gitmanJoeh_238702_im04

Part TwoImportant Conceptual Tools

Part Two Includes

Chapter 4 Return and Risk

Chapter 5 Modern Portfolio Concepts

49 Gitman/Joehnk • Fundamentals of Investing, Ninth Edition

Chapter 4Return and Risk

Outline

Learning Goals

I. The Concept of Return

A) Components of Return

1. Current Income2. Capital Gains (or Losses)

B) Why Return Is Important

1. Historical Performance2. Expected Return

C) Level of Return

1. Internal Characteristics2. External Forces

D) Historical Returns

Concepts in Review

II. The Time Value of Money

A) Interest: The Basic Return to Savers

1. Simple Interest2. Compound Interest

B) Computational Aids for the Use of Time Value Calculations

1. Financial Calculators2. Computers and Spreadsheets

C) Future Value: An Extension of Compounding

D) Future Value of an Annuity

E) Present Value: An Extension of Future Value

F) The Present Value of a Stream of Income

1. Present Value of a Mixed Stream2. Present Value of an Annuity

G) Determining a Satisfactory Investment

Concepts in Review

III. Measuring Return

A) Real, Risk-Free, and Required Returns

B) Holding Period Return

1. Understanding Return Components2. Computing the Holding Period Return (HPR)3. Using the HPR in Investment Decisions

C) Yield: The Internal Rate of Return

1. Yield for a Single Cash Flow2. Yield for a Stream of Income3. Interest-on-Interest: The Critical Assumption

D) Finding Growth Rates

Concepts in Review

IV. Risk: The Other Side of the Coin

A) Sources of Risk

1. Business Risk2. Financial Risk3. Purchasing Power Risk4. Interest Rate Risk5. Liquidity Risk6. Tax Risk7. Market Risk8. Event Risk

B) Risk of a Single Asset

1. Standard Deviation: An Absolute Measure of Risk2. Coefficient of Variation: A Relative Measure of Risk3. Historical Returns and Risk

C) Assessing Risk

1. Risk-Return Characteristics of Alternative Investment Vehicles2. An Acceptable Level of Risk3. Steps in the Decision Process: Combining Risk and Return

Concepts in Review

Summary

Putting Your Investment Know-How to the Test

Discussion Questions

Problems

Case Problems4.1. Solomon’s Decision4.2. The Risk-Return Tradeoff: Molly O’Rourke’s Stock Purchase Decision

Excel with Spreadsheets

Trading Online with OTIS


Key Concepts

Chapter 4 Return and Risk 52

1. The concept of return, its component parts, and the forces that affect the level of return realized by an investor. Historical returns are reviewed.

2. Interest income and the concept of time value, its underlying future and present value computations, and its use in the investment decision-making process.

3. Usage of financial calculators, computers, and spreadsheets in measuring risk and return.

4. Real, risk-free, and required returns on investments.

5. The computation and use of the holding period return and the internal rate of return, and the role yield can play in the investment decision.

6. The sources and basic types of risk, the concept of risk, its positive relationship to return, and its role in investment decision-making.

7. The basic steps involved in evaluating the risk-return characteristics of an investment.

Overview

The concepts of return and risk are developed in this chapter. The chapter is conceptually more demanding than the preceding one, so the instructor should plan to spend more class time on it.

1. Returns are rewards for investing. The components of total return are current income and capital gains (or losses). Current income is income received in cash or near-cash, whereas capital gains refers to income that is attributed to an increase—realized or unrealized—in the value of the investment.

2. Expected return motivates a person to invest in a particular vehicle. Expectations of returns are based on the past returns of that vehicle. Measuring the historical return of a particular investment reveals its average return as well as the trend of its returns. The instructor may demonstrate this by discussing Table 4.3 in class.

3. The level of returns for a particular investment vehicle depends on internal characteristics, such as type of investment and issuer characteristics, and external forces, such as war, political events, and the level of price changes (inflation or deflation).

4. The vitally important concepts of the time value of money, future value, and present value are covered next. These concepts are best explained by working through a few examples that deal 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.

5. A satisfactory investment is one in which the present value (PV) of benefits (discounted at the appropriate rate) equals or exceeds the PV of costs. The instructor may indicate that NPV (PV of benefits minus PV of costs) measures the same thing.


6. The required return of an investment is the rate which compensates the investor for its risk. It is composed of the real rate of return—the return earned in a perfect world with no risk—plus the expected inflation premium, which is called the risk-free rate, plus the risk premium for the investment.

7. The holding period return (HPR), defined next, is useful in making investment decisions. The instructor may show the class how HPR is computed in Table 4.11, stressing that the HPR from identical periods should be used when comparing two investments.

8. Yield, also called the internal rate of return (IRR), represents the annual rate of return earned on a long-term investment. A satisfactory investment is one that has a yield equal to or greater than the appropriate discount rate. Some instructors may want to spend time discussing the critical assumption—interest-on-interest—that underlies yield. Others may choose to skip this technical, but important, discussion. Those who cover interest-on-interest should find Figure 4.3 helpful in explaining it. At this point, the instructor should have made it clear to the class that the returns from an investment may be measured either in dollar or percentage terms.

9. The calculation of growth rates for streams of dividends or earnings, a present value application that is an important part of common stock valuation, is covered next.

10. The concept of risk is next introduced and defined. The possible sources of risk include business risk, financial risk, purchasing power risk, interest rate risk, liquidity risk, tax risk, market risk, and event risk. The basic types of risk include diversifiable, nondiversifiable, and total. Next introduced is the risk of a single asset, the standard deviation as a measure of absolute risk, and the coefficient of variation as a relative measure of risk. The risk-return tradeoff is also discussed.

11. The text’s description of the four steps in the investment process are useful and should be highlighted.

Answers to Concepts in Review

1. The return on investment is the expected profit that motivates people to invest. It includes both current income and/or capital gains (or losses). Without a positive expected return, there is no benefit to investing and individuals have no reason to save and invest. Since net demanders are willing to pay net savers a positive return for their funds, the opportunity to earn a positive return exists.

Return on investment can come from either current income or capital gains, or both. Current income, most commonly, is periodic payments, such as interest received on bonds, dividends on stock, or rent received from real estate. To be considered income, these payments must be received in cash or be readily convertible to cash. Capital gain refers to the change in the market value of the investment. The amount by which the proceeds from the sale of an investment exceed the original purchase is called a capital gain. If it is sold for less than the original purchase price, a capital loss is realized.

The use of percentage returns is generally preferred to dollar returns to allow investors to directly compare different sizes and types of investments.

2. Although future returns are not guaranteed by past performance, historical data often gives the investor a meaningful basis on which to form future expectations. Past return data, such as average returns or trends seen in certain time periods, can be used along with the investor’s insights about future prospects of the investment. Together the historical data and future prospects help the investor to formulate an expected return on the investment.


The level of expected returns depends on many internal characteristics of the investment and the external forces on the investment. Internal characteristics include the type of investment, the quality of management, the method by which the investment is financed, and the customer base of the issuer. External forces include war, shortages, price controls, Federal Reserve actions, and political events, among others. External forces, unlike internal characteristics, cannot be controlled by the issuer of the investment. Investment vehicles are affected differently by these forces—the expected return of one investment may increase while the expected return of another may decrease in response to external forces.

History tells us that stock market returns have averaged well above the interest rates payable on savings accounts at banks. In recent years especially during the latter part of the 1990’s the returns were well above the stock market averages for the earlier part of the century. Unfortunately, the article does not provide a clear message for investors other than “history can repeat itself.” If the investor is looking for short-term gains over a year or two, the probabilities are mixed depending on what time periods are cited. The best advice, given these statistics, is to invest to hold for the long term in order to ride out the market’s twists and turns.

3. 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 is expected to be received. Thus, the sooner one receives money the better—the more valuable is that money.

Because money has time value, people willing to invest their money should be able to earn a positive return. For example, an investor expecting to receive a $100 interest payment for 2 different securities doesn’t necessarily value them equally. If the first investment pays the interest at the end of one year, but the second investment pays the interest at the end of two years, the first investment will be more valuable. The $100 can be reinvested to earn interest for an entire year. At the end of Year 2, the first investment has returned $100 interest, the second has returned $100 to the investor. The $100 reinvestment has earned a positive return; the other $100 has not had a chance to accumulate interest.

4. (a) Interest is the current income you receive from placing available funds in a savings account, CD, bond, or by making a loan. It is in effect the “rent” paid on your money by those who obtain use of it.

(b) Simple interest is interest paid (earned) only on the initial balance for the actual amount of time it is on deposit. With simple interest, the stated interest rate is always equal to the true rate of interest (or return).

(c) Compound interest is interest paid not only on the initial deposit but also on interest accumulated from one period to the next. This is the method savings institutions generally employ. When interest is compounded annually, the simple, compound, and true rates of interest are the same.

(d) The true rate of interest (or return) takes the concept of compounding into account. When interest is compounded annually, the stated and true interest rates are equal. For more frequent compounding, the “true” rate of interest would be higher than the stated rate. Hence an APR of 15% on a credit card which is compounded daily has a true interest rate of 16.18%.

5. The true rate of interest rises as interest is compounded more frequently than annually. The true and stated rates are the same when interest is compounded annually. Continuous compounding occurs when interest is compounded over the smallest possible time period.


6. 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 concerned with finding the current value of a future sum, given that the investor earns a stated return—the discount rate (or opportunity cost)—on similar investments. The discount rate is the rate at which future sums are discounted to find their present values. The present value concept is the inverse of the future value concept.

7. 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 occur at the end of each year. To simplify the calculation of the future value of an annuity, one can use the future value interest factor of the annuity table included in Appendix A, Table A.2. The present value of an annuity can be found similarly in the present-value interest factor of the annuity table in Appendix A, Table A.4. Calculators are also designed with annuity “PMT” keys.

8. 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 mixed stream.

9. Ignoring risk, a satisfactory investment is one for which the present value of benefits (discounted) equals or exceeds the present value of costs. If the present value of benefits exceeds the cost, the investor would earn more than the discount rate i.e. the return on the investment is greater than the discount rate.

10. (a) The real rate of return is the return earned in a certain, risk-free world. It would equal the nominal rate of return on a risk-free security less inflation. Historically, it has been relatively stable in the range of 0.5 to 2%.

(b) The expected inflation premium represents the expected average future rate of inflation. It is the compensation that investors demand for future expected inflation; that is, the decline in the purchasing power of the dollar.

(c) The risk premium varies for different security issues and represents the additional return required to compensate an investor for the risk characteristics of the issue and the issuer. It is the return on a risk security (e.g. stocks, bonds) minus the risk-free rate of return, which is the rate on a 90-day T-Bill.The risk-free rate of return equals the real rate of return plus the expected inflation premium: RF r* IP. It is the return on a riskless security as measured by the 90-day T-Bill.The required rate of return equals the real rate of return plus the expected inflation premium (together, the risk-free rate) and the risk premium: ri RF IP. Alternatively, it equals the risk-free rate of return plus the risk premium.

11. The holding period is simply the period of time over which the investor wishes to measure the return on an investment. In comparing alternative investment vehicles, it is essential to use equal-length holding periods so that the two vehicles being compared are judged under identical conditions. This adds objectivity to the comparison. Most interest rates are quoted on an annual basis, so it is generally convenient to use a one-year holding period.

The holding period return (HPR) is the total return earned from holding an investment for a specified period of time. To calculate HPR, all that is needed is the beginning- and end-of-period investment values along with the value of current income received by the investor. Because HPR doesn’t account for the time value of money, the holding period is usually one year or less.


12. The yield, or IRR, is the annual rate of return earned by a long-term investment. It is also defined as the discount rate that produces a present value of benefits received equal to the present value of costs/investments. Unlike the HPR, it takes into account the time value of money and can be used to calculate the return on investments held for over one year. The HPR is inappropriate for investments held for more than one year.

13. The critical assumption underlying the use of yield as a return measure is an ability to earn a return equal to the calculated yield on all income received from the investment during the holding period. If you earn 10 percent on all income received from an investment during the holding period, your yield on the investment will be 10 percent. On the other hand, if you earn 0 percent on the income received, your rate of return on the investment would actually be less than 10 percent. If the interest-on-interest earned from the investment is less than its calculated yield, the investment’s return will fall below the yield. Clearly when using yield as a measure of investment return, the validity of this assumption must be recognized and evaluated. If the interest-on-interest assumption does not hold, use of the calculated yield could lead to poor investment decisions. (Note: The instructor may want to use the discussion of interest-on-interest and Figure 4.3 to demonstrate this somewhat complex, but extremely important, yield concept.)

14. If the present value of returns from an investment is greater than the initial cost of the investment, it is a satisfactory investment and should be acceptable. If the yield from an investment is greater than the appropriate discount rate for that investment, the present value and yield provide the same conclusion regarding acceptability.

In the example given, Investment A is clearly acceptable since its yield (8 percent) is greater than the appropriate discount rate (7 percent). Investment B, on the other hand, is not acceptable since its present value of returns ($150) is $10 less than its cost ($160). Investment C is not acceptable since its yield (8 percent) is lower than the appropriate discount rate (9 percent).

15. Risk is the chance that the actual return from an investment may differ from what is expected. The standard deviation is the statistic used to measure risk. The risk-return tradeoff is the relationship between the expected returns from an investment and the risk associated with them. The required returns from an investment increase as risk increases to provide an incentive for him or her to take higher risks i.e. in order to accept higher risks, the investors have to be compensated with higher returns.

16. (a) Business risk is concerned with the degree of uncertainty associated with an investment’s earnings and the investment’s ability to pay investors interest, dividends, and other returns owed them. Business risk is usually related to the firm’s line of business.

(b) Financial risk is the risk associated with the mix of debt and equity (capital structure) used to finance the firm. The greater the firm’s debts and interest obligations, the greater its financial risk.

(c) Purchasing power risk arises because of uncertain inflation rates and price-level changes in the future. When prices rise, each dollar invested has less value—it can buy less, and vice versa.

(d) Interest rate risk is risk associated with changes in the prices of fixed-income securities resulting from changing market interest rates. As the market interest rates change, the prices of these securities change in the opposite direction, thereby changing the level of return that an investor can expect to obtain from them. Another important aspect of interest rate risk involves the ability to reinvest income received at the initial rate of return in order to earn the fully compounded rate of return.

(e) Liquidity risk is the risk of not being able to liquidate an investment conveniently and at a reasonable price. In general, investment vehicles traded in markets with small demand and supply characteristics tend to be less liquid than those traded in broad markets.


(f) Tax risk is the risk that tax laws enacted by Congress will adversely affect certain types of investments and decrease their after-tax returns.


(g) Market risk is the risk of changes in investment returns caused by factors independent of the given investment vehicle. It results from factors such as political, economic, and social events, or changes in investor tastes and preferences.

(h) Event risk is the risk that comes from a largely or totally unexpected event which has a significant and usually immediate effect on the underlying value of an investment. The effect of this risk seems to be isolated in most cases, affecting only certain companies and properties.

17. Standard deviation is the most common measure of an asset’s risk. It measures the dispersion of returns around an asset’s average or expected return. Standard deviation is an absolute measure of risk, and thus can be used to compare the riskiness of competing investments with the same expected return. The coefficient of variation (CV) measures the relative dispersion of an asset’s average or expected returns. Like standard deviation, the higher the CV, the higher the risk. CV differs from standard deviation because it is a relative measure of risk and can be used to compare the riskiness of competing investments with different expected returns.

18. Investors’ attitudes toward risk or their risk-return tradeoffs may be classified as one of the following:

Risk-indifferent investors do not require a greater return in exchange for each unit of additional risk.

Risk-averse investors require greater return in exchange for each unit of additional risk. The trade-off here is positive; return must increase as risk increases.

Risk-taking investors accept a lower return in exchange for greater risk. This tradeoff is negative; such investors enjoy risk and are therefore willing to accept lower returns for increasing levels of risk.

In general, most investors are risk-averse. They require increased returns from an investment as its risk increases. The risk preference of an investor is an important determinant of his/her investment decisions. Risk-averse investors may not make speculative investments, while risk-taking investors may. Thus, an investment that is considered unsatisfactory by a risk-averse investor may be deemed satisfactory by a risk-taking investor.

19. The investment process can be summarized in four steps:

(1) Estimate the expected return over a given holding period using historical data or projected return data, or both. The time value of these returns must be considered for long-term investments.

(2) Assess the risk of the investment returns through the subjective (judgmental) evaluation of historical returns and by using beta (for securities).

(3) Evaluate the risk-return behavior of each alternative investment. The expected return must be “reasonable” given the level of risk possessed by the investment. Only investments offering the highest expected return for a given level of risk are considered reasonable.

(4) Select the vehicles with the highest return for the level of risk the investor is willing to take. These fit the definition of a good investment.


Suggested Answers to Discussion Questions

Answers will vary according to student’s selections, tastes, and preferences.

Solutions to Problems

1. The investor would earn $8.25 on a stock that paid $3.75 in current income and sold for $67.50. Part of the total dollar return includes a $4.50 capital gain which is the difference between the proceeds of the sale and the original purchase price ($67.50 – $63.00) of the stock.

2. The investor had interest income of $900 (three payments of $300 each), and a capital loss of $500.

3. (a) Current income: $2.70

(b) Capital gain: $60 – $50 $10(c) Total return:

(1) In dollars: $2.70 $10.00 $12.70(2) As a percentage of the initial investment: $12.70 0.25 or 25%.

50.00

4. (a) Current income is the interest income, which is equal to $900 (three payments of $300).

(b) Capital gain is $10,000 – $9,500 $500(c) Total return $900 $500 $1,400. $1,400/$9,500 14.7%.

5. (a) Total return Current income Capital gains (or losses) where:

Capital gains (or losses) Ending price – Beginning price

(1) (2) (3) (4) (5)(1) – (2) (3) (4)

YearEndingPrice –

BeginningPrice

CapitalGain

CurrentIncome

TotalReturn

2001 $32.50 $30.00 $2.50 $1.00 $3.502002 35.00 32.50 2.50 1.20 3.702003 33.00 35.00 –2.00 1.30 –0.702004 40.00 33.00 7.00 1.60 8.602005 45.00 40.00 5.00 1.75 6.75

(b) Of course, there is no correct answer here, but one might forecast using the arithmetic average or the average one-year holding period return.

(i) The arithmetic average:

(ii) The average holding period return (HPR):


(1) (2) (3)

YearTotal

Return*Beginning

Price(1) (2)

HPR2001 $3.50 $30.00 11.7%2002 3.70 32.50 11.42003 –0.70 35.00 –2.02004 8.60 33.00 26.12005 6.75 40.00 16.9*From part (a) in the previous page.

(b)(i) (ii)

Forecasts for:

Based onArithmetic Average

Based onAverage HPR

2006 $4.37 ($45.00) * 0.128 $5.76

2007 $4.37 ($49.00)** 0.128 $6.27

*End of 2005 price gain in original data.**For lack of information, we are assuming the 2006 return is $4.00 from capital gains and $1.76 from current income.

(c) Students should be made aware of the fact that many other forecasts are possible. Other factors may be relevant here: Will the pattern of two good years followed by a bad one continue? Do future prospects seem bright? (We will discuss forecasting returns on specific investment vehicles in later chapters.)

6. Total return current income plus capital gains.

Current income 100 ($1 $1.2 $1.3) $350.Capital gain $100 ($33 – $30) $300.00.Total return $350 $300 $650.As a percent of the original investment: $650/(100 $30 $3,000) 21.7%.


8. (a) Future value of $300 in twelve years at 7 percent annual compound interest:

(b) The future value at the end of 6 years of an $800 annual end-of-year deposit at 7% interest:

Note: For simplicity, the problems in the rest of the chapter use the abbreviations FV, FVIF, FVIFA, PV, PVIF, PVIFA, and k (interest rate/rate of return), n (number of years/investment period).

9. Future Value of an Investment: FVn Investment Amount (FVIFk,n)

InvestmentA FV20 PV FVIF5%, 20 yrs.

FV20 $200 2.653FV20 $530.60Calculator solution: $530.66

B FV7 PV FVIF8%, 7 yrs.

FV7 $4,500 1.714FV7 $7,713Calculator solution: $7,712.21

C FV10 PV FVIF9%, 10 yrs.


D FV12 PV FVIF10%, 12 yrs.


E FV5 PV FVIF11%, 5 yrs.


10. This is a future value equation. Calculate the future value of $10,000 using a FVIF1%, 24 periods value. $10,000 1.270 $12,700.


11. Future Value of an Annuity Investment: FVAk,n Annual Deposit FVIFAk,n

InvestmentA FVA8%,10 yrs. $2,500 14.487

$36,217.50Calc. Sol’n. $36,216.41

B FVA12%,6 yrs. $500 8.115 $4,057.50

Calc. Sol’n. $4,057.59

C FVA20%,5 yrs. $1,000 7.442 $7,442

Calc. Sol’n. $7,441.60

D FVA6%,8 yrs. $12,000 9.897 $118,764

Calc. Sol’n. $118,769.61

E FVA14%,30 yrs. $4,000 356.778 $1,427,112

Calc. Sol’n. $1,427,147.39

(FVIFA from Appendix A, Table A.2)

12. Future value of an annuity of $1,000 for five years at 6%. $1,000 5.637 $56,370.

13. The least you would accept for each investment is its future value at the end of six years:

(a) Future Value of an Investment: FVn Investment Amount (FVIFk,n)

Calc. Sol’n $8,385.50(b) Future Value of an Annuity Investment:

Calc. Sol’n. $15,046.67(c) FV of $3,000 at 9% for 6 years FVA of $1,000 deposit at 9% at end of each of the next five

years:(1) FV9%,6 yrs. $3,000 1.677 (From App. A, Table A.1)

$5,031Calc. Sol’n $5,031.30

(2) FVA9%,6 yrs. $1,000 7.523 (From App. A, Table A.2) $7,523

Calc. Sol’n. $7,523.33(3) Total $5,031 $7,523

$12,554


(d)

YearEnd-of Year

Deposit

Numberof Years

to Compound FVIF, 9%FutureValue

1 $900 5 1.539 $1,385.103 900 3 1.295 1,165.505 900 1 1.090 981.00

Total FV $3,531.60

14. Present Value: PV FVn (PVIFk,n)

Investment FV PVIFPresentValue

CalculatorSolution

A PV12%, 4 yrs.* $7,000 .636

$4,452 $4,448.63

B PV8%, 20 yrs.* $28,000 .215

$6,020 $6,007.35

C PV14%, 12yrs.* $10,000 .208

$2,080 $2,075.59

D PV11%, 6yrs. $150,000 .535

$80,250 $80,196.13

E PV20%, 8yrs $45,000 .233

$10,485 $10,465.56

PVIF from Table A.3, Appendix A.

15. 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 6%:

16. You are trying to find the present value of $1,000 in 8 years using an 8% discount rate. PVIF 8%, 8 periods $1000 0.540 $1000 $540. The price of the bond is lower at 8%. This is because it is discounted at a higher rate (8% vs. 6%).

17. This is a present value question. Calculate the present value of $10,000 using a PVIF 3%, 10 periods value. $10,000 0.744 $7,440.


18.

IncomeStream

End ofYear Income PVIF, 12%

PresentValue

A 1 $2,200 0.893 $1,9652 3,000 0.797 2,3913 4,000 0.712 2,8484 6,000 0.636 3,8165 8,000 .567 4,536

$15,556Calculator solution $15,555.51

B 1 $10,000 0.893 $ 8,930 2–5 5,000 2.712* 13,560

6 7,000 .507 3,549$26,039

Calculator solution $26,034.59

C 1–5 $10,000 3.605** $36,0506–10 8,000 2.045*** 16,360

$52,410Calculator Solution $52,411.34

* Sum of PV factors for years 2–5.PVIF from Table A.3, Appendix A.** PVIFA for 12%, 5 years*** (PVIFA for 12%, 10 years) – (PVIFA for 12%, 5 years)PVIFA from Table A.4, Appendix A.

19. (a)

IncomeStream

End ofYear Income PVIF,

15%

PresentValue

A 1 $4,000 0.870 $3,4802 3,000 0.756 2,2683 2,000 0.658 1,3164 1,000 0.572 572


B 1 $1,000 0.870 $8702 2,000 0.756 1,5123 3,000 0.658 1,9744 4,000 0.572 2,288



(b) Income Stream A, with a present value of $7,636, is higher than Income Stream B’s present value of $6.644 because the larger cash inflows occur in A in the early years when their present value is greater. The smaller cash flows are received further in the future.


20. The present value of an investment with annual income streams is calculated using the formula: PVAn

Annual returns (PVIFAk,n)

Investment Calculation Present Value Calc. Sol’nA PVA7%, 3 yrs. $1,200 2.624 $3,148.80 $3,149.18B PVA12%, 15

yrs

5,500 6.811 37,460.50 37,459.75

C PVA20%, 9

yrs.

700 4.031 2,821.70 2,821.68

D PVA5%, 7 yrs. 14,000 5.786 81,004.00 81,009.23E PVA10%, 5 yrs 2,200 3.791 8,340.20 8,339.73

21. Find the present value of the annuity and compare it to the lump sum payment. PVIVA 8%, 20 periods $1,000,000 9.818 $9,818,000. Since this is less than $15 million, you should take the $15 million today.

22. (a) Present value of $500 to be received in four years at an 11 percent discount rate:

PVIF from Table A.3, Appendix A.(b) The present value of the income from Stream A is the present value of an annuity.

PVIFA from Table A.4, Appendix A.The present value at the start of 2006 of the income from Stream B is the present value of a mixed stream—the present value of each benefit summed.

(1) (2) (3)(1) (2)

Year Benefit PVIF, 9% Present Value2006 $140 0.917 $128.382007 120 0.842 101.042008 100 0.772 77.202009 80 0.708 56.642010 60 0.650 39.002011 40 0.596 23.842012 20 0.547 10.94

Total $437.04


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 in Stream B are worth $437.04. The difference is attributable to the fact that Stream B has larger benefits or cash flows earlier, thereby causing its present value at the 9 percent rate to be higher than Stream A.


23. The analysis of Terri’s investment opportunities uses the formula:

FVn PV (FVIFk,n)

Investment Calculation DecisionA $30,000 $18,000 FVIFk,5

yrs.

1.667 FVIFk,5 yrs.

10% < k < 11% Invest

B $3,000 $600 FVIFk, 20 yrs.

5 FVIFk, 20

8% < k < 9% Forgo

C $10,000 $3,500 FVIFk, 10

yrs.

2.857 FVIFk, 10 yrs.

11% < k < 12% Invest

D $15,000 $1,000 FVIFk, 40

yrs.

15 FVIFk, 40 yrs.

7% < k < 8% Forgo


An alternative approach accurately answering the same questions would be the calculation of the present value of cash inflows and comparing the results to the investment’s cost.

Investment A

$30,000 0.621 $18,630 (using PVIF10%,5 years)Cost of Investment $18,000Return exceeds cost—Invest

Investment B

$3,000 0.149 $447 (using PVIF10%,20 years)Cost of Investment $600Cost exceeds return—forgo

Investment C

$10,000 0.386 $3,860 (using PVIF10%,10 years)Cost of Investment $3,500Return exceeds cost—Invest

Investment D

$15,000 0.022 $330 (using PVIF10%,40 years)Cost of Investment $18,000Cost exceeds return—forgo

24.

IncomeStream

End ofYear Income PVIF, 17%, n

Present Value

A 1 $2,500 0.855 $2,137.502 3,500 0.731 2,558.503 4,500 0.624 2,808.004 5,000 0.534 2,670.005 5,500 0.456 2,508.00

Total PV $12,682.00Calculator solution $12,680.08

B 1 $4,000 0.855 $3,420.002 3,500 0.731 2,558.503 3,000 0.624 1,872.004 1,000 0.534 534.005 500 0.456 228.00

Total PV $8,612.50Calculator solution $8,610.42



At 17% required return, the present value of the income from Investment A, $12,682, is less than the $13,000 purchase price. Investment B’s present value is $8,612.50, above the purchase price. Therefore, Kent should purchase Investment B.

25. $15,000 PVIFA1%, 50 payments

$15,000 39.196$15,000/39.196 $382.69

26. Balance is the present value of the payments at 12%. Payments are $382.69 PVIFA1%, 40 periods $382.69 32.835 $382.69 $12,565.62

27. (a) Using the notation given in the chapter, the risk-free rate of interest for both Investments is:

(b) The required returns for each investment are calculated as follows:

r1 r* IP RPi or RF RPi

rA 3% 5% 3% 8% 3% 11%

rB 3% 5% 5% 8% 5% 13%

28. The risk-free rate real rate expected inflation premium. If the expected inflation premium increases by 1%, then the risk-free rate will increase by 1% to 8%.


29. Holding period return (HPR)

If the investments are held beyond a year, the capital gain (loss) component would not be realized and would likely change. Assuming they are of equal risk, Investment Y would be preferred since it offers the higher return (16.0% for Y versus 11.67% for X).

30. HPR (current income over the period capital gains)/beginning investment value

First investment: ($1.00 2.00) / $25 12%. Since this is a six-month investment, the annualized return is two times the HPR (12/6), or 24%.

Second investment: ($2.40 $3.00) / $27 20%.

The first investment provides the higher annualized return.

31. HPR ($50 ($1,000 – $950))/$950 $100/$950 10.5%

32. The present value is $5,000. The value in 10 years will be $9,000.

(a) Using present value, the yield is calculated as:

From Table A.3, Appendix A, at 10 years the PVIF of 6% is 0.558, which is very close to 0.556. The yield, then, is estimated to be 6%.

(b) If a minimum return of 9% is required, this investment would not be recommended because it only yields about 6%.

33. Using PFIV of 4%:

$65 0.962 $62.53 $70 0.925 $64.75 $70 0.889 $62.23$7,965 0.855 $6,810.08

$6,999.59


34. Interest on the investment in year “five”

(10,000 FVIF8%, 5 years) – $10,000 ($10,000 1.469) – $10,000 $4,690

Yield:

$10,000 (PVIF?%, 5 years $4690) (PVIF ?%, 6 years $14,500)

Using 12%:

(0.567 $4690) (0.507 $14,500) $2,659.23 $7,351.50 $10,010.73

Since this total is close to $10,000, the yield is approximately 12%. (Since the value is above $10,000, the true yield is slightly higher than 12%).

35.

(1) (2) (3) (4) (5)Initial Future (1) (2) Approximate

Investment Investment Value Years Discount Rate Yield*A $1,000 $1,200 5 0.833 4% (0.822)B 10,000 20,000 7 0.500 10% (0.513)C 400 2,000 20 0.200 8% (0.215)D 3,000 4,000 6 0.750 5% (0.747)E 5,500 25,000 30 0.220 5% (0.231)

* From Table A.3, Appendix A.

36. (a)

The closest PVIF for 8 years is 0.404, for a yield of 12%.

(b) Rosemary should make the proposed investment because the yield, in both the present value and approximate yield calculations, is greater than her 10% required return.

37. The yield for these investments is the discount rate that results in the stream of income equaling the initial investment.

Investment A: Using the present value of an annuity formula:

Looking at Table A.4, Appendix A, the closest factor for five years occurs at 14% (3.433); therefore, this investment yields about 14%.


Investment B: It is necessary to try several different discount rates to determine the yield for Investment B. One way to estimate a starting point is to use the average annual income in the formula used in Part A and adjusting it based on whether the larger cash flows are received in the earlier or later years. The Internal Rate of Return (IRR) function on a business calculator makes the task easier.

The closest interest rate to 3.167 in Table A.4, Appendix A, is 17%. Because the larger cash flows are received in the later years, 16% is a good starting point.

(1) (2) (3) (4) (5)(1) (4)

Year Income PVIF, 16% PV at 16% PVIF, 15% PV at 15%

1 $2,000 0.862 $1,724.00 0.870 $1,740.002 2,500 0.743 1,857.50 0.756 1,890.003 3,000 0.641 1,923.00 0.658 1,974.004 3,500 0.552 1,932.00 0.572 2,002.005 4,000 0.476 1,904.00 0.497 1,988.00

PV of Income $9,340.50 $9,594.00 Calculator Solution $9,341.49 $9,591.88

The discount rate that results in a present value closest to $9,500 is 15%.Calculator solution for IRR 15.36

38. (a) Using the same technique as shown in the prior question, we find that 7% is a possible discount rate. Because the larger cash flows occur in the early years, 8% is a good starting point.

(1) (2) (3) (4) (5)(1) (2) (1) (4)

Year Income 8% PVIF PV at 8% 9% PVIF PV at 9%1 $6,000 0.926 $5,556 0.917 $5,5022 3,000 0.857 2,571 0.842 2,5263 5,000 0.794 3,970 0.772 3,8604 2,000 0.735 1,470 0.708 1,4165 1,000 0.681 681 0.650 650

PV of Income $14,248 $13,954

The discount rate that results in a present value closest to $14,000 is 9%.Calculator solution for IRR 8.85%.

(b) Elliott should not make the proposed investment because the yield in the present value calculation is less that the 11% required return.


39.

(4) (5)End of (1) (2) (3) (1) (2) (1) (4)Year Income 10% PVIF PV at 10% 11% PVIF PV at 11%

$0 1 –$1000 1 –$10002006 140 0.909 127.26 0.901 126.142007 120 0.826 99.12 0.812 97.442008 100 0.751 75.1 0.731 73.102009 80 0.683 54.64 0.659 52.722010 60 0.621 37.26 0.593 35.582011 40 0.564 22.56 0.535 21.42012 1220 0.513 625.86 0.482 588.04

PV of Income $41.80 –$5.58

The Yield is very close to 11% on this investment.

Since the yield of 11% is greater than the minimum required return of 9.0%, the investment is recommended. This project would result in positive Net Present Value to the investor.

40. Growth rates are calculated using the present value formula PV FVn PVIFk,n

Investment

A n 2005 – 1991 4PV FV4 PVIFk,4 yrs.

$5.00 $8.00 PVIFk,4 yrs

0.625 PVIFk,4yrs

12% < k < 13%Calculator Solution 12.47%

B n 2005 – 1996 9PV FV9 PVIFk,9 yrs.

$1.50 $2.28 PVIFk,9 yrs.

0.658 PVIFk,9yrs.

4% < k < 5%Calculator solution 4.76%

C n 2005 – 1999 6PV FV6 PVIFk,6 yrs.

$2.50 $2.90 PVIFk,6 yrs.

0.862 PVIFk,6yrs.2% < k < 3%Calculator solution 2.50%

41. 2004 – 1997 7 years

$1 FVIF ?%, 7 years $2.21FVIF $2.21/$1 2.21FVIF 12%, 7 years 2.211, so the yield is about 12%


42. 2004 – 2000 4 years

350 FVIF ?%, 4 years, 441.7FVIF ?%, 4 years 441.7/350 1.262FVIF 6%, 4 years 1.262, so the yield is about 6%

43. (a) Investment A, with returns that vary widely—from 1% to 26%—appears to be more risky than Investment B, whose returns vary from 8% to 16%.

(b)

Investment A:

(1) (2) (3) (4)Return Average (1) – (2) (3)2

Year r i Return, r ri – r (ri – r)2

2001 19% 12% 7% 49%2002 1 12 –11 1212003 10 12 –2 42004 26 12 14 1962005 4 12 –8 64

434

Investment B:

(1) (2) (3) (4)Year Return Average (1) – (2) (3)2

r i Return, r ri – r (ri – r)2

2001 8% 12% –4% 16%2002 10 12 –2 42003 12 12 0 02004 14 12 2 42005 16 12 4 16

40


(c) Investment A, with a standard deviation of 10.42, is considerably more risky than Investment B, whose standard deviation is 3.16. This confirms the conclusions reached in Part A.

(d) Because the real benefit of calculating the coefficient or variation is in comparing investments that have different average returns, the standard deviation is not improved upon.

44. Since Investment A returns are much more variable, Investment A should carry a higher risk premium. Investment A’s required return is 14%.

Solutions to Case Problems

Case 4.1 Solomon’s Decision

This case introduces the student to the concepts of opportunity cost and required rate of return. It further requires students to compute present values and select investments using the “reasonable return” approach discussed in the chapter.

(a) Using the present value technique for the equally risky projects, we select the one with the highest present value (net of the purchase price of the investment—$1,050 for both investments) if the present value exceeds the investment cost.Present value of A: This is the present value of a nine-year $150 annuity plus 1 payment in year 10 of $1,150, so we will use the present-value interest factor for an annuity from Table A.4, Appendix A, and the present-value interest factor for a dollar from Table A.3.

Present value of B: This is the present value of a mixed stream, so we use the present-value interest factors for one dollar from Table A.3.

(1) (2) (3)(1) (2)

Year Benefit PVIF (12%) Present Value

2006 $100 0.893 $89.302007 150 0.797 119.552008 200 0.712 142.402009 250 0.636 159.002010 300 0.567 170.102011 350 0.507 177.452012 300 0.452 135.602013 250 0.404 101.002014 200 0.361 72.202015 150 0.322 48.30

Total PV $1,214.90

Each investment is acceptable because its present value is greater than its initial cost of $1,050. However, the present value of B is higher than A, and since both cost $1,050, B would be the preferred investment. If both are equally risky, B has a higher return ($1,214.90) for its level of risk than A ($1,169.50).


(b) For projects of unequal risk, we must evaluate each at its required rate of return (adjusted for its level of risk). Using a 16 percent interest factor (from Table A.3, Appendix A), the present value of Investment B is:

(1) (2) (3)(1) (2)


2006 $100 0.862 $86.202007 150 0.743 111.452008 200 0.641 128.202009 250 0.552 138.002010 300 0.476 142.802011 350 0.410 143.502012 300 0.354 106.202013 250 0.305 76.252014 200 0.263 52.602015 150 0.227 34.05

Total PV $1,019.25

Since $1,019.25, the present value of investment B, does not remain greater than its cost ($1,050), it is no longer an acceptable investment. Investment A is the only one of the two earning a reasonable (i.e., acceptable) return.

(c) Since Investment A was acceptable (had a positive present value) at a discount rate of 12 percent, its yield must be more than 12 percent. Investment B was acceptable at a discount rate of 12 percent, so its yield is also greater than 12%. However, Investment B’s yield is between 12 and 16 percent. The present value of B at 12 percent equaled $1,214.90, and at 16 percent it was $1,019.25. Since the present value at 16 percent is closer to the cost than the present value at 12 percent, the actual yield is closer to 16 percent. It appears to be about 15.25 percent. (The actual yield computed using a calculator is 15.31 percent.)

(d) Investment A:Present value technique:Step 1: Find the present value of $150 income for 9 years using the present value of an annuity formula.Step 2: Find the PV of $1,150 in year 10.Step 3: Add the two amounts to get the total PV. Try different discount rates to determine the yield (IRR). Because we know from question 1 that the present value at 12% is $1,196.50, just above the initial investment amount, start with 13%:


Try 14%:

Yield 14%

Investment B: Because the present value of the income from Investment B is $1,019.25 at 16% (Question 2), try a 15% discount rate:

(1) (2) (3)(1) – (2)


2006 $100 0.870 $87.002007 150 0.756 113.402008 200 0.658 131.602009 250 0.572 143.002010 300 0.497 149.102011 350 0.432 151.202012 300 0.376 112.802013 250 0.327 81.752014 200 0.284 56.802015 150 0.247 37.05

Total PV $1,063.70

Yield is 15% (Calculator solution: 15.31%)

(e) Because Investment A is acceptable at a 12 percent discount rate while Investment B is unacceptable at a 16 percent discount rate, Investment A is recommended. Although it seems as if investment B is being penalized with a higher discount rate, due to its greater risk it must earn a 16 percent yield to be acceptable.

(f) His initial investment of $50 would have grown to $81.45 at the end of the ten years assuming that he makes no withdrawals from his savings account. This is calculated using a future-value interest factor from Table A.1, as illustrated below:

FV5%, 10yrs. $50 1.629 $81.45


Case 4.2 The Risk-Return Tradeoff: Molly O’Rourke’s Stock Purchase Decision

The title of this case clearly states its objective. It requires students to review and apply the concept of the risk-return trade-off.

(a)

HPR for Stock X:

(1) (2) (3) (4)(2) – (3)

(5)[(1) (4)]/(3)

YearCurrentIncome

EndingPrice

BeginningPrice

CapitalGain HPR

1996 $1.00 $22.00 $20.00 $2.00 15.00%1997 1.50 21.00 22.00 –1.00 2.271998 1.40 24.00 21.00 3.00 20.951999 1.70 22.00 24.00 –2.00 –1.252000 1.90 23.00 22.00 1.00 13.182001 1.60 26.00 23.00 3.00 20.002002 1.70 25.00 26.00 –1.00 2.692003 2.00 24.00 25.00 –1.00 4.002004 2.10 27.00 24.00 3.00 21.252005 2.20 30.00 27.00 3.00 19.26

Average (expected) HPR for stock X 11.74%

HPR for Stock Y:

(1) (2) (3) (4)(2) – (3)

(5)[(1) (4)]/(3)

YearCurrentIncome

EndingPrice

BeginningPrice

CapitalGain HPR

1996 $1.50 $20.00 $20.00 $0.00 7.50%1997 1.60 20.00 20.00 0.00 8.001998 1.70 21.00 20.00 1.00 13.501999 1.80 21.00 21.00 0.00 8.572000 1.90 22.00 21.00 1.00 13.812001 2.00 23.00 22.00 1.00 13.642002 2.10 23.00 23.00 0.00 9.132003 2.20 24.00 23.00 1.00 13.912004 2.30 25.00 24.00 1.00 13.752005 2.40 25.00 25.00 0.00 9.60

Average (expected) HPR for stock Y 11.14%


(b)

Investment X:

(1) (2) (3) (4)Return Average (1) – (2) (3)2

Year ri Return, r ri – r (ri – r)2

1996 15.00% 11.74% 3.26% 10.63%1997 2.27 11.74 –9.47 89.681998 0.95 11.74 9.21 84.821999 –1.25 11.74 –12.99 168.742000 13.18 11.74 1.44 2.072001 20.00 11.74 8.26 68.232002 2.69 11.74 –9.05 81.902003 4.00 11.74 –7.74 59.912004 21.25 11.74 9.51 90.442005 19.26 11.74 7.52 56.55

712.97

Coefficient of variation 8.9%/11.74% 0.76

Investment Y:

(1) (2) (3) (4)Return Average (1) – (2) (3)2

Year ri Return, r ri – r (ri – r)2

1996 7.50% 11.14% –3.64% 13.25%1997 8.00 11.14 –3.14 9.861998 13.50 11.14 2.36 5.571999 8.57 11.14 –2.57 6.602000 13.81 11.14 2.67 7.132001 13.64 11.14 2.50 6.252002 9.13 11.14 –2.01 4.042003 13.91 11.14 2.77 7.672004 13.75 11.14 2.61 6.812005 9.60 11.14 –1.54 2.37

Coefficient of variation 2.78%/11.14% 0.25


(c) and (d)

Summary Statistics:

Investment X Investment YExpected Return 11.74% 11.14%Standard Deviation 8.82 2.78Coefficient of Variation 0.76 0.25

Comparing the expected returns calculated in question 1, Stock X provides a return of 11.74 percent, which is only slightly above the expected return of Y (11.14 percent). Whether the higher return on Stock X is sufficient to compensate for the higher risk would be determined by the “price of risk” in the financial markets.

As can be seen, Standard Deviation and Coefficient of variation of Stock X is higher than the corresponding values of stock Y. This might be a signal to prefer stock Y. But these numbers have to be interpreted with caution as one of the above stocks is to be added to a well diversified portfolio.

This part of the case problem anticipates the use of beta and CAPM, which will be covered in the next chapter on modern portfolio concepts. The calculation of required return provides an objective approach to assess the investment risk.

Using the capital asset pricing model, the required return on each stock is as follows:

Capital Asset Pricing Model: ri Rf [bI (rm RF)]

Investment r Rf [bI (rm RF)]

A 11.8% 7% [1.6 (10% 7%)B 10.3% 7% [1.1 (10% 7%)

From the calculations in question 1, Stock X has an expected return of 11.74% and a required return of 11.8%. On the other hand, Stock Y has an expected return of 11.14% and a required return of only 10.3%.

So while we concluded that it would be difficult to make a choice between X and Y because the additional return on X may or may not provide the needed compensation for the extra risk, we see that by calculating a required rate of return, it is easy to reject X and invest in Y. The required return on Stock X is 11.8%, which is higher than expected return (11.74%); therefore Stock X is unattractive. Using the same logic, Stock Y is a better investment vehicle.

Clearly, Molly is better off using beta rather than a strictly subjective approach to assess investment risk. The use of beta and CAPM allows risk to be quantified and converted into a required return that can be compared to the expected return to draw a definitive conclusion about investment acceptability. The student will meet these concepts in the next chapter.

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