When calculating future values with compounded interest, we need three parts - the present value or principal amount, an interest rate, and the number of time periods in which the money is invested. The formula takes whatever amount of money you are investing today and adds it to the interest which is compounded over time. The formula is shown below:

Where:

**FV**= Future Value**PV**= Present Value**i**= period interest rate expressed as a decimal**n**= number of compounding periods

The present value is simply the amount of money that will be invested, *i* is the interest rate for each time interval, and *n* is the number of compounding intervals. The formula can be used when compounding annually, monthly, or at whatever time interval over which you wish to compound. The only thing you must remember is that the interest rate must match your time period. If you are compounding daily, for example, then be sure that you are working with a daily interest rate, or if you are compounding monthly, be sure that you are working with a monthly interest rate. Let’s take a look at an example.

#### Example

Let’s say you invest $1,000 in an account that pays 4% interest compounded annually. How much will you have after five years? In order to calculate the future value of our $1,000, we must add interest to our present value. Because we are compounding interest, we must reinvest our interest earned so that our interest earned also earns interest. Herein lies the power of compounding! Let’s look at the calculation.

The first step in the calculation is exactly the same as with calculating future values with simple interest. The only difference here is that rather than sticking that interest in your pocket, you are reinvesting it. Just like with calculating simple interest, after one year, you will end up with $1,040 in your account because you have earned $40 in interest ($1,000 * 4%). Now though, we will choose to reinvest this interest, so in the second year, you will earn 4% interest on $1,040, which is the amount you will have after the first year. So, after the second year, you have $1,040 * (1 + 4%), which is $1,081.60. This means that you earned $41.60 in interest in the second year because you earned 4% on $1,040. We are earning interest on our previously earned interest rather than earning the same amount of interest each year. We can replicate this same process over the course of a five-year period to see how things progress. The process is shown in the chart below.

Year | PV | Interest | FV |
---|---|---|---|

1 | $1,000.00 | $40.00 | $1,040.00 |

2 | $1,040.00 | $41.60 | $1,081.60 |

3 | $1,081.60 | $43.26 | $1,124.86 |

4 | $1,124.86 | $45.00 | $1,169.86 |

5 | $1,169.86 | $46.79 | $1,216.65 |

As you can see, the amount of interest increases each year as the balance of the account at the beginning of each year increases. After five years, you will have $1,216.65 in your account. What has happened here is that we have added our interest (the sum of the dollar amounts in Column “Interest”) to our initial principal or present value amount. Now, you may be thinking that this seems complicated to compute and that it takes a lot of steps in order to arrive at what your $1,000 will be worth in five years, but thankfully, we have our formula to help us with this. The future value is calculated easily with our formula below:

While our formula computes the future value, finding the interest portion is only one more step. All we have to do is subtract our present value from our future value because the future value is simply the present value plus interest. In this case, our total accumulated interest is $216.65 (once again, this is the sum of interest earned each year).

One thing to note is that, because we were given an annual rate and were compounding annually, we were able to plug *i* and *n* into the formula directly. Let’s take a look at what to do when the rate given is not the rate per compound period.

### What to do when the Compounding Bases aren’t the Same

The formula for calculating compound interest is time agnostic, meaning that we can use the formula for compounding over any length time interval, but we must make sure that the rate represents how much our principal is compounding each period, i.e. that our rate and period length are of the same basis. Say in our previous example that we earned interest semiannually rather than annually. How will *n* and *i* change? Because *n* represents the number of compounding periods, and we are compounding semiannually for five years, there will be 10 compounding periods. We multiply five years by a compounding frequency of two (twice per year) to arrive at the number of compounding periods. Now we also can’t use the same rate, because if we have *n* as 10, and we used our annual rate, then this would be compounding annually for ten years. In order to adjust the rate, we must divide it by 2, since we are now earning 2% per period rather than 4%. This may seem a little confusing, but just remember that no matter how many periods over which your principal is compounding, your compounding rate must match the length of the period. Let’s walk through an example.

#### Example

Suppose we stick with an example similar to the one above. You invest $1,000 in an account at a bank, but this time the bank is promising to pay you an annual interest rate of 4%, compounded semiannually, for five years. This means that the bank will pay you twice per year, and each time you will reinvest your interest. What will be the future value of your principal after five years? Because the interest is paid twice a year (i.e. semiannually), the interest rate *i* will be cut in half. This is also the reason that our original number of periods is multiplied by 2 since the money invested is compounded twice per year. If you earn interest twice per year over a five-year period, you will earn interest at ten different times. If you start the year with $1,000, then after six months, the bank will pay you 2% (half of 4%) on your $1,000, which is $20, so you now have $1,020. At the end of the year, the bank will pay you 2% interest again, but this time it will pay you interest on your $1,020 that you had after six months. This equates to $1,020 * (1 + 2%) = $1.040.40. Again, we can show this process for five years in the chart below.

Year | PV | Interest | FV |
---|---|---|---|

0.5 | $1,000.00 | $20.00 | $1,020.00 |

1 | $1,020.00 | $20.40 | $1,040.40 |

1.5 | $1,040.40 | $20.81 | $1,061.21 |

2 | $1,061.21 | $21.22 | $1,082.43 |

2.5 | $1,082.43 | $21.65 | $1,104.08 |

3 | $1,104.08 | $22.08 | $1,126.16 |

3.5 | $1,126.16 | $22.52 | $1,148.69 |

4 | $1,148.69 | $22.97 | $1,171.66 |

4.5 | $1,171.66 | $23.43 | $1,195.09 |

5 | $1,195.09 | $23.90 | $1,218.99 |

We ended with a slightly higher future value than what we would have had if our interest were compounded annually. Now again, this requires a lot of steps to do by hand, but thankfully, we can also calculate this future value using our formula. We must be careful with what we input into the formula. Here, *n* will be twice the number of years, since we are compounding ten times, and *i* will be half of our annual rate, since we are compounding twice per year. The work is shown below:

Once again, our formula calculates a future value, but we are only one step away from calculating interest. All we have to do is subtract our principal from our future value. In our example, the accumulated interest is $218.99, which is our future value of $1,218.99 minus our principal of $1,000 (remember that this interest is the sum of all the interest payments each year).

### Compounded Interest with Deposits

In our examples before, we put money in an account and watched it grow, which is great, yet many times investors will add more money to an account on a regular basis. That fixed sum that is deposited regularly is called an annuity. One important point to note is that, in order to use the formula below, these fixed payments must be made at equal intervals. We have two different formulas to calculate the future values of the deposits. Why? Investors may add deposits at the beginning of each deposit period (called an “annuity due”), or they may add deposits at the end of each deposit period (called an “ordinary annuity”). The two formulas are similar, yet there is a subtle difference: the annuity due is simply compounded over one more period of time. The formula for the future value of an ordinary annuity is shown first, followed by the formula for an annuity due.

Ordinary Annuity

Annuity Due

Where:

**FV**= Future Value**PMT**= Payment each deposit period**i**= period interest rate expressed as a decimal**n**= number of compounding periods

#### Example of an Ordinary Annuity

Suppose you deposit $135 into an account every quarter and the bank promises to pay you interest of 6% compounded quarterly. You want to see how much you will have in the account at the end of three years. The way this works is that after the first quarter of the first year, you add $135 into your account. That amount then accrues interest over each quarter until the end of the three years. It compounds according to the compound interest formula eleven times. Recall that the exponent on that formula is the number of compounding periods. Here, we have three years, and the interest compounds four times a year, but because we deposit the money at the end of each quarter, the initial $135 that you deposit will compound only eleven times instead of twelve (the case if you would deposit the $135 initially). Now let’s take a look at what happens at the end of the second quarter. Now, you deposit $135 again, but this time, this deposit will accrue interest using the compound interest formula ten times. The process repeats until at the end of three years, you deposit your last $135 that will not accrue interest since you are depositing it on the same day you are checking the balance in your account. Remember also that, because you are compounding quarterly, the annual rate must be divided by four since your deposits are earning interest every quarter. A diagram may be useful to illustrate this concept.

We see here that each period you add $135. That amount is compounded quarterly for the number of quarters remaining before the end of the three-year period. Think of this as twelve different compound interest calculations, one for each quarter that you deposit $135. At the end of three years, simply add up each compound interest calculation to get your total future value. The chart below demonstrates how this works.

Year | Periods | PMT | Interest Factor | FV |
---|---|---|---|---|

0.25 | 11 | $135.0 | 1.1779489374 | $159.023 |

0.5 | 10 | $135.0 | 1.160540825 | $156.673 |

0.75 | 9 | $135.0 | 1.1433899754 | $154.358 |

1 | 8 | $135.0 | 1.1264925866 | $152.076 |

1.25 | 7 | $135.0 | 1.1098449129 | $149.829 |

1.5 | 6 | $135.0 | 1.0934432639 | $147.615 |

1.75 | 5 | $135.0 | 1.0772840039 | $145.433 |

2 | 4 | $135.0 | 1.0613635506 | $143.284 |

2.25 | 3 | $135.0 | 1.045678375 | $141.167 |

2.5 | 2 | $135.0 | 1.030225 | $139.080 |

2.75 | 1 | $135.0 | 1.015 | $137.025 |

3 | 0 | $135.0 | 1 | $135.000 |

Total: |
$1,760.56 |

Here, we multiply the payment each period by the second half of the compound interest formula, noted here as “Interest Factor.” From here, we compute the future value after each period, and we sum all future values to get our future value at the end of three years. Now, yes, this is a lot of steps, but thankfully we have our formula to calculate that same value in just a few basic algebraic steps. The details are shown below.

As we have done previously, if we want to calculate interest earned, we simply subtract out the raw amounts that we added each period, which in total equates to $135 * 12 = $1620. Therefore, interest accumulated is equal to $1760.56 - $1620 = $140.56

#### Example of an Annuity Due

Luckily, we don’t have to work too hard for this example. Let’s again assume that you are depositing $135 quarterly for three years, that compounds at 6%. We still want to know how much money we will have at the end of three years, but what happens if we deposit that money at the beginning of each period? All that happens is that in that three-year period, each deposit accrues interest for one more period. Because you deposit $135 right at the beginning, that amount compounds for all twelve periods, and your last deposit of $135 will have the chance to earn interest for the last period. The diagram below demonstrates how this works.

Here, we have the same number of payments, but each deposit accumulates interest for one more period than when the deposits are made at the end of the period, hence the reason why on our Annuity Due formula, we have to tack on the bit at the end that compounds our future value for one more period. Again, we calculate twelve different future values, and we sum those future values to get the value in the account at the end of three years. The chart below demonstrates these steps.

Year | Periods | PMT | Interest Factor | FV |
---|---|---|---|---|

0.25 | 12 | $135.0 | 1.1956181715 | $161.408 |

0.5 | 11 | $135.0 | 1.1779489374 | $159.023 |

0.75 | 10 | $135.0 | 1.160540825 | $156.673 |

1 | 9 | $135.0 | 1.1433899754 | $154.358 |

1.25 | 8 | $135.0 | 1.1264925866 | $152.076 |

1.5 | 7 | $135.0 | 1.1098449129 | $149.829 |

1.75 | 6 | $135.0 | 1.0934432639 | $147.615 |

2 | 5 | $135.0 | 1.0772840039 | $145.433 |

2.25 | 4 | $135.0 | 1.0613635506 | $143.284 |

2.5 | 3 | $135.0 | 1.045678375 | $141.167 |

2.75 | 2 | $135.0 | 1.030225 | $139.080 |

3 | 1 | $135.0 | 1.015 | $137.025 |

Total: |
$1,786.97 |

Notice that our future value is greater than if we had started depositing at the end of each period, since we do earn interest on each deposit one more time. Now this method is just as many steps as before, but thankfully we can use our trusty formula. The steps are provided below:

So, our value after three years is $1,786.97. In order to calculate accumulated interest, we once again must subtract out the sum of our deposits, which is still $1,620, so we now arrive at total interest of $1,786.97 - $1,620 = $166.97.