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?

Compounding
Individuals hope to have more future dollars by deferring use of their current ones. They will do this only if future growth in their funds is possible; in other words, if they can compound their money.
Discounting
The counterpart to compounding, going from future value down to a present one, a process known as discounting.

Even Cash Flows

Payments that are equal (as a loan, mortgage, or lease payment), they are referred to as an annuity.
  • If each payment is made at the beginning of the year, the series is called an annuity due.
  • If each payment is made at the end of the year, the series is referred to as an ordinary 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?

The answer is $110—$100 plus $10 interest.

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.

Now, however, what happens at the end of the second year, presuming this same 10% interest payment?

The answer is now $121 or again, the interest rate of 10% times the new principal of $110 added to the new principal.

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.

Using a financial function or business calculator to determine the amount. Through the inputting of variables for N (the number of years), I/YR (the annual interest rate), and PV (the original principal or present value), we can determine the amount that will have accrued at a set future date, or FV (future value).

Many financial function calculators assume a certain amount of logic when solving time value of money problems.

  • Whenever you invest (or spend) money, your calculator needs to know that it is an “outflow” of funds. You alert your calculator that it is an outflow by entering a payment (or investment) as a negative number.
  • When you receive money, your calculator reports the result as a positive number. In short, enter payments as negative numbers and receipts as positive numbers.

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.

Getting back to the example, $100 (the PV or beginning value) after 20 years (N) at 10% interest per year (I/YR) will grow to an amount of $672.75 (the FV or ending value). This figure is derived by inputting the given numbers for PV, N and I/YR on those specified keys and solving for FV.

If you entered the $100 as a positive number for PV, your calculator gave you -$672.75 as your answer. Common sense tells you the answer must be positive, so what happened? When you entered $100 as the PV, you told the calculator that you’re going to receive $100 now. Since the calculator has a receipt of money, it will assume there must be a payment (or investment) of money as well. Since you gave it a positive number for a receipt, it assumes the answer you are solving for must be a negative number (simply representing an outflow of funds). Try entering -$100 as the PV to see if it results in a positive number (representing a receipt of funds).

Another view that shows the importance of understanding Present Value versus Future Value is this:

Assume that you received a certificate of deposit that was purchased five years ago for $1,000. The certificate says it earned 6.8% per year, and you want to know what it will be worth tomorrow, at the end of the five years. The beginning deposit was $1,000 so that is entered as the PV, or present value, as a negative number. (In this case, it is very important to remember that the PV on the calculator has nothing to do with the present day. It is simply the beginning value). Five is N, since that is the number of years for the certificate, and 6.8 is I/YR. To find out what it will be worth tomorrow, press FV to calculate the future value (also called the ending value). You will find out that your certificate will be worth $1,389.49.


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.

As with the compounding (or future value) process, the present value of a dollar depends on:

  • The length of time before it will be received
  • The annual (or some other time period) interest rate.

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.

For example, a dollar to be received after 20 years is worth considerably less than one to be received after five years when both are discounted at the same rate.Using the $672.75 future value (FV) determined in the previous example, compute the present value of this same amount using an N of 5 rather than 20 at the discount rate (I/YR) of 10%. Your answer should be $417.72. Since we know that the $672.75 amount is worth $100 twenty years back (at a 10% discount rate), we have just proven the accuracy of the previous statement.

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%.

Remember, while a financial function calculator eases the computation of the arithmetic, you must first still determine whether the problem concerns a future or present value computation. It is here that a thorough understanding of the compounding and discounting concept is critical.


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.)

  • The accumulation of funds to meet the financial goal is then referred to as the future sum of an annuity.
  • If each payment is made at the beginning of a year, the series is called an annuity due.
  • If the payments are made at the end of the year, the series is known as an ordinary annuity.

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:

Setting the calculator mode to BEGIN and then inputting -$100 into the PMT key, 10 (for %) in I/YR., 3 for N, and then solving for FV.

In the next case (i.e., the ordinary annuity), the FV will be $331.00. To derive this, use the same keystrokes as before, but make sure your calculator is no longer in the BEGIN mode.

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.

Accordingly, there are two points to be made here:

  • The longer the time period and the higher the interest rate (or rate of return), the greater the sum that is accumulated in the future.
  • The sooner dollars are invested, the greater their accumulated value will be at the end of a specified period of time.

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.

If the annuity is an ordinary annuity, the present value of the future payments could be determined by obtaining the present value of each payment and adding them together.Using the financial function calculator can simplify it.

Consider an example of an individual who wants to withdraw $10,000 annually from her savings account at the end of each year for the next ten years. She can earn 8% on these funds over this time period, so use this amount as the applicable discount rate. What is the present value of that future cash flow? Another way to ask it is: How much does she need in her savings account before she starts the withdrawals? The answer is $67,100.81. Using the calculator, input -$10,000 as PMT, 8 (for %) in I/YR, and 10 for N. Then solve for PV.

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.

As with the present value of a dollar, the present value of an annuity is related to the interest rate and the length of time over which the annuity payments are made. The lower the interest rate and/or the longer the period of the annuity, the greater the present value of the annuity.


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 one-time 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).

Consider the example of funding a college education. If the problem is asking you to compute the cost of this education using some presently known figure but over some time period in the future at a given rate of increase, you are being asked to compute the future value of a lump sum (or a dollar in the above explanation). Conversely, if you want to calculate the amount of life insurance you would need to pay for the future cost of a child's college education if your client were to die tomorrow, you are being asked to compute a present value amount. Alternatively, if you are being asked to save for the estimated cost of this education at some point in the future (for example, beginning saving for a one-year old child to attend college when he or she is age 18), you are being asked to compute an annuity amount.

In working time value of money problems, trust your intuition as to what the question is asking you. You will usually be correct. Thinking about the kind of answer you are looking for will usually tell you what kind of calculation is needed. The important thing is to think the problem through. Once you understand the time value of money concepts, you will be able to look at your answer and know if it makes sense. You will not likely be able to know if it is exactly right, but when you look at your answer, you can tell if it is reasonable considering the facts.


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.

Fortunately, adjusting for these payments that are made other than annually is quite simple, particularly with the use of your financial function calculator. First, you must tell the calculator the number of payments there are each year. For example, if you have 12 payments per year, press: 12 SHIFT P/YR. It's that easy. To check to see the number of payments per year for which the calculator is set, press: SHIFT CLEAR ALL or C ALL and hold down this second key for a moment while you look at the display. It will show the number of payments per year. With this done, when you enter the annual interest rate, the calculator will automatically adjust the interest rate for compounding periods that are less than a year in length.
When you need to enter the number of compounding periods, enter the number of years, press SHIFT and then the x P/YR key (same key as N). This multiplies the number of years times the number of compounding periods per year to get the total number of compounding periods. Let's look at what happens if you are investing $100 at the beginning of each month for 15 years at 8%:

  • BEGIN mode
  • 12 SHIFT P/YR
  • 100 +/- PMT
  • 8 I/YR
  • 15 SHIFT x P/YR
  • Solve for FV

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 5-8) involve the consideration of inflation and an inflation-adjusted 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.

Example: George has an IRA with a current balance of $4,000. How long will it take for this account to grow to $20,000 at a 12% annual rate of return?

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.

Example: The Barrons would like to accumulate $50,000 for a down payment on a new home. If they are able to save $500 at the end of each month and these funds earn 10%, how long will it take to accumulate the $50,000?

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.

Example: Jack and Jane would like to save $10,000 for a down payment on a boat they would like to buy in 3 years. They think they can earn 14% on their savings. How much will they need to save at the end of each year?

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.

Example: Joyce wants to add some money to the college education fund she has begun for her son. She wants to save an additional $40,000 over the next 10 years and believes she can earn 12% on her money. How much does she need to save at the beginning of each year to accumulate the $40,000?

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.

Example: Susan wants to save over the next 20 years by depositing $2,000 at the beginning of each year in an investment account returning her 10% per year. Susan expects an inflation rate of 3% per year and will be increasing her annual contributions by that rate. How much will she accumulate over the 20 years?

BEGIN mode
1 P/YR

2000 PMT
20 N
1.10 / 1.03 - 1 x 100 = I/YR

(this calculation determines the inflation-adjusted rate of return for the annual investments)
PV
(this gives us a lump sum present value of all 20 years of inflation-adjusted 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 inflation-adjusted payments made at the end of a period over successive years.

Example: Susan wants to save over the next 20 years by depositing $2,000 at the end of each year in an investment account returning her 10% per year. Susan expects an inflation rate of 3% per year and will be increasing her annual contributions by that rate. How much will she accumulate over the 20 years?

BEGIN mode
1 P/YR

2000 PMT
20 N
1.10 / 1.03 - 1 x 100 = I/YR

(this calculation determines the inflation-adjusted rate of return for the annual investments)
PV
(this gives us a lump sum present value of all 20 years of inflation-adjusted 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.

Example: Ray wants to establish a separate fund for his daughter’s college education for the next five years. She starts this education later this month. He figures that the $10,000 annual cost today will increase by 6% annually, but he can earn 12% on funds set aside on her behalf. How much money does Ray need to set aside today to fund his daughter’s college education?

BEGIN mode
1 P/YR
10000 +/- PMT
5 N
1.12 ÷ 1.06 - 1 x 100 = I/YR
(this calculation determines the inflation-adjusted 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.

Example: Ray wants to establish a separate fund for his daughter’s college education for the next five years. She starts this education 1 year from now (in other words, Ray doesn't have to make each year's payment until the end of the year). He figures that the $10,000 annual cost today will increase by 6% annually, but he can earn 12% on funds set aside on her behalf. How much money does Ray need to set aside today to fund his daughter’s college education?

BEGIN mode
10000 +/- PMT
5 N
1.12 ÷ 1.06 - 1 x 100 = I/YR
Solve for PV
Divide by 1.12
(which is 1 + the 12% growth rate on his investment; you divide the PV by 1.12 to take into account the end of the year payments; in essence, the entire cost of his daughter's college education will cost less than Example 7, since Ray doesn't have to make each year's payment until the end of the year.)

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.

Example: Bill and Sue want to add $150 at the beginning of each month to their current $7,000 education fund for their 4-year-old daughter, Megan. If they can earn 8% per year, how much will it be worth when she is ready for college at age 18?

BEGIN mode
12 P/YR
7000 +/- PV
150 +/- PMT
14 Shift xP/YR
8 I/YR
Solve for FV

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 tax-deductible. 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.

(a) What is the amount of their monthly payment?

Set to END mode
12 SHIFT P/YR
200000 PV
30 SHIFT xP/YR
7.75 I/YR
Solve for PMT.
Note: The HP calculators display the answer as a negative number, since each month’s payment amount is a cash outflow. Parts (a) through (d) of this example are all part of one continuous series of calculations using the AMORT key starting in part (b). Be sure that you do not clear your calculator between each part.

Answer: $1,432.82

(b) After making their 18th monthly payment, how much will they have repaid on their original mortgage balance?

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

(c) After making their 18th monthly payment, how much total interest will they have paid through that point?

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

(d) What will be their unpaid principal balance on this mortgage after they make the 18th payment?

Without clearing the calculator,
=
Answer: $197,314.75

Maximum Purchase Price Example and Application

George has $20,000 that he has saved for the purchase of a new home. He figures that he can afford a monthly payment of $900. Interest rates are currently 8.5% on a 30-year mortgage note. What is the maximum purchase price of the home that George can currently afford? (Assume he has other funds for closing costs, property taxes and insurance.)

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

Jessica wishes to lease an automobile valued at $30,000 over a three-year period. The lease provides for an option to buy the auto at the end of three years for $18,000. If the dealer needs to yield a return of 14 percent on this lease, what should be the amount of the monthly payment (due on the first of each month) incurred by Jessica?

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