## Quantitative Methods

An introduction to the basic structure and technique of determining the time value of money, consider that you have deposited $1,000 in a savings account at the bank today and that you will leave the money with the bank for one year. You might ask yourself if you would be willing to leave the money with the bank for the year, then at the end of the year, withdraw the funds ($1,000), with no additional compensation. If not, why not? Reflecting on your response may help you to understand the introductory paragraph.

No, you probably expect to receive more than $1,000 from the bank at the end of the year. There are many ways to view the subtleties of this event. One perspective directs you to recognize that the bank is borrowing your money for one year - and because they are “using” your money, you expect to receive “rent” for the use of the funds for the year. Another way of thinking about this transaction is to consider that you are “investing” your funds in a relatively low-risk investment (the savings account) and that you expect to receive a “return” on your investment. In either case, you expect a reward for placing your funds at the disposal of the bank, thus the bank pays you a fee, called “interest”. The interest fee or “rent” is usually quoted as a “rate” or interest rate and refers to the percentage payment that will be paid on the principal for a period of time. Unless stated otherwise, the conventional means of quoting interest rates is to state the interest rate for an annual period or one year. Thus, if our $1,000 deposited at the bank earned interest at 9%, it would be assumed that the bank is paying us $90 for use of the money deposited for one year.

To place the above scenario into a more structured argument and contemporary syntax, if we deposit $1,000 today (the present value) which will earn 9% per year (the compounding rate of interest per period); the funds on deposit in one year (the future value) total $1,090.

This “future value” is calculated as follows

Future value (FV) = Present value (PV) + [Present value (PV) * Interest rate ( r )]orFV = PV + [PV * r]orFV = $1,000 + [$1,000 * .09] = $1,090

Suppose we decided to leave the funds on deposit for two years instead of one year. This could be calculated as follows:

FV = PV + [PV * r] + {[PV + (PV * r)] * r}

= $1,000 + [$1,000*.09] + {[$1,000 + ($1,000 * .09)] * .09}

= $1,000 + [$90] + [$1,090 * .09]

= $1,000 + $90 + $98.10

= $1,188.10

If you are comfortable with algebra, you might notice that the right hand side of the preceding equation can be simplified by factoring.

Begin with:

FV = PV + [PV * r] + {[PV + (PV * r)] * r}

Clear the brackets, which produces:

FV = PV + PVr + PVr + PVr2

Then factor the term PV from the restated equation and collect terms:

FV = PV (1+ r + r + r2)

FV = PV (1+ 2r + r2)

Then reduce the result to its simplest form

FV = PV (1+ r )2

If we substitute the information from our earlier example for $1,000 earning 9% interest compounded annually for two periods, we get the result

FV = PV (1+ r )2

FV = $1,000 (1+ .09 )2

FV = $1,000 (1.09 )2

FV = $1,000 (1.1881)

FV = $1,188.81

This is equivalent to the extended calculations rendered above.

As you might have suspected, the calculation of the future value of a present amount is a geometric series and the formula can be generalized as follows

FV = PV (1 + r)n

where:

FV = Future value of a present amount

PV = Present value of amount

r = Interest rate per period

n = Number of compounding periods

This generalized form serves as the basis of all calculations involving the time value of money. For instance, suppose that we were interested in determining the present value of an amount we wanted or expected to receive in the future. By solving the future value equation for the unknown PV, we can determine the present value.

Consider that you want to buy a car in 4 years and you want to pay cash for the vehicle. You expect the car to cost $10,000. If you could earn 8% per year on a Certificate of Deposit at the local bank, how much would you have to deposit today in order to accumulate the $10,000?

We can solve this problem by manipulating the equation for future value.

We know that:

FV = PV (1 + r)n

$10,000 = PV (1 + .08)4

$10,000 = PV (1.08)4

PV = $10,000/(1.08)4

PV = $10,000/(1.08)4

PV = $10,000/1.3604889

PV = $7,350.30

The result is that if you deposit $7,350.30 today in an interest-bearing investment that earns 8% annually for four years, you would accumulate $10,000 by the end of four years. If you wish to evaluate this, you might consider reviewing the table that reflects the extended form of calculation.

Date Interest Rate Interest Earned Balance

1/1Year 1 Deposit 7,350.30

12/31Year 1 8% 588.02 7,938.32

12/31 Year 2 8% 588.02 7,938.32

12/31 Year 3 8% 635.07 7,938.32

12/31 Year 4 8% 740.74 10,000.00

As may be obvious, we could simply restate the future value equation to solve for the present value, which would produce

FV = PV (1 + r)n

PV = FV /(1 + r)n

The term, (1 + r)n , is usually referred to as a time value of money factor and is the variable that relates the future value to the present value. You are probably familiar with tables of present value factors and future value factors. The equations above reflect the information contained in the tables; that is, the tables represent the calculation of the factor for various combinations of interest rates and time periods. Further, our formulas tell us that the present value factors are directly related to the future value factors. The relationship is evident - the factor for the future value table, (1 + r)n , is the inverse of the factor found in present value tables 1/(1 + r)n

To affirm that this is the case, consider the factors used to solve the previous problem. If we wanted to know the future value of $7,350.30 deposited in an interest bearing investment, earning 8% per year for four years, we would use the future value equation, as follows

FV = PV (1 + r)n

FV = $7,350.30 (1 + .08)4

FV = $7,350.30 (1 + .08)4

FV = $7,350.30 (1.3604889)

FV = $10,000.00

The factor for this future value calculation is (1.3604889). From the present value calculation, the factor was (1/1.3604889), the inverse of the future value factor.

#### Time Value Of Money Calculations