Time Value of Money
by Binam Ghimire
1
Learning Objectives
 Concept of TVM
 Represent the cash flows occurred in different time
periods using cash flow time line
 Calculate the present value and future value of given
streams of cash flows with and without using table
 Identify the impact of time period and required rate of
return on present value and future value
 Prepare amortised schedule for amortised term loan
 Compare interest rates quoted over different time
intervals
 Understand the real and nominal cash flows
2
Source: Total Access
http://www.totalaccess.co.uk/Case_Studies/big_ben
3
Warm Up: The School Maths
 Adding a percent to an amount
 A computer costs £ 420 + VAT 15%. What is the cost
after VAT is added?
 Reverse percentage
 The price of a computer is £230 after VAT of 15%. What
is the price before VAT is added?
4
Time value of Money:
Concept and Significance
 Theory behind TVM: “One pound today is worth more
than one pound tomorrow”
 Deals with values of cash flows occurred at different
points in time
 Every sum of money received now has a value for
reinvestment
5
Time value of Money:
Concept and Significance
 Why is APR a legal requirement?
Image source: google
6
Time value of Money:
Concept and Significance
 Having Cash NOW is valuable
Inflation
People’s preference to current consumption
 TVM Usage: Firms, Households, Banks, Insurance Cos.
 TVM in Corporate Finance: Economic Welfare of
Shareholders
 Investment Decisions
 Financing Decisions
7
Cash Flow Time Line
 Cash Flow Time Line is an important tool used to
understand the timing of cash flows. It is a graphical
presentation of cash flows occurring at different points
of time, and is helpful for analysing the time value of
cash flows
Time
0
1
2
3
4
5
8
Cash Flow Time Line
 The corresponding cash flows are placed below the
scale as shown in the following time line of cash flows:
Time
1
2
3
4
5
10
50
70
100
90
0
Cash Flows -100
 The time line of cash flow is also used to denote the
interest rate that each cash flow earns
Time
0
8%
1
2
3
4
5
Cash flow – 100
9
Future Value
 Future Value (FV) is the sum of investment amount and
interest earned on the investment
 Interest may be earned simple and compound
 Accordingly FV can be calculated using simple and
compound interest rate methods
10
Future Value:
Simple Interest
 Interest earned only on the original investment is
Simple Interest. For example, a savings account with
a bank in which bank posts the interest every year. The
interest for principle £P deposited at r % interest p.a.
for t number of years will equal to I (where I = p x n x r
÷ 100). So the FV of an investment will be P + I
 We aim to maximise the benefit from any monetary
transaction. So in real world, interest is paid (savings) or
charged (loan) more than once during the tenure of the
savings or loan. As such the Simple Interest does not
exist
11
Future Value :
Compound Interest
 Compound Interest is interest earned on principle
plus interest from the previous period
r
FV=P x 1+
100
t
 FV of £ 100,000 investment for one year, rate of
interest: 12% and when interest is 1) simple interest 2)
compounded monthly and daily are shown in figure next
12
Future Value :
Compound Interest Effect
112,747
Simple Interest
Interest compounded
monthly
112,000
Interest compounded daily
112,683
100,000
Jan
Feb
Mar
2011 Beginning
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
2011 End
13
Future Value :
Compound Interest different intervals
 Hence, with Compound interest, FV grows when the
frequency of interest payment is higher. (i.e. total
amount of interest is growing every next period).
 Note that when interest is per annum and to be
calculated for mth times within a year the formula
changes to:
m
r 


FV  P x 1  m 
 100 


 And if it is for more than one year, the formula is
r 


FV  P x 1  m 
 100 


mx n
Where, n=no. of years
14
Compound Interest different intervals
has offered 5 %
interest charged half
yearly
has offered 4.5%
interest charged every
3 months
The name of the banks are used only to illustrate the APR and the numbers are imaginary. Logo source: website
of respective banks.
 You want to borrow money
15
Future Value :
Compound Interest Table
 Below FV of £ 1 at the end of various numbers of years
have been calculated for different rates of interest:
Interest Rate per annum
Year
12%
13%
14%
15%
16%
1
1.12
1.13
1.14
1.15
1.16
2
1.25
1.28
1.30
1.32
1.35
3
.
.
.
10
.
.
.
1.40
.
.
.
1.44
.
.
.
1.48
.
.
.
1.52
.
.
.
1.56
3.11
3.39
3.71
4.05
4.41
20
9.65
11.52
13.74
16.37
19.46
50
289.00
450.73
700.23
1083.66
1670.70
 The table is known as FV Compound Interest table
[(1+r)t]
16
Future Value :
Graphical View
Future value of £ 1
4.0
3.0
i = 15%
2.0
i = 10%
i = 5%
1.0
i = 0%
Time periods
0
2
4
6
8
10
FV of a sum of money has positive relation with the
time period and interest rate
17
Future Value:
FV, Time Period and Interest rate
 FV increases with rate of interest and time
 The higher the interest rate higher will be the FV
 More the number of years the more will be the FV
 Higher the no. of frequency of payment (within the
year) higher will be the FV
18
Future Value :
Compound Interest in Excel
 In Microsoft Excel, FV function wizard can be used to
calculate FV
 Check the relationship between FV and r and t by
making line chart in Microsoft Excel for r=5, 10 & 15
percent for t= 0,2,4,6,8,10,12 & 14 years
19
Present Value
 Present value (PV) is today’s value of a future cash flow
 FV is only available in the future
 Can we get the FV now? i.e. can we convert the FV into
PV?
 You may if you sacrifice 1) the interest that you could
have earned had you invested the money and 2) adjust
for the time period that you don’t want to wait. Hence,
FV
PV=
1+r t
 The sacrifice is reflected in the formula above. When t
and r get bigger PV gets smaller
20
Present Value:
Discount Rate
You bought a television. The payment term is single
payment of £ 2,000 to be made after 2 years. If you
can earn 8% on your money how much money
should you set aside today in order to make final
payment when due in 2 years?
£ 2,000
PV=
= £1,714.68
2
1+0.08
Note that to calculate PV, we discounted FV at the
interest rate r. The calculation is therefore termed a
discounted cash flow. The interest rate is known as
Discount rate
21
Present Value:
Discount Factor
The PV formula may be converted as follows
1
PV=FV x
1+r
t
The expression 1/(1+r)t basically measures the PV of
£1 received in year t. The expression is known as
Discount Factor. Using the PV of £ 1, a table can
be constructed for FV to be received after t years at
different discount rates
22
Present Value :
Discount Factor Table
 Below PV of £ 1 at the end of various numbers of years
have been calculated for different rates of interest:
Year
Interest / Discount Rate per annum
12%
13%
14%
15%
16%
1
0.893
0.885
0.877
0.870
0.862
2
0.797
0.783
0.769
0.756
0.743
3
.
.
.
10
0.712
.
.
.
0.693
.
.
.
0.675
.
.
.
0.658
.
.
.
0.641
.
.
.
0.322
0.295
0.270
0.247
0.227
20
0.104
0.087
0.073
0.061
0.051
50
0.003
0.002
0.001
0.001
0.001
 The table is known as PV Discount Factor table
[1/(1+r)t]
23
Present Value:
Graphical View
Present Values of £ 1
1.00
i = 0%
0.75
i = 5%
0.50
i = 10%
0.25
i = 15%
Periods
0
2
4
6
8
10
The PV of a sum of money has inverse relation with
the time period and interest rate
24
Present Value:
PV, Time Period and Interest rate
 PV increases when rate of interest falls. The lower the
interest rate higher will be the PV
 Similarly, PV increases when the time period decreases
 In the example of TV purchase above, note that
£1,714.68 invested for 2 years at 8% will prove just
enough to buy your computer
FV=£ 1,714.68 x 1.08 2 =£ 2,000
25
Present Value:
PV, Time Period and Interest rate
 The longer the time period available for payment the
less you need to invest today (i.e. lower PV value). For
example, if the payment for TV was only required in the
third year, the PV would be:
£2,000
PV=
1+0.08
3
= £1,587.66
 The relationship between PV and Interest Rate is also
the same i.e. higher interest rate lower PV
26
Multiple Cash Flow:
FV and PV for uneven Cash Flows
 Most real word investments will involve many cash flows
over time. This is also known as Stream of cash
flows. Calculation of FV and PV of a stream of cash
flows is more important and common in finance
 To calculate the FV of a stream of uneven cash flows,
we may calculate what each cash flow is worth at that
future date, and then add up the values
 To find the PV of a stream of uneven cash flows, we
may calculate what each FV is worth today and then add
up the values
27
Multiple Cash Flow:
Future Value of Uneven Cash Flow
 A security provides the following Cash Flows
End of Year
Cash Flow (£ )
1
100
2
150
3
200
4
250
5
400
 If the interest rate is 10% the FV will be
Year
Cash Flows (£)
10% FV £1
FV
1
100
(1.1)4 = 1.4641
£ 146.41
2
150
(1.1)3
= 1.3310
199.65
3
200
(1.1)2 = 1.2100
242.00
4
250
(1.1)1 = 1.1000
275.00
5
400
(1.1)0 = 1.0000
400.00
FV of uneven CF stream
£ 1,263.06
 The third column above is the factor from FV of £ 1
table
28
Multiple Cash Flow:
Present Value of Uneven Cash Flow
 The PV for the same Cash Flow above assuming same
rate as discount rate will then be
Year
Cash Flows
PV 10%
PV
1
100
0.9091
90.91
2
150
0.8264
123.96
3
200
0.7513
150.26
4
250
0.6830
170.75
5
400
0.6209
248.36
PV of uneven CF stream
£ 784.24
 The third column above is the factor from PV of £ 1
table
29
Multiple Cash Flow:
FV and PV for even Cash Flows
 In finance, it is more common to find a stream of
EQUAL cash flows at same time intervals. For example,
a home mortgage or a car loan might require to make
equal monthly payments for the life of the loan. Any
such sequence of equally spaced, level cash flows is
called an Annuity
30
Multiple Cash Flow:
Annuity and Annuity Due
 Annuity may be Ordinary Annuity and Annuity Due
 In case of an ordinary annuity, each equal payment is
made at the end of each interval of time throughout the
period
 In case of Annuity Due, equal payments are made at the
beginning of each interval throughout the period
31
Multiple Cash Flow:
Annuity and Annuity Due
 For example, if an individual promises to pay £ 1,000 at
the end of each of three years for amortisation of a
loan, then it is called an ordinary annuity. If it were the
annuity due, each payment would be made at the
beginning of each of the three years
Time
0
8%
Ordinary annuity
Time
Annuity due
0
1,000
8%
1
2
1,000
1,000
1
2
1,000
1,000
3
1,000
3
32
Multiple Cash Flow:
Future Value of an Annuity
 Future Value for the Ordinary Annuity (FVA) will be
Time
0
Ordinary annuity
8%
1
2
3
1,000
1000
1,000
1000 × 1.08
1,000 × 1.082
1,080
1,166.4
FVA3 = £ 3,246.4
33
Multiple Cash Flow:
Future Value of an Annuity
 Formula,
C[(1  r)  1]
FVA t 
r
t
where C is the Cash payment from annuity
 If payment value is £ 1, it would be:
[(1  r)  1]
FVA t 
r
t
 The table showing future value of £ 1 for various years
at different interest rates is called FV Annuity table
34
Multiple Cash Flow:
Future Value of an Annuity Due
 Future value for the Annuity Due (FVAD) will be
Time
0
8%
Annuity due 1,000
3
1
2
1,000
1,000
1000 × 1.08
1000 × 1.082
1000 × 1.083
1,080
1,166.4
1,259.71
FVA3 = £ 3,506.11
 Hence,
C[(1  r) t  1]
FVADt 
x (1  r)
r
35
Multiple Cash Flow:
Perpetuity, Delayed Annuity &Annuity
Cash Flow
Year
1
2
3
4
5
6
Investment A
£1
£1
£1
£1
£1
£1…
£1
£1
£1…
Investment B
Investment C
£1
£1
£1
36
Multiple Cash Flow:
Perpetuity
 If the payment stream lasts forever, it is Perpetuity
 Example: Consol
 For a perpetual stream of £C payment every year the PV
= C/r
 So a perpetual stream of £1 payment every year the PV
= 1/r
37
Multiple Cash Flow:
Delayed Perpetuity
 Delayed perpetuity is one in which the payment (C)
only starts after some time t
 PV of a delayed perpetuity for £ 1 payment starting after
time t will be
1
1 
PV of Delayed Perpetuity (£1)   x
t 
 r (1  r) 
38
Multiple Cash Flow:
Present Value of an Annuity
 Annuity is basically a difference between an Annuity
(immediate) and a delayed perpetuity
 PV of an Annuity (t year) will therefore be PV of
immediate Perpetuity – PV of Delayed Perpetuity
starting after t year
1
1
PV of Annuity (£1)   
 r r(1  r)
 1  1  r  
 or 

r

 
t
t
 This is also called t-year annuity factor. The PV table
constructed for various time t and interest rate r for
payment £1 each year is called PV Annuity factor
table
39
Multiple Cash Flow:
PV of an Annuity and Annuity Due
 For Payment £ C every year, PV of t year annuity is
therefore Payment x annuity factor or
1
1
c r r 1+r
t
 Or:
1  1  r  
c

r


t
40
Multiple Cash Flow:
PV of an Annuity and Annuity Due
 For Payment £ C every year, PV of t year annuity is
therefore Payment x annuity factor or
1
1
c r r 1+r
t
 Present value of Annuity Due simply requires the use of
the formula:
PV of ordinary annuity x (1+r)
41
Multiple Cash Flow:
PV of an Annuity and Annuity Due
Cash Flow
Year
1
2
3
4
5
6
Investment A
£1
£1
£1
£1
£1
£1…
1/r
£1
£1
£1…
1/r(1+r) 3
Investment B
Investment C
£1
£1
£1
PV
1/r - 1/r(1+r) 3
42
Multiple Cash Flow:
Amortised Loans
 Loan that is to be repaid in equal periodic installments
including both principal and interest is known as
Amortised Loan
 Let us suppose a loan of £ 10,000 is to be repaid in four
equal installments including principal and 10 percent
interest per annum
 The lender needs to set the payments so that a present
value of £ 10,000 is received
 Present Value = Mortgage Payment x t year
annuity Factor
43
Multiple Cash Flow:
Amortised Loans
 Mortgage Payment =
£10,000
 £ 3,155
 1

1
 .10  .10(1.10) 4 


 The loan amortisation schedule is shown next
44
Multiple Cash Flow:
Amortised Loan Schedule
Year
(1)
1
2
3
4
Beginning
Amount
(2)
£ 10,000.00
7,845.30
5,475.16
2,868.01
Payment
(3)
£ 3,154.67
3,154.67
3,154.67
3,154.67
Interest
(4) = (2) x 0.10
£ 1,000.00
784.53
547.52
286.66*
Repayment of
Principal
(5) = (3) – (4)
£ 2,154.67
2,370.14
2,607.15
2,868.01
Ending Balance
(6) = (2) – (5)
£ 7,845.33
5,475.16
2,868.01
0
45
Cash Flow and Inflation
 Note that cash flows are affected by the level of
inflation in the country
 Generally, the rate of interest quoted in the market are
nominal interest rate which does not take into
consideration the effect of price changes. The real
interest rate can be calculated using the formula:
1  Nominal Interest Rate
1  Real Interest Rate 
1  Inflation Rate
46
Cash Flow and Inflation:
Example
 What is the real interest rate when you deposit £ 1,000
in a bank at interest rate of 5%. Suppose that the
Inflation is 5% as well
 What will it be if interest rate is 6% and inflation is only
2 %?
 Discounting real payment by real interest rate and
Nominal Payment by Nominal interest rate will always
give the same answer. We must not mix up real and
nominal
47
Thank You
48