Finance, Financial Markets, and NPV
First Principles
Finance
• Most business decisions can be looked at as a
choice between money now versus money later.
• Finance is all about how special markets, the
financial markets, help people make themselves
better off by moving money across time.
• As simple as this sounds, the related concepts
seem complex and the markets appear
complicated enough to require some introduction.
• We will develop the important concept that helps
us “keep score” in an honest way while we think
about moving money across time.
Example
• Suppose right now I have $100 but I am not planning to
use it until dinner tomorrow.
• You on the other hand have the good fortune of having a
dinner date tonight but the misfortune of not getting paid
until tomorrow. Oh the humiliation.
• You offer a solution to the terrible problem.
• You suggest I give you the $100 today and tomorrow night
you give me $100 back.
• This makes you better off by avoiding the humiliation and
allowing you to engage in a desired activity, but what
about me?
Example cont…
• One thought is of course that idiot professors don’t
really matter in the face of your humiliation. But
let’s put that aside for now.
• How can we make it so we are both better off and
what do we call such an arrangement?
• What if you can’t find me or someone as kind and
generous?
• How can we make the process easier?
– Can we keep lots of people from wasting lots of time
looking for partners?
– Can we balance those who want to borrow and those
who want to lend?
Example Concluded
• Simple as it was our example enabled us to
introduce the following fundamental ideas.
–
–
–
–
–
–
Financial market.
Time value of money.
Interest rate (the price in this market).
Financial Intermediaries.
Market clearing.
Equilibrium interest rates.
Maintained Assumptions
• For the moment, in order to simplify the analysis, we will
assume:
– Perfect certainty
– Perfect capital markets
•
•
•
•
Information freely available to all participants.
Equal access.
All participants are price takers.
No transactions costs or taxes.
– Investors are rational.
– We are in a one-period world.
• The last assumption will be dropped quickly with the first
to follow soon.
Money & Time
• An important message of the example is that
money must be thought of as having two “units.”
• Currency ($, £, ¥) is of course the commonly
identified unit but time (date received) must also
be established before we can determine value.
• Example: Your employer offers you a bonus for
excellent performance. You may choose between
$10,000 today or $12,500 in one year (after the
firm does its IPO and has more liquidity).
– Compare future values.
Present Value
• Compare $100 today versus $107 in one
year if you can earn 6% interest.
$100(1  r )  $100(106
. )  $106 in one year
• Compare them today instead of in one year.
$today(1  r )  $100(106
. )  $tomorrow  $106
• Rearrange this to find:
$tomorrow $106
$107
$today 

 $100 and so
 $100.94 today
(1  r )
(106
. )
(106
. )
Present Value Examples
• You just won the new Colorado lottery scratch
game. The lottery office offers you $50,000 today
or $55,000 if you wait a year. The current interest
rate is 7%, what do you do? How much money
(present value) will a poor choice cost you?
• Your rather odd uncle Ralph has set up a trust in
your name that will pay you $1,300,000 in one
year. How much can you borrow against this trust
if the current interest rate is 9%?
The First Principle
• The financial markets provide information that
enables us to evaluate choices (investment
opportunities) both as individuals and as
corporations.
– An investment opportunity is a way that individuals or
firms adjust their consumption (spending) across time.
– Since financial markets also represent a way to
accomplish this important task we know that:
– An investment project can be worth undertaking only if
it represents a better option than is available in the
financial markets.
• Close substitutes provide the basis for comparison.
• Lottery example and opportunity costs.
Net Present Value
• In order to determine whether you are better off making an
investment or not we can use the idea of discounting future
cash flows and comparing the present value of the future
cash in-flows to the current cost.
NPV  C0 
C1
(1  r )
• This is net present value. It is a powerful decision making
tool. If NPV is positive what does that tell us? If it is
negative?
– The interest rate that sets the NPV equal to zero is called the
internal rate of return or the yield of the investment. (More on this
later.)
Net Present Value Example
• Do you take a riskless investment that requires $217 to
undertake and will payout $230 in one year if the bank is
offering you a 5% CD?
• NPV: -$217 + $230/1.05 = $2.05 ($, time 0) > 0.
• What if you put the $217 in the CD: $217(1.05) = $227.85
so the comparable alternative has a lower future payout.
• Comparing directly: $230 - $227.85 = $2.15 ($, time 1).
• Note: $2.05(1.05) = $2.15, i.e., the approaches are making
exactly the same comparison, NPV does it at time zero.
Comparing future values just compares value at time one.
The Two-Period Case
• One payment two years from now:
– We talked about getting cash next year, what if
it doesn’t come till two years from now?
C
– One illustration: if (1  r ) is the time 0 value of
a cash flow at time 1, C (1  r ) is the time 1 value
of a time 2 cash payment. We already know
how to change a time 1 value to a time 0 value:
1
2
PV (C2 ) 
V1
C (1  r )
C2
 2

(1  r )
(1  r )
(1  r ) 2
A Two-period Example
• A second view:
– If you have $100 cash today, a bank will give you 7%
interest per year, and you leave the money in the bank
for two years, how much will you have?
– Answer: $100(1.07)(1.07) = $114.49. So $114.49 is the
future (2 year) value of $100 of current cash (at 7%).
Algebra tells us that the present value of the future
$114.49 must be $100. Calculate this as
$114.49/(1.07)2 = $100.
– Notation: PV(C2) = C2/(1+r)2
– Generally: PV(Ct) = Ct/(1+r)t and FVt(C0) = C0(1+r)t
Multi-period Examples
• If you invest $15 for 20 years at 9% with no
withdrawals what will be the final balance
(future value)?
$15(1.09)20 = $84.07
• If you will receive $25,000 in 6 years and
the relevant interest rate is 11%, what is the
present value of this future payment?
$25,000
 $13,366.02
6
(1.11)
Simple vs. Compound Interest
• Suppose that I have had some finance training and I know
better than to stuff my $100,000 under my mattress.
Instead I put it in the bank for 12 years at an 8% interest
rate. Not having stayed till the end of the course, however,
at the end of each year I withdraw the interest I earn and
stuff it under my mattress. How much will I have at the
end of the 12 years?
– I’ll still have my $100,000 of principal and at the end of each of
the 12 years I will have put $100,000(.08) = $8,000 under the
mattress, leaving $100,000 + 12*$8,000 = $196,000.
• If I made no withdrawals during the 12 years I’d have
$100,000(1.08)12 = $251,817.01
– What drives the $55,817 difference?
The Present Value of a Series of
Future Cash Flows
• What happens if we have an investment that
provides cash flows at many future dates?
• Its very easy, discount each of the future cash
flows to the present, then just add them up.
– We can (and should) do this because once we have
discounted them, their present values all represent cash
values today. Since all the values are as of the same
date they can be directly compared (added).
– In other words, proper discounting restates the future
cash flows as their equivalent amounts at a common
point in time. They are (only) then directly comparable.
Present Value of a Series of
Future Cash Flows
• Those are the words, here are the symbols:
PV 
C3
C1
C2
CT





(1  r1 ) (1  r2 ) 2 (1  r3 ) 3
(1  rT )T
T

t 1
Ct
(1  rt ) t
• For NPV the adjustment is obvious:
NPV  C0 
C3
C1
C2
CT





(1  r1 ) (1  r2 ) 2 (1  r3 ) 3
(1  rT )T
T
 C0  
t 1
Ct
(1  rt ) t
Multi-Period NPV Example
• Ralph, your brother-in-law, has offered you an
investment opportunity. For an investment of
$117,000 you will own half of a ferret ranch
located outside of Flagstaff AZ.
• You are convinced that the ranch will generate
enough income to payout a total of $80,000 in one
year, $95,000 in two years and $150,000 in three
years.
• The current rate of interest is 5%.
• What is the wise investment decision?
Alternate Compounding Periods


Interest is sometimes “compounded” over periods other
than a year. In terms of “bank account” examples, this
simply means that interest is credited to the account more
frequently than once a year.
Caveat: All of the time value of money formulas we will
see use the implicit assumption that the compounding
interval is the same as the payment interval. e.g.:
• Mortgage loans call for monthly payments.
• Bonds make coupon payments semiannually.
• If this is not true you must make adjustments.
Alternate Compounding Periods (Cont.)
• Let m denote the number of compounding intervals per
year, n the number of years, and r be the stated annual rate
of interest.
• The relation between present and future values is given as:
FVn = PV(1 + r/m)n×m
E.g., if PV = 1000, r = .12 and m = 1 then FV2 is:
FV2 = 1000(1 + .12)2×1 = $1254.40,
while if m = 4 (quarterly compounding), then
FV2 = 1000(1 + .12/4)2×4
FV2 = 1000(1 + .03)8 = $1266.77
Example
• Find the PV of $500 to be received in 5 years, with:
• 12% stated annual rate, annual compounding,.
500
PV 
 $283.71
5
1  .12
 12% stated annual rate, semiannual compounding,
PV 
500
1  (.12 / 2)
10
 $279.20
 12% stated annual rate, quarterly compounding,
500
PV 
 $276.84
20
1  (.12 / 4) 
Stated And Effective Annual Rates



Notice that the use of more frequent compounding acts as
if to (or effectively) increase(s) the interest rate.
The Effective Annual Rate (EAR) is the annual interest
rate that would produce the same answer, with annual
compounding, as is obtained with more frequent
compounding. It can be obtained by:
EAR = (1 + r/m)m - 1
so if r = .12 and m = 4, then EAR = (1.03)4 - 1 = .1255.
The effective annual rate is the rate that if you earned it for
a year with annual compounding, you end up with the
same money as you would under frequent compounding.

Note the appropriate discount rate in any application is always an
effective rate. At times this may also be a stated rate.
Example
• A bank quotes a mortgage rate of 8% (the stated
annual rate), but will compute monthly loan
payments using standard time value formulas.
This implies monthly compounding. What is the
effective annual interest rate on the loan?
12
 0.08 
EAR   1 
  1  0.0830

12 
So the loan effectively costs you 8.30% per year for
every dollar you borrow for a year.
Valuing Streams of Structured
Future Cash Flows
• Now we are going to discuss the valuation of
certain highly structured cash flow streams.
• The resulting valuation formulas are useful for
simplifying the analysis of certain situations.
• Pay attention to the exact timing of the cash flows,
the formulas don’t work unless you get this right.
– Drawing diagrams of the cash flows can be useful. 
• These formulas can make life easier and so are
worth understanding.
Perpetuity
• A stream of equal payments, starting in one
period, and made each period, forever. Forever??
C
C
C
…
0
1
2
3
C
PV 
r
• Remember, this gives the value of this stream of
cash flows as of time 0, one period before the first
payment arrives.
Growing Perpetuity
• A growing perpetuity is a stream of periodic payments that
grow at a constant rate and continue forever.
C1
C2 = C1(1+g) C3 = C1(1+g)2
…
0
1
2
3
• The present value of a perpetuity that pays the amount C1
in one period, grows at the rate g indefinitely, when the
discount rate is r, is:
C1
PV 
rg
Examples
Perpetuity: $100 per period forever discounted at 10% per period
$100
$100
$100
…
0
1
2
3
PV = C/r = $100/0.10 = $1,000
Growing perpetuity: $100 received at time t = 1, growing at
2% per period forever and discounted at
10% per period
$100
0
1
$102
2
$104.04
3
PV = C1/(r –g ) = $100/(0.10 – 0.02) = $1,250
…
Verification of the Perpetuity Example Answers
Place the present value in a bank account, and recreate the
payments. Let’s stop at 4 years since “forever” would take a while.
Level Perpetuity
Year
BofY Bal INT@10% Payment EofY Bal
1
1000
100
100
1000
2
1000
100
100
1000
3
1000
100
100
1000
4
1000
100
100
1000
Growing Perpetuity
Year
BOY Bal. INT@10% Payment EOY Bal.
1
1250
125
100
1275
2
1275
127.5
102
1300.5
3
1300.5
130.05
104.04
1326.51
4
1326.51
132.651 106.1208
1353.04
Note that the account balance is growing. At what rate? Why must this happen?
Annuities
• An annuity is a series of equal payments, starting next
period, and made each period for a specified number (3) of
periods.
2
3
1
0
C
C
C
– If payments occur at the end of each period (the first is one period
from now) it is an ordinary annuity or an annuity in arrears.
– If the payments occur at the beginning of each period (the first
occurs now) it is an annuity in advance or an annuity due.
0
1
2
C
C
C
3
Valuing Annuities
• We can do a lot of grunt work or we can notice that a T
period annuity is just the difference between a standard
perpetuity and one whose first payment comes at date T+1.
• The present value of a T period annuity paying a periodic
cash flow of C, when the discount rate is r, is:
C
1

C/r  C 
1 
1
  1 
  C  

PV   
T 
T 
T 
 r (1  r )  r  (1  r ) 
 r r (1  r ) 
• If we have an annuity due instead, the net effect is that
every payment occurs one period sooner, so the value of
each payment (and the sum) is higher by a factor of (1+r).
• Or we can add C to the value of a T-1 period annuity.
Annuity Example
• Compute the present value of a 3 year ordinary
annuity with payments of $100 at r = 10%.
PV  $100
1
1
1
 $100 2  $100 3  $248.68
1.1
1.1
1.1
or,
 1

1
  248.68
PV  $100

3 
 0.1 0.1(1.1) 
Annuity Due Example
• What if the last example had the payments
at the beginning of each period not the end?
1
1
PV  $100  $100
 $100 2  $273.55
1.1
1.1
• Or,
• Or,
PV  $248.68(1.1)  $273.55
 1

1
  $273.55
PV  $100  $100

2 
 0.1 0.1(1.1) 
Example: A five year annuity paying
$2000 per year, with r = 5%.
• Valuing the payments individually we get:
1
2
3
4
5
2,000.00
2,000.00
2,000.00
2,000.00
2,000.00
1,904.76
1,814.06
1,727.68
1,645.40
1,567.05
_________
8,658.95

Using the annuity formula we get:
 1

1
 = $8,658.95
PV = 2000
5
 0.05 0.05(1.05 ) 
Alternatively, suppose you were given
$8,658.95 today instead of the annuity
year
1
2
3
4
5
principal
interest
PMT
$ 8,658.95 $ 432.95 $ (2,000.00)
$ 7,091.90 $ 354.60 $ (2,000.00)
$ 5,446.50 $ 272.32 $ (2,000.00)
$ 3,718.82 $ 185.94 $ (2,000.00)
$ 1,904.76 $ 95.24 $ (2,000.00)
Ending Bal
$ 7,091.90
$ 5,446.50
$ 3,718.82
$ 1,904.76
$
0.00
• Notice that you can exactly replicate the annuity cash
flows by investing the present value to earn 5%.
• This again demonstrates that present value calculations
provide a literal equality, in that the future cash flows can
be converted into the present value and vice versa, if (and
only if) the selected discount rate is representative of
actual capital market conditions.
Growing Annuities
• A stream of payments each period for a fixed
number of periods where the payment grows each
year at a constant rate.
C1
0
1
C1(1+g)
…
2
C1(1+g)T-2 C1(1+g)T-1
T-1
T
T
 1
1
1

g

 
PV  C1 


 
 r  g r  g  1 r  
T

C1   1  g 
PV 
1 


r  g   1 r 




Example
• What is the present value of a 20 year annuity with
the first payment equal to $500, where the
payments grow by 2% each year, when the interest
rate is 10%?
500
0
1
500(1.02) 500(1.02)18 500(1.02)19
…
2
19
T=20
20

1
1
1  0.02  

PV  500


 
 0.10  0.02 0.10  0.02  1  0.10  
 $4,869.52
A Valuation Problem
What is the value of a 10-year annuity that pays $300 a year at
the end of each year, if the first payment is deferred until 6
years from now, and if the discount rate is 10%?
0
1
2
3
4
5
6
7
8
300 300 300
9
•
10
11
12
13
•
•
•
•
14
15
• 300
The value of the annuity payments as of five years from now
is:
1
 1

PV5  300

 $1,843.37
10 
 0.10 0.10(1  0.10) 
Now discount this equivalent payment back 5 years to time
zero:
1843.37
PV0 
 $1144.58
5
(1  0.10)
Application: Retirement Planning
• You have determined that you will require $2.5 million
when you retire 25 years from now. Assuming an interest
rate of r = 7%, how much should you set aside each year
from now till retirement?
– Step 1: Determine the present equivalent of the targeted $2.5
million.
PV = $2,500,000/(1.07)25
PV = $2,500,000/5.42743 = $460,623
– Step 2: Determine the annuity that has an equivalent present value:
 1

1

$460,623  C 

25 
 .07 .07(1.07) 
$460,623  C 11.65358
$39,526  C
Retirement Planning cont…
• Now suppose that you expect your income to grow at 4%
and you want to let your retirement contributions grow
with your earnings. How large will the first contribution
be? How about the last?
25
 C1   1.04  
$460,623  
1  
 
 .07  .04   1.07  
C 
$460,623   1 .50882185
 .03 
$460,623  C1 16.960728
$27,158  C1 , and
C25  $27,1581.04   $69,614.
24
A College Planning Example – Outside Class
• You have determined that you will need $60,000 per year
for four years to send your daughter to college. The first of
the four payments will be made 18 years from now and the
last will be made 21 years from now. You wish to fund
this obligation by making equal annual deposits at the end
of each of the next 21 years. You expect to earn 8% per
year on the deposits.
– Step 1: Determine the t = 17 value of the obligation.
 1

1
  60000(3.312127)  $198,727
PV17  60000

4 
 0.08 0.08(1  0.08) 
– Step 2: Determine the equivalent t = 0 amount.
198727 198727

 $53,710
17
(1.08)
3.70002
College Planning cont…
• Step 3: Determine the 21-year annuity that is
equivalent to the stipulated present value.
 1

1

$53,710  C 

21 
 .08 .08(1.08) 
$53,710  C 10.016803 
$5,362  C
Present Value Homework Problem
• Your child will enter college 5 years from now.
Tuition is expected to be $15,000 per year for
(hopefully) 4 years (t=5,6,7,8).
• You plan to make equal yearly deposits into an
account at the end of each of the next 4 years
(t=1,2,3,4) to fund tuition. The interest rate is 7%.
• How much must you deposit each year?
• What if tuition were growing over the 4 years?
• Think about:
– How to decide whether/when to refinance your house?
Leasing vs. Buying a Car – Outside Class
• Saab 9-3 five-door/five-speed, CD, Air, Prestige
• Lease Terms (Source WSJ 8/6/98)
• Up front fees $1,748 including down payment
(due at t=0).
• Refundable security deposit of $300
• 38 monthly payments of $299 (t=1, 2, …, 38)
• Residual value of $16,454
• Annual interest rate: 8%
T=0
T=1
T=2
T=35
T=38
...
-1,748
-300
-2,048
-299
-299
-299 -299 -299
Payments are an annuity:
-299
-16,454
300
-16,154
 1

1
 = -$10,001
PVA = -299
38 
 0.0067 0.0067(1.0067 ) 
Present value of residual value and security deposit:
 16,154
PV 
 $12,534
38
(1.0067)
Lease vs. Buy cont…
• The present value of the lease payments is:
-$2,048 + (-$10,001) + (-$12,534) =
-$24,583
• What does this number mean?
• If we could purchase the car for less than $24,583
we are better off buying.
• When considering the alternative of purchasing
the car, does whether we pay cash or borrow to
make the purchase affect the lease/buy decision?