Discounted Cash Flow Valuation
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Be able to compute the future value of
multiple cash flows
Be able to compute the present value of
multiple cash flows
Be able to compute loan payments
Be able to find the interest rate on a loan
Understand how loans are amortized or paid
off
Understand how interest rates are quoted
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Amount to which an investment will grow
after earning interest.
FV = C0×(1 + r)T
Where C0 is cash flow today (time zero) and
r is the appropriate interest rate.
Ex: Suppose you invest $500 in a bank today
and $600 in one year. If the fund pays 9%
annually, how much will you have in two
years?
FV=500(1.09)2 + 600(1.09)=1,248.05
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Value today of a future cash flow
PV=Future Value of t Period/(1 + r)T
Ex: 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=3,000/(1 + 0.08)2 = $2,572
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Annuity is the finite series of equal payments
that occur at regular intervals
-If the first payment occurs at the end of the
period, it is called an ordinary annuity
-If the first payment occurs at the beginning
of the period, it is called an annuity due.
C
1 
PV  1 
r  (1  r )T 
 (1 r ) 1 

FV  C 


r


t
A constant stream of cash flows with a fixed maturity.
C
C
C
C

0
1
2
3
T
C
C
C
C
PV 



2
3
(1  r ) (1  r ) (1  r )
(1  r )T
The formula for the present value of
an annuity is:
C
1 
PV  1 
r  (1  r )T 
Ex1: You are purchasing a car. You are
scheduled to make 3 annual installments of
$4,000 per year. Given a rate of interest of
10%, what is the price you are paying for the
car (i.e. what is the PV)?
Ex2: You plan to save $4,000 every year for 20
years and then retire. Given a 10% rate of
interest, what will be the FV of your
retirement account?
A constant stream of cash flows that lasts forever.
0
C
C
C
1
2
3
C
C
C
PV 



2
3
(1  r ) (1  r ) (1  r )
The formula for the present value of a
perpetuity is:
C
PV 
r
…
Ex1: An investment offers a perpetual cash flow of
$500 every year. The return you require on such
an investment is 8 percent. What is the value of
this investment?
Ex2: Suppose the Fellini Co wants to sell preferred
stock at $100 per share. A very similar issue of
preferred stock already outstanding has a price
of $40 per share and offers a dividend of $1
every quarter. What dividend will Fellini have to
offer if the preferred stock is going to sell?
The issue that is already out has a present value of
$40 and a cash flow of $1 every quarter forever.
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EAR is the actual rate paid (or received) after
accounting for compounding that occurs during the
year.
If you want to compare two alternative investments
with different compounding periods you need to
compute the EAR and use that for comparison.
Compounding is the process of calculating interest
on an investment overtime to earn more interest.
Stated or Quoted Interest Rate: is the interest rate
expressed in terms of the interest payment made
each period. Also, quoted interest rate.
Where m is the number of periods per year
 You are looking at two saving accounts. One
pays 5.25%, with daily compounding. The
other pays 5.3% with semiannual
compounding. Which account should you use?
First account:
EAR= (1+ 0.0525/365)365=5.39%
Second account:
EAR= (1+ 0.053/2)2-1=5.37%
Which account should you choose and why?
Let’s verify the choice. Suppose you invest $100 in
each account. How much will you have in each
account in one year?
First account:
Daily rate= 0.0525/365=0.00014383562
FV=100(1.00014383562)365=105.39
Second account:
Semiannual rate=0.0530/2=0.0265
FV=100(1.0265)2 =105.37
You have more money in the first account.
Annual percentage rate (APR) is the interest
rate charged per period multiplied by the
number of periods per year. The quoted rate
is the same as an APR.
For example. An APR of 12 percent on a loan
calling for monthly payments is really 1
percent per month. The EAR on such a loan is
thus:
Ex: Suppose if you put it in another account,
you earn 3% per quarter.
What is the APR? 3(4)=12%
How much are you effectively earning?
FV=(1.03)4 =1.1255
Rate=(1.1255-1)/1=0.1255=12.55%
If you have effective rate, how can you
compute the APR?
Rearrange the EAR equation you get:
APR=m[(1+EAR)1/m -1]
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Pure Discount Loans
Borrower receives money today and repays a
single lump sum at some time in the future.
Treasury bills are a common example of pure
discount loans.
Interest-Only Loans
Borrower pays interest only each period and
the entire principal at maturity. Corporate
bonds are a common example of interestonly loans.
Consider a 5-year, interest only loan with a 7%
interest rate. The principal amount is
$10,000. Interest is paid annually.
What would the stream of cash flows be?
Years 1-4: Interest payments of
0.07(10,000)=700
Year 5: Interest + Principal=10,700
Borrowers repay part or all of principal over the
life of the loan. Two methods are (1) fixed
amount of principal to be repaid each period,
which results in uneven payments, and (2)
fixed payments, which results in uneven
principal reduction. Traditional auto and
mortgage loans are example of the second
type of amortized loans.
For example. suppose a business takes out a
$5,000, five-year loan at 9 percent. The loan
agreement calls for the borrower to pay the
interest on the loan balance each year and to
reduce the loan balance each year by $1,000.
Since the loan amount declines by $I,000
each year, it is fully paid in five years.
Probably the most common way of amortizing
a loan is to have the borrower make a single
fixed payment every period.
 For example, suppose our five-year, 9
percent, $5,000 loan was amortized this way.
How would the amortization schedule look?
In the first year. the interest is $450. Since the
total payment is $1,285.46. the principal paid
in the first year must be:
Principal paid =$1,285.46 - 450 = $835.46
The ending loan balance is thus:
Ending balance =$5,000 - 835.46 = $4,164.54
End of Chapter 5