UNSW Business School
School of Banking and Finance
FINS5514: Capital Budgeting and Financial
Decisions
Lecture 2: The Investment Decision I.
Time Value of Money, Annuities, Perpetuities and
Comparing Interest Rates
The Time Value of Money
• A dollar received today is worth more than a
dollar received in the future.
– You could invest the dollar today and it would
generate a return, meaning that you would get
more than a dollar in the future.
• Imagine that you have $100 today and invest
it at a yearly interest rate of 10%.
• In one year’s time you will have $110
– Thus $100 today is worth $110 in one year’s time.
Time Lines
• 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 below 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 (continued)
• 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
Time
0
Cash flows ‐100
10%
1
5%
2
3
?
The Time Value of Money (continued)
• This introduces some important concepts:
• Present Value (PV). The amount today
• Interest rate (r ). The amount the bank pays on
money invested each year.
• Interest received (INT). The dollar amount that
you receive during the year.
• Future Value (FV). The amount in the future.
• Number of periods involved (t). How many
periods in the analysis.
Simple and Compound Interest
• When calculating future values you need to
consider what type of interest you are receiving.
• Compound interest: Here we assume that
interest is reinvested so you are receiving interest
on both the principal amount and the previous
interest.
• Simple interest: Here interest is not reinvested so
the interest in each period is on the principal
amount only.
Future Value, 1 Period Ahead
• Using these concepts the future value one period
ahead can be estimated as:
FVt  PV  INT  PV  (PV  r)
FVt  PV 1  r 
•
•
•
•
Where PV is the present value,
INT is the dollar amount of interest received
r is interest rate and
t is the number of time periods
Future Value, 1 Period Ahead
(continued)
• For example, assume: t =1, r =10%, PV=$100.
• The future value (FV) 1 period ahead (as t=1) is:
FVt  PV  INT  PV  (PV  r)
FVt  PV 1  r   100(1.1)  $110
• This is simple interest as it is calculated on the
principal amount only.
Future Value over Multiple Periods
• When there are several periods to consider,
the calculation of the future value can be
expanded.
• Imagine that $100 has been invested for 3
years at an annual rate of 10%.
• The future value can then be estimated using
the time line:
Future Value Over Multiple Periods
(continued)
Time
Cash flows
0
‐100
Interest earned
Amount at end of
period, FVt
10%
1
2
3
?
?
?
100 x 10%
=10
100+10
=110
Future Value Over Multiple Periods
(continued)
Time
Cash flows
0
‐100
Interest earned
Amount at end of
period, FVt
10%
1
?
100 x 10%
=10
2
3
?
110 x 10%
=11
?
100+10
=110
The interest in each period is calculated using the final amount
from the previous period. This is compound interest
Future Value Over Multiple Periods
(continued)
Time
Cash flows
0
‐100
Interest earned
Amount at end of
period, FVt
10%
1
2
3
?
?
?
100 x 10% 110 x 10%
=10
=11
100+10
=110
110+11
=121
121 x 10%
=12.1
121+12.1
=133.1
Future Value over Multiple Periods
(continued)
• The formula for the future value can be expanded
to cover several periods.
• The formula is:
FVt  PV 1  r 
t
• Where PV is the present value,
• r is interest rate and
• t is the number of time periods
Comparing Simple and Compound
Interest
• If $100 is invested for 4 years at an interest
rate of 10% per annum, what is the future
value?
• Using simple interest, the future value is
• Using compound interest the future value is
Comparing Simple and Compound
Interest (continued)
• Compound interest is always larger than
simple interest for multiple period investments
• The difference may be small to begin with but
it gets bigger and bigger over time
• For example, if $100 is held for 20 years with
10% interest, what is the future value?
– With simple interest, it is $300.00
– With compound interest, it is $672.75
Present Value
• The present value of a cash flow which will
pay off t periods in the future is the amount
that, if it were invested today, would grow into
the future value.
• The PV of an investment can be calculated if
the interest rates and the FV are known.
• The process for calculating the present value is
known as discounting.
– This is, essentially, the opposite of compounding
Present Value (continued)
• Consider the time line:
Time
0 10%
1
2
4
3
FV=146.41
PV=?
• The present value can be found using this
formula
FVt
PVt 
t
1  r 
Present Value (continued)
• Using the numbers in the time line above, the
present value can be estimated as follows:
FVt
PVt 
t
1  r 
146.41
 $100
PV4 
4
1  10%
Solving for the Interest Rate
• Often we will want to know what the implied interest rate is
in an investment
• To find the interest rate the relationship
between the FV and PV is rearranged to give:
Solving for the Interest Rate
(continued)
• Consider an investment that has a present
value of $100 and a future value, in time 4, of
$146.41
• Assuming that the cash flows are constant, the
rate of interest paid by this investment can be
found as:
 FVt
r  
 PVt
1
t
1
4

146.41


  1  0.1  10%
  1  
 100 

Solving for Time
• To find the time the relationship between the
FV and PV is rearranged to give:
 FVt 
l n

PVt 

t
l n1  r 
Solving for Time (continued)
• Consider an investment that has a present value
of $100 and a future value of $146.41. The
interest rate is 10%
• Assuming that the cash flows are constant, the
time period is found as:
 FV 
l n t  ln 146.41 

PVt 
0.38
100





t

4
l n1 r  l n1 0.10 0.095
Simplifications
• We now consider multiple cash flows
• Annuity
– A stream of constant cash flows that lasts for a fixed
number of periods
• Perpetuity
– A constant stream of cash flows that lasts forever
• Growing annuity
– A stream of cash flows that grows at a constant rate for
a fixed number of periods
• Growing perpetuity
– A stream of cash flows that grows at a constant rate
forever
23
Annuities
• An annuity is a sequence of equal payments
made at fixed times for a specified number of
periods.
• There are several types of annuities:
– Ordinary annuities
– Annuities due
– Deferred annuities
Ordinary Annuities
• An ordinary annuity is one in which the
payments are made at the end of each time
period.
• The future value (FVAt) of an ordinary annuity
is estimated by working out the compounded
value of each payment and adding together
these sums.
• Imagine an annuity pays $200 at the end of
each period and the interest rate is 10%
Ordinary Annuities (continued)
• First we calculate the interest earned:
Time
0
Payment (C)
Interest earned
(Compound)
10%
1
$200
$20
2
$200
3
$200
The first $200 payment
earns $20 in the first
compounding period (to
t=2) giving a total of $220.
Ordinary Annuities (continued)
• Now the next period:
Time
0
Payment (C)
Interest earned
(Compound)
10%
1
$200
$20
$22
$42
2
$200
3
$200
The first payment is now $220
and earns a further $22 in
interest in the next period (to
t=3). The total interest on this
payment is $42 over the life of
the investment.
Ordinary Annuities (continued)
• The total interest on the first payment is:
Time
0
Payment (C)
Interest earned
(Compound)
Amount after
interest
10%
1
2
3
$200
$200
$20
$22
$42
Thus the final value of the
first $200 is $242. This is
the same as calculating
the FV of $200, using
r=10%, t=2
$242
$200
Ordinary Annuities (continued)
• Now the second payment:
Time
0
Payment (C)
Interest earned
(Compound)
Amount after
interest
10%
1
$200
2
$200
$20
$22
$42
$20
$242
$220
3
$200
Repeat this
process for
the second
$200
payment.
Ordinary Annuities (continued)
• And the last payment:
Time
0
Payment (C)
Interest earned
(Compound)
Amount after
interest
10%
1
$200
2
$200
The last $200
payment does not
earn any interest as
the investment ends
at this point.
3
$200
$0
$200
Ordinary Annuities (continued)
• So the complete future value is:
Time
0
Payment (C)
Amount after interest
10%
1
2
3
$200
$200
$200
$242
$220
$200
The total FV is the sum of these amounts after interest
Future value =
$242+$220+$200 = $662
Ordinary Annuities (continued)
• The equation for the future value of an ordinary
annuity is:
FVAt  C 1  r 
t 1
 C 1  r 
t 2
 ...
t
 C 1  r   C  1  r 
t n
0
n1
• Where C is the payment in each period and all
the other terms are defined as before.
Ordinary Annuities (continued)
• A more concise alternative is the formula:
 1  r t  1 
FVAt  C 

r


Future Value of an Ordinary Annuity, An
Example
• Calculate the future value of an ordinary
annuity in four periods time. The annuity has
payments of $200 and the interest rate is 2%
 1  r t  1 
FVAt  C 

r


 1  0.024  1 
 200
  $824.32
0.02


Ordinary Annuities (continued)
• The present value (PVAt) of an ordinary
annuity is estimated by discounting the value
of each payment and adding together these
sums.
• Imagine an annuity pays $200 at the end of
each period and the interest rate is 10%.
Ordinary Annuities (continued)
• The time line for this annuity is:
Time 0
10%
Payment (C)
1
$200
2
3
$200
$200
Ordinary Annuities (continued)
• Calculating the PV for the first amount:
Time 0
10%
C
Discounted value
1
$200
$181.82
2
3
$200
$200
This is the amount
(PV) that you would
have to invest at time
0 to have $200 at time
1
Ordinary Annuities (continued)
• Now the PV for the second amount:
Time 0 10%
C
Discounted value
1
$200
$181.82
2
3
$200
$200
$165.29
This is the amount (PV) that you would have to invest at time 0 to
have $200 at time 2
Ordinary Annuities (continued)
• And the PV for the final amount:
Time 0 10%
C
Discounted value
1
$200
$181.82
2
3
$200
$200
$165.29
$150.26
Repeat this calculation for the final cash flow
Ordinary Annuities (continued)
• The total PV for the investment is:
Time 0
10%
C
Discounted value
1
$200
$181.82
2
3
$200
$200
$165.29
$150.26
The PVA is the sum of the discounted values
Present value of the
annuity (PVAt )
$181.82+$165.29+$150.26 = $497.37
Ordinary Annuities (continued)
• The equation for the present value of an
ordinary annuity is:
1
2
 1 
 1 
PVAt  C 
  C
  ...
1r 
1r 
t
1 
1 


 C

 C 

1r 
n1  1  r 
t
n
Ordinary Annuities (continued)
• An alternative approach is to use this formula:
1 
1
PVAt  C 

t
 r r 1  r  
Present Value of an Ordinary Annuity, An
Example
• Calculate the present value of an ordinary
annuity that pays $150 each period. The
annuity lasts for 3 periods and the interest
rate is 2%
Annuities Due
• An annuity due is one in which the payments
are made at the beginning of each time
period.
Time
Payments
0
$C
r%
1
2
$C
$C
3
Annuities Due (continued)
• The future value of an annuity due is
estimated by working out the compounded
value of each payment and adding together
these sums.
• Imagine an annuity pays $200 at the
beginning of each period and the interest rate
is 10%
Annuities Due (continued)
• Calculating interest payments
Time 0 10%
C
Interest earned
(Compound)
Totals after interest
3
1
2
$200
$200
$200
$20
$22
$24.2
$66.2
$20
$22
$42
$20
$0
$220.0
$0
$266.2
$242.0
Annuities Due (continued)
• The future value is then:
Time 0 10%
C
$200
1
2
$200
$200
3
Interest earned
$66.2
$42
$20
$0
Totals after interest
$266.2
$242.0
$220.0
$0
Future value, FVAt =
$266.2+$242+$220 = $728.20
Annuities Due (continued)
• Another way to find the future value is to use
the formula given as:
 1  r t  1 
1 r 
FVAt Annuity Due  C 
r


Future Value of an Annuity Due, An
Example
• Calculate the future value of an annuity due in
4 periods time, with payments of $200 and an
interest rate of 2%.
t
1


r


1


FVAt Annuity Due  C 
1  r 


r


 1  0.024  1 
1  0.02  $840.81
 200
0.02


Annuities Due (continued)
• The present value of an annuity due is
estimated by discounting the value of each
payment and adding together these sums.
• Imagine an annuity pays $200 at the
beginning of each period and the interest rate
is 10%.
Annuities Due (continued)
• Consider the first payment:
Time 0 10%
C
Discounted
value
$200
$200.00
1
2
$200
$200
3
The Annuity Due pays at
the beginning of the period
so this payment is already
at time 0 and does not
have to be discounted
Annuities Due (continued)
• The complete time line for this annuity is:
Time 0
C
Discounted
value
1
2
$200
$200
$200
$200.00
$181.82
$165.29
Present value of the
annuity PVAt
10%
200+181.82+165.29 = $547.11
3
Annuities Due (continued)
• The present value of an annuity due can also
be found using this equation:
1
1 
1  r 
PVAt Annuity Due  C 
t 
 r r 1  r  
Present Value of an Annuity Due, An
Example
• Calculate the present value of an annuity due
that pays $150 each period, lasts for 3 periods
and has an interest rate of 2%
• The formula gives:
Linking Ordinary Annuities and
Annuities Due
• The future value of the annuity due is larger
than the future value of the ordinary annuity.
– This is because the payments are made earlier and
so they earn more interest.
• To convert from an ordinary annuity to an
annuity due, it is necessary to increase the
value by (1+r):
Value of Annuity Due
 Value of Ordinary Annuity  1  r 
Deferred Annuities
• A deferred annuity is one in which the stream
of fixed payments are delayed until some
future time
– An example of this is a pension scheme which
does not pay out until the recipient is 60 years old.
Deferred Annuities (continued)
• Finding the PV of an annuity deferred for b
periods is a two stage process:
1. Calculate the PV of the ordinary annuity at time
b‐1 using the usual formula
2. Discount back to the present day by multiplying
the PV calculation by:
1  r 
b+1
•
Where b is the number of periods until payment
starts.
Deferred Annuity
• You receive a four-year annuity of $500 per year
beginning at date 6. r=10%. What’s PV of the annuity?
9 10
8
6
7
0 1
5
2
3 4
58
Perpetuity
• A constant stream of cash flows that lasts forever
0
1
2
3
C
C
C
…
C
C
C



PV 
2
3
(1  r ) (1  r ) (1  r )
• The formula for the present value of a perpetuity is:
PVPerpetuity
C

r
59
Perpetuities, An Example
• Calculate the present value of a preference
share with a dividend of $5. The interest rate
is 4%. This gives:
PVPerpetuity
C
5
 
 $125
r 0.04
Constant Growth Rates
• 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
Growing perpetuity
• A growing stream of cash flows that lasts forever
0
1
C
2
C×(1+g)
3
…
C ×(1+g)2
C
C  (1  g ) C  (1  g )



PV 
2
3
(1  r )
(1  r )
(1  r )
2
• The formula for the present value of a growing
perpetuity is (if g<r):
C
PVGrowing Perpetuity 
r g
62
Growing perpetuity - example
• 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?
0
1
2
3
…
63
Growing annuity
• A growing stream of cash flows with a fixed maturity
0
1
2
T
3

C
C×(1+g)
C ×(1+g)2
C×(1+g)T-1
C
C  (1  g )
C  (1  g )
PV 


2
T
(1  r )
(1  r )
(1  r )
T 1
• The formula for the present value of a growing
annuity is:
T
C  1 g 
PV 

1  
r  g   1  r 



64
Growing annuity - example
• 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

65
Uneven Multiple Cash Flow Streams
• So far it has been assumed that the cash flows
have been the same in each period (annuities)
• This is unrealistic. Many investments will not
generate constant cash flows.
• The present value of an investment with uneven
cash flows is estimated by calculating the present
value of each of the cash flows and then adding
all of these individual answers together.
Uneven Cash Flow Streams
(continued)
• Imagine an investment has the following time
line
Time
0
10%
CF
Discounted
value
Present value, PV
4
1
2
3
$200
$150
$200
$300
$123.97
$150.26
$204.9
$181.82
181.82+123.97+150.26+204.90 = $660.95
Uneven Cash Flow Streams
(continued)
• The same estimation could have been
completed using this general formula for the
present value
1
2
 1 
 1 
PV  CF1 
  CF2 
 
1r 
1r 
1 

... CFn 
1r 
n
Uneven Cash Flow Streams
(continued)
• Substituting in the values from the timeline
gives:
1
2
 1 
 1 
PV  200
  150

 1  10% 
 1  10% 
3
1 
1 


 200
  300

 1  10% 
 1  10% 
PV  181.82  123.97  150.26  204.90
PV  660.95
4
Uneven Cash Flow Streams
(continued)
• The future value of an investment with
uneven cash flows is estimated by calculating
the future value of each of the cash flows and
then adding all of these individual answers
together.
Uneven Cash Flow Streams
(continued)
• Using the same cash flows over 5 periods:
Time
0 10%
CF
Compound
interest
Future value =
1
2
3
4
$200
$150
$200
$300
$20.00
$22.00
$24.20
$26.62
$92.82
$15.00
$16.50
$18.15
$49.65
$20
$22
$42
$30
292.82+199.65+242+330=$1064.47
5
Uneven Cash Flow Streams
(continued)
• The same estimation could have been done by
compounding each cash flow to the end of the
stream and then summing these values
• Substituting in the values gives:
FV  2001  10%   1501  10% 
4
3
 2001  10%   3001  10% 
2
1
FV  292.82  199.65  242 330  1064.47
Compounding periods
• So far it has been assumed that cash flows are
yearly (annual compounding).
• However, cash flows occur more often and there
are many different compounding periods used:
– Bonds generally pay interest semi‐annually
– Dividends on shares are often paid quarterly
– Banks pay often pay interest on a daily basis
• Comparisons between investments that use
different rates are difficult
– A common base must be used.
Compounding periods (continued)
• For more frequent rates some changes need to
be made:
1. The annual interest rate must be changed to a
periodic rate
– Divide the annual rate by the number of payments per
year
2. The number of time periods must be altered to
represent the number of periods under
examination
– Multiply the number of years by the number of
periods in each year.
Compounding periods (continued)
• Consider a problem involving annual
compounding.
• If $100 is invested for three years with an
annual interest rate of 10%, what will the
future value be?
FV3  1001  10%   133.1
3
Changing the Compounding Period
(continued)
• Now what if the interest is paid semi‐annually?
• This means that instead of 3 periods at 10%, we
have 6 periods at 5%. This gives:
FV6  1001  5%  134.01
6
• This semi‐annual result is higher.
– The semi‐annual investment pays interest relatively
more frequently so the amounts accumulate more
quickly.
Quoted Interest Rates
• The quoted (or nominal) interest rate is the rate
that is quoted by financial institutions.
– For this interest rate to be meaningful, it is also
necessary to know how many compounding periods
there are in a year.
• Once the nominal interest rate is known, it can
be used to find other interest rates.
Effective Annual Interest Rates
• The Effective Annual Interest Rate is the rate
of interest actually being earned by the
investor.
– This is the interest rate that would generate the
same future value if annual compounding had
been used.
– If you are comparing two investments with
different compounding periods, then calculate the
EAR for both investments and use that to make
the comparison
Effective Annual Interest Rates
(continued)
• The Effective Annual Interest Rate is linked to
the nominal interest rate by
m
EAR   1  Q   1
m

– Where EAR is the effective annual interest rate,
– Q is the quoted interest rate and
– m is the number of compounding periods per year.
Effective Annual Interest Rates
(continued)
• Consider the previous example where the quoted
interest rate was 10% compounded semi‐annually.
• In this case, the effective rate is:
Annual Percentage Rates
• The annual percentage rate (APR) is a form of
quoted rate and it the rate charged by a
lender in each period.
– This may be the same as the EAR, but not always.
• The APR is the interest rate per period
multiplied by the number of periods in one
year.
– For example, consider a loan charging 3% a month
has an APR of 3x12 = 36%
Interest Rates and Compounding
• Changing the compounding period changes the
EAR.
– Making the period smaller increases the EAR.
• For example, consider a loan paying annually with
a 10% EAR. What if we change the period?
– Quarterly compounding gives EAR = 10.38%
– Weekly gives: EAR = 10.506
– Daily gives: EAR = 10.515
Interest Rates and Compounding
(continued)
• However, there is a limit to how high the EAR
can go.
• This limit is given by:
EAR  expq  1
• For this 10% loan this is:
EAR  exp q  1  exp 0.1  1
 1.105  1  10.517%
Loans and Amortization
• 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
Pure Discount Loans
• 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:
Interest Only Loans
• 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.
Amortized Loans
• 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.
Summary
• Future value, present value
• Four simplifying formulas:
C
Perpetuity : PV 
r
C
Annuity : PV 
r

1 
1  (1  r )T 


C
Growing Perpetuity : PV 
rg
T

 1 g  
C
 
Growing Annuity : PV 
1  
r  g   (1  r )  


88