Recipe 5.16 Calculating Asset Appreciation (Future Value)

5.16.1 Problem

You want to know how much something will be worth in the future, given its value today and an expected interest rate, i (for example, calculating the amount of money you can accumulate in a savings account given an initial deposit).

5.16.2 Solution

You must calculate the appreciation of an asset over time at the assumed interest rate. This is often referred to as the future value of an asset. For a given interest rate, i, an asset appreciates by a factor of (1 + i) for each period (such as a year).

5.16.3 Discussion

For the purposes of illustration, let's calculate the future value of an asset using the brute-force method.

If you deposit $1 in a bank account that earns 5% interest per year, one year from now, you will have earned 5 cents in interest. The total account, including principal and interest, will be worth $1.05. The math for this is:

FV = PV * (1 + i);

where FV is the future value (the amount of money you'll have next year, sometimes called FV1), PV is the present value (the amount of money you deposited initially, sometimes called FV0), and i is the interest rate (expressed as a decimal, such as 0.05).

So if you deposit $100 at a 5% interest rate, the future value at the end of one year is $105, which is determined as follows:

FV1 = 100 * (1 + .05);

Typically, interest is compounded over time, so at the end of the second year, the value is:

FV2 = FV1 * (1 + i);

Here is a function that calculates the future value by brute force for an arbitrary number of periods. We name the parameter that represents the interest rate interest (instead of i) so as not to mistake it for an index variable (and we'll use periods as our loop variable to avoid the reverse confusion):

Math.bruteFutureValue = function (interest, n, PV) {
  // PV       = initial deposit (present value)  
  // interest = periodic interest rate
  // n        = number of periods
 
  // Start with the future value equal to the present value.
  FV = PV;
  
  // Now compound it over n periods at an interest rate of interest.
  for (var periods = 1; periods <= n; periods++) {
    FV = FV * (1 + interest);
  }  
  return FV;
};
// Example usage: 
// How much will $500 appreciate in 10 years at a 5% rate?
trace ("It will be worth: " + Math.bruteFutureValue(.05, 10, 500));

You can use the custom CurrencyFormat( ) method from Recipe 5.6 to format the result for display.

It doesn't matter whether the interest is compounded daily, weekly, monthly, or annually, provided that the specified interest rate is correct for the number of periods. For example, if compounding monthly, be sure to multiply the number of years by 12 and divide the annual interest rate by 12.

// How much will $500 appreciate in 10 years at a 5% rate with monthly compounding?
trace ("It will be worth: " + Math.bruteFutureValue(.05/12, 10*12, 500));

So the brute-force method appears to work, but it has two drawbacks. One drawback is that it can be slow if you are doing a lot of calculations. The other problem is that it can't be used to calculate so-called continuous compounding.

Let's revisit this formula for the value of the account at the end of the second year:

FV2 = FV1 * (1 + i);

Substituting PV * (1 + i) for FV1, we get:

FV2 = (PV * (1 + i)) * (1 + i);

This can also be written as:

FV2 = PV * ((1 + i) * (1 + i));

The last portion is (1 + i) squared, so the formula can be written as:

FV2 = PV * Math.pow((1 + i), 2);

where 2 is the number of periods.

Extrapolating, you can guess that the future value after n periods would be:

FVN = PV * Math.pow((1 + i), n);

So we can speed up our calculation by using the following implementation:

Math.futureValue = function (interest, n, PV) {
  // PV       = initial deposit (present value)  
  // interest = periodic interest rate
  // n        = number of payment periods  
  multiplier = Math.pow ((1 + interest), n);
  return (PV * multiplier);
};

You'll see that you get the same result as with our earlier brute-force approach:

// How much will $500 appreciate in 10 years at a 5% rate with monthly compounding?
trace ("It will be worth: " + Math.futureValue(.05/12, 10*12, 500));

but our new code will be much faster for a large number of periods (when n is large).

However, to perform continuous compounding, the number of periods, n, becomes infinite. Although the derivation for the formula is beyond the scope of this book, continuous compounding can be implemented as follows:

Math.continuousCompounding = function (interest, n, PV) {
  // PV       = initial deposit (present value)  
  // interest = periodic interest rate
  // n        = number of payment periods  
  multiplier = Math.pow (Math.E, n*interest);
  return (PV * multiplier);
};
// How much will $500 appreciate in 10 years at a 5% rate with continuous
// compounding?
trace ("It will be worth: " + Math.continuousCompounding (.05, 10, 500));

The above calculations assume an interest rate, i, which may not be known or even predictable in many scenarios. For example, you can't predict the exact return of the stock market over any given period (although you can make educated guesses based on historic data if the time period is long enough). So, typically, you should allow the user to specify an interest rate and perform the calculations under various scenarios.

Remember to represent the interest rate as a decimal, such as 0.01. If you specify an interest rate of 1.0, it is equivalent to 100% interest! And be sure to specify the correct interest rate for the chosen number of periods (such as the monthly interest rate rather than the annual interest rate).

If you are calculating the future value of a fixed investment (such as a bond or certificate of deposit), you would know the fixed rate it returns. However, you might not be able to predict the rate of inflation. Everyone knows that inflation reduces the effective value of money in the future. To calculate the so-called real rate of interest, subtract the rate of inflation from the stated interest rate. For example, if your bond earns 5%, but you expect inflation to be 4%, you can use the above examples with an interest rate of 1% to calculate how much your money will be worth in the future relative to today's dollars.

5.16.4 See Also

Recipe 5.6, Recipe 5.17, Recipe 5.18, and Recipe 5.19. For the exact derivation of continuous compounding and other financial formulas, see any competent college-level financial accounting or applied mathematics book. You can find much of the interesting math online by Googling for "continuous compounding derivation." For example, see http://ljsavage.wharton.upenn.edu/~waterman/Teaching/IntroMath99/Class04/Notes/node13.htm.