Can you explain the difference between "simple" and "compound" interest? Please provide some of the uses of compound interest in business

Simple interest is interest on the principle amount while compound interest is when your principle and any earned interest earned interest. The interest rate is applied to the original principle and any accumulated interest.
Compound interest is critical to investment growth. With simple interest, interest is paid just on the principal. With compound interest, the return that you receive on your initial investment is automatically reinvested. In other words, you receive interest on the interest.

Future Value = Present Value (1+r)^n
r= interest rate
n= time period

Future value of today's Rs 100 @10% per annum after one year will be Rs. 110/-
Thus compounding technique is use to find the future value of the investment

Use of Compounding in business:

1) Future value of Annuity

WE can also use the compounding technique to find out the future value of annuity in the following manner:

Example:
What's the future value in 10 years of $1,000 payments received at the beginning of each year for the next 10 years? A 5.625% interest rate is assumed.

Here we have to find out the compounded value of annuity
F=A*((1+r)^n-1)/r
F=Future value, A= Annuity r= rate of interest n=duration

A= 1000, r= 5.625%, n=10= 12950.96

2. This is also used in taking the investment decisions of a firm. The firm's investment decisions would generally include expansion, acquisition, modernization and replacement of the long-term assets. Sale of a division or business (divestment) is also as an investment decision.
Decisions like the change in the methods of sales distribution, or an advertisement campaign or research and development programs have long-term implications for the firm's expenditures and benefits, and therefore, they should also be evaluated as investment decisions.
The methods such as Net Present value, IRR use the concept of Compounding to find out the present value of the project and also incorporates time value of money. This is described below.

2. What is "present value"? What is an example of the "present value" concept? How does a single cash flow present value example differ from an annuity calculation?
PRESENT VALUE

Present value is the future value being discounted by an appropriated capitalization rate. The value of sum of money received today is more than its value received after some time.
In other words present worth of rupee received after some time will be less than the rupee received today. This is because of time value of money. The investor has time preference of money because he has reinvestment opportunities for funds, which are received early.
(I.M. Pandey)
The present value of a future cash flow is the nominal amount of money to change hands at some future date, discounted to account for the time value of money. A given amount of money is always more valuable sooner than later since this enables one to take advantage of investment opportunities.

1)What's the present value of a. $9,000 in 7 years at 8 percent? Present value= 5251.41

P= present value, F= Future value r= rate of interest n= duration

P=F/(1+r)^n

Present value of Annuity:
Annuity involves payment or receiving of the same amount of cash each year. Thus the present value of annuity is:
P=A*((1/r)-((1/(r*((1+r)^n)))
P= present value, A= Annuity r= rate of interest n= duration

Example

John Smith will receive $8,500 a year for the next 15 years from her trust. If a 7 percent interest rate is applied, what is the current value of the future payments?

Here we have to find out the present value of annuity and by applying y the above formulae the value is $77417.27

Finding the capital recovery annuity
The compounding value technique is also used to find the capital recovery annuity

Example:
To help you get through college, your parents made a deposit of $25,000 into a bank account paying 8% interest. Starting next year, you plan to withdraw equal amounts for the account at the end of each of the next four years. What is the most you can withdraw annually?
Here we have to find capital recovery annuity
A=P*((r*(1+r)^n)/(1+r)^n-1)
P=present value, A= Annuity r= rate of interest n=duration A=$7548

Thus unlike single cash flow, the annuity involves regular payment or receipt of the cash flow at every fixed period of interval.

3. How is a home mortgage an example of TVM? How can you show that more interest is paid at the beginning of a loan period than at the end?

Home mortgage is an example of TVM. This is because it is an annuity. As this involves payment of the same amount of cash after each fixed period. Thus the present value of annuity is:
P=A*((1/r)-((1/(r*((1+r)^n)))
P= present value, A= Annuity r= rate of interest n= duration
The interest is paid more at the beginning of the loan period as the principal in the beginning is more. Gradually the portion of principal is paid back and hence the payment of the interest reduces gradually.