Session 4: Time Value of Money

Learning Objective

Explain to the learner on the concept of time value of money.

Important Terms

• Time line
• Discounting
• Compounding
• Principal amount
• Simple interest
• Annuity
• Amortized loan

Time Value of Money

To make itself a valuable as possible to stock holders; an enterprise must choose the best combination of decisions on investment, financing and dividends. In any economy in which individuals, firm and governments have the time preference, the time value of money is an important concept. Stockholders will pay more for an investment that promises returns over years 1 to 5 than they will pay for an investment that promises identical returns for 6 years through 10.
The decision to purchase new plant and equipment or to introduce a new product in the market requires using capital allocating or capital budgeting techniques. Essentially we must determine whether future benefits are sufficiently large to justify current outlays. It is important that we develop the mathematical tools of the time value of money as the first step towards making capital allocating decisions.

Principal amount (P)
This is the amount of money that is initially being considered. It might be an amount to be invested or loaned or it may refer to the initial value or cost of plant or machinery. Thus if the company was considering a bank loan of say K500,000, this would be referred to as the principal amount borrowed.

Accrued amount (A)
This term is applied generally to a principal amount after some time has elapsed for which interest has been calculated and added.

Simple and Compound Interest

When an amount of money is invested over a number of years, the interest earned can be dealt with in two ways.

SIMPLE INTEREST

This is where any interest earned is NOT added back to the principal amount invested.

For example, suppose that K200,000 is invested at 20% simple interest per annum. The following table shows the state of the investment, year by year:

COMPOUND INTEREST

The notion of compound interest is central to understanding the mathematics of finance. The term itself merely implies that interest paid on loan or an investment is added to the principle. As a result, interest is earned on interest.
Compounding is the arithmetic process of determining the final value of a cash flow or series of cash flow or series of cash flows when compound interest is applied.

The difference between the two methods can easily be seen by comparing the above two tables. Notice that the amount on which simple interest is calculated is always the same.

Time Line

An important tool used in time value of money analysis and graphically shows the timing of cash flows. In the above example for the simple interest, the time line can be produced as:

Discounting

The process of determining the present value of future cash flows. It is an important concept, which is used in project appraisals. The opportunity cost rate is the rate available on the next best alternative with same equal risk as the current investment.

Suppose money can be invested at 10%. The K200, 000 could be invested and be worth K220,000 in one years time. Put another way, the value K200,000 in one years time is exactly the same as K200,000 now.( if the investment rate is 10%). Similar K200,000 now has the value as K200,000(1.1)2 = K242,000 in two years time. To state the above ideas more precisely, if the current investment rate is 10%, then:

Appendix 1 gives tables showing the present values of the discount factor (these are two rows marked D) for a wide range of values of i and n. these are known as discounting tables.
Suppose we wanted to find the present value of K15,000 in 6 years time, subject to a discount rate of 19%.

The discount factor (from the table, with D = 19% and N = 6) is 0.3521.

Therefore the present value = K15,000 (0.3521) = K5281.5.

Annuity

Annuity is a sequence of fixed equal payments (or receipts) made over uniform time intervals. Some common examples of annuities include: weekly wages, monthly salaries, insurance premiums, hire purchase payments.
Annuities are used in all areas of business and commerce. Loans are normally repaid with an annuity, investment funds are set up to meet fixed future commitments (for example, asset replacement) by the payment of an annuity.

Annuities may be paid:

1. At the end of payment intervals( called an ordinary annuity)
2. At the beginning of payment intervals (called annuity due).

The terms of an annuity may:

1. Begin and end on fixed dates ( a Certain annuity)
2. Depend on some event that cannot be fixed ( a Contingent annuity)

A perpetuity annuity is one that carries on indefinitely.

The most common form of annuities are certain and ordinary. That is the annuity is paid at the end of the payment interval and will begin and end on fixed dates. Personal loans and most domestic hire purchase are paid off in a similar manner but normally without the initial deposit.
Annuities that are being invested however are often due, that is paid of ‘in advance’ of the intervals
The present value (PV) of an annuity could be found as for any cash flow by discounting each return individually, but there is a more economical method. Consider the case of an annuity of K10,000 that runs for four years at 10% interest. Assume that the first payment will be made after one year. Using the discount factor table the PV is:

Sinking Fund

A sinking fund can be defined as an annuity invested in an order to meet a known commitment at some future date. Sinking funds are usually used for the following purposes:

1. Repayment of debts.
2. To provide funds to purchase a new asset when the existing asset is fully depreciated.

Example of debt repayment using a sinking fund:

Here a debt is incurred over a fixed period of time, subject to a given interest rate. A sinking fund must be set up to mature to the outstanding amount of the debt.
For example, if K250,000 is borrowed over three years at the rate of 12% compounded, the value of the outstanding debt at the end of third year will be K250,000(1.12)3 = K351,232. If money can be invested at 9.5%, we need to find the value of the annuity, A, which must be paid into the fund in order that it matures to K351,123. Assuming that payments into the funds are in arrears, we need:

That is the annual payment into the sinking fund is K106,627.8 (which will produce, 9.5%, K251,232 at the end of 3 years).

Perpetuities

A special case of an annuity is where a contract runs indefinitely and there is no end to the payments. This is called a perpetuity. Steam of equal payments expected to continue forever.

Semi annual and other compounding periods semi-annual compounding is the arithmetical process of determining the final value of determining the final value of cash flows when interest is added twice a year.

A Mortised Loan

Loan repaid in equal payments over its life. Installment prepayments are prevalent in mortgage loans, auto loan and consumer loans and in certain business loans. The distinguishing feature is that the loan is repaid in equal periodic payments that embody both interest and principal. These payments can be made monthly, quarterly, Semi-annually or annually. The debt is said to be amortized if this method is used.

Examples:

A company negotiates a loan of K200,000 over 15 years at 10.5% per annum. Calculate the annual payment necessary to amortize the debt.

Interest Rates

1. Nominal rates
   The rate which is quoted or stated on loan or investment.
2. Effective annual rate
   The rate, which would produce the same ending (future), values if annual compounding had been used.
3. Periodic rate
   The rate charged by a lender or paid by a borrower each period. It can be rate per year, per six-month period, per quarter, per month or per day.