1. Time Value of Money
$$$ n $ %
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2. What is Time value of money?
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Worth of a rupee received today is different from the worth of rupee to be received in future.
We say that money has a time value because that money can be invested with the expectation of earning a positive rate of return

3. Time lines
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Time lines visualize the what's happening in particular point of time Periods 0 5% Cash PV= $100 FV=? 0 to 1 , 1 to 2 and 2 to 3 are time interval, time 0 is today , and it is beginning of the period; time 1 is period one from today; time 2 is period 2 from today and so on.

4. Future Value
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A dollar in hand today is worth more than a dollar to be received in the future. Because , if you had it now, you could invest it, earn interest and end up with more dollar in future. For e.g. if you plan to deposit $100 in a bank that pays a guaranteed 5% interest each year. How much would you have at the end of year 3?

5. Step-by-Step Approach
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In this approach time line is used to find future value
Time 0 5%
Amount of beginning period $100  $  $  $115.50
At beginning, Present value (PV)= 100, time (t)= 0 , interest rate (I) = 5% multiply the initial amount, and each succeeding amount by (1+I) i.e. ( = )
Amount of interest earned in first year is, $100(0.05)=$5 , so the amount at the end of year 1 (t=1) is,
FV1 = PV+INT
= PV +PV(I)
= PV(1+I) = $100( ) = $100(1.05) = $105

6. In second year, interest amount is $105(0. 05)= $5
In second year, interest amount is $105(0.05)= $5.25 Now, FV2 = FV1(1+I) = PV(1+I)(1+I) = PV(1+I)2 = 100(1.05)2 = $ If N=3 , then we multiply PV by (1+I) three different times , FV3 = FV2(1+I) = PV(1+I)(1+I)(1+I) = PV(1+I)3 = 100(1+0.05)3 = $ Now we can define Future value as, FVn = PV(1+I)n (1)
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7. Formula Approach
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We can apply the equation -1 above in this approach, in our example for third year, FV3 = $100(1+0.05)3 = $100(1.05)3 = $115.76

8. Present Value
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The present value of cash flow due n years in the future is the amount which , if it were on hand today, would grow to equal the future amount.
Finding the present value is discounting and it is reverse of compounding
Formula for present value can be develop simply by solving equation 1 above;
Future Value = FVn = PV(1+I)n
Present Value = PV = FVn (1+I)n

9. For example Suppose Us government bond promises to pay $ 115
For example Suppose Us government bond promises to pay $ three years from now. If the interest rate on 3 –years government bond is 5% , how much is the bond worth today? Solution, Given, Future value (FV) = $ Interest Rate (I) = 5% No. of Periods (n) = 3 years Periods: Cash flow time line PV =?  ?  ?  $115.76
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10. By formula , Present Value (PV) = FVn (1+I)n = 115.76 (1+0.05)3 = 115.76 (1.05)3 = $100
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11. How to find Interest Rate?
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FVn = PV(1+I)n The equation above has 4 variables, if we know three of them we can solve one. If we know PV, I and n , we can solve for FV , now suppose we know PV, FV and n , we need to find I. For example, a security has cost of $100 and that it will return $150 after 10 years. What is interest rate? Solution, Given , Present value (PV) = $100 Future Value (FV) = $150 No. of Years (n) = 10 years Interest Rate (I) = ?

12. From equation, FVn = PV(1+I)n $150 = $100(1+I)10 $150/$100 = (1+I)10 (1+I)10 = 1.5 (1+I) = 1.5(1/10) (1+I) = I = – 1 = 4.14%
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13. How to Find Number of Years?
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We sometimes need to know how long it will take to accumulate a required sum of money.
For example,
How long would it take $1000 to be double if it were invested in a bank that pays 6% per year?
Solution,
Given, Present value (PV) = $1000
Future Value (FV) = $2000
Interest rate (I) = 6% p.a.
No. of Years (n) = ?

14. from equation 1 FVn = PV(1+I)n $2000 = $1000(1+0
from equation 1 FVn = PV(1+I)n $2000 = $1000(1+0.06)n $2000/$1000 = (1+0.06)n 2 = (1+0.06)n By taking log both sides, we can find value of n, ln (2) = n [ln(1.06)] n = ln (2)/ln(1.06) n = / n = 11.9 years Therefore, it takes $1000 about 11.9 years to grow to $2000 if the interest rate is 6%
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15. Annuities
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Many assets provides the series of cash inflow over time and many obligations like auto loan, students loan requires a series of payment.
If payments are equal and are made at fixed intervals , then the series is an annuity.
If the payment occur at the end of each period then it is ordinary annuity. For example: Student loan, Car loans etc.
Here is the time line for a $100, 3-years, 5% ordinary annuity
Periods 0 5%
Payments $ $ $100

16. here is the time line for annuity due periods 0 1 2 3
If the payments are made at the beginning of each period then it is annuity due. For example: life insurance premiums, rental payments for apartments
here is the time line for annuity due
periods
Payments -$100 -$100 -$100
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17. Future Value of an Ordinary Annuity
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An ordinary annuity is a series of equal payments, with all payments being made at the end of each and successive period. The future value of annuity due formula is used to calculate the ending value of a series of payments or cash flows where the first payment is received immediately. An example of an ordinary annuity is a series of rent or lease payments. For example, if we deposit $100 at the end of the each year for 3 years and earn 5% per year. What is the future value? Payment amount (PMT) = $100 Interest Rate (I) = 5% Number of periods (n) = 3

18. Periods: 0 1 2 3 Cash flow time line -$100 -$100 -$100 -$100. 00 -$105
Periods: Cash flow time line -$100 -$100 -$100 -$ $ $ As we can see from the time line diagram, with step by step approach, FVA = PMT(1+I)n-1 + PMT(1+I)n-2 + PMT(1+I)n-3 = $100(1.05)2 + $100(1.05)1 + $100(1.05)0 = $315.25
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19. The formula for calculating the future value of an ordinary annuity is: FVAn = PMT (1+I)n I − 1 I where, FVA = The future value of the annuity stream to be paid in the future PMT = The amount of each annuity payment I = The interest rate n = The number of periods over which payments are to be made For example, if we deposit $100 at the end of the each year for 3 years and earn 5% per year Payment amount (PMT) = $100 Interest Rate (I) = 5% Number of periods (n) = 3 From formula, FVA = $100 (1+0.05) − = $315.25
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20. Present Value of an Ordinary Annuity
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The present value calculation for an ordinary annuity is used to determine the total cost of an annuity if it were to be paid right now.
We discount each payment back to time 0, then sum those discounted values to find annuity’s present value
For example, if we deposit $100 at the end of the each year for 3 years and earn 5% per year. What is the present value?
Payment amount (PMT) = $100
Interest Rate (I) = 5%
Number of periods (n) = 3

21. Periods: 0 1 2 3 Cash flow time line -$100 -$100 -$100 $95. 24 $90
Periods: Cash flow time line -$100 -$100 -$100 $95.24 $90.70 $86.38 $ As we can see from the time line diagram, with step by step approach, PVA = PMT/(1+I)1 + PMT/(1+I)2 + PMT/(1+I)3 = $100/(1.05)1 + $100/(1.05)2 + $100/(1.05)3 = $272.32
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22. The formula for calculating the present value of an ordinary annuity is: PVAn = PMT 1 I − 1 I(1+I)n where, PVA = The future value of the annuity stream to be paid in the future PMT = The amount of each annuity payment I = The interest rate n = The number of periods over which payments are to be made For example, if we deposit $100 at the end of the each year for 3 years and earn 5% per year. What is present value? Payment amount (PMT) = $100 Interest Rate (I) = 5% Number of periods (n) = 3 From formula, PVA = $ − (1+0.05)3 = $272.32
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23. Perpetuities
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A perpetuity is simply an annuity with an extended life. Since the payment go on forever.
we can find PV of perpetuity with following formula
PV of perpetuity = PMT I
Where, PMT= regular payment
I = Interest rate
For example: Find the value of British consol with face value of $1000 that pays $25 per in perpetuity. In 1888, the going interest in financial market place was 2.5%, so at the time of consol’s value was $1000.
Consol value1888 = $ = $1000

24. consol value if rates fall to 2% = $25 0.02 = $1250
In 2006, 118 years later, annual payment was still $25 , but going interest rate had risen to 5.2% ,
Consol’s value2006 = $ = $480.77
If the interest rates decline in future say, 2% , the value of consol will rise:
consol value if rates fall to 2% = $ = $1250
We can conclude that, when interest rates change the price of outstanding bonds will also change. Bond prices decline if rates rise and increase if rates fall.
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25. Future value of an uneven cash flow stream
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Future value of uneven cash flow stream is the sum of the future value of each cash flow compounded to the end of the stream at required rate of return. It is calculated by using the following relationship:
FVn = CF1(1+i)n-1 + CF2(1+i)n-2 + CF3(1+i)n CFn(1+i)0

26. For instance, let us suppose a security provides the following stream of cash flows till its maturity in five years.
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End of year 1 2 3 4 5
Cash flow ($)
100
150
200
250
400

27. In above future value calculation the first year end cash flow is compounded for four years the second year end cash flow is compounded for three years and so on.
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year
Cash flows($)
10%FVIF
FV($) 1
100
(1.1)4 =
146.41 2
150
(1.1)3 =
199.41 3
200
(1.1)2=1.2100
242.00 4
250
1.1)1 =
275.00 5
FV OF UNEVEN CASH FLOW
400
(1.1)0= 1
$400 $

28. Compounding Periods of Interest Rate
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Annual compounding(simple or quoted rate):interest is compounding once a year.
Semiannual compounding: if interest is paid semiannually, it is compounded two times during the year.
Quarterly compounding: interest is compounded four times during the year.
The number of compounding periods during the year is denoted as ‘m’.
For any compounding period less than one year, we make following two changes in all our present and future value calculation:
first, the rate of interest (i) is divided by the number of compounding periods (m) during the year;
second, the number of years that cash flow occurs (n) is multiplied by the number of compounding periods during the year.

29. The Result of Frequent Compounding
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Simply interests are compounded annually, semi annually, quarterly and monthly. Then what will happen to the future value of investment if interest is compounded more frequently then once a year? To determine the variations that occur with different compounding assumption. For example: 30th January, 2016 a single deposit of $10000 is made for two years at an interest rate of 12%.what will be the result n investment after maturity period?(if the interest is compounded annually, semi annually, quarterly and daily)

30. Compounding annually:
It receives two interest deposits- one at the end of each year(n)=2,the annual interest rate(r)=12% per year, present value of investment (PV)=10000.
therefore, future value of investment is equal to
FV = PV (1 + i)n
=10000(1+0.12)2 = $12544
Compounding semi annually:
It receives four interest deposits-one at the end of each six- month period(n = 4),annual interest rate =12% so, semi- annual interest rate = 6%
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31. Compounding quarterly:
Future value of investment when interest compounding semiannually = (1+0.06)^4=$
Compounding quarterly:
It receives eight interest deposits- one at the end of each 3 months(n)=8,the quarterly interest rate (i)=3% ,present value of investment (PV)=10000.
therefore, future value of investment is equal to
FV = PV (1 + i)n
= $10000(1+0.03)8
= $
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32. therefore, future value of investment is equal to FV = PV (1 + i)n
Compounding monthly:
It receives 24 interest deposits- one at the end of each months (n)=24,the monthly interest rate(r)=1% per ,present value of investment (PV)=10000.
therefore, future value of investment is equal to
FV = PV (1 + i)n
= $10000(1+0.01)24
= $
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33. The Impact of Frequent Compounding
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Compounding
In one year (n)
Interest per period
Future value of invested for 2 years($)
Annual 1
12%
11200
Semiannual 2 6%
11236
Quarterly 4 3%
11255
Monthly 12 1%
11268

34. More frequently then once a year?
Then question is arise why the future value of investment is greater if interest is compounded more frequently then once a year
More frequently then once a year?
Because interest will be earned on interest more often, so the future value of investment increase as the frequency of compounding increase
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35. Types of Interest Rates
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1. Simple or quoted interest rate:
The rate of interest, which is quoted by borrowers and lenders, is known as simple or quoted interest rate
The practitioners in the stock, bond, commercial loan, banking and finance company’s loan express all financial contracts in terms of simple loan.
2. Periodic rate:
The rate of interest charged by lender or paid by borrower at each interest period is known as periodic rate of interest.
It can be stated as interest rate per year or interest rate per six month, or per quarter or per month and so on.

36. Periodic Rate (iPER) = i simple/M ( M=NO.OF COMPOUNDING IN A YEAR)
Periodic rate is calculated as simple interest rate divided by number of period in a year as given in equation (3.23).
Periodic Rate (iPER) = i simple/M
( M=NO.OF COMPOUNDING IN A YEAR)
3. Effective annual rate:
This is the annual rate that produces the same result as if we had compounded at a given period rate M times per year
Effective annual rate(Eff%)=(1+Inom/m)^m-1
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37. Amortized loan
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Amortized loan refers to the loan that is to be repaid in equal periodic installments including both principal and interest. The concepts of present value and compound interest rate are used to amortize a loan over the time in equal installments.
Let us suppose a loan of Rs 10,000 is to be repaid in 4 equal installments including principal and 10 percent interest per annum.
We apply the following steps to determine the annual payment and set up an amortization schedule of the loan.

38. Determining annual payment PMT = Amount of loan/(PVIFA i,n)
= $10‚000/3.1699
= $3,154.67
It means an installment of $3, paid annually for four years will pay off both principal and interest of the loan.
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39. Setting Loan Amortization Schedule
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Once the annual amount of installment is determined, the loan amortization schedule could be set up as follows:
Loan amortized schedule,Rs10000 at 10% for 4 year
Year
(1)
Beginning amount
(2)
Payment
(3)
Interest
(4)=(2)*0.10
Repayment of principal
(5)=(3)-(1)
Ending balance
(6)=(2)-(5) 1 Rs Rs 2
784.53 3
754.52 4
286.10
0.00

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