Concept 4 - Interest

CONCEPT OF TIME VALUE OF MONEY

The value of money is not constant. This is one of the principal facts on which the entire economic world is based. A rupee today will not be equal to a rupee tomorrow. Hence, a rupee borrowed today cannot be repaid by a rupee tomorrow. This is the basic need for the concept of interest. The rate of interest is used to determine the difference between what is borrowed and what is repaid.

There are two basis on which interests are calculated:

Simple Interest

It is calculated on the basis of a basic amount borrowed for the entire period at a particular rate of interest. The amount borrowed is the principal for the entire period of borrowing.

Compound Interest 

The interest of the previous year/s is/are added to the principal for the calculation of the compound interest.

This difference will be clear from the following illustration:

A sum of ` 1000 at 10% per annum will have

Simple interest                                                             Compound interest

      ` 100                                  First year                                  ` 100

      ` 100                                  Second year                              ` 110

      ` 100                                  Third year                                 ` 121

      ` 100                                  Fourth year                               ` 133.1

Note that the previous years’ interests are added to the original sum of ` 1000 to calculate the interest to be paid in the case of compound interest.

Terminology Pertaining to Interest

The man who lends money is the Creditor and the man who borrows money is the Debtor.

The amount of money that is initially borrowed is called the Capital or Principal money.

The period for which money is deposited or borrowed is called Time.

The extra money, that will be paid or received for the use of the principal after a certain period is called the Total interest on the capital.

The sum of the principal and the interest at the end of any time is called the Amount.

So, Amount = Principal + Total Interest.

Rate of Interest is the rate at which the interest is calculated and is always specified in percentage terms.

SIMPLE INTEREST

The interest of 1 year for every ` 100 is called the Interest rate per annum. If we say “the rate of interest per annum is r%”, we mean that ` r is the interest on a principal of ` 100 for 1 year.

Relation Among Principal, Time, Rate Percent of Interest Per Annum and Total Interest

Suppose, Principal = ` P, Time = t years, Rate of interest per annum = r% and Total interest = ` I,

Then I = ( P x t x r ) / 100

i.e. Total interest = ( Principal x Time x Rate of Interest per annum ) / 100

Since the Amount = Principal + Total interest, we can write,

Amount A = P + ((P x t x r) / 100)

Time = (Total Interest / Interest on the Principal for 1 year) years

Thus, if we have the total interest as ` 300 and the interest per year is ` 50, then we can say that the number of years is 300/50 = 6 years.

Note: The rate of interest is normally specified in terms of annual rate of interest. In such a case we take the time t in years. However, if the rate of interest is specified in terms of 6-monthly rate, we take time in terms of 6 months. Also, the half-yearly rate of interest is half the annual rate of interest. That is if the interest is 10% per annum to be charged six-monthly, we have to add interest every six months @ 5%.

COMPOUND INTEREST

In monetary transactions, often, the borrower and the lender, in order to settle an account, agree on a certain amount of interest to be paid to the lender on the basis of specified unit of time. This may be yearly or half-yearly or quarterly, with the condition that the interest accrued to the principal at a certain interval of time be added to the principal so that the total amount at the end of an interval becomes the principal for the next interval. Thus, it is different from simple interest. In such cases, the interest for the first interval is added to the principal and this amount becomes the principal for the second interval, and so on. The difference between the amount and the money borrowed is called the compound interest for the given interval.

Formula

Case 1: Let principal = P, time = n years and rate = r% per annum and let A be the total amount at the

end of n years, then

Case 2: When compound interest is reckoned half-yearly. If the annual rate is r% per annum and is to be calculated for n years. Then in this case, rate = (r/2)% half-yearly and time = (2n) half-years.

Case 3: When compound interest is reckoned quarterly. In this case, rate = (r/4)% quarterly and time = (4n) quarter years.

Note: The difference between the compound interest and the simple interest over two years is given by 

DEPRECIATION OF VALUE

The value of a machine or any other article subject to wear and tear, decreases with time. This decrease is called its depreciation.

Thus if V0 is the value at a certain time and r% per annum is the rate of depreciation per year, then the value V1 at the end of t years is

POPULATION

The problems on Population change are similar to the problems on Compound Interest. The formulae applicable to the problems on compound interest also apply to those on population. The only difference is that in the application of formulae, the annual rate of change of population replaces the rate of compound interest.

However, unlike in compound interest where the rate is always positive, the population can decrease. In such a case, we have to treat population change as we treated depreciation of value illustrated above. In the case of compound interests, the percentage rule for calculation of percentage values will be highly beneficial. 

Thus, a calculation: 4 years increase at 6% pa CI on ` 120 would yield an expression: 120 × 1.064. It

would be tedious. So, we should look at it from the following percentage change graphic perspective:

If you try to check the answer on a calculator, you will discover that you have a very close approximation. Besides, given the fact that you would be working with options and given sufficiently comfortable options, you need not calculate so closely; instead, save time through the use of approximations.

APPLICATIONS OF INTEREST IN D.I.

The difference between Simple Annual Growth Rate and Compound Annual Growth Rate:

The Measurement of Growth Rates is a prime concern in business and Economics. While a manager might be interested in calculating the growth rates in the sales of his product, an economist might be interested in finding out the rate of growth of the GDP of an economy.

In mathematical terms, there are basically two ways in which growth rates are calculated. To familiarize yourself with this, consider the following example.

The sales of a brand of scooters increase from 100 to 120 units in a particular city. What does this mean to you? Simply that there is a percentage increase of 20% in the sales of the scooters. Now read further:

What if the sales moves from 120 to 140 in the next year and 140 to 160 in the third year? Obviously, there is a constant and uniform growth from 100 to 120 to 160 – i.e. a growth of exactly 20 units per year.

In terms of the overall growth in the value of the sales over there years, it can be easily seen that the sale has grown by 60 on 100 i.e. 60% growth.

In this case, what does 20% represent? If you look at this situation as a plain problem of interests 20% represents the simple interest that will make 100 grow to 160.

In the context of D.I., this value of 20% interest is also called the Simple Annual Growth Rate. (SAGR)

The process for calculating SAGR is simply the same as that for calculating Simple Interest.

Suppose a value grows from 100 to 200 in 10 years – the SAGR is got by the simple calculation 100%/10 = 10%

What is Compound Annual Growth Rate (CAGR)?

Let us consider a simple situation. Let us go back to the scooter company.

Suppose, the company increases it’s sales by 20% in the first year and then again increases its’ sales by 20% in the second year and also the third year. In such a situation, the sales (taking 100 as a starting value) trend can be easily tracked as below:

As you must have realised, this calculation is pretty similar to the calculation of Compound interests. In the above case, 20% is the rate of compound interest which will change 100 to 172.8 in three years. This 20% is also called as the Compound Annual Growth Rate (CAGR) in the context of Data interpretation. Obviously, the calculation of the CAGR is much more difficult than the calculation of the SAGR and the Compound Interest formula is essentially a waste of time for anything more than 3 years. (upto three years, if you know your squares and the methods for the cubes you can still feasibly work things out – but beyond three years it becomes pretty much infeasible to calculate the compound interest). 

So is there an alternative? Yes there is and the alternative largely depends on your ability to add well. Hence, before trying out what I am about to tell you, I would recommend you should strengthen yourself at addition. Suppose you have to calculate the C.I. on ` 100 at the rate of 10% per annum for a period of 10 years. You can combine a mixture of PCG used for successive changes with guesstimation to get a pretty accurate value.

In this case, since the percentage increase is exactly 10% (Which is perhaps the easiest percentage to

calculate), we can use PCG all the way as follows:

Thus, the percentage increase after 10 years @ 10% will be 159.2 (approx).

However, this was the easy part. What would you do if you had to calculate 12% CI for 10 years. The percentage calculations would obviously become much more difficult and infeasible. How can we tackle this situation?

In order to understand how to tackle the second percentage increase in the above PCG, let’s try to evaluate where we are in the question. We have to calculate 12% of 112, which is the same as 12% of 100 + 12% of 12. But we have already calculated 12% of 100 as 12 for the first arrow of the PCG. Hence, we now have to calculate 12% of 12 and add it to 12% of 100. 

Hence the addition has to be: 12 + 1.44 = 13.44

Take note of the addition of 1.44 in this step. It will be significant later. The PCG will now look like:

We are now faced with a situation of calculating 12% of 125.44. Obviously, if you try to do this directly, you will have great difficulty in calculations. We can sidestep this as follows:

12% of 125.44 = 12% of 112 + 12% of 13.44.

But we have already calculated 12% of 112 as 13.44 in the previous step.

Hence, our calculation changes to:

12% of 112 + 12% of 13.44 = 13.44 + 12% of 13.44

But 12% of 13.44 = 12% of 12 + 12% of 1.44. We have already calculated 12% of 12 as 1.44 in the

previous step.

Hence 12% of 13.44 = 1.44 + 12% of 1.44 = 1.44 + 0.17 = 1.61 (approx)

Hence, the overall addition is 13.44 + 1.61 = 15.05

Now, your PCG looks like:

You are again at the same point—faced with calculating the rather intimidating looking 12% of 140.49

Compare this to the previous calculation:

The only calculation that has changed is that you have to calculate 12% of 15.05 instead of 12% of 13.44. (which was approx 1.61). In this case it will be approximately 1.8. Hence you shall now add 16.85 and the PCG will look as:

If you evaluate the change in the value added at every arrow in the PCG above, you will see a trend—

The additions were: +12, +13.44 (change in addition = 1.44), +15.05(change in addition = 1.61), +16.85 (change in addition = 1.8)

If you now evaluate the change in the change in addition, you will realize that the values are 0.17, 0.19. This will be a slightly increasing series (And can be easily approximated).

Thus, the following table shows the approximate calculation of 12% CI for 10 years with an initial value of 100. Thus, 100 becomes 309.78 (a percentage increase of 209.78%)

Similarly, in the case of every other compound interest calculation, you can simply find the trend that the first 2 – 3 years interest is going to follow and continue that trend to get a close approximate value of the overall percentage increase. Thus for instance 7% growth for 7 years at C.I. would mean:

An approximate growth of 60.24%

The actual value (on a calculation) is around 60.57% – Hence as you can see we have a pretty decent approximation for the answer.

Note: The increase in the addition will need to be increased at a greater rate than as an A.P. Thus, in this case if we had considered the increase to be an A.P. the respective addition would have been:

+7, +7.49, +8.01, +8.55, +9.11, +9.69, +10.29.

However +7, +7.49, +8.01, +8.55, +9.11, +9.75, +10.35 are the actual addition used. Notice that using 9.75 instead of 9.69 is a deliberate adjustment, since while using C.I. the impact on the addition due to the interest on the interest shows an ever increasing behavior.