134 Part 2 Fundamental Concepts in Financial Management
5-5 FINDING THE NUMBER OF YEARS, N
We sometimes need to know how long it will take to accumulate a certain sum of
money, given our beginning funds and the rate we will earn on those funds. For
example, suppose we believe that we could retire comfortably if we had $1 mil-
lion. We want to! nd how long it will take us to acquire $1 million, assuming we
now have $500,000 invested at 4.5%. We cannot use a simple formula—the situ-
ation is like that with interest rates. We can set up a formula that uses logarithms,
but calculators and spreadsheets! nd N very quickly. Here’s the calculator setup:N I/YR PV PMT FV4.5 –500000 0 100000015.7473Enter I/YR! 4.5, PV! $500000, PMT! 0, and FV! 1000000. Then when you
press the N key, you get the answer, 15.7473 years. If you plug N! 15.7473 into the
FV formula, you can prove that this is indeed the correct number of years:FV! PV(1 " I)N! $500,000(1.045)15.7473! $1,000,000You would also get N! 15.7473 with a spreadsheet.SELF^ TEST The U.S. Treasury o# ers to sell you a bond for $585.43. No payments will be
made until the bond matures 10 years from now, at which time it will be re-
deemed for $1,000. What interest rate would you earn if you bought this
bond for $585.43? What rate would you earn if you could buy the bond for
$550? for $600? (5.5%; 6.16%; 5.24%)
Microsoft earned $0.33 per share in 1997. Ten years later in 2007 it earned
$1.42. What was the growth rate in Microsoft’s earnings per share (EPS) over
the 10-year period? If EPS in 2007 had been $0.90 rather than $1.42, what
would the growth rate have been? (15.71%; 10.55%)SELF^ TEST How long would it take $1,000 to double if it was invested in a bank that
paid 6% per year? How long would it take if the rate was 10%? (11.9 years;
7.27 years)
Microsoft’s 2007 earnings per share were $1.42, and its growth rate during
the prior 10 years was 15.71% per year. If that growth rate was maintained,
how long would it take for Microsoft’s EPS to double? (4.75 years)5-6 ANNUITIES
Thus far we have dealt with single payments, or “lump sums.” However, many
assets provide a series of cash in" ows over time; and many obligations, such as
auto, student, and mortgage loans, require a series of payments. When the pay-
ments are equal and are made at! xed intervals, the series is an annuity. For exam-
ple, $100 paid at the end of each of the next 3 years is a 3-year annuity. If the pay-
ments occur at the end of each year, the annuity is an ordinary (or deferred)Annuity
A series of equal payments
at fixed intervals for a
specified number of
periods.Annuity
A series of equal payments
at fixed intervals for a
specified number of
periods.