpoint: Alternatives to the NPV Rule

Material relevant to this section:

BMA: chapter 5.

— Problem set: 1, 4, 5, 7, 8, 11, 13, 14

Bulk pack problem set #4, questions 7-9

What is next?

The Placement Exam

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Summary of Key Takeaways

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Summary of Key Takeaways

The following slides are the key takeaways for each of the sections ofthe course

These key takeaways are already at the end of the respective sectionsof the bulk pack. I am repeating them here for easy reference —byputting them all in one place, this may help with revision and alsowith looking up formulas in the exam

Note that there is also a formula sheet at the end of the “ProblemSets and Solutions”bulk pack

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Key Takeaways: Constant Interest Rate

Compounding factor gives future value of $1

— CFT =(1+ r

m

)mT= erT if m→ ∞

Effective annual rate gives equivalent rate under once-a-yearcompounding

— r =(1+ r

m

)m − 1Discount factor gives present value of $1

— DFT = 1/CFT

Growing perpetuity: PV0 = C1/ (r − g)Annuity factor gives PV of $1 for T years. aT |r =

1r

[1− 1

(1+r )T

]Real rate of return R = 1+r

1+i − 1

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Key Takeaways: Term Structure

Spot rate rT is the rate per year for T years starting today

The term structure is the graph of r1, r2, ... for different T

With a non-flat term structure, PV = ∑Tt=1

Ct(1+rt )t

Forward rate ft is the rate for 1 year starting at t − 1 and ending at t:— ft =

(1+rt )t

(1+rt−1)t−1− 1

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Key Takeaways: Pricing Bonds

Treasury bills pay no coupon, and are quoted in terms of a discountrate: P = F

(1− d N

360

)Treasury notes and bonds pay a semiannual coupon (C2 every 6months), and are quoted in units of 1/32

The price of a bond is P = ∑Tt=1

C(1+rt )t

+ F(1+rT )T

where rt is thet-year spot rate

The yield of a bond is the single constant discount rate y that solvesP = ∑T

t=1C

(1+y )t +F

(1+y )T . It is a (weighted geometric) average ofthe spot rates

— For Treasuries, quoted yields are semiannual APRs: y given by(1+ y

2

)2= 1+ y

The coupon of a bond is fixed. Its yield depends on marketconditions, and is inversely related to its price

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Key Takeaways: Capital Budgeting Under Certainty

The required rate of return on a project is determined exclusively bythe rate of return available elsewhere in the capital market

— It is independent of shareholders’preferences for consumption today vs.consumption tomorrow

A financial manager can therefore ignore shareholder preferences.His/her goal is simply to maximize shareholder value, by takingpositive-NPV projects and rejecting negative-NPV projects

— Once shareholder value is maximized, shareholders can use borrowingand lending to choose whatever consumption pattern suits theirindividual preferences

The Fisher separation theorem assumes capital markets are complete,effi cient and perfect

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Key Takeaways: Using the NPV Rule

Free Cash Flow = Cash From Operations - Capex

CFO is not profit. It is after tax, ignores sunk costs, and includesopportunity costs. 3 ways to calculate:

1 (R − E −D + X ) (1− tc ) +D2 R − E + X − tc (R − E −D + X )3 (R − E + X ) (1− tc ) + tcD

Discount nominal cash flows at a nominal discount rate; discount realcash flows at a real discount rate

If different lives, calculate Equivalent Annual Cash Flow: the cashflow of an annuity with the same life and PV as the investment.Equals the annual lease rate

If resource constraints, calculate Profitability Index: PI = NPV−C0

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Key Takeaways: Alternatives to the NPV Rule

Investment rule should:

— Consider time value of money— Consider all of a project’s cash flows, and only these cash flows: ignoreexisting projects.

— Have a clear benchmark

Payback period: number of years to recover initial investment

Accounting return: net income / average investment

Internal rate of return: constant discount rate which makes NPV = 0.Like bond yield. Accept if IRR > r . Problems if:

— Non-flat term structure: what r to compare against?— Lending or borrowing: if latter, accept if IRR < r— Multiple IRRs or no IRRs— Mutually exclusive projects

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