Reading this chapter will enable you to:
3–7 Calculate the present value for an inflation-adjusted payment.
Serial Payments
his chapter discusses the concept of a serial payment. Unlike an annuity, which provides a series of regular equal payments, a serial payment provides a series of regular payments that increase periodically with inflation. Many clients depend upon a fixed income stream at certain points throughout their lifetimes, for example, at retirement. However, in an inflationary environment, a fixed-income annuity will not allow a retired client to maintain a constant standard of living. A more acceptable means of providing an income stream over a period of time is to have the stream of income increase annually as inflation increases. This chapter describes how to determine the present value of a serial payment and how to determine the serial payment needed to attain a goal. The former (PV of a serial payment) is used again in planning insurance needs and retirement needs; the latter (serial payment for a future sum) is used again for planning retirement needs.
Present Value of a Serial Payment
The present value of a serial payment is computed by compounding the periodic payment at the inflation rate and then discounting the payment for the return on investments. Assume a client wants to receive an equivalent of $10,000 in today’s dollars at the end of each year for the next four years (i.e., the desired annual payment amount, in the future, will buy what $10,000 will purchase today). He assumes inflation will average 5% over the long run and that he can earn an 8% compound annual return on investments. He wants to invest a lump sum today to fund this need, and he wants to dissipate the fund entirely at the beginning of the fourth year, when the last payment is received.
For the client to receive the equivalent of $10,000 in today’s dollars at the end of each year, the payment first must be adjusted annually for the inflation rate. The client needs $10,500 at the end of the first year, $11,025 at the end of the second year, $11,576.25 at the end of the third year, and $12,155.06 at the end of the fourth year to maintain a constant standard of living.
The client wants to invest a lump sum today to fund these distributions, and he wants to dissipate the principal entirely at the end of the fourth year. The receipts (that is, $10,500, $11,025, $11,576.25, and $12,155.06) must be discounted back to present value to reflect the client’s ability to earn an 8% return on investments. The exhibit below shows the amounts received at the end of each of the next four years, discounted annually for the return on investments.
Exhibit 1 PV 0 1 2 3 $ 9,722 9,452 9,190 $10,500 $11,025 $11,576.25 8,934 $ 37,298 4 $12,155.06
The client needs to deposit $37,298 today to attain his goal of receiving the equivalent of $10,000 in today’s dollars at the end of each of the next four years.
Inflation-adjusted interest rate. The calculation of the unknown present value includes three known variables: the initial payment or receipt in today’s dollars ($10,000), the annual inflation-adjusted interest rate (used in the previous calculations, but not as it is illustrated below), and the number of periods. (For the HP 10BII+ calculator, the number of compounding periods per year also must be entered.) The interest rate used for this calculation is not simply the difference between the inflation rate and the rate of return. The appropriate method to calculate the interest rate when simultaneously compounding a payment is based on discounting an inflation rate based on a return. If the inflation-adjusted interest rate is not used as the interest rate to calculate the amount needed to fund serial payments, the result will be inaccurate; with an inaccuracy magnified as the difference between the inflation rate and the return increases or the number of payments increases. Following is the equation for the inflation-adjusted interest rate. 100 1 inflation of Rate 1 return of Rate 1 × − + +
Therefore, for the previous example, the inflation-adjusted interest rate is 2.85714%.
(
1.0285714 1)
100 2.85714%or2.8571%(
rounded)
1.0285714 1.05 1.08 = × − =In the following keystroke sequence, the calculation of the inflation-adjusted interest rate is bracketed. (As with other time value of money problems, you should clear your calculator prior to performing this calculation.)
Calculator: HP 10BII+1 HP 12C2 Keystrokes: (Set for 1 P/YR)
1.08, ÷, 1.05, –,1 ×, 100, =, 10000, 4, 1.08, ENTER, 1.05, ÷ 1, – 100, ×, 10000, 4,
1 With the 10BII+, calculation of the inflation-adjusted interest rate may be simplified by entering 1 +
the inflation rate, then pressing the INPUT key, entering 1 + the rate of return, pressing the [SHIFT] key, and then pressing the % CHG key. For this example, the keystrokes for this problem would be 1.05, INPUT, 1.08, SHIFT, % CHG, I/YR.
2 With the HP 12C, the calculation of the inflation-adjusted interest rate may be simplified by entering
1 + the inflation rate, then pressing the ENTER key, entering 1 + the rate of return, and then
pressing the Δ % key.
For this example, the keystrokes for this problem would be 1.05, ENTER, 1.08, Δ %, i.
The display will show the answer—in this case, $37,298.32.
This answer is identical to the answer arrived at using the long-hand calculation method in Exhibit 1. Here however, the calculated inflation-adjusted interest rate is used just as it is illustrated in the preceding calculation.
The present value serial payment calculation may be used in a variety of
applications. As presented in this course, primary applications are those where the desire is to make one deposit that will grow, when invested, to completely fund a future need. Core examples of such needs include life insurance needs and college funding (illustration follows). (The Retirement Planning course uses the same process, but focuses on making payments to fund the future need. This
adds/modifies a few steps in the process presented here, and is discussed after the College Funding Example.)
College Funding Example
One application of the present value of a serial payment is to determine funding for a child’s college education. The calculation process involves three steps: 1. Determine how many years in the future the first tuition payment will be
needed, then calculate the future value of one year’s (current) tuition using only the rate of inflation.
2. Calculate the present value (annuity due) of a serial payment using
the inflated tuition amount from step one as PMT.
the inflation-adjusted rate of return as I/YR using the following formula:
100 1 inflation of Rate 1 return of Rate 1 × − + + and
the number of years the child will attend college as N.
3. Discount the amount from the second step back to “today” using only the rate of return.
The following example shows how to apply each of the three steps. Your answer may vary slightly, as a result of rounding. For this example, assume that one year’s college tuition is $10,000 today; education inflation is 6%; and the rate of return is 8%. Further assume that Mary is three years old, and will begin a four- year college program at age 18. Calculate the amount required to provide higher- education funds for Mary.
CHILD: Mary
Annual college costs $ 10,000
STEP 1
Serial Payment Adjustment Inflation Calculation:
Number of periods until student begins college 15
% inflation 6%
$ 23,966 STEP 2
Serial Payment Calculation:
Number of years child will attend college 4
% inflation 6%
% after-tax return 8%
Calculate the present value of the annuity due $ 93,234 STEP 3
Discount Calculation:
Number of periods until student begins college 15
% after-tax return 8%
Calculate the present value of the above PVAD $ 29,391 needed income when serial payments
begin
Note: Present value of serial payment calculations are used in the life insurance needs determination process. You can find additional practice questions in Module 6, The Life Insurance Selection Process.
Calculating for payments in Step 3 (rather than a lump sum)
NOTE: YOU WILL NOT BE REQUIRED TO CALCULATE PAYMENTS FOR STEP 3 ON THE COURSE EXAM. THE INFORMATION IS
INCLUDED HERE AS GENERAL INFORMATION ONLY (the process is used and tested in the Retirement Planning course).
When using the serial payment process to determine a series of payments to fund the future need, rather than a lump sum, the calculation steps are a little different. As previously discussed, payments come in two types: level (periodic) and serial (increasing). When determining both types of payment, the first two steps in the
serial payment process (as described above) remain the same. Step 3 gets modified.
If the goal is to use a level or periodic payment to fund the future need (rather than the lump sum identified previously in Step 3), simply solve for PMT instead of PV. So, using the ending value in Step 2 (College Funding example above), the keystrokes to determine a level payment for Step 3 are (in END mode; ordinary annuity):
15 N, 8 I/YR, (0 PV), 93,234 FV, PMT = $3,433
Note that the only difference between calculating a lump sum or a payment for Step 3 is pressing the PMT key rather than the PV key.
When the goal is to make serial payments to fund the future need, the process is more complex, requiring several additional steps. We will not go into detail on this process, but a general overview is worthwhile.
Once you have determined the Step 2 amount (e.g., $93,234 in the example above), you must bring that amount back to today’s dollars. This means you deflate the future sum. To do this, instead of using the investment rate for I/YR (e.g., 8%), you use the inflation rate (e.g., 6%). Expanding the example above, the keystrokes to do this are:
15 N, 6 I/YR, 0 PMT, 93,234 FV, PV = $38,903.29 Now that you have a new starting point (e.g., 38,903), you can proceed to calculate the serial payment needed to fund the future need. To do this, you need to learn another serial payment calculation: serial payment for a future sum.
Reading the next part of this chapter will enable you to:
3–8 Calculate the inflation-adjusted payment for a future sum.
Serial Payment for a Future Sum
NOTE: THE SERIAL PAYMENT FOR A FUTURE SUM CALCULATION MAY BE TESTED ON THE COURSE EXAM. AS SUCH, YOU NEED TO KNOW THE INFORMATION FOR THIS COURSE (as well as for the Retirement Planning course).
This chapter describes how to determine the periodic savings needed to attain a future goal when an increasing payment is chosen. A serial payment may be calculated to determine the annual savings needed to attain a financial goal. For example, to fund a future retirement income objective, the financial planner might recommend that the client save a certain amount in today’s dollars each year. The savings dollar amount would increase annually with the rate of inflation to maintain a constant value to finance a constant standard of living. Assume a client wants to retire in five years. In terms of today’s dollars, he will need an additional $100,000 in five years to have sufficient funds to finance his retirement. He assumes inflation will average 4% over the long run and that he can earn a 7% annual return on investments. He wants to determine a series of payments that will add up to $121,665 in five years. (The future value of $100,000 inflated by 4% annually for five years is $121,665.29.)
Note: In order to perform the required calculation on a financial calculator (e.g., HP 10BII+), the $100,000 future need—as stated in today’s dollars—must be entered as a future value. Why? The calculator is programmed to inflate the $100,000 at the same time it is calculating the serial payment. This is why you do not enter the $100,000 as a present value. It may not make logical sense (i.e., “Shouldn’t the $100,000 stated in today’s dollars be entered as a present
value?”), but this is the way the calculator is programmed, so this is the required entry.
In performing this calculation on a financial calculator, the steps are identical to those for calculating the payment for an ordinary annuity, except that the inflation-adjusted interest rate is used. (Clear your calculator and set it to calculate for an ordinary annuity.)
The keystrokes are as follows:
Calculator: HP 10BII+ HP 12C Keystrokes: (Set for 1 P/YR)
1.07, ÷, 1.04, –,1 ×, 100, =, 100000, 5, 1.07, ENTER, 1.04, ÷ 1, – 100, ×, 100000, 5,
The answer to the above calculation, $18,878.96, must be adjusted annually for inflation because it represents the current value of the payment. In other words, while this calculation was made as if the first payment would be made “today,” the first payment will actually be made in the future. This means that inflation must be added to the answer in order to arrive at the correct future payment amount. This is true even though inflation was factored into the initial
calculation. Again, the reason to add inflation is that the calculated payment will not be made immediately, but at some point in the future. For these calculations, the future payment date is assumed to be one year from “today,” so a year’s worth of inflation must be added to the initial calculation in order to arrive at the correct answer.
As shown in the following exhibit, the first payment will not be made until one year from now, at point A. Thus, the deposit needed at the end of the first year is $19,634.11 ($18,878.96 × 1.04); at the end of the second year, $20,419.48; at the end of the third year, $21,236.26; at the end of the fourth year, $22,085.71; and at the end of the fifth year, $22,969.14. This exhibit illustrates the timing of the annual deposits and the accumulated amount at the end of the five years using 7% discounting. The goal of $100,000 in today’s dollars ($121,665.29) will be reached with this stream of payments, representing an adjustment for inflation to our future sum.
Exhibit 2 FV 2 1 3 4 $25,736.31 25,014.74 24,313.39 $121,665.29 $20,419.48 $19,634.11 23,631.71 22,969.14 0 5 $22,085.71 $21,236.26
Reading the next part of this chapter will enable you to:
3–9 Determine the general result when one parameter in a time value of money calculation is changed.
Calculations Involving Single Sums
Combined with Annuities
Many calculations involve both single sums and annuities. For instance, an investor may make an initial deposit into a mutual fund of $2,000 [PV] and subsequently invest $250 at the end of each year for 15 years [PMT]. In this case, the initial deposit of $2,000 is treated as a single sum, whereas the $250 annual
payment stream is treated as an annuity. Calculator instructions for several types of combined single sum/annuity problems are provided below.
Future value calculation problem. An investor makes an initial deposit of $20,000 into a mutual fund. Each subsequent year he deposits an additional $2,500 into the fund. What will be the value of the account in eight years if the fund earns 9% annually?
As pointed out previously, the initial deposit is treated as a single sum, while the $2,500 annual payment stream is treated as an annuity, with the first payment being made one year from today.
The time line below illustrates the pertinent information presented in the
problem. (Cash inflows for this time line and the time lines that follow are shown above the line, whereas cash outflows are shown below the line.)
$2,500 $2,500 $2,500 $2,500 1N = 1 Year 0 1 2 3 4 5 6 $2,500 $2,500 ? $2,500 7 $2,500 8 $20,000
A time-consuming method of dealing with this problem would be first to
calculate the future value of $20,000 in eight years at 9% annual earnings, then to calculate the future value of the annuity stream in eight years at 9% annual earnings, and finally to add the two future values together. However, the financial calculator is capable of accommodating both the initial deposit and the subsequent stream of payments in one calculation. Keystrokes are shown as follows:
(Clear your calculator and set for an ordinary annuity.)
Calculator: HP 10BII+ HP 12C Keystrokes: (Set for 1 P/YR)
20000, , 2500, , 9, 8, 20000, , 2500, , 9, 8,
The future value in this case is $67,422.44.
Present value calculation problem. A client would like to accumulate $300,000 for retirement, which will begin in 10 years. She can invest $10,000 at the end of each year toward this goal in an account earning 8% annually. What initial lump- sum deposit, in addition to the payment stream, is required for her to be able to meet this goal?
This problem is illustrated below on the time line.
$10,000 $10,000 $10,000 $10,000 1N = 1 Year 0 1 2 3 4 5 6 $10,000 $10,000 $300,000 $10,000 7 ? $10,000 9 8 $10,000 10 $10,000
Keystrokes are as follows:
(Clear your calculator and set for an ordinary annuity.)
Calculator: HP 10BII+ HP 12C Keystrokes: (Set for 1 P/YR)
300000, 10000, , 8, 10, 300000, 10000, , 8, 10,
The initial deposit required to meet the client’s goal is $71,857.23.
Compounding periods calculation problem. A client wants to save $125,000 to achieve a future goal. He has $26,000 to invest currently and can invest $10,000 at the end of each year toward his goal. If the investment vehicle selected earns 10% annually, how many years will it take to achieve his goal?
This problem is illustrated on the time line below. $10,000 $10,000 1N = 1 Year 0 1 2 3 $10,000 $125,000 ? $26,000
Keystrokes are as follows:
(Clear your calculator and set for an ordinary annuity.)
Calculator: HP 10BII+ HP 12C Keystrokes: (Set for 1 P/YR)
125000, 26000, , 10000, , 10, 125000, 26000, , 10000, , 10,
The answer is 6.08 years. (On the HP 12C, this response is rounded up
to 7.) The correct answer on a test question would be seven years, as at the end of six years he has not met his goal. He would only have $123,216.69, $1,783.31 short of his goal.
Periodic payment problem. A client wishes to accumulate $90,000 for a future goal in seven years. She can deposit $32,000 today in an account earning 11% annual interest and plans to make an additional payment into the account at the end of each year. What periodic payment will be required at the end of each year to meet her goal?
? ? ? ? 1N = 1 Year 0 1 2 3 4 5 6 ? ? ? 7 $32,000 $90,000
Keystrokes are as follows:
(Clear your calculator and set for an ordinary annuity.)
Calculator: HP 10BII+ HP 12C Keystrokes: (Set for 1 P/YR)
90000, 32000, , 7, 11, 90000, 32000, , 7, 11,
The periodic payment required each year is $2,408.49.
Rate of return problem. Six years ago a client invested $5,000 in a mutual fund. He made additional investments of $300 at the end of each year. Yesterday the client redeemed all fund shares and received $8,500. What was the rate of return on this investment?
This problem is illustrated on the time line below.
$300 $300 $300 $300 1N = 1 Year 0 1 2 3 4 5 6 $300 $300 $5,000 $8,500
Keystrokes are as follows:
(Clear your calculator and set for an ordinary annuity.)
Calculator: HP 10BII+ HP 12C Keystrokes: (Set for 1 P/YR)
5000, , 300, , 8500, 6, 5000, , 300, , 8500, 6,
The rate of return has been 4.44% annually.
When the problem involves compounding and payments that happen more frequently than annually, the adjustments discussed in Chapters 1 and 2 must be made. (For the HP 12C, the periodic interest rate and the number of