iso-885dfdfd9-1__time value of money 2

7/29/2019 Iso-885dfdfd9-1__time Value of Money 2

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TIME VALUE OF MONEY:

Money is an arm or leg. You either use it or lose it. ----- Henry Ford.

Ramchandra, an woodcutter, had handed over all his savings Rs.700 to his 3 year old

son, Arjun in his deathbed. Instead of spending his entire savings for his medicines he

decided to give up his life. Arjun kept Rs.700 safely & has grown up in his uncles

house. When Arjun was 10 years old, still he can remember his fathers pain before hisdeath, one urgency came and that was his uncle was fallen in same disease and he

wanted to follow the same route of his elder brother. But this time Arjun was at least

having some maturity, he protested, & tried to go for medicines with his fathers

lifelong savings. Still he couldnt survive his uncles life, because this time by spending

Rs.700 anybody can survive from a fever but not from a critical disease. Because

within 7 years, money value or purchasing power has gone down.

That means, money value can be increased if it keeps on rolling. There is another story

also in the same line. This story is taken from the Living Bible, Mathew chapter 25.

The story is as follows:


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A man, going off to another country, called together to his servants and loaned

them money to invest for him, while he was gone. He gave 500 to one, 200 toanother and 100 to the last dividing it in proportion to their abilitiesand then

left on his trip. The man who received the 500 began immediately to buy and

sell with it and soon earned another 500. The man with 200 went right to work,

too, and earned another 200.

But the man who received the 100 dug a hole in the ground and hid the money for

safe keeping. After a long time their master returned from his trip and called them tohim to account for his money.

Then the man with the 100 came and said, Sir, I knew you were a hard man, and I

was afraid you would rob me what I earned, so I hid your money in the earth and here

it is!

But his master replied, You lazy rouge! Since you knew I would demand your profit,

you should at least have put my money into the bank so I could have some interest.

Hence this is time value of money


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Basis of Time Value:

An important concept in Financial Management is that the cash flow is more genuine

term than profit for understanding business. Cash flow is Kingthis seems to be

slogan for current business. The cash inflow or cash outflow usually occur at

different times and over a period of time. This leads us to consider the Time-Value of

Money.

Hence value of money depends on when the cash flow occursRs.100 now is more

worthy than Rs.100 at a future date. Because earlier one is more certain & obvious

than the later one. There are some other reasons also:

1. Interest or rent:

Money like any other commodities has a price. If you own it, you can rent it or

deposit it in your bank and earn some money or interest of that. The rent or interest

of money is the investors return, which reflects the time value of money. It

comprises:

a) Risk-free rate of return rewarding investors for forgoing immediate consumption,b) Compensation for risk and loss of purchasing power.

Money can be invested productively to generate real returns. For example, if a sum

of Rs.100 invested in raw material and labor results in finished goods worth Rs.107,

we can say that the investment of Rs.100 has earned a rate of return of 7%.


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2. Uncertainty: Rs.100 now is more certain than Rs.100 at a future date. This bird-

in-the-hand principle affects many aspects of financial management. That is why,

individuals prefer current consumption to future consumption.

3. Inflation: Under inflationary conditions, the value of money, expressed in terms of

its purchasing power over goods and services, declines.

Nominal or Market Interest Rate = Real rate of interest or return + Risk

premiums + Expected rate of inflation

The basis of finding time value of money is normally specified by the rate per period

usually denoted in percentage terms. Normally the chosen period is annual more by

convention than by rule.

Basically there are two methods by which the time value of money can be taken care

of--- process of compounding and process of discounting. To understand the basic

idea of Time value of money let us consider a venture which requires an immediate

outlay of Rs.1000 and the subsequent inflows of Rs.250 in each following 4 years.

The time line of above cash flows are as follows:


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Year 0 1 2 3 4

Rs. -1,000 250 250 250 250

Compounding and Discounting: Cash flows are occurring at the different point of

time. For meaningful comparison, all these cash flows should be assessed on the basis

of a same point of time. Either the cash flow occurring today has to be converted into

its equivalent at a future date or the cash flow occurring later has to be converted back

to todays value.

The future value of money that is available today can be calculated using the concept

ofCompounding and that value is known as Future Value ( FV) or Compounded

Value ( CV). shown in the given table 5.1.


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0 1 2 3 4

-1000 250 250 250 250

+FV(250)

+FV(250)

+FV(250)

+FV(-1000)

Table 1

Process of Compounding

The present value of money accruing later is estimated by the process ofDiscounting

and the value is known as Present Value ( PV) or Discounted Value (DV). shown in

the given table 5.2


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0 1 2 3 4

-1000 250 250 250 250

+PV(250)

+PV(250)

+PV(250)

+PV(250)

Table 2: Process of Discounting

Compounded or Future Value:

Simple and Compound Interest: The future value (FV) of a sum of money invested at

a given annual rate of interest will depend on whether the interest is paid only on the

original investment ( called Simple Interest), or whether it is calculated on the originalinvestment plus accrued interest ( called Compound Interest). In the case of compound

interest, there is a further factor affecting the future value, namely the frequency with

which interest is paid (e.g. monthly, quarterly or annually).


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With simple interest, the future value is determined by:FVn = V0(1+in),

where , FVn = future value at time n,V0 = original sum invested or the principal value

i = annual rate of simple interestSay, you win Rs.100,000 from a Realty Show of a television channel programme

and decide to invest in Fixed deposit of a commercial bank at the rate of 10% for

five years, simple interest. The future value will be the original of Rs.100000 plus

five years interest, giving a total of Rs.150,000.FV5 = Rs.100,000 [1+ (0.10)(5)] = Rs.150,000.

If i is compound interest, in subsequent years the interest is paid on the original

capital plus accrued interest. The process of compounding provides a convenient

way of adjusting for the time value of money. An investment made now in the

capital market of V0

gives rise to a cash flow of V0(1+i)2 after 2 years, and so on. In

general, the future value (FV) of V0 invested today at a compound rate of interest

of i% for n years will be:


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FV(i,n) = V0(1+i)n

Suppose your Rs.100000 investment was put into a building society paying a fixed

10% a year compound interest. What will your investment be worth after five years? In

one years time, the investment will be worth

100,000(1+0.10) = Rs.110,000

After 2 years it will be worth 100,000(1+0.10)2 = 121,000

After 5 years it will be worth:

FV5 = 100,000(1+0.10)5 = 161,000

Hence the effect of compound interest yields a much higher value than simple interest,

which yielded 150000.


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Period of deposit Rate per annum

Upto 6 months 10%

More than 6 months to 1 year 10.5%

1 year and above 11%

Example 1:

The fixed deposit scheme of Canara Bank offers the following interest rates. Given in

the table 5.3.

Table 3

Change of Interest Rate

Example 2:

Banks usually offer variable, rather than fixed, rates of interest. Assume the rate of 10%remains for the first 2 years then fails to 8% for the years 3 to 5. The calculation now

has 2 elements:

FV5 = Rs.100,000 (1+0.10)2(1+0.08)3 = Rs.152,400.

More Frequent Compounding


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Unless otherwise stated, it is always assumed that compounding or discounting is an

annual process; cash payments of benefits arise either at the start or the end of the year.

But government bonds pay interest semi-annually or quarterly. Interest charged on credit

cards is applied monthly. To compare the true costs or benefits of such financial

contracts, it is necessary to determine the effective rate of interest, termed the annual

percentage rate (APR), or effective interest rate.

Returning to our earlier example of Rs.100,000 invested for five years at 10% compound

interest, we now assume 5% payable every six months. After the first six months, the

interest is Rs.5,000 which is reinvested to give interest for the second half year ofRs.105,000 * (10/2) % = Rs.5,200. The end of year value is therefore Rs.(105,000 +

5,200) = Rs.110,200.

We can still use the compound interest formula but with I as the six-monthly interest rate

and n the six-monthly, rather than annual, interval. In converting the annual

compounding formula to another interest payment frequency the trick is simply to dividethe annual rate of interest (i) and multiply the time period (n) by the number of payments

each year.

The generalized formula for these shorter compounding period isFVn = V0( 1+ k/m)m*n,where m is no. of years compounding is done.


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Example 3:

Suppose you deposit Rs.20,000 with an investment company which pays 12 & interestwith quarterly compounding. How much will this deposit grow in 5 years?FV = 20,000 * (1+.12/4)4*5 = 20,000(1+.03)20 =Rs.36,120The following table 5.4 calculates the APRs based on a range of interest payment

frequencies for a 22% per annum loan. It can be seen that by charging compound

interest on a daily basis, the effective annual rate is 24.6%, some 2.6% higher than on an

annual basis.Annual Percentage rate for a Loan:

Annually (1+0.22) -1 0.22 or 22%

Semi-annually (1+0.22/2)21 0.232 or 23.2%

Quarterly (1+0.22/4)41 0.239 or 23.9%

Monthly (1+0.22/12)121 0.244 or 24.4%

Daily (1+0.22/365)3651 0.246 or 24.6%

Table 4


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Hence for more frequent compounding we have to calculate effective interest rate from

nominal interest rate with corresponding frequency per year. The general relationship

between the effective and nominal interest rate is as follows:

r = (1+k/m)m -1,

Where r = effective interest rate,k = nominal interest rate,m = frequency of compounding per year

Example 4:

Find out the effective interest rate, if the nominal interest rate is 10% and frequency of

compounding is half yearly.

r = (1+k/m)m -1 = ( 1+ .10/2)21 = .1025 or 10.25%


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Doubling Period(DP):

A frequent question asked by the investor is, How long it will take to make an amountdouble for a certain rate ofinterest?

This question can be answered by a thumb rule known as rule of 72 orrule of 69.

By rule of 72 , DP = 72/i for an interest rate = i%.

By rule of 69, DP = 0.35 + 69/i for an interest rate = i%.

Example 5:

Find out the doubling period for an interest rate of 10% by applying the two rules.

By rule of 72 , DP = 72/i for an interest rate = i% = 72/10 = 7.2 years.

By rule of 69, DP = 0.35 + 69/i for an interest rate = i% = 0.35 + 69/10 = 0.35 + 6.9 =

7.25 years.


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Figure 1: Future Value


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5.1.3. Discounted or Present Value:

An alternative way of assessing the worth of an investment is to invert the compounding

process to give the present value of the future cash flows. This process is called

discounting. Todays value of any future cash flow is known as Discounted orPresent

Value.

Choice of Rs.100 now or the same amount in one years time, it is always preferable to

take the Rs.100 now because it could be invested over the next year at, say, a 10% interest

rate to produce Rs.110 at the end of one year. If 10% is the best available annual rate ofinterest, then one would be indifferent to receiving Rs.100 now or Rs.110 after one year.

Hence the present value of Rs.110 received one year is Rs.100.

We can obtain present value simply by dividing the future cash flow by 1 plus the rate of

interest, i, i.e.

PV = 110/(1+1.1) = 100

Discounting is the process of adjusting future cash flows to their present values. It is, in

effect, compounding in reverse.

Recall that earlier we specified the future value as:


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n

PV = FV/ (1+i) n Dividing both sides by (1+i)n,

Example 6:Compute the present value of Rs.1,611 receivable after 5 years at the rate of 10% .

PV = 1611/(1+0.1) 5

The message is: Do not pay more than Rs.1,000 today for an investing offering a certain

return of Rs.1,611 after 5 years, assuming a 10% market rate of interest.


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Effect of discounting:

It is useful to see how the discounting process affects present values at different rates of

interest. This is seen in the following figure that the value of Rs.1 decreases verysignificantly as the rate and period increases from 0 to 20% and within 10 years

respectively. The following table 5.5 summarizes the discount factors for three rates of

interest. Investment surveys (e.g. Pike 1988) suggest that 15% discount is the most

popular and useful discount rate used in evaluation of capital projects. In this case every

5 years the discounted value halves approximately. Thus, after 5 years the value of Rs.1

is 50 paisa, after 10 years 25 paisa and so on.

year 10% 15% 20%

0 1.0 1.0 1.0

5 0.6 0.5 0.4

10 0.4 0.25 0.16

15 0.24 0.12 0.06

20 0.15 0.06 0.03

25 0.19 0.03 0.01

Table 5Present value of a single future sum Rs.1


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Present value(Rs) 0 %

1.0

0.75 5%

0.5 10%

0.25 15%

0.0 20%

Period(years) 2 4 6 8 10

Table 6

The relationship between present value of Rs.1 and interest over time


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Discount Table and Present value formula:

Much of the tedium of using formulae and power functions can be eased by using discount

tables or computer based spreadsheet packages. The discount factor or interest factor

Rs.1 for a 10% discount rate in three years time is:1/(1.10)3= 1/1.33 = 0.751

This can be found in Appendix X by locating the 10% column and the 3 years row. We call

this the Present Value Interest Factor (PVIF) and express it as PVIF(10%,3yrs)

or

PVIF(10,3)

.

PVIF(i, n)= 1/(1+i)n


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Therefore to determine the present value of a future sum, we have to just locate the PVIF

factor for the given values of i and n and multiply the factor value with the given sum.

Since PVIF(i,n) represents the present value of Rs.1 receivable after n years at a rate of

interest of i%. It is obvious that PVIF values can not be greater than one. The PVIF valuesfor different combinations of i and n are given in Appendix X at the end of this book.

Compounded table and Future value formulae:

The inverse of PVIF is Future Value Interest Factor (FVIF). The above equation can be

written as follows:

PVn = Present Value = FVn = FVn * PVIF(i,n)(1+i)n

Therefore, Compounded value or Future value = FVn = PVn* (1+i)n = PVn* FVIF(i,n).

FVIF(i,n) = (1+i)n

Therefore, PVIF(i,n) = 1/ FVIF(i,n)

The FVIF values for different combinations of i and n are given in Appendix Y at the end

of this book.


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Example 7:

Rama raju promises you to give you Rs.5,000 after 10 years in exchange for Rs.1,000

today. What interest is implicit in his offer?

Let i be the rate of interest.

It is given that Rs.1000* FVIF(i,10) = 5000, FVIF(i,10) = 5

From the tables we find FVIF(16,10) = 4.411And FVIF(18,10)= 5.234

Applying a linear approximation in the interval, we get

i = 16% + 2% * (5.04.4111) = 17.4%.(5.2344.411)


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Present Value


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Annuity:

An annuity is an investment paying a fixed sum each year(A) for a specified period of

time(n) at the rate of interest k. Examples of annuities include many credit agreements

and house mortgages.

Present Value of an annuity

The present value of a regular annuity can be represented in terms of the symbols defined

in the table as follows:

PVAn = A/(1+i) + A/(1+i)2 + A/(1+i)3+ -------------+A/(1+i)n ------- (1)

Multiplying both sides by (1+i) we get,

PVAn(1+i) = A + A/(1+i) + A/(1+i)2+ ----------+A(1+i)n-1 ------------(2)

Subtracting equation (1) from equation (2), we get

PVAni = AA/(1+i)n = A[ 1 1/(1+i)n]

PVAn = A [(1+i)n1] = A*PVIFA(i,n)

i* (1+i)n

The PVIFA(i,n) values for different combinations of k and n are given in Appendix Z1 atthe end of this book.


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Example 8:Suppose an annuity of Rs.1000 is issued for 20 years at 10%. Using the table in Appendix

Z1, we can find the present value as follows:PVA(10,20) = Rs.1,000 * PVIFA(10,20) = Rs.1,000 * 8.5136 = Rs.8,513.60

Future value of an annuity

The future value of a regular annuity can be expressed as follows:

FVAn = A(1+i)n-1 +A(1+i)n-2 + ----- A --------(3)

Multiplying equation (1) by (1+i) on both sides, we get

FVAn(1+i) = A(1+i)n +A(1+i)n-1 + ----- A(1+i) ---------(4)

Subtracting equation (3) from equation (4), we get

FVAn

= A[(1+i)n1]/i = A* FVIFA(i,n)

Therefore, PVIFA(i,n) = FVIFA(i,n)*PVIF(i,n)

The FVIFA(i,n) values for different combinations of i and n are given in Appendix Z2 at

the end of this book.


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Example 9:

You can save Rs.6,000 a year for 5 years. What will be these savings cumulate to at the

end of 10 years, if the rate of interest is 10%?

The accumulated value after 10 years will be as follows:

= Rs.[6,000*FVIFA(10,5)*FVIF(10,5)] = Rs.[6000*6.105*1.611] = Rs.59010.93

Time Amount Amount Amount Amount Amount

1

2

3

4

5

MV

1,000

1,000(1.1)5

1,000

+1,000(1.1)4

1,000

+1,000(1.1)3

1,000

+1,000(1.1)2

1,000

+1,000(1.1)


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Time Amount Amount Amount Amount Amount

1

2

3

4

5

MV

1,000

1,000

1,000

+1,000(1.1)-1

1,000

+1,000(1.1)-2

1,000

+1,000(1.1)-3

1,000

+1,000(1.1)-4


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Annuity Due

In case of annuity, it is generally assumed that payment has been made at the end of

each year. If the payment is made at the beginning of each year, then this is called

annuity due and its value is found by the product of the value (either present orfuture) of a regular annuity and the factor (1+i).

FVAn(due) = A*FVIFA(i,n)*(1+i)

PVAn(due) = A*PVIFA(i,n)*(1+i)

Example 10:

What is the present value of a 5 year annuity due of Rs. 5,000 at 10%?

PVAn(due) = A*PVIFA(i,n)*(1+i) = Rs.5,000*PVIFA(10,5)*(1+0.1) = Rs.2,0850.5

Perpetuity

Frequently, an investment pays a fixed sum each year for a specified number of years.

A series of annual receipts or payments is termed an annuity. The simplest form of an

annuity for an infinite series is called Perpetuity.


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For example, certain government stocks offer a fixed annual income, but there is no

obligation to repay the capital. The present value of such stocks (called irredeemables)

is found by dividing the annual sum received by the annual rate of interest:

PVperpetuity = A/I

Example 11:

Uncle Shyam wishes to leave you in his will an annual sum of Rs.10,000 a year starting

next year. Assuming an interest rate of 10%, how much of his estate must be set aside

for this purpose?

PVperpetuity = A/i

PV = Rs.10,000/0.1 = Rs.100,000

Let your benevolent uncle now wishes to compensate for inflation estimated to be at

6% per annum (say).

Then PV = A/(i-g) = Rs.10,000/(0.1-0.06) = 250,000.

Similarly, Present value of growing perpetuity = A/ (i-g) where, g = growth rate.


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Calculating interest rates

We know that PV = FV* PVIF(i,n)

Therefore, PVIF(i,n) = PV/FV = 1/(1+i)n

Or, (1+i)n = FV/PV

i = (FV/PV)1/n -- 1

Example 12:

A credit company may offer to lend you Rs.1,000 today on condition that you repay Rs.

1,643 at the end of three years. Then what is the compound interest rate for this offer?

i = (1643/1000)1/31 = 18%


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Numerical Problems

1. Assume that it is now January 1 2003. On January 1, 2004, you will deposit Rs 1,000

into a savings account that pays 9%.

i) If the bank compounds interest annually, how much will you have in your

account on January 1, 2007?

ii) Suppose you deposited four equal payments in your account on January 1 of

2004, 2005, 2006 and 2007. Assuming a 9% interest rate, how much would each ofyour payments have to be for you to obtain the same ending balance as you

calculated in part (i) ?

2. Assume that it is now January 1, 2009 and you will need Rs 1 lakh on January 1,

2013. Your bank compounds interest at a 9% annual rate.

i) How much must you deposit on January 1, 2010 to have a balance of Rs 1 lakh onJanuary 1, 2013.

ii) If you want to make equal payments on every January 1 from 2010 to 2013 to

accumulate Rs 1 lakh, how much must each of the four payments be?


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3. Bank MBI pays 6% interest, compounded quarterly, on its money market account. The

managers of the bank NBI want its money market account to equal bankMBIs

effective annual rate, but interest is to be compounded on a monthly basis. What

nominal interest rate must bank NBI set?

4. Use equations and a financial calculator to find the following values :

i) An initial Rs 1000 compounded for 10 years at 9%.

ii) The present value of Rs 1000 due in 10 years at a 9% discount rate.

5. To the closest year how long will it take Rs 4000 to double if it is deposited and earnsfollowing interest rates?

i) 9%

ii) 10%

iii) 100%

6. While Nagarjuna was a student at the University of Chennai, he borrowed Rs 120,000in student loans at an annual interest rate of 9.5 %. If he repays Rs 15000 per annum,

how long to the nearest year will it take him to repay the loan?

7. Which amount is worth more at 14% - Rs 1,000 in hand today or Rs 2,000 due in 6

years?


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8. X Ltd invests Rs 8 lakhs to clear a tract of land and to plant some young mahogany

trees. The trees will mature in 10 years, at which time X Ltd plans to sell the forest at

an expected price of Rs 80 lakhs. What is X Ltds expected rate of return?

9. Find the present values of the following cash flow streams. The appropriate interest rate

is 9%.

Year Cash flow

0 -1000

1 150

2 200

3 200

4 300

iso-885dfdfd9-1__time value of money 2

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