Time Value of Money
An Introduction to the Time Value of Money 
Money has time value, a dollar today is worth more than a dollar to be received in the future. In an inflationary environment (which has been the state of the economy certainly since the end of World War II in 1945), the “real” value of the dollar declines over time. This, combined with other factors, such as the imposition of taxes, means that the value of that dollar today is as great as it will ever be. Conversely, since a future dollar is worth less, why would an investor ever trade the present dollar for one to be received in the future?

Even Cash Flows 
Payments that are
equal (as a loan, mortgage, or lease payment), they are
referred to as an annuity.
The time at which the payment is made has a considerable effect on the total sum that is accumulated. 
The Future Value of a Dollar 
If $100 is deposited into a savings account at the beginning of a year, and this account pays an interest rate of 10% annually, how much will the investor have in the account at the end of that same year?
This simple calculation may be expressed arithmetically, but suffice
it to say that the 10% interest rate is taken times the principal of
$100 and added to the original principal with the total as the correct
answer.
The $1 over and above the $10 on the original principal of $100 illustrates
the principle of the compounding of one’s money. By continuing
with these calculations, it is possible to determine the amount that
will be in the account at the end of a considerable number of years,
but it will obviously take some time and work.
If you are investing money now (as a lump sum or as a series of payments
or deposits), enter a negative number. If you will receive money in
the future, enter a positive number.
Another view that shows the importance of understanding Present Value versus Future Value is this:

The Present Value of a Dollar 
The preceding considered the question of how much one dollar is worth,
compounded over time. Now consider the reverse. How much is a dollar
that will be received in the future worth today? It should be intuitive
that, in the first preceding example, the $672.75 to be received 20
years from now using a 10% annual reduction factor equals $100 today.
This annual reduction factor is referred to by the term discounting
in the time value of money process. Discounting determines the
worth of money that is to be received in the future in terms of its
current or present value.
The further into the future the dollar will be received, and/or the higher the interest (or discount) rate, the lower the present value of the dollar.
Now, using your calculator, what is $10,000 to be received in 12 years
worth today discounted at an annual interest rate of 8%? Your answer
should be a present value of $3,971.14. In layman’s terms, you
are saying that receiving $10,000 12 years from now would be like having
$3,971.14 today, at an interest rate of 8%. 
The Future Sum of an Annuity 
Of course, as an individual identifies a financial goal, many times the implementation of a systematic savings program is necessary to meet this goal. Examples of financial goals that typically require the investment of monthly or annual payments are the funding of a college education for a child or saving for one’s own retirement. If these payments are equal, the series of payments is called an annuity. (Note: You should distinguish this term for time value of money purposes from that of commercially sold insurance products that also use this name.)
Example: Remember our individual who has deposited in a savings account that pays 10% interest per year. Let us vary these facts slightly and say that he not only deposits this $100 in one year but the same amount for the next two years as well. If he deposits these amounts at the beginning of each year, how much will he have at the end of the three years? How much is this amount if he deposits these amounts at the end of each year? In the first case, the answer is $364.10. Using the calculator, this answer is derived by:
The difference between the ending values of the two types of annuity
payments will be quite substantial as the number of years increases
or the interest rate rises. Consider a savings account where the individual
deposits $1,800 annually for 25 years at 8% interest. If the deposits
are made at the end of the year (an ordinary annuity), the ending amount
is $131,590.69. If these deposits are made at the beginning of the year
(an annuity due), the ending amount is $142,117.95, a difference of
$10,527.26. Now, use these same savings amounts over the same period
of time, but increase the interest rate to 12%. The future sum of the
ordinary annuity is now $240,000.97, while the annuity due sum is $268,801.08,
a difference of $28,800.11.

The Present Value of an Annuity 
Just as with the future value and present
value of a dollar analysis, it is often necessary to reverse
the future sum of an annuity calculation to that of the present value
of the annuity. Oftentimes, a potential investor is not so
much interested in a future value of his or her periodic payments but
its present value. This is particularly the case where the investor
wants to compare several investment alternatives, such as whether to
buy one bond over another.
Compare this amount to payments to be withdrawn at the beginning of
the year, i.e., the present value of an annuity due. The keystrokes
on the calculator are exactly the same, but you must be in the BEGIN
mode. The answer will be $72,468.88. Again, notice the difference in
amount that time makes. By withdrawing these payments at the beginning
of the year rather than at the end, our individual will need an additional
$5,368.07 in his or her savings account. 
The Applications of Concepts 
All time value of money problems
use the concepts of compounding and discounting, although there
are variations on the theme. In answering time value problems, you must
first determine whether the problem involves a lump sum or onetime
payment, or a series of payments (i.e., an annuity). After
that, you need to ascertain if the problem concerns going from
the present to the future (i.e., a future value computation) or from
the future back down to the present (a present value computation). 
Compounding and Payments Made Other than Annually 
All of the previous examples used interest
rates or series of payments that were made only once per year. Of course,
compounding can and does often occur more than once per year. For example,
interest on a savings account may be paid monthly or at a frequency
of 12 times per year.
The answer is $34,834.51. When you entered 15 SHIFT x P/YR, the display showed 180, or 15 x 12, which is the total number of compounding periods. When you use this function of the calculator, you realize very soon how important it is to clear the calculator between calculations. Clearing the calculator is the way you find out for how many payments per year the calculator is set. 
Variations on the Theme 
As noted, once the basics of present and future value concepts have been mastered, it is then possible to solve calculations that employ additional variations. This section of the reading presents several of these variations that you will be expected to understand in order to successfully complete the CFP Board exam. The last of these variations (Examples 58) involve the consideration of inflation and an inflationadjusted interest rate calculation. This will be particularly important when computing life insurance needs and performing a retirement savings analysis. 
Number of Years for a Present Value to Grow to a Future Value 

The number of years it takes for a given sum to increase to another specified sum in the future. 


4000 +/ PV 20000 FV 12 I/YR Solve for N 
Answer: 14.2 years 
Number of Years for Payments to Grow to a Future Value 

The number of years for payments to accumulate to a future value. 


Set to END mode 12 Shift P/YR 50000 FV 500 +/ PMT 10 I/YR Solve for N to obtain the number of months required. (However, then divide this number by 12 to convert back from monthly payments to equivalent years.) 
Answer: 6.09 years 
Annuity Payment for a Future Sum, Ordinary Annuity 

The payment needed at the end of a period to accumulate to a future value. 


END mode 1 Shift P/YR 10000 FV 14 I/YR 3 N Solve for PMT 
Answer: $2,907.31 
Annuity Payment for a Future Sum, Annuity Due 

The payment needed at the beginning of a period to accumulate to a future value. 


BEGIN mode 1 Shift P/YR 40000 FV 12 I/YR 10 N Solve for PMT 
Answer: $2035.15 
Future Value of an Increasing Annuity, Annuity Due 

A future sum that reflects a series of payments that increase at a specified rate and earn interest at a specified rate. It results from a series of payments that are made at the beginning of a period over successive years. Note that payments in subsequent years include an inflation factor so as to keep the purchasing power of the sum the same. 


BEGIN mode 1 P/YR 2000 PMT 20 N 1.10 / 1.03  1 x 100 = I/YR (this calculation determines the inflationadjusted rate of return for the annual investments) PV (this gives us a lump sum present value of all 20 years of inflationadjusted payments in today’s dollars; now we are ready to take this PV and simply compound it for 20 years at 10% to its future value) 10 I/YR (by the way, we do not need to reenter 20 for N, since the calculator still has that value stored) 0 PMT (this clears out the previously stored value of 2000) Solve for FV 
Answer: $154,672.22 
Future Value of an Increasing Annuity, Ordinary Annuity 

A future sum that results from a series of inflationadjusted payments made at the end of a period over successive years. 


BEGIN mode 1 P/YR 2000 PMT 20 N 1.10 / 1.03  1 x 100 = I/YR (this calculation determines the inflationadjusted rate of return for the annual investments) PV (this gives us a lump sum present value of all 20 years of inflationadjusted payments in today’s dollars; now we are ready to take this PV and simply compound it for 20 years at 10% to its future value) 10 I/YR (by the way, we do not need to reenter 20 for N, since the calculator still has that value stored) 0 PMT (this clears out the previously stored value of 2000) Solve for FV Divide by 1.10 (which is 1 + the 10% growth rate on her investment; you divide the FV by 1.10 to take into account her end of the year deposits; in essence, she has lost 1 year of potential return by waiting until the end of the year to make her deposits.) 
Answer: $140,611.11 
Present Value of an Increasing Annuity, Annuity Due 

The value today of a series of payments that are to be received in the future at the beginning of each period and increases by the rate of inflation in future years. 


BEGIN mode 1 P/YR 10000 +/ PMT 5 N 1.12 ÷ 1.06  1 x 100 = I/YR (this calculation determines the inflationadjusted rate of return for the annual investments) Solve for PV 
Answer: $44,922.24 
Present Value of an Increasing Annuity, Ordinary Annuity 

The value today of a series of payments that are to be received in the future, but at the end of each period. An inflation factor is also taken into account. 


BEGIN mode 
Answer: $40,109.14 
Future Value of a Lump Sum and Annuity Due Payment 

Future value of a series of payments added to an existing lump sum. 


BEGIN mode 
Answer: $67,885.80 
Loan, Mortgage, and Lease Payment Calculations 
Now that you have mastered
the concepts of present and future valuation and the variations on this
theme, this introduction to the time value of money will conclude with
some extremely practical applications. Specifically, we will compute
payments that are to be made when entering into a term loan (such as
when purchasing a new automobile), a mortgage loan, and when entering
into a consumer lease. While it is not covered here, you should also
be aware of and understand the term amortization involved in paying
off a loan (typically a mortgage). Amortization is the process of dividing
a payment into amounts for principal and interest. This is typically
important for tax reasons, as most interest payments on a mortgage incurred
to purchase a primary residence are taxdeductible. Amortization tables
are readily available from mortgage brokers and other lenders involved
in the financing of a home.

Loan Example and Application 

John and Susan have recently financed the purchase of a new home with a $200,000 mortgage note at 7.75 percent annual interest over 30 years. 


Set to END mode 
Answer: $1,432.82 

Without clearing the calculator, 1 INPUT 18 SHIFT AMORT = 
Answer: $2,685.25 

Without clearing the calculator, = 
Answer: $23,105.60 

Without clearing the calculator, = 
Answer: $197,314.75 
Maximum Purchase Price Example and Application 


END mode 12 P/YR 30 Shift xP/YR 8.5 I/YR 900 +/ PMT Solve for PV 
Answer: $137,048.28, consisting of a $117,048.28 loan plus the $20,000 down payment. 
Lease Payment Example and Application 


BEGIN mode 12 P/YR 14 I/YR 30000 +/ PV 18000 FV 3 Shift x P/YR Solve for PMT 
Answer: $612.98 