Chapter 5
Discounted Cash Flow Valuation
1
Overview








Important Definitions
Finding Future Value of an Ordinary Annuity
Finding Future Value of Uneven Cash Flows
Finding Present Value of an Ordinary Annuity
Finding Present Value of Uneven Cash Flows
Valuing Level Cash Flows: Annuities and
Perpetuities
Comparing Rates: The Effect of Compounding
Periods
Loan Types and Loan Amortization
2
Important Definitions

Annuity: A level stream of cash flows for a fixed period of
time.




Ordinary Annuity: Cash flows take place at the end of each period.
Annuity Due: Cash flows takes place at the beginning of each
period.
Perpetuity: A level stream of cash flows forever.
Note: For now computations will be shown using
formulations and calculator entries. I recommend
focusing on calculator entries because it is much more
efficient in getting answers. Formulations are provided for
demonstration purposes. You should also review Excel file
to see how excel functions can be used.
3
Finding Future Value of an Ordinary
Annuity
 1  r t  1
 1  0.10 5  1
FVt  C 
  2,000 
  12,210.20
r
0.10




N
I/Y
P/Y
PV
PMT
FV
5
10
1
0
2,000
-12,210.20
MODE
4
Finding Future Value of Uneven Cash
Flows

Suppose you plan to deposit $100 into an account in one
year and $300 into the account in three years. How much
will be in the account in five years if the interest rate is 8%?


FV can be computed for each cash flow at a common future period
(in this case 5 years from today)
FV = 100(1.08)4 + 300(1.08)2 = 136.05 + 349.92 = 485.97
0
1
100
2
3
4
5
300
136.05
349.92
485.97
5
Finding Future Value of Uneven Cash
Flows – Continued


Calculator solution of the previous problem would
involve determining FV of each payment in five years
from today and then adding them
N
I/Y
P/Y
PV
PMT
FV
4
8
1
-100
0
-136.05
2
8
1
-300
0
-349.92
MODE
Total FV = 136.05 + 349.92 = 485.97
6
Finding Present Value of an Ordinary
Annuity
1

1

 1  r t
PVt  C 
r


1



1


 1  0.065 
  1,000
  4,212.37
0.06






N
I/Y
P/Y
PV
PMT
FV
5
6
1
-4,212.37
1,000
0
MODE
7
Finding Present Value of Uneven
Cash Flows
Years
Cash Flow







0
1
200
2
400
3
600
4
800
Find the PV of each cash flow and add them
Year 1 CF: 200 / (1.12)1 = 178.57
Year 2 CF: 400 / (1.12)2 = 318.88
Year 3 CF: 600 / (1.12)3 = 427.07
Year 4 CF: 800 / (1.12)4 = 508.41
Total PV = 178.57 + 318.88 + 427.07 + 508.41
Total PV = 1,432.93
8
Finding Present Value of Uneven
Cash Flows – Continued
0
1
200
2
3
4
400
600
800
178.57
318.88
427.07
508.41
1,432.93
9
Finding Present Value of Uneven
Cash Flows – TI BA II PLUS

When using CF make sure to clear previous entries

CF  2ND CLR WORK
CF
CF0
0
ENTER
C01
200
ENTER
F01
1.00
ENTER
C02
400
ENTER
F02
1.00
ENTER
C03
600
ENTER
F03
1.00
ENTER
C04
800
ENTER
F04
1.00
ENTER
NPV
12
ENTER
CPT
1,432.93
10
Example


You are offered the opportunity to put some money away
for retirement. You will receive five annual payments of
$25,000 each beginning in 40 years. How much would
you be willing to invest today if you desire an interest rate
of 12%?
This cash flow set has two parts



PV of annuity payments 39 years from today
PV of a single payment in 39 years
See the timeline below
0 1 2
…
39
0 0 0
…
0
40
41
42
25K 25K 25K
43
44
25K 25K
11
Example – Continued

PV of annuity 39 years from today
1

1  1  r t
PVt  C 
r



1




1  1  0.125 
  25,000
  90,119.41
0.12






PV of single payment in 39 years
90,119.41
PVt 
 1,084.71
39
1  0.12
12
Example – Continued – TI BA II PLUS


PV of annuity 39 years from today
N
I/Y
P/Y
5
12
1
PV
PMT
FV
-90,119.41 25,000
MODE
0
PV of single payment in 39 years
N
I/Y
P/Y
PV
PMT
FV
39
12
1
-1,084.71
0
90,119.41
MODE
13
Example – Continued – TI BA II PLUS
CF
CF0
0
ENTER
C01
0
ENTER
F01
39.00
ENTER
C02
25,000
ENTER
F02
5.00
ENTER
NPV
12
ENTER
CPT
1,084.71
14
Finding Future or Present Value of an
Annuity Due



Annuity Due cash flows take place at the beginning of each
period – this means that there is a payment now
The immediate payment requires that we account its impact
on FV and PV correctly
The basic adjustments




Treat the annuity as if it were an ordinary annuity and determine FV
or PV
Then multiply the answer by (1 + r)
FV or PV of Annuity Due = (FV or PV of Ordinary Annuity) x (1 + r)
If you are computing PMT then the adjustment is slightly
different



PMT of Annuity Due = (PMT of Ordinary Annuity) / (1 + r)
This is because immediate payment with annuity due reduces amount
borrowed leading to lower overall payment
I suggest using calculator settings for PMT, N, and I/Y computations 15
Finding Future or Present Value of an
Annuity Due – Continued

Consider the above annuity due.



This is a five year annuity due.
When you compute FV it is computed at the end of year 5.
Assume 10% rate.
 1  r t  1
 1  0.105  1
FVt  C 
 1  r   400
 1  0.10  2,686.24
r
0.10




1

1

 1  r t
PVt  C 
r


1



1


 1  0.105 
1  r   400
1  0.10  1,667.95
0.10






16
Finding Future or Present Value of an
Annuity Due – TI BA II PLUS

To compute FV and PV of an annuity due you need to change MODE
to BGN






2ND BGN
2ND SET
2ND QUIT
Once complete you will see BGN on the display
To return to END follow the same steps
END mode is not displayed
N
I/Y
P/Y
PV
PMT
FV
MODE
5
10
1
0
400
-2,686.24
BGN
N
I/Y
P/Y
PV
PMT
FV
MODE
5
10
1
-1,667.95
400
0
BGN
17
Example

Suppose you win the Publishers Clearinghouse $10
million sweepstakes. The money is paid in equal
annual installments of $333,333.33 over 30 years. If
the appropriate discount rate is 5%, how much is the
sweepstakes actually worth today?
N
I/Y
P/Y
30
5
1
PV
PMT
-5,124,150.29 333,333.33
FV
MODE
0
18
Non Annual Interest Earnings
(Compounding of Interest Rate) and
Payments


Remember that FVt = PV0 x (1 + r)t
When dealing with non annual interest and payment you
should make two adjustment



Formulations should be adjusted as follows




Interest rate should be periodic
Number of periods should reflect the total number of periods given
number of years and frequency of interest earnings in a year
If “m” is the number of compounding in a year then where you see
“r” divide it by “m” and multiply “t” by “m”
This assumes that “t” is the number of years
FVt = PV0 x (1 + r/m)t X m
Excel adjustment are same as formulations that can be
made in a function dialog box or data cells
19
Non Annual Interest Earnings
(Compounding of Interest Rate) and
Payments – Continued



FVt = PV0 x (1 + r/m)t X m
Entering the above equation into TI BA II PLUS
Method 1:





Keep P/Y = 1 (so is C/Y = 1)
N = t X m (Number of periods)
I/Y = r/m (Periodic interest rate)
Rest of the information is entered as before
Method 2:





Change P/Y = m (so is C/Y = m)
N = t X m (Number of periods)
I/Y = r (Annual interest rate)
Rest of the information is entered as before
Note that P/Y has to be checked for consistency every time
20
Example
Buying a Car

Suppose you want to borrow $20,000 for a new car.
You can borrow at 8% per year, compounded
monthly. If you take a 4-year loan, what is your
monthly payment?
1

1

t m

r


 1  

m
PVt  C  
r

m



1



1

412 48 


0
.
08



 1 



 

12 

20
,
000

C
 C  488.26



0.08



12









21
Example
Buying a Car – TI BA II PLUS

Note the adjustment
2ND P/Y
12 ENTER
2ND QUIT
N
I/Y
P/Y
PV
PMT
FV
48
8
12
-20,000
488.26
0
MODE
22
Example
Credit Card


Note: Finding interest rate and number of periods in annuity
formulations are relatively time consuming. By this point
you should be more comfortable with your calculator. From
now on, solutions will be shown based on calculator entries.
You ran a little short on your spring break vacation, so you
put $1,000 on your credit card. You can only afford to make
the minimum payment of $20 per month. The interest rate
on the credit card is 1.5 percent per month (18% per year
compounded monthly). How long will you need to pay off
the $1,000.
N
I/Y
P/Y
PV
PMT
FV
93.11
18
12
-1,000
20
0
MODE
23
Example
Borrowing Money from Unlikely Resources

Suppose you borrow $10,000 from your parents
to buy a car. You agree to pay $207.58 per
month for 60 months. What is the annual
interest rate compounded monthly?
N
I/Y
P/Y
PV
PMT
FV
60
9.00
12
-10,000
207.58
0
MODE
24
Present Value of a Perpetuity


Suppose you will receive a fixed payment every period
(month, year, etc.) forever. This is an example of a
perpetuity
You can think of a perpetuity as an annuity that goes on
We know the equation for PV as :
1

1

 1  r t
PV  C 
r







1
approaches zero.
t
1  r 
Therefore the PV equation reduces to
If t gets large then the term
PV 
C
r
25
Example
Perpetuity


What should you be willing to pay in order
to receive $10,000 annually forever, if you
require 8% per year on the investment?
PV = $10,000 / 0.08 = $125,000
26
Effective Annual Rate (EAR) vs.
Annual Percentage Rate (APR)

Effective Annual Rate (EAR)




Annual Percentage Rate (APR)






This 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.
Sometimes it is also called Annual Percentage Yield (APY) or Effective Rate
This is the annual rate that is quoted by law
By definition APR = periodic rate times the number of periods per year
Consequently, to get the periodic rate we rearrange the APR equation:
Periodic rate = APR / number of periods per year
Sometimes it is also called Stated Rate, Quoted Rate or Nominal Rate
You should NEVER divide the effective rate by the number of periods
per year – it will NOT give you the periodic rate
27
Rate Conversions

APR to EAR
APR 

EAR  1 
m 


m
1
EAR to APR
1

APR  m 1  EAR  m - 1



Calculator convention



NOM
EFF
C/Y
= APR
= EAR
=m
28
Example
Rate Conversion

You are looking at two savings accounts. One pays 5.25%, with daily
compounding. The other pays 5.3% with semiannual compounding.
Which account should you use?


5.25% compounded daily





2nd ICONV
NOM
5.25
C/Y
365
CPT
5.39
ENTER
ENTER
2nd ICONV
NOM
5.30
C/Y
2
CPT
5.37
ENTER
ENTER
(EFF)
(EFF)
5.30% compounded semiannually





Compare EAR and choose the larger one since you are investing. If you
were borrowing then you would choose the lower EAR alternative
(EFF)
(EFF)
Choose 5.25% compounded daily
29
Example
Rate Conversion – Continued

Let’s verify the choice. Suppose you invest
$100,000 in each account. How much will you have
in each account after 5 years?

N
I/Y
P/Y
PV
PMT
FV
1,825
5.25
365
-100,000
0
130,015.19


Daily Compounded Account:
MODE
Semiannually Compounded Account:
N
I/Y
P/Y
PV
PMT
FV
10
5.30
2
-100,000
0
129,894.13
MODE
You have more money in the first account.
30
Amortized Loan with Fixed Payment


Each payment covers the interest expense plus
reduces principal
Consider a 4-year loan with annual payments. The
interest rate is 8% and the principal amount is
$5,000.
N
I/Y
P/Y
PV
PMT
FV
4
8
1
-5,000
1,509.60
0
MODE
31
Amortization Table
Year
Beginning
Balance
Interest
Paid
Principal
Paid
Ending
Balance
1
5,000.00
1,509.60
400.00
1,109.60
3,890.40
2
3,890.40
1,509.60
311.23
1,198.37
2,692.02
3
2,692.02
1,509.60
215.36
1,294.24
1,397.78
4
1,397.78
1,509.60
111.82
1,397.78
0.00
6,038.42
1,038.41
4,999.99
Totals

Total
Payment
Computations:



Interest Paid = Beginning Balance X Interest Rate
Principal Paid = Total Payment – Interest Paid
Ending Balance = Beginning Balance – Principal Paid
32
Amortization Table – Continued

If you want to determine the outstanding
balance of the loan after 2 years:


2nd AMORT
2 ENTER
2 ENTER


This will allow you to see loan information at that
point in time.
You can change P1 and P2 to get the data
for the specified payment range
33