FINS5514: Capital Budgeting and
Financing Decisions
Lecture 2: Investment Decision-I
Topics Covered Today
_______________________________________________________________________________
• The Time Value of Money
– Future Value and Compounding
– Present Value and Discounting
– Future and Present Values of Multiple Cash Flows
– Valuing Level Cash Flows: Annuities and Perpetuities
– Comparing Rates: The Effect of Compounding
2-2
2-3
Basic Definitions
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• Present Value – earlier money on a time line (ie value
today)
• Future Value – later money on a time line (ie value at
some time in the future)
• Interest rate – “exchange rate” between earlier money
and later money
– Discount rate
– Cost of capital
– Opportunity cost of capital
– Required return
• time value of money
• On a time line:
– Time 0 is today,
– Time 1 is one period from today or the end of Period 1
and the start of Period 2, and so on.
• Information on a time line is written as:
– Cash flows are written above the tick marks
– Unknown cash flows are denoted with a question mark
– Interest rates are written above the line and between
the tick marks.
Time lines
_________________________________________________________________________________
• Here there is a cash outflow of $100 at time 0 (note
the minus sign),
• The interest rate is 10% in period 1 (t=0 to t=1) and
5% in both period 2 and period 3.
• There are no cash flows at times 1 and 2. There is a
cash flow at time 3 exists but the value is unknown
0 1 2 3
-100 ?
Time
Cash flows
10% 5%
2-6
Future Values
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• Suppose you invest $1000 for one year at 5% per
year. What is the future value in one year?
– Interest = 1000(.05) = 50
– Value in one year = principal + interest = 1000 + 50 = 1050
– Future Value (FV) = 1000(1 + .05) = 1050
• Suppose you leave the money in for another year.
How much will you have two years from now?
– FV = 1050 (1.05) = 1102.50
– ie 1000(1.05)(1.05) = 1000(1.05)2 = 1102.50
2-7
Future Values: General Formula
_________________________________________________________________________________
FV = PV(1 + r)t
– FV = future value
– PV = present value
– r = period interest rate, expressed as a decimal
– T = number of periods
• Future value interest factor = (1 + r)t
2-8
Effects of Compounding
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• Simple interest
– assumes that interest rate paid is a flat percentage of the
principal (P) each period
• Compound interest
– interest is earned/paid on the principal plus any
accumulated interest determined since the start of the
deposit/loan.
• Consider the previous example
– FV with simple interest = 1000 + 50 + 50 = 1100
– FV with compound interest = 1102.50
– The extra 2.50 comes from the interest of .05(50) = 2.50
earned on the first interest payment
2-9
Future Values – Example 2
_________________________________________________________________________________
• Suppose you invest the $1000 from the previous
example for 5 years. How much would you have?
– FV = 1000(1.05)5 = 1276.28
• The effect of compounding is small for a small number
of periods, but increases as the number of periods
increases. (Simple interest would have a future value of
$1250, for a difference of $26.28.)
2-10
Future Values – Example 3
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• Suppose you had a relative deposit $10 at 5.5% interest
200 years ago. How much would the investment be
worth today?
• What is the effect of compounding?
2-11
Present Values
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• How much do I have to invest today to have some
amount in the future?
– FV = PV(1 + r)t
– Rearrange to solve for :
PV = FV / (1 + r)t
• Discounting ≈ present value of some future amount
• “value” generally refers present value unless we
specifically indicate that we want the future value.
2-12
Present Value – One Period Example
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• Suppose you need $10,000 in one year for the down
payment on a new car. If you can earn 7% annually,
how much do you need to invest today?
PV = 10,000 / (1.07)1 = 9345.79
2-13
Present Values – Example 2
_________________________________________________________________________________
_
• Your parents set up a trust fund for you 10 years ago
that is now worth $19,671.51. If the fund earned 7%
per year, how much did your parents invest?
PV = 19,671.51 / (1.07)10 = 10,000
2-14
Present Value – Important Relationship I
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• For a given interest rate – the longer the time
period, the lower the present value
– What is the present value of $500 to be received in 5
years? 10 years? The discount rate is 10%
5 years: PV = 500 / (1.1)5 = 310.46
10 years: PV = 500 / (1.1)10 = 192.77
2-15
Present Value – Important Relationship II
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• For a given time period – the higher the interest
rate, the smaller the present value
– What is the present value of $500 received in 5 years
if the interest rate is 10%? 15%?
Rate = 10%: PV = 500 / (1.1)5 = 310.46
Rate = 15%: PV = 500 / (1.15)5 = 248.59
2-16
Discount Rate
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_
• Often we will want to know what the implied
interest rate is in an investment
• Rearrange the basic PV equation and solve for r
FV = PV(1 + r)t
r = (FV / PV)1/t – 1
2-17
Discount Rate – Example 1
_________________________________________________________________________________
• You are looking at an investment that will pay $1200
in 5 years if you invest $1000 today. What is the
implied rate of interest?
r = (1200 / 1000)1/5 – 1 = .03714 = 3.714%
2-18
Discount Rate – Example 2
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• Suppose you have a 1-year old son and you want to
provide $75,000 in 17 years towards his college
education. You currently have $5000 to invest. What
interest rate must you earn to have the $75,000 when
you need it?
2-19
Finding the Number of Periods
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• Start with basic equation and solve for t (remember
your logs)
FV = PV(1 + r)t
t = ln(FV / PV) / ln(1 + r)
2-20
Number of Periods – Example 1
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• You want to purchase a new car and you are willing
to pay $20,000. If you can invest at 10% per year
and you currently have $15,000, how long will it be
before you have enough money to pay cash for the
car?
t = ln(20,000 / 15,000) / ln(1.1) = 3.02 years
2-21
Multiple Cash Flows –Future Value Example 1
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• You are able to deposit $4000 at the end of the next 3
years in a bank account paying 8% interest. You
currently have $7000 in the account. How much will you
have in 3 years? In 4 years?
• Find the value at year 3 of each cash flow and add them
together.
2-22
cash flow 7000 4000 4000 4000
|______|_____|_____|_____|
period 0 1 2 3 4
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 = 8817.98 + 4665.60 + 4320 + 4000
= 21,803.58
• Value at year 4 = 21,803.58(1.08) = 23,547.87
2-23
Multiple Cash Flows – FV Example 2
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• 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?
2-24
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 = 1616.26
• Second way – use value at year 2:
– FV = 1248.05(1.09)3 = 1616.26
2-25
Multiple Cash Flows – Present Value Example 3
• You are offered an investment that will pay you
$200 in one year,$400 the next year, $600 the next
year and $800 at the end of the fourth year. You can
earn 12% on very similar investments. What is the
most you should pay for this one?
cash flow 200 400 600 800
|______|_____|_____|_____|
period 0 1 2 3 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 = 1432.93
2-24
2-24
2-27
Annuities and Perpetuities Defined
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• 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 or annuity in
advance
• Perpetuity – infinite series of equal payments
2-28
Ordinary annuity
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• Annuity paid over n periods.
• The first cash flow (R1) occurs at the end of the first
period.
• All cash flows in the annuity need to be discounted to
t=0.
Cash flow R1 R2 R3 R4 ... Rn-1 Rn
|___|___|___|___|___|____________|___|
Period 0 1 2 3 4 n-1 n
2-29
Annuities Basic Formulas
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• Annuities:






−+
=












+
−
=
r
rCFV
r
rCPV
t
t
1)1(
)1(
11
2-30
To get the formulas :
• Assume an annuity of $1 for n periods, where the
periodical compound interest rate is r.
Cash flow $1 $1 $1 $1 ... $1 $1
|___|___|___|___|___|____________|___|
Period 0 1 2 3 4 n-1 n
• Then the future value of this stream of cash flows is
FV = 1 + 1(1 + r)1 + 1(1 + r)2 + . . . . . + 1(1 + r)n-1 (A)
• Multiply each side of (A) by (1+r):
FV(1+r) = (1+r)1 + (1+r)2 + (1+r)3 + . . . + (1+r)n (B)
• Subtract (A) from (B)
FV(1 + r) - FV = (1 +r)n - 1
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• and simplifying:
• Using a similar technique we find that the present
value of a series of cash flows of $1 for n periods, at a
rate of r
]1 - ) + 1 ( [1=
n
r
rFV
2-32
Annuity – Sweepstakes Example
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• 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
2-33
Buying a House
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• 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?
2-34
Buying a House – Continued
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• 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 – 5604 = 14,396
Total Price = 140,105 + 14,396 = 154,501
2-35
Finding the Payment
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• 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
2-36
Finding the Number of Payments
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• Suppose you borrow $2000 at 5% and you are going to
make annual payments of $734.42. How long before you
pay off the loan?
2000 = 734.42(1 – 1/1.05t) / .05
0.136161869 = 1 – 1/1.05t
1/1.05t = 0.863838131
1.157624287 = 1.05t
t = ln(1.157624287) / ln(1.05) = 3 years
2-37
Future Values for Annuities
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• Suppose you begin saving for your retirement by
depositing $2000 per year in an IRA. If the interest
rate is 7.5%, how much will you have in 40 years?
FV = 2000(1.07540 – 1)/.075 = 454,513.04
2-38
Annuity due
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• The first cash flow occurs at the beginning of the
first period. Notice that R1 occurs at t=0.
• It is already discounted or expressed in net present
value terms.
• When calculating the present value of the annuity
only the remaining n-1 payments need to be
discounted then added to the value of the first cash
flow.
cash flow R1 R2 R3 R4 ... Rn
|___|___|___|___|___|____________|___|
period 0 1 2 3 4 n-1 n
2-39
Annuity Due- FV
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• 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?
cash flow 10000 10000 10000
|______|_____|_____|
period 0 1 2 3
FV = (10,000[(1.083 – 1) / .08]) *(1.08) = 35,061.12
• If you use the regular annuity formula, the FV will occur
at the same time as the last payment (ie year 2).
• To get the value at the end of the third period, you have
to take it forward one more period.
x (1 + r)
2-38
2-38
• Calculate the present value of an annuity due that
pays $150 each period, lasts for 3 periods and has
an interest rate of 2%
cash flow 150 150 150
|______|_____|_____|
period 0 1 2 3
PV = 150 1 – 1 x (1+0.02) = $441.23
[0.02 * (1.02)3]
0.02
Annuity Due- PV
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2-42
Deferred Annuity
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• A deferred annuity refers to a stream of fixed payments
which begin sometime in the future, i.e. the first
payment is delayed.
• A typical case would be a pension which does not begin
until age 60 or an inheritance paid in fixed instalments
beginning after a particular date.
2-43
Methods
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• Method 1
– Let b be the delay period and assume that the annuity is
an ordinary annuity
– Calculate the present value of the annuity at the beginning
of period b. To calculate the PV of this stream at time zero
(today) discount the above annuity to time 0, by dividing
by (1+r)b-1
cash flow R1 R2 ... Rn
|___|___|___|___|___|____________|
period 0 1 … b-1 b n
• Method 2
– Take the present value of annuity paid for n years and
subtract an annuity paid over b-1 periods
cash flow R1 R2 ... Rn
|___|___|___|___|___|____________|
period 0 1 … b-1 b n
2-41
2-44
2-45
Deferred Annuity example
_________________________________________________________________________________
• Determine the present value of a $250 annuity, payable
annually for five years when the first payment occurs at
the end of the seventh year. Use a discount rate of 11%
p.a.
cash flow 250 250 250 250 250
|___|___|___|___|___|___|___|
period 0 1 … 6 7 8 9 10 11
PV = [250[1 – 1/1.115] / .11]*(1.11)-6 or
PV = [250[1 – 1/1.1111] / .11]-[250[1 – 1/1.116] / .11]
2-46
Perpetuities
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• A perpetuity involves the payment of a stream of fixed
payments where cash flows continue forever.
PV = C / r
• The formula is determined by considering that the
number of compounding periods is infinite
(continuously compounded) and finding the limits as n
goes to infinity
2-47
Perpetuity – Example
_________________________________________________________________________________
• Company B wants to sell preference shares at $100
per share. A similar issue of shares already issued
has a price of $40 and offers a dividend of $1 per
quarter. What dividend does the company have to
offer if the shares are going to sell?
PV = $40, CF = $1 per quarter
• Current required return:
40 = 1 / r
r = .025 or 2.5% per quarter
• Dividend for new preferred:
100 = C / .025
C = 2.50 per quarter
• Often annuities and perpetuities grow over
time.
• For example, income from an investment
could start at a set amount $C and then grow
by a fixed percentage (g) every year.
• In this situation, the growth rate must be
taken into account
Constant Growth Rates
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62
• A growing stream of cash flows that lasts forever
0 1 2 3
……
C C×(1+g) C ×(1+g)2
+ + +L
C ×(1+ g)2
(1+ r) (1+ r)2 (1+ r)3
C C ×(1+ g)
PV =
• The formula for the present value of a growing
perpetuity is (if gPVGrowingPerpetuity =
C
r -g
Growing Perpetuity
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63
Growing perpetuity - example
0
…
1 2 3
$1.30 $1.30×(1.05) $1.30 ×(1.05)2
= $26.00
0.10 − 0.05
$1.30PV =
• The expected dividend of Company XYZ next year is
$1.30 and dividends are expected to grow at 5%
forever. If the discount rate is 10%, what is the present
value of this dividend stream?
64
0 1
C
C×(1+ g)T −1C C×(1+ g)
(1+ r) (1+ r)2 (1+ r)TPV = + +L+
r − g
PV =
C
( 1+ r)
(1+ g)T
1−
L
2 3
C×(1+g) C ×(1+g)2 C×(1+g)T-1
T
• A growing stream of cash flows with a fixed maturity
• The formula for the present value of a growing annuity is:
Growing Annuity
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65
Growing annuity - example
$20,000
= $265,121.57$20,000
(1.10 )
(1.03) 401−
0.10−0.03
PV =
$20,000×(1.03)
• A retirement plan offers to pay $20,000 the first year, and to
increase the annual payment by 3% each year until the person
dies. Assume the person will die in 40 years. What is the present
value at retirement if the discount rate is 10%?
0 1 2 40
L
$20,000×(1.03)39
2-53
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.
2-54
Annual Percentage Rate
_________________________________________________________________________________
• This is the annual rate that is quoted by law-
nominal or quoted rate
• 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
2-55
Computing APRs
_________________________________________________________________________________
• What is the APR if the monthly rate is 0.5%?
0.5(12) = 6%
• What is the APR if the semiannual rate is 0.5%?
0.5(2) = 1%
• What is the monthly rate if the APR is 12% with
monthly compounding?
12 / 12 = 1%
– Can you divide the above APR by 2 to get the
semiannual rate? NO!!! You need an APR based on
semiannual compounding to find the semiannual rate.
2-56
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
2-57
Computing EARs – Example
_________________________________________________________________________________
• 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?
– How much are you effectively earning?
2-58
EAR – Formula
_________________________________________________________________________________
1
m
APR
1 EAR
m
−



+=
- APR is the quoted rate
- m is the number of compounding periods per year
2-59
Decisions, Decisions II
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• 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?
2-60
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.
2-61
Computing APRs from EARs
_________________________________________________________________________________
• If you have an effective rate, how can you compute the
APR?
• Rearrange the EAR equation and you get:



 += 1 - EAR) (1 m APR m1
2-62
APR – Example
_________________________________________________________________________________
• 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?
= 12 (1 + .12)/ − 1
= .1138655152 or 11.39%
2-63
Computing Payments with APRs
_________________________________________________________________________________
• Suppose you want to buy a new computer system
and the store is willing to sell it to allow you to make
monthly payments. The entire computer system
costs $3500. 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
3500 = C[1 – 1 / 1.01408333333)24] / .01408333333
C = 172.88
2-64
Future Values with Monthly Compounding
____________________________________________________________________________________
• 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
2-65
Continuous Compounding
_________________________________________________________________________________
• Sometimes investments or loans are figured based on
continuous compounding
EAR = eq – 1
– The e is a special function on the calculator normally
denoted by ex
• Example: What is the effective annual rate of 7%
compounded continuously?
– EAR = e.07 – 1 = .0725 or 7.25%
• When loans are made, the borrower is obliged to
repay both the principal amount and to pay
interest.
• There are many different ways this can be done.
• Here we will consider three popular approaches:
– Pure discount loans
– Interest only loans
– Amortized loans
Loans and Amortization
_________________________________________________________________________________
2-66
• A pure discount loan is one in which the
principal is repaid in one lump sum.
• Valuing a pure discount loan requires
calculating its present value.
• Calculate the present value of a $50,000 pure
discount loan with a ten year life span and a 5%
interest rate:
Pure Discount Loans
_________________________________________________________________________________
2-67
• An interest only loan is one in which interest is paid in
each period and then the principal is repaid in one lump
sum at the end of the loan.
• Consider a $20,000 loan with interest of 10% and a 3
year life span.
– In each year, the interest is $2000
– At the end of year 3, the $20,000 is repaid.
Interest Only Loans
_________________________________________________________________________________
2-68
• In an amortized loan, interest if paid every
period and the principal is also paid off at
regular intervals.
– Most amortized loans have a single fixed payment
each month – a form of ordinary annuity
• Valuing these loans uses the same formula for
the present value of an ordinary annuity as
before.
Amortized Loans
_________________________________________________________________________________
2-69