Time value
of money: Valuing cash flow streams
“Compound interest is the eighth
wonder of the world.
He who understands it, earns it ...
he who doesn't ... pays it.”
Albert Einstein
Learning objectives
• Value a series of many cash flows.
• Value a perpetual series of regular cash flows
called a perpetuity.
• Value a common set of regular cash flows
called an annuity.
• Value both perpetuities and annuities when
the cash flows grow at a constant rate.
• Calculate the number of periods, cash flow, or
rate of return of a loan or investment.
2
Finance way of thinking
• Problem solving process
– Timelines
– Positing, doing, checking
• Disciplinary understandings
– Interpreting your results / making sense of your
answer
• Rearranging formulae
– Very basic level
Terms
• Interest rate, discount rate, yield, return
r
• Value, cash flow, cost, inflow
• Term, time, period
t
Time Line of Cash Flows:
Making it easy
•Tick marks at ends of periods
• Time 0 is today;
• Time 1 is the end of Period 1
0
CF0
r%
1
2
CF1
CF2
Copyright  2011 McGraw-Hill Australia Pty Ltd
PPTs t/a Essentials of Corporate Finance 2e by Ross et al
3
4-7
Looking forward
• What is the future value of an amount?
– Suppose you invest $100 for one year at 10% (r)
per year. What is the future value in one year?
• Interest = 100(.1) = $10
• Value in one year = principal + interest
= 100 + 10 = $110
• Future Value (FV) = 100(1 + 0.1) = $110
– Suppose you leave the money in for another year. How
much will you have two years from now?
• FV = 100(1.1)(1.1)
= 100(1.1)2
= $121
Future Values:
Effects of Compounding
• Simple interest
– Interest earned only on the original
principal
• Compound interest
– Interest earned on principal and on interest
received
– “Interest on interest” – interest earned on
reinvestment of previous interest payments
Copyright  2011 McGraw-Hill Australia Pty Ltd
PPTs t/a Essentials of Corporate Finance 2e by Ross et al
4-9
Future Values:
Effects of Compounding
• Consider the previous example
– FV w/simple interest
= 100 + 10 + 10 = 120
– FV w/compound interest
=100(1.10)2 = 121.00
– The extra 1.00 comes from the interest of
.10(10) = 1.00 earned on the first interest
payment
Copyright  2011 McGraw-Hill Australia Pty Ltd
PPTs t/a Essentials of Corporate Finance 2e by Ross et al
4-10
10%
Amount
Year
invested Interest balance
0
100.00
10.00
110.00
1
110.00
11.00
121.00
170,00
2
121.00
12.10
133.10
160,00
3
133.10
13.31
146.41
4
146.41
14.64
161.05
150,00
140,00
130,00
120,00
110,00
Interest
100,00
Amount invested
90,00
1
2
3
4
5
The effect on FV of increasing r
• Lets do a few computations
– What is the value of $100 in 10 years time at:
•
•
•
•
5%
10%
15%
20%
Future values
$100 Discount rate
Time
0%
5%
0
100
100
1
100
105.00
2
100
110.25
3
100
115.76
4
100
121.55
5
100
127.63
6
100
134.01
7
100
140.71
8
100
147.75
9
100
155.13
10
100
162.89
10%
100
110.00
121.00
133.10
146.41
161.05
177.16
194.87
214.36
235.79
259.37
15%
100
115.00
132.25
152.09
174.90
201.14
231.31
266.00
305.90
351.79
404.56
20%
100
120.00
144.00
172.80
207.36
248.83
298.60
358.32
429.98
515.98
619.17
700
Future value of $1
600
500
0%
5%
10%
15%
20%
400
300
200
100
0
0
5
10
15
Future Value Interest Factors
Page 562 Appendix A
Future value – single sums
If you deposit $100 in an account earning 6% pa, how
much would you have in the account after 1 year?
PV =
100
0
FV = ???
106
1
Future value – single sums
If you deposit $100 in an account earning 6% pa, how
much would you have in the account after 5 years?
PV =
100
FV = ???
133.82
0
5
solution
FV = PV (1 + r )t
= 100 ( 1 + 0.06 )5
= 100 x 1.3382
= 133.82
Future value – quarterly compounding
If you deposit $100 in an account earning 6% pa
with quarterly compounding, how much would you
have in the account after 5 years?
PV =
100
134.69
FV = ???
0
5
solution
t = 5 x 4 = 20 periods
r = 6 / 4 = 1.5% per period
FV = 100*(1+0.015)^20
= 134.69
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
FVIF(r,t) = Future Value Interest Factor for $1
invested at r% for t periods.
Copyright  2011 McGraw-Hill Australia Pty Ltd
PPTs t/a Essentials of Corporate Finance 2e by Ross et al
4-18
Present Value
FV = PV(1 + r)t
FV / (1+r)t = PV
PV = FV / (1+r)t
Which is equivalent to
PV = FV(1+r)-t
• “Discounting” = finding the present value of one
or more future amounts
• (1+r)-t is the present value factor
Copyright  2011 McGraw-Hill Australia Pty Ltd
PPTs t/a Essentials of Corporate Finance 2e by Ross et al
4-19
Single sums
If you will receive $100 in one year from now, what
is its present value if your opportunity cost is 6%?
PV = ???
94.34
0
FV = 100
1
solution
or
PV = FV / ( 1 + r )t
PV = FV * ( 1 + r )-t
=100 / ( 1 + 0.06 )1
=100 * ( 1.06 )-1
= 100 / 1.06
= 100 * (0.9434)
= 94.34
= 94.34
Single sums
If you will receive $100 in one year from now, what
is its present value if your opportunity cost is 6%
pa compounded quarterly?
PV = ???
94.34
0
solution
PV = FV / ( 1 + r )t
=100 / ( 1 + 0.015 )4
= 100 / 1.061364
= $94.22
FV = 100
1
Bringing future amounts into todays
terms
• Finding the present value of some future
amount
To reach a savings target of $3,000 after two
years at 8% p.a. compounded quarterly, how
much would we have to invest today?
Draw a timeline and write what we know, r, t, …
To reach a savings target of $3,000 after two
years at 8% p.a. compounded quarterly, how
much would we have to invest today?
PV = $3,000 / (1.02)8
= $3,000 / 1.17166
= $2,560.47
Present Value:
Important Relationship I
For a given interest rate:
– The longer the time period,
– The lower the present value
FV
PV 
t
(1  r )
For a given r, as t increases, PV decreases
Copyright  2011 McGraw-Hill Australia Pty Ltd
PPTs t/a Essentials of Corporate Finance 2e by Ross et al
4-24
Page 564
Discount Rate
• To find the implied interest rate, rearrange
the basic PV equation and solve for r:
FV = PV(1 + r)t
FV/PV = (1 + r)t
(FV/PV) 1/t = ((1 + r)t) 1/t
r = (FV / PV)1/t – 1
• If using formulas with a calculator, make sure you
can use both the yx and the 1/x keys
Copyright  2011 McGraw-Hill Australia Pty Ltd
PPTs t/a Essentials of Corporate Finance 2e by Ross et al
4-27
If you sell a block of land for $11,933 that you bought
5 years ago for $5,000, what is your annual rate of
return?
PV = ???
5,000
0
FV = 11,933
5
Calculating rates of return
If you sell land for $11,933 that you bought 5 years ago for
$5,000, what is your annual rate of return?
Solution
FV = PV * ( 1 + r )t
11933 = 5000 * ( 1 + r )5
11933 / 5000 = (1 + r )5
2.3866 = (1 + r )5
2.38661/5 =1 + r
1 + r = 1.19
r = 0.19
Calculating rates of return
If you sell land for $11,933 that you bought 5 years ago for
$5,000, what is your annual rate of return?
Using FV table to check
FV = PV * ( 1 + r )t
11933 = 5000 * ( 1 + r )5
11933 / 5000 = (1 + r )5
2.3866 = (1 + r )5
Table a1 across row n=5 we find
2.3866 is between 18 and 20
r = 19% sounds ok
Calculating rates of return
If you sell land for $11,933 that you bought 5 years ago for
$5,000, what is your annual rate of return?
Using PV table to check
PV = FV ( PVIFr,t)
5000 = 11933 ( PVIFr,5 )
PVIFr,5 = 5000 / 11933
PVIFr,5 = 0.419
Table a2 across row n=5 we find 0.419
is between 18 and 20
r = 19% sounds ok
Finding the Number of Periods
• Start with basic equation and solve for t:
FV = PV(1 + r)t
FV/PV = (1 + r)t
ln(FV/PV) = t ln (1 + r)
 FV 
ln

PV 

t
ln(1  r )
4-32
Calculating n
Suppose you placed $100 in an account that pays
9.6% interest, compounded monthly. How long will it
take for your account to grow to $500?
PV = ???
100
0
FV = 500
?
Calculating t
solution
PV = 100, FV = 500,
r = .096/12 = 0.008 per month
FV=PV(1+r)t
500 = 100(1.008)t
500/100 = (1.008)t
ln(5) = t ln(1.008)
t = 202
ln(5) / ln(1.008) = t
months
t = 201.98
Check your understanding
1. If you had $10,000 now and could invest it
for five years at 5%, and then a further 5
years at 7%, how much would you have at
the end?
2. If your firm had to repay $7m in 25 years
time, using a discount rate of 6%, what is the
debt worth today?
Check your understanding
3. You are offered an investment opportunity
involving a return of $100,000 in 15 years
time, on an initial investment of $50,000.
What rate of return is implied?