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Chapter 3
The Time
Value
of Money
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 1 of 48
The Role of Time Value in Finance
• Most financial decisions involve costs & benefits that
are spread out over time.
• Time value of money allows comparison of cash flows
from different periods.
Which investment would you choose?
(a)An investment of €100,000 that would return
€200,000 after one year
(b)An investment of €100,000 that would return
€220,000 after two years
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 2 of 48
The Role of Time Value in Finance
• In general, all else being equal, the sooner a € is
received, the more quickly is re-invested
• However a short-term investment is not necessarily
more valuable because this depends on the re-
investment interest rate
Answer
The investment (a) is more valuable if it can re-
invested at an annual interest rate >10%,
otherwise the investment (b) is more valuable
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 3 of 48
Cash Flow Time Lines
• The cash flow time lines help to visualize the timing of
the cash flows associated with a particular situation
•The construction of a cash flow time line is fairly easy:
Time 0 1 2 3 4
k = 10%
Cash Flows -500 FVn = ?
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 4 of 48
Difference between simple interest and
compounded interest
With simple interest, an investor doesn’t earn
interest on interest
• Year 1: 5% of €100 = €5 + €100 = €105
• Year 2: 5% of €100 = €5 + €105 = €110
• Year 3: 5% of €100 = €5 + €110 = €115
With compounded interest, an investor earns
interest on interest
• Year 1: 5% of €100.00= €5.00 + €100.00 = €105.00
• Year 2: 5% of €105.00= €5.25 + €105.00 = €110.25
• Year 3: 5% of €110.25= €5.51+ €110.25= €115.76
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 5 of 48
Future Value
Definition and Formula
• Future Value (FV)—determine to what amount an
investment will grow over a particular time period
– re-invested interest (earned in previous periods) earns
interest
– compounding—interest compounds or grows the investment
• FVn = PV0(1+k)n = PV(FVIFk,n)
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 6 of 48
Effects of compounding
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 7 of 48
Future Value - Example
• Suppose an investment of €100 for one year at 5% per
year. What is the future value in one year?
– The compounding rate is given as 5%. Hence the value of
current Euros in terms of future Euros is 1.05 future Euros
per current Euro. Hence future value is 100(1.05) = €105.
•Suppose that money is left in for another year. What is
the future value in two years from now?
– Assume that the money in one year as present value and
the money in two years as future value. Therefore the price
of one-year-from-now money in terms of two-years-from-now
money is 1.05. Therefore 105 of one-year-from-now Euros in
terms of two years-from-now Euros is 105(1.05) = 100
(1.05)(1.05) = 100(1.05)2 = 110.25
– By making the same assumptions, the FV in 3 years would
be 115.76
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 8 of 48
Future Value– Example using Excel
An investment of €100 is made today at 5% interest.
How much money will this investment yield in 3 years?
Excel Function
PV 100 (assumes compounded interest
k 5,00% as capital is being re-invested)
n 3
=FV (interest, periods, pmt, PV)
FV? 115,76
=FV (.05, 3, , 100)
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 9 of 48
Financial Calculator Solution
In the previous example: PV = €100, k =
5.0%, n = 3
3 5 -100 0 ?
N I PV PMT FV
115,76
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 10 of 48
A Graphic view of Future Value
Relationship among Future Value, Growth or
Interest Rates and Time
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 11 of 48
Present Value
Definition and Formula
• Present value (PV)—determine the current value of an
amount that will be paid, or received, at some time in the
future
– PV is the future amount restated in current dollars; future
interest has not been earned, thus it is not included in the
PV
– discounting—deflate, or discount, the future amount by
future interest that can be earned
• PV0 = FVn[1/(1+k)n] = FV(PVIFk,n)
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 12 of 48
Present Value - Example
• Suppose an investor needs €10,000 in two years for
the down payment on a new car. If the investor can earn
6% annually, how much does he need to invest today?
– PV = 10,000 / (1.06)2 = 8899.96
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 13 of 48
Present Value– Example using Excel
How much is needed for an investment today in
order to have €10,000 in 2 years if the investor can
earn 6% interest on his investment?
FV 10.000 Excel Function
k 6,00%
=PV (interest, periods, pmt, FV)
n 2
PV? 8.899,96 =PV (.06, 2, , 10,000)
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 14 of 48
Financial Calculator Solution
In the previous example: FV = €10000, k =
6.0%, n = 2
2 6 ? 0 10000
N I PV PMT FV
-8899,96
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 15 of 48
A Graphic view of Present Value
Relationship among Present Value, Growth or
Interest Rates and Time
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 16 of 48
Solving for interest rates - Example
• If a mutual fund investment that was bought six years
ago at a price of €1000 is now worth €5525, what rate of
return (k) has the investor already earned today?
– FV = PV (1+k)n
– 5525 = 1000 (1+k)6
– and hence k = 33%.
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 17 of 48
Solving for interest rates with Excel
What rate of return has an investor earned from a
€1000 investment bought 6 years ago that is worth
today €5525?
1999 1.000
2000 1.127
PV 1.000 Excel Function
=Rate(periods, pmt, PV,
2001 1.158 FV 5.525 FV)
2002 2.345
2003 3.985 n 6 =Rate(6, ,1000, 5525)
2004 4.677
2005 5.525 k? 33,0%
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 18 of 48
Financial Calculator Solution
In the previous example: PV = €1000, FV =
€5525, n = 6
6 ? -1000 0 5525
N I PV PMT FV
33
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 19 of 48
Solving for period - Example
• If a security worth €712 is invested at 6 percent, how
long will it take to grow to €848?
– FV = PV (1+k)n
– 848 = 712 (1+0.06)n
– and hence n = 3.
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 20 of 48
Financial Calculator Solution
In the previous example: PV = €712, FV =
€848, k = 6%
? 6 -712 0 848
N I PV PMT FV
3
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 21 of 48
Relationship between interest rates
and present value
• For a given interest rate – the longer the time period,
the lower the present value
• For a given time period – the higher the interest rate,
the smaller the present value
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 22 of 48
Future Value of an annuity
Definition and Formula
• Annuity—a series of equal payments that are made at
equal intervals
– Ordinary annuity—has cash flows that occur at the end of
each period
– Annuity due—has cash flows that occur at the beginning of
the period
•The future value of an annuity, FVA, can be computed
by solving for the future value of a lump-sum amount
• FVAn = A (1+k)n - 1 = A(FVIFAk,n)
k
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 23 of 48
FV of Ordinary Annuity - Example
• Suppose an equal cash flow of deposits of €100 at the
end of each year for five years at 3% per year. How
much will these deposits grow?
– The growth rate is given as 3%. Therefore FVA = 100(1.03)0
+ 100(1.03)1+ 100(1.03)2 + 100(1.03)3 + 100(1.03)4 = 530,91
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 24 of 48
Future Value of an Ordinary Annuity
– using Excel
How much will the deposits grow if the initial
deposit is €100 at the end of each year at 3%
interest for five years.
PMT 100 Excel Function
k 3,0% =FV (interest, periods, pmt, PV)
n 5
FV? 530,91 =FV (.03, 5,100, )
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 25 of 48
Financial Calculator Solution
In the previous example: PMT = €100, k =
3%, n=5
5 3 0 -100 ?
N I PV PMT FV
530,91
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 26 of 48
FV of an Annuity Due - Example
• Suppose an equal cash flow of deposits of €100 at the
beginning of each year for five years at 3% per year.
How much will these deposits grow?
– The growth rate is given as 3%. Therefore FVA = 100(1.03)1
+ 100(1.03)2+ 100(1.03)3 + 100(1.03)4 + 100(1.03)5 = 546,84
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 27 of 48
Future Value of an Annuity Due –
using Excel
How much will the deposits grow if the initial
deposit is €100 at the beginning of each year at
3% interest for five years.
PMT 100 Excel Function
k 3,00% =FV (interest, periods, pmt, PV)
n 5
=FV (.03, 5,100, )
FV 530,91
FVA? 546,84 =530.91*(1.03)
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 28 of 48
Financial Calculator Solution
In the previous example: PMT = €100, k =
3%, n=5 (switch calculator to BEGIN)
5 3 0 -100 ?
N I PV PMT FV
546,84
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 29 of 48
Present Value of an annuity
Definition and Formula
• The present value of an annuity, FVA, can be
computed by solving for the future value of a lump-sum
amount
•Annuity due is an annuity with cash flows that occur at
the beginning of the period.
• PVA0 = A 1 - [1/(1+k)n] = A(PVIFAk,n)
k
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 30 of 48
PV of an Ordinary Annuity -
Example
• Suppose an equal cash flow of payments of €1000 at
the end of each year. How much could an investor
borrow if he could afford annual payments of $1,000
(which includes both principal and interest) at the end of
each year for five years at 10% interest?
– The present value of the annuity is calculated as follows
: PVA = 1000/(1.1)1 + 1000/(1.1)2+ 1000/(1.1)3 + 1000/(1.1)4
+ 1000/(1.1)5 = 3790,79
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 31 of 48
Present Value of an Ordinary Annuity
– using Excel
•How much could an investor borrow if he could
afford annual payments of €1,000 (which includes
both principal and interest) at the end of each
year for five years at 10% interest?
PMT 1000 Excel Function
I 10,0% =PV (interest, periods, pmt, FV)
n 5
PV? 3.790,79 =PV (.10, 5, 1000, )
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 32 of 48
Financial Calculator Solution
In the previous example: PMT = €1000, k =
10%, n=5
5 10 ? -1000 0
N I PV PMT FV
3790,79
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 33 of 48
PV of an Annuity Due- Example
• Suppose an equal cash flow of payments of €1000 at
the beginning of each year. How much could an
investor borrow if he could afford annual payments of
$1,000 (which includes both principal and interest) at the
end of each year for five years at 10% interest?
– The present value of the annuity is calculated as follows
: PVA = 1000/(1.1)0 + 1000/(1.1)1+ 1000/(1.1)2 + 1000/(1.1)3
+ 1000/(1.1)4 = 4169,87
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 34 of 48
Present Value of an Annuity Due–
using Excel
•How much could an investor borrow if he could
afford annual payments of €1,000 (which includes
both principal and interest) at the beginning of
each year for five years at 10% interest?
PMT 1000 Excel Function
k 10,00%
=PV (interest, periods, pmt, FV)
n 5
PV 3790,79 =PV (.10, 5, 1000, )
PVA? 4169,87
= 3790,79*(1,1)
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 35 of 48
Financial Calculator Solution
In the previous example: PMT = €1000, k =
10%, n=5 (switch calculator to begin)
5 10 ? -1000 0
N I PV PMT FV
4169,87
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 36 of 48
Solving for interest rates with
annuities- Example
• If an investor pays €846,80 for an investment that
promises to pay €250 per year for the next four years,
what rate of return (k) will the investor earn on the
investment?
– Assuming that payments are made at the end of each year,
this is an ordinary annuity. The solution from the annuity
equation provides k=7%. Beware that trial and error process
should be used.
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 37 of 48
Financial Calculator Solution
In the previous example: PV = €846,80, PMT
= €250, n = 4
4 ? -846,80 250 0
N I PV PMT FV
7
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 38 of 48
Solving for interest rates - Example
• If an investor pays €1685 for an investment that
promises to pay him back €400 per year, how many
payments must he receive to earn a 6% return?
– Assuming that payments are made at the end of each year,
this is an ordinary annuity. The solution from the annuity
equation provides n=5%. Beware that trial and error process
should be used.
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 39 of 48
Financial Calculator Solution
In the previous example: PV = €1685, PMT =
€400, k = 6
? 6 -1685 400 0
N I PV PMT FV
5
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 40 of 48
Present Value of a Perpetuity
• A perpetuity is a special kind of annuity.
• With a perpetuity, the periodic annuity or cash flow
stream continues forever.
PV = Annuity/k
• For example, how much would an investor have to
deposit today in order to withdraw €1,000 each year
forever if the investor can earn 8% return?
PV = €1,000/.08 = $12,500
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 41 of 48
Uneven Cash Flow Streams
• In an uneven cash flow stream, the cash flows are not
the same (equal).
• Simplifying techniques, i.e. the use of a single
equation to compute PV cannot be used
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 42 of 48
Present Value of an uneven
Cashflow Stream - Example
• Calculate the present value of the following uneven
cashflow stream assuming a required return of 9%.
Year Cash Flow PVIF9% ,N PV
1 400 0,917 366,80
2 800 0,842 673,60
3 500 0,772 386,00
4 400 0,708 283,20
5 300 0,650 195,00
PV 1.904,60
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 43 of 48
Present Value of an uneven
Cashflow Stream
- Example using Excel
• Find the present value of the following uneven
cashflow stream assuming a required return of 9%.
Ye ar Cas h Flow
1 400
Excel Function
2 800
3 500 =NPV (interest, cells containing CFs)
4 400
5 300
=NPV (.09,B3:B7)
NPV 1.904,76
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 44 of 48
Compounding More Frequently
than Annually
• Interest is compounded more than once per year—
quarterly, monthly, or daily. The more frequent the
compounding, the more the investor earns because he
is earning on interest more frequently
• Therefore, the effective interest rate (the rate of return
per year considering interest compounding) is greater
than the nominal (annual) interest rate.
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 45 of 48
Annual and Semi-annual
compounding
• For example, what would be the difference in future
value if the depositors puts €100 for 5 years in the
bank and earns 3% annual interest compounded (a)
annually, (b) semiannually?
Annually: 100 x (1 + .03)5 = €115.92
Semiannually: 100 x (1 + .015)10= €116.05
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 46 of 48
Nominal & Effective Rates
• The nominal interest rate is the stated or contractual
rate of interest charged by a lender or promised by a
borrower.
• The effective interest rate is the rate actually paid or
earned.
• In general, the effective rate > nominal rate whenever
compounding occurs more than once per year
EAR = (1 + k/m) m -1
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 47 of 48
Nominal & Effective Rates
• Example: Paul bought a vacation package for winter
holidays and charged it to his credit card. What is the
effective rate of interest on Paul’s credit card if the
nominal rate is 18% per year, compounded monthly?
EAR = (1 + .18/12) 12 -1
EAR = 19.56%
Essentials of Managerial Finance by S. Besley & E. Brigham Slide 48 of 48
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