This is an introduction to the basic concepts of the time value of money, an important fundamental topic within mathematics of finance.

Time Value of Money

One of the most fundamental concepts in finance is that money has a “time value.” One dollar that you have today can be invested such that you will have more than one dollar at some future time. Your investment earns interest. Also, with the effects of inflation, $1000 a year from now will not have as much purchasing power as it does today.

Future Value

If you invest $1000 for one year at an annual interest rate of 6%, what will you receive after one year?  Before continuing, some variables will be defined. Present Value (PV) is what you have now.  Future Value (FV) is what you will have after a period of time.  The interest rate (r) is the rate of interest for the period of time in question.  The number of periods of time (n) is the number of times the interest earned gets re-invested.

In our example the FV is $1000 + $1000 x 0.06, which equals $1060. You will have $1060 one year from now if you invest $1000 today at an annual interest rate of 6%.
The general formula is FV = PV + PV r.  This can be rewritten as FV = PV (1 + r).

If we let that investment ride for another year (totaling 2 years), the future value would be FV =  PV + PV r + (PV + PV r) r.  From algebra we can expand this formula and get the following equation: FV = PV + 2 PV r + PV r ².  From this we get the formula FV = PV (1 + r)².  We can generalize this by introducing the variable n that was mentioned above, to get the general formula: FV = PV (1 + r)ⁿ.

This calculation was made assuming that the interest is compounded yearly.  In other words, r is the annual interest rate and n is the number of years.  To do this same calculation on a monthly basis, simply divide the annual rate r by twelve and use that number for r in the equation and multiply the number of years by twelve and use that number for n.

Present Value

If we have the future value formula above, we can rearrange the terms to derive the present value formula: PV = FV / (1 + r)ⁿ.  As an example, suppose
someone was to offer to pay you $1000 for work done, but could not pay you until two years from now.  What is $1000 two years from now worth right now assuming an annual
interest rate of 5%?  PV = 1000 / (1 + 0.05)² =  1000 / 1.1025 = 907.01. Therefore, $1000
two years from now is worth $907.01 today, assuming an annual interest rate of
5% that is reinvested. To check the answer of $907.01, multiply $907.01 by 1.05
to calculate the amount after one year. Take that amount (which includes the
interest earned), and invest it again for another year at 5% interest (multiply
it by 1.05). The result should be $1000.

Annuities

A series of payments made over time. When payments are made at the end of each period, it is known as an ordinary annuity.  Annuities with payments made at the beginning of the period are called annuities due or annuity in advance.  If the payments are equal and the time periods are also the same, a formula could be used instead of calculating it manually or on a spreadsheet.

Present Value of an Annuity

If you were offered an investment that paid you $1000 at the end of each year for the next 3 years, what would you be willing to pay for it today?  The total amount that the investment pays you is $3000 but since it is not paid to you in full until the 3 years are up, you would only be willing to pay some amount that is less than $3000.  We can calculate that exact amount you’d be willing to pay if we have an interest rate to work with.  Let’s say the annual interest rate is 5%. Let the payments of $1000 be
represented by the letter A.

Using our PV formula above we can calculate the present value of the ordinary annuity as follows: PV_A =  FV / (1 + r) + FV / (1 + r)² + FV / (1 + r)³.  Notice
that we take the first $1000 (FV) and discount it one year, the second $1000 is discounted two years and the third $1000 is discounted for 3 years.  We can simply add these together to get
the present value of these three payments.  The general form of this equation is PV_A = (A / r) + A / [ r (1 + r)ⁿ ].  In the example above: A = 1000, r = 0.05 and n = 3. 

An Annuity Due payments take place at the beginning of the period. The formula is PV_AD =  A (1 + r) /r + r (1 + r) / r (1 + r)ⁿ.

Future Value of an Annuity

The formula is FV_A = A [ ((1 + i)ⁿ – 1) / i ].  An example of this would be an investment that you pay into at regular intervals where the interest earned is compounded (put back into the investment).  The formula for the future value of an Annuity Due is FV_AD = (1 + r) [ (1 + r)ⁿ (A / r) – (A / r) ].

Amortizing a Loan

Suppose you were to borrow $1000 today and arrange to make payments on the loan to retire it after 3 years.  You will make 3 equal payments.  The first payment will be made one year from now. Interest is compounded each year. The interest rate is 6%.  First, the calculation will be shown in a table then the formula will be given. Notice that as the balance of the loan is reduced, the interest portion of the payment decreases and the principal portion of the payment increases.

If P is the amount of your payments, A is the amount of the loan, r is the interest rate and n is the number of periods then the formula to calculate the amount of the equal payment is: P = A r / [ 1 – ( 1 / [1 + r] )ⁿ ]. Suppose you wanted to know what the balance of the loan will be at the end of the second year.  From the table you can see that the answer is $352.93.  We will let t be the time for which we want the balance.  In our example, t = 2. The formula for the balance of a loan at time t is: Bal = A [ [ (1 + r)ⁿ – (1 + r)^t ] / [(1 + r)ⁿ -1 ] ].

Perpetuities

A perpetuity is a series of equal payments over an infinite time period into the future. In this section the steps to derive this formula are shown below.
Because this given cash flow C continues forever, the present value is given by an infinite series:
PV = C / ( 1 + i ) + C / ( 1 + i )2 + C / ( 1 + i )3 + . . .
This can be reduced using algebra to the formula below.
The present value of a perpetuity can be calculated with the simple formula: PV = C / i

Growing Perpetuities

Sometimes the payments in a perpetuity are not constant but rather, increase at a certain growth rate g.
PV yields the expression for the present value of a growing perpetuity:
PV = C / (i – g), and for this to be a valid formula, the growth rate must be less than the interest rate: g < i.