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 lines0 1 2 3

10%

100

FV = ?

5

Types of InterestxS i m

p l I te re st e n

Compound Interest

I te re st p a i ( e a rn e d ) o n o n l th e n d y o ri i a la m o u n t, o r p ri ci a l g n n p , b o rro w e d ( l n t). e 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 ExampleAssume that you deposit Rs1,000 in

SI

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

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

= =

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

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) of the PV

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 of a Single $1,000 DepositFuture Value (U.S. Dollars)

20000 15000 10000 5000 0 1st Year 10th Year 20th Year 30th Year 10% Simple Interest 7% Compound Interest 10% Compound Interest

Compound Interest Graphically4500 4000 3500 3000 2500 2000 l a V e r t u F 1500 1000 500 0 0 1 2 3 4 5 6 7 8 9 10 Years 11 12 13 14 15 16 17 18 195% 10% 15% 20%

3833.7

1636.6

672.75

265.3

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 rateThe 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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AbbreviationsPV - 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 FormulaFV n = PV ( 1 + i n ) Where FVn = the future of the investment at end of n years n = number of years PV = the present value, or original amount invested at the beginning of the first year20

the

i= the annual interest (or discount) rate

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

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Future Value Using TablesFVn = 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 periods22

Future Value using TablesWhat 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 85723

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)

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.

Future value of a Annuity

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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%06%

1

2 500

3 500

4 500

5 50 0

500

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FV Annuity ExampleWhat 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 FormulaWhat 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 TablesI 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 AnnuityYear 1% 2% 3% 4% 5%

6%

1 2 3 4 5

1.000 2.010 3.030 4.060 5.101

1.000 2.020 3.060 4.122 5.204

1.000 2.030 3.091 4.184 5.309

1.000 2.040 3.122 4.246 5.416

1.000 2.050 3.153 4.310 5.526

1 2 3 4 5.

Applications of Future value of AnnuityWhat 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 FundSinking 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 (1 + i) n 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 ExcelPresent Value(PV) Future Value(FV) Equal Periodic receipt/payment(pmt) Number of periods(N per)

Interest/Discount Rate(Rate)

Present valuePresent 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. Lets examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually. PV = FV2 / (1+i)2 = Rs.1,000 / (1.07)2 0

= FV2 / (1+i)2

= Rs.873.44

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Pre se n t V a l e U si g Ta b l s u n eP V n = FV ( PVIF in ) ,W h e re PV n = th e p re se n t va l e o f a fu tu re su m o f u money

FV = th e fu tu re va l e o f a n i ve stm e n t a t u n the end of an investment period P V IF = Pre se n t V a l e i te re st fa cto r o f $ 1 u n i = th e i te re st ra te n n = n u m b e r o f co m p o u n d i g p e ri d s n o40

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 streamAny 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 seriesIn 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

neven Ex. U

C flowduringvarious years. ash

0 10 %

1

2

3

4

10 0

30 0

30 0

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Present Value of an AnnuityPensions, 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 AnnuityUsing the example, and assuming a discountP 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 .8 0 11 .0 11 .0 11 .0 11 .0 11 .0 ( ) ( ) ( ) ( ) ( )

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

10 0 0 1

10 0 2

10 0 3

10 0 4

10 0 5

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

Present Value of an AnnuityOR

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%

PV of an A n n u i3% ty 2%0.9804 1.9416 2.8839 3.8077 0.9709 1.9135 2.8286 3.7171

4%

5%

1 2 3 4

0.9901 1.9704 2.9410 3.9020

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 How much can u borrow for an item Period of loan Amortization Determining the periodic withdrawal Finding the Interest Rate

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Amortized LoansLoans 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 LoansThe 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 Loan1. 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 FormulaFinding 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 ScheduleYr. 1 2 3 4 Annuity Rs 2,101.58 Rs2,101.58 2,101.58 2,101.58 Interest Principal Balance Rs4,798.42 3,416.60 1,827.51 -0-

Rs900.00 Rs1,201.58 719.76 512.49 274.07 1,381.82 1,589.09 1,827.51

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