TVM Concepts and Calculations

Learning Objectives
 Understand the concept of the time value of
money.
 Be able to determine the time value of money:
 Future Value.
 Present Value.
 Present Value of an Annuity.
 Future Value of an Annuity.

The Time Value of Money
Would you prefer to
have 1 million dollar now
or
1 million dollar 10 years
from now?
Of course, we would all
prefer the money now!
This illustrates that there
is an inherent monetary
value attached to time.

What Does Time Value of
Money Mean?
 The concept that money available today is worth
more than the same amount of money in the
future
 This preference rests on the Time value of
money.
 Thus, a money received today is worth more
than a money received tomorrow, why?

 This is because that
 a money received today can be invested to earn
interest
 Due to money's potential to grow in value over
time.
 Because of this potential, money that's available
in the present is considered more valuable than
the same amount in the future

 Time Value of Money is dependent not only on
the time interval being considered but also the
rate of discount used in calculating current or
future values.
 Based on this, we can use the time value of
money concept to calculate how much you need
to invest or borrow now to meet a certain future
goal

 Take a simple example:
 You need to sell your car and you receive offers from
three different buyers.
 The first offer gives 4,000 Birr to be paid now
 The second offer gives 4,100 Birr to be paid one year from
now
 The third offer gives 4,600 Birr to be paid after five years.
 Assume that the second and third offers have no credit
risk.
 The risk free interest rate is 5%.
 Which offer would you accept?

Interest
 A rate which is charged or paid for the use of money.
An interest rate is often expressed as an annual percentage of the
principal.
 a nominal interest rate, is one where the effects of inflation have
not been accounted for.
 Real interest rates are interest rates where inflation has been
accounted for
 Changes in the nominal interest rate often move with changes in
the inflation rate, as lenders not only have to be compensated for
delaying their consumption, they also must be compensated for the
fact that a dollar will not buy as much a year from now as it does
today.

What are the components of interest rate?
 Real Risk-Free Rate - This assumes no risk or uncertainty, simply
reflecting differences in timing.
 Expected Inflation - The market expects aggregate prices to rise, and the
currency's purchasing power is reduced by a rate known as the inflation
rate.
 Default-Risk Premium - What is the chance that the borrower won't
make payments on time, or will be unable to pay what is owed? This
component will be high or low depending on the creditworthiness of the
person or entity involved.
 Liquidity Premium- Some investments are highly liquid, meaning they
are easily exchanged for cash. Other securities are less liquid, and there
may be a certain loss expected if it's an issue that trades infrequently.
 Holding other factors equal, a less liquid security must compensate the
holder by offering a higher interest rate.
 Maturity Premium - Other things being equal, a bond obligation will be
more sensitive to interest rate fluctuations, if maturity period is longer.

Simple Interest and Compound Interest
When you deposit money into a bank, the bank
pays you interest.
When you borrow money from a bank, you pay
interest to the bank.
Simple interest is
money paid only on
the principal.
Principal is the amount of
money borrowed or invested.
Rate of interest is the
percent charged or
earned.
Time that the money
is borrowed or
invested (in years).
I = P  r  t

Compound interest
Compound interest is interest paid not only on
the principal, but also on the interest that has
already been earned. The formula for compound
interest is below.
“A” is the final dollar value, “P” is the principal,
“r” is the rate of interest, and “t” is the number
of compounding periods per year.
 t
r
p
A 
 1

Uses of Time Value of Money
 Time Value of Money, or TVM, is a concept that is
used in all aspects of finance including:
 Bond valuation
 Stock valuation
 Accept/reject decisions for project management
 Financial analysis of firms
 And many others!

Types of TVM Calculations
 There are many types of TVM calculations
 The basic types that will be covered are:
 Future value of a lump sum
 Present value of a lump sum
 Present and future value of cash flow stream
 Present and future value of annuities

Basic Rules
 The following are simple rules that you should always
use no matter what type of TVM problem you are trying
to solve:
1. Stop and think: Make sure you understand what the
problem is asking.
2. Draw a representative timeline and label the cash
flows and time periods appropriately.
3. Write out the complete formula using symbols first
and then substitute the actual numbers to solve.
4. Check your answers using a calculator.

Example
 How much money will you have in 5 years if
you invest $100 today at a 10% rate of return?
 Draw a timeline
 Write out the formula using symbols:
FVt = CF0 * (1+r)t
0 1 2 3
$100 ?
i = 10%
4 5

Future Value
 Future value is the value of an asset at a
specific date in the future.
 It measures the nominal future sum of money
that a given sum of money is "worth" at a
specified time in the future assuming a certain
interest rate, or more generally, rate of return.
 Actually, the future value does not include
corrections for inflation or other factors that
affect the true value of money in the future.

Future Value Equation
 FVn = PV(1 + i)n
 FV = the future value of the investment at
the end of n year
 i = the annual interest rate
 PV = the present value, in today’s dollars,
of a sum of money
 This equation is used to determine the value
of an investment at some point in the future.

Future Value of a Lump Sum
 Future value determines the amount that a sum
of money invested today will grow to in a given
period of time
 You can think of future value as the opposite of
present value
 The process of finding a future value is called
“compounding”

Example of FV of a Lump Sum
 How much money will you have in 5 years if you invest $100 today
at a 10% rate of return?
1. Draw a timeline
2. Write out the formula using symbols:
FVt = CF0 * (1+r)t
0 1 2 3
$100 ?
i = 10%
4 5

Example of FV of a Lump Sum
3. Substitute the numbers into the formula:
FV = $100 * (1+.1)5
4. Solve for the future value:
FV = $161.05

Future Value of a Cash Flow
Stream
 The future value of a cash flow stream is equal to the
sum of the future values of the individual cash flows.
 The FV of a cash flow stream can also be found by
taking the PV of that same stream and finding the FV
of that lump sum using the appropriate rate of return
for the appropriate number of periods.

 The following equation can be used to find the
Future Value of a Cash Flow Stream at the end
of year t.
 where
 FVt = the Future Value of the Cash Flow Stream at the end of
year t,
 CFt = the cash flow which occurs at the end of year t,
 r = the discount rate,
 t = the year, which ranges from zero to n, and
 n = the last year in which a cash flow occurs

 For example, consider an investment which
promises to pay $100 one year from now, $300
two years from now, 500 three years from now
and $1000 four years from now. How much will
be the future value of the cash flow streams at
the end of year 4, given that the interest rate is
10%,?
 The Future Value at the end of year 4 of the
investment can be found as follows in the next
slide:

1. Draw a timeline:
0 1 2 3 4
$100 $300 $500 $1000
i = 10%
?
?
?
?

2. Write out the formula using symbols
n
FV = S [CFt * (1+r)n-t]
t=0
OR
FV = [CF1*(1+r)n-1]+[CF2*(1+r)n-2]+ [CF3*(1+r)n3] +
[CF4*(1+r)n-4]
3. Substitute the appropriate numbers:
FV = [$100*(1+.1)4-1]+[$300*(1+.1)4-2]+[$500*(1+.1)4-3]
+[$1000*(1+.1)4-4]

4. Solve for the Future Value:
FV = $133.10 + $363.00 + $550.00 + $1000
FV = $2046.10
5. Check using the calculator:
 Make sure to use the appropriate interest rate, time
period and present value for each of the four cash
flows. To illustrate, for the first cash flow, you should
enter PV=100, n=3, i=10, PMT=0, FV=?. Note that
you will have to do four separate calculations.

Exercise
 Find the Future Value at the end of year 4 of
the following cash flow stream given that the
interest rate is 10%.

 Solution:


Annuities
 An annuity is a cash flow stream in which the cash
flows are all equal and occur at regular intervals.
 To considered as annuity the following conditions must
be present
 The periodic payment must be equal in amount
 The time period between payments should be constant
 The interest rate per year remains constant
 The interest is compounded at the end of each time period

 There are different types of annuities, among
these are
1. An ordinary annuity is one where the payments are
made at the end of each period
2. An annuity due is one where the payments are made
at the beginning of each period
3. A deferred annuity is one where the payments do not
commence until period of times have elapsed
4. A perpetuity is an annuity in which the payments
continue indefinitely
Types of Annuities

Ordinary Annuity
 Ordinary Annuity
 The amount of an ordinary annuity (annuity in
arrears) consists of the sum of the equal periodic
payments and compounded interest on the payments
immediately after the final payments.
 The future value (or accumulated value) of an annuity
is the amount due at the end of the term
 FV of ordinary annuity, it is the sum of all the
periodic payments made and interest accrued up to
and including the final payment period

 Where
 CF = Cash flow per period
 r = interest rate
 n = number of payments
 





 


r
1
r
1
CF
FV
n
nuity
ordianryAn
  r
n
A ACF
PMT
FV ,
 Using table

 The formula for future value of annuity is derived as follows:
FVAn = A(1+r)n-1+A(1+r)n-2+……A(1+r) + A…....1
 Multiply both sides by (1+r), it gives
 FVAn(1+r) = [A(1+r)n-1+A(1+r)n-2+……A(1+r) + A](1+r)
 FVAn + FVAnr = A(1+r)n + A(1+r)n-1+A(1+r)n-2…A(1+r)….2
 Subtract 1 from 2, the result will be
 FVAn + FVAnr = A(1+r)n + A(1+r)n-1+A(1+r)n-2……A(1+r)
 - FVAn = A(1+r)n-1 - A(1+r)n-2+……-A(1+r) - A
 FVAnr = A(1+r)n – A
 Divide both sides by r
 FVAnr = A(1+r)n – A =
 r r
 





 


r
r
A
FVA
n
n
1
1

 Example: What amount will accumulate if we deposit $5,000
at the end of each year for the next 5 years? Assume an
interest of 6% compounded annually
PV = 5,000
i = .06
n = 5
Year 1 2 3 4 5
Begin 0 5,000.00 10,300.00 15,918.00 21,873.08
Interest 0 300 618 955.08 1,312.38
Deposit 5,000.00 5,000.00 5,000.00 5,000.00 5,000.00
End 5,000.00 10,300.00 15,918.00 21,873.08 28,185.46

 





 


06
.
0
1
06
.
0
1
5000
5
oa
FV
 





 

06
.
0
1
06
.
1
5000
5
oa
FV





 

06
.
0
1
3382255776
.
1
5000
oa
FV







06
.
0
3382255776
.
0
5000
oa
FV
 
63709296
.
5
5000

oa
FV
4648
.
28185

oa
FV
 





 


r
1
r
1
CF
FV
n
nuity
ordianryAn

Annuity Due
 A form of annuity where periodic receipts or
payments are made at the beginning of the period
and one period of the annuity term remains after
the last payment.

 The Future Value of an Annuity Due is identical to an
ordinary annuity except that each payment occurs at the
beginning of a period rather than at the end.
 Since each payment occurs one period earlier, we can calculate
the present value of an ordinary annuity and then multiply the
result by (1 + i).
Where:
FVad = Future Value of an Annuity Due
FVoa = Future Value of an Ordinary Annuity
i = Interest Rate Per Period
 
i
FV
FV oa
ad 
 1
 
i
FV
FV oa
ad 
 1

 Example: What amount will accumulate if we deposit $5,000 at
the beginning of each year for the next 5 years? Assume an
interest of 6% compounded annually.
 PV = 5,000, i = .06, n = 5
 FVoa = 28,185.46 (1.06) = 29,876.59
Year 1 2 3 4 5
Begin 0 5,300.00 10,918.00 16,873.08 23,185.46
Interest 0 300 618 955.08 1,312.38
Deposit 5,000.00 5,000.00 5,000.00 5,000.00 5,000.00
End 5,000.00 10,300.00 15,918.00 21,873.08 28,185.46
Begin 5,000.00 10,300.00 15,918.00 21,873.08 28,185.46
Deposit 5,000.00 5,000.00 5,000.00 5,000.00 5,000.00
Interest 300 618 955.08 1,312.38 1,691.13
End 5,300.00 10,918.00 16,873.08 23,185.46 29,876.59

 





 


06
.
0
1
06
.
0
1
5000
5
ad
FV
 
06
.
1
 





 

06
.
0
1
06
.
1
5000
5
ad
FV  
06
.
1





 

06
.
0
1
3382255776
.
1
5000
ad
FV  
06
.
1







06
.
0
3382255776
.
0
5000
ad
FV  
06
.
1
 
63709296
.
5
5000

oa
FV  
06
.
1
  
06
.
1
4648
.
28185

ad
FV
59
.
876
,
29

ad
FV

 Example: What amount will accumulate if we deposit
$1,000 at the beginning of each year for the next 5
years? Assume an interest of 5% compounded
annually.
 PV = 1,000
i = .05
n = 5

 Or
= $1000*5.53*1.05
= $5801.91

Future value of Deferred Annuity
 When the amount of an annuity remains on
deposit for a number of periods beyond the
final payment, the arrangement is known as a
deferred annuity.
 When the amount of an ordinary annuity continues
to earn interest for an additional one year period we
have an annuity due situation
 When the amount of an ordinary annuity continues
to earn interest for more than one additional periods,
we have a deferred annuity situation.

 Where
 FV = Future Value
 P = regular Payments
 ACF = Annuity cash flow
 SCF= Single cash flow
 N = the Number of compounding periods
 M = The number of deferred periods
 










1
1
m
n
i
i
P
FV
 








 1
1
i
i
p
m
    i
m
i
n SCF
ACF
P
FV ,
,

   
 
i
m
i
m
n ACF
ACF
P
FV ,
, 
 
Formula to calculate future value of deferred annuity

 Example: What amount will accumulate if we deposit
$5,000 at the beginning of each year for
the next 5 years and wait to get the amount
for additional 2 years. Assume an interest
of 6% compounded annually.
PV = 5,000
i = .06
n = 5
m=2

 







 

06
.
0
1
06
.
1
5000
7
da
FV
 







 

06
.
0
1
06
.
1
5000
2





 

06
.
0
1
503630
.
1
5000
da
FV 




 

06
.
0
1
1236
.
1
5000














06
.
0
1236
.
0
5000
06
.
0
503630
.
0
5000
da
FV
   
06
.
2
5000
3938
.
8
5000 

da
FV
10300
41969

da
FV
31669

da
FV
 





 



i
i
P
FV
m
n
1
1  





 


i
i
p
m
1
1
   





 









 



06
.
0
1
06
.
0
1
5000
06
.
0
1
06
.
0
1
5000
2
2
5
FV

Application
 The future value annuity formula can be applied in
different contexts.
 Its important applications are
 To know how much we have in the future
 FV= A(1+r)n
 To know how much to save annually
 To find out the interest rate
 To know how long should wait to get the accumulated
money
  1
1 

 n
r
r
FV
A
n
FV
i 
n
FV
i 
r
A
FV
r n
log
)
1
log( 


What does mean present
value of money?
• It is a process of discounting the
future value of money to obtain its
value at zero time period (at
present)
• Present values tell you the amount
you must invest today to
accumulate a certain amount at
some future time

• To determine present values, we need to know:
– The amount of money to be received in the future
– The interest rate to be earned on the deposit
– The number of years the money will be invested
 Discounting and Compounding
The mechanism for factoring in the present value of
money element is the discount rate.
The process of finding the equivalent value today of a
future cash flow is known as discounting.
Compounding converts present cash flows into future cash flows.

Calculating the Present
Value
• So far, we have seen how to calculate
the future value of an investment
• But we can turn this around to find
the amount that needs to be invested
to achieve some desired future value:

• Using the Present Value Table
– Present value interest factor (PVIF):
a factor multiplied by a future value to
determine the present value of that
amount (PV = FV(PVIFA)
– Notice that PVIF is lower as the number
of years increases and as the interest
rate increases
• It can also be calculated using a
financial calculator
PV = [FV/(1+r)n]

Cash Flow Types and
Discounting Mechanics
• There are five types of cash flows -
– single cash flows (Lump sum cash
flows),
– Cash flows stream
– annuities,
– growing annuities and
– perpetuities

I. Single Cash Flows
• A single cash flow is a single cash
flow in a specified future time
period.
• Cash Flow: CFt
_______________________|
Time Period: t
• The present value of this cash flow
is-
PV of Single Cash Flow =
  








 t
t
r
CF
PV
1

• You would like to accumulate Birr
50,000 in five years by making a single
investment today. You believe you can
achieve a return from your investment
of 8% annually.
• What is the amount that you need to
invest today to achieve your goal?
Example: Present Value of a
single cash flow:

16
.
34029
)
08
.
1
(
000
,
50
5


PV
Or
We can use table
PV = 50,000(PVIF)5,8%
=50,000 x 0.681
= 34050

II. Valuing a Stream of Cash
Flows
• Valuing a lump sum (single) amount is easy to
evaluate because there is one cash flow.
• What do we need to do if there are multiple cash
flow?
– Equal Cash Flows: Annuity or Perpetuity
– Unequal/Uneven Cash Flows
     
 














N
t
t
t
N
N
i
FV
PV
i
FV
i
FV
i
FV
PV
1
2
2
1
1
1
1
1
1

Valuing a Stream of Cash
Flows
0 1 2 3
? 1000 2000 3000
• Uneven cash flows exist when there are
different cash flow streams each year
• Treat each cash flow as a Single Sum
problem and add the PV amounts together.

• What is the present value of the preceding
cash flow stream using a 12% discount rate?
=
Yr 1 $1,000 / (1.12)1 = $ 893
Yr 2 $2,000 / (1.12)2 = 1,594
Yr 3 $3,000 / (1.12)3 = 2,135
$4,622
 




n
t
n
t
r
FV
PV
0 1      
 
4622
2135
1594
893
12
.
0
1
300
12
.
0
1
200
12
.
0
1
100
3
0
3
2
1

 


 













PV
PV
PV
n
t

III. Annuities
• The present value of an annuity is the
value now of a series of equal amounts to
be received (or paid out) for some
specified number of periods in the future.
• It is computed by discounting each of the
equal periodic amounts.
• An annuity is a series of nominally equal
payments equally spaced in time
• The timeline shows an example of a 5-
year, $100 annuity
0 1 2 3 4 5
100 100 100 100 100

Present Value of an Annuity
• The present value of an annuity can be
calculated by taking each cash flow and
discounting it back to the present, and adding
up the present values.
• We can use the principle of value additivity to
find the present value of an annuity, by simply
summing the present values of each of the
components:
       
PV
Pmt
i
Pmt
i
Pmt
i
Pmt
i
A
t
t
t
N
N
N






  


 1 1 1 1
1
1
1
2
2

(cont.)
• Alternatively, there is a short cut that can
be used in the calculation [A = Annuity; r =
Discount Rate; n = Number of years]
• Thus, there is no need to take the present
value of each cash flow separately
• The closed-form of the PVA equation is:















r
r
FV
PV
N
)
1
(
1
1

(cont.)
• Using the example, and assuming a
discount rate of 10% per year, we find
that the present value is:
         
PVA      
100
110
100
110
100
110
100
110
100
110
379 08
1 2 3 4 5
. . . . .
.
0 1 2 3 4 5
100 100 100 100 100
62.09
68.30
75.13
82.64
90.91
379.08

• We can use this equation to find the present
value of our annuity example as follows:















1
.
0
)
1
.
0
1
(
1
1
100
5
PV











 

1
.
0
)
1
.
1
(
1
1
100
5
PV







1
.
0
3791
.
0
100
PV  
791
.
3
100

PV
1
.
379

PV
 This equation works for all regular (ordinary)
annuities, regardless of the number of payments

Annuities Due
• The annuities that we begin their
payments at the end of period 1 are
referred as regular annuities (ordinary
annuities)
• An annuity due is the same as a regular
annuity, except that its cash flows occur at
the beginning of the period rather than at
the end
0 1 2 3 4 5
100 100 100 100 100
100 100 100 100 100
5-period Annuity Due
5-period Regular Annuity

Present Value of an Annuity Due
• The formula for the present value of an annuity due,
sometimes referred to as an immediate annuity, is used to
calculate a series of periodic payments, or cash flows, that
start immediately
• We can find the present value of an annuity due in the same
way as we did for a regular annuity, with one exception
• Note from the timeline that, if we ignore the first cash flow,
the annuity due looks just like a four-period regular annuity
• Therefore, we can value an annuity due with:
CF
i
i
CF
PV
n
















1
)
1
(
1
1

Due (cont.)
• Therefore, the present value of our
example annuity due is:
 Note that this is higher than the PV of the
regular annuity
  98
.
416
100
1
.
0
1
.
0
1
1
1
100
1
5


















AD
PV

Deferred Annuities
• A deferred annuity is the same as
any other annuity, except that its
payments do not begin until some
later period
• The timeline shows a five-period
payment deferred annuity
0 1 2 3 4 5
100 100 100 100 100
6 7

PV of a Deferred Annuity
• We can find the present value of a
deferred annuity in the same way as any
other annuity, with an extra step required

PV deferred annuity formula
   







































i
i
i
i
CF
PV
m
m
n
DA
1
1
1
1
1
1
OR
 
  




























 m
n
DA
i
i
i
CF
PV
1
1
1
1
1

PV of a Deferred Annuity
(cont.)
• To find the PV of a deferred annuity, we
can first calculate the ordinary PVA
equation, and then discount that result
back to period 0
• Here we are using a 10% discount rate
0 1 2 3 4 5
0 0 100 100 100 100 100
6 7
PV2 = 379.08
PV0 = 313.29

PV of a Deferred Annuity (cont.)
 
  




























 2
5
1
.
0
1
1
1
.
0
1
.
0
1
1
1
100
DA
PV
  










 
 2
1
.
1
1
1
.
0
6209
.
0
1
100
DA
PV
 












21
.
1
1
1
.
0
3791
.
0
100
DA
PV
  
8264
.
0
791
.
3
100

DA
PV
  
29
.
313
8264
.
0
1
.
379


AD
AD
PV
PV

Or we can calculate with two steps
Step 1:
Step 2:
 























1
.
0
1
.
0
1
1
1
100
5
2
PV





 

1
.
0
6209
.
0
1
100
2
PV







1
.
0
3791
.
0
100
2
PV
1
.
379
2 
PV
 
29
.
313
21
.
1
1
.
379
1
.
1
1
.
379
2
0 








PV

Uneven Cash Flows
• Very often an investment offers a
stream of cash flows which are not
either a lump sum or an annuity
• We can find the present or future
value of such a stream by using the
principle of value additivity

Uneven Cash Flows: An
Example (1)
• Assume that an investment offers the
following cash flows. The required return
is 7%, what is the maximum price that
you would pay for this investment?
0 1 2 3 4 5
100 200 300
     
PV    
100
107
200
107
300
107
51304
1 2 3
. . .
.

Uneven Cash Flows: An
Example (2)
• Suppose that you need to deposit the
following amounts in an account paying 5%
per year. What should be the balance of the
account at the end of the third year?
0 1 2 3 4 5
300 500 700
   
FV   
300 105 500 105 700 155575
2 1
. . , .

Present value of a Growing
Annuity
• A cash flow that grows at a constant
rate for a specified period of time is a
growing annuity
• A time line of a growing annuity is as
follows
     
n
g
A
g
A
g
A
n
......
2
1
0
1
.....
1
1
2




• The present value of a growing annuity can be
calculated using the following formula
• The above formula can be used when
– The growing rate is less than the discount rate
(g<r) or
– The growing rate is more than the discount
rate (g>r)
– However, it doesn’t work when the growing
rate is equal to the discount rate (g=r)
 
 
 


















g
r
r
g
g
A
PV
n
n
ga
1
1
1
1

Example:
• Suppose you have the right to harvest a
tea plantation for the next 20 years
over which you will get 100,000 tons of
tea per year. The current price per ton
of tea is Birr 500, but it is expected to
increase at a rate of 8% per year. The
discount rate is 15% per year.
• Then, how much is the present value of
the tea that you can harvest?

Solution:
  
 
 
683
,
736
,
551
08
.
0
15
.
0
15
.
0
1
08
.
0
1
1
08
.
0
1
000
,
100
500
20
20
Birr
PV
X
Birr
PV
ga
ga




















IV. Present value of perpetuity annuity
• A perpetuity is an annuity of with an infinite
duration.
• Hence the present value of perpetuity may be
expressed as follows
PV∞ = CF x PVIFA
• Where PV∞ = present value of a perpetuity
• CF = constant annual cash flows
• PVIFA = present value interest factor for
perpetuity (an annuity of infinite
duration)
– The value of PVIFA is
  r
r
PVIFA
t
t
1
1
1
1


 



TVM Concepts and Calculations

Recommended

Recommended

More Related Content

Similar to TVM Concepts and Calculations

Similar to TVM Concepts and Calculations (20)

Recently uploaded

Recently uploaded (20)

TVM Concepts and Calculations