Compound Interest

What is the Difference Between Future Value and Present Value?

In many investment problems, you are given an amount of money and asked what will it accumulate to in a certain amount of time at some interest rate. Essentially, these problems are asking you to find the future value of the amount of money. Depending on how that money accumulates, you might use one of several different formulas.

If the problem specifies simple interest, you would use

where an amount PV accumulates to a future value FV in t years at a rate of r.

If the problem specifies compound interest, you would use

where i is the interest rate per period and n is the number of periods.

It is also possible for a problem to specify continuous compounding in which case you would use

where r is the continuous rate and t is the number of years.

For a future value problem, the quantities on the right side of each of these equations will be specified so that you can calculate the future value FV.

In a present value problem, you will be given the amount in the future FV and asked to find the amount you would start with to get to that amount. Let’s look at two possible problems.

Problem What would need to be deposited today to reach a value 0f 100,000 dollars in 10 years at a rate of 3% per year compounded annually?

Solution The first thing we need to realize is which formula we need to use. Since it says the interest is compounded annually, we need

In the compound interest formula we are given that the future value is 100,000 dollars, the interest rate per period is i = 0.03, and the money will accumulate over 10 periods. Putting these values into the compound interest formula yields

Solving this equation for PV yields

This means that the present value of 100,000 dollars is approximately 74,409.39 dollars when compounded annually for 10 years at an annual rate of 3%.

Now let’s change the problem up and see how this affects the solution process.

Problem What is the present value of 100,000 dollars compounded continuously at a rate of 3% per year?

Solution On the surface, this is almost the same problem as the one above except that it specifies continuous interest. This means we need to start from

instead of the compound interest formula. Putting in the values in the problem yields

Isolating the present value PV gives

In each of these problems, we need to solve for the present value PV but we start from a different formula that depends on the how interest is earned.

How Do You Solve for the Number of Years in the Compound Interest Formula?

In another MathFAQ, I looked at how you can find the rate in the compound interest formula. Now let’s look at an example where we solve for the number of years n. This problem is different because what we are looking for appears in a power.

Problem Suppose $5000 is deposited in an account that earns 2% compound interest that is done annually. In how many years will there be $6000 in the account.

Solution This problem requires the use of the compound interest formula,

This formula applies when interest is earned on an annual basis and the interest is earned once a year.

Let’s look at the quantities in the problem statement:

$5000 is deposited in an account > P = 5000
that earns 2% compound interest that is done annually > r = 0.02
Will there be $6000 in the account > A = 6000

Putting these values into the formula above gives us

Unlike other problems where we solve for P or r, here we need to solve for the power in the right hand side, n. Solving for a value in the power requires the property of logarithms, log(y^x) = x logy. It allows us to move the n in the power and change it to a multiplier. But before we can apply this property, we isolate the factor containing the n:

Now take the logarithm of both sides of the equation:

This gives us

or n ≈ 9.21 years.

In WolframAlpha, we could evaluate the logs as follows.

How Do You Solve For The Rate In The Compound Interest Formula?

Problems that ask you to solve for the rate r in the compound interest formula require the use of roots or creative use of exponents. Let’s look at an example.

Problem Suppose 5000 dollars is deposited in an account that earns compound interest that is done annually. If there is 7000 dollars in the account after 2 years, what is the annual interest rate?

Solution The easiest way to approach this problem is to use the compound interest formula,

This formula applies when interest is earned on an annual basis and the interest is earned once a year.

Let’s look at the quantities in the problem statement:

5000 dollars is deposited in an account > P = 5000
If there is 7000 dollars in the account after 2 years > A = 7000 and n = 2

Putting these values into the formula above gives us

We need to find the annual interest rate r. Since the r is hidden in the parentheses, we start by isolating the parentheses.

To get at the r, we need to remove the square on the parentheses.

Using a calculator to do the square root, we get r ≈ 0.183 or 18.3%.

Although most calculators have a square root key, when removing powers it is often useful to raise both sides to a power. For instance, we could remove the square by raising both sides to the ½ power.

When you raise a power to another power, you multiply the exponents 2 ∙ ½ = 1. The right side simply becomes 1 + r. Now we can solve for r:

Using the power key on your calculator gives the same answer as before. Make sure the 1/2 power is entirely in the power. You can make sure this happens using parentheses: (7000/5000)^(1/2)-1.

Now what if the interest is earned over six years instead of two years? Instead of a square on the parentheses we now have a sixth power.

To solve for r in this equation, we follow similar steps.

The root can be computed on a graphing calculator using the MATH button or put into WolframAlpha:

Either method gives r ≈ 0.577 or 5.77%. Notice that the annual interest is lower when it is earned over a longer period of time.

If we use a ¹/₆ power to solve for r, we would carry out the steps below:

Using a ¹/₆ power on your calculator gives the same answer as above.

How Do You Find Compound Interest Future Value In Google Sheets?

Spreadsheets have several built in functions for working with compound interest and annuities. To use these functions, we’ll start with a standard sheet.

This worksheet contains the variables used throughout Chapter 8. These variables correspond to these letter used in the text.

Number of periods is n
Annual interest rate is r
Payment is R
Present value is P
Future value is A
Periods per year is m

Values given in a problem will be entered in column B. Values calculated by the spreadsheet will be entered in column C. We will also assume that amounts paid out are negative and amounts received are positive.

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