OVERVIEW
A dollar in the hand today is worth more than a dollar to be received in the future because, if you had it now, you could invest that dollar and earn interest. Of all the techniques used in finance, none is more important than the concept of the time value of money, or discounted cash flow (DCF) analysis. The principles of time value analysis that are developed in this chapter have many applications, ranging from setting up schedules for paying off loans to decisions about
whether to acquire new equipment.
Future value and present value techniques can be applied to a single cash flow (lump sum), ordinary annuities, annuities due, and uneven cash flow streams. Future and present values can be calculated using a regular calculator or a calculator with financial functions. When compounding occurs more frequently than once a year, the effective rate of interest is greater than the quoted rate.
The cash flow time line is one of the most important tools in time value of money analysis.
Cash flow time lines help to visualize what is happening in a particular problem. Cash flows are placed directly below the tick marks, and interest rates are shown directly above the time line; unknown cash flows are indicated by question marks. Thus, to find the future value of
$100 after 5 years at 5 percent interest, the following cash flow time line can be set up:
Time: 0 1 2 3 4 5
| | | | | |
Cash flows: -100 FV5 = ?
A cash outflow is a payment, or disbursement, of cash for expenses, investments, and so on.
A cash inflow is a receipt of cash from an investment, an employer, or other sources.
Compounding is the process of determining the value of a cash flow or series of cash flows some time in the future when compound interest is applied. The future value is the amount to which a cash flow or series of cash flows will grow over a given period of time when compounded at a given interest rate. The future value can be calculated as
T
HET
IMEV
ALUE OFM
ONEYOUTLINE
5%
FVn = PV(1 + k)n,
where PV = present value, or beginning amount; k = interest rate per period; and n = number of periods involved in the analysis. This equation can be solved in one of two ways:
numerically or with a financial calculator. For calculations, assume the following data that were presented in the time line above: present value (PV) = $100, interest rate (k) = 5%, and number of years (n) = 5.
Compounded interest is interest earned on interest.
To solve numerically, use a regular calculator to find 1 + k = 1.05 raised to the fifth power, which equals 1.2763. Multiply this figure by PV = $100 to get the final answer of FV5 =
$127.63.
With a financial calculator, the future value can be found by using the time value of money input keys, where N = number of periods, I = interest rate per period, PV = present value, PMT = annuity payment, and FV = future value. By entering N = 5, I = 5, PV = -100, and PMT = 0, and then pressing the FV key, the answer 127.63 is displayed.
Some financial calculators require that all cash flows be designated as either inflows or outflows, thus an outflow must be entered as a negative number (for example, PV = -100 instead of PV = 100).
Some calculators require you to press a “Compute” key before pressing the FV key.
A graph of the compounding process shows how any sum grows over time at various interest rates. The greater the rate of interest, the faster is the rate of growth.
The interest rate is, in fact, a growth rate.
The time value concepts can be applied to anything that is growing.
Finding the present value of a cash flow or series of cash flows is called discounting, and it is simply the reverse of compounding. In general, the present value is the value today of a future cash flow or series of cash flows. By solving for PV in the future value equation, the present value, or discounting, equation can be developed and written in several forms:
PV =
n n n
n
k) (1 FV 1 k)
(1
FV .
To solve for the present value of $127.63 discounted back 5 years at a 5% opportunity cost rate, one can utilize either of the two solution methods:
Numerical solution: Divide $127.63 by 1.05 five times to get PV = $100.
Financial calculator solution: Enter N = 5, I = 5, PMT = 0, and FV = 127.63, and then
press the PV key to get PV = -100.
The opportunity cost rate is the rate of return on the best available alternative investment of equal risk.
A graph of the discounting process shows how the present value of any sum to be received in the future diminishes and approaches zero as the payment date is extended farther into the future. At relatively high interest rates, funds due in the future are worth very little today, and even at a relatively low discount rate, the present value of a sum due in the very distant future is quite small.
The compounding and discounting processes are reciprocals, or inverses, of one another. In addition, there are four variables in the time value of money equations: PV, FV, k, and n. If three of the four variables are known, you can find the value of the fourth.
If we are given PV, FV, and n, we can determine k by substituting the known values into either the present value or future value equations, and then solving for k. Thus, if you can buy a security at a price of $78.35 which will pay you $100 after 5 years, what is the interest rate earned on the investment?
Numerical solution: Use a trial and error process to reach the 5% value for k. This is a tedious and inefficient process. Alternatively, you could use algebra to solve the time value equation.
Financial calculator solution: Enter N = 5, PV = -78.35, PMT = 0, and FV = 100, then press the I key, and I = 5 is displayed.
Likewise, if we are given PV, FV, and k, we can determine n by substituting the known values into either the present value or future value equations, and then solving for n. Thus, if you can buy a security with a 5 percent interest rate at a price of $78.35 today, how long will it take for your investment to return $100?
Numerical solution: Use a trial and error process to reach the value of 5 for n. This is a tedious and inefficient process. The equation can also be solved algebraically.
Financial calculator solution: Enter I = 5, PV = -78.35, PMT = 0, and FV = 100, then press the N key, and N = 5 is displayed.
An annuity is a series of equal payments made at fixed intervals for a specified number of periods. If the payments occur at the end of each period, as they typically do, the annuity is an ordinary, or deferred, annuity. If the payments occur at the beginning of each period, it is called an annuity due.
The future value of an ordinary annuity, FVAn, is the total amount one would have at the end of the annuity period if each payment were invested at a given interest rate and held to the end of the annuity period.
Defining FVAn as the future value of an ordinary annuity of n years, and PMT as the periodic payment, we can write
FVAn = PMT
n 1 t
t
)n
k 1
( = PMT
1 n
0 t
)t
k 1
( = PMT
k 1 ) k 1
( n
.
Using a financial calculator, enter N = 3, I = 5, PV = 0, and PMT = -100. Then press the FV key, and 315.25 is displayed.
For an annuity due, each payment is compounded for one additional period, so the future value of the entire annuity is equal to the future value of an ordinary annuity compounded for one additional period. Thus:
FVA (DUE)n = PMT
) k 1 k (
1 ) k 1
( n
.
Most financial calculators have a switch, or key, marked “DUE” or “BEG” that permits you to switch from end-of-period payments (an ordinary annuity) to beginning-of- period payments (an annuity due). Switch your calculator to “BEG” mode, and calculate as you would for an ordinary annuity. Do not forget to switch your calculator back to “END” mode when you are finished.
The present value of an ordinary annuity, PVAn, is the single (lump sum) payment today that would be equivalent to the annuity payments spread over the annuity period. It is the amount today that would permit withdrawals of an equal amount (PMT) at the end (or beginning for an annuity due) of each period for n periods.
Defining PVAn as the present value of an ordinary annuity of n years and PMT as the periodic payment, we can write
PVAn = PMT
n
1 t
)t
k 1 (
1 = PMT
k
) k 1 (
1 1 n
= PMT
k ) k 1 (
1 n
.
Using a financial calculator, enter N = 3, I = 5, PMT = -100, and FV = 0, and then press the PV key, for an answer of $272.32.
One especially important application of the annuity concept relates to loans with constant payments, such as mortgages and auto loans. With these amortized loans the amount borrowed is the present value of an ordinary annuity, and the payments constitute the annuity stream.
The present value for an annuity due is
PVA (DUE)n = PMT
) k 1 k (
) k 1 (
1 1 n
.
Using a financial calculator, switch to the “BEG” mode, and then enter N = 3, I = 5, PMT = -100, and FV = 0, and then press PV to get the answer, $285.94. Again, do not forget to switch your calculator back to “END” mode when you are finished.
You can solve for the interest rate (rate of return) earned on an annuity.
To solve numerically, you must use the trial-and-error process and plug in different values for k in the annuity equation to solve for the interest rate.
You can use the financial calculator by entering the appropriate values for N, PMT, and either FV or PV, and then pressing I to solve for the interest rate.
You can solve for the number of periods (N) in an annuity.
To solve numerically, you must use the trial-and-error process and plug in different values for N in the annuity equation to solve for the number of periods.
You can use the financial calculator by entering the appropriate values for I, PMT, and either FV or PV, and then pressing N to solve for the number of periods.
A perpetuity is a stream of equal payments expected to continue forever.
The present value of a perpetuity is:
PVP =
k PMT rate
Interest Payment
.
For example, if the interest rate were 12 percent, a perpetuity of $1,000 a year would have a present value of $1,000/0.12 = $8,333.33.
A consol is a perpetual bond issued by the British government to consolidate past debts; in general, any perpetual bond.
The value of a perpetuity changes dramatically when interest rates change.
Many financial decisions require the analysis of uneven, or nonconstant, cash flows rather than a stream of fixed payments such as an annuity. An uneven cash flow stream is a series of cash flows in which the amount varies from one period to the next.
The term payment, PMT, designates constant cash flows, while the term CF designates cash flows in general, including uneven cash flows.
The present value of an uneven cash flow stream is the sum of the PVs of the individual cash flows of the stream.
The PV is found by applying the following general present value equation:
PV =
n
1 t
t t
) k 1 (
CF 1 .
With a financial calculator, enter each cash flow (beginning with the t = 0 cash flow) into the cash flow register, CFj, enter the appropriate interest rate, and then press the NPV key to obtain the PV of the cash flow stream.
Be sure to clear the cash flow register before starting a new problem.
Similarly, the future value of an uneven cash flow stream, or terminal value, is the sum of the FVs of the individual cash flows of the stream.
The FV can be found by applying the following general future value equation:
FVn =
n
1 t
t n t(1 k)
CF .
Some calculators have a net future value (NFV) key which allows you to obtain the FV of an uneven cash flow stream.
We generally are more interested in the present value of an asset’s cash flow stream than in the future value because the present value represents today’s value, which we can compare with the price of the asset.
Once we know its present value, we can find the future value of an uneven cash flow stream by treating the present value as a lump sum amount and compounding it to the future period.
If one knows the relevant cash flows, the effective interest rate can be calculated efficiently with a financial calculator. Enter each cash flow (beginning with the t = 0 cash flow) into the cash flow register, CFj, and then press the IRR key to obtain the interest rate of an uneven cash flow stream.
IRR stands for internal rate of return, which is the return on an investment.
Annual compounding is the arithmetic process of determining the final value of a cash flow or series of cash flows when interest is added once a year. Semiannual, quarterly, and other compounding periods more frequent than on an annual basis are often used in financial transactions. Compounding on a nonannual basis requires an adjustment to both the compounding and discounting procedures discussed previously. Moreover, when comparing securities with different compounding periods, they need to be put on a common basis. This requires distinguishing between the simple, or quoted, interest rate and the effective annual rate.
The simple, or quoted, interest rate is the contracted, or quoted, interest rate that is used to calculate the interest paid per period.
The periodic rate is the interest rate charged per period.
Periodic rate = Stated annual interest rate/Number of periods per year.
The annual percentage rate, APR, is the periodic rate times the number of periods per year.
The effective annual rate, EAR, is the rate that would have produced the final compounded value under annual compounding. The effective annual rate is given by the following formula:
Effective annual rate (EAR) = 1.0, m
1 k
m SIMPLE
where kSIMPLE is the simple, or quoted, interest rate (that is, the APR), and m is the number of compounding periods (interest payments) per year. The EAR is useful in comparing securities with different compounding periods.
For example, to find the effective annual rate if the simple rate is 6 percent and semiannual compounding is used, we have:
EAR = (1 + 0.06/2)2 – 1.0 = 6.09%.
For annual compounding use the formula to find the future value of a single payment (lump sum):
FVn = PV(1 + k)n.
When compounding occurs more frequently than once a year, use this formula:
FVn = PV
n m SIMPLE
m 1 k
.
Here m is the number of times per year compounding occurs, and n is the number of years.
The amount to which $1,000 will grow after 5 years if quarterly compounding is applied to a nominal 8 percent interest rate is found as follows:
FVn = $1,000(1 + 0.08/4)(4)(5) = $1,000(1.02)20 = $1,485.95.
Financial calculator solution: Enter N = 20, I = 2, PV = -1000, and PMT = 0, and then press the FV key to find FV = $1,485.95.
The present value of a 5-year future investment equal to $1,485.95, with an 8 percent nominal interest rate, compounded quarterly, is found as follows:
. 000 , 1 ) $
02 . 1 (
95 . 485 , 1 PV $
/4) 08 . 0 1 ( PV 95 . 485 , 1
$
20
(4)(5)
Financial calculator solution: Enter N = 20, I = 2, PMT = 0, and FV = 1485.95, and then press the PV key to find PV = -$1,000.00.
In general, nonannual compounding can be handled one of two ways.
State everything on a periodic rather than on an annual basis. Thus, n = 6 periods rather than n = 3 years and k = 3% instead of k = 6% with semiannual compounding.
Find the effective annual rate (EAR) with the equation below and then use the EAR as the rate over the given number of years.
EAR = 1.0.
m 1 k
m SIMPLE
An important application of compound interest involves amortized loans, which are paid off in equal installments over the life of the loan.
The amount of each payment, PMT, is found using a financial calculator by entering N (number of years), I (interest rate), PV (amount borrowed), and FV = 0, and then pressing the PMT key to find the periodic payment.
Each payment consists partly of interest and partly of repayment of the amount borrowed (principal). This breakdown is often developed in a loan amortization schedule.
The interest component is largest in the first period, and it declines over the life of the loan as the outstanding balance of the loan decreases.
The repayment of principal is smallest in the first period, and it increases thereafter.
The text discussion has involved three different interest rates. It is important to understand their differences.
The simple, or quoted, rate, kSIMPLE, is the interest rate quoted by borrowers and lenders.
This quotation must include the number of compounding periods per year.
This rate is never shown on a time line, and it is never used as an input in a financial calculator unless compounding occurs only once a year.
kSIMPLE = Periodic rate m = Annual percentage rate = APR.
The periodic rate, kPER, is the rate charged by a lender or paid by a borrower each interest period. Periodic rate = kPER = kSIMPLE/m.
The periodic rate is used for calculations in problems where two conditions hold:
(1) payments occur on a regular basis more frequently than once a year, and (2) a payment is made on each compounding (or discounting) date.
The APR, or annual percentage rate, represents the periodic rate stated on an annual basis without considering interest compounding. The APR never is used in actual calculations; it is simply reported to borrowers.
The effective annual rate, EAR, is the rate with which, under annual compounding, we would obtain the same result as if we had used a given periodic rate with m compounding periods per year.
EAR is found as follows:
EAR = 1.0.
m 1 k
m SIMPLE
