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CHAPTER 4
THE TIME VALUE OF MONEY
TOPIC OUTLINE, KEY LECTURE CONCEPTS, AND TERMS
SPECIAL NOTICE: More complete time value of money tables than those in the text
are located at the Online Learning Centre, www.mcgrawhill.ca/college/brealey.
4.1
FUTURE VALUES AND COMPOUND INTEREST
A. Cash flows occurring in different time periods are not comparable unless adjusted
for time value.
B. The future value is the amount to which an investment will grow after earning
interest. Future value = investment × (1+r)t.
C. The expression, (1+r)t, refers to compound interest or interest earned on interest
at the rate, r, for t periods. An investment of $100 for five years at 6 percent
interest, compounded annually would be $100 × (1.06)5 = $133.82, with the
$33.82 representing the accumulated interest.
D. If the $100 investment above earned 6 percent simple interest, or annual interest
on the original investment, the sum of the original $100 plus accumulated simple
interest of $100 × (.06) = $6.00 × five years = $30.00 would be $130. Note that
with compound interest an additional $3.82 is earned in the five-year period. See
Table 4.1 and Figure 4.1 for arithmetic and graphical analyses, respectively.
Year
1
2
3
4
5
Balance at
Start of Year
$100.00
$106.00
$112.36
$119.10
$126.25
Interest Earned
during Year
.06 × $100.00 = $6.00
.06 × $106.00 = $6.36
.06 × $112.36 = $6.74
.06 × $119.10 = $7.15
.06 × $126.25 = $7.57
Balance at
End of Year
$106.00
$112.36
$119.10
$126.25
$133.82
E. In addition to the future value factors in Table 4.2, future values for a larger range
of years and interest rates are found in Table A.1 at the Online Learning Centre,
www.mcgrawhill.ca/college/brealey.
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Copyright © 2006 McGraw-Hill Ryerson Limited
4.2
PRESENT VALUES
A. The value today of a future cash flow is called the present value. The present
value computation solves for the original investment at a certain rate when one
knows the future value. The present value is the reciprocal of the future value
calculation. Present value (1+r)t = future value, while the present value (PV) =
1/(1+r)t × future value.
B. The interest rate used to compute present values of future cash flows is called the
discount rate. This will be an important variable when value determination is
studied in the next chapter.
C. Present values are directly related to the future cash flows and inversely related to
the discount rate, r, and time, t. The higher the future cash flows, the higher the PV;
the higher the discount rate and longer the term, the lower the PV.
D. The expression, 1/(1+r)t, is called a discount factor, which is the PV of a $1
future payment. Discount factors for whole number discount rates and years are
calculated and available for use in Table 4.3 and in Table A.2 at the Online
Learning Centre, www.mcgrawhill.ca/college/brealey.
E. Cash flows occurring at different time periods are not comparable for financial
decision-making. The cash flows must be time adjusted at an appropriate discount
rate, usually to the “present” for comparison, summation, or other analysis. A time
line presentation helps students visualize the concept. See Figure 4.4.
Finding the Interest Rate
A. In the expression, PV = FV×(1+r)t, when the PV, FV, and t are known, (1+r) may
be solved arithmetically. The discount rate calculated is also called the annual
interest rate, growth rate, and internal rate of return, depending on the situation.
Finding the Investment Period
A. The expression PV = FV×(1+r)t can also be solved for t, the investment period,
when the other values are known.
4.3
MULTIPLE CASH FLOWS
Future Value of Multiple Cash Flows
A. Many finance situations involve more than one cash flow. Whether they are equal,
consecutive payments or irregular, unequal cash flows over time, they are referred
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to as a stream of cash flows.
B. To find the value at some certain date of a stream of cash flows, calculate what
each cash flow will be worth at that future date, and sum up these future values.
Present Value of Multiple Cash Flows
A. Calculating the present value of an unequal series of future cash flows is
determined by summing the present values of each discounted single future cash
flow.
4.4
LEVEL CASH FLOWS: PERPETUITIES AND ANNUITIES
A. A future stream of cash flows associated with an investment may be compared or
summed if adjusted to a common time period, usually the present (PV).
B. The multiple cash flows may be the same amount and be equally spaced over the
term, called an annuity, may be an annuity with cash flows assumed to be
received forever, called a perpetuity, or the future cash flow stream may be
unequal and intermittent over some future period.
C. The future value of a sum of equal, annual (or every period) cash flows is the
future value of an annuity. Annuity refers to equal, consecutive payments. An
ordinary annuity assumes cash flows occur at the end of each period; an annuity
due assumes they are paid at the beginning of the period.
D. The present value of a sum of equal, annual (or every period) cash flows is the
present value of an annuity.
E. The key point is that when discounted to the present, all future cash flows are
standardized for comparison, for summing, and other analysis, such as net present
value studied later.
How to Value Perpetuities
A. The present value of a never-ending equal stream of cash flows is called a
perpetuity.
B. The PV of a perpetuity is equal to the periodic cash flow divided by the
cash payments C
appropriate discount rate, or PV (perpetuity) = discount rate = r .
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Copyright © 2006 McGraw-Hill Ryerson Limited
How to Value Annuities
A. An annuity is an equally spaced, level stream of cash flows, such as $50 per year
for ten years. If cash flows occur at the end of each period, it is called an ordinary
annuity. If cash flows occur at the beginning of each period, it is called an annuity
due. Figure 4.10 assumes an ordinary annuity.
B. The present value of an annuity is the difference between an immediate perpetuity
and a delayed perpetuity. The delayed perpetuity’s cash flows start the first period
after the end of the relevant annuity period. Arithmetically, the PV of a t year
1
1

annuity is C   r  r ×(1 + r)t , where C represents the annuity cash flows per


period, and r is the appropriate discount rate. The bracketed quantity in the
formula above is called a present value annuity factor. Tables 4.4 and A.3,
found at the Online Learning Centre, www.mcgrawhill.ca/college/brealey, have
present value annuity factors with varied whole numbers r and t.
Annuities Due
A. A financial calculator refers to annuity cash flows as payments. The annuity
default mode in the financial calculator is the ordinary, end-of-period, annuity. To
switch to annuity due, there is usually a “due” or “beg” key. When in “due” mode,
it will indicate so in the display. Remind students to check the display before the
next problem to make sure the correct mode has been selected. See time line,
Figure 4.12, for a comparison of an ordinary annuity versus an annuity due
annuity.
B. The present value sum of a series of consecutive, equal payments, C, paid at the
beginning of each period, is called the present value of an annuity due
1
1

Present value of a $C annuity due = C × [1 +  r  r ×(1 + r)t-1


]
C. A loan amortization problem uses the formula for the present value of an annuity,
solving for C, the loan payment per period. If loan payments are made at the end
of the month, use the formula for the present value of a regular (deferred) annuity.
If loan payments are made at the beginning of the month, use the formula for the
present value of an annuity due.
D. Often loans are paid monthly. The adjustment of annual to more frequent
payments or compounding is to multiply the annual t by the number of payments
per year and use the appropriate interest rate for the period. A ten-year loan with
monthly payments would have t = 12 × 10 = 120 payments and the monthly
interest rate would have to be determined. We return to issue of determining
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interest rates when payments are made more frequently that once per year in
Section 3.6, at the end of the chapter.
Future Value of an Annuity
A. The present value sum of a series of consecutive, equal payments, C, is called the
present value of an annuity or:
1
1

Present Value of an Annuity = C   r  r ×(1 + r)t


B. The future value sum of a series of consecutive, equal payments, C, is called the
future value of an annuity, calculated by multiplying the present value of annuity,
above, by (l+r)t or:
1
1

Future Value of an Annuity = C   r  r × (1 + r)t × (1+r)t


=C×
(1 + r)t - 1
r
C. With an ordinary annuity the cash flows (PMT or payments in a financial
calculator) are assumed to flow at the end of the year.
Cash Flows Growing at a Constant Rate – Variations on Perpetuities and Annuities
A. When cash flows are not level over time but are growing at a constant rate,
variations of the annuity formulas can be used.
B. The present value of a perpetual stream of cash flows growing at constant rate g
is:
first cash payment
C1
Present value of a growing perpetuity = discount rate - growth rate = r - g ,
where C1 is the first payment, r is the discount rate and g is the constant growth
rate of the cash flows.
C. The present value of a finite stream of payments growing at constant rate g for T
periods is:
C1
1+g T
Present value of a growing annuity = r - g × 1 - 1 + r
[ (
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Copyright © 2006 McGraw-Hill Ryerson Limited
)]
4.5
INFLATION AND THE TIME VALUE OF MONEY
A. Inflation is an overall general rise in the price level for goods and services.
B. In the time value of money analysis above, interest rates were assumed to be
“real” rates, and the cash flows over the time line were assumed to have the same
purchasing power. With inflation the purchasing power of cash flows over a time
line declines at the rate of inflation.
Real Versus Nominal Cash Flows
A. One measure of inflation is the Consumer Price Index (CPI). The annualized
percentage increases in the CPI are a measure of the rate of inflation.
B. Consumers and investors are concerned about the real value of $1 or the
purchasing power of the dollar or investment return in a period of time.
Inflation and Interest Rates
A. Actual dollar prices or interest rates are called nominal dollars or interest rates.
Bonds, loans, and most financial contracts are quoted in nominal interest rates.
B. Nominal rates, adjusted for inflation in a period, are real interest rates, or the rate
at which the purchasing power of an investment increases.
C. The real rate of interest is calculated as follows:
1 + nominal rate
1 + real interest rate = 1+ inflation rate
D. The approximate real rate is the nominal rate minus the inflation rate for the
period:
real interest rate ≈ nominal interest rate – inflation rate
The approximation is most accurate when both the nominal interest rate and the
inflation rate are low.
E. Investors and lenders include expected inflation rates in nominal rates to
compensate for the loss of purchasing power.
F. Nominal rates include expected real rates of return plus expected inflation rates.
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Valuing Real Cash Payments
A. Since nominal rates include real rates plus expected inflation, discounting nominal
future cash flows by nominal rates will give the same answer as discounting real,
expected-inflation-adjusted cash flows by the real interest rate.
B. Current dollar cash flows must be discounted by the nominal interest rate; real
cash flows must be discounted by the real interest rate.
C. Expected inflation is a significant variable in retirement planning, tuition savings
plans, choice of vocation, or any long-term financial planning. Even a low rate of
inflation can have a major negative effect on people who will receive relatively
fixed nominal income or returns.
D. The actual purchasing power rate of return (real rate) on an investment is the
nominal expected rate of return, 1+r, divided by 1 + the expected inflation rate.
With high inflation, the realized real rate may be negative.
Real or Nominal?
Most financial analyses in this text will assume nominal rates and will discount
nominal cash flows. When one set of cash flows are presented in real terms and
another in nominal terms, you cannot combine or compare them directly. You must
first convert one of the cash flows to match the other in order to compare, contrast,
and mix the cash flows. Same holds for interest rates: real and nominal rates cannot
be directly compared. Adjust one of them for inflation. Do not mix nominal and real
or you will have garbage!
4.6
EFFECTIVE ANNUAL INTEREST RATES
A. The effective annual interest rate (EAR) is the period interest rate annualized
using compound interest. If the three-month period interest rate is 5 percent, the
effective annual rate is 5 percent to the fourth power, for there are four threemonth periods in a year or, (1.05)4 -1 = 21.55 percent. The exponent used is the
number of periods, in this case three months, equal to one year.
B. The effective annual interest rate is the annual interest rate equivalent to the
period interest rate. In other words, investing at the period rate will give the exact
same future value as investing at its effective annual equivalent rate. Use the EAR
to ensure that the same interest rate is being used to value cash flows with
different frequency of payment.
C. The annual percentage rate (APR) is the period rate times the number of periods
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to complete a year or the interest rate that is annualized using simple interest. In
the case above the three-month period rate of 5 percent times the number of threemonth periods in a year, four, equals 20 percent, an annual percentage rate (APR).
Another way, for any investment, 5 percent per three-month period using simple
interest will equal the amount, say $100, times 5 percent times four to calculate
the annualized amount of simple interest of $20.00. That works out to a $20/$100
= 20 percent APR.
D. To convert an annual percentage rate (APR) to an effective annual rate, divide the
APR, using our 20 percent rate above, by the number of annual interest periods
(4), and annualize that period rate or 1.054 -1 = 21.55 percent. See Table 4.6
which calculates the effective rates for several compounding periods using a 6
percent APR.
Compounding
Period
1 year
Semianually
Quarterly
Monthly
Weekly
Daily
Periods
Per Year
1
2
4
12
52
365
Per-Period
Interest Rate
6%
3
1.5
.5
.11538
.01644
Growth Factor of
Invested Funds
1.06
1.032 = 1.0609
1.0154 = 1.061364
1.00512 = 1.061678
1.001153852 = 1.061800
1.0001644365 = 1.061831
Effective
Annual Rate
6.0000%
6.0900
6.1364
6.1678
6.1800
6.1831
E. To summarize: If you have an APR and the number of payments per year is m, the
effective annual equivalent is:
m
APR

EAR = 1 + m  - 1


Likewise, if the per period rate is i and there are m payment (or compounding)
periods per year, the effective annual equivalent formula is also:
m
EAR = [1 + i] - 1
If you have an EAR and want to determine the equivalent period rate, i, where
interest is paid m times per year, the formula is:
Period rate equivalent to an annual rate = i = (1 + EAR)1/m - 1
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