Time Value of Money and NPV

1
Time Value of Money and NPV

• We introduce the time value of money
• This leads to the opportunity cost of capital,
• Which in turn is used to calculate (net) present
  value and future value

2
The Money Market

• Individuals and institutions have different
  income streams and different intertemporal
  consumption preferences.
• Because of this, a market has arisen for money.
  The price of money is the interest rate.

3
Security Markets

• Individuals and Institutions also have different
  risk preferences and disagree about the prospects
  of industries and corporations.
• This gave rise to security markets, where risky
  assets are traded.

4
The Efficient, Competitive Market

• In a competitive market
• Trading is costless.
• Information is available to all participants
• There are many traders no individual can move
  market prices.
• In an efficient market, all similar securities
  promise the same expected return. Otherwise
  potential for arbitrage.
• Financial markets are the most efficient and
  competitive of all markets.

5
The Opportunity Cost of Capital I

• Assume that you as CFO have 100 million
  available cash. You could
• Return the cash to investors, who can trade in
  financial markets.
• Buy financial securities such as debt and equity
  of other companies OR
• Engage in a real investment project.

6
The Opportunity Cost of Capital II

• When should you undertake a project?
• Only if you can achieve a higher return than what
  you could get for similar financial assets.
• The rate of return you could obtain from similar
  financial assets is called the (opportunity) cost
  of capital.

7
The Opportunity Cost of Capital III

• What is a similar asset? Similar in
• Liquidity?
• Firm size?
• Firm age and maturity?
• Yes, all of these, but most important
• SIMILAR IN RISKYNESS
• In the case of debt, the cost of capital equals
  the interest you pay

8
Net Present Value

• We can calculate how much better off in todays
  dollar the investment makes us by calculating the
  Net Present Value.

9
Corporate Investment Decision Making

• In reality, shareholders do not vote on every
  investment decision faced by a firm and the
  managers of firms need decision rules to operate
  by.
• All shareholders of a firm will be made better
  off if managers follow the NPV ruleundertake
  positive NPV projects and reject negative NPV
  projects.

10
The Separation Theorem

• The separation theorem in financial markets says
  that all investors will want to accept or reject
  the same investment projects by using the NPV
  rule, regardless of their personal preferences.
• Logistically, separating investment decision
  making from the shareholders is a basic
  requirement of the modern corporation.

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Summary

• Financial markets exist because people want to
  adjust their consumption over time. They do this
  by borrowing or lending.
• An investment should be rejected if a superior
  alternative exists in the financial markets.
• If no superior alternative exists in the
  financial market, an investment has a positive
  net present value.

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In this course

• we will first see how much mileage we get as
  long as we know the opportunity cost of capital
• Corporations typically have a good estimate of
  their cost of capital
• We will later on talk about how the cost of
  capital is determined

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Lets go into the details

• and look at some examples of
• present values and
• future values
• We start out simply and then move to multiple
  periods.
• You should already be familiar with these
  please review the concepts!

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The One-Period Case Future Value

• In the one-period case, the formula for FV can be
  written as
• FV C1(1 r)
• Where C1 is cash flow at date 1 and
• r is the cost of capital.

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The One-Period Case Future Value

• If you were to invest 10,000 at 5-percent
  interest for one year, your investment would grow
  to 10,500
• 500 would be interest (10,000 .05)
• 10,000 is the principal repayment (10,000 1)
• 10,500 is the total due. It can be calculated
  as
• 10,500 10,000(1.05).
• The total amount due at the end of the investment
  is called the Future Value (FV).

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The One-Period Case Present Value

• If you were to be promised 10,000 due in one
  year when interest rates are at 5-percent, your
  investment be worth 9,523.81 in todays dollars.

The amount that a borrower would need to set
aside today to to able to meet the promised
payment of 10,000 in one year is call the
Present Value (PV) of 10,000.
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The One-Period Case Present Value

• In the one-period case, the formula for PV can be
  written as

Where C1 is cash flow at date 1 and r is the
opportunity cost of capital.
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The One-Period Case Net Present Value

• The Net Present Value (NPV) of an investment is
  the present value of the expected cash flows,
  less the cost of the investment.
• Suppose an investment that promises to pay
  10,000 in one year is offered for sale for
  9,500. Cost of Capital is 5. Should you buy?

Yes!
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The One-Period Case Net Present Value

• In the one-period case, the formula for NPV can
  be written as

If we had not undertaken the positive NPV
project considered on the last slide, and instead
invested our 9,500 elsewhere at 5-percent, our
FV would be less than the 10,000 the investment
promised and we would be unambiguously worse off
in FV terms as well 9,500(1.05) 9,975 10,000.
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The Multiperiod Case Future Value

• The general formula for the future value of an
  investment over many periods can be written as
• FV C0(1 r)T
• Where
• C0 is cash flow at date 0,
• r is the appropriate interest rate, and
• T is the number of periods over which the cash is
  invested.

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The Multiperiod Case Future Value

• Suppose that Jay Ritter invested in the IPO of
  the Modigliani company. Modigliani pays a current
  dividend of 1.10, which is expected to grow at
  40-percent per year for the next five years.
• What will the dividend be in five years?
• FV C0(1 r)T
• 5.92 1.10(1.40)5

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Future Value and Compounding

• Notice that the dividend in year five, 5.92, is
  considerably higher than the sum of the original
  dividend plus five increases of 40-percent on the
  original 1.10 dividend
• 5.92 1.10 51.10.40 3.30
• This is due to compounding.

23
Future Value and Compounding
24
Present Value and Compounding

• How much would an investor have to set aside
  today in order to have 20,000 five years from
  now if the current rate is 15?

20,000
PV
25
How Long is the Wait?

• If we deposit 5,000 today in an account paying
  10, how long does it take to grow to 10,000?

26
What Rate Is Enough?

• Assume the total cost of a college education will
  be 50,000 when your child enters college in 12
  years. You have 5,000 to invest today. What rate
  of interest must you earn on your investment to
  cover the cost of your childs education?
  Answer 21.15.

27
Compounding Periods

• Compounding an investment m times a year for T
  years provides for future value of wealth

For example, if you invest 50 for 3 years at 12
compounded semi-annually, your investment will
grow to
28
Effective Annual Interest Rates

• A reasonable question to ask in the above example
  is what is the effective annual rate of interest
  on that investment?

The Effective Annual Interest Rate (EAR) is the
annual rate that would give us the same
end-of-investment wealth after 3 years
29
Effective Annual Interest Rates (continued)

• So, investing at 12.36 compounded annually is
  the same as investing at 12 compounded
  semiannually.

30
Continuous Compounding (Advanced)

• The general formula for the future value of an
  investment compounded continuously over many
  periods can be written as
• FV C0erT
• Where
• C0 is cash flow at date 0,
• r is the stated annual interest rate,
• T is the number of periods over which the cash is
  invested, and
• e is a transcendental number approximately equal
  to 2.718. ex is a key on your calculator.

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Simplifications

• Perpetuity
• A constant stream of cash flows that lasts
  forever.
• Growing perpetuity
• A stream of cash flows that grows at a constant
  rate forever.
• Annuity
• A stream of constant cash flows that lasts for a
  fixed number of periods.
• Growing annuity
• A stream of cash flows that grows at a constant
  rate for a fixed number of periods.

32
A Perpetuity is a

• constant stream of cash flows that lasts forever.

The formula for the present value of a perpetuity
is
33
Perpetuity Example

• What is the value of a century-old British
  consol that promises to pay 15 each year
  forever?
• The interest rate is 10-percent.

34
A Growing Perpetuity is a

• growing stream of cash flows that lasts forever.

The formula for the present value of a growing
perpetuity
35
Growing Perpetuity Example

• The expected dividend next year is 1.30 and
  dividends are expected to grow at 5 forever.
• If the discount rate is 10, what is the value of
  this promised dividend stream?

36
An Annuity is a

• constant stream of cash flows with a fixed
  maturity.

The formula for the present value of an annuity
is
37
Annuity Example

• If you can afford a 400 monthly car payment,
  how much car can you afford if interest rates are
  7 on 36-month loans?

38
And finally A Growing Annuity
Formula for the present value of a growing
annuity
39
Growing Annuity

• A defined-benefit retirement plan offers to pay
  20,000 per year for 40 years and increase the
  annual payment by 3-percent each year. What is
  the present value at retirement if the discount
  rate is 10-percent?

40
What Is a Firm Worth?

• Conceptually, a firm should be worth the present
  value of the firms cash flows.
• The tricky part is determining the size, timing
  and risk of those cash flows.

41
Summary and Conclusions

• We introduced future value and present value.
• Interest rates are commonly expressed on an
  annual basis, but semi-annual, quarterly, monthly
  and even continuously compounded interest rate
  arrangements exist.
• The formula for the net present value of an
  investment that pays C for N periods is

42
Summary and ConclusionsFour Useful Formulas