Unit 2 

CONCEPTS IN VALUATION

2.1 INTRODUCTION

In our economics life, money is not free. Money has time value. Interest rates give money its time value. If the investor has some spare cash or funds, he can invest it in savings deposit in a bank and receive more money later. If the investor wants to borrow money, he must repay a larger amount in the future due to interest. The result is that Rs. 100 in hand today, is worth more than Rs. 100 to be received a year from now. This is because Rs. 100 today can be invested to provide Rs. 100 plus interest after a year. The interest rates in the economy provide money with its time value. There are two types of decisions which requires some consideration of time value. The first decision involves investing money now in order to receive future cash benefits. The other decision involves borrowing now to take current expenditure at a cost of having less money in the future. The intelligent investor requires familiarity with the concepts of compound interest.

2.2 TIME VALUE OF MONEY

In the world of finance and investment, time does have a value, Rs. 100 today are more valuable than Rs. 100 a year later. This is because capital can be employed productively to generate positive returns. Again, individuals normally prefer current consumption to future consumption. Even in case of inflation, Rs. 100 today represents greater real purchasing power as compared to Rs. 100 one year later. The longer the term of a loan, the greater the amount that must be paid due to interest. Bonds are worthless to an investor, if the maturity is longer. Therefore, this makes sense under the general framework of the time value of money.

2.3 BASIC CONCEPTS

(a)               PRESENT VALUE: A present value is the discounted value of one or more future cash

Flows.

(b)               FUTURE VALUE: A future value is the compounded value of a present value.

(c)               DISCOUNT FACTOR: The discount factor is the present value of a rupee received in

The future.

(d)               COMPOUNDING FACTOR: The compounding factor is the future value of a rupee.

Discount and compounding factors are functions of two things: (i) the interest rate used, and (ii) the time between the present value and the future value. The discount factor decreases as time increases. The discount factor also decreases as interest rate increases.

2.4 TECHNIQUES OF VALUATION AND TIME VALUE OF MONEY

RELATIONSHIP

There are two techniques of adjusting the time value of money:

I. Compounding Technique

II. Discounting or present Value Technique

I. Compounding Technique

The time preference for money encourages person to receive the money at present instead of waiting for future. But he may like if he is duly compensated for the waiting time by way of ensuring more money in future.

For example, a person being offered Rs. 100 today may wait for a year if he is ensured of Rs. 100 at eh end of one year (taking his preference for an interest of 10% p.a).

The future value at the end of period 1 can be calculated by a simple formula given below:

FV = PV (1+i)n

Where, FV = Future value at the end of period 1

PV = Value of money at time 0 i.e. original sum of money

I = Rate of interest

n = Number of years

Taking the example given above, the value of money after 2 years is given below:

FV = PV (1+i)n

= 100 (1+.10)2

= Rs. 121

II. Discounting or Present Value Technique

Present value is the exact opposite of compound or future value. While future value shows how much a sum of money becomes at some future period, present value shows what the value is today of some future sum of money. The present value of money to be received on future date will be less because we have lost the opportunity of investing it at some interest.

For example, there is an opportunity to buy a debenture today and we get back Rs. 1000 after one year. What will you be willing to pay for the debenture today if your time preference for money is 10% p.a? We can calculate the present value of Rs. 1000 to be received after one year at 10% time preference rate as below:

PV = FV / (1+i)t

The basic time value of money relationships is presented in the following equations:

PV = FV x DF

FV = PV XCF

Where, PV = Present value

FV = Future value

DF = Discounting factor =   1    

(1 + R)t

CF = Compounding factor = (1 + R)t

Where,  R = Rate of interest

t = time in years.

2.5 FUTURE VALUE OF SINGLE AMOUNT

The future value of an amount invested or borrowed at a given rate of interest can be calculated if the maturity period is given. Suppose, a deposit of Rs. 5,000 gets 10 percent interest compounded annually for a period of 3 years, the future value will be:

PV X CF = 5,000 (1.10)3 = 5,000 x 1.311 = Rs. 6,655.

Illustration 1:

Shashikant deposit Rs. 1, 00,000 with a bank which pays 10 percent interest compounded annually, for a period of 3 years. How much amount he would get a maturity?

Solution:

FV  = PV X CF

= 1,00,000 x (1.10)3

= 1,00,000 x 1.331

= Rs. 1,33,100

Mr. Shashikant will get Rs. 1,33,100 after 3 years.

2.6 FUTURE VALUE OF ANNUITY

An annuity is a series of payments of a fixed amount for a specified number of periods. When payment is made at the end of each year, it is called ordinary annuity. On the other hand, when the payments are made at the beginning of the year, it is called an annuity due. Normally, it is assumed that the first annuity payment occurs at the end of the first year.

FVa = A x  (1 + R)t – 1

R

Where,A = Periodic cash payments

R = Annual interest rate

t = time in years / duration of annuity

The value of (1+R)t – 1 can be determined by using the Time value of money tables.

R

The Future Value Interest Factors (FVIFA) for various years are a shown in table:

Illustration 2:

Four equal annual payments of Rs. 5,000 are made into a deposit account that pays 8 percent interest per year. What is the future value of this annuity at the end of 4 years?

Solution

The future value of annuity FVa  = A x (1 + R) t – 1

R

= Rs. 5,000 x FVIFA @ 8%

= Rs. 5,000 x 4.5061

= Rs. 22530.50

2.7 DOUBLING PERIOD

Sometimes, investor should know how long it will take to double his money at a given rate of interest. In this case, a rule of thumb called the rule of 72, can be used. This rule works pretty well for most of the interest rates. The rule of 72 says that it will take seventy-two years to double your money at 1 percent interest. You can calculate the doubling by dividing 72 by the interest rate. You can also estimate the interest rate required to double your money in the given number of years by dividing number of years into 72.

For example, if the interest rate is 12 percent, it will take 6 years to double your money (72+23). On the other hand, if you want to double your money in 6 years, the interest rate should be 12 percent (72+6).

A more accurate method used for doubling your money is using the rule of 69. According to this rule, the doubling period of an investment is = 0.35 + 69 Thus the doubling period of Interest rate investment of different rates of interest can be determined as follows:

(1)               Interest rate 12%

0.35 + 69 = 0.35 + 5.75 = 6.1 years

12

(2)               Interest rate 15%

0.35 + 69 = 0.35 + 4.60 = 4.95 years

15

Illustration 3:

If the interest rate is 10%, what are the doubling periods of an investment at this rate?

Solution

(a)               As per rule of 72, the doubling period will be

72 = 7.2 years

10

(b)              As per the rule of 69, the doubling period will be

= 0.35 + 69

10

= 0.35 + 6.9 = 7.25 years

PRESENT VALUE:

Many times, investors like to know the present value which grows to a given future value. Suppose, you want to save some money from your salary to but a scooter after 5 years. You should know how much money should be put into bank now in order to get the future value after 5 years. The present value is simply the inverse of compounding used in determining future value. The general relationship between future value and present value is given in the following formula:

PV = FV x DF

= FV x       1     

(1+R)

Illustration 4:

Find the present value of Rs. 50,000 to be received at the end of four years at 12 percent interest compounding quarterly.

Solution:

PV = FV x PVIF at 12%

= Rs. 50,000 x 0.623

= Rs. 31,150

2.7 PRESENT VALUE OF AN UNEVEN SERIES OF PAYMENTS

The annuity includes the constant amount in which cash flows are identical in every period. Many financial decisions involve constant cash flows, however, some important decisions are concerned with uneven cash flows. For example, investment in shares is expected to pay an increasing series of dividends over time. The capital budgeting projects also do not normally provide constant cash flows.

In order to deal with uneven payment streams, we have to multiply each payment by the appropriate PVIF and then sum these products to obtain the present value of an uneven series of payments.

Illustration 5:

Mr. Shah has invested Rs. 50,000 on Xerox machine on 1st Jan. 2002. He estimates net cash income from Xerox machine in next 5 years as under.

At the end of 5th year Machine will be sold at Scarp value of Rs. 5,000. Advise him whether his project to viable, considering interest rate of 10% p.a.

Solution:

Calculation of Present Value of Future Cash Flows:

Note: It is assumed that the net cash income is received at the end of the year.

Considering 10% interest rate, the net present value of all future cash flows is Rs. 75,635 which is higher than present net cash flow of Rs. 50,000. Thus, the project is viable.

2.8 PRESENT VALUE OF ANNUITY

Many times investors want to know the present value which must be invested today in order to provide an annuity for several future periods. For example, a grandfather wants to deposit enough money today to meet the tuition fees of his grand-son for the next three years. The interest rate is 8%. The present value of this annuity is the sum of the present values of all the future inflow of the annuities. The present value of an annuity can be expressed in the following formula:

PVA1  = A x 1    +    1    +    1    

(1+R)3          (1+R)2          (1+R)1

=  (1+R)t-1

R(1+R)t

Where  PVA1  = Present value of an annuity with a duration of t’ periods

A  = Constant periodic flow R = Interest Rate

The present value interest factors for an annuity (PVIF) can be determined by using the Time Value of Money Tables. The (PVIF) for various years are given below:

For all positive interest rates, PVIFA for the present value of an Annuity is always less than the number of periods the annuity runs, whereas FVIFA for the future value of an annuity is equal to or greater than the number of periods.

Illustration 6:

What is the present value of a 4 years’ annuity of Rs. 8,000 at 12% interest?

Solution:

PVA   = (1+R)t-1

R(1+R)t

The value of (1+R)t – 1 as per table is 3.0373

R(1+R)t

= Rs. 8,000 x PVIF at 12%

= Rs. 8,000 x 3.0373

= Rs. 24.298

2.9 NET PRESENT VALUE

Net Present Value (NPV) is the most suitable method used for evaluating the capital investment projects. Under this method, cash inflow and outflows associated with each project are worked out. The present value of cash inflows is calculated by discounting the cash flows at the rate of return acceptable to the management. The cash outflows represent the investment and commitments of cash in the project at various points of time. It is generally determined on the basis of cost of capital suitably adjusted to allow for the risk element involved in the project. The working capital is taken as a cash outflow in the initial year. The cash inflow represents the net profit after tax but before depreciation. Depreciation is a non-cash expenditure hence it is added back to the net profit after tax in order to determine the cash inflows. The Net Present Value of cash inflows and the present value of cash outflows. If the NPV is positive the project is accepted, and if it is negative, the project is rejected.

Discounted cash flow is an evaluation of the future net cash flows generated by a project. This method considers the time value of money concept and hence it is considered better for evaluation of investment proposals. If these are mutually exclusive projects, this method is more useful. The Net Present Value is determined as follows:

NPV = Present value of future cash inflows – Present value of cash outflows.

Illustration 7:

An investment of Rs. 40,000 made on 1/4/2002 provides inflows as follows:

Which alternative would you prefer in the investor’s expected return is 10%? Give reason(s) for your preference.

Solution:

Calculation of Present Values:

Alternative I

Alternative II

The net present value of all future cash flows is Rs. 40,789 in case of Alternative I and Rs. 39,962 in case of II. The NPV in case of alternative is higher at 10% discounting factor. Hence, alternative I is preferred for investment.

Illustration 8:

A Finance company has introduced a scheme of investment of Rs. 40,000. The returns would be Rs. 8000, 10000, 11000 and 12000 in the next five years. The indicated rate of interest is 10%. Compute the present value of the investment and advice regarding the investment.

Solution:

(i)                 Present value of investment = Rs. 40,000.

(ii)               Present value of returns:

(iii)      Present value of investment is Rs. 37,188 which is lower than investment of Rs. 40,000. The net present value (i.e. 37,188 - 40,000 = Rs. 2,812) is negative. Hence the investment is not profitable at 10% interest.

Illustration 9:

The share of Ridhi Ltd (Rs.10) was quoting at Rs. 102 on 01.04.2002 and the price rose to Rs. 132 on 01.04.2005. Dividends were received at 10% on 30th June each year. Cost of funds was 10%. Is it a worth-while investment, considering the time value of money? (Present value factor at 10% were 0.909, 0.826 and 0.751).

Solution:

Calculation of Present Value of Cash inflows:

Considering the time value of money, the NPV is negative, hence, it is not a wise investment.

Illustration 10:

XYZ & Co. Is considering investing in a project requiring a capital outlay of Rs. 2,00,000. Forecast for annual income after tax is as follows:

Evaluate the project on the basis of Net Present Value taking 14% discounting factor and advise whether XYZ & Co. Should invest in the project or not? The Present value of Re. 1 at 14% discounting rate are 0.8772, 0.7695, 0.6750, 0.5921 and 0.5194.

Solution:

Depreciation = 20% of 2,00,000 = Rs. 40,000 Profit after tax is given.

The cash inflow after tax (CFAT) = Profit After Tax (PAT) + Depreciation.

Net Present Value is positive, hence XYZ & Co should invest in the project.

Illustration 11:

Find out the present value of a debenture from the following:

(Present values of Re. 1 at 12% are, 0.8929, 0.7972, 0.7118, 0.6355 and 0.5674)

Solution:

PVd  = I (PVAF) + FV (DF)

= I (PVAF 12% for 5 years) + FV (DF 12% for 5 years)

= 150 (3,6048) + 1,000 (0.5674)

= Rs. 540.72 + 567.40

= Rs. 1108.12

Illustration 12:

Mr. Vishwanathan is planning to buy a machine which would generate cash flow as follows:

If discount rate is 10%, is it worth to invest in machine?

Solution:

Calculation of Net Present Value

As the NPV is positive, it is worth investing in the machine.

Illustration 13:

A machine cost Rs. 80,000 and is expected to produce the following cash flows:

If the cost of capital is 12 percent, is it worth buying the machine?

Solution:

Calculation of Net Present Value

As the Net Present Value is negative, it is not worth buying the machine.

Illustration 14:

Find the compounded value of annuity where three equal yearly payments of Rs. 2000 are deposited into an account that yields 7% compound interest.

Solution:

The future value of annuity FVa (for 3 years) = A x (1 + R)t – 1

R

= Rs. 2,000 (FVAFA @ 7%

= Rs. 2,000 x 3.215

= Rs. 6,430

Illustration 15:

Calculate the compound value when Rs. 10,000 are invested for 3 years and the interest on it is compounded at 10% p.a semi-annually.

Solution:

FV  = PV x CF

FV  = PV x (1 + R)t

= 10,000 X (1 + 2 ) 2 x 3

= 10,000 (1.05) 6

= Rs. 10,000 x 1.340

= Rs. 13,400

2.10 INTERNAL RATE OF RETURN

When the present value of cash inflows is exactly equal to the present value of cash outflows we are getting a rate of return which is equal to our discounting rate. In this case the rate of return we are getting is the actual return on the project. This rate is called the IRR.

In the net present value calculation we assume that the discount rate (cost of capital) is known and determine the net present value of the project. In the internal rate of return calculation, we set the net present value equal to zero and determine the discount rate (internal rate of return) which satisfies this condition.

Both the discounting methods NPV and IRR relate the estimates of the annual cash outlays on the investment to the annual net of tax cash receipt generated by the investment. As a general rule, the net of tax cash flow will be composed of revenue less taxes, plus depreciation. Since discounting techniques automatically allow for the recovery of the capital outlay in computing time-adjusted rates of return, it follows that depreciation provisions implicitly form part of the cash inflow.

Internal rate of return method consists of finding that rate of discount that reduces the present value of cash flows (both inflows and outflows attributable to an investment project to zero. In other words, this true rate is that which exactly equalises the net cash proceeds over a project's life with the initial investment outlay.

If the IRR exceeds the financial standard (i.e. cost of capital), then the project is prima facie acceptable. Instead of being computed on the basis of the average or initial investment, the IRR is based on the funds in use from period to period.

The actual calculation of the rate is a hit-and-miss exercise because the rate is unknown at the outset, but tables of present values are available to aid the analyst. These tables show the present value of future sums at various rates of discount and are prepared for both single sums and recurring annual payments.

What Does IRR Mean?

There are two possible economic interpretations of internal rate of return: (i) Internal rate of return represents the rate of return on the unrecovered investment balance in the project. (ii) Internal rate of return is the rate of return earned on the initial investment made in the project.

Formula for calculating IRR:

NPV i.e 0 =  CF0 +   CF1 +   CF2 +   CF3 +   CFn 

(1+IRR)1        (1+IRR)2        (1+IRR)3        (1+IRR)n

Where,

CF0   = Initial Investment

CF1, CF2, CF3, CFn = Cash flows of respective years

n   = each period

NPV   = Net Present Value

IRR   = Internal Rate of Return

Evaluation

A popular discounted cash flow method, the internal rate of return criteria has several virtues:

•         It takes into account the time value of money.

•         It considers the cash flow stream in its entirety.

•         It makes sense to businessmen who want to think in terms of rate of return and find an absolute quantity, like net present value, somewhat difficult to work with.

The internal rate of return criteria, however, has its own limitations.

•         It may not be uniquely defined. If the cash flow stream of a project has more than one change in sign, there is a possibility that there are multiple rates of return.

•         The internal rate of return figure cannot distinguish between lending and borrowing and hence a high internal rate of return need not necessarily be a desirable feature.

The internal rate of return criterion can be misleading when choosing between mutually exclusive projects that have substantially different outlays. Consider projects P and Q

Both the projects are good, but Q, with its higher net present value, contributes more to the wealth of the stockholders. Yet from an internal rate of return point of view P looks better than Q. Hence, the internal rate of return criterion seems unsuitable for ranking projects of different scale.

2.11 MATHEMATICAL TABLES

Table A- 1 Present Value of Re. 1:

PVIF =      1    

(1 + k)n

Table A-1 (continued)

Table A- 2 Present Value of an Annuity of Re. 1 per period for n periods:

1 -   1    

(1+k)n

k

PVIFA =    1    

(1+k)1

Table A-2 (continued)

Table A-3 Future Value of Re. 1 at the end of n Periods.

FVIF = (1+k)n