Política de formación

[

]

Po PVo = = +

1

To illustrate using the example where FV8 = N18,800

i = 10% and n = 6, if we do not know Po or Vo, that can be determined as follows:

[

]

Po

PVo = =

= +

=

Therefore, the present value of N18,800 at 11 percent simple interest in an 8 year period is N10,000.

3.1.2 Compound Interest

Compound interest is defined as amount earned (or paid) on any previous interest earned, as well as on the principal lent (or borrowed).

periodically added to the principal; and interest is earend on the previous interest as well as on the original principal. Generally in dealing with problems of compound interest, we are either interested in future value (FV) or present value (PV) rather than the naira amount of the compound interest alone. The formula for calculating future (compound) (value at the end of n period at i rate of interest is:

FVn = Po(1 + i)ⁿ Example Two

Supposing you deposit N12000 in a savings account paying percent compound interest and is being kept for 5 years. Determine the future value (V) of the amount at the end of the 5 year period.

Solution

FV5 = N12,000 (1 + 0.9)5

= N12,000(1.09)(1.09)(1.09)(1.09)(1.09).

= N12,000 91.5386) :. FV5 = N18,463.49

This implies that at the end of five years, the account would be worth N18,463.49.

In general, FVn, the future (compound) value of a deposit at the end of n periods is:

FVn = Po (1 +i)ⁿ

Time preference for money requires that value of future cash flows should be determined in relation to their present value in order to make rational financial decision. Thus, the process of converting future worth of an amount to the present value is referred to as discounting which is the inverse of compounding. For instance, if you are to collect N20,000 at the end of 7years from today, if your opportunity cost of fund or interest rate compound annually is 12 percent; what is the value of that amount equated to the present? In solving this type of problem, the interest rate is known as the discount rate (or capitatlization rate).

Thus to solve the above exercise of finding present value (or discounting) we simply invert the compounding formula; we already know that:

( ) ⁱ

Po FVn = 1 +

To determine the Present Value (PVo) or (Po) we make PVo the subject of the formula.

 interest factor at i% for n periods this reciprocal is referred to as the present value interest factor at i% for n periods. The use of a calculator can alternatively ease the computation.

:. ^9,046.98

This implies that the value of #20000 which will be collected in a 7 year period from today is N9,046.98 in present day value, taking 12 percent discounting factor. Alternatively if we keep N9,046.98 in a fixed deposit account that gives 12 percent compound interest per annum, the amount will accrue to N20,000 in 7 years.

Compounding More Than Once a Year

Sometimes interest may be compounding more than once in a year.

Some banks compound interest weekly, monthly, quarterly or semi-annually. There are some incremental benefits to a depositor or non-annual interest compounding. For instance support that interest is paid quarterly. If you then deposit N10,000 in a savings account at a nominal 12 percent annual interest rate, the future value at the end of three

In other words, at the end of the first quarter of the year you would receive 3 percent in interest, not 12 percent. At the end of a year the future value of the deposit would be:

09 once a year; the N55.09 different is caused by interest being earned in the second, third and fourth quarters on the interest paid at the end of each quarter. The more times during the year that interest is paid, the greater the future value at the end of a given year.

The general formula for solving the future value at the end of n years where interest is paid m times a year is given by:

mn

To illustrate, assume that interest is paid monthly and that you wish to know the future value of N10,000 at the end of one year where the nominal annual interest rate is 12percent. The future value would be:

25

The future value at the end of five years for example with monthly compounding is:

Compared to a future value with quarterly compounding is

)

Also with seminannual compounding is:

)

Also with annual compounding is:

)

Thus, the more frequently interest is paid, the grater the future value.

Present Value of Uneven Sum

In most cases the future sum may not be just one lump sum. The future cash flow may be uneven sums or not uniform Naira values. For instance it is very common when one makes investment the benefits, i.e., the cash flows to be generated over the life of the project will be fluctuating. As such to find the present value of an uneven cash flow stream, we have to find the present value of each of the future receipts and then sum such present values. For example, if future uneven cash receipts are denoted as Rn(n=1,2,3,4,5…), then the present value of such cash receipts series can be determined as follows:

n

R = the periodic/yearly receipts

i = the annual rate of interest/discounting rate n = the number of periods/years

Example Three

Suppose an investment opportunity promises the following cash flows in future years:

Find the sum of the present values of these cash receipts if the firm’s discount rate is 13 percent.

Solution:

Year Cash Flow (N) PVIF13,n PV (N)

1 250,000 0.88496 221,240.00

2 230,000 0.78135 180,124.50

3 200,000 0.69305 138,610.00

4 240,000 0.61332 147,196.80

5 275,000 0.54276 149,259.00

6 300,000 0.48032 144,096.20

7 220,000 0.42506 93,513.20

Present value of the future cash flows (discounted at 13%) = N1,074,039.50

This sum of the present value of the future cash flows to be generated by the project (investment opportunity) can be compared with the expected Naira cost of the investment to find Net present value (NPV). If in the above illustration the Naira cost of the investment is N1 million, the Net Present Value (NPV of the project is given by:

NPV = N1,074,039.50 – N1,000,000 = N74,030.50

This amount (N74,039.50) is the Net Present Value which is the benefit to be generated by the project within its 7-year life span equated to the present value.

3.2 Annuities

An annuity is a series of payments or receipts occurring over a specified number of periods. An annuity can either be classified as an ordinary annuity or an annuity due. In an ordinary annuity receipts or payments are made at the end of each period, whereas in an annuity due the receipts or payments are made at the beginning of each period.