INVESTMENT ANALYSIS TECHNIQUES
Expenditures in business are classified into two major groups – Revenue Expenditures, and Capital Expenditures Revenue Expenditures refers to annual operating inputs the benefit of which does .riot extend, beyond one accounting period, usually one year. Examples of revenue expenditures are such inputs as fertilizer, feed, seed, fuel, electricity and so on. On the other hand capital expenditures refer to the input of long lasting capital assets such as land, machinery, building, breeding stock and so on. The benefit of these assets beyond one accounting period.
Investment analysis involves the entire process of planning expenditures whose returns are expected to extend beyond one year.
Investment analysis is one of the most important decisions in any business organization for the following reasons:
It involves committing large funds for future benefits, making the decision make a hostage of future events.
Availability of capital assets should be properly timed.
The capital assets purchased must be of good quality
Asset expansion involves substantial expenditures.
Capital budgeting requires the ability to compete.
That is the decision makers must understand business investment behavior and factors that motivate firms to undertake investment programmes.
Investment analysis/an application of the economic theory which states that a firm should operate at the point where its marginal revenue is just equal to its marginal cost. In budgeting however marginal revenue is taken to be the percentage rate of return, IRR on investment while marginal cost is the firm’s marginal cost of capital MCC. That is, a firm should operate at the point where IRR = MCC.
The task of investment analysis is not only to generate ideas of investment opportunities but also to choose among them. Ideas of many good investment opportunities result from hard thinking careful planning and large funds for research and development. After generating the ideas, we assemble a list of the proposed new investments together with the data necessary to appraise them. There are two main types of investment proposals and they are:
Replacement and Expansion.
Out of the investment proposals available to a farm business, some may be good, while others are poor. Futhermore, a farm business may have more projects than it is able to finance or willing to finance. In such cases, ranking the projects will be very helpful in deciding the projects to choose. The choice may be among independent projects, the firm through investment analysis determines how many projects should be selected. And for mutually exclusive projects, the firm determines which one of the several projects should be selected. The three important methods for ranking investment proposals are as follows:
We have earlier listed the three methods of ranking a project as:
Net present value
Internal rate of return. Each of these methods will be discussed and illustrated with the same investment proposal.
Payback period is the number of years it takes a firm to recover its original investment from net cash flows. The formula for payback period is as follows:
But this formula should not be applied when the expected annual returns vary.
Ide Agrofeed Ltd. Has two investment proposals, project A and B which have initial outlay of N10000 respectively and a life of 6 years each. The expected income streams are shown below:
Calculate the payback period of project A and project B.
We cannot apply the formula for payback period since the expected income streams vary over the years. Therefore we shall get the payback period by simple proportion as shown below:
N9000 is recovered in 2 years.
The remaining N1000 is recovered in 1/3 years i.e.
Therefore N10,000 is recovered in 21/3 years.
The whole N10,000 is recovered in 4 years.
Our calculation has shown that project A has a short payback period than project B. therefore project A should be preferred to project B.
It should be noted however that payback period ranking method has several disadvantages, primary among which is that it does not take into account the time value of money.
Net Present Value (NPV)
This method of ranking is also known as the discounted cashflow method. It involves finding the present value of the expected net cash flows of a project, discounted at the cost of capital and substracting from it the initial cost outlay of the project. If the present value is positive the project should be accepted but if it is negative, the project should be rejected. The formula for finding Net Present Value is given
t = time in years showing from 1 to n years
where NCF = Net cash flow
K = marginal cost of capital
I = Initial cost of the project
Find the NPV of projects A and B proposed by Ide Agrofeeds assuming that the cost of capital is 15%.
PV of inflows =
Less Project Cost =
Note that independent projects with positive NPV are accepted while those with negative NPV are discarded. On the other hand, for mutually exclusive projects we accept only the one with the highest NPV. Based on this explanation we can now interprete the above result for project A and B. If they are independent projects then both will be accepted because their NPVs are positive. But if they are mutually exclusive projects, then project B which has a higher NPV is accepted as a better alternative.
Internal Rate of Return (IRR)
Where NCF = Net cashflows
I = Initial cost of project
T = Time in years starting from 1 to n years.
But when the annual cash flows from the investment is uniform, then we apply the following simple formula:
In each case we know the initial cost outlay, I and the net cashflows of the investment, but we do not know the value of n. According to the definition, IRR is the value of r that satisfies equation (1) above.
Find the IRR of projects A and B proposed by the Ide Agro feeds Limited assuming that the cost of capital is 15%.
Note: Since the cashflows from projects A and B are not uniform, we cannot solve the problem by applying equation (2) above, rather we resort to interpolation. Interpolation involves trial and error until we arrive at the value of r that satisfies equation (1). The procedure for interpolation is as follows:
Find a rate that will give a positive NPV for the project and call that rlow or rl;
Find another rate that will give a negative NPV for the project and call this rhigh or rh;
The negative NPV is designated NPV1 while the positive NPV is designated NPVh.
Substitute NPVh, NPV1, rh and r1 in the equation given below:
For interpolation we use the equation:
A useful hint is that the cost of capital of most firms is in the range of 10 to 15%. For this reason, it is advisable to start the trial from 10 percent. Then from that point we move a “little bit to the right and a little bit to the left” depending on the outcome of each trial.
Project A (with r1 = 15, and rh = 16).
Discounted Net Cashflow
PV of Inflows =
Less project cost =
10 (+ve NPV)
Discounted Net Cashflow
PV of Inflows =
Less project cost =
– 153 (-ve NPV)
So far, we have obtained r1 = 15, rh = 16, NPV1 = -153 and NPVh = 10. We then interpolate with this data to get IRR for project A as follows:
Project B (with rh = 20% and r1 = 18%).
Discounted Net Cashflow
PV of Inflows =
Less Project cost =
581 (+ve NPV)
Discounted Net Cashflows
PV of Inflows =
Less Project Cost =
-94 (-ve NPV)
So far we have obtained r1 = 10%, rh = 20%, NPV1 = -94 and NPVh = 581.
From calculations, the IRR of project B is higher than the IRR of project A. therefore project B is preferred to project A.
Summary of the Analysis
Net present value
Internal Rate of Return
The selection of projects depends on whether they are independent projects or mutually exclusive projects. For the above illustration we shall select from projects A & B (i) when they are independent and (ii) when they are mutually exclusive.
If projects A & B are independent we shall select as follows:
May be accepted
May be accepted
If projects A & B are mutually exclusive we shall select as follows
Whenever there is a conflict among the methods, we rest our judgement on the result of NPV. This is because the NPV represents the value added by the firm if the project is executed. But IRR measures only rate and not value added. Note that the IRR is the rate of return that comes to firm beyond the firm’s cost of capital. The IRR should be greater than the cost of capital for the project to be profitable.
So if projects A and B are independent then both are accepted. But if they are mutually exclusive then only B is accepted.
In the above illustration, the expected cashflows from projects A and B varied over the years. But for some proposals. The expected annual cashflows are the same throughout the life of the projects. In the following illustration, we shall survey proposals in which the expected annual cashflows do not vary.
There are two projects A and B with initial capital outlays of N9000 and N21000 respectively. Project A has an economic life of 3 years while B has 10 years. The benefits of each project are:
Direct labour savings
Indirect labour savings
Maintenance & Repairs Savings
The projects are mutually exclusive. Which of the projects will the firm accept? Use straightline method of depreciation. Tax rate is 40%
We shall take the following steps:
Find the annual depreciation
Find the net cashflows by applying the formula, net cashflows = (TR-CE) (1-T) + (D) (T)
Rank the projects by the three methods.
Annual depreciation of project A =
Annual depreciation of project R =
Calculating net cashflows:
Total or Annual Benefits
Less operating cash expenses (or CE) 0 0
Less annual depreciation 3000 2625
Less Tax 40% (i.e. 0.4 x 2500) 1000 3350
Net income after tax 1500 5025
Add depreciation 3000 2625
Net cashflows 4500 7650
Payback period – The annual cashflows in this case do not vary so we shall apply the following formula:
LOAN INTEREST RATES AND LOAN REPAYMENTS
Loan refers to the amount borrowed by a debtor from a creditor, with the promise to return the amount and pay interest for its use. Borrowed money may help to start a business, grow, expand and make additional profit. Farmers and ranchers can and do borrow money from a number of different services. In this section, we shall discuss:
Loan repayment plans
Interest rate determination
Factors affecting the cost of credit
LOAN REPAYMENT PLAN
There are various forms of loan repayment schedules but the two common ones are: i. Single-payment loan schedule and ii. Amortised loan schedule
Single-payment loan schedule – In this schedule all the principal is repaid in one lump sum when the loan is due while interest is paid as agreed with the creditor – either at the beginning or at the end of the payment period, or even annually. This type of repayment schedule is made for short-term loans and some intermediate loans. It requires a good cash flow planning to ensure that sufficient cash will be available when the loan is due.
Amortised Loan Schedule: An authorized loan is one which has periodic interest and principal payments. So amortised loan schedule is a plan to make periodic interest and principal payments. The payment may be equal principal payment or equal total payment.
For equal principal payment, the schedule has the same amount of principal due on each payment date plus interest on the unpaid balance. The first payment is usually the largest with successive payments, becoming progressively smaller. This is because since the unpaid balance decreases with each principal payment the interest payments are decreasing.
For equal total payment, smaller principal payments are made towards the early years but larger principal payments towards the end. A large portion of the total loan payment is interest in the early years, but the interest decreases and the principal increases with each payment, making the last payment mostly principal.
The amortised loan schedule is a repayment plan usually made for long-term loans and some intermediate loans.
Interest Rate Determination
We have learnt that firms may borrow money from financial sources for use in their businesses, and that the loan is returned with interest. There can be a variety of quoted or stated interest rates and different finance charges of other types. But the stated interest is not necessarily the effective interest. The borrower often is made to pay more than the stated rate of interest. Determination of the effective, or true, rate of interest on a loan depends on the stated rate of interest and the method of charging interest by the lender. One method of charging interest is called the regular interest in which interest is paid at maturity of the loan. The other method is the discounted interest in which the interest is paid in advance. A firm planning to borrow from financial sources is advised to look for the best combination of interest rate and loan terms or repayment schedule. In the remaining part of our discussion, we shall show how to determine the true or effective rate of interest on a loan under.
Single payment loan and 2. Amortised loan. The effective rate is what the borrower will use to compare loan offers.
Single Payment Loan:
Regular Interest: In this case the loan is paid at maturity together with interest. The stated rate of interest is the effective rate of interest. For example, on a N10,000 loan for one year at 7%, the interest is N700. This stated rate of interest (7%) can be shown to be the same as the effective rate of interest in the following way:
Interest paid at maturity =
Discounted Interest (Single payment) – In this case the interest is paid in advance. This makes the stated rate of interest different from the effective rate. On the N10000 loan for one year at 7% the discount is N700, and the borrower obtains the use of N9300. The effective rate of interest is calculated as follows:
Effective rate =
So while the stated rate of interest is 7%, the effective rate is 7.5%.
Amortised Loan – Instalment loan has been given as an example of amortised loan. Consider the N10000 loan for one year at 7% which is repaid in 12 monthly instalments. We can calculate the effective rate of interest as shown below:
Regular interest on instalment loan:
If the loan is repaid in 12 monthly instalments but the interest is calculated on the original balance, then the effective rate of interest will be quite high. The reasoning is as follows: The borrower has the full amount in the first month. As from the second month repayment commences and so the amount available to the borrower decreases and in the last month only 1/12 of the total loan will be available. On the average the borrower makes use of only about half of the loan or N5000. Based on this N5000.
b. Discounted interest on instalment loan:
If the loan is repaid in 12 monthly instalments but the interest is calculated on the original loan balance then the effective interest is even higher than that of regular interest. The reasoning is the same as in calculating the regular interest on instalment loan above except that in this case the interest is paid in advance. That is the borrower has N10,000 – N700 or N9300 in the first month. The average amount used by the borrower throughout the period is N9300÷2 or 4650. The effective rate of interest is calculated as shown below:
Interest rate on discounted instalment loan =
In our example note that interest is paid on the original amount (N10000) of the loan, not on the amount actually outstanding i.e. the declining balance. This causes the effective interest rate to be approximately double the stated rate. To illustrate this point further consider the following example on a N10000 loan for 1 year at 8% which is repaid in 2 instalments of 6 months each what is the effective rate when interest is repaid:
On the original amount?
On the unpaid balance or declining balance.
Interest paid on original amount:
Interest on original amount for 1 year = 10000 x 0.08 = N800
Average amount used during the period = N7500 (see 2(a)?
Effective rate of interest =
Interest paid on declining balance:
N10000 at 8% for ½ year = N400
N5000 at 8% for ½ year = N200
Notice that when interest is paid only on the unpaid balance (i.e. declining balance), the stated rate of interest is the same as the effective rate of interest. But when interest is paid on the original balance, the effective rate of interest is higher than the stated rate.
FACTORS AFFECTING THE COST OF CREDIT
Cost of credit which is the interest and other financial charges paid by borrowers is affected by a number of factors.
These factors include the following:
Leaders’ Cost of Fund – This refers to the cost paid by lenders for keeping and using the funds in their custody. Lenders cost of fund is a function of some variables two of which are listed below:
Source of Fund – commercial banks for example are lenders and their source of fund is mainly customers’ deposits. The banks pay interest which is a cost on customers deposits. Other lenders get their funds from other sources and their cost of fund depend on the sources.
Demand and Supply conditions in the capital market – The interaction of forces of demand and supply in the capital market will determine the rate of interest to be paid by the lenders.
Risk Premium – Lenders face two kinds of risks:
Default risk and b. Market or interest rate risk
Default risk is the possibility that the loan will not be repaid at all or in full in accordance with the loan contract. The unpaid balance is bad debt. Lenders anticipate the possibility of bad debts and include them in the interest charged to borrowers.
On the other hand market default is the possibility of charge in the general level of interest rate after a loan is made.
Administration and Service Cost:
These are many and are affected by the size and terms of the loan. They include overhead coasts, cost of application, investigation, correspondence costs and so on.
This tends to reduce the real value of loan repayments. The real value of loan repayment depends on rate of interest charged, on the loan as well as inflation. Suppose a loan of N100 is given at 10% interest rate for one year. At 0% inflation N100 is repaid with N10 interest at the end of one year. But at 6% inflation. The N110 to be repaid is discounted at 6% to get the present value of repayment as follow:
At 6% inflation the present value =
Where n = Number ofyears
This calculation shows that the value of amount of repayment is reduced by inflation i.e. N103.80 instead of N110. Lenders are aware of this inflation and include it in the cost of credit.
This can also affect the cost of credit.
Lenders cost of fund
Administration and service cost
COST OF CAPITAL
Capital is a basic requirement of any business and as a factor of production it has a cost. The cost of capital has an effect in virtually every investment and financing decisions made in farm business. It is therefore important that farms and ranches will calculate accurately their cost of capital and use it in their business. But the task of calculating the cost of capital is difficult and this is because capital has a composite natures. The components of capital are as follows:
Each of these components of capital has a cost called the component cost. Example, the component cost of debt, component cost of preferred stock and so on. In estimating the cost of capital of projects, the component costs must be ascertained and taken into account. To this effect, a firm is viewed as an on-going concern and its cost of capital should be calculated as a weighted average, or composite of the various types of funds it uses. In the remaining part of this discussion we shall explain each component cost. And then we shall go on to show how to calculate cost of capital as a weighted average of the component costs. This weighted average cost of capital as a weighted average of the component costs. This weighted average cost of capital represents the actual cost of capital which the firm must use in its financial decision making.
Preferred stock is made up of preferred shares, also known as preference shares. This constituent part of the capital structure of a firm is not difficult to evaluate. When a firm issues preferred shares, it agrees to pay a fixed dividend to shareholders, assuming of course that there are profits available for distribution. Preferred stock is expensive and therefore an popular part of the capital structure. The reason why preferred stock is expensive is because the firm’s income is taxed before preferred dividends are evicted and this has the effect of raising the tax liability of the firm. The component cost of preferred stock is approximately equal to the returns investors receive on the shares. For example, if the preferred shares a dividend rate of 8%.
A complication may arise with the price of preferred shares, in that the shares may be issued at an amount which is lower than the par value. For example, suppose a firm issues preferred stock which sells at N100 a share and pays N8 annual dividend. However, the firm must incure a selling or floation cost of N4 a share. In other words, buyers of the preferred issue will pay N100 a share, but brokers charge a selling commission of N4 a share, so the firm nets N96 a share. In this case we shall calculate the component cost of preferred capital by applying the following formula:
Where Kp = component cost of preferred capital
D = Dividends per annum
S = Net proceeds of shares (or price).
For the example above the component cost of preferred capital is as follows:
A similar example may be given in percentage as follows:
A N1 preferred share is issued at N0.96 and carry a dividend of 8%. To calculate the component cost of preferred capital we apply the same formula as above.
Debt is another constituent of capital structure. It is rather similar to preferred share capital in the sense that the firm pays a fixed amount as interest to the investors. But unlike preferred shares, debt interest is tax, deductible under the current tax laws. This means that interest on debt is deducted from income of the firm before tax is applied thus reducing tax payable but preferred share dividend is deducted after taxation. The effect ofthis is that it is more economical for a firm to raise funds through debt capital than through preferred share capital. Since tax payable is reduced when interest on debt is charged against profit, stockholders are therefore concerned with after tax rather than before tax earning. For this reason, only the component cost of debt capital after taxes should be used.
Where Kd = Component cost of debt
b = before tax cost (or interest rate)
T = Tax rate
A firm pays interest on debt at the rate of 6%. If the tax rate is 48%, what is the firm’s component cost of debt?
Kd = b(1 – T)
Kd = 6% (1 – 48%)
= 0.06 (1 – 0.48)
Retained earnings represent profits which have been retained in the company rather than being distributed to the shareholders. Although retained earnings have not incurred a cost to the firm, there is an opportunity cost in respect of this constituent of capital structure. The opportunity cost arises because the shareholders have forgone dividends and could reasonably expect to obtain future income from these investments.
The component cost of retained earnings Kr, is defined as the rate of return shareholders require on the firm’s common stock. The famous Gordon-Shapiro model was developed to provide a simple model for calculating the component cost of retained earnings based on dividends yield and growth. The model is expressed as follows:
Where P0 = Current price of the stock
D1 = dividend expected to be paid per share.
g = expected growth rate of stock price.
In equilibrium the expected and required rates of return must be equal so we can solve for Kr.
A firm expected to earn N2 a share and to pay a N1 dividend during the coming year. The firm’s earnings, dividends, and stock price have all been growing at about 5% a year, and this growth is expected to continue indefinitely. The stock is in equilibrium and currently sells for N20 a share. Using this information we can calculate the required rate of return on the stock in equilibrium which by definition is the component cost of retain earnings.
This refers to newly issued common stock, or ordinary shares. A firm may decide to raise more funds by issuing new common stock. Selling new common stock involves floatation costs. For this reason the component cost of common stock is higher than that of retained earnings.
Assuming d1 = dividend yield, Po – Current price of the stock, f = percentage cost of selling the issue, g = growth rate then the component cost of new common stock, Kc is given as follows:
Where Pn = Po(1 – f), which is the net price received by the firm.
A firm expected to earn N2 a share and to pay a N1 divided during the coming year. The firm’s earnings, dividends and stock price have been growing at about 5% a year, and this growth rate is expected to continue indefinitely. The stock is in equilibrium and currently sells for N20 a share. Assume that the percentage cost of selling the issue is 10%. Then the component cost of common stock can be calculated as follows:
But P.A = P(1 – f) = 20
Then Kc =
So far we have learnt how to compute the various component costs of capital. Now we shall use these component costs to determine the weighted average cost of capital.
WEIGHTED AVERAGE COST OF CAPITAL:
If a company has only one source of capital, it would be relatively easy to calculate the cost of capital. However, most companies have a mixed capital structure, so the cost of capital has to be calculated by taking into consideration each constituent part of capital.
The first step in calculating the weighted average cost of capital, K is to determine the (1) cost of individual capital components as already described and illustrated. The second step is to establish the proper set of weights to be used in the averaging process. In doing this usually we assume that the present capital structure of the firm is at an optimum, where optimum is defined as the capital structure that will produce the minimum average cost of capital for raising a given amount of fund or a minimum cost of incremental capital. The third step is to find the proportion of each component with respect to the required fund. Then finally we compute the average cost of capital using the individual capital components and established set of weights.
The liability side of Coleman Tobacco Farms Limited is shown below:
Nm % of total assets
Payables and accurals 186 19.4
Tax accria;s 44 4.6
Total ‘free’ current funds 230 24.0
Interest bearing debt 160 16.7
Preferred stock 7 0.8
Common equity 560 58.5
Non-free fund 727 750
We can find the cost of capital for Coleman Tobacco farms in the following way:
Step I – Determining the cost of individual capital components.
Details of the calculations involved in determining the cost of individual capital components will not be shown here since we have already discussed them. So we shall assume that the same methods have been applied to obtain the following results:
Capital Components Cost
Preferred Stock 7.5%
Common Stock 11%
Step II – establishing the proper set of weights.
The proper weights are established by finding the amount of each capital component as a percentage of the total non-free fund as follows:
E.g. for preferred stock the weight is
Interest-bearing debts 160 22
Preferred Stock 7 1
Common Equity 560 77
This means that each N1 of new capital is raised as N0.22 of debt, N0.01 of preferred stock, and N0.77 as common equity (retained earnings or new stock).
Step III – Calculating the proportion of each component.
To maintain the target capital structure, the N20m must be raise follows
Non-free fund Computation Proportion
Preferred Stock N0.2m
Step IV – Calculating the weighted average cost of capital, K.
Amount of capital (Nm)
Therefore the weighted average cost of capital, K, for raising N20m by Coleman Tobacco farms is 9.5%
A firm paid an ordinary dividend of N900,000 this year. It is the firms policy to distribute 60% of its earnings as dividend to its shareholders and to re-invest the remainder. Market value of the company (i.e. present value of future earnings, which have been capitalized at the cost of capital) is N10,000,000. Calculate:
Total earnings fo the firm for this year
Component cost of retained earnings
Forecast next years earnings
Amount paid is dividend = N900,000 = 60% according to the firm’s policy.
60% represents N900,000
But the total earnings will be 100%
Total earnings = N1500000
Retained earnings = total earned – dividend distributed
= 1500000 – 900,000
Growth rate is obtained by multiplying earnings rate by the proportion of re-investment
Earnings rate = 15%
Earnings = 100%
Less: Dividend = 60%
Therefore Growth rate = 15% x 40%
Component cost of retained earning capital
= 0.09 + 0.06
= 0.15 or 15%
Notice that the component cost of retained earning capital is equal to the earnings rate showing an equilibrium situation.
Next year’s earnings represent a growth rate of 6% on present year’s earning.
This year’s earning = N1500000
Amount added this year at 6% growth rate = 1500000 x 6%
Then, best year’s earnings = N1500000
PRINCIPLES OF COMPOUNDING AND DISCOUNTING
Time value of money is built into the principle of compounding and discounting. The concept of time value of money recognizes that one naira today is worth more than one naira tomorrow. This is explained by the fact that one naira received today can be invested to earn interest and therefore increase to one naira plus the interest earned. In the following discussion about the principles of compounding and discounting, it is assumed that money is always invested at an interest rate.
Principle of Compounding: – Compounding is the arithmetic process of determining the final value of a payment or series of payments when compound interest is applied. So the future (or compound) value of money is determine by compounding. Note that future value of money refers to the value of an investment at a specified date in the future. This concept assumes that the investment will earn interest which is reinvested at the end of each time period to also earn interest. In other words, the future value includes the original investment, the interest it earns, and interest on the accumulated interest.
Compounding can be applied to a one-time lump sum investment or to an investment which takes place periodically over time. In the following part we shall see how compound or future values of investment are calculated by using formulae and interest factor tables.
Consider this question: Udoka farm business deposits N2000 in a savings and loan association that pays 4% interest compounded annually. How much will Udoka farm business have at the end of one year?
This is a simple problem that requires elementary solution by applying the simple interest to the principle as follows:
Principal + Interest = N2000 + N80 = N2080
But the problem will become cumbersome as the compounding periods increase. For example it will not be easy to find how much Udoka farm business above, will have at the end of 10 years. To make the task easier, we shall apply the formula; Cv = P(1 + r)n
Where Cv = Compound value
P = Principal or beginning amount at time zero
R = Interest rate
Cv = 2000 (1 + 0.04)10
= 2000 x 1.4802 = N2960.40
An amount may be compounded annually as in the above case or semi-annually or even quarterly. When the compounding period is more than once a year, then the following additional formulae are used to adjust the interest rate and number of compounding periods.
n = N x m
where R = unadjusted interest rate
r = adjusted interest rate
m = number of periods in a year i.e. semi-annual = 2, quarterly = 4etc
N = unadjusted compounding periods
n = adjusted number of compounding periods
Example, find how much Udoka farm business will have at the end of 10 years if the amount of N2000 is deposited at the rate of 4% compounded semi-annually.
First we adjust the interest rate and compounding periods as follows:
Interest rate, r. =
Compounding period, n = m x N = 2 x 10 = 20 periods
CV = P (1 + r)n
= 2000 (1 + 0.02)20
= 2000 x 1.4859 = N2971.80
Notice that the amount obtained by compounding semi-annually (i.e. N2971.80) is more than that of compounding annually (i.e. N2960.40). This is because in the semi-annual (i.e. multiple compounding the saver) (Udoka farm business) earns interest on interest more frequently.
Apart from using formula, compound values can be calculation by the use of interest factor tables. Recall the formula CV = P(1 + r)n which is then multiplied by P to give the compound value. For example, suppose N2000 is invested for 10 years at 4% interest compounded annually. The interest factor of 10 years or periods at 4% is given in the interest factor table as 1.4802. therefore the compound value becomes 2000 x 1.4802 = N2960.40. Compare the second example on Udoka farms.
So far, we have discussed compounding with respect to one-time lump sum investment or single payment. Now let us consider an investment which takes place over time or a series of payments. A series of payments of a fixed amount for a specified number of years is known as annuity.
The compound value of an annuity, Sna, can be found by applying the appropriate formula. For ordinary annuity in which payment is made at the end of the payment period the formula is given as:
Where Sna = sum of annuity
a = the fixed amount
r = interest rate
n = number of periods or years.
Find the annuity of N1000 compounded annually at 10% interest for 5 years.
This problem can be explained using the following diagram
The interest factor to use in funding the compound value of the annuity is 6.1051. Alternatively we can obtain this figure 6.1051 directly from the compound value interest factor table of annuities (CVIFA). The interest factor represents in the formula given above.
Therefore Sna = 1000 x 6.1051 = N6105.10
Using the formula we can also substitute the relevant data and obtain the correct answer as follows:
1000 x 6.1051
Principles of Discounting – Discounting is the process of finding the present value of a series of future cashflows. Notice that discounting is simply the reverse of compounding. This means that a present sum is compounded to find its future value and a future sum is discounted back to the present to find its current or present value. This discounting is done because a sum to be received in the future is worth somewhat less now because of the time difference assuming a positive interest rate; recall the concept of time value of money. A present value can be interpreted as the sum of money which would have to invested now at a given rate of interest to equal the future sum on the same date.
Suppose you are offered the alternative of either N5000 today or N6085 at the end of five years, which option will you choose? A correct choice must be based on the concept of time value of money. To make the choice, you should find the present value of N6085 at the prevailing interest rate. Suppose the prevailing interest rate in the economy is 4%, then the present value of N6085 is N5000. This means that you should be indifferent about the choice since N5000 today is the same as N6085 at the end of the next 5 years.
The formula for finding present value is given as follows
Where PV = present value
N = number of years
CV = Sum of the end of n years
R = discount rate or interest rate
The above figures may now be substituted to show that the present value of N6085 is N5000 i.e.
= 6085 x 0.8219 = N5000
Consider another illustration. Find the present value of N3600 at 20% interest rate for 5 years compounded annually.
= 3600 x 0.4019
Present value can also be calculated when the compounding period is multiple e.g. semi-annually or quarterly. For example, if in the above example the interest rate is 20% for 5 years compound semi-annually then the present value of the sum of N3600 will be:
n = N x m = 5 x 2 = 10 periods
= 3600 x 0.3855 = N1387.8
Note that the present value is less (N1387.8) when compound period is semi-annually than when it is annually. Recall that the reverse is the case with compound value; present value is the reverse of compound value.
In order to simplify calculations involving present value we can use the present value interest factor tables.
Recall that the formula for present value is
We can find from the table and then multiply the result by the compound value (CV), to get the present value.
Example use the interest factor table to find the present value of N121.67 at the discount rate of 4% for 5 years.
Solution – Use the PVIF table to trace the interest factor that corresponds to 5 periods at 4% i.e.
PV = CV (PVIF) = 121.67 (0.8219) = N100
We can find the present value, PVa of an annuity by applying the appropriate formula or using the table of present value interest factor of annuity, PVIFA.
Find the present value of the ordinary annuity of N1000 compounded annually at interest rate of 10% for 5 years.
Solution – We can explain this problem diagrammatically as follows:
The present value interest factor of the annuity, PVIFA is 3.7907 which can also be obtained from the interest factor table.
The formula to be used is
Where PVa = present value of the annuity
a = the fixed amount
r = rate
n = number of compounding periods.
= 1000 (3.7907) = N3790.70
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