Let’s
begin with interest calculations. Let’s suppose you open a savings account, and
you deposit $5,000 today. We call this $5,000 the principal or present value (PV).
The ** compounding rate** or

The
interest rate per compounding period is usually abbreviated by the letter “** i”.
**Sometimes the interest rate is represented by the “%” symbol. Usually, you
have to enter the interest rate per compounding period. So, for example, if
your annual interest rate is 5% and you have four compounding periods per year
you would enter 1.25% into your calculator.

There
are two ways in which a bank might pay or credit you interest. One way they
could pay you interest is simply to compute it based on your original $5,000
deposit. This is called ** simple interest**. In the case of a 5%
annual rate, you’d earn $250 a year. If you didn’t add to the principal, each
year you’d get $250 added to your account. The following year, the interest
would still be calculated only on the original principal.

The
more typical way banks credit interest is the ** compound interest**
method. In the compound method, interest is credited based on the original
principal

..

Almost
all computations of interest involving financial instruments and accounts
utilize compounding computations. So, it is important to note the difference
between what is called ** nominal **versus

This next table shows the relationship between the nominal and effective interest rate holding the number of compounding periods constant at 12. You can see that the effective interest increases with increasing nominal interest rate.

A very
interesting pattern emerges when a nominal interest rate is held constant and
we greatly increase the number of compounding periods, not just within one year
but over several years. In this next table we assume $1 is put in a bank
account and will earn 5% nominal interest over 20 years. In the last column you
begin to see the accumulated balance get closer and closer to a very important
number known as “** e**”, or Euler’s constant. As the
compounding periods increase the accumulated balance will get closer and closer
to 2.71828 which is

Finally, the last column of this table contains the general compound interest formula for computing the future value of a lump sum. This lump sum we have been calling the present value or initial deposit. The accumulated balance we will refer to as the future value.

Again, the above formula and example assumes that all interest is reinvested or compounded.

Importantly, the above formula allows us to convert one lump sum payment in the future to an equivalent present value and vice versa. By rearranging terms, we arrive at the formula for converting a future value into a present value:

Now
these formulas can be expanded to cover cases of multiple future payments and a
short cut can be applied ** if **the future payments are equal. If we
apply the above PV formula to each payment, we can see how this works:

There is no need to memorize these formulas because even the simplest of calculators and every spreadsheet program has built in functions that will perform the computations. All you need to be careful of is to apply the functions in the right order. You also need to be aware of when to use these formulas…and when not to.

One thing you should be aware of is that these formulas are extremely sensitive to the choice of interest rates so perhaps the most important question to be answered is ‘how do we choose an appropriate interest rate?” Before continuing with the mechanics and application of present value and compound interest formulas, in the next page I will discuss how interest rates are established.

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Copyright 2018 Michael Sack Elmaleh