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P.V. VISWANATH
FOR A FIRST COURSE IN FINANCE
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 NPV and IRR
 How do we decide to invest in a project or not?
 Using the Annuity Formula
 Valuing Mortgages and Similar payment plans
 Valuing Simple Financial Securities
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 The NPV decision rule says: Accept a project if NPV>0.
 There is another decision rule based on the Internal Rate

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of Return (IRR).
The IRR is the rate of return that makes the NPV = 0.
To understand what the IRR is, we can use the concept of
the NPV profile.
The NPV profile is the function that shows the NPV of the
project for different discount rates.
Then, the IRR is simply the discount rate where the NPV
profile intersects the X-axis.
That is, the discount rate for which the NPV is zero.
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 Suppose we are looking at a new project and you have
estimated the following cash flows:




Year 0:
Year 1:
Year 2:
Year 3:
-165,000 (required initial investment)
63,120
70,800
91,080
 Suppose the required rate of return is 12%. Then we can
compute the NPV as the sum of the discounted present
values of these cashflows.
 NPV = -165000 + 63120/(1.12) + 70,800/(1.12)2 +
91080/(1.12)3 = 12,627.42
 If we use different discount rates, we will get different
NPVs, as shown in the next graph.
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70,000
60,000
50,000
NPV
40,000
30,000
20,000
10,000
0
-10,000 0
0.02 0.04 0.06 0.08
0.1
0.12 0.14 0.16 0.18
0.2
0.22
-20,000
Discount Rate
 Clearly, the IRR decision rule corresponding to the NPV rule
is: Accept a project if the IRR > Required Rate of Return.
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 Present Value – earlier money on a time line
 Future Value – later money on a time line
100
100
100
100
100
0
1
2
3
4
5
100
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 If a project yields $100 a year for 6 years, we may want to know
the value of those flows as of year 1; then the year 1 value would
be a present value.
 If we want to know the value of those flows as of year 6, that year
6 value would be a future value.
 If we wanted to know the value of the year 4 payment of $100 as
of year 2, then we are thinking of the year 4 money as future
value, and the year 2 dollars as present value.
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 Interest rate – “exchange rate” between earlier money and later money
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
(normally the later money is certain).
Discount Rate – rate used to convert future value to present value.
Compounding rate – rate used to convert present value to future value.
Required rate of return – the rate of return that investors demand for
providing the firm with funds for investment. This is from the investors’
point of view. The higher the rate of return available, the more investors
are willing to supply.
Cost of capital – the rate at which the firm obtains funds for investment;
this is from the firm’s point of view. The lower the rate that firms have to
pay, the more funds they will demand since more investment projects
will meet the hurdle rate of return, i.e. the cost of firms’ funds.
The total amount of funds that will be lent will be equal to the amount at
which the investors’ required rate of return will equal the amount that
the firms is willing to pay. Hence in equilibrium, the cost of capital will
be equal to the investors’ required rate of return.
Opportunity cost of capital – the rate that the firm has to pay investors
in order to obtain an additional $ of funds, i.e. this is the marginal cost
of capital. This is the rate that investors demand from this firm because
if the firm doesn’t pay this this much, they can get that return from other
demanders of capital.
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 If capital markets are in equilibrium, the rate that
the firm has to pay to obtain additional funds will
be equal to the rate that investors will demand
for providing those funds. This will be “the”
market rate.
 Hence this is the single rate that should be used
to convert future values to present values and
vice-versa.
 Hence this should be the discount rate used to
convert future project (or security) cashflows into
present values.
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 Suppose you invest $1000 for one year at 5% per year. What is
the future value in one year?


The compounding rate is given as 5%. Hence the value of current
dollars in terms of future dollars is 1.05 future dollars per current
dollar.
Hence the future value is 1000(1.05) = $1050.
 Suppose you leave the money in for another year. How much will
you have two years from now?


Now think of money next year as present value and the money in two
years as future value. Hence the price of one-year-from-now money
in terms of two-years-from-now money is 1.05.
Hence 1050 of one-year-from-now dollars in terms of two yearsfrom-now dollars is 1050(1.05) = 1000 (1.05)(1.05) = 1000(1.05)2 =
1102.50
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 FV = PV(1 + r)t
 FV = future value
 PV = present value
 r = period interest rate, expressed as a decimal
 T = number of periods
 Future value interest factor = (1 + r)t
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 Simple interest
 Compound interest
 The notion of compound interest is relevant when money is invested for more than
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
one period.
After one period, the original amount increases by the amount of the interest paid
for the use of the money over that period.
After two periods, the borrower has the use of both the original amount invested and
the interest accrued for the first period. Hence interest is paid on both quantities.
This is why if the interest rate is r% per period, then a $1 today grows to $(1+r)
tomorrow and $(1+r)2 in two periods.
(1+r)2 = 1+2r+r2 . The 2r is the “simple” interest for each of the two periods and the
r2 = r x r is the interest for the second period on the $r of interest earned in the first
period.
This computation is done automatically when we use the formula FV(C in t periods)
= C(1+r)t
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From Brealey, Myers and Allen, “Principles of Corporate Finance”
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From Brealey, Myers and Allen, “Principles of Corporate Finance”
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 Suppose you invest the $1000 from the previous
example for 5 years. How much would you have?

FV = 1000(1.05)5 = 1276.28
 The effect of compounding is small for a small
number of periods, but increases as the number
of periods increases. (Simple interest would have
a future value of $1250, for a difference of
$26.28.)
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 Suppose you had a relative deposit $10 at 5.5%
interest 200 years ago. How much would the
investment be worth today?

FV = 10(1.055)200 = 447,189.84
 What is the effect of compounding?
 Without compounding the future value would have been
the original $10 plus the accrued interest of
10(0.055)(200), or 10 + 110 = $120.
 Compounding caused the future value to be higher by an
amount of $447,069.84!
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 Suppose your company expects to increase unit
sales of books by 15% per year for the next 5
years. If you currently sell 3 million books in one
year, how many books do you expect to sell in 5
years?

FV = 3,000,000(1.15)5 = 6,034,072
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 How much do I have to invest today to have some
amount in the future?
 FV = PV(1 + r)t
 Rearrange to solve for PV = FV / (1 + r)t
 When we talk about discounting, we mean
finding the present value of some future amount.
 When we talk about the “value” of something, we
are talking about the present value unless we
specifically indicate that we want the future
value.
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 Suppose you need $10,000 in one year for the
down payment on a new car. If you can earn 7%
annually, how much do you need to invest today?
 PV = 10,000 / (1.07)1 = 9345.79
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 You want to begin saving for your daughter’s
college education and you estimate that she will
need $150,000 in 17 years. If you feel confident
that you can earn 8% per year, how much do you
need to invest today?

PV = 150,000 / (1.08)17 = 40,540.34
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 Your parents set up a trust fund for you 10 years
ago that is now worth $19,671.51. If the fund
earned 7% per year, how much did your parents
invest?

PV = 19,671.51 / (1.07)10 = 10,000
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 For a given interest rate – the longer the time
period, the lower the present value
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What is the present value of $500 to be received in 5
years? 10 years? The discount rate is 10%
5 years: PV = 500 / (1.1)5 = 310.46
10 years: PV = 500 / (1.1)10 = 192.77
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 For a given time period – the higher the interest
rate, the smaller the present value

What is the present value of $500 received in 5 years if the
interest rate is 10%? 15%?
Rate = 10%: PV = 500 / (1.1)5 = 310.46
 Rate = 15%; PV = 500 / (1.15)5 = 248.58

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 PV = FV / (1 + r)t
 There are four parts to this equation
 PV, FV, r and t
 If we know any three, we can solve for the fourth
 FV = PV(1+r) t
r = (FV/PV)1/t – 1
t = ln(FV/PV)  ln(1+r)
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 You are looking at an investment that will pay
$1200 in 5 years if you invest $1000 today. What
is the implied rate of interest?

r = (1200 / 1000)1/5 – 1 = .03714 = 3.714%
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 Suppose you are offered an investment that will
allow you to double your money in 6 years. You
have $10,000 to invest. What is the implied rate
of interest?

r = (20,000 / 10,000)1/6 – 1 = .122462 = 12.25%
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 Suppose you have a 1-year old son and you want
to provide $75,000 in 17 years towards his
college education. You currently have $5000 to
invest. What interest rate must you earn to have
the $75,000 when you need it?

r = (75,000 / 5,000)1/17 – 1 = .172688 = 17.27%
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 Start with basic equation and solve for t
(remember your logs)


FV = PV(1 + r)t
t = ln(FV / PV) / ln(1 + r)
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 You want to purchase a new car and you are
willing to pay $20,000. If you can invest at 10%
per year and you currently have $15,000, how
long will it be before you have enough money to
pay cash for the car?

t = ln(20,000/15,000) / ln(1.1) = 3.02 years
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 Suppose you want to buy a new house. You
currently have $15,000 and you figure you need
to have a 10% down payment plus an additional
5% in closing costs. If the type of house you want
costs about $150,000 and you can earn 7.5% per
year, how long will it be before you have enough
money for the down payment and closing costs?
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 How much do you need to have in the future?



Down payment = .1(150,000) = 15,000
Closing costs = .05(150,000 – 15,000) = 6,750
Total needed = 15,000 + 6,750 = 21,750
 Using the formula

t = ln(21,750/15,000) / ln(1.075) = 5.14 years
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 The present value of an annuity can be calculated
by taking each cash flow and discounting it back
to the present, and adding up the present values.
Alternatively, there is a short cut that can be used
in the calculation [A = Annuity; r = Discount
Rate; n = Number of years]
A
1 
PV of an Annuity  PV ( A, r , n)  1 
n 
r  (1  r ) 
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 The present value of an annuity of $1,000 at the
end of each year for the next five years, assuming
a discount rate of 10% is PV of $1000 each year for next 5 years
1 
1 
(1.10)5 
= $1000
 $3,791
 .10



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 The reverse of this problem, is when the present
value is known and the annuity is to be estimated
- A(PV,r,n).
Annuity given Present Value

r

= A(PV, r,n) = PV
1
1 (1 + r)n





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 Suppose you borrow $200,000 to buy a house on
a 30-year mortgage with monthly payments. The
annual percentage rate on the loan is 8%.
 The monthly payments on this loan, with the
payments occurring at the end of each month,
can be calculated using this equation:

Monthly interest rate on loan = APR/12 = 0.08/12 =
0.0067
Monthly Payment on Mortgage


0.0067


= $200,000
 $1473.11
1
1 
(1.0067)360 

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 The future value of an end-of-the-period annuity
can also be calculated as follows(1 + r)n - 1 
FV of an Annuity = FV(A,r,n) = A 

r


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 Thus, the future value of $1,000 at the end of
each year for the next five years, at the end of the
fifth year is (assuming a 10% discount rate) FV of $1, 000 each year for next 5 years
(1.10)5 - 1 
= $1000 
= $6,105

.10


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 If you are given the future value and you are
looking for an annuity, you can use the following
formula:
Annuity given Future Value

r

= A(FV, r,n) = FV 
(1+ r)n - 1 

Note, however, that the two formulas, Annuity,
given Future Value and Present Value, given
annuity can be derived from each other, quite
easily. You may want to simply work with a single
formula.
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 Assume that you want to send your newborn child to a private
college (when he gets to be 18 years old). The tuition costs are
$16000/year now and that these costs are expected to rise 5% a
year for the next 18 years. Assume that you can invest, after taxes,
at 8%.


Expected tuition cost/year 18 years from now = 16000*(1.05)18 =
$38,506
PV of four years of tuition costs at $38,506/year = $38,506 * PV(A
,8%,4 years) = $127,537
 If you need to set aside a lump sum now, the amount you would
need to set aside would be 
Amount one needs to set apart now = $127,357/(1.08)18 = $31,916
 If set aside as an annuity each year, starting one year from now 
If set apart as an annuity = $127,537 * A(FV,8%,18 years) = $3,405
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 You are trying to value a straight bond with a fifteen year
maturity and a 10.75% coupon rate. The current interest rate
on bonds of this risk level is 8.5%.
PV of cash flows on bond = 107.50* PV(A,8.5%,15 years) +
1000/1.08515 = $ 1186.85
 If interest rates rise to 10%,
PV of cash flows on bond = 107.50* PV(A,10%,15 years)+ 1000/1.1015
= $1,057.05
Percentage change in price = -10.94%
 If interest rate fall to 7%,
PV of cash flows on bond = 107.50* PV(A,7%,15 years)+ 1000/1.0715 =
$1,341.55
Percentage change in price = +13.03%
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 A consol bond is a bond that has no maturity and
pays a fixed coupon. Assume that you have a 6%
coupon console bond. The value of this bond, if
the interest rate is 9%, is as follows Value of Consol Bond = $60 / .09 = $667
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 A growing perpetuity is a cash flow that is expected to
grow at a constant rate forever. The present value of a
growing perpetuity is PV of Growing Perpetuity
CF1
=
(r - g)
where



CF1 is the expected cash flow next year,
g is the constant growth rate and
r is the discount rate.