Time Value of Money

FINANCE MANAGEMENT

TIME VALUE OF MONEY

TIME VALUE OF MONEY

MONEY HAS TIME VALUE BECAUSE INDIVIDUALS PREFER

CURRENT CONSUMPTION TO FUTURE CONSUMPTION

CAPITAL CAN BE EMPLOYED PRODUCTIVELY TO GENERATE POSITIVE RETURNS

TIME VALUE OF MONEY

An investment of one rupee today would grow to (1+r) after a year.

Hence ‘r’ is the rate of return earned on the investment

In an inflatory period, a rupee today represents a greater purchasing power than a rupee a year hence

FUTURE VALUE OF A SINGLE AMOUNT

FUTURE VALUE OF AN ANNUITY PRESENT VALUE OF A SINGLE

AMOUNT PRESENT VALUE OF AN ANNUITY

Suppose you have invested Rs 1000 today and deposited with financial institution which pays 10% interest compounded annually for a period of 3 years

Rs 1000 today and deposited with financial institution which pays 10%interest compounded annually for a period of 3 years.

FIRST YEAR Principal at the beginning 1000 Interest for the year (1000x0.10) 100 Principle at the end 1100

SECOND YEAR Principal at the beginning 1100 Interest for the year (1000x0.10) 110 Principle at the end 1210

THIRD YEAR Principal at the beginning 1210 Interest for the year (1000x0.10) 121 Principle at the end 1331

FORMULA The process of investing money as well as reinvesting the interest

earned thereon is called compounding. The future value or compounded value of an investment after n years

when the interest rate is r percent is

FVn = PV (1+r)n

(1+r)n Is called the future value interest factor or future value factor which can be found as follows

Multiply 1.10 ie(1+r), 3 times (this is tedious when period of investment is so long

BY CALCULATOR Check you have key labeled Yx. Enter1.10 Press the key labeled yx. Enter3 Press= Get the answe

FORMULA FOR FUTURE AVLUE OF A SINGLE AMOUNT

The general formula for the future value of a single amount is FVn = PV (1+r)n

Where FVn = future value n years hencePV = Cash today (present value)r = number of years for which

compounding is done

Value of FVIFr,n for various combinations of ‘r’ and ‘n’

n/r 6% 8% 10% 12% 14%

2 1.124 1.166 1.210 1.254 1.300

4 1.262 1.360 1.464 1.574 1.689

6 1.419 1.587 1.772 1.974 2.195

8 1.594 1.851 2.144 2.476 1.853

10 1.791 2.159 2.594 3.106 3.707

12 2.012 2.518 3.138 3.896 4.817

FVIF TABLE Alternatively you can consult a future

value interest factor table Suppose you deposit Rs 1000/- today in

a bank that pays 10% interest compounded annually. How much the deposit grow after 8 years and 12 years

After 8 years Rs 1000(1.10)8 = Rs 1000(2.144) =Rs 2144/-

After 12years Rs 1000(1.10)12= Rs 1000(3.138) =Rs 3188/-

COMPOUND AND SIMPLE INTEREST

In compound interest each payment is reinvested to earn further interest for future period

In simple interest, no interest is earned on interest

Example for simple interest

FUTURE VALE = PV[1+no of yrs x int.rate] Rs 1000 invested at 10%for simple interest for

100 yrs 1000x[1+100x .10] = 1000 x[ 1+10] = Rs

11,000/- Example for compound interest 1000[1+0.10] 100

SEE THE DIFFERENCE !!!

Rs 1,37,80,612orRs 137.8 lakhsOrRs 1.38 crores

DOUBLING PERIOD

INVESTORS USUALLY ASK -When my money will be doubled?

To answer this, we may look at the future value interest factor table A

We can see that when interest rate is 12%, it takes about 6 yrs to double the amount . It will take 12 yrs at 6%

RULE OF 72

According to this rule, the doubling period is obtained by dividing 72 by interest rate.

Say, interest rate is 8%, the doubling period is 9 years.(72/8)

Rule of 69

According to this rule of thumb, the doubling period is equal to

0.35 +69/int rate say int rate is 10%, doubling period

is 0.35 + 69/10 = 7.25

Finding growth rate-no of employees

How many employees your company will have in 10 years, if the present strength is 5000 and expected to grow by 5%

5000 X (1.05)10 = 5000 X 1.625 = 8149 ABC Ltd had a revenue of Rs 100 M in 1990 which

increased to Rs 1000M in 2000. Find growth in Revenue.

What was the compound growth in revenue?

100 (1+g)10 =1000

(1+g)10 =1000/100 = 10

1+g = 101/10

g = 101/10 – 1

=1.26-1=0.26 = 26%

PRESENT VALUE OF A SINGLE AMOUNT

Suppose some one promise Rs 1000/- a year hence. The value will be definitely less than 1000

we already know the formula for future value - FVn = PV (1+r)n.

Dividing both sides by (1+r)n we get PV = FVn[ 1/ (1+r)n]

The factor [1/ (1+r)n] is called the present value index factor for different combinations of r and n.

Table for PVIF for different r,n

n/r 6% 8% 10% 12% 14%

2 0.890 0.857 0.826 0.797 0.770

4 0.792 0.735 0.683 0.636 0.592

6 0.705 0.6 30 0.565 0.507 0.456

8 0.626 0.540 0.467 0.404 0.351

10 0.558 0.463 0.386 0.322 0.270

12 0.497 0.397 0.319 0.257 0.208

[ 1/ (1+r)n]

PROBLEM-PRESENT VALUE

What is the present value of Rs1000/- receivable 6 years hence if the rate of discount is 10%

Rs 1000 x PVIF (1O%,6) = Rs 1000 x (0.565) = Rs 565/-

PROBLEM-PRESENT VALUE

What is the present value of Rs 1000 receivable 20 yrs hence if the discount rate is 8%

Suppose the table is not having value for 20 yrs, we get as below

1000 x (1/1.08)20 = 1000 (1/1.08)10 x (1.08)10

1000 x (0.463) x (0.463) = 214/=

Present value of an uneven series

In financial analysis we often come across uneven cash flow.

In such cases, calculate individual cases and add

The formula is PVn = A1/(1+r) + A2/(1+r)2 +.. An/(1+r)n

Present value of an uneven series

year Cash flow PVIF 12%n PV of individual cash flow

1 1000 0.893 893

2 2000 O.797 1594

3 2000 0.712 1424

4 3000 0.636 1908

5 3000 0.567 1701

6 4000 0.507 2028

7 4000 0.452 1808

8 5000 0.404 2020

TOTAL 13376

annuity

Future value

FUTURE VALUE OF AN ANNUITY An annuity is a stream of constant

cash flow occurring at regular intervals of time

When cash flow occurs at the end of the period, the annuity is called an ordinary annuity or a deferred annuity(LIC Premium)

If it occurs at the beginning of each period, annuity is called annuity due

Suppose you invest Rs 5000 annually in a bank for 5 yrs at

10 %, what will be the value of this series of deposit after 5 years.

Assuming that each deposit occurs at the end of each year, the future value of each annuity will be

1000(1.10)4+1000(1.10)3+1000(1.10)2+1000(1.10)1+1000 1000x1.465+1000X1.331+1000X1-21+1000X1.10+1000 =RS 6105

Future value of an annuity

TIME LINE FOR ANNUITY

1 2 3 4 5

10001000 1000 1000 1000 110012101331

1464

--------------6105

----------------

Value of FVIFArn for various combinations of r and n

n/r 6% 8% 10% 12% 14%

2 2.060 2.080 2.100 2.120 2.140

4 4.375 4.507 4.641 4.779 4.921

6 6.975 7.336 7.716 8.115 8.536

8 9.897 10.636 11.436 12.299 13.232

10 13.187 14.487 15.937 17.548 19.337

12 16.869 18.977 21.384 24.133 27.270

FORMULA

The future value of an annuity is given by the following formula

FVAn = A (1+r)n-1 r

Where FVAn is the future value of an annuity which has a duration of n yrs.

A= constant periodic flowr = interest rate per periodn = duration of an annuity

The term (1+r)n-1 is future value interest factor r

FUTURE VALUE OF AN ANNUITY

APPLICATIONS

Knowing what lies in store for you

Suppose you have deposited Rs 30000/year in your PPF account for 30 years. What will be accumulated amount in your PPF at the end of 30 years if the interest rate is 11%

= Rs 30000(FVIFA 11%30YRS)

=30000X (1+r)n-1 = 30000x(1.11)30-1 r 0.11

= 30000x199.03 = Rs 59,70,600

How much should you save annually You want to buy a house after 5

years when it is expected to cost Rs 2m. How much should you save annually if your savings earn a compound rate of 12%

FVIFA (n=5, r=12%)= (1+0.12)5-1 0.12

= Rs 2000000

6.53

= Rs 314812

Annual deposit in a sinking fund Abc ltd has an obligation to redeem Rs

5000m bonds 6 years hence. How much the company deposit annually in the fund account where in it earns 14% interest to accumulate Rs 500m in 6years time.

FVIFA n=6,r=14 = (1+r)n-1 = (1+0.14)6-1 r 0.14

= 8.536THE ANNUAL SINKING FUND DEPOSIT WILL

BE Rs 500M/8.536 = Rs 58.575m

Finding interest rate

A finance coy advertise that it will pay a lump sum of Rs 8000 at the end of 6 years to investors who deposit annually Rs 1000 for 6 years. What interest rate is implicit in this offer.

A finance coy advertise that it will pay a lump sum of Rs 8000 at the end of 6 years to investors who deposit annually Rs 1000 for 6 years. What interest rate is implicit in this offer.

The interest rate may be calculated in 2 stages 1ST STEP find FVIFA,r6 for this contract as follows Rs 8000 = Rs 1000xFVIFAr6 FVIFA, r6= Rs Rs8000/Rs1000 = 8

2nd STEP Look at FVIFAr,n table and read the row corresponding to 6

years until you find close to 8.00 FVIFA 12% ,6 IS 8.115 SO CONCLUDE THE RATE OF INTEREST -12%

HOW LONG SHOULD U WAIT

You want to take a trip abroad which

costs Rs 1000000/- You can save annually Rs 50000/-to

full fill the desire. How long will have to wait if your savings earn an interest of 12%

You want to take a trip abroad which costs Rs 1000000/-You can save annually Rs 50000/-to full fill the desire. How long will have to wait if your savings earn an interest of 12%

The future value of an annuity of Rs 50000/- that earns 12% is equal to Rs 1000000/-

50000xFVIFA n=?,12% = 1000000=50000 x(1+r)n-1 = 1000000

r=50000 x1.12n-1 = 1000000

0.12=1.12n-1 = 1000000 X 0.12 = 2.4

500000

=1.12n-1= 2.4 +1 = 3.4=n log 1.12 = log 3.4n x 0.0492 = 0.5315

n = 0.5315/0.0492 = 10.8 yrs

Will have to wait for 11 years

annuity

present value

Present value of an annuity

Suppose you expect to receive Rs 1000/- annually for 3 years, each receipt occurring at the end of the year. What is the present value of this stream of benefits if the discount rate is 10%

The present value is the sum of the present values of all inflows of this annuity

Rs 1000(1/1.10) +Rs 1000(1/1.10)2 +Rs 1000(1/1.10)3

=Rs 1000x0.9091+Rs1000x0.88+Rs 1000x0.7513 =Rs 2478.8

The time line for Rs 1000/-

0 1 2 3

901.1 826.4 751.3 2478.8 =present value

Formula In general terms, the present value of

an annuity may be expressed as follows PVAn = A + A + ----A + A

1+r (1+r)2 (1+r)n-1 (1+r)n

A 1 + 1 + ----1 + 1

1+r (1+r)2 (1+r)n-1 (1+r)n

A 1 1 (1+r)n

r

formula

A 1 1 (1+r)n

r

Is referred as present value interest factor for an annuity(PVIFA r,n)

A-Constant periodic flow

Table for value of PVIFAr,n for different combinations of r and n

n/r 6%

2 1.833 1.783 1.737 1.690 1.647

4 3.465 3.312 3.170 3.037 2.914

6 4.917 4.623 4.355 4.111 3.889

8 6.210 5.746 5.335 4.968 4.639

10 7.360 6.710 6.145 5.650 5.216

12 8.384 7.536 6.814 6.194 5.660

APPLICATIONS

1. How much can you borrow for a car2. Period of loan amortation3. Determining the loan amortation

schedule4. Determining periodic withdrawal5. Finding interest rate

How much can you borrow

You can afford to pay per Rs 12000/- per month for 3 years for a new car. Interest rate advised by the company is 1.5% per month for 36 months. How much can you borrow.

To determine how much you can borrow, you have to calculate the present value of Rs 12000/-month for 36M at 1.5%

PVIFAr,n = 1-1/1/(1+r)n/r 1-1/1/(1.05)36/0.015 = 27.70 Present value = Rs 12,000x27.70 You can borrow = Rs 332400

PERIOD OF LOAN AMMORTATION

You want to borrow Rs 10,80,000/- to buy a flat. You approach a housing finance company which charges 12.5 interest. You can pay Rs 1,80,000 per year towards loan ammortation. What should be the maturity period of loan

You want to borrow Rs 10,80,000/- to buy a flat. You approach a housing finance company which charges 12.5 interest. You can pay Rs 1,80,000 per year towards loan ammortation. What should be the maturity period of loan

The present value of an annuity Of Rs 180000/- is set equals to 1080000

180000 x PVIF n,r = 1080000 180000xPVIFn=?r=12.5%=1080000 180000[ 1-1/(1.125)n/0.125 ] = 1080000 Given this equality, the value of n is [ 1-1/(1.125)n/0.125 ] = 1080000/180000=6 1-1/(1.125)n = 0.75 1/(1.125)n = 0.25 1= 0.25 x (1.125)n 1.125n = 4 n log 1.125 = log4 n x 0.0512 = 0.6021 n= 0.6021/0.0512 = 11.76 = maturity is 12 years

Determining the loan ammortization schedule

Most of the loans are paid in equal periodic installments(monthly, quarterly, annually), which cover interest as well as principal repayment. Such loans are called amortized loans.

For an amortized loan we should like to know (a) the periodic installment payment and (b) the loan amortization schedule showing break up of periodic installment between the interest component and principal repayment component.

Determining the loan ammortization schedule

Suppose a firm borrow 1000000 at an interest of 15% and loan is to be paid in 5 equal installments, payable at the end of next 5 years.

The annual installment payment A is obtained by solving the following equation

Loan amount = A X PVIFA n=5,r=15% 1000000 = A X 3.3522 Hence A = 298312. The ammortization schedule is shown in the next slide (NB – interest is calculated by multiplying the

beginning loan balance by interest rate.- principal repayment is equal to annual installment minus interest)

Ammortization Schedule

yr Beginning amt(1)

Annual installment(2)

Interest

(3)

PrincipalRepayment2-3 =4

Remaining balance1-4 =5

1 1000000 298312 150000 148312 851688

2 851688 298312 127753 170559 681129

3 681129 298312 102169 196143 484986

4 484986 298312 727482 225564 259422

5 259422 298312 38913 259399 23

Determining the periodic withdrawal

Your father deposit Rs 3,00,000 on retirement in a bank which pays 10% annual interest. How much can be withdrawn annually for a period of 10 years.

300000 = A X PVIFA 10%, 10 yrs A = 300000/6.145 = Rs 48819

Finding interest rate

Suppose someone offers you the following financial contract.If you deposit Rs 10,000 with him he promises to pay Rs 2500/- annually for 6 years. What interest rate do you earn on this depositRefer next slide

Finding interest rate ?Suppose someone offers you the following financial contract.If you deposit Rs 10,000 with him he promises to pay Rs 2500/- annually for 6 years. What interest rate do you earn on this deposit

The interest rate may be calculated in two steps Step 1 – find PVIFr,6 for the contract by dividing Rs

10,000 by Rs 2,500 PVIFA r,6 = Rs 10000/2500 = 4 Step 2 – look at the PVIFA table and read the row

corresponding to 6 yrs until you find a value close to 4 Doing so, you will find PVIFA 12%6 = 4.111 & PVIFA 14%6 = 3.889 Since 4 lies in the middle of these values, interest rate

lies (approx) in the middle. So interest rate is 13%

Present value of a growing annuity

A cash flow that grows at constant rate for a specified period of time is a growing annuity

The time line of the growing annuity is shown below

A(1+g) A(1+g)2 A(1+g)n

0 1 2 n The present value of a growing annuity can

be determined using the following formula PV of the growing annuity is A(1+g) ------------------(formula)

PV of growing annuity

Suppose you have the right to harvest a teak plantation for next 2o years over which you expect to get 100000/- cubic feet of teak/year. The current price per cubic feet is Rs 500/= but is expected to grow (increase)at the rate of 8% per year. The discount rate is 15%. The present value of teak that you can harvest from the teak forest can be determined as follows

PV of teak is Rs 500x100000(1.08)(formula) Rs 55,17,36,683

A note on annuity due

So far we have discussed ordinary annuities in which cash flows occur at the end of each period.

In the case of annuity due, cash flows occur at the beginning of each period.

Eg, lease for an appartment

Time line for ordinary annuity and annuity due. Ordinary annuity

A A A A

0 1 2 n-1 n Annuity due

A A A A 0 1 2 n-1 n Since cash flows of an annuity due occur one period

earlier in comparison to cash flows on an ordinary annuity, the following relationship holds

Annuity due value = Ord. annuity value x (1+r) So first calculate present and future values as though it

were ordinary annuity. Second, multiply your answer by (1+r)

Present value of a perpetuity

A perpetuity is an annuity of infinite duration

Formula is P<> = A X PVIF r, <> Where P<> = present value of a

perpetuity A = constant annual payment PVIFA r <> = present value interest

factor for a perpetuity –

Present value of a perpetuity

Present value interest factor of a perpetuity is 1 divided by the interest rate expressed in decimal form. Hence, the present value of a perpetuity is simply equal to the constant annual payment divided by the interest rate .

For example, the present value of a perpetuity is Rs 10,000 and interest rate is 10% is equal to 10000/0.10=100000.

This is quite convincing because an initial sum of Rs 100000 would if invested at the rate of interest of 10% provide a constant annual income of Rs 10000 for ever.

INTRA-YEAR COMPOUNDING & DISCOUNTING

So far we assumed that compounding is done annually and now consider the case where compounding is done more frequently.

Intra year compounding

Eg- deposit Rs 1000/- at 12% semi annual First 6 months Principal at beginning = 1000 Int for 6m(1000x0.12/2) = 60 Principal at end = 1060 Second six months Principal at beginning = 1060 Int for 6m(1060x0.12/2) = 63.6 Principal at end = 1123.6 If the compounding is done annually, the principal at

the end of one year would be 1000 (1.12) = 1120 The difference 3.6 represents interest on interest

Intra year compounding

The general formula for future value of a single cash flow after n years when compounding is done m times a year is

FVn = PV [ 1+r/m] m x n

Suppose you deposit Rs 5000 in a bank for 6 yrs and its interest rate is 12% and the frequency of compounding is 4 times a year, your deposit after 6 years will be

5000 x [ 1 + 0.12/4] 4x6

5000(1.03)24

5000 x 2.0328 = Rs 10164/=