CONCEPTS & MODES OF ANALYSIS, simple interest, time value of money.

CONCEPTS & MODES OF ANALYSIS

What is Simple Interest?

Simple Interest: Simple Interest is the interest paid only on the principal amount borrowed. No interest is paid on the interest accrued during the term of the loan.
There are three components to calculate simple interest: principal, interest rate and time.

Formula for calculating simple interest:

I = Prt
Where,
I = interest
P = principal
r = interest rate (per year)
t = time (in years or fraction of a year)

Example:
Mr. X borrowed Rs. 10,000 from the bank to purchase a household item. He agreed to repay the amount in 8 months, plus simple interest at an interest
rate of 10% per annum (year).
If he repays the full amount of Rs. 10,000 in eight months, the interest would be:
P = Rs. 10,000 r = 0.10 (10% per year) t = 8/12 (this denotes fraction of a
year)
Applying the above formula, interest would be:
I = Rs. 10,000*(0.10)*(8/12) = Rs. 667.

This is the Simple Interest on the Rs. 10,000 loan taken by Mr. X for 8 months.
If he repays the amount of Rs. 10,000 in fifteen months, the only change is with time.

Therefore, his interest would be:
I = Rs. 10,000*(0.10)*(15/12) = Rs. 1,250

What is Compound Interest?

Compound interest means that, the interest will include interest calculated on interest. The interest accrued on a principal amount is added back to the principal sum, and the whole amount is then treated as new principal, for the calculation of the interest for the next period.

For example, if an amount of Rs. 5,000 is invested for two years and the sinterest rate is 10%, compounded yearly:

* At the end of first year the interest would be (Rs. 5,000*0.10) or Rs.500
* In the second year the interest rate of 10% will applied not only to Rs. 5000but also to the Rs. 500 interest of the first year.
Thus in the second year the interest would be (0.10*Rs. 5500) or Rs. 550

For any loan or borrowing unless simple interest is stated, one should always assume interest is compounded. When compound interest is used we must always know how often the interest rate is calculated each year. Generally the interest rate is quoted annually. E.g. 10% per annum.

Compound interest may involve calculations for more than once a year, each using a new principal, i.e. (interest + principal). The first term we must understand in dealing with compound interest is conversion period. Conversion period refers to how often the interest is calculated over the term of the loan or investment. It must be determined for each year or fraction of a year.

E.g.: If the interest rate is compounded semiannually, then the number of conversion periods per year would be two. If the loan or deposit was for five years, then the number of conversion periods would be ten. Formula for calculating Compound Interest:

C = P (1+i)n

Where
C = amount
P = principal
i = Interest rate per conversion period
n = total number of conversion periods

Example:

Mr. X invested Rs. 10,000 for five years at an interest rate of 7.5%
compounded quarterly

P = Rs. 10,000
i = 0.075 / 4, or 0.01875
n = 4 * 5, or 20, conversion periods over the five years

Therefore, the amount, C, is:
C = Rs. 10,000(1 + 0.01875)^20
= Rs 10,000 x 1.449948
= Rs 14,499.48

So at the end of five years Mr. X would earn Rs. 4,499.48 (Rs.14,499.48 – Rs.10,000) as interest. This is also called as Compounding.   Compounding plays a very important role in investment since earning a simple interest and earning an interest on interest makes the amount received at the end of the period for the two cases significantly different.
If Mr. X had invested this amount for five years at the same interest rate offering the simple interest option, then the amount that he would earn is calculated by applying the following formula:

S = P (1 + rt),
P = 10,000
r = 0.075
t = 5

Thus, S = Rs. 10,000[1+0.075(5)]
= Rs. 13,750
Here, the simple interest earned is Rs. 3,750. A comparison of the interest amounts calculated under both the method   indicates that Mr. X would have earned Rs. 749.48 (Rs.4,499.48 – Rs. 3,750) or nearly 20% more under the compound interest method than under the simple interest method.

Simply put, compounding refers to the re-investment of income at the same rate of return to constantly grow the principal amount, year after year. Should one care too much whether the rate of return is 5% or 15%? The fact is that with compounding, the higher the rate of return, more is the income which keeps getting added back to the principal regularly generating higher rates of return year after year.

The table below shows you how a single investment of Rs 10,000 will grow at various rates of return with compounding. 5% is what you might get by leaving your money in a savings bank account, 10% is typically the rate of return you could expect from a one-year company fixed deposit, 15% - 20% or more is what you might get if you prudently invest in mutual funds or equity shares.

The Impact of Power of Compounding: The impact of the power of compounding with different rates of return and
different time periods:

What is meant by the Time Value of Money?

Money has time value. The idea behind time value of money is that a rupee now is worth more than rupee in the future. The relationship between value of a rupee today and value of a rupee in future is known as ‘Time Value of Money’. A rupee received now can earn interest in future. An amount invested today has more value than the same amount invested at a later date because it can utilize the power of compounding. Compounding is the process by which interest is earned on interest. When a principal amount is invested, interest is earned on the principal during the first period or year. In the second period or year, interest is earned on the original principal plus the interest earned in the first period. Over time, this reinvestment process can help an amount to grow significantly.

Let us take an example:

Suppose you are given two options:
(A)Receive Rs. 10,000 now   OR
(B) Receive Rs.10,000 after three years.

Which of the options would you choose?

Rationally, you would choose to receive the Rs. 10,000 now instead of waiting for three years to get the same amount. So, the time value of money demonstrates that, all things being equal, it is better to have money now rather than later.

Back to our example: by receiving Rs.10,000 today, you are poised to increase the future value of your money by investing and gaining interest over a period of time. For option B, you don't have time on your side, and the payment received in three years would be your future value. To illustrate, we have provided a timeline:

If you are choosing option A, your future value will be Rs. 10,000 plus any interest acquired over the three years. The future value for option B, on the other hand, would only be Rs. 10,000. This clearly illustrates that value of money received today is worth more than the same amount received in future since the amount can be invested today and generate returns.
Let us take an another example:

If you choose option A and invest the total amount at a simple annual rate of 5%, the future value of your investment at the end of the first year is Rs. 10,500, which is calculated by multiplying the principal amount of Rs. 10,000 by the interest rate of 5% and then adding the interest gained to the principal amount.

Thus, Future value of investment at end of first year:
= ((Rs. 10,000 X (5/100)) + Rs. 10,000
= (Rs.10,000 x 0.050) + Rs. 10,000
= Rs.10,500

You can also calculate the total amount of a one-year investment with asimple modification of the above equation: Original equation: (Rs.10,000 x 0.050) + Rs.10,000 = Rs.10,500 Modified formula: Rs.10,000 x [(1 x 0.050) + 1] = Rs.10,500 Final equation: Rs. 10,000 x (0.050 + 1) = Rs. 10,500 Which can also be written as:

S = P (r+ 1)
Where,
S = amount received at the end of period
P = principal amount
r = interest rate (per year)

This formula denotes the future value (S) of an amount invested (P) at a simple interest of (r) for a period of 1 year.

How is time value of money computed?

The time value of money may be computed in the following circumstances:

1. Future value of a single cash flow
2. Future value of an annuity
3. Present value of a single cash flow
4. Present value of an annuity

(1)   Future Value of a Single Cash Flow

For a given present value (PV) of money, future value of money (FV) after a period ‘t’ for which compounding is done at an interest rate of ‘r’, is given

by the equation
FV = PV (1+r)t

This assumes that compounding is done at discrete intervals. However, in case of continuous compounding, the future value is determined using the formula
FV = PV * ert

Where ‘e’ is a mathematical function called ‘exponential’ the value of exponential (e) = 2.7183. The compounding factor is calculated by taking natural logarithm (log to the base of 2.7183).

Example 1: Calculate the value of a deposit of Rs.2,000 made today, 3 years hence if the interest rate is 10%.

By discrete compounding:
FV = 2,000 * (1+0.10)3 = 2,000 * (1.1)3 = 2,000 * 1.331 = Rs. 2,662
By continuous compounding:
FV = 2,000 * e (0.10 *3) =2,000 * 1.349862 = Rs.2699.72

Annuity (FVIFA). The same can be applied in a variety of contexts. For e.g. To know accumulated amount after a certain period, to know how much to Save annually to reach the targeted amount, to know the interest rate etc.

Example 1: Suppose, you deposit Rs.3,000 annually in a bank for 5 years and your deposits earn a compound interest rate of 10 per cent, what will be value of this series of deposits (an annuity) at the end of 5 years? Assume
that each deposit occurs at the end of the year.
Future value of this annuity is:
=Rs.3000*(1.10)4+ Rs.3000*(1.10)3+ Rs.3000*(1.10)2+ Rs.3000*(1.10)
+ Rs.3000
=Rs.3000*(1.4641)+Rs.3000*(1.3310)+Rs.3000*(1.2100)+Rs.3000*(1.10)
+ Rs.3000
= Rs. 18315.30

4.   Present Value of an Annuity

The present value of annuity is the sum of the present values of all the cashinflows of this annuity.
Present value of an annuity (in case of discrete discounting)
PVA = FV [{(1+r)^t - 1 }/ {r * (1+r)^t}]

The term [(1+r)^t - 1/ r*(1+r)^t] is referred as the Present Value Interest factor for an annuity (PVIFA).

Present value of an annuity (in case of continuous discounting) is calculated as:
PV_a = FV_a * (1-e^-rt)/r
Example 1:What is the present value of Rs. 2000/- received at the end of
each year for 3 continuous years
= 2000*[1/1.10]+2000*[1/1.10]^2+2000*[1/1.10]^3
= 2000*0.9091+2000*0.8264+2000*0.7513
= 1818.181818+1652.892562+1502.629602
= Rs. 4973.704

3.   Present Value of a Single Cash Flow

Present value of (PV) of the future sum (FV) to be received after a period ‘t’ for which discounting is done at an interest rate of ‘r’, is given by the equation
In case of discrete discounting: PV = FV / (1+r)^t
Example 1: What is the present value of Rs.5,000 payable 3 years hence, if the interest rate is 10 % p.a.
PV = 5000 / (1.10)3        i.e. = Rs.3756.57
In case of continuous discounting: PV = FV * e^-rt

Example 2: What is the present value of Rs. 10,000 receivable after 2 years
at a discount rate of 10% under continuous discounting?
Present Value = 10,000/(exp^(0.1*2)) = Rs. 8187.297