The value of money can be expressed as the present value (discounted) or future value (compounded). A $100 invested in bank @ 10% interest rate for 1 year becomes $110 after a year. From the example, $110 is the future value of $100 after 1 year and similarly, $100 is the present value of $110 to be received after 1 year. They are just reciprocal of each other.

## FUTURE VALUE

### Definition of Future (or Compounded) Value

It can be defined as the rising value of a today’s sum at a specified future date given at a specified rate of interest. It is calculated by compounding technique.

### Future Value Example with Compounding of Money

Compounding of money is the value addition in the initial principal amount after defined intervals at a given rate of interest. For example, if Mr. A invests $1,000 for say 3 years @ 10% interest rate compounding annually then the income will rise as follows:

First Year: | Principal at the beginning | 1,000.00 |

Interest for one year (1000*0.10) | 100 | |

Principal at the end | 1,100.00 | |

Second Year: | Principal at the beginning | 1,100.00 |

Interest for the year (1100*0.10) | 110 | |

Principal at the end | 1,210.00 | |

Third Year: | Principal at the beginning | 1,210.00 |

Interest for the year (1210*0.10) | 121 | |

Principal at the end | 1,331.00 |

This process of calculation is known as compounding and the sum arrived at after compounding of initial amount is known as Future Value. In our example, the future value of $1000 is $1331 after 3 years @ 10% interest rate compounding annually. Similarly, a present value of $1331 is $1000 under same conditions.

### Future Value Formula and its Explanation

This was a very simple example. In practical use, there can be 20 years in place of just 3 and more frequent compounding as compared to annual. Following formula helps in determining the future value of any sum very easily.

FV = PV (1+r)n

Where,

PV = Present value or the principal amount

FV = FV of the initial principal n years hence

r = Rate of Interest per annum

n = a number of years for which the amount has been invested.

In this equation, (1+r)n is the compounding factor which calculates the principal amount along with interest and interest on interest. It is called “Future Value Interest Factor”

Now if we solve the above example with the given formula, we get

=1000(1+0.10)3 = Rs. 1, 331/-

The formula is helpful to calculate amount invested for longer maturity periods say 10-20 years very quickly and easily.

### Multiple Compounding Periods

Compounding period refers to the no. of years/months for which the interest is made due. These can be monthly, quarterly, half yearly and annually etc. For example, if the interest is charged on a monthly basis, then annual interest rate ‘r’ shall be divided by 12 and no. of years ‘n’ shall be multiplied by 12. So, when a frequency of compounding is more, the effective interest amount is also more.

### Rule of 72 Trick

There is a general question in an investor’s mind that How many years will it take to double my money? ‘Rule of 72’ is a user-friendly mathematical rule used to quickly estimate the ‘rate of interest’ required to double your money given the ‘number of years’ of investment and vice versa. It is specifically called the rule of 72 because the number 72 is used in its formula.

### Formula for Rule of 72

For calculating the number of years required to double your money given the rate of interest, the formula is:

Number of years = 72 / Interest

Example: At 8%, money takes 72/8 or 9 years to double it.

Or

For calculating the rate of interest required to double your money given the number of years of investment, the formula is:

Rate of Interest = 72 / Number of years

Example: For a 10-year investment, it requires 72/10 or 7.2% rate of interest p.a. to double itself.

## PRESENT VALUE

### Definition of Present (or Discounted) Value

It can be defined as today’s value of a single payment or series of payment to be received at a later date, given at a specified discount rate. The process of determining the present value of a future payment or a series of payments or receipt is known as discounting.

### Present Value Example with Discounting of Money

In absolute terms, discounting is the opposite of compounding. It is a process for calculating the value of money specified at a future date in today’s terms. The interest rate for converting the value of money specified at a future date in today’s terms is known as the discount rate.

We will take a reverse example of future value as explained above. Mr. A has an offer to get $1331 after 3 years if he pays $975 today. A market interest rate is 10% with annual compounding. What should Mr. A do? To pay $ 975 or not. To decide whether he should pay $975 or not, he should be able to compare his proposed outflow of today with the today’s value of $1331 to be received after 3 years. Refer table above, we know that the present value of $1331 after 3 years is $1000. So, Mr. A should definitely pay $975 because there is a clear-cut benefit of $25 over and above the interest earnings.

This process of calculation of present value is known as discounting and the sum arrived at after discounting of a future amount is known as Present Value.

### Present Value Formula and its Explanation

The formula to calculate the present value is as follows:

PV = FV / (1+r)n

Or

PV = FV * 1/(1+r)n

Where,

PV=Present value or the principal amount

FV= FV of the initial principal n years hence

r= Rate of Interest per annum

n= number of years for which the amount have been invested.

In this equation, ‘1/(1+r)n’ is the discounting factor which is called “Present Value Interest Factor”.

Our current example can be easily solved with the formula –

PV = FV * 1/(1+r)n

PV = 1331 * 1/(1+10%)3

PV = 1331 * 1/(1+0.10) 3

PV = 1331/1.13

PV = 1331/1.331

PV = 1000

### PRESENT VALUE INTEREST FACTOR (PVIF) AND FUTURE VALUE INTEREST FACTOR (FVIF) TABLES

PVIF and FVIF tables are available to facilitate the ease of calculations. Following is an example of FVIF table with various periods and percentage of interest. For example in our case, we have to look for r =10 and n= 3, the value is 1.331. From this only we can find PVIF just by dividing it by 1, 0.751 (1/1.331) or a separate table is also available.

FVIF | Rate of Interest (r) | |||||||

1% | 2% | 3% | 4% | 5% | 6% | 7% | ||

Period (n) | 1 | 1.010 | 1.020 | 1.030 | 1.040 | 1.050 | 1.060 | 1.070 |

2 | 1.020 | 1.040 | 1.061 | 1.082 | 1.103 | 1.124 | 1.145 | |

3 | 1.030 | 1.061 | 1.093 | 1.125 | 1.158 | 1.191 | 1.225 | |

4 | 1.041 | 1.082 | 1.126 | 1.170 | 1.216 | 1.262 | 1.311 | |

5 | 1.051 | 1.104 | 1.159 | 1.217 | 1.276 | 1.338 | 1.403 | |

6 | 1.062 | 1.126 | 1.194 | 1.265 | 1.340 | 1.419 | 1.501 | |

7 | 1.072 | 1.149 | 1.230 | 1.316 | 1.407 | 1.504 | 1.606 |

PVIF | Rate of Interest (r) | |||||||

1% | 2% | 3% | 4% | 5% | 6% | 7% | ||

Period (n) | 1 | 0.99 | 0.98 | 0.971 | 0.962 | 0.952 | 0.943 | 0.935 |

2 | 0.98 | 0.961 | 0.943 | 0.925 | 0.907 | 0.89 | 0.873 | |

3 | 0.971 | 0.942 | 0.915 | 0.889 | 0.864 | 0.84 | 0.816 | |

4 | 0.961 | 0.924 | 0.888 | 0.855 | 0.823 | 0.792 | 0.763 | |

5 | 0.951 | 0.906 | 0.863 | 0.822 | 0.784 | 0.747 | 0.713 | |

6 | 0.942 | 0.888 | 0.837 | 0.79 | 0.746 | 0.705 | 0.666 | |

7 | 0.933 | 0.871 | 0.813 | 0.76 | 0.711 | 0.665 | 0.623 |

### Calculator Trick

This table can be easily made over calculator. Let the interest rate be 1%. The steps are:

1. First convert the percentage to decimals: 1/100=0.01

2. Add 1: 1+0.01= 1.01

3. Now Divide 1 by the step 2 result: 1/1.01=0.990

4. Now press equal to sign(=) on the calculator so many times as the number of years, and you will get the series on factors year wise. You can match the results with the table.

5. Prepare your own table as per the rate of interest.

Thus, the above concepts enable us to judge in certain terms whether it is beneficial to receive or spend money now or later. This concept is widely used for project decisions and evaluation. One can make general decisions for projects by calculating its payback period. But accurate decisions call for calculating the present value of future income so that we know the exact returns the project will give and thus can decide upon the viability of the project. Similarly, if future value of a certain amount is calculated it adds attractiveness to the investment proposals.