Time Value of Money Concepts

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

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

1. Formula and Definition

The equation below calculates the current value of a single sum to be paid at a specified date in the future. This value is referred to as the present value (PV) of a single sum.

If we remember that 1/xn can be written as x-n, then a more compact form of the equation can be written as:

The PV of a single sum formula is used as a valuation mechanism. It tells us how much an amount to be transacted in the future is worth today (or some date prior to the receipt or payment date).

For example, if we are want to buy a $1,000 zero coupon bond that matures in 10 years, how much is it worth right now? In other words, how much should we pay for that bond today?

The PV of a single sum formula can value liabilities as well as assets. For example, if we owe a debt that obligates us to pay $5,000 in 3 years time, what is a fair amount we could offer to settle that debt today?

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

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


2. Present Value (PV) of a Single Sum Illustrated

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

What is the current value (PV) of a CD that will pay $100 in 3 years if the prevailing interest rate is 5% compounded annually? In other words, how much do I need to deposit to have $100 in 3 years?

Drawn from the the perspective of the investor, the problem is illustrated below. The investor will receive $100 in three years time (the FV) and this amont is "discounted" back to today at 5% in order to calculate the required deposit (the PV)...

The arrow drawn pointing away from the time line (labeled "100.00") represents a cash inflow to the investor. The arrow drawn pointing to the time line (labeled "?PV") represents a cash outflow from the investor, in this case it is the amount invested in the CD. 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 how much we will receive at maturity, the term of the investment, and the interest rate, we can summarize our inputs to the PV of a single sum equation as...

FV = 100.00
i = 0.05
n = 3

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

...we calculate a PV of $86.38.

The mechanics of the calculation are illustrated below...

So what does it mean when we say that the present value of $100 in 3 years at 5% is $86.38?

In essence it means that the receipt of $100 in three years is worth the same as the receipt of $86.38 today. The logic behind this assertion is that if we deposited $86.38 into an investment account paying 5% annually, it would grow to $100 in three years. In this case we should be indifferent as to our preference for one option over the other because $86.38 today or $100 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 PV of a single sum, there are times when we need to know the value of one of the other variables (n, i, or FV) .

For single sums, solving for any of the other TVOM variables is simply a 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 PV 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 identities are 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 PV of $100 in 3 years at 5% under monthly compounding is $86.10. 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. Future Value (FV)

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

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

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


4. Compounding Frequency

The PV 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 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 present 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 present value PV is calculated as...

Changing the compounding period from annual to quarterly reduces the present value by $0.23 over the 3 year period ($86.38 - $86.15). More frequent compounding means less money is required up front (i.e., at "present") in order to grow to a specified amount in the future.

Under monthly compounding, the PV is even smaller...

...and smaller still under daily compounding...

...and smallest under continuous compounding...

The PV formula used for continuous compounding looks a little strange. However, it is derived directly from the standard PV of a single sum equation. The concept of continuous compounding and derivation of the formula is 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 PV of a single sum in Excel:

a. PV function

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

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

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

b. Discount Schedule

We can compute the PV without the aid of a special function by creating a discount schedule as shown here...

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

The schedule simply discounts the FV balance one period at a time. In other words, the "beg" balance is the FV and the "end" balance is the PV. The "int" amount is equal to the difference between the two and is calculated as...


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 present 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.