Time Value of Money Concepts

This page covers the following topics regarding the calculation of the future value of a single sum:

  1. Formula and Definition
  2. FV of a Single Sum Illustrated
  3. Solving for Other Variables in the FV Equation
  4. Compounding Frequency
  5. Excel
  6. HP-12C
  7. Programming Languages

1, Formula and Definition

The equation below calculates how large a single sum will become at the end of a specified period of time. This value is referred to as the future value (FV) of a single sum.

Observe from the formula that the future value (FV) consists of both a present value (PV) piece - an initial lump sum - and an accumulated interest piece. Thus, we start with a fixed amount and calculate how large it will grow (i.e., accumulate or compound) over the specified period of time and interest rate.

The FV of a single sum formula serves as a means of valuation. It tells us what something will be worth at a future date. This "something" can be an asset or a liability. For example, if we borrow $1,000 today and don't make any payments until the loan comes due two years from now, how much will we owe at that time? Likewise, if we deposit $5,000 into a bank account today, how much will it be worth in 180 days?

Note the distinction between the FV of a single sum and the PV of a single sum. The FV of a single sum answers the question "How much will it be worth then?" while the PV of a single sum answers the question "What is it worth now (or before 'then')?". The PV of a single sum is discussed separately here.

Also note that the formula above gives us the FV of a single sum; in other words, a fixed, lump sum amount. The future value of an annuity formula gives us the FV of a series of periodic payments. The FV of an annuity is discussed separately here.


2. Future Value (FV) of a Single Sum Illustrated

The following simplified example illustrates the basic operation of the FV of a single sum formula.

How much will I receive at the end of 3 years if I invest a single sum of $50 today at 8% interest compounded annually? In other words, at 8%, how large will my $50 grow in 3 years.

Drawn from the the perspective of the investor, the problem is illustrated below. The investor deposits $50 (the PV) and this amount "accumulates" over 3 years at 5% to some larger amount (the FV)...

The arrow drawn pointing toward the time line (labeled "50.00") represents a cash outflow from the investor. The arrow drawn pointing from the time line (labeled "?FV") represents a cash inlow to the investor, in this case it is the accumulated amount of the CD which the investor may withdraw at maturity. The question mark denotes the fact that this is the unknown amount whose value is the object of our calculation. (For additional assistance reading a cash flow diagram, click here.)

So now that we have identified our initial deposit, the term of the investment, and the interest rate, we can summarize our inputs to the FV of a single sum equation as...

FV = 50.00
i = 0.08
n = 3

...and plugging these values into the equation...

...we calculate a FV of $62.99.

The mechanics of the calculation are illustrated below...

So what does it mean when we say that the future value of $50 in 3 years at 8% is $62.99?

It means that since $50 today will grow (accumulate) to $62.99 in three years time, we should be indifferent as to our preference for one option over the other. In other words, $50.00 today and $62.99 in three years are financially equivalent.

Or at least they are according to TVOM principals and a set of assumptions discussed more fully here.

In reality, there are other factors that need to be taken into consideration (taxes, default risk, cash flow, etc.) before we can really declare "equivalence." Still, TVOM theory and its associated calculations provide a powerful tool for analyzing financial alternatives by providing a mechanism for placing cash flows at different time periods on a comparable basis.


3. Solving for Other Variables

While the equation discussed above allows us to calculate the FV of a single sum, there are times when we need to know the value of one of the other variables (n, i, or PV) .

For a single sum, solving for any of the other TVOM variables is a simple matter of rearranging the basic formula to isolate the variable being sought.

a. Compounding periods (n)

Knowledge of the following algebraic identity is necessary for isolating the exponent n...

Now by rearrangement of the FV of a single sum equation we can find the number of compounding periods (n) in our original example as...

b. Interest rate (i)

The following algebraic identity is helpful when solving for i...

Now we can solve for the interest rate (i) in our original example as...

If the compounding frequency is something other than annual, the interest rate (i) determined above would need to be multiplied by the number of compounding periods per year (m) in order to return the annual interest rate.

For example, the FV of $50 in 3 years at 8% under monthly compounding is $63.51. In this case we would calculate an annual i as...

See the discussion on "Compounding Frequency" that follows for more information on adjustments made to the values of i and n under non-annual compounding frequencies.

c. Present Value (PV)

Rearranging to solve for the PV of a single sum is fairly straight forward...

...and using the values from our original example, we confirm the PV as...

The PV of a single sum is discussed in more detail here.


4. Compounding Frequency

The FV of a single sum equation at the top of the page assumes annual compounding.

But what if in our original example we were compounding quarterly rather than annually?

In this case we must "synchronize" the values for i and n in order to accommodate the non-annual compounding frequency.

We start by defining n, the number of compounding periods in the term, as being equal to the product of two numbers: the number of compounding periods in the year (m) and the number of years in the term (Y)...

Thus for a three year term (Y=3) with quarterly compounding (m=4), the number of compounding periods (n) is 12 (4 x 3).

Now that we have modified n, we must adjust i.

i is almost always given as a annual nominal rate. If the compounding frequency is something other than annual, then i must be made proportional to the the period in which it is being applied. Typically this is accomplished by dividing i by m. Since here we are compounding quarterly, i would be divided by 4.

Taking all of this into account, if we rewrite the standard future value of a single sum equation to incorporate the synchronization process, it looks like this...

...and if in our original example above we had used quarterly rather than annual compounding, the FV is calculated as...

Changing the compounding period from annual to quarterly increases the future value by $0.42 over the 3 year period ($63.41 - $62.99). More frequent compounding means more interest is being earned on interest resulting in a greater accumulation (future value).

Under monthly compounding, the FV is even larger...

...and larger still under daily compounding...

...and largest under continuous compounding...

The FV formula used for continuous compounding looks a little strange. However, it is derived directly from the standard FV of a single sum equation. The concept of continuous compounding and derivation of the formula are discussed in more detail at Continuous Compounding.

Additional information on the impact of frequency and term on TVOM calculations can be found at Miracle of Compounding.


5.Excel

There are two approaches to solving for the FV of a single sum in Excel:

a. FV function

If all we want is the FV of a single sum, we can use Excel's FV function as shown here...

...where the cell formulas look like this...

...and the input parameters to the function are defined as follows...

Note that this function can be used for both single sum and annuity calculations depending on the parameters supplied.

b. Accumulation Schedule

We can compute the FV of a single sum without the aid of a special function by creating a accumulation schedule as shown here...

...where the cell contents look like this...

The schedule simply accumulates the balance one period at a time. In other words, the "beg" balance is the PV and the "end" balance is the FV. The "int" amount is equal to the interest earned on the PV for the current period.


6. HP-12C


7. Programming Languages

Practical application of TVOM concepts often involves using a programming language to code the calculations.

Listed below are some very simple illustrations of how the standard TVOM equation for the future value of a single sum can be coded in four different programming languages:

C#:

JavaScript:

VB Script:

T-SQL:

Note on SQL. Be careful about performing this type of math calculation using SQL because the code executes on the database server. As a general rule, the processing power of the database server is best reserved for performing large scale data modification and retrieval operations rather than arithmetic calculations. In a production environment such calculations are typically performed in a COM object on a middle tier server or perhaps by a VB Script in an Active Server Page on the web server or even by JavaScript on the client's browser. I included this example only to show that such TVOM calculations are possible using Microsoft's implementation of the SQL language (called T-SQL or Transact-SQL). The reader should be aware that doing so can make for a very expensive query. Be sure to consider all of the options before including such functionality in your production SQL code.