Time value of money and types of interest rates

Time Value of Money:

It is defined as the time-dependent value of money stemming both from changes in purchasing capacity of money (inflation or deflation) and from the real earning potential of alternative investments over time. Since money has the capacity to earn interest, its value increases with time. Hence it is the relationship between interest and time.

Money has a time value. The economic value of an amount depends on when it is received. Money has earning power over time.

A rupee received today has more value than a rupee received at some future time.

The cost of money is measured by an interest rate, a percentage that is periodically applied and added to an amount of money over a specified length of time. Although the money left in the savings account earns interest so that the balance over time is greater than the sum of the deposits but whenever we go for any investment, we will have to consider the following three factors:

1. Liquidity: It is the reward for not being able to use your money while you are holding the stock or mortgage or promise.
2. Risk premium: It is the reward for any chance that you would not get your money back or that it will have declined in value while invested.
3. Inflation factor: It is compensation for the decrease in purchasing power between the time you invest it and time it is returned to you.

Hence interest may be defined as the cost of having money available for use.

Elements of transaction involving interest:

1. An initial amount of money that, in transaction involving debt or investment, is called the
2. The interest rate that measures the cost or price of money and that is expressed as a percentage per period of time.
3. A period of time called interest period that determines how frequently interest is calculated.
4. A stated length of time that makes the duration of the transaction and thereby establishes a certain number of interest periods.
5. A future amount of money that results from the cumulative effects of the interest rate over a number of interest periods.

Simple Interest:

It uses fixed percentage of the principal i.e the amount of interest earned is directly proportional to the initial principal amount. Under simple interest, the interest earned during each interest period will not earn an additional interest on interest amount in the remaining periods.

Simple interest earned by depositing principal ‘P’ at interest rate ‘i’ over the ‘N’ years is given by

I = (I × P)N

The total amount available at the end of ‘N’ years is given by

F = P + I = P. (1 + i.N)

Compound Interest:

When the total time period is subdivided into several interest periods (one year, half-yearly, quarterly, monthly, weekly), interest is endorsed at the end of each interest period and is permitted to accumulate from one interest period to next, then the interest is said to be compounded.

Under the compound interest, the interest earned in each period depends on the total amount owed at the end of the previous period. This means interest will be earned on the interest charged in the previous period as well.

At the end of first year, F = P + i.P = P.(1 + i)

At the end of second year, F = P.(1 + i) + i.[P.(1 + i)] = P.(1 + i)²

At the end of third year, F = P.(1 + i)² + i.[ P.(1 + i)²] = P.(1 + i)³

Continuing this for ‘n’ interest periods, we get

At the end of n periods, F = P.(1 + i)ⁿ

This can be represented functionally as F = P (F/P, i%, n) where factor in the bracket is called the Single payment compound factor.

Nominal and Effective Interest Rates

Generally, payments are made annually but some of the familiar financial transactions, both personal and in engineering economic study implicate payments that are not made annually such as monthly mortgage payments and quarterly earnings on the savings account. The need of evaluation and compare of different cash flows with different compounding periods led to the concepts of nominal and effective interest rates.

Nominal Interest Rates:

If a financial institution uses a unit of time other than a year i.e. a quarter, a month, half year, then it quotes interest rate on the annual basis such as r% compounded monthly, quarterly or half yearly. The interest rate or Annual Percentage Rate (APR) is called the nominal interest rate.

Suppose a bank expresses the interest rate as ‘18% compounded monthly’, we say that that 18% is the nominal interest rate or APR and the compounding frequency is monthly i.e. number of compounding period is 12 per year so each month the bank will charge 1.5% (= 18%/12 ) interest on an unpaid balance. Similarly, 18% compounded quarterly means 4.5% (= 18%/4) per 3 months, 18% compounded semiannually means 9% (= 18%/2) per 6 months.

Although APR is commonly used by the financial institution and is familiar to many customers, it does not clarify accurately the amount of interest that will accumulate in a year. This disadvantage can be explained by Effective Interest Rate.

Effective Interest Rates:

The actual or the exact rate of interest earned on the Principal in a year. The effective interest rates are always expressed on an annual basis unless specifically stated otherwise.

Suppose Rs.10000 is deposited in a savings account that pays interest of 9% compounded quarterly. The interest rate per quarter is 2.25%. Here, P = Rs.10000, i = 2.25% and N = 4 so by using the compound interest formula, F = P(1 + i)^N, we get, F = Rs.10,930.83. This gives Rs.930.83 is the annual interest earned that means the effective annual interest rate (i_a) = 930.83/10000 = 0.093083 or 9.3083%.

Certainly, more frequent compounding increases the amount of interest paid over a year at the same nominal interest rate.

In general concept, the effective annual interest rate is expressed as:

i_a = (1+r/M)^M – 1

where, r = nominal interest rate

r/M = interest rate per compounding period

M = number of compounding periods per year

Effective Interest Rate per payment period:

If the transaction occurs more than one time a year then the effective interest rate on the basis of the payment period is computed. Effective interest rate per payment period is expressed as:

i = (1 + r/CK)^C – 1

where, C = number of interest periods per payment period

K = number of payment periods per year

r/CK = interest rate per compounding period

Continuous compounding:

Some financial institutions offer frequent compounding in order to be competitive on the financial market or to entice potential depositors the interest rate per compounding period (r/M) becomes very small as the number of compounding periods (M) becomes very large. As ‘M’ approaches infinity then ‘r/M’ approaches zero, so we approximate the condition of continuous compounding.

Considering the limit ‘CK’ tends to infinity in the above equation of effective interest per payment period, we get the effective interest rate per payment period for continuous compounding expressed as:

i_c = e^r/K – 1

where, r = nominal interest rate

K = number of payment periods per year

BIBLIOGRAPHY:

Chan S.Park, Contemporary Engineering Economics, Prentice Hall, Inc.
E. Paul De Garmo, William G.Sullivan and James A. Bonta delli, Engineering
Economy, MC Milan Publishing Company.
James L. Riggs, David D. Bedworth and Sabah U. Randhawa,Engineering
Economics, Tata MCGraw Hill Education Private Limited.