Lecture 6: Evaluation of
Multiple Cash Flows
Prof. Victor Glass

Key Concepts and Skills
6F-2
• 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 interest rates are quoted
• Understand how loans are amortized or paid off
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Chapter Outline
• Future and Present Values of Multiple Cash Flows
• Valuing Level Cash Flows: Annuities and Perpetuities
• Comparing Rates: The Effect of Compounding
• Loan Types and Loan Amortization
6F-3
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Chapter Outline
• Derive Simplifying Formulas to calculate present value
• Will assume fixed future inflows or inflows that grow at a steady rate
• Will assume a fixed required return on investment

Problem taxonomy
• Typical valuation problems
• Initial outlay, generates periodic payments, ends with a lump sum payment
• Initial outlay, generates periodic payments, no lump sum ending payment
• A fixed payout forever
• An initial outlay then periodic payments grow at a fixed percent
• Determining a periodic mortgage payment
• Other mortgages: interest only, balloon, amortized etc.

Problem Taxonomy
• Interest rate problems
• The effects of faster compounding
• Annual Percentage Rate versus Effective Percentage Rate

Future Value Problems
• Initial Deposit
• Periodic Deposits of Equal value for the next several periods
• Two approaches
• Treat each future cash flow as a single problem and sum results
• Use an Excel formula shortcut – if the future deposits are equal and end in
the final period

Multiple Cash Flows – FV
Example 1
• You think you will be able to deposit $4,000 at the end of each of the
next three years in a bank account paying 8 percent interest.
• You currently have $7,000 in the account.
• How much will you have in three years?
• How much will you have in four years?
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Multiple Cash Flows – FV
Example 1
• Find the value at year 3 of each cash flow and
add them together
 Today (year 0): FV = 7000(1.08)3 = 8,817.98
 Year 1: FV = 4,000(1.08)2 = 4,665.60
 Year 2: FV = 4,000(1.08) = 4,320
 Year 3: value = 4,000
 Total value in 3 years = 8,817.98 + 4,665.60 + 4,320 +
4,000 = 21,803.58
• Value at year 4 = 21,803.58(1.08) = 23,547.87
6F-9
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Multiple cash flows: future value using Excel and
treating each cash deposit as a separate problem
• You think you will be able to deposit $4,000 at the end of each of the
next three years in a bank account paying 8 percent interest.
• You currently have $7,000 in the account.
• How much will you have in three years?
• How much will you have in four years?
• Ans:
• Type .08 in a cell, then type 7000, 4000, 4000, and 4000 in rows
• Assume .08 is in cell a1. 7000 is in a2, 4000 in a3, 4000 in a4, and 4000 in a5
• =a2*(1+$a$1)^3 + a3*(1+$a$1)^2 +a4*(1+$a$1)^1+ a5
• Note: a1 becomes $a$1 by hitting F4 after typing in a1

Excel
Future Value interest rate = 0.08
Unequal Deposits
t Deposits FV
0 7000 8817.98
1 4000 4665.60
2 4000 4320.00
3 4000 4000.00
sum 21803.58
Future Value interest rate = 0.08
Unequal Deposits
t Deposits FV
0 7000 =C361*(1+$F$358)^3
1 4000 =C362*(1+F358)^2
2 4000 =C363*(1+$F$358)
3 4000 =C364
sum =SUM(E361:E364)

Using Excel’s FV function
• -FV(rate, period, payment, PV) =-FV(.08,3,4000,7000)
• Note the negative sign
PV 7000
Payment 4000
period 3
r 0.08
FV Formula $21,803.58

Multiple Cash Flows – FV Example 2 – limit
of the FV Function
• Suppose you invest $500 in a mutual fund 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
 CAUTION: Can’t use FV formula because you
didn’t put in $600 at the end of year 2.
6F-13
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Multiple Cash Flows –
Example 2 Continued
• How much will you have in 5 years
if you make no further deposits?
• First way:
 FV = 500(1.09)5 + 600(1.09)4 = 1,616.26
• Second way – use value at year 2:
 FV = 1,248.05(1.09)3 = 1,616.26
6F-14
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Multiple Cash Flows – FV Example 3 – a
continuation problem
• 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 = 100(1.08)4 + 300(1.08)2 = 136.05 + 349.92 =
485.97
6F-15
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Present Value of Multiple Cash Flows
• Two methods
• Treat each inflow as a separate problem
• Use the special Excel PV formula if the inflows are equal

Multiple Cash Flows – PV Example 4 – treat
unequal cash flows as separate problems and
sum
• Receive $200 at end of year 1, $400 at end of
year 2, $600 at end of year 3, and $800 at end of
year 4
• Find the PV of each cash flows and add them
 Year 1 CF: 200 / (1.12)1 = 178.57
 Year 2 CF: 400 / (1.12)2 = 318.88
 Year 3 CF: 600 / (1.12)3 = 427.07
 Year 4 CF: 800 / (1.12)4 = 508.41
 Total PV = 178.57 + 318.88 + 427.07 + 508.41 =
1,432.93
6F-17
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In Excel
Unequal Cash flow payments discount rate= 0.12
t Payments
1 200 178.6
2 400 318.9
3 600 427.1
4 800 508.4
1432.9
Unequal Cash flow payments discount rate= 0.12
t Payments
1 200 =C352/(1+$F$350)^A352
2 400 =C353/(1+$F$350)^A353
3 600 =C354/(1+$F$350)^A354
4 800 =C355/(1+$F$350)^A355
=SUM(E352:E355)

Example 4 Timeline
0 1 2 3 4
200 400 600 800
178.57
318.88
427.07
508.41
1,432.93
6F-19
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Multiple Cash Flows
Using a Spreadsheet
• Receive 200 after 1 year, 400 after 2 years, 600
after 3 years, and 800 after 4 years
6F-20
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Multiple Cash Flows –Example 5
• You are considering an investment that will pay you $1,000 in one
year, $2,000 in two years, and $3000 in three years.
• If you want to earn 10% on your money, how much would you be willing to
pay?
 PV = 1000 / (1.1)1 = 909.09
 PV = 2000 / (1.1)2 = 1,652.89
 PV = 3000 / (1.1)3 = 2,253.94
 PV = 909.09 + 1,652.89 + 2,253.94 = 4,815.92
6F-21
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Proof that PV matches future cash
flows
Ending
t Principal Interest Withrawal
s
Balance
0 4815.93
1 4815.93 481.59 1000.00 4297.52
2 4297.52 429.75 2000.00 2727.27
3 2727.27 272.73 3000.00 0.00

Decisions, Decisions
• Your broker calls you and tells you that he has
this great investment opportunity.
• If you invest $100 today, you will receive $40 in one
year and $75 in two years.
• If you require a 15% return on investments of this
risk, should you take the investment?
• PV = 40/(1.15)+ 75/(1.15)^2 = 34.78+56.71= 91.49
• Conclusion: bad deal, losing 8.51 in opportunity cost
6F-23
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Saving For Retirement: Example 6
• 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%?
• Excel
• =25000/(1.12)^40 + 25000/(1.12)^41 + …. +
25000/(1.12)^44 = 1084.71
6F-24
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Saving For Retirement Timeline
0 1 2 … 39 40 41 42 43 44
0 0 0 … 0 25K 25K 25K 25K 25K
Notice that the year 0 cash flow = 0 (CF0 = 0)
The cash flows in years 1 – 39 are 0 (C01 = 0; F01 =
39)
The cash flows in years 40 – 44 are 25,000 (C02 =
25,000; F02 = 5)
6F-25
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Quick Quiz – Part I
• Suppose you are looking at the following possible
cash flows:
• Year 1 CF = $100;
• Years 2 and 3 CFs = $200;
• Years 4 and 5 CFs = $300.
• The required discount rate is 7%.
• What is the value of the cash flows at year 5?
• What is the value of the cash flows today?
• What is the value of the cash flows at year 3?
6F-26
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Annuities and Perpetuities
• These are special cases of PV and FV problems

Annuities and
Perpetuities Defined
• Annuity – 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
• Perpetuity – infinite series of equal payments
6F-28
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Annuities and Perpetuities
Basic Formulas
• Perpetuity: PV = C / r
• Annuities:





 

















r
r
C
FV
r
r
C
PV
t
t
1
)
1
(
)
1
(
1
1
6F-29
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Examples 7 and 8
• C= 100 per year
• R = .1 per year
• T for annuity is 10 years
• Perpetuity PV = C/r = 100/.1 =1000
• Annuity PV = C/r*[1-1/(1+r)^t]
• =(100/.1)*[1- 1/(1+.1)^10]
• =$614.46
• Annuity chops off all cash flows after the end of year 10; therefore, it
has a smaller PV than the perpetuity
Could use:
=-PV(.1,10,100) = 614.46

The PV annuity formula again

Annuity (Borrowing) – Example 9
• After carefully going over your budget, you have determined you can
afford to pay $632 per month toward a new sports car.
• You call up your local bank and find out that the going rate is 1 percent per
month for 48 months.
• How much can you borrow?
• To determine how much you can borrow, we need to calculate the
present value of $632 per month for 48 months at 1 percent per
month.
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Annuity – Example 9
• You borrow money TODAY so you need to compute the present value.
• Formula:
54
.
999
,
23
01
.
)
01
.
1
(
1
1
632
48















PV
6F-33
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=-PV(.01,48,632) =
23,999.54

Annuity -
Sweepstakes Example 10
• 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?
 PV = 333,333.33[1 – 1/1.0530] / .05 = 5,124,150.29
6F-34
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Buying a House: Example 11
• You are ready to buy a house, and you 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 6% per year with
monthly compounding (.5% per month) for a 30-
year fixed rate loan.
• How much money will the bank loan you?
• How much can you offer for the house?
6F-35
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Buying a House - Continued
• Bank loan
 Monthly income = 36,000 / 12 = 3,000
 Maximum payment = .28(3,000) = 840
 PV = 840[1 – 1/1.005360] / .005 = 140,105
• Total Price
 Closing costs = .04(140,105) = 5,604
 Down payment = 20,000 – 5,604 = 14,396
 Total Price = 140,105 + 14,396 = 154,501
6F-36
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Annuities on the Spreadsheet -
Example
• The present value and future value formulas in
a spreadsheet include a place for annuity
payments
• Click on the Excel icon to see an example
6F-37
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Quick Quiz – Part II
• You know the payment amount for a loan, and
you want to know how much was borrowed.
Do you compute a present value or a future
value?
• You want to receive 5,000 per month in
retirement.
• If you can earn 0.75% per month and you expect to
need the income for 25 years, how much do you
need to have in your account at retirement?
6F-38
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Finding the Payment; Another Common
Problem. Example 12
• Suppose you want to borrow $20,000 for a new car.
• You can borrow at 8% per year, compounded monthly (8/12 = .66667% per
month).
• If you take a 4- year loan, what is your monthly payment?
 20,000 = C[1 – 1 / 1.006666748] / .0066667
 C = 488.26
6F-39
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Finding the Payment on a Spreadsheet
• Another special formula in Excel for an equal
payments problem is
 PMT(rate,nper,pv,fv)
 The same sign convention holds as for the PV and
FV formulas
6F-40
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Excel Payment Formula
• =PMT (rate, nper, pv, [fv], [type])
• Arguments
• rate - The interest rate for the loan.
• nper - The total number of payments for the loan.
• pv - The present value, or total value of all loan payments now.
• fv - [optional] The future value, or a cash balance you want after the last
payment is made. Defaults to 0 (zero).
• type - [optional] When payments are due. 0 = end of period. 1 = beginning of
period. Default is 0.

Using excel pmt formula: Example 13
PV = 250,000 Cell B3
NPER = 360 (30 years * 12 months per year)
RATE = 0.58% (Same as .0058)
Monthly Payment = ($1,656.55)
Formula
=PMT(B5,B4,B3)
= PMT(.0058,360, 250,000)
The payment was left as negative to indicate that it is a cash outflow.

Finding the Number of Payments –
Example 13
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Correction: t=-ln(1-rPV/C)/ln(1+r)

Finding the Number of Payments – Example
13
 t = ln(1/.25) / ln(1.015)
 = 93.111 months = 7.76 years
6F-44
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1000= (20/.015)* [1-1/1.015^T]
1000*.015/20 = 1 - 1/1.015^T
1000*.015/20 -1 =-1/1.015^T
1/1.015^T=1-1000*.015/20 = 1-.75
1/1.015^T=.25
1/.25=1.015^T
4=1.1015^T
ln4=T*ln(1.015)
T= ln(4)/ln(1.015)

Special excel formula nper
• Another way of solving this problem is to use the NPER formula
• NPER(rate, payment, PV)
• Make sure you give the payment a negative sign
• NPER(.015,-20,1000)= 93.11

Finding the Number of Payments – Another
Example 14
• Suppose you borrow $2,000 at 5%, and you are
going to make annual payments of $734.42.
• How long before you pay off the loan?
 2,000 = 734.42(1 – 1/1.05t) / .05
 .136161869 = 1 – 1/1.05t
 1/1.05t = .863838131
 1.157624287 = 1.05t
 t = ln(1.157624287) / ln(1.05) = 3 years
6F-46
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Finding the Rate: Another Basic
Problem. Example 15
• 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 monthly interest rate?
• =Rate(nper, pay, PV, guess at rate)
• =Rate(60,-207.58,10,000, .05) =..75% =
.0075/month
6F-47
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Annuity – Finding the Rate Without a Financial
Calculator
• Trial and Error Process
 Choose an interest rate and compute the PV of the
payments based on this rate
 Compare the computed PV with the actual loan amount
 If the computed PV > loan amount, then the interest rate
is too low
 If the computed PV < loan amount, then the interest rate
is too high
 Adjust the rate and repeat the process until the
computed PV and the loan amount are equal
6F-48
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Quick Quiz – Part III
• You want to receive $5,000 per month for the
next 5 years.
• How much would you need to deposit today if you
can earn 0.75% per month?
• What monthly rate would you need to earn if you
only have $200,000 to deposit?
• Suppose you have $200,000 to deposit and can
earn 0.75% per month.
• How many months could you receive the $5,000
payment?
• How much could you receive every month for 5
years?
6F-49
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Future Values for Annuities: Example
16
• Suppose you begin saving for your retirement by
depositing $2,000 per year in an IRA.
• If the interest rate is 7.5%, how much will you have in 40
years?
 FV = 2,000(1.07540 – 1)/.075 = 454,513.04
 FV =(C/r) * [(1+r)^T-1]
6F-50
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Special formula for FV of an annuity
• = -FV(rate,period,payment)
• =-FV(.075,40,2000) = $454,513.04

Annuity Due: Example 17
• You are saving for a new house, and you put $10,000 per year in an
account paying 8%. The first payment is made today.
• How much will you have at the end of 3 years?
• FV ={(C/r) * [(1+r)^T-1]]*1.08 shifts the timing of interest payment by one
period
 FV = 1.08*10,000[(1.083 – 1) / .08] = 35,061.12, then add $10,000 for last
period
 Note: since you deposited funds at the beginning of the first period, you need
to add that extra interest payment
 =FV(.08,3,10000, 10000) “FV(rate,nper,pmt,PV) = 45,061
6F-52
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Annuity Due Timeline
0 1 2 3
10000 10000 10000
32,464
35,016.12
6F-53
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Perpetuity – Example 18
• Suppose the Fellini Co. wants to sell preferred stock at $100 per
share.
• A 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?
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Perpetuity – Example 18
• Perpetuity formula: PV = C / r
• Current required return:
 C=1
 r = .025 or 2.5% per quarter
 PV =40
• Dividend for new preferred:
 100 = C / .025
 C = 2.50 per quarter
6F-55
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Quick Quiz – Part IV
• You want to have $1 million to use for
retirement in 35 years.
• If you can earn 1% per month, how much do you
need to deposit on a monthly basis if the first
payment is made in one month?
• What if the first payment is made today?
• You are considering preferred stock that pays a
quarterly dividend of $1.50.
• If your desired return is 3% per quarter, how much
would you be willing to pay?
6F-56
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Work the Web Example
• Another online financial calculator can be found
at MoneyChimp.
• Click on the web surfer and work the following
example:
 Choose calculator and then annuity
 You just inherited $5 million. If you can earn 6% on
your money, how much can you withdraw each year
for the next 40 years?
 MoneyChimp assumes annuity due!
 Payment = $313,497.81
6F-57
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Table 6.2
6F-58
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Excel Special Functions for constant
multiple payments
 -FV(rate, nper, payment, PV) -- present value PV is typically 0 in Ch. 5
 -PV(rate, nper, payment, FV) -- future value FV is typically 0 in Ch. 5
 NPER(rate, payment, PV) payment must be negative Solving for t
 -PMT(rate,nper,pv) Solving for C
 Rate(nper, pay, PV, guess at rate) pay must be negative Solving for r
 Link: https://exceluser.com/1024/excels-five-annuity-functions/

Growing Annuity
A growing stream of cash flows with a fixed
maturity
t
t
r
g
C
r
g
C
r
C
P V
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
1
2


































t
r
g
g
r
C
PV
)
1
(
)
1
(
1
6F-60
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Growing Annuity: Example 19
A defined-benefit retirement plan offers to pay
$20,000 per year for 40 years and increase the
annual payment by three-percent each year. What
is the present value at retirement if the discount
rate is 10 percent?
57
.
121
,
265
$
10
.
1
03
.
1
1
03
.
10
.
000
,
20
$
40


















PV
6F-61
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Growing Perpetuity
A growing stream of cash flows that lasts forever











 3
2
2
)
1
(
)
1
(
)
1
(
)
1
(
)
1
( r
g
C
r
g
C
r
C
P V
g
r
C
PV


6F-62
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Example 20: Growing Perpetuity
The expected dividend next year is $1.30, and
dividends are expected to grow at 5% forever.
If the discount rate is 10%, what is the value of this
promised dividend stream?
00
.
26
$
05
.
10
.
30
.
1
$



P V
6F-63
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Effective Annual Rate (EAR)
• 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.
6F-64
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Annual Percentage Rate
• This is the annual rate that is quoted by law
• By definition APR = period rate times the
number of periods per year
• Consequently, to get the period rate we
rearrange the APR equation:
 Period rate = APR / number of periods per year
• You should NEVER divide the effective rate by
the number of periods per year – it will NOT
give you the period rate
6F-65
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Computing APRs: Example 21
• What is the APR if the monthly rate is .5%?
 .5(12) = 6%
• What is the APR if the semiannual rate is .5%?
 .5(2) = 1%
• What is the monthly rate if the APR is 12% with
monthly compounding?
 12 / 12 = 1%
6F-66
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Things to Remember
• You ALWAYS need to make sure that the interest
rate and the time period match.
 If you are looking at annual periods, you need an annual
rate.
 If you are looking at monthly periods, you need a
monthly rate.
• If you have an APR based on monthly
compounding, you have to use monthly periods
for lump sums, or adjust the interest rate
appropriately if you have payments other than
monthly
6F-67
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Computing EARs – Example 22
• Suppose you can earn 1% per month on $1 invested
today.
 What is the APR? 1(12) = 12%
 How much are you effectively earning?
• FV = 1(1.01)12 = 1.1268
• Rate = (1.1268 – 1) / 1 = .1268 = 12.68%
• 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(1.03)4 = 1.1255
• Rate = (1.1255 – 1) / 1 = .1255 = 12.55%
6F-68
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EAR - Formula
Remember that the APR is the quoted rate, and
m is the number of compounding periods per year
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1
m
APR
1
EAR
m









Remember that the APR is the quoted rate, and
m is the number of compounding periods per year
Copyright © 2016 by McGraw-Hill Global Education LLC. All rights reserved.
Special Excel formula: Effect(APR,nper)

Decisions, Decisions II
• 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?
 First account:
• EAR = (1 + .0525/365)365 – 1 = 5.39%
 Second account:
• EAR = (1 + .053/2)2 – 1 = 5.37%
• Which account should you choose and why?
6F-70
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=Effect(.0525,365)
=.0539

Decisions, Decisions II, Continued
• 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 = .0525 / 365 = .00014383562
• FV = 100(1.00014383562)365 = 105.39
 Second Account:
• Semiannual rate = .0539 / 2 = .0265
• FV = 100(1.0265)2 = 105.37
• You have more money in the first account.
6F-71
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Computing
APRs from EARs
• If you have an effective rate, how can you compute
the APR? Rearrange the EAR equation and you get:





 
 1
-
EA R)
(1
m
A P R m
1
6F-72
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Example 23: APR
• Suppose you want to earn an effective rate of 12%
and you are looking at an account that compounds on
a monthly basis. What APR must they pay?
 
11.39%
or
8655152
113
.
1
)
12
.
1
(
12 1 2
/
1




A P R
6F-73
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Computing Payments
with APRs: Example 24
• Suppose you want to buy a new computer
system and the store is willing to allow you to
make monthly payments. The entire computer
system costs $3,500.
• The loan period is for 2 years, and the interest rate is
16.9% with monthly compounding.
• What is your monthly payment?
 Monthly rate = .169 / 12 = .01408333333
 Number of months = 2(12) = 24
 3,500 = C[1 – (1 / 1.01408333333)24] / .01408333333
 C = 172.88
6F-74
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Future Values with
Monthly Compounding: Example 25
• Suppose you deposit $50 a month into an
account that has an APR of 9%, based on
monthly compounding.
• How much will you have in the account in 35 years?
 Monthly rate = .09 / 12 = .0075
 Number of months = 35(12) = 420
 FV = 50[1.0075420 – 1] / .0075 = 147,089.22
6F-75
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Present Value with
Daily Compounding: Example 26
• You need $15,000 in 3 years for a new car.
• If you can deposit money into an account that pays
an APR of 5.5% based on daily compounding, how
much would you need to deposit?
 Daily rate = .055 / 365 = .00015068493
 Number of days = 3(365) = 1,095
 FV = 15,000 / (1.00015068493)1095 = 12,718.56
6F-76
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Continuous Compounding: Example 27
• Sometimes investments or loans are figured
based on continuous compounding
• EAR = eq – 1
 The e is a special function in Excel =exp( )
• Example: What is the effective annual rate of
7% compounded continuously?
 EAR = e.07 – 1 = .0725 or 7.25%
6F-77
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Quick Quiz – Part V
• What is the definition of an APR?
• What is the effective annual rate?
• Which rate should you use to compare alternative investments or
loans?
• Which rate do you need to use in the time value of money
calculations?
6F-78
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Pure Discount Loans – Example 6.12
• Treasury bills are excellent examples of pure
discount loans. The principal amount is repaid
at some future date, without any periodic
interest payments.
• If a T-bill promises to repay $10,000 in 12
months and the market interest rate is 7
percent, how much will the bill sell for in the
market?
 PV = 10,000 / 1.07 = 9,345.79
6F-79
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Example: Interest-Only Loan
• 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 .07(10,000) = 700
• Year 5: Interest + principal = 10,700
• This cash flow stream is similar to the cash
flows on corporate bonds, and we will talk
about them in greater detail later.
6F-80
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Example: Amortized Loan with Fixed Principal
Payment
• Consider a $50,000, 10-year loan at 8%
interest. The loan agreement requires the firm
to pay $5,000 in principal each year plus
interest for that year.
• Click on the Excel icon to see the amortization
table
6F-81
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Example: 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.
 What is the annual payment?
• 4 N
• 8 I/Y
• 5,000 PV
• CPT PMT = -1,509.60
• Click on the Excel icon to see the amortization
table
6F-82
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Work the Web Example
• There are websites available that can easily
prepare amortization tables
• Click on the web surfer to check out the
Bankrate.com website and work the following
example
• You have a loan of $25,000 and will repay the
loan over 5 years at 8% interest.
 What is your loan payment?
 What does the amortization schedule look like?
6F-83
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Quick Quiz – Part VI
• What is a pure discount loan?
• What is a good example of a pure discount loan?
• What is an interest-only loan?
• What is a good example of an interest-only loan?
• What is an amortized loan?
• What is a good example of an amortized loan?
6F-84
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Ethics Issues
• Suppose you are in a hurry to get your income tax
refund.
• If you mail your tax return, you will receive your refund in 3 weeks.
• If you file the return electronically through a tax service, you can get the
estimated refund tomorrow.
• The service subtracts a $50 fee and pays you the remaining
expected refund.
• The actual refund is then mailed to the preparation service.
• Assume you expect to get a refund of $978.
• What is the APR with weekly compounding?
• What is the EAR?
• How large does the refund have to be for the APR to be
15%?
• What is your opinion of this practice?
6F-85
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Comprehensive Problem
• An investment will provide you with $100 at the
end of each year for the next 10 years. What is the
present value of that annuity if the discount rate is
8% annually?
• What is the present value of the above if the
payments are received at the beginning of each
year?
• If you deposit those payments into an account
earning 8%, what will the future value be in 10
years?
• What will the future value be if you open the
account with $1,000 today, and then make the
$100 deposits at the end of each year?
6F-86
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Summary: How to do Valuation Problems
• Valuation problems have five elements: PV, FV, C, r, and t. With constant growth add g
• PV problem – given FV, C, r, and t. Example: How much would you be willing to pay now for the opportunity
to receive $1000neach year (C) for 3 years (t) and a lump sum payment at the end of three years (FV), if your
required return is 8% (r)?
• FV problem – You has $10,00 in the bank now (PV) and will deposit $1000 each year (C) for three years (t) if
you can earn 5% in the bank (r)?
• R problem – You borrow $10,000 (PV) and repay $2,000 each year(C) for 12 years (t) what interest rate are
you being charged?
• C problem – You borrow $10,000 (PV) and must pay it off in four years (t). The interest rate charged is
10%.(r), what is your annual payment?
• If the problem is a monthly payment, divide r/12 and multiply 12*t
• You can only use the special Excel functions if C is constant.