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UMGC Module 2: Financial Securities

Topics

The Time Value of Money

The Time Value of Money

Arguably, the time value of money concept is the most fundamental concept in the study of

finance. It basically states that a dollar received today is worth more than a dollar received

tomorrow. The underlying reason for this statement is that it is assumed that the dollar received

today can be invested and earn some interest before the dollar received tomorrow, whereas the

dollar to be received tomorrow obviously cannot earn interest today. Logically, this means a

dollar to be received in the future must somehow be discounted, or adjusted in value, to be

financially comparable to a dollar received in the present.

Extrapolating on this principal, logically, if we are to compare the expected return of alternative

projects, strategies, and investments over varying periods of time, we must first convert all the

dollars received at the various times in the future back to their present-day value. This process is

called present value.

Alternatively, we could achieve the same desired comparable results by moving all the dollars

received at various times out to some common future date. This process is called future value.

Only when the alternative investment cash flows are compared in dollar values of the same date

is the financial evaluation of alternative returns meaningful.

The Concept of Compound Interest and Future Value

Although most financial analysis is conducted from the perspective of present value, we will

begin our discussion of the time value of money concept with future value because it is easier to

understand. We start by considering the term compound interest. It occurs when the interest paid

on an investment during the first period is added to the principal, and during the second period

interest is earned on both the principal and the prior period interest.

There are three methods of calculating compound interest—longhand, equation, and financial

tables. Using the following scenario, we'll go through each method. Consider $100 invested for

three years earning compounded interest at an annual rate of 6% per year.

1. Longhand—This method can be tedious, especially if you're calculating the compound

interest over 10 or 20 years.

Beginning of

year 1 = $100 = $100.00

End of year 1

(FV1) = $100 + $100(.06) = $106.00

End of year 2

(FV2) =

[$100 + $100(.06)] + [$100 + $100 (.06)] (.06)

or $106.00 + $106 (.06) =

$112.36

$112.36

End of year 3

(FV3) = $112.36 + $112.36 (.06) = $119.10

The above mathematical progression can be generalized into the following formula for

calculating the compound or future value (FV) on any present value (PV) amount given

these two variables, the discount rate per period (i) and the number of periods (n).

2. Compound interest or future value equation—This method is much cleaner and

quicker than the longhand method.

FVn = PV(1 + i)n

where:

FVn = future value at the end of n periods

PV = present value, or the original amount, deposited at the beginning of

period

n = number of periods of compounding = 3

i = interest rate per period = 6%

Please note that in the generalized formula, we specifically used the term periods and not

years because mathematically the general formula derived applies for any given period of

time (years, quarters, weeks, or days). Financially, we typically see compounding

annually, semiannually, or quarterly.

3. Financial tables—You can find these future value (compound sum of $1) tables at the

back of your textbook. They may make compounding interest easier than do the other

two methods.

Also note that the (1 + i)n factor in the tables will work for any amount PV. Simply

multiply the amount by the factor (l + i)n. Given this relationship, financial tables can be

constructed for all reasonable combinations of interest rate per period and number of

periods. We recommend that you look at financial tables for the compound sum of $1 for

various interest rates and periods.

If you check the table for the interest rate of 6% for years 1, 2, and 3, you will find the

following factors:

i = 6%, n = 1 —– = 1.060

i = 6%, n = 2 —– = 1.124

i = 6%, n = 3 —– = 1.191

Multiply these factors by the PV of $100, shown in our above-illustrated example, and you get

the following amounts, which, adjusted for rounding, match the solutions in our compounding

example:

i = 6%, n = 1 —– = 1.060 ($100) = $106.00

i = 6%, n = 2 —– = 1.124 ($100) = $112.40 *difference caused by rounding

i = 6%, n = 3 —– = 1.191 ($100) = $119.10

This example illustrates how the compounding formula developed for future value calculations is

used to construct the standard financial compounding tables found in all finance textbooks. The

compounding formula rewritten to allow easy use of the data in the compound financial tables is:

where FVIFi, n represents the appropriate financial table compounding factor for interest rate i

and periods n.

It is important for you to be able to solve future-value financial problems by financial calculator,

mathematical calculation, or financial tables. We recommend making the investment, in cash and

time, in a financial calculator. It is a significant time saver in this course and may have uses in

your future professional and personal endeavors.

Also, you should note that most time-value-of-money problems, including compound-interest

problems, can be solved by spreadsheet formulas, such as those included in Microsoft Excel.

Because computers are not allowed in the proctored final examinations, however, the spreadsheet

method of solution is not emphasized in this course.

Moving from the Future-Value to the Present-Value Concept

As stated earlier, in practice financial management generally has a greater use for present value

than future value, and typically we discount all future cash flows back to the present for proper

comparative financial analysis and decision making. Determining the present value—that is, the

value in today's dollars of a sum (or stream) of cash flows to be received in the future—is

nothing other than the inverse of compounding. The difference in these techniques comes about

merely from the investor's perspective in time.

Mathematically, the present value (PV) formula can be derived by algebraically rearranging the

compounding formula developed above to solve for the PV:

FVn = PV(1 + i)n

Rearrange the future value formula to algebraically solve for PV, and you obtain:

PV = FVn/(1 + i)n

where again:

FVn = future value at the end of n periods

PV = present value, or original amount, at the beginning of period 1

n = number of periods of discounting

i = interest rate per period

Again, similar to the compound-value formula, PV financial tables can be constructed using this

formula. They can be constructed in two ways, either by using the present value (PV) formula or

by dividing the FVIF table values into 1, because, by definition, the PV is the reciprocal of the

future value (FV). Once these tables are constructed, they can be used to solve PV discounting

problems with the following formula:

PV = FVn (PVIF i,n)

where PVIFi,n represents the appropriate financial table discounting factor for interest rate i, and

periods n. These PV factors are shown in the present value of $1 table in your textbook.

Future-Value/Present-Value Relationship to Variables

It is important that you understand that compounding (future value) and discounting (present

value) are reciprocals of each other. If you know either factor, you can calculate the other by

taking the reciprocal. To illustrate the inverse relationship between compounding and

discounting, the following presentation will show the graphical effect of compounding and

discounting $100.00 for five years at interest rates of 0%, 5%, and 10%.

Future-Value/Present-Value Relationship to Variables

The following presentation will show the graphical effects of first compounding and then

discounting $100.00 over a five-year period at interest rates of 0 percent, 5 percent, and 10

percent.

You should now set up and solve several simple financial problems for both present value and

future value to understand clearly what happens to each of these values as the interest rate (i) and

number of periods (n) change. One way to do so is to use the above graphical problem and solve

two consecutive years (3, 4) for all three rates (0%, 5%, and 10%). Be sure to change only one

variable at a time. Make a table of the results, and you should see clearly the inverse relationship

of present value to future value, as shown in the graphical presentation above.

Annuities (Future Value and Present Value)

With the present/future value concepts understood, we can now discuss another common

financial application that is used in bonds, interest payments, pensions, and so forth—the

annuity. An annuity is the special case cash flow where a series of equal dollar payments is made

for a specified number of periods. Typically, financial management is interested in determining

either the future value or present value of an annuity.

The Future Value of an Annuity

It is important to remember that the period payment amount in an annuity must be a constant for

all periods. The development of the general annuity formula for the future-value case (a

compound annuity) is based on the constant-payment amount (PMT) and the previously

developed formula for calculating future value (1 + i)t. This is illustrated below for the future

value of a three-year annuity of $500 for three years at 6%:

FV3 = (PMT)(1 + i)3 + (PMT)(1 + i)2 + (PMT)(1 + i)1 + (PMT)

Obviously, this formula can be reduced to a general formula by factoring out the period payment,

PMT, and summing the discounted year payments from t = 1 to n – 1 periods, as shown below:

The future value formula for an annuity.

where:

FVn = future value of the annuity

(PMT) = constant payment deposited or received each period

i = interest rate per period

n = number of periods

The future value of an annuity can also be expressed in the alternative financial table format:

FVn = PMT (FVIFAi, n)

where:

The Future value of an Annuity Factor

The Present Value of an Annuity

The development of a general formula for the present value of an annuity is the reciprocal of the

future value of an annuity and follows the same logical progression to arrive at:

where:

PV = present value of the future annuity

(PMT) = constant payment deposited or received each period

i = interest rate per period

n = number of periods

Or expressed in financial table format:

PV = PMT (FVIFA i, n) The present value of an annuity factor.

Present Value of an Uneven or Complex Cash Flow

In the real world, the majority of cash flows that must be financially analyzed are typically not

single cash flows or equal-payment annuities, but rather cash streams with varying amounts per

period. This type of cash flow is called complex cash flow and requires a special approach to

determine its present value. The present value of a complex cash flow stream can always be

found by discounting the cash flows for each individual year by multiplying the present value

interest factor for that year (PVIFi, n) times the cash flow amount. The advantages of this

approach are that it is simple, accurate, always works, and does not require memorization of any

complex formulas.

For example, let's compute the PV of the following complex cash stream, assuming a discount

rate of 6%.

Year $ Amount Factor (6%)* Discounted Amount 0.00 1.000 0.00

1 500.00 0.943 471.70

2 200.00 0.890 178.00

3 (400.00) 0.840 (335.84)

4 500.00 0.792 396.04

5 300.00 0.747 224.17

Total 1,100.00

Net present value $993.60

*From financial tables

Note that this answer developed using financial table values differs slightly from the

mathematical formula approach:

Manual method (above) ———- $933.50

Financial calculator ————— $934.07

This is not an unusual case, and slight differences between methods occur frequently because of

the number of significant digits used in the various methods of calculation. Both answers are

acceptable in this course.

Although this is a laborious method if calculated by use of either the financial tables or a regular

calculator, it becomes a relatively simple and quick method if one uses the cash-flow feature of

the financial calculator. We recommend taking the time to learn this method on your financial

calculator because you will encounter complex cash flows several times in this course.