Ch 06
Jan 27 & Feb 3, 2012
Ch. 6 Discounted Cash Flow Valuation
I. Valuation of Multiple Uneven Cash Flows
a) FV of Multiple Uneven Cash Flows
b) PV of Multiple Uneven Cash Flows
II. Annuities
a) PV of Annuities
b) FV of Annuities
III. Perpetuities
IV.
Effective Annual Rates (EAR) & Annual Percentage Rates
(APR)
V.
Loan types and loan amortization
Welcome to Ch 6 - Discounted Cash Flow Valuation. I this chapter we will
discuss the Valuation of Multiple Cash Flows
How do we handle several (or multiple) cash flows rather than one lump
sum?
Use a time line!!
6.3 example for PV calculation of multiple cash flows occurring in future:
three methods: a) discount back one period at a time
b) calculate the PV individually for each cash flow and add them up (use
this method!!)
c) use financial calculator
You are offered an investment that will pay you $400 in one year, $600 in
two years, $1000 in three years, and $500 in four years. If you can earn
11% on a similar investment how much would pay for the investment at the
most? (or what is the PV of these cash flows?)
0-------------1--------------2------------3-------------4
$400
$600
$1000
$500
400
360.36
(1.11) 1
486.97
600/(1+.11) 2
1000/(1+.11) 3
731.19
500/(1.11) 4
329.37
$1907.89
You would be willing to pay 1907.89 at the most.
1
USE CF function:
A lot less work: enter CFs, Frequency, rate, and calculate the NPV
CF0 0
CF1 400 ENTER 
F 1 1
CF2 600 ENTER 
F 2 1
CF 3 1000 ENTER 
F3 1
CF4 500 ENTER 
F4 1
NPV
I 11 ENTER
CPT
USE for FV also, calculate the PV first and then the FV.
Year 0
0
1
400
2
400
3
600
4
5
1000 500
2
400
3
500
4
600
2
400
3
600
4
5
1000 500
Change F01 to 2 enter:
NPV = $2,079.18
Year 0
0
1
400
5
6
1000 500
Go to C02
INS
$500 enter
F02 = 1 enter
NPV = $2,306.62
Year 0
1
0
400
DEL C02
NPV = $2,079.18
2
6.4 example for FV calculation of multiple cash flows:
you deposit $2000 in one year $4000 in two years $1000 in three years and
$900 in four years. How much money will you have in five years? Assume
you earn 8% interest!
Three ways: a) calculate the balance a year at a time
b) calculate the FV of each cash flow and add them up
c) use financial calculator
0-------------1--------------2------------3-------------4-------------5
2000
4000
1000
900
900 x (1.08) 1 =972
1000 x (1.08) 2 =1166.4
4000 x (1.08) 3 =5038.85
2000 x (1.08) 4 =2720.98
FV =9898.23
Note: use your memory function:
STO 1 (where 1 is location, locations: 0-9)
STO 2
When done:
RCL 1+ RCL 2 + =
2nd MEM , 2nd CLR WORK, 2nd quit
Use financial calculator:
CF
2nd CLR WORK
Cf0 = 0 
C01 =2000 enter 
F01 = 1 
C02 = 4000 enter 
F02 = 1 
C03 = 1000 enter 
F03 = 1
C04 = 900 enter
F04 = 1
NPV
I=8% enter 
CPT
NPV = 6736.57
CE/E 6736.57 PV, 5 N, 8 I/Y, CPT FV = $9898.23
3
II. Annuities
annuity: multiple cash flows that have the same amount over a fixed period
of time.
Note: ordinary annuity (1st payment occurs in one year, or end of period
payments)
Annuity due (1st payment occurs today, or beginning of period payments),
change calculator to BEG
0
1
2
3
4
Ordinary annuity: the PV is calculated as of time O for CF starting at time 1.
If CF starts at time 0, the formula calculates the PV as of t = -1. Therefore
we need to bring PV forward one period by multiplying times (1+r)
If first CF starts at beg of period, there is one additional period for
compounding to get FV and one less discounting for PV
(multiply times (1+r)).
Financial calculator: C = PMT
PV & PMT have opposite sign or FV & PMT have opposite sign
a) PV of annuity
(for example a car loan requires the borrower to make equal payments)
you could use method just discussed but there is a short-cut. Mortgage
loan could have 360 payments. Instead of making 360 calculations, only
make one using this formula:
1  presentvaluefactor
1  {1 / (1  r ) t }
]
] =C x [
r
r
where C is are the constant cash flow
PV of annuity = C x [
(PV of annuity = C x PVIFA(r,t) ) p. A-3
calculate the present value factor and plug into formula.
6.7 => example for PV of annuity:
You want to buy a car by borrowing from your bank. You can afford to
spend $540 a month for four years. The bank charges interest rate of 2%
per month for 48 months. How much can you borrow to buy your car?
C= PMT +/- $540
t= N = 48
r= I/Y = 2%
PV of annuity = CPT PV = 540 x [1-(1/(1.02) 48 )]/.02
=540 x [1-.3865]/.02=$16,563.48
4
assume the fist payment occurs today:
2nd BEG, 2nd SET, 2nd QUIT:
$16,894.75 ( = $16,563.48 x 1.02)
set calculator back to END
6.8 => How to get monthly payments (C)
Your interest rate is .5% per month, you borrow $20,000 and make 48
payments to repay the loan, how high are your payments?
I/Y = .5% per month
N=48 months
PV=$20,000 amount you will borrow to buy your new car
CPT PMT= $469.70
6.9 => How to get t (# of payments)
Your interest rate is 1% per month, you borrow $10,000 and you can afford
$600 per month. How long does it take to repay your loan?
I/Y= 1% per month
PV= $10,000
PMT=+/$600 monthly payment
CPT N = 18.32 months
Watch out!!!
How to find the rate (r )
trial-and-error (r ) because 2 rs in formula
Remember: PV & r are inversely related!!!
use any rate and calculate the PV.
If PV is too high, increase rate so that PV will decrease.
We will do such an trial-and-error procedure when we calculate IRR.
5
b. Future Value of an Annuity
FV of an Annuity= C x [(1+r) t -1]/ r
Annuity of Future Value = C x (Future value factor - 1)/ r
Annuity of Future Value = C x FVIFA(r,t) FVIFA is given in tables (A-4)
but we use calculator
calculate future value factor first and plug into formula.
6.10 => Example for calculating the FV of an annuity:
You make 20 payments of $1,000 at the end of each period at 15% per
period, how much will your account grow to be?
PMT=$1,000
I/Y= 15%
N= 20
CPT FV = $102,443.58
FV of annuity = 1000 x [(1.15) 20 -1]/.15
=1000 x [(16.37-1)/.15]
=1000 x 102.44
=$102,443.58
[ Do not cover: More complicated practice problem:
You are ready to buy a house and have $20,000 for a down payment and
closing costs.
Closing costs are estimated to be 4% of the loan value.
You have an annual salary of $36,000 and the bank is willing to allow your
monthly mortgage payment to be equal to 28% of your monthly income.
The interest rate on the loan is .5% per month for a 360-month (30-year)
fixed rate loan. How much money will the bank loan you?
How much money can you offer for the house?
Bank loan:
Monthly income: $36,000/12 = $3,000
Maximum payment = .28($3,000) = 840
Maximum loan amount:
360 N
.5% I/Y
840 +/- PMT
CPT PV = $140,105
Total price you can offer for the house:
Closing costs: .04(140,105) = $5,604
Down payment = 20,000 – 5,604 = $14,396
Total price = $140,105 + $14,396 = $154,501 ]
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III. Perpetuities
is an annuity for which the stable cash flows continue forever
PV of Annuity = C x (1- present value factor)/r
if t gets large present value factor becomes very small
PV of Perpetuity = C x 1/r = C/r
Example for calculating PV of perpetuity
preferred stock is an example of a perpetuity
A firm that sells preferred stock to investor promises to pay a fixed cash
dividend every period forever.
6.11=>example Suppose a firm sells a share of pref. stock for $200/share.
What dividend does the firm have to offer per quarter if a comparable
preferred stock offers 3% per quarter?
PV=$200
r=.03
PV of perpetuity = C/r
200 = C/.03
C=$200 x .03=$6 dividend per quarter
6.12 => Another example: Company pays $10 div per quarter. Quarterly
rate=4%
PV = 10/.04 = $250
Growing Perpetuity
is an annuity for which the cash flow grows every period at rate forever
do not need to apply in this chapter, but in the stock valuation chapter, just
want to mention it now.
PV of Growing Perpetuity = C/(r-g)
CF
Single
PV, FV, N, I/Y
multiple
Unequal
CF & NPV
equal
perpetuity (PV = C/r)
Or
Annuity
PV, PMT, N, I/Y
Or
FV, PMT, N, I/Y
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NOT COVERED:
  1  g T 
 
1  
  1  r  

PV of growing Annuity = C
rg
Note: you can use TVM functions with 2 adjustments:
1 r 

 1 100
I/Y
1  g 
PMT
PMT

1 g
Growing perpetuity:
PV of Perpetuity = C x 1/(r-g) = C/r-g ]
IV. Effective Annual Rates (EAR) & Annual Percentage Rates (APR)
interest rates are stated in different ways
when we want to compare interest rates and investments, we need to state
them in terms of effective interest rates. (=EAR)
until now: r=10% means that money is compounded annually. If money
compounded annually (=once a year) than this is effective rate.
But sometimes money can be compounded semi-annually, quarterly, or
daily.
10% annual, compounded semi-annually = quoted interest rate (in terms of
total interest payment made each period) 5% in first half year and 5% in
second half (APR)
effective annual rate = interest rate in terms of amount the investor will
actually earn (is expressed as if interest was compounded once per year).
Annual Percentage Rate (APR) on a loan
In the US lenders have to display an APR
APR=interest rate per period x #of periods per year
is an annual quoted rate
ER = rate compounded ONCE per period
EAR = annual rate compounded ONCE per year
Quoted rate = rate compounded MORE THAN ONCE per period
APR =annual rate compounded MORE THAN ONCE per year
8
EAR = [1+ (APR)/m] m -1
m=number of times the interest is compounded during one year
semiannual = 2
quarterly = 4
monthly=12
6.14 Example: A bank is charging 3% per month on a car loan. =>transform
to rate per year = APR (but which is still compounded monthly)
APR=.03 x 12=.36
The EAR = [1+(.36/12)] 12 -1=42.58%
6.15 => example for calculating EAR from APR: annual interest rate is 18%
compounded monthly; what is EAR?
EAR= [1+ (.18/12)] 12 -1=19.56%
Or use ICONV
6.16=> example for calculating APR from EAR: You want to actually earn
15% per year on a loan. If you want to quote the rate as an APR
compounded quarterly what rate do you quote?
.15 =[1+(APR/4)] 4 -1
1.15 =[1+(APR/4)] 4
1.15 1/ 4 =1+(APR/4)
1.0356=1+(APR/4)
.0356 = APR/4
.1422=14.22% quoted rate, compounded quarterly
Or use ICONV
FOR SIMPLE CF, IT DOES NOT MATTER WHETER YOU USE A YEARLY
RATE OR MONTHLY OR OTHER PERIOD RATE, JUST MAKE SURE THAT
THE RATE MATCHES THE TIME:
6.17 => Example: You invest $5,000 at 6% APR, compounded monthly.
How much will you have in 4 years? Match rate to periods:
1) monthly: 6/12=.5% I/Y, 5000PV, 4x12 = 48N, CPT FV = $6,352.45 or
2) yearly: [1+ (.06/12)] 12 -1 = 6.16778 I/Y, 5000 PV, 4 N, CPT FV = $6,352.45
(or use ICONV)
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NOTE: FOR ANNUITY THE WAY THE PAYMENT IS MADE WILL DETERMINE
THE PERIOD: MATCH THE “EFFECTIVE” RATE AND T TO PYMT
INTERVAL.
6.18 => Example: You borrow $10,000. The loan calls for monthly payments
for 3 years. The APR is 9%, compounded monthly, what are the monthly
payments?
You are forced to use monthly rate and periods:
9/12 = .75% I/Y, 3x12 = 36 N, 10,000 PV, CPT PMT $-318
Continuous compounding:
As the # of periods get larger (time intervals get smaller), the EAR gets
larger and approaches:
EAR=e quotedrate -1 e=2.71828
continuous compounding; time intervals are infinitely small; interest is
credited the moment it is earned.
FV=PV(1+r)t
FV=PV(1+e quoted rate-1) t
FV=PV e quoted rate x t
PP 6.20 example
what is the largest EAR for a 12% quoted rate with (i.e. continuous
compounding)?
EAR=e .12 -1=2.71828 .12 -1=1.1275-1=12.75%
.12 [ex]
What is the FV of 1000 in 2 years with 12% interest under continuous
compounding?
1) 1000 FV, 2 N, 12.75% I/Y, CPT FV = 1,271.25
Or 1) FV = 1000 e.12 x 2 = 1000 e.24 = 1,271.25
PP 6.21 example
What is the APR if the EAR is 14% under continuous compounding?
EAR = eAPR-1
.14 = eAPR-1
1.14 = eAPR
Ln 1.14 = APR x Ln(e)
Ln 1.14 = APR x 1
=> .131 = APR
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SKIP? (Loan types & loan amortization:
Pure discount loans: borrower receives a certain amount today and repays
principal & interest in one single lump sum in the future (example: T-bills,
government borrows short-term)
PV
FV
Interest-only loans: borrower receives principal today, interest only is paid
each period, at end of loan principal is paid back (example: most corporate
bonds)
PV
I
I
I
FV
+I
Amortized loans: borrower repays parts of principal (loan amount) over
time. Regular principal payments pay off loan = amortization
Total payment = principle + interest
PV
I+P
I+P
I+P
I+P
Sometimes an amortized loan calls for fixed PRINCIPAL payments
Note: in this case interest payments go down, and total payments go down
fixed
PMT = P + I
Most common type: borrower pays fixed TOTAL PAYMENT each period
(examples: car loan & mortgage loans)
Note: Interest part of payment declines; principle part of payment increases
each period
fixed
PMT=P
+I
Use AMORT worksheet on your calculator or Excel spreadsheet to
calculate amortization schedules.
Example:
$1,500 loan, r = 10%, fixed principal payments of $500: 1500/3 = 500
(prin bal x int)
(beg-end)
Year Beg princ balance Prin paid
Int paid
tot paym
end bal
1
1,500
500
150
650 1,000
2
1,000
500
100
600 500
3
500
500
50
550 0
1,500
300
1,800
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Now: fixed total payments: calculate the PMT of annuity:
1500 PV, 10 I/Y 3 N CPT PMT -603.17
Year
1
2
3
Beg princ balance
1,500
1,046.83
548.34
tot pmt
603.17
603.17
603.17
1,809.5
(prin bal x int) (pmt-int)
(beg-end)
Int paid
150
104.68
54.83
309.51
end bal
1,046.83
548.34
0
princ paid
453.17
498.49
548.34
1,500
Do Review
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