Future Value of A Series Of Payments : -

Future Value of a Series of Payments

This article demonstrates how to calculate the future value of an uneven series of payments. We will find the total value of these payments at a future point in time, considering a specific interest rate.

The Problem

Calculate the future value at the end of 5 years of the following series of payments, assuming a 10% annual rate of interest.

  • Year 1: ₹1,000
  • Year 2: ₹2,000
  • Year 3: ₹3,000
  • Year 4: ₹2,000
  • Year 5: ₹1,500

The Formula

To find the future value (FV) of an uneven series of payments, we calculate the future value of each individual payment at the end of the term and then sum them up. The general formula is:

$$ FV = \sum_{t=1}^{n} R_t(1+i)^{n-t} $$

Where:

  • \(FV\) is the Future Value
  • \(R_t\) is the payment in period \(t\)
  • \(i\) is the interest rate per period
  • \(n\) is the total number of periods
  • \(t\) is the period in which the payment is made

For our specific 5-year problem, the formula expands to:

$$ FV = R_1(1+i)^{4} + R_2(1+i)^{3} + R_3(1+i)^{2} + R_4(1+i)^{1} + R_5(1+i)^{0} $$

Step-by-Step Calculation

1. Substitute the known values into the formula:

$$ FV = 1000(1+0.10)^{4} + 2000(1+0.10)^{3} + 3000(1+0.10)^{2} + 2000(1+0.10)^{1} + 1500 $$

2. Simplify the terms inside the parentheses:

$$ FV = 1000(1.10)^{4} + 2000(1.10)^{3} + 3000(1.10)^{2} + 2000(1.10) + 1500 $$

3. Calculate the value of the compounded factors:

$$ FV = 1000(1.4641) + 2000(1.331) + 3000(1.21) + 2000(1.10) + 1500 $$

4. Perform the multiplications:

$$ FV = 1464.10 + 2662.00 + 3630.00 + 2200.00 + 1500.00 $$

Final Result

The total Future Value at the end of 5 years is: ₹11,456.10

Difficult Words Explained

  • Future Value (FV): The total worth of an investment or series of payments at a specific date in the future.
  • Series of Payments: A number of payments made over a set period of time, like monthly deposits into a savings account.
  • Rate of Interest: The percentage charged on a loan or paid on an investment for a certain period, usually one year.
  • Compounding: The process of earning interest on both your initial investment and on the accumulated interest from previous periods. It makes your money grow faster.
  • Principal: The original amount of money you invest or borrow, before any interest is added.
  • Annuity: A series of *equal* payments made at regular intervals (our example is a non-annuity because the payments are unequal).