Principles of Accounting 2
Present Value of Future Cash Flows
Introduction
The purpose of this handout is to teach you how to calculate the value today of either (a) a single future cash flow or (b) a series of future cash flows (each of the same amount, at regular intervals). We call this the present value. This is an important concept and technique. It has many applications in business. The applications that we will see in this course are (a) bond pricing, in Chapter 14, and (b) net present value, which relates to capital budgeting and decision-making, in Chapter 25.
We will begin by defining some key terms, and explaining some underlying concepts and tools. We will then demonstrate a straightforward technique to perform each of the key calculations, which relate to two situations:
1. You will receive or pay a single amount on a specific day in the future. You wish to know its value today. [“Present value of a single sum”]
2. You will receive or pay the same amount on more than one specific day(s) in the future. These cash flows will occur at regular intervals. You wish to know their aggregate value today. [“Present value of an annuity”]
Note that in each case, whether we are dealing with a single cash flow or a series of cash flows, the solution we seek is a single amount that expresses the value of the cash flow(s).
Basic Present Value Concepts
A dollar received today is worth more than a dollar to be received at some future time. This is because today’s dollar can be invested, and will therefore produce returns, and increase to some greater amount in the future.
We call the returns on investment interest. Interest is payment for the use of money. It is the excess cash received or paid over the amount loaned or borrowed.
Ex.: We loan or borrow $1,000. Later we receive or repay $1,150. The $150 “excess return” is interest.
The amount received or paid is generally stated as percentage “rate of interest.”
Ex.: 150/1,000 = 15%
This is generally expressed in terms of annual interest rate. IMPORTANT: When you see an interest rate, assume it refers to an annual rate unless otherwise explicitly stated.
Calculating Interest
Inputs needed to compute the amount of interest
Principal—amount borrowed or invested
Interest rate—expressed as a percentage rate
Time—the amount of time that the principal will be outstanding (in terms of periods)
Simple interest
Simple interest is calculated on the amount of principal only. It is the return on (growth of) the principal for one time period. The formula is:
Interest = principal x interest rate x number of periods
Ex.: $1000 @ 12% for 2 years
= $1000 x 12% x 2 = $240
Compound interest
Compound interest is calculated on principal and on any interest that has not been withdrawn or paid. This represents the return on (growth of) the principal for two or more periods.
Ex.:
Simple interest calculation / Simple interest amount / Accumulated year-end balance / Compound interest calculation / Compound interest amount / Accumulated year-end balance1,000 x 10% / 100 / 1,100 / 1,000 x 10% / 100 / 1,100
1,000 x 10% / 100 / 1,200 / 1,100 x 10% / 110 / 1,210
1,000 x 10% / 100 / 1,300 / 1,210 x 10% / 121 / 1,331
Total / 300 / 331
· Virtually all interest computations in business use compound interest.
· The returns are greater using compound interest.
Also note that calculating compound interest is more difficult than calculating simple interest, because you must proceed step by step. The techniques you will learn here will simplify this calculation by avoiding the need to calculate each period. They provide a one-step solution. You can see that in a situation with many years, this can save a great deal of work.
Compound interest tables
We use tables to help us calculate compound interest. They are provided near the end of your textbook after page B-11.
- Table B.1—Present Value of 1—the amount which must be deposited now to equal $1 in the future (Question: Will the amounts in the table be greater than or less than 1?)
- Table B.3—Present Value of an Annuity of 1—the amount that must be deposited now to be able to withdraw $1 at the end of each period
The tables each have the same format:
· Each row represents number of periods
· Each column represents a rate of interest
Use of term “periods” vs. “years”
The interest rate is usually expressed in years, but compounding often occurs more frequently. To address this issue, we will always work with compounding periods, rather than years. The period is simply the amount of time that elapses before compounding occurs. To solve present value of money problems, we must be able to convert annual interest rates to period interest rates, and the number of years to the number of periods.
· We convert the annual rate to the period rate by dividing the annual rate by the number of compounding periods each year:
Ex.: 10% annual interest, compounded twice each year = 5% per period
· We convert the number of years to the number of periods by multiplying the number of years by the number of compounding periods each year
Ex.: 4 years, compounded twice each year = 8 periods
Reading the tables
- Table B.1—Present Value 1—The numbers you will look up in this table are called “present value factors.” [Note that they are always less than one, because the amount will grow to $1 using the interest rate and number of periods selected.]
· We will represent the present value factor for a n periods at i interest rate as
PVFn,i . The present value factor for 4 periods at 7% interest per period is therefore represented as PVF4,7%. If you look in the table, selecting the row for 4 periods and the column for a rate of 7%, you will find the factor is 0.7629.
- Table B.3—Present Value of an Annuity of 1—The numbers you will look up in this table are called “present value factor of an ordinary annuity.”
· We will represent the present value factor of an ordinary annuity for n periods at i interest rate as PVF-OAn,i . This factor for 8 periods at 4% interest per period is therefore represented as PVF-OA8,4%. If you look in the table, selecting the row for 8 periods and the column for a rate of 4%, you will find the factor is 6.7327.
Important terms
· Stated (nominal, or face) rate—the annual rate of interest
· Effective rate—the rate for a year, including the effect of compounding
If compounded more than once per year, the effective rate is greater than stated rate
General procedure
For all of these problems, you will:
· Read the problem and draw a time line to represent the situation given.
· Write the formula that solves for the unknown value in the situation.
o There are two formulas, one for the present value of a single sum and one for the present value of an annuity.
· Plug the numbers from both the stated problem and the appropriate table into the formula and solve.
That’s all there is to it!
Time lines
Interest
0 1 2 3 4 5
Present number of periods Future
Value value
We will use a time line to represent each present value situation. Note that today (present value) is represented on the left by time zero, and the future value is at the right, its number depending on the number of periods in the situation as given.
Fundamental variables
You will work with four quantities in all present value problems. Three will always be given, and your job will be to solve for the fourth. The purpose of the time line is to show clearly which values are given and which one is unknown in the situation given. The values are:
· Rate of interest—unless otherwise stated, this is an annual rate. It will need to be converted to a period rate if compounding occurs more than once per year.
· Number of periods—number of compounding periods (not necessarily years). Again, you must convert years to periods if necessary.
· Future Value—The value at a future date of a given amount(s) invested, with compound interest.
· Present Value—The value now of a future amount(s) “discounted” with compound interest.
Single-sum problems
Important: the present value will always be less than the future value.
Rate and number of periods given
Formulas
Find the present value (future value given): PV = FV (PVFn,i)
· You are solving for the present value, so you use the present value factor (and the future value is given).
Steps
1. Set up the problem
a. Read the problem, draw the time line.
b. Place future value on the time line.
c. Write the formula.
2. Find the factor
a. Based on the number of years and the number of periods per year, determine the number of periods.
b. Based on the annual rate and the number of periods per year, determine the rate per period.
c. Look up factor in the table.
3. Plug in the values and calculate.
4. Check to be sure your answer makes sense. Is the present value less than the future value?
Example
Present value
How much may I borrow it I promise to pay back $30,000 in 3 years and the lender requires a 10% rate of interest?
PV $30,000
3 years
PV = FV (PVFn,i)
30,000 x PVF (3, 10%)
30,000 x 0.7513
$22,539 Answer: I can borrow $22,539.
Annuity problems
An annuity is a series of cash flows of equal amounts, at equal intervals, compounded once each interval. In an ordinary annuity, the cash flows occur at end of each period.
T 0 1 2 3 4
cash flow cash flow cash flow cash flow
We use the term ordinary annuity to distinguish from an annuity due, in which the cash flows occur at the beginning of each period. We will work exclusively with ordinary annuities in this course.
In working with annuities, we use the term rents to refer to the cash flows. For any given annuity, all rents will be of the same amount. There will be one rent at the end of each period. Since the last rent (in an ordinary annuity) occurs on the future value date, it will not grow at all, as we just received or paid it before calculating the future value.
The value of an annuity could be calculated piecemeal using the present value table.
T 0 1 2 3 4
PV1
PV2
PV3
PV4
PV-OA
The sum of the present values of all the rents is the present value of the annuity.
It’s less work to calculate this using the annuity tables.
Formulas
Find the present value of an annuity: PV-OA = R x PVF-OA (n, i)
Examples
Present value of an annuity
I promise to pay $10,000 at the end of each year for 10 years. The lender requires 8% interest. How much may I borrow if the first payment will be one year from today?
0 1 2 9 10
10,000 10,000 10,000 10,000
PV-OA = ?
PV-OA = R x PVF-OA (n, i)
PV-OA = 10,000 x PVF-OA (10, 8%)
PV-OA = 10,000 x 6.7101
PV-OA = $67,101 Answer: I can borrow $67,101.
Present value of an annuity (semiannual compounding)
I promise to pay $5,000 at the end of each six-month period for 10 years. The lender requires 8% interest. How much may I borrow if the first payment will be six months from today?
(Note that the total of the payments is the same as in the previous example. The payments, however, are half the amount and twice as often, paid for 20 periods rather than 10. The stated interest rate is 8%; but because the payments are made twice each year, the interest rate used in the calculation is 4% per semiannual period.)
0 1 2 19 20 (periods)
5,000 5,000 5,000 5,000
PV-OA = ?
PV-OA = R x PVF-OA (n, i)
PV-OA = 5,000 x PVF-OA (20, 4%)
PV-OA = 5,000 x 13.5903
PV-OA = $67,952 Answer: I can borrow $67,952.
Assignment:
1. For practice, do the following quick studies and exercises from the end of Appendix B in your text:
QS B-4 and QS B-6
Exercises B-7 and B-10
The solutions to these are posted on the web page in Excel format. The Excel file contains the solutions using the same method as presented in this paper and that we will use in class.
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