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Appendix 4A The Time Value of Money 67
Appendix 4A
The Time Value of Money
Outline
4A-I. Interest: The Basic Return to Savers
A. Simple Interest
B. Compound Interest
4A-II. Computational Aids for Use in Time Value Calculations
A. Financial Calculators
B. Computers and Spreadsheets
4A-III. Future Value: An Extension of Compounding
A. Calculator Use
B. Spreadsheet Use
4A-IV. Future Value of an Annuity
A. Calculator Use
B. Spreadsheet Use
4A-V. Present Value: An Extension of Future Value
A. Calculator Use
B. Spreadsheet Use
4A-VI. Present Value of a Stream of Returns
A. Present Value of a Mixed Stream
1. Calculator Use
2. Spreadsheet Use
B. Present Value of an Annuity
1. Calculator Use
2. Spreadsheet Use
Concepts in Review
Overview
The vitally important concepts of the time value of money, future value, and present value are covered in
an appendix to Chapter 4. These concepts are best explained by working through a few examples that deal
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68 Smart/Gitman/Joehnk • Fundamentals of Investing, Thirteenth Edition
with single sums, annuities, and mixed streams. The instructor should emphasize that present value
calculations provide a dollar value (in today’s terms) of future cash flows. The present value concept is a
powerful tool that makes it possible to compute the dollar value of any asset. Some assets that might be
profitably considered in class are stocks, bonds, other financial assets, physical assets (machines), real
estate, and even companies themselves.
Answers to Concepts in Review
4A.1. Time value of money refers to the fact that, with the opportunity to earn interest on funds, the value
of money depends on the point in time when the money received. Thus, the sooner one receives
money the better—the more valuable is that money.
Because money has time value, people who are willing to invest their money should be
able to earn a positive return. For example, an investor expecting to receive a $100 interest
At the end of year 2, the first investment has returned $100 interest; the second has
4A.2. a. Interest is the income you receive from placing available funds in a savings account, CD,
d. The true rate of interest (or return) takes the concept of compounding into account. When
4A.3. The true rate of interest rises as interest is compounded more frequently than annually. The true
4A.4. The future value of a cash flow represents the amount to which a current deposit will grow over a
given time period if it is placed in an account paying compound interest. Present value is
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Appendix 4A The Time Value of Money 69
4A.5. An annuity is a stream of equal cash flows that occur in equal intervals over time. These cash
flows can be paid out or received. An ordinary annuity has cash flows that occur at the end of each
4A.6 A mixed stream of returns is a series of returns that exhibits no pattern. To find the present value of
a mixed stream, calculate the present value of each component of the mixed stream. The
summation of the present value of individual components gives us the present value of the entire
Solutions to Problems
NOTE: Solutions are shown as input to a typical financial calculator and/or as Excel formulas. Because
of the way calculators and Excel compute TVM answers, cash outflows and inflows have opposite signs.
The solutions given here ignore the signs.
4A-1. The simple interest calculations for parts a. and b. can be presented in tabular form:
b. a.
Date
Beginning
Balance*
Annual
Interest
Ending
Balance**
1/1/17 $5,000 $5,000 0.06 $300 $5,000
Assuming all transactions occur at the beginning of the period
Since all interest earned is withdrawn, the ending account balance equals the
beginning account balance.
4A.2. a. Future value of $350 in 10 years at 6% annual compound interest:
b. The future value at the end of five years of an $700 annual end-of-year deposit at 8% interest:
Note: For simplicity, the problems in the rest of the chapter use the abbreviations FV, PV, I (interest
rate/rate of return), and N (number of years/investment period).
4A.3. Future value of an investment: FVn Investment amount FVIFk, n
Investment
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70 Smart/Gitman/Joehnk • Fundamentals of Investing, Thirteenth Edition
4A.4.
4A.5. Future value of an annuity investment: FVAk, n
Calc: N=number of periods, i=rate of return per period,
PV=0, PMT = investment per period, FV= future value
Excel: =FV(periodic rate, number of periods, periodic payment, present value)
Investment
4A.6. Future value of an annuity of $1,000 for 10 years at 2%.
4A.7. The amount of money you would have after six years is:
a. Future value of an investment: FVn
b. Future value of an annuity investment:
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Appendix 4A The Time Value of Money 71
c. FV of $3,000 at 9% for six years FVA of $1,000 deposit at 9% at end of each of the next five
years:
d. This problem is best solved as three FV of a sum problems, then totaled.
N i PV PMT FV
5 9 900 0 $1,384.76
4A.8.
Answer
Investment N i PV PMT FV
A 4 7% $ 5,340.27 0 7,000.00
4A.9. This problem uses present value to solve an investment problem. The amount at which the
bond will sell today is the value today of its value at maturity (in eight years), given an interest
rate of 2%:
4A.10.
N=7, i=4%, PMT=0. FV=1000, PV=$759.92 ans
4A.11.
4A.12. Most financial calculators have built in routines for calculating net present value, but they differ
widely in their approaches. The easiest way to solve this problem is with a spreadsheet. Merely
4A.13. a. a. It is best to set up a spreadsheet to solve this problem. Using a 1% discount rate,
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72 Smart/Gitman/Joehnk • Fundamentals of Investing, Thirteenth Edition
b. Although both streams pay out $10,000 in total, the present value of the first income
stream is higher, no matter what the discount rate is, because the income is higher in the
4A.14. Present value of annuity problems can easily be solved with any financial calculator.
The answers shown below would appear as negative numbers. For EXCEL, use =PV(rate,
nper,pmt) for example A =PV(.07,3,1200)
Answers
Investment N i PV PMT FV
A 3 7% $3,149.18 1,200 0
4A.15. Find the present value of the annuity and compare it to the lump sum payment.
4A.16. a.Present value of $500 to be received in four years at an 11% discount rate:
b. The present value of the income from Stream A is the present value of an annuity.
The problem is most easily solved with a spreadsheet using the formula =npv(.09, range of
cells containing income streams).
Income Stream
End of Year A B
PV at 9% $402.64 $437.12
Note: These streams may be used to illustrate the time value of money. Both streams have
$560 in total benefits, but the benefits in Stream A are presently worth $402.64, while the benefits
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Appendix 4A The Time Value of Money 73
4A.17. The best way to solve this problem is to calculate the IRR or the growth rate of each investment
and compare it to the rate that Terri could earn on other similar investments (10%). Any
investment that earns more than 10% is acceptable.
Answer
Investment NIRR PV PMT FV
A 5 10.8% –18,000 0 30,000
Note that for financial calculators and Excel, PV and FV must have opposite signs.
Excel: =rate(nper,pmt,pv,fv) for example =rate(5,0,-18000,30000) result 10.8%
Ms. Allessandro should accept investments A and C and forgo B and D.
Alternative approach: discount the FVs at 10% and reject the investments with a PV lower
than the purchase price.
Answer
Investment N i PV PMT FV
A 5 10.0% $18,627.64 0 30,000
She would still accept A and C (because for those the PV is greater than the purchase
price) and forgo B and D (because for those the PV is less than the purchase price).
4A.18. This problem is most easily solved using Excel’s NPV formula to compute the present value of the
Income Stream
End of year A B
1 2,500 4,000
2 3,500 3,500
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74 Smart/Gitman/Joehnk • Fundamentals of Investing, Thirteenth Edition
4A.19. Monthly interest rate is 12%/12 or 1%.
4A.20. Balance is the present value of the 40 remaining payments discounted at 12% (1% per month).
©2017 Pearson Education, Inc.
