The Time Value Of Money

The Time Value of Money 

Which would you rather have --Rs 1,000 today or Rs1,000 in 5 years?

Obviously, Rs1,000 today. today Money received sooner rather than later allows one to use the funds for investment or consumption purposes. This concept is referred to as the TIME VALUE OF MONEY!! MONEY 



What is time value of money?   It is the value of money figuring in a given

amount of interest earned over a given amount of time.

Why TIME? 





TIME allows one the opportunity to postpone consumption and earn INTEREST. INTEREST NOT having the opportunity to earn interest on money is called OPPORTUNITY COST. 

How can one compare amounts in different time periods? One can adjust values from different time

periods using an interest rate.

 Remember, one CANNOT compare numbers

in different time periods without first adjusting them using an interest rate.

Time lines 0

1

2

3

10%

100

FV = ?

5

Types of Interest ◆S im

p le In te re st

In te re st p a id ( e a rn e d ) o n o n ly th e o rig in a la m o u n t, o r p rin cip a l, b o rro w e d ( le n t).

Compound Interest 

Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent).

Simple Interest Formula 

Formula    

SI = P0(i)(n)

SI: Simple Interest P0: Deposit today (t=0) i: n:

Interest Rate per Period Number of Time Periods

Simple Interest Example Assume that you deposit Rs1,000 in

an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?

SI

= P0(i)(n) Rs1,000(.07)(2) Rs140

= =

Simple Interest (FV) What is the Future Value (FV) FV of the

deposit?

FV = P0 + SI = Rs 1,000 + $140 = Rs 1,140 Future Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate. 

Simple Interest (PV) What is the Present Value (PV) PV of the

previous problem?

The Present Value is simply the Rs1,000 you originally deposited. That is the value today! Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. 

Why Compound Interest? Future Value (U.S. Dollars)

Future Value of a Single $1,000 Deposit 20000 10% Simple Interest 7% Compound Interest 10% Compound Interest

15000 10000 5000 0

1st Year 10th Year

20th Year

30th Year

Compound Interest Graphically 4500

3833.7

4000 5%

3500

10%

3000

15%

2500

20%

2000 al treV u F

1636.6

1500 1000

672.75

500

265.3

0 0

1

2

3

4

5

6

7

8

9

10 Years

11

12

13

14

15

16

17

18

19

 



Do You want to Double Your Money? 



How long does it take to double Rs.5,000 at a compound rate of 12% per year? 

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The “Rule of 72” & “Rule of 69” By rule 72 Years to Double = 72 / i% 





By rule 69 Years to Double = 0.35+(69 / i%) 



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Actual time- 6.12 years By rule of 72- 6 years By rule of 696.10 years 

 

Doubling period: It is a period in which the amount invested becomes double.

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Finding the growth rate The rate of interest at which the amount is invested, is called the growth rate.  We can find the growth rate by using the formula of future value. Question: A bank offers you to deposit Rs.100 and promises to pay Rs.112 after 1 year. What rate of interest would you earn? Ans:-12 



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Future value   Future value of a single amount Future value of an Annuity

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Future Value: is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate. Present Value: is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. Compounding: The process of calculating future values of cash flows. Discounting: The process of calculating present values of the cash flows. 

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Abbreviations PV - Present value FV - Future value Pmt - Per period payment amount N - Either the total number of cash flows or

the number of a specific period i - The interest rate per period

Future Value – using Formula FV n = PV ( 1 + i) n Where FVn = the future of the investment at end of “n” years

the

i= the annual interest (or discount) rate n = number of years PV = the present value, or original amount invested at the beginning of the first year 20

Future Value Example 

Example: What will be the FV of Rs1000 in 8 years at interest rate of 10%?



FV2= PV(1+i)n = 1000 (1+.1)8 Rs100 (1.10)8

= Rs2144



21

Future Value – Using Tables FVn = PV (FVIFin, )

Where FVn = the future of the investment at the end of n year

PV = the present value, or original amount invested at the beginning of the first year FVIF = Future value interest factor or the compound sum of Rs1 i = the interest rate n = number of compounding periods 22

Future Value – using Tables What is the future value of Rs500

invested at 8% for 7 years? (Assume annual compounding) Using the tables, look at 8% column, 7 time periods to find the factor 1.714  FVn = PV (FVIF8%,7yr )  

= Rs500 (1.714) = Rs 857 23

Table for Future Value Y ear 1% 2% 3% 4% 5% 6% 1 2 3 4 5 6 7

1.010 1.020 1.030 1.041 1.051 1.062 1.072

1.020 1.040 1.061 1.082 1.104 1.126 1.149

1.030 1.061 1.093 1.126 1.159 1.194 1.230

1.040 1.082 1.125 1.170 1.217 1.265 1.316

1.050 1.103 1.158 1.216 1.276 1.340 1.407

1.060 1.124 1.191 1.262 1.338 1.419 1.504

7% 1.070 1.145 1.225 1.311 1.403 1.501 1.606

8 1 1 1 1 1 1 1

Future Value-Using Excel =FV(Rate,years,pmt)

Future value of a Annuity

Annuity: An Annuity is a stream of constant cash flow occurring at regular intervals of time. The premium payments of a life insurance policy. Deferred Annuity: When the cash flow occur at the end of each period the annuity is called deferred or ordinary annuity. Annuity Due: When the cash flow occur at the beginning of each period the annuity is called annuity due. 

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Examples of Annuities     

Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings

Growth of a 5yr $500 Annuity Compounded at 6% 0

1

2

3

4

5

500

500

500

500

50 0

6%

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FV Annuity – Example What will be the FV of 5-year Rs500

annuity compounded at 6%?



FV5 = 500 (1+.06)4 + 500 (1+.06)3 +500(1+.06)2+





500 (1+.06) + 500

   



= 500 (1.262) + 500 (1.191) + 500 (1.124)+ 500 (1.090) + 500 = 631.00 + 595.50 + 562.00 + 530.00 + 500 = Rs.2,818.50 29



Future value of annuity: The compound value of annuity. 



FVA = A

( [(1+i)n - 1] / i )

The term within the brackets is the compound value factor for an annuity.



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FV of an Annuity – Using Formula What will Rs500 deposited in the bank every year

for 5 years at 10% be worth?

FV = PMT {(FVIFi,n -1)/ i }

Simplified form of this equation is:





 



FV5 = PMT (FVIFAi,n )

= PMT [ (1+0.10)5-1 ]/i = Rs500 (5.637) = Rs2,818.50





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Future Value of Annuity— Using Tables I deposit Rs 1000 annually in a bank for 5 years

and my deposits earn a compound interest of 6%? What will be the value of these series of deposits at the end of 5 years?

FVAn=A[(1+r)n-1]/r [(1+r)n-1]/r=Future value of interest factor for an annuity  =FVIFAr,n  =FVIFA6%,5  =1000(5.637) (From table)  =Rs 5637 

Future Value of Annuity Year

1 2 3 4 5

1%

1.000 2.010 3.030 4.060 5.101

2%

1.000 2.020 3.060 4.122 5.204

3%

1.000 2.030 3.091 4.184 5.309

4%

1.000 2.040 3.122 4.246 5.416

5%

1.000 2.050 3.153 4.310 5.526

6%

1 2 3 4 5.

Applications of Future value of Annuity What lies in store for you?

Finding the accumulated PPF Annual Deposit in a Sinking Fund? Finding the Interest Rate? How long should you wait? 



Sinking Fund Sinking fund is a fund, which is created out

of fixed payments each period to accumulate to a future sum after a specified period. For example, companies generally create sinking funds to retire bonds (debentures) on maturity. The factor used to calculate the annuity for a given future sum is called the sinking fund factor (SFF).

  i A = Fn  n  (1 + i ) − 1  

Futura Limited has an obligation to redeem Rs

500 million bonds 6 years hence.How much should the company deposit annually in a sinking fund account wherein it earns 14 % interest to cumulate 500 million in 6 years time?



A=500[0.14/ {(1+0.14)6-1}]  =58.575 million 

 

Using Excel Present Value(PV) Future Value(FV) Equal Periodic receipt/payment(pmt) Number of periods(N per) 

 

Interest/Discount Rate(Rate)

Present value Present value of a Single Amount Present value of an Uneven series Present value of an Annuity 

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 

Present value of a Single Amount General formula: PV0 = FVn / (1+i)n



Q. Assume that you need Rs1,000 in 2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually. 2 2 PV = FV / (1+i) = Rs.1,000 / (1.07) 0 2 



= FV2 / (1+i)2

= Rs.873.44

 

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Pre se n t V a lu e – U sin g Ta b le s P V n = FV ( PVIF in , ) W h e re PV n = th e p re se n t va lu e o f a fu tu re su m o f money

FV = th e fu tu re va lu e o f a n in ve stm e n t a t the end of an investment period P V IF = Pre se n t V a lu e in te re st fa cto r o f $ 1 i = th e in te re st ra te n = n u m b e r o f co m p o u n d in g p e rio d s 40

Present Value Tables

–

Using

What is the present value of Rs 100 to be

received in 10 years if the discount rate is 6%? Find the factor in the table corresponding to 6% and 10 years  PVn = FV (PVIF6%,10yrs. )  

= Rs100 (.558) = Rs55.80

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Year 1 2 3 4 5 6 7 8 9 10

1% 0.990 0.980 0.971 0.961 0.951 0.942 0.933 0.923 0.914 0.905

2% 0.980 0.961 0.942 0.924 0.906 0.888 0.871 0.853 0.837 0.820

3% 0.971 0.943 0.915 0.888 0.863 0.837 0.813 0.789 0.766 0.744

4% 0.962 0.925 0.889 0.855 0.822 0.790 0.760 0.731 0.703 0.676

5% 0.952 0.907 0.864 0.823 0.784 0.746 0.711 0.677 0.645 0.614

6% 0.943 0.890 0.840 0.792 0.747 0.705 0.665 0.627 0.592

0.558

Uneven cash flow stream Any series of cash flow that does not conform to the definition of an annuity  is considered to be an uneven cash flow stream. Eg. A series such as: Rs 1000/-,Rs 1000/, Rs 1000/-, Rs 2000/- ,Rs 2000/- , Rs 2000/would be considered an uneven cash flow stream . We might consider it as a series of two consecutives annuities. 

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Present value of an uneven series In financial analysis we often come across uneven cash flows streams then to calculate the present value we use the  PV= A1/(1+r) + A2/(1+r)2+……An/ (1+r)n 



Ex. Uneven

Cash flowduringvarious years.

  

0

1

2

3

4

10 0

30 0

30 0

50

10 %

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Present Value of an Annuity Pensions, insurance obligations, and

interest owed on bonds are all annuities. To compare these three types of investments we need to know the present value (PV) of each. PV can be computed using calculator, tables, spreadsheet or formula.

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Present Value of an Annuity Using the example, and assuming a discount P V

rate of 10% per year, we find that the present value is: A

1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 = 1 + 2 + 3 + 4 + 5 = 3 7 9 .0 8 1 . 1 0 1 .1 0 1 . 1 0 1 . 1 0 1 . 1 0 ( ) ( ) ( ) ( ) ( )

62 . 09 68 . 30 75 . 1 82 3 . 6 90 .. 08 379 4

0

10 0

10 0

10 0

10 0

10 0

1

2

3

4

5

Present Value of an Annuity

General formula: PV= A/(1+r) + A/(1+r)2+……A/ (1+r)n 

OR



PV = A (PVIFA)

 

Ques: Suppose you expect to receive Rs 1,000/- annually for 3 years , at the end of each of the year . What will be the present value of this stream if the discount rate is 10%? Ans : Rs 2486.8/

 

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PV of an Annuity – Using Table § Calculate the present value of a $500

annuity received at the end of the year annually for four years when the discount rate is 6%.



PV = PMT (PVIFAi,n )  = Rs500(3.465) (From the table)  = Rs 1732.50 

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Year

1%

1 2 3 4

0.9901 1.9704 2.9410 3.9020

PV of an A n2%n u ity 3% 0.9804 1.9416 2.8839 3.8077

0.9709 1.9135 2.8286 3.7171

4%

5%

0.9615 1.8861 2.7751 3.6299

0.9524 1.8594 2.7232 3.5460

0 1 2 3

APPLICATIONS OF PRESENT VALUE OF ANNUITY Ø How much can u borrow for an item Ø Period of loan Amortization Ø Determining the periodic withdrawal Ø Finding the Interest Rate

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Amortized Loans Loans paid off in equal installments over

time are called amortized loans.

 For example, home mortgages and auto loans.  Reducing the balance of a loan via annuity payments is called amortizing.  



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Amortized Loans The periodic payment is fixed. However,

different amounts of each payment are applied towards the principal and interest. With each payment, you owe less towards principal. As a result, amount that goes toward interest declines with every payment (as seen in figure 5-3).

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Amortized Loans

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Steps to Amortizing a Loan 1. 2.

Calculate the payment per period. Determine the interestin Period t. (Loan Balance at t-1) x (i% / m) 3. Compute principal payment in Period t. (Payment - Interest from Step 2) 4. Determine ending balance in Period t. (Balance - principal payment from Step 3) 5. Start again at Step 2 and repeat. 

Amortization Example 

Example: If you want to finance a new machinery with a purchase price of $6,000 at an interest rate of 15% over 4 years, what will your annual payments be?



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Payments – Using Formula Finding Payment: Payment amount can be

found by solving for PMT using PV of annuity formula. PV of Annuity =PMT [1-(1+i)-1 ]  I  6,000 = PMT (2.855)  PMT = 6,000/2.855  = Rs2,101.58

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Amortization Schedule Yr.

Annuity

Interest

Principal

Balance

1

Rs 2,101.58

Rs900.00 Rs1,201.58

2

Rs2,101.58

719.76

1,381.82

3,416.60

3

2,101.58

512.49

1,589.09

1,827.51

4

2,101.58

274.07

1,827.51

Rs4,798.42

-0-

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