Last Study Topics
• Present Value (PV)
– A dollar today is worth more than a dollar
tomorrow
• Net Present Value (NPV)
– NPV = PV – INV
• NPV Rule
– Accept the project that makes a net contribution
to value.
• Rate Of Return Rule
– Rate of Return is > Cost of Capital
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Topics covered
• Opportunity Cost of Capital
• Investment vs. Consumption
• Managers and the Interests of Shareholders
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Instructor: Mr. Wajid Shakeel Ahmed
Opportunity Cost of Capital
• The opportunity cost of capital is such an
important concept.
• Theory of valuation endorses that rate of
return of the project must always be greater
to the rate of return of an alternative
opportunity.
• This alternative rate becomes an opportunity
cost of capital.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Opportunity Cost of Capital
Example
You may invest $100,000 today. Depending on the
state of the economy, you may get one of three
possible cash payoffs:
Economy
Payoff
Slump
Normal
Boom
$80,000 110,000 140,000
80,000  100,000  140,000
Expected payoff  C1 
 $110,000
3
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• But what’s the right discount rate?
• You search for a common stock with the same
risk as the investment.
• Stock X turns out to be a perfect match i.e., X’s
price next year, given a normal economy, is
forecasted at $110.
• The stock price will be higher in a boom and
lower in a slump, but to the same degrees as your
investment ($140 in a boom and $80 in a slump).
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
Example - continued
The stock is trading for $95.65. Depending on the
state of the economy, the value of the stock at the
end of the year is one of three possibilities:
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Economy
Slump
Normal
Boom
Stock Pric e
$80
110
140
Instructor: Mr. Wajid Shakeel Ahmed
Continue
Example - continued
The stocks expected payoff leads to an expected
return.
80  110  140
Expected payoff  C1 
 $110
3
expected profit 110  95.65
Expected return 

 .15 or 15%
investment
95.65
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• The amount it would cost investors in the
stock market to buy an expected cash flow is
$110,000. (They could do so by buying 1,000
shares of stock X @ $110 each)
• It is, therefore, also the sum that investors
would be prepared to pay you for your
project.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Opportunity Cost of Capital
Example - continued
Discounting the expected payoff at the expected
return leads to the PV of the project.
110,000
PV 
 $95,650
1.15
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• To calculate net present value, deduct the
initial investment:
NPV = 95,650 - 100,000 = -$4,350
• The project is worth $4,350 less than it costs.
• It is not worth undertaking.
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Instructor: Mr. Wajid Shakeel Ahmed
Alternative
• Compare the expected project return with the
cost of capital:
expected profit 110  100
Expected return 

 .10 or 10%
investment
100
• The 10 % return on project < 15 % return on
stock of equal risk.
• The project is not worthwhile.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Discount rate equals the
opportunity cost of capital
• The opportunity cost of capital is the return
earned by investing in the best alternative
investment.
• This return will not be realized if the investment
under consideration is undertaken.
• Thus, the two investments must earn at least the
same return.
• This return rate is the discount rate used in the
net present value calculation.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
A source of Confusion
• Bank A will lend you the $100,000 that you
need for the project at 8 percent. Does that
mean that the cost of capital for the project is
8 %? Or does not?
• Does Not - First, the interest rate on the loan
has nothing to do with the risk of the project.
• Second, whether you take the loan or not, you
still face the choice between the projects.
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Instructor: Mr. Wajid Shakeel Ahmed
Case: Norman
• Norman Gerrymander has just received a $2
million bequest. How should he invest it?
• There are four immediate alternatives.
– Which of these investments have positive NPVs?
– Which would you advise Norman to take?
Lets have a look each of them!
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Alternative A
• A - Investment in one-year U.S. government
securities yielding 5 percent.
• Step 1: calculate the payoff after 1 year.
= 2.0M X 5% = 2.1 M.
• Step 2: Discounting the expected payoff at the
expected return leads to the PV of the project.
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
2.1 M
PV 
 $ 2 .0 M
1.05
• Step 3: To calculate net present value, deduct the
initial investment:
NPV = $2.0M - $2.0M = $0
• The ‘A’ investment has not contributed to the net
value.
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Instructor: Mr. Wajid Shakeel Ahmed
Alternative B
• B - A loan to Norman’s nephew Gerald, who
has for years aspired to open a big Cajun
restaurant in Duluth.
• Gerald had arranged a one-year bank loan for
$900,000, at 10 percent, but asks for a loan
from Norman at 7 percent.
• Step 1: calculate the payoff after 1 year.
= 0.9 M X 7% = 0.96 M.
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• Step 2: Discounting the expected payoff at the
expected return leads to the PV of the project.
0.96 M
PV 
 $0.88M
1.10
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• Step 3: To calculate net present value, deduct
the initial investment:
NPV = $0.88M - $.9M = -$0.02M
• The project is worth $0.02M less than it costs.
• It is not worth undertaking.
• The ‘B’ investment has brought decline to the
net value.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Alternative C
• C - Investment in the stock market. The
expected rate of return is 12 percent.
• Step 1: calculate the payoff after 1 year.
= 2.0 M X 12% = 2.24 M.
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Instructor: Mr. Wajid Shakeel Ahmed
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• Step 2: Discounting the expected payoff at the
expected return leads to the PV of the project.
2.26 M
PV 
 $ 2 .0 M
1.12
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• Step 3: To calculate net present value, deduct
the initial investment:
NPV = $2.0M - $2.0M = $0
• The ‘C’ investment has not contributed to the
net value nor a decline.
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Instructor: Mr. Wajid Shakeel Ahmed
Alternative D
• B - Investment in local real estate, which
Norman judges is about as risky as the stock
market.
• The opportunity at hand would cost $1 million
and is forecasted to be worth $1.1 million
after one year.
• Step 1: calculate the payoff after 1 year.
= $ 1.1 M.
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• Step 2: Discounting the expected payoff at the
expected return leads to the PV of the project.
1.1 M
PV 
 $0.98 M
1.12
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• Step 3: To calculate net present value, deduct
the initial investment:
NPV = $0.98M - $1.0M = -$0.02M
• The project is worth $0.02M less than it costs.
• It is not worth undertaking.
• The ‘D’ investment has brought decline to the
net value.
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• Results from four Alternatives;
– The ‘A’ investment has not contributed to the net value
nor a decline.
– The ‘B’ investment has brought decline to the net value.
– The ‘C’ investment has not contributed to the net value
nor a decline.
– The ‘D’ investment has brought decline to the net value.
• Decision;
– Norman should invest in either the risk-free government
securities or the risky stock market, depending on his
tolerance for risk.
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Instructor: Mr. Wajid Shakeel Ahmed
Investment vs. Consumption
• Some people prefer to consume now. Some
prefer to invest now and consume later.
• Borrowing and lending allows us to reconcile
these opposing desires which may exist within
the firm’s shareholders.
• Could a positive-NPV project for Ms. X be a
negative-NPV proposition for Mr. Y? Could they
find it impossible to agree on the objective of
maximizing the market value of the firm?
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• This could be answer;
• Well functioned capital markets allows
investors with different time patterns of
income and desired consumption to agree on
whether investment projects should be
undertaken.
• Lets consider an example !
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Instructor: Mr. Wajid Shakeel Ahmed
Ant vs Grasshopper
• A is an ant, who wishes to save for the future;
G is a grasshopper, who would prefer to spend
all his wealth.
• Suppose both of them can earn a return of
about 14 percent on every $100 investment.
The interest rate is 7 percent.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Investment vs. Consumption
The grasshopper (G) wants to
consume now. The ant (A) wants to
wait. But each is happy to invest. A
prefers to invest 14%, moving up the
red arrow, rather than at the 7%
interest rate. G invests and then
borrows at 7%, thereby transforming
$100 into $106.54 of immediate
consumption. Because of the
investment, G has $114 next year to
pay off the loan. The investment’s
NPV is $106.54-100 = +6.54
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Investment vs. Consumption
•
Dollars
Later
A invests $100 now
and consumes $114
next year
114
107
The grasshopper (G) wants to consume now.
The ant (A) wants to wait. But each is happy
to invest. A prefers to invest 14%, moving up
the red arrow, rather than at the 7% interest
rate. G invests and then borrows at 7%,
thereby transforming $100 into $106.54 of
immediate consumption. Because of the
investment, G has $114 next year to pay off
the loan. The investment’s NPV is $106.54100 = +6.54
G invests $100 now,
borrows $106.54 and
consumes now.
100
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106.54
Dollars
Now
Instructor: Mr. Wajid Shakeel Ahmed
Calculation
• Return of A: INV = C0;
– r = C1 -C0 / C0
*
C1 = Payoff;
100 =
= 14%
• NPV for G: INV = C0;
C1 = Payoff; INT = r;
– PV = C1/ (1+r)t =
NPV =
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= $106.54
PV – C0 =
Instructor: Mr. Wajid Shakeel Ahmed
Explanation
• In our example the ant and the grasshopper
placed an identical value on same project and
were happy to share in its construction.
• They agreed because they faced identical
borrowing and lending opportunities.
• Whenever firms discount cash flows at capital
market rates, they are implicitly assuming that
their shareholders have free and equal access
to competitive capital markets.
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• Despite their different tastes, both A and G
are better off by investing in the project and
then using the capital markets to achieve the
desired balance between consumption today
and consumption at the end of the year.
• If A and G were shareholders in the same
enterprise, there would be no simple way for
the manager to reconcile their different
objectives.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Summary
• Opportunity Cost of Capital
– Rate of an Alternative investment opportunity
having a similar risk.
• Investment vs. Consumption
– Manager finds it difficult to reconcile the different
objectives of the shareholders.
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Instructor: Mr. Wajid Shakeel Ahmed
Calculation of NPV and ROR
• The opportunity cost of capital is 20 percent for all four
investments.
Initial Cash
Cash Flow
Investment
Flow, C0
in Year 1, C1
1
10,000
18,000
2
5,000
9,000
3
5,000
5,700
4
2,000
4,000
a. Which investment is most valuable?
b. Suppose each investment would require use of the same
parcel of land. Therefore you can take only one. Which one?
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Continue
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Continue
• Answer to question ‘a’;
– Investment 1, since this investment with respect
to others has generate highest NPV with Max rate
of return.
• Answer to question ‘b’;
– Investment 1, since this investment with respect
to others has highest contributed net value.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Decision Rules
• Here then we have two equivalent decision
rules for capital investment;
– Net present value rule; Accept investments that
have positive net present values.
– Rate-of-return rule; Accept investments that offer
rates of return in excess of their opportunity costs
of capital.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Fundamental Result
• Our justification of the present value rule was
restricted to two periods and to a certain cash
flow.
• However, the rule also makes sense for
uncertain cash flows that extend far into the
future. The argument goes like this:
• 1- A financial manager should act in the
interests of the firm’s owners, its stockholders.
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• 2- Stockholders do not need the financial
manager’s help to achieve the best time
pattern of consumption.
• 3- How then can the financial manager help
the firm’s stockholders?
– There is only one way: by increasing the market
value of each stockholder’s stake in the firm.
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• Despite the fact that shareholders have
different preferences, they are unanimous in
the amount that they want to invest in real
assets.
• This means that they can cooperate in the
same enterprise and can safely delegate
operation of that enterprise to professional
managers.
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• These managers do not need to know
anything about the tastes of their
shareholders and should not consult their own
tastes.
• Their task is to maximize net present value. If
they succeed, they can rest assured that they
have acted in the best interest of their
shareholders.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Managers and Shareholder Interests
• Do managers really looking after the interests
of shareholders?
• This takes us back to the principal–agent
problem.
– Several institutional arrangements that help to
ensure that the shareholders’ pockets are close to
the managers’ heart.
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• Utilizing their Voting power;
– If shareholders believe that the corporation is
underperforming and that the board of directors
is not sufficiently aggressive in holding the
managers to task, they can try to replace the
board in the next election.
• E.g; chief executives of Eastman Kodak,
General Motors, Xerox, Lucent, Ford Motor,
etc were all forced to step aside.
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Instructor: Mr. Wajid Shakeel Ahmed
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• As a result the stock price tumbles.
– This damages top management’s reputation and
compensation.
• Part of the top managers’ paychecks comes
from bonuses tied to the company’s earnings
or from stock options, which pay off if the
stock price rises but are worthless if the price
falls below a stated threshold.
– This should motivate managers to increase
earnings and the stock price.
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• If managers and directors do not maximize
value, there is always the threat of a hostile
takeover.
• The further a company’s stock price falls, due
to lax management or wrong-headed policies,
the easier it is for another company or group
of investors to buy up a majority of the shares.
– The old management team is then likely to find
themselves out on the street and their place is
taken by a fresh team.
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• Should managers look after the interests of
shareholders?
• For this question we need to understand that
In most instances there is little conflict
between doing well (maximizing value) and
doing good.
– Profitable firms are those with satisfied customers
and loyal employees and vice versa;
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• Ethical issues do arise in business as in other
walks of life.
– when we say that the objective of the firm is to
maximize shareholder wealth, we do not mean that
anything goes.
• In business and finance, as in other day-to-day
affairs, there are unwritten, implicit rules of
behavior.
• To work efficiently together, we need to trust
each other.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Summary
• Opportunity Cost of Capital
– Rate of an Alternative investment opportunity
having a similar risk.
• Investment vs. Consumption
– Manager finds it difficult to reconcile the different
objectives of the shareholders.
• Managers and the Interests of Shareholders
– Are managers taking care of the interests of the
Shareholders or should they?
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Principles of Corporate Finance
Brealey and Myers

Sixth Edition
How to Calculate Present Values
Chapter 3
11/16/2014
Irwin/McGraw Hill
Instructor: Mr. Wajid Shakeel Ahmed
©The McGraw-Hill Companies, Inc., 2000
Topics Covered
•
•
•
•
•
Valuing Long-Lived Assets
PV Calculation Short Cuts
Compound Interest
Interest Rates and Inflation
Example: Present Values and Bonds
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Present Values
Discount Factor = DF = PV of $1
 Discount Factors can be used to compute
the present value of any cash flow.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Present Values
Discount Factor = DF = PV of $1
DF 
1
t
(1 r )
 Discount Factors can be used to compute
the present value of any cash flow.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Present Values
C1
PV  DF  C1 
1  r1
DF 
1
(1 r ) t
 Discount Factors can be used to compute
the present value of any cash flow.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Present Values
Ct
PV  DF  Ct 
1  rt
 Replacing “1” with “t” allows the formula
to be used for cash flows that exist at any
point in time.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Present Values
Example
Suppose you will receive a certain cash inflow
of $100 next year (C1 = $100) and the rate of interest on
one-year U.S. Treasury notes is 7 percent (r1 = 0.07). What
would be the present value of Cash inflow ?
PV 
11/16/2014
C1
(1r )

100
(1.07)
 $94
Instructor: Mr. Wajid Shakeel Ahmed
Present Values
Example
Suppose you will receive a certain cash inflow
of $100 in two year (C2 = $100) and the rate of interest on
two-year U.S. Treasury notes is 7.7 percent (r1 = 0.077).
What would be the present value of year 2 Cash inflow ?
PV 
11/16/2014
C1
(1r )

100
(1.077)
 $86.21
Instructor: Mr. Wajid Shakeel Ahmed
Present Values
Example
You just bought a new computer for $3,000. The payment
terms are 2 years same as cash. If you can earn 8% on
your money, how much money should you set aside today
in order to make the payment when due in two years?
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Present Values
Example
You just bought a new computer for $3,000. The payment
terms are 2 years same as cash. If you can earn 8% on
your money, how much money should you set aside today
in order to make the payment when due in two years?
PV 
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3000
2
(1.08 )
 $2,572.02
Instructor: Mr. Wajid Shakeel Ahmed
Present Values
 PVs can be added together to evaluate
multiple cash flows.
PV 
11/16/2014
C1
(1 r )
 (1 r ) 2 ....
C2
1
Instructor: Mr. Wajid Shakeel Ahmed
Present Values
 PVs can be added together to evaluate
multiple cash flows, and called as
discounted cash flow or (DCF) formula;
PV  
11/16/2014
Ct
t
(1 rt )
Instructor: Mr. Wajid Shakeel Ahmed
Present Values
• If a dollar tomorrow is worth less than a
dollar today, one might suspect that a dollar
the day after tomorrow should be worth
even less.
• But, lets assume - given two dollars, one
received a year from now and the other two
years from now, the value of each is
commonly called the Discount Factor.
Assume r1 = 20% and r2 = 7%.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Present Values
• Given two dollars, one received a year from now
and the other two years from now, the value of
each is commonly called the Discount Factor.
Assume r1 = 20% and r2 = 7%.
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DF1 
1.00
(1.20)1
DF2 
1.00
(1.07 ) 2
 $.83
 $.87
Instructor: Mr. Wajid Shakeel Ahmed
Continue
• If First we lend $1,000 for one year at 20
percent, we have;
– FV = PV (1+rt)t
=
• Go to the bank and borrow the present value
of this $1,200 at 7 percent interest, we have;
– PV = FV / (1+rt)t
=
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• If we going to find out the net present value of
our investment we get,
– NPV = PV – Investment
=
“Just imagine in this game of lending and borrowing
How much money you can earn without taking risks”
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
• Of course this story is completely fanciful.
– “There is no such thing as a money machine.”
• In well-functioning capital markets, any
potential money machine will be eliminated
almost instantaneously by investors who try to
take advantage of it.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Present Values
Example
Assume that the cash flows
from the construction and sale
of an office building is as
follows. Given a 7% required
rate of return, create a present
value worksheet and show the
net present value.
Year 0
Year 1
Year 2
 150,000  100,000  300,000
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Instructor: Mr. Wajid Shakeel Ahmed
Present Values
Example - continued
Assume that the cash flows from the construction and sale of an office
building is as follows. Given a 7% required rate of return, create a
present value worksheet and show the net present value.
Period
Discount
0
Factor
1. 0
1
1
1.07
2
1
1.07 2
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 .935
 .873
Cash
Present
Flow
 150,000
Value
 150,000
 100,000
 93,500
 300,000
 261,900
NPV  Total 
$18,400
Instructor: Mr. Wajid Shakeel Ahmed
NPV Formula
Mathematically; PV @ 7%
∑PV = PV of C1+ PV of C2
= -$93,500 + $261,900
= $168,400
NPV = ∑ PV - INV
= $168,400 - $150,000
= $18,400
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Instructor: Mr. Wajid Shakeel Ahmed
Short Cuts
• Sometimes there are shortcuts that make it
very easy to calculate the present value of an
asset that pays off in different periods.
• These tolls allow us to cut through the
calculations quickly.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Short Cuts
Perpetuity - Financial concept in which a cash
flow is theoretically received forever.
cashflow
Return 
present va lue
C
r
PV
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Short Cuts
Perpetuity - Financial concept in which a cash
flow is theoretically received forever.
cash flow
PV of Cash Flow 
discount rate
C1
PV 
r
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Case : Investment A
• An investment costs $1,548 and pays $138 in
perpetuity. If the interest rate is 9 percent,
what is the NPV?
– PV = C / r
= $1533.33
– NPV = PV - INV
= $1533.33 - $1548
= -$ 14.67
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Short Cuts
Annuity - An asset that pays a fixed sum each
year for a specified number of years.
1
1 
PV of annuity  C   
t
 r r 1  r  
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Case: leasing a car
Example
You agree to lease a car for 4 years at $300 per month.
You are not required to pay any money up front or at the
end of your agreement. If your opportunity cost of capital
is 0.5% per month, what is the cost of the lease?
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Instructor: Mr. Wajid Shakeel Ahmed
Continue
Example - continued
You agree to lease a car for 4 years at $300 per
month. You are not required to pay any money up
front or at the end of your agreement. If your
opportunity cost of capital is 0.5% per month,
what is the cost of the lease?
 1

1
Lease Cost  300  

48 
 .005 .0051  .005 
Cost  $12,774.10
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Case: Endowment Funds
Example - Suppose, for example, that we begins to wonders
what it would cost to contribute in a endowment fund with
an amount of $100,000 a year for only 20 years?
 1

1
Lease Cost  100,000  

20 
 .10 .101  .10 
Cost  $851,400
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Alternatively
• We can simply look up the answer in the
annuity table given on the next slide.
• This table gives the present value of a dollar
to be received in each of t periods.
• In our example t = 20 and the interest rate r
= .10, and therefore;
• We look at the twentieth number from the top
in the 10 percent column. It is 8.514. Multiply
8.514 by $100,000, and we have our answer,
$851,400.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Compound Interest
• There is an important distinction between
compound interest and simple interest.
– When money is invested at compound interest,
each interest payment is reinvested to earn more
interest in subsequent periods.
– In contrast, the opportunity to earn interest on
interest is not provided by an investment that
pays only simple interest.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
18
16
14
12
10
8
6
4
2
0
10% Simple
Number of Years
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
30
27
24
21
18
15
12
9
6
10% Compound
3
0
FV of $1
Compound Interest
Compounding Interest
Suppose that the bank starts with $10 million of automobile loans
outstanding. This investment grows to ;
$10 * 1.005 = $10.05 million after month 1,
$10 * 1.0052 = $10.10025 million after month 2, and
$10 1.00512 $10.61678 million after 12 months.
Thus the bank is quoting a 6 percent APR but actually earns
6.1678 percent if interest payments are made monthly.
In general, an investment of $1 at a rate of r per annum
compounded m times a year amounts by the end of the year to
[1 + (r/m)]m, and the equivalent annually compounded rate of
interest is [1 + (r/m)]m - 1.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Compound Interest
i
ii
Periods Interest
per
per
year
period
iii
APR
(i x ii)
iv
Value
after
one year
v
Annually
compounded
interest rate
1
6%
6%
1.06
2
3
6
1.032
= 1.0609
6.090
4
1.5
6
1.0154 = 1.06136
6.136
12
.5
6
1.00512 = 1.06168
6.168
52
.1154
6
1.00115452 = 1.06180
6.180
365
.0164
6
1.000164365 = 1.06183
6.183
11/16/2014
6.000%
Instructor: Mr. Wajid Shakeel Ahmed
Explanation
• 1$ at the end of the year can be calculated as;
–
[1 + (r/m)]m ;
• Since, we have r = 6%; periods = m = 12, 52;
– =
– = $1.06168
;
=
= $1.06180
• Annually compounded rate;
– [1 + (r/m)]m – 1
– =
11/16/2014
;
=
Instructor: Mr. Wajid Shakeel Ahmed
Continuous Compounding
• Eventually one can quote a continuously
compounded rate, so that payments were assumed to
be spread evenly and continuously throughout the year.
• In terms of our formula, this is equivalent to letting m
approach infinity.
As m approaches infinity [1 + (r/m)]m approaches
(2.718)r or er = (2.718)r
•By the end of t years ert = (2.718)rt
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Example 1
• Suppose you invest $1 at a continuously
compounded rate of 11 percent (r=.11) for one year
(t =1). The end-year value is e =.11, which can be
calculated as;
ert = (2.718).11*1 = $1.116
•In other words, investing at 11 percent a year
continuously compounded is exactly the same as
investing at 11.6 percent a year annually
compounded.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Example 2
• Suppose you invest $1 at a continuously
compounded rate of 11 percent (r=.11) for two year
(t =2). The end-year value is e =.11, which you can be
calculated as;
ert = (2.718).11*2 = $1.246
• In other words, investing at 11 percent for 2 years
continuously compounded is exactly the same as
investing at 24.6 percent a year annually
compounded.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Example 3
• Suppose that we start thinking more seriously
and decided to found a home for us, which
will cost $100,000 a year, starting and spread
evenly over 20 years. Annual compounding
rate is 10%. What sum should we set aside?
• Here we going to apply Annuity formula
– Step 1: calculate continuously compounded rate
– r = 9.53% or ( e .0953 = 1.10)
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Continue
• Step2: calculate the PV of the annuity;
– PV = C [1/r – (1/r*1/ert)]
=
= $893,200
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Alternatively
• we could have cut these calculations short by
using PV of annuity Table, given on the next
slide.
• This shows that, if the annually compounded
return is 10 percent, then $1 a year spread
over 20 years is worth $8.932.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Inflation
Inflation - Rate at which prices as a whole are
increasing.
Nominal Interest Rate - Rate at which money
invested grows.
Real Interest Rate - Rate at which the
purchasing power of an investment
increases.
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Inflation
1+nominal interest rate
1  real interest rate =
1+inflation rate
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Inflation
1+nominal interest rate
1  real interest rate =
1+inflation rate
approximation formula
Real int. rate  nominal int. rate - inflation rate
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Inflation
Example
If the interest rate on one year govt. bonds is 5.9%
and the inflation rate is 3.3%, what is the real
interest rate?
Savings
Bond
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Inflation
Example
If the interest rate on one year govt. bonds is 5.9%
and the inflation rate is 3.3%, what is the real
interest rate?
1 + real interest rate =
1 + real interest rate =
real interest rate
11/16/2014
=
1+.059
1+.033
1.025
.025 or 2.5%
Instructor: Mr. Wajid Shakeel Ahmed
Savings
Bond
Inflation
Example
If the interest rate on one year govt. bonds is 5.9%
and the inflation rate is 3.3%, what is the real
interest rate?
1+.059
1 + real interest rate = 1+.033 Savings
1 + real interest rate =
real interest rate
=
1.025
Bond
.025 or 2.5%
Approximation =.059-.033 =.026 or 2.6%
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Valuing a Bond
Example
If today is October 2000, what is the value of the following
bond?
 An IBM Bond pays $115 every Sept for 5 years. In Sept
2005 it pays an additional $1000 and retires the bond.
 The bond is rated AAA (WSJ AAA YTM is 7.5%).
Cash Flows
Sept 01 02 03 04 05
115
115 115 115 1115
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Valuing a Bond
Example continued
If today is October 2000, what is the value of the following bond?
 An IBM Bond pays $115 every Sept for 5 years. In Sept 2005 it pays an
additional $1000 and retires the bond.
 The bond is rated AAA (WSJ AAA YTM is 7.5%).
115
115
115
115
1,115
PV 




2
3
4
1.075 1.075 1.075 1.075 1.0755
 $1,161.84
11/16/2014
Instructor: Mr. Wajid Shakeel Ahmed
Bond Prices and Yields
1600
1400
Price
1200
1000
800
600
400
200
0
0
2
4
6
5 Year 9% Bond
11/16/2014
8
10
1 Year 9% Bond
Instructor: Mr. Wajid Shakeel Ahmed
12
14
Yield