Lecture 04 NPV Rule

8/17/2019 Lecture 04 NPV Rule

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Professor Sang Byung [email protected]

NPV rule



Last class

• The time value of money (2nd part)• PV short cuts

• Annuity

• Perpetuity

• Growing annuity/perpetuity

• Delayed annuity/perpetuity



Today

•Finish the remainder of Lecture 03• Compounding frequencies

• Investment criteria• NPV rule



Net Present Value

•NPV of a project or investment• The difference between

• PV of its future benefits

• PV of its costs

• In many cases, simply

• PV of future cash flows net of an initial investment

0 1 2 T-1 T ……3

−

where

(initial investment) is negative



NPV calculation

•Nothing special!

+ 1 + + 1 + + ⋯+ 1 +

+ =

= 1 +

0 1 2 T-1 T

……3

−



Example

• Assume that the cash flowsfrom the construction and

sale of an office building is

as follows.

• Given a 7% required rate of

return, what is the NPV?

150,000 100,0001.07 + 300,000

1.07 $18,573.67



NPV rule

•NPV rule• Accept projects with NPV>0

• Reject projects with NPV<0

• Result

• Following the NPV rule maximize the value of the firm.



NPV rule

•Is this rule really obvious?• What if a project has the positive NPV, but

• All cash flows occur 10 years from now?

• Shareholders are really old?

• Do you still think that you need to always accept a

project with the positive NPV?

• Do individuals’ preferences matter?

– Today’s consumption versus future consumption?



Illustrative example

•

Michael was endowed with $1M.• He can borrow or lend from a bank at 20%.

• He allocates wealth between youth and old age

• What should he do?

1. He can go on a trip around the world, and then live inpoverty in old age.

2. He can go on a smaller trip, have a moderate lifestyle in hisyouth (spending $0.5M) and still have 0.5(1.2)=0.6M for hisold age.

3. He can put it all in the bank and go for an even better triparound the world in old age ($1.2M)



• Choose one point on the

line depending on hispreferences

• Slope is -1.2

• Trade-off between spending

in youth vs in old age

• Every $1 dollar you do not

spend, you get $1.2 in old

age

Today vs Future

•

All these possibilities are equally “correct.”• The choice just depends on Michael’s preferences!

1.2 1 1.2 1.2



Opening a restaurant

•Start-up cost would be $0.7M.

• It will yield $0.8M in old age.

• Should he invest?

• No! If he puts the $0.7M in the bank, he will have

• 0.7(1.2) $0.84 > $0.8• In other words,

• 0.7 + .. 0.03 < 0

• We don’t need to know anything about Michael’s

preferences to make this decision.



Opening a bar

•Start-up cost would be $0.7M.

• It will yield $0.9M in old age.

• Should he invest?

• It seems that the answer should be yes.

• 0.7 + .. 0.05 > 0

• But, spending $0.7M today leaves him only $0.3M

• If he wants to spend less than $0.3M in youth, it is fine.

• But what if he wants to spend more than $0.3M in youth?

–

For example, he might want to buy a nice house.



After investing in a bar

•

Michael wants to consume $ in youth.• If > $0.3, he needs to borrow $(x-0.3) from the bank.

• In old age

– Earn $ 0.9M from the bar

– Need to pay back the loan

y 0.9 – 1.2( 0.3)• If < $0.3, he deposits the remainder, which is $(0.3-x).

• In old age

– Earn $ 0.9M from the bar

– Need to collect the deposit

0.9 + 1.2(0.3 )• He spends $(1.26 – 1.2x) million in old age.



1.2 1.2

1.26 1.2

y intercept: 1.26

x intercept: 1.05

Comparison

•

After investing a bar• Michael has better choices!

• If he wants to spend all inyouth, he can spend $1.05$,

which is $0.05M larger thanbefore.

• Recall that the NPV = $0.05M!

•

NPV rule• Accepting a positive NPV

project is equivalent toreceiving their NPV in cashtoday!



More than one shareholder

•

Should corporations follow the same rule?• Not one shareholder, but many

•

For example, assume two shareholders1. Grandma (100 years old and wants money now)

2. Child’s investment trust (wants money later)

• The company has two projects:

1. Produce sports cars (income now but NPV<0)

2. Develop electric cars (income later but NPV>0)



Conflict btw shareholders?

•Is there a conflict between the two shareholders?• NO! Both are better off if the firm develops electronic

cars.

• Why? Grandma can borrow, and then she (or her heirs)

can repay the loan using the project’s cash flows.

• Both shareholders, regardless of their preferences forcurrent or future income, want the firm to choose the

positive NPV project.



Fisher Separation Theorem

•Investment decision does not depend onpreferences of individual investors for current

versus future income.

• All shareholders want the firm to use the NPV rule, which

maximizes their share value.

• Then, they will use the capital market (e.g. banks) toobtain their desired consumption.



Importance of this theorem

•

Without it, the CFO’s job would be impossible!• The CFO would have to poll thousands of people to see when

they want their consumption, and somehow aggregate thosepreferences.

• On the other hand, individuals would have toalways be lobbying for their wishes!

• Because the NPV rule works,

• We can diversify our wealth across corporations.

• Corporations can draw from the huge amount of resourcesavailable in capital markets.



What have we assumed?

•This rule will be the basis of all we do.

• But, we made some simplifying assumptions.

1. The borrowing rate is equal to the lending rate.

2. All investors have the same information (or they might

disagree on NPV or even r).

3. Markets are competitive (no firm affects interest rates).



Another example

•Time Inc will make a profit of $45M at the endof this year, and $90M at the end of next year.

•

Assume its discount rate is 10%. What is thepresent value of these cash flows? (That is,

what is Time Inc worth today?)

• Answer:

45

1.1

+ 90

1.1 115.3



Example (cont’d)

• Jane Fonda is the only shareholder of Time Inc.

• All profits Time Inc generates are paid out as

dividends.

• She wants to donate $100M to the UN next year.

• Does this mean that Time Inc reconsider its

future business?



Example (cont’d)

•The answer is no!

• Then, how can she achieve her goal?

• Method 1: Sell shares

•

She can sell her shares for $115.3M today, invest it at 10% to get$126.8M next year, pay $100M to the UN, and reinvest the

remaining $26.8M at 10% to end up with $29.5M.

• Method 2: Dividend payment + a loan from a bank

•

She can keep her shares, get a dividend of $45M next year,borrow an additional $55M, and pay $100M to the UN. At the

end of the second year, she will get a dividend of $90M, pay

$55M*1.1=$60.5M to her lenders to repay the loan, and again be

left with $29.5M.



Takeaways

•Lessons for Time Inc• The management does not have to worry about Jane

Fonda’s need for cash in making their decisions about

investments.

• It does not need to generate income by speeding up

production or taking on short-term projects instead of

making long-term R&D investments.

• It does not have to worry about Jane Fonda’s desire to

save the rain forests…



Takeaways

•How is this possible?• Once Time Inc maximizes the value of its shares, the

capital markets allow Jane Fonda to sell or borrow

against them and do whatever she wants.

• In fact, the more the value of Time Inc’s shares, the more

Fonda can give to the UN, and the more rain forests can

be saved!

Lecture 04 NPV Rule

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