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What is time value of money?
The idea that money available at the present time is worth more than the same amount in the future due to its potential earning capacityTime value of money means that the value of a sum of money received today is more than its value received after a year sometime. Conversely, the sum of money received in future if less valuable than it is today. -M.Y. Khan & P.K. Jain
Example
Would you prefer to have $1 million now or$1 million 10 years from now?
Of course, we would Of course, we would all prefer the money all prefer the money now!now!
This illustrates that This illustrates that there is an inherent there is an inherent monetary value monetary value attached to timeattached to time..
Benefit of Time Value of money
• Investment benefit– This is because a dollar received today can be invested to earn interest– The amount of interest earned depends on the rate of return that can
be earned on the investment
• Future Uncertainty In future the price of money decreases, so if we received a dollar
today then we do not have to face future uncertainty.• Inflation Inflation means increasing amount but decreasing purchasing power.
If we prefer the present consumption, then do not have to face inflation.
Present value and future value
• Present value The total monetary value today that will be received or paid in future is
termed as present value. Example-If you received $10,000 today, the present value would of course
be $10,000 because present value is what your investment gives you now if you were to spend it today.
• Future value The value that will be received or paid in future for a value paid or received
today is termed as future value. Example-By investing $10,000 today, we are poised to increase the future
value of our money by gaining interest over a period of time.
SIMPLE INTEREST
• Simple Interest The interest that is charged only on the principle amount is
called simple interest.Example-If we invest $100000 with 10% interest rate, then after
2 years the amount will be -
Principle$ $100000*10%
Year -1 10000
Year-2 10000
Total 120000
Compound Interest
• Compound Interest When interest is paid on not only the principal amount
invested, but also on any previous interest earned, this is
called compound interest.Example-If we invest $100000 with 10% , then after 2 years the
amount will be-
Principle amount $100000 * 10%
Year-1 10000
Year-2 11000
Total 121000
Time line
• The line that represents the monetary value of any particular amount according to the time is called time line.
FV=1331
0 1 2 3i=10%
PV=1000
TIME LINE
Compounding cash flows process- The process of finding future value is often referred to
as compounding cash flows. That means to go from present value to future value is called compounded cash flows.
Discounting cash flows process-The process of finding present value is often referred to
as discounting cash flows. That means to go from future value to present value is called discounting cash flows.
Patterns Of Cash FlowThe cash flow-both inflows and outflows-of afirm can be described by general pattern. It can be defined as follows- Single amount Annuity :- Ordinary Due Mixed stream Beginning of the period End of the period
Single amount
The amount that will be received or paid just for a single time period is called single amount.
Example-Include $1000 today paid and $700 received at the end of 5 years.
Formula for single amount-
1.PV= 2.FV=PV(1+i)nFV
(1+i)n
Example of PV
Mr. X wants to know the present value of $1000 that will be received 3 years from now. The interest rate is 10%. So the present value is-
PV=1000/(1+0.1)3
=$970.60
Time line of PV
i=10%
1000
0 1 2 3
PV = ?
Example of FV
If Mr. X places $1ooo in saving accounts paying 10% interest at annually, at the end of the 3 years what will be the amount he will receive?
FV=1000(1+0.1)3 =$1331
Time line of FV
FV = ?
0 1 2 3i=10%
1000
AnnuityThe amount of money will be paid or received after equal time period and in equal amount of money is called annuity. Annuity is two types – 1.Annuity due (Beginning of the period)
A
ii
APVN
AD
1111
ii
iAFV
n
AD
1
)1(
Example of PVAD
• An insurance agent is trying to sell an immediate retirement annuity, which for a single amount paid today will provide with $12000 at the beginning of each year for the next 3 years. Currently earn 9% on low-risk investments comparable to the retirement annuity. Ignoring taxes ,what is the most you would pay for this annuity?
33.33109$
1200009.009.01
1112000
13
ADPV
Time line of PVAD
1 2 30
PV=33109.33
12000 12000 12000 12000
Example of FVAD
Amount of Annuity Interest Rate Deposit Period
$2500 8% 3 Years
Find the future value of annuity-
28.8765$
08.0108.0
)08.01(2500
3
ADFV
Ordinary Annuity
• Ordinary annuity (End of the year):
ii
APVN
OA
111
i
iAFV
n
OA
)1(
Amount of annuity Interest rate Deposit period
$1000 10% 3 years
Example Of PVOA
85.2486$
1.01.01
111000
3
OAPV
So the present value of ordinary annuity-
Time line Of PVOA
1 2 30
10001000 1000
PV= 2486.85
Example of FVOA
• Mr. X plan to make equal, annual , end of year deposits into an account paying 8% annual interest . If he deposit only $600 per year into the account how much will he have accumulated by the end of the 3 years.
84.1947$
08.0
)08.01(600
3
OAFV
Ordinary Annuity Of FVOA
$600$600 $600$600 $600$600
FV=$1947.84
$600 $600 $600
0
600
1 2 3
Mixed stream A stream of cash flow that is not an
annuity, a stream of unequal periodic cash flows that reflect no particular patterns.
Beginning of the year:-
nn
i
FV
i
FV
i
FV
i
FVPV
)1(.........
)1()1()1( 23
12
01
04
13
22
31 )1()1()1()1( iPViPViPViPVFV
Example of PV mixed stream
• For the following mixed stream cash flows shown in the table, determine the present value at the beginning of the 3 year if deposits are made into an account paying annual interest of 8%. Find the present value of mixed stream.
321 )08.01(
1200
)08.01(
1000
)08.01(
900
PV
Year cash flow
1 900
2 1000
3 1200
=$2643.27
Time Line of PV mixed stream
1 2 30
9001000 1200
PV=$2643.27
Example of FV mixed stream
123 )08.01(1000)08.01(700)08.01(500 FV
• For the following mixed stream cash flows shown in the table, determine the future value at the beginning of the 3 year if deposits are made into an account paying annual interest of 8%. Find the future value of mixed stream.
Year cash flow
1 500
2 700
3 1000
=$2706.05
Time Line of FV mixed stream
00
i=8%i=8%
$600$600 $600$600 $600$600
FV =$2706.05
$500 $700 $1000
1 2 3
Mixed stream
• End of the year-
................)1()1()1( 33
22
11
i
FV
i
FV
i
FVPV
05
14
23
32
41 )1()1()1()1()1( iPViPViPViPViPVFV
Example of PV mixed stream
Find the present value at the end of the 3 years with 8% interest rate where mixed stream amount are $600, $800, $1000 respectively
Year Cash flows No. of year of i
PV=FV/(1+i)n
1 600 3-1=2 514.402 800 1 555.563 1000 0 1000
Total amount=$2069.96
Time Line of PV mixed stream
1 2 30
1000 800 600
PV=$2069.96
Example of FV mixed stream
Find the present value at the end of the 3 years with 8% interest rate where mixed stream amount are $2000, $4000, $6000 respectively
Year Cash flows No. of year of i
PV=FV/(1+i)n
1 2000 3-1=2 2289.8
2 4000 1 4280
3 6000 0 6000
Total Amount=$12569.80
Time Line of FV mixed stream
$4000 $6000
FV =$12569.80
$2000
0 1 2 3
Loan Amortization & Schedule
Loan Amortization-• The term loan amortization refers to the determination of
equal periodic loan payments.Amortization Schedule-• A schedule of amortization of payments to repay a loan. It
shows the allocation of each loan payment to interest and principal
Example of Amortization Schedule
• Mr. X borrowed $15000 at a 14% annual rate of interest to be repaid over 3 years. The loan is amortized into three equal, annual, end of year payment. Calculate annual, end of year loan payment and prepare a loan amortization schedule.
)1(1
*
in
iPVA
)14.01(3
1
14.0*15000
=6460.97
Amortization Schedule
End of the year
Loan payment
Beginning principal
Interest payment
Principal payment
Ending principal amount
1 6460.97
15000 2100 4360.97
10639.03
2 6460.97
10639.03
1489.46
4971.51
5667.52
3 6460.97
5667.52
793.45
5667.52
0
