Time Value Of Money

TIM E VAL UE OF MONE Y FIN AN CE

MAN AGEMENT

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



Interest for the year (1000x0.10)



Principle at the end

1000 100 1100

 SECOND YEAR 

Principal at the beginning



Interest for the year (1000x0.10)



Principle at the end

1100 110 1210

 THIRD YEAR 

Principal at the beginning



Interest for the year (1000x0.10)



Principle at the end

1210 121 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=

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 hence PV = Cash today (present value) r

= number of years for which compounding is done

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

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/-

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  Exam pl e f or sim ple inter est 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/  Exam pl e f or compound inter est

SEE THE DIFFERENCE !!!

 Rs 1,37,80,612  or  Rs 137.8 lakhs  Or

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

[ 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

annuity Future value

FUTURE VALUE OF AN  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

Future value of an annuity  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



TIME LINE FOR ANNUITY 1 2 3 4 5 10011000 1000 1000 1100

1210

1000

1331

1464 6105 ----------------

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

Value of FVIFArn for various combinations of r and n

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 flow r = interest rate per period n = duration of an annuity 

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

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

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

= 8.536 THE ANNUAL SINKING FUND DEPOSIT

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 500000

=1.12n-1

= 2.4 +1 = 3.4

=n log 1.12

= log 3.4

n x 0.0492

= 0.5315

=

2.4

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

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

APPLICATIONS 1. How much can you borrow for a car 2. Period of loan amortation 3. Determining the loan amortation schedule 4. Determining periodic withdrawal 5. 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

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

Ammortization Schedule

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 deposit Refer 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/-

 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

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)

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

 0 1 2  Annuity due

A

A

A

n-1 A

A

n 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.

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  Second six months

= 1060

 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/=