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Chapter 5Modern Portfolio Concepts

.1 Outline

Learning Goals

I. Principles of Portfolio Planning

A) Portfolio Objectives

B) Portfolio Return and Standard DeviationC) Correlation and Diversification

1. Correlation

2. Diversification

3. Impact on Risk and Return

D) International Diversification

1. Effectiveness of International Diversification

2. Methods for International Diversification

3. Benefits of International Diversification

Concepts in Review

II. The Capital Asset Price Model (CAPM)

A) Components of Risk

B) Beta: A Popular Measure of Risk

1. Deriving Beta

2. Interpreting Beta

3. Applying Beta

C) The CAPM: Using Beta to Estimate Return

1. The Equation

2. Historical Risk Premiums3. The Graph: The Security Market Line (SML)

4. Some Closing Comments

Concepts in Review

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80 Gitman/Joehnk • Fundamentals of Investing, Ninth Edition

III. Traditional Versus Modern Portfolio Theory

A) The Traditional Approach

B) Modern Portfolio Theory

1. The Efficient Frontier

2. Portfolio Betas

a. Risk Diversificationb. Calculating Portfolio Betas

c. Using Portfolio Betas

d. Interpreting Portfolio Betas

3. The Risk-Return Tradeoff: Some Closing Comments

C) Reconciling the Traditional Approach and MPT

Concepts in Review

IV. Constructing a Portfolio Using an Asset Allocation Scheme

A) Investor Characteristics and Objectives

B) Portfolio Objectives and Policies

C) Developing an Asset Allocation Scheme

1. Approaches to Asset Allocation

a. Fixed Weightings

b. Flexible Weightings

c. Tactical Asset Allocation

2. Asset Allocation Alternatives

3. Applying Asset Allocation

Concepts in Review

Summary

Putting Your Investment Know-How to the Test

Discussion Questions

Problems

Case Problems

5.1 Traditional Versus Modern Portfolio Theory: Who’s Right?

5.2 Susan Lussier’s Inherited Portfolio: Does It Meet Her Needs?

Excel with Spreadsheets

Trading Online with OTIS

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Chapter 5 Modern Portfolio Concepts 81

.2 Key Concepts

1. The concept of a portfolio, the importance of portfolio objectives, and the calculation of the returnand standard deviation of a portfolio.

2. The effect of positive and negative correlation and diversification on portfolio return and risk.

Demonstrating that diversification’s advantages are greater where correlation is lower.

3. The aspects of international diversification including effectiveness, methods, and benefits.

4. Modern risk concepts and the use of beta to measure the relevant risk in order to assess potentialinvestments. Contrasting CAPM and APT.

5. The two basic approaches to portfolio management—traditional portfolio management versusmodern portfolio theory (MPT).

6. The role of investor characteristics and objectives and portfolio objectives in planning and building aportfolio.

7. Procedure for building a portfolio using an asset allocation scheme that considers investorcharacteristics and objectives as inputs to the establishment of portfolio objectives and policies.

.3 Overview

This chapter discusses the fundamentals of planning and building a portfolio, with special attention paid toreturn correlation and systematic risk.

1. The chapter begins with the definition and possible objectives of a portfolio. The instructor shouldstress the concept of a risk-return tradeoff—in order to get more return, an investor must bear morerisk. The chapter emphasizes that one of the major benefits of owning a portfolio is risk reduction

through diversification. The student learns to calculate portfolio returns and the standard deviation ofa portfolio.

2. Using correlation, a statistical measure of the relationship between securities in a portfolio, anddiversification to reduce risk and increase return are discussed.

3. The opportunities for international investment are numerous, thus the effectiveness, methods andbenefits of international diversification are discussed.

4. Beta is a modern measure of risk. The graphic derivation of beta is demonstrated and can be used todiscuss the interpretation and use of beta. The instructor may wish to indicate some sources forobtaining beta and demonstrate the computation of the required return in class.

5. While beta is a measure of risk, the link between risk and return is made using beta and the capitalasset pricing model (CAPM). The CAPM is graphically presented by the security market line (SML).Understanding this model should enhance the student’s ability to grasp the true significance of the risk-return trade-off among assets. In addition, knowledge of differing investor risk preferences—risk-indifferent, risk-averse, and risk-taking—should further enhance their understanding of the risk-return trade-off.

6. Special attention is paid to the varying risk premiums across asset classes and how arbitrage pricingtheory might be used to explain risk premium differences.

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7. The next section compares traditional portfolio management with modern portfolio theory. Thetraditional approach to portfolio construction emphasizes balancing the portfolio by selectinginvestments from a broad cross-section of industries, while modern portfolio theory relies on suchstatistical concepts as expected returns, standard deviation, correlation, portfolio betas, and R2. Itmight be helpful to note that MPT postulates a specific mathematical relationship between risk andreturn. The beta equation shows such a relationship, where the bi measures the beta coefficient (thenon-diversifiable or systematic risk) for company i. The risk-return tradeoff bears the samerelationship.

8. The fourth section of the chapter provides basic guidelines for building a portfolio using an assetallocation scheme. In addition to portfolio objectives, an individual’s level and stability of income,family factors, net worth, experience and age, and disposition toward risk are key factors to considerduring portfolio construction. The instructor should mention that tax and liquidity considerationsshould also be taken into account when constructing a portfolio. The logic as well as generalprocedures involved in developing an asset allocations scheme consistent with the investor’s needs isdemonstrated. All these discussions focus on the chapter’s key idea: the individual investor shouldassemble a portfolio that will yield maximum expected returns commensurate with the level of riskhe or she is willing to assume.

.4 Answers to Concepts in Review

1. A portfolio is simply a collection of investment vehicles assembled to meet a common investmentgoal. An efficient portfolio is a portfolio offering the highest expected return for a given level of riskor the lowest level of risk for a given level of expected return.

In trying to create an efficient portfolio, an investor should be able to put together the best portfoliopossible, given his risk disposition and investment opportunities. When confronted with the choicebetween two equally risky investments offering different returns, the investor would be expected tochoose the alternative with the higher return. Likewise, given two investment vehicles offering thesame returns but differing in risk, the risk averse investor would prefer the vehicle with thelower risk.

2. The return of a portfolio is calculated by finding the weighted average of returns of the portfolio’scomponent assets:

1

n

p j j

j

r w r =

= ×∑

where n = number of assets, w j = weight of individual assets, and r j = average returns.The standard deviation of a portfolio is not the weighted average of component standard deviations;the risk of the portfolio as measured by the standard deviation will be smaller. It is calculated byapplying the standard deviation formula (Equation 4.10a) to the portfolio assets, rather than just thereturns for one asset:

2

1

( ) ( 1)n

p p

i

s r r n=

= − ÷ −

∑

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3. Correlation refers to the statistical measure of the relationship, if any, between a series of numbers.The correlation between asset returns is important when evaluating the effect of a new asset on theportfolio’s overall risk. Once the correlation between asset returns is known, the investor can choosethose that, when combined, reduce risk.

(a) Returns on different assets moving in the same direction are positively correlated ; if they movetogether exactly, they are perfectly positively correlated .

(b) Negatively correlated returns move in opposite directions. Series that move in exactly oppositedirections are perfectly negatively correlated . (See Figure 5.1)

(c) Uncorrelated returns have no relationship to each other and have a correlation coefficient ofclose to zero.

4. Diversification is a process of risk reduction achieved by including in the portfolio a variety ofvehicles having returns that are less than perfectly positively correlated with each other.Diversification of risk in the asset selection process allows the investor to reduce overall risk bycombining negatively correlated assets so that the risk of the portfolio is less than the risk of theindividual assets in it. Even if assets are not negatively correlated, the lower the positive correlationbetween them, the lower the resulting risk.

5. Combining assets with high positive correlation increases the range of portfolio returns; combiningassets with high negative correlation reduces the range of portfolio returns. When negativelycorrelated assets are brought together through diversification, the variability of the expected returnfrom the resulting combination can be less than the variability or risk of the individual assets. Whenone asset has high returns, the other’s returns are low and vice versa. Therefore, the result ofdiversification is to reduce risk by providing a pattern of stable returns.

(a) When two assets are perfectly positively correlated, both the range of returns and of risk will bebetween the return/risk of the two assets.

(b) With two uncorrelated assets, the range of return will be between the two assets’ returns and therisk, between the risk of the most risky and the risk of the least risky, but greater than zero.

(c) The range of return for two perfectly negatively correlated assets will be between the returns ofthe two assets. The range of risk will be between the risk of the most risky and zero.

6. International diversification can provide the benefits of higher returns and reduced risk. However,whether an individual investor ultimately benefits from this kind of diversification depends on factorssuch as resources, goals, sophistication, and psychology of the investor.

There are several methods for achieving international portfolio diversification. Internationaldiversification can be achieved by investing directly abroad in either U.S. dollars or in foreigncurrencies securities. International diversification can also be achieved domestically in the U.S. byinvesting in foreign companies listed and sold on U.S. exchanges or over the counter.

Because investing abroad is less convenient, more expensive, and riskier than investing domestically,investors should avoid directly investing in foreign-currency-denominated instruments. Investors willprobably do better choosing foreign investment vehicles available in the U.S. such as international

mutual funds and ADRs.Some of the newer international investment strategies involve diversifying by country or regionrather than in a continent. Others believe in investing in U.S. as well as foreign multinationalcorporations. Still another strategy calls for investing in individual company shares. Some evenadvocate mutual funds in a global industry sector.

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7. (a) Diversifiable (unsystematic) risk the part of an investment’s risk that the investor can eliminatethrough diversification. This kind of risk is also called firm-specific risk. This kind of risk can beeliminated by holding a diversified portfolio of assets.

(b) Nondiversifiable (systematic) risk refers to events or forces such as war, inflation, or politicalevents and effects all investments. Nondiversifiable risk, which cannot be eliminated by holdinga diversified portfolio, is considered the only relevant risk. This is because the “smart” investor is

expected to remove unsystematic risk through diversification. Hence the market will reward aninvestor for only the systematic risk.

8. Beta is a measure of systematic or non-diversifiable risk. It is found by relating the historical returnson a security with the historical returns for the market. In general, the higher the beta, the riskier thesecurity.

The relevant risk measured by beta is the nondiversifiable risk of an investment. It is relevant sinceany intelligent investor can eliminate unsystematic risk by holding a diversified portfolio ofsecurities.

The market return is typically measured by the average return of all (or a large sample of) stocks.Usually the Standard & Poor’s 500 stock composite index or some other broad index is used tomeasure market return.

The beta for the overall market is the benchmark beta—it is 1.0 and other betas are viewed in relationto this benchmark. The positive or negative sign on a beta indicates whether the stock’s returnchanges in the same direction as the general market (positive beta) or in the opposite direction(negative beta). In terms of the size of beta, the higher the stock’s beta, the riskier the security. Stockswith betas greater than 1.0 are more responsive to changes in market returns, and stocks with betasless than 1.0 less responsive than the market.

9. Betas are typically positive and range in value between 0.5 and 1.75. Most securities have positivebetas. This means that the returns on most stocks move in a direction (though not in magnitude)similar to the market as a whole. This is quite intuitive to understand as macro economic factorsaffect most securities in a similar manner. Hence the betas tend to be positive.

10. The capital asset pricing model (CAPM) links together risk and return to help investors makeinvestment decisions. It describes the relationship between required return and systematic risk, asmeasured by beta. The equation for the CAPM is:

[ ( )]i F m F

r R b r R= + × −

As beta increases, so does the required return for a given investment. The risk premium,

[b × (r m – RF )], is the amount by which return increases above the risk-free rate to compensate for theinvestment’s nondiversifiable risk, as measured by beta. Risk premiums range from over thirteenpercent for small company stocks to under two percent for long-term government bonds. Investors inTreasury bills do not earn a risk premium.

The security market line (SML) is a graphic representation of the CAPM and shows the required

return for each level of beta.

11. CAPM provides only a rough forecast of future returns, because it is based on historical data. Thoseusing CAPM typically adjust return forecasts for their expectations of future returns.

Arbitrage pricing theory (APT) suggests that the market risk premium is better explained by anumber of underlying factors that influence share price. While beta measures systematic risk, APTidentifies systematic factors. As such, beta can be derived from the influences described by APT.Investor attention remains focused on the CAPM because it provides a simply means to link risk andreturn.

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12. Traditional portfolio management emphasizes “balancing” the portfolio. The traditional portfolioincludes a wide variety of stocks and/or bonds which emphasize interindustry diversification. Thesecurities selected are usually high-quality and issued by stable, established companies and/orinstitutions. Traditional portfolio managers typically invest in well-established companies for avariety of reasons. First, well-established companies probably will continue to be successful in thefuture, i.e., there is less risk. Second, the securities of these firms are more liquid and are available inlarge quantities. Since a security that is readily marketable has low marketability risk, traditionalportfolio managers like to hold this type of security. Third, it is easier to convince clients to invest inportfolios made up of well-known corporate securities.

13. Modern portfolio theory (MPT) is based on the use of statistical measures including mathematicalconcepts such as correlation (of rates of return) and beta. Combining securities with negative or lowpositive correlation reduces risk through statistical diversification. By analyzing securities usingcorrelation and beta (which is a statistical measure of the relative volatility of a security or portfolioreturn as compared to a broadly derived measure of stock market return), the investor attempts tocreate a portfolio with minimum diversifiable risk that provides the highest return for a given level ofacceptable diversifiable risk.

The feasible or attainable set of all possible portfolios refers to the risk-return combinations

achievable with all possible portfolios. It is derived by first calculating the return and risk of allpossible portfolios and plotting them on a set of risk-return axes (see Figure 5.7).

14. The efficient frontier is the site of all efficient portfolios (those with the best risk-return tradeoff). Allportfolios on the efficient frontier are preferable to the others in the feasible or attainable set.

Plotting an investor’s utility function or risk indifference curves on the graph with the feasible orattainable set of portfolios will indicate the investor’s optimal portfolio—the one at which anindifference curve meets the efficient frontier. This represents the highest level of satisfaction for thatinvestor.

15. The two kinds of risk associated with a portfolio are diversifiable (or unsystematic) risk andnondiversifiable (or systematic) risk. Diversifiable (unsystematic) risk is the risk unique to each

investment vehicle that can be eliminated through diversification, by selecting stocks possessingdifferent risk-return characteristics. Nondiversifiable risk is possessed by every investment vehicle. Itis the risk that general market movements will alter a security’s return. One cannot eliminatenondiversifiable risk through diversification. It is this type of risk that represents the contribution ofan asset to the risk of the portfolio and is therefore the relevant risk. The total risk of a portfolio is thesum of its nondiversifiable and diversifiable risk. A fully diversified portfolio will possess onlynondiversifiable risk.

16. Beta is an index that measures the expected change in a security’s or portfolio’s return relative to achange in the market return. For example, if a security has a beta of 2.0 and the market return movesup by 10 percent, the security return increases by 2.0 times that amount—that is, 20 percent. Betameasures only the nondiversifiable, or relevant, risk of a security or portfolio. Typical beta values fall

between 0.5 and 1.75. The portfolio beta is the weighted average of the betas of the individual assetsin the portfolio.

17. The coefficient of determination (R2) is used to statistically identify the relevance of a betacoefficient. It indicates the percentage of an individual security’s return that can be explained by itsrelationship with the market return. Securities that are highly correlated with the market will havebetas with high R2 values. Likewise, if securities are combined into well-diversified portfolios, theexplanatory power of the portfolio’s beta coefficient (its R2) will be higher.

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18. Modern portfolio theory requires the use of sophisticated computer programs and a mathematicalfacility that is beyond the reach of the average individual investor. On the other hand, the traditionalapproach seems very subjective and does not have strong theoretical underpinnings. However, bothstrategies require diversification in order to ensure satisfactory performance. The text suggests afour-stage procedure for use by the individual investor in order to reconcile these approaches:

(1) Determine how much risk he or she is willing to bear.

(2) Seek diversification among different types of securities and across industry lines, payingattention to the correlation of returns between securities.

(3) Using beta, assemble a diversified portfolio consistent with an acceptable level of risk.

(4) Evaluate alternative portfolios in order to make sure that the chosen portfolio provides thehighest return for the given level of acceptable risk.

19. An investor’s personal characteristics are important inputs to an investment policy. In particular,there are five factors to consider: (1) Level and stability of income; (2) family factors; (3) net worth;(4) investor experience and age; and (5) investor disposition toward risk. The first factor determineswhether or not the investor wants high dividend paying stocks or stocks with good capitalappreciation potential. The next two reflect to what extent the investor wants to take risk. Forexample, a person with a family having moderate net worth might take less risk that an unmarried

person with a sizable net worth. The investor’s experience and age also determine whether or not theinvestor wishes to take high or low risk and whether or not the person seeks high current income orhigh capital appreciation potential. Needless to say, the investor’s disposition toward risk ultimatelydetermines the type of portfolio he or she will choose. Given an acceptable level of risk, the investorshould select that portfolio offering the highest expected return in a fashion consistent with thefactors addressed above.

20. Portfolio objectives can fall into five major categories: current income, capital preservation, capitalgrowth, tax considerations, and risk. An investor’s portfolio strategy will be guided by his or herparticular portfolio objectives, which are in turn based on his or her needs and attitudes toward risk.Normally, a person with current needs and a motive for capital preservation would choose low-beta(low-risk) securities. An investor whose main objective is capital growth would make investments

with higher risk, such as growth stocks, options, commodities and financial futures, gold, real estate,and other more speculative investments. High-income investors generally wish to defer taxes andearn investment returns in the form of capital gains. This implies a strategy of higher-risk investmentsand a longer holding period. All investors must consider the risk-return tradeoff when makinginvestment decisions. Ultimately, the amount of risk an investor is willing to take and the risk-returntradeoff will determine the kind of vehicles he or she will include in a portfolio.

21. An asset allocation scheme is an investment strategy that involves dividing one’s portfolio intovarious asset classes to preserve capital. It seeks to protect against negative developments while stilltaking advantage of positive developments. It is based on the belief that the total return of a portfoliois influenced more by the way investments are allocated than by the actual investments. Furthermore,researchers have found that asset allocation has a much greater impact on reducing total risk exposure

than picking an investment vehicle in any single category. Clearly, asset allocation is an importantaspect of portfolio management. An example of an asset allocation would be to put 30 percent of theportfolio in common stock, 50 percent in bonds, 5 percent in short-term securities, and 15 percent inreal estate.

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22. There are three basic approaches to asset allocation.

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(a) Fixed weightings involve allocating a fixed percentage of the portfolio to each of the (typically 3to 5) asset categories. Under this approach the weights do not change over time. Because ofshifting market values, the portfolio using this approach may have to be revised annually or aftermajor market moves in order to maintain the fixed percentage allocations.

(b) Flexible weightings involve periodic adjustments of the weights for each asset category basedeither on market analysis or technical analysis (i.e., market timing). The use of flexible weights is

often called strategic asset allocation. The weights under this approach are generally changed inorder to capture greater returns in a changing market.

(c) Tactical asset allocation is a sophisticated approach that uses stock index futures and bondfutures to change a portfolio’s asset allocation. When stocks seem less attractive than bonds, thisstrategy involves selling stock index futures and buying bond futures; and, when bonds seem lessattractive than stocks, the strategy results in buying stock index futures and selling bond futures.Because this approach relies on a large portfolio and the use of quantitative models for cues, it isgenerally only appropriate for large institutional investors rather than individual investors.

23. An asset allocation plan should consider the investor’s investment, savings and spending patterns,the economic outlook, tax situations, return expectations, risk tolerance, and so forth. Age will alsohave an effect; younger investors are often willing to accept greater risk than those at or near

retirement. Such plans must be formulated for the long run, stress capital preservation, and providefor periodic revision in order to maintain consistency with changing investment goals.

To decide the appropriate asset mix, investors must evaluate each asset category relative to: currentreturn, growth potential, safety, liquidity, transaction costs (brokerage fees), and potential tax savings.Frequently, mutual funds are employed to diversify within each asset category; a family of funds canbe used to permit switching among categories by phone. As an alternative to building his or her ownportfolio, an investor can buy shares in an asset allocation fund , a mutual fund that seeks to reducevolatility by investing in the right assets at the right time. These funds, like asset allocation schemes,emphasize diversification and perform at a relatively consistent level. They pass up the potential forspectacular gains in favor of predictability. Generally only those with less than about $25,000 and/orlimited time will find asset allocation funds most attractive. Those with between $25,000 and$100,000 to invest and adequate time can use mutual funds to create a workable asset allocation, and

those with more than $100,000 and adequate time can justify do-it-yourself asset allocation.

.5 Suggested Answers to Investing in Action Questions

Student-Managed Portfolios Earn Top Grades

Develop a brief proposal including an asset allocation strategy and two other parameters.

Answer:

The article mentions the fact that student-managed portfolios allocate funds to stocks and bonds. Theytypically consider both mid-cap and large-cap securities. Students are assigned to a variety of industries.At one school, students must limit higher-risk, small-cap stocks to 10% of the portfolio and hold ten of the

fifteen largest companies.

Keep Your Balance

Develop an appropriate personal allocation scheme.

Answer:

The scheme should begin with an analysis of personal investment goals and risk tolerance. Within broadasset categories, one will have to identify allocations, such as income stocks, growth stocks, and foreigncompany stocks. Once the framework is in place, make individual selections. Rebalance annually.

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.6 Suggested Solutions to Discussion Questions

Answers will vary with student responses.

.7 Solutions to Problems

1.

Beginning Value Ending Value Return %

2002 $50,000.00 $55,000.00 10.0%

2003 $55,000.00 $58,000.00 5.5%

2004 $58,000.00 $65,000.00 12.1%

2005 $65,000.00 $70,000.00 7.7%

Average 8.8%

2.

Beginning Value Ending Value Return % Difference Squared

2002 $50,000.00 $55,000.00 10.0% 1.10 1.21

2003 $55,000.00 $58,000.00 5.5% 3.30 10.89

2004 $58,000.00 $65,000.00 12.1% 3.30 10.89

2005 $65,000.00 $70,000.00 7.7% 1.10 1 .21

Average 8.8% Total 24.2

Div n –1 (3) 8.067

Sp 2.840

3. (a) Average portfolio return for each year: r p = (w L × r L) + (w M × r M )

Year

Asset L

(wL×

rL)

Asset M

(wM×

rM ) =

Expected

Portfolio Return

rp2006 (14% × 0.40 = 5.6%) + (20% × 0.60 = 12.0%) = 17.6%2007 (14% × 0.40 = 5.6%) + (18% × 0.60 = 10.8%) = 16.4%2008 (16% × 0.40 = 6.4%) + (16% × 0.60 = 9.6%) = 16.0%2009 (17% × 0.40 = 6.8%) + (14% × 0.60 = 8.4%) = 15.2%2010 (17% × 0.40 = 6.8%) + (12% × 0.60 = 7.2%) = 14.0%2011 (19% × 0.40 = 7.6%) + (10% × 0.60 = 6.0%) = 13.6%

(b) Portfolio Return:

1

17.6 16.4 16.0 15.2 14.0 13.615.467

6

n

j j

j

p

r w r n

r

=

= × ÷

+ + + + += =

∑ p

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(c) Standard Deviation: 2

) ( 1)n

p p

i=1

s (r r n= − ÷ −√ ∑

2 2 2

2 2 2

2 2 2 2 2 2

[(17.6% 15.5%) (16.4% 15.5%) (16.0% 15.5%)

(15.2% 15.5%) (14.0% 15.5%) (13.6% 15.5%) ]

6 1[(2.1%) (0.9%) (0.5%) ( 0.3%) ( 1.5%) ( 1.9%) ]

5

(4.41% 0.81% 0.25% 0.09% 2.25% 3.61%

5

p

p

p

p

s

s

s

s

= − + − + −√

+ − + − + −

−+ + + − + − + −√=

+ + + + +√=

=11.42

2.284 1.5115

√ = =√

(d) The assets are negatively correlated.

(e) By combining these two negatively correlated assets, overall portfolio risk is reduced.

4.

ASSET L Weight W × r ASSET M Weight W × r

ExpectedPortfolio Return

rp

2006 14% 60% 8.40% 20% 40% 8.00% 16.40%

2007 14% 60% 8.40% 18% 40% 7.20% 15.60%

2008 16% 60% 9.60% 16% 40% 6.40% 16.00%

2009 17% 60% 10.20% 14% 40% 5.60% 15.80%

2010 17% 60% 10.20% 12% 40% 4.80% 15.00%

2011 19% 60% 11.40% 10% 40% 4.00% 15.40%

Average Return 15.7%

Return Avg. Return Difference Squared

16.40% 15.7% 0.70 0.49

15.60% 15.7% 0.10 0.01

16.00% 15.7% 0.30 0.09

15.80% 15.7% 0.10 0.01

15.00% 15.7% 0.70 0.49

15.40% 15.7% 0.30 0.09

Sum 1.18

Div by 5 0.24

Sp (square root) 0.49

The average return is almost the same in each case (15.47 vs. 15.7), but the standard deviation ismuch lower in this portfolio because less weight is given to the more variable asset.

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5. (a) Average portfolio return:n

p j jr w k = ×∑j=1

Alternative 1: 100% Asset F

16% 17% 18% 19%17.5%

4 p

r + + +

= =

Alternative 2: 50% Asset F 50% AssetG

Year

Asset F

(wF×

rF)

Asset G

(wG×

rG) =

Portfolio

Return

rp

2006 (16% × 0.50 = 8.0%) + (17% × 0.50 = 8.5%) = 16.5%2007 (17% × 0.50 = 8.5%) + (16% × 0.50 = 8.0%) = 16.5%2008 (18% × 0.50 = 9.0%) + (15% × 0.50 = 7.5%) = 16.5%2009 (19% × 0.50 = 9.5%) + (14% × 0.50 = 7.0%) = 16.5%

66 16.5%4

pr = =

Alternative 3: 50% Asset F 50% Asset H

Year

Asset F

(wF × rF)

Asset H

(wH × rH ) =

Portfolio

Return

rp

2006 (16% × 0.50 = 8.0%)

+ (14% × 0.50 = 7.0%)

= 15.0%

2007 (17% × 0.50 = 8.5%)

+ (15% × 0.50 = 7.5%)

= 16.0%

2008 (18% × 0.50 = 9.0%)

+ (16% × 0.50 = 8.0%)

= 17.0%

2009 (19% × 0.50 = 9.5%)

+ (17% × 0.50 = 8.5%)

= 18.0%

6616.5%

4 p

r = =

(b) Standard Deviation:2

1

( ) ( 1)n

P i

i

s r r n=

= − ÷ −√ ∑(1)

2 2 2

2 2 2 2

[(16.0% 17.5%) + (17.0% 17.5%) (18.0% 17.5%) (19.0% 17.5%)]

4 1

[( 1.5%) + ( 0.5%) (0.5%) (2.5%) ]

3

3

F

F

F

F

s

s

s

s

− − + − + −√=−

− − + +√=

(2.25%) + 0.25% + 0.25% + 2.25%)√=

√ 5= = √1.667 =1.291

3

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(2)

2 2 2

2 2 2 2

[(16.5% 16.5%) (16.5% 16.5%) (16.5% 16.5%) (16.5% 16.5%)]

4 1

[(0) (0) (0) (0) ]0

3

FG

FG

s

s

− + − + − + −√=−

+ + +√= =

(3)

2 2 2 2

2 2 2 2

[(15.0% 16.5%) (16.0% 16.5%) (17.0% 16.5%) (18.0% 16.5%) ]

4 1

[( 1.5%) (0.5%) (0.5%) (1.5%) ]

3

(2.25%+ 0.25% + 0.25% + 2.25%)

3

51.667 =1.291

3

FH

FH

SH

SH

s

s

s

s

− + − + − + −√=−

− + + +√=

√=

√= = √

(c) Summary: r p: Average

Portfolio Return s p

Alternative 1 (F ) 17.5% 1.291Alternative 2 (FG) 16.5% 0Alternative 3 (FH ) 16.5% 1.291

Since the assets have different average returns, the standard deviation and the correlation patternsshould be used to determine the best portfolio. Alternative 3, with positively correlated assets, istherefore the most risky. Alternative 1 has the highest average return but does not offer theopportunity to reduce risk; it has a standard deviation equal to Alternative 3. Alternative 2 is the

best choice, it is perfectly negatively correlated and has the least risk.

6. (a) Average return,returns

3r = ∑

12% 14% 16% 42%14%

3 3

16% 14% 12% 42%14%

3 3

12% 14% 16% 42%14%

3 3

A

B

C

r

r

r

+ += = =

+ += = =

+ += = =

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(b) Standard Deviation:2

1

( ) ( 1)n

i i

i

s r r n=

= − ÷ −√ ∑

2 2 2

2 2 2

2 2 2

[(12.0% 14%) (14% 14%) (16% 14%) ]

3 1

[(4%) (0) (4%)]

4 = 2%2

[(16% 14%) (14% 14%) (12% 14%) ]

3 1

[(4%) (0) (4%)]4 2%

2

[(12% 14%) (14% 14%) (16% 14%) ]

3 1

[(4%) (0) (4%)]4 2%

2

A

A

B

B

C

C

s

s

s

s

s

s

− + − + −√=−

+ +√= =

− + − + −√=−

+ +√= = =

− + − + −√=−

+ +√= = =

(c)

Portfolio Return

Year Portfolio AB Portfolio AC

2006 (12% × 0.50) + (0.50 × 16%) = 14%

(12% × 0.50) + (12% × 0.50) = 12%

2007 (14% × 0.50) + (0.50 × 14%) = 14%

(14% × 0.50) + (14% × 0.50) = 14%

2008 (16% × 0.50) + (0.50 × 12%) = 14%

(16% × 0.50) + (16% × 0.50) = 16%

14% 14% 14% 42% 12% 14% 16% 42%14% 14%3 3 3 3

AB AC r r + + + += = = = = =

(d) Portfolio AB is perfectly negatively correlated.

Portfolio AC is perfectly positively correlated.

(e) Standard deviation of portfolios:

2 2 2

2 2 2

[(14% 14%) (14% 14%) (14% 14%) ]

3 1

[(0%) (0) (0%)] 0%0%

2 2

[(12% 14%) (14% 14%) (16% 14%) ]3 1

[(4%) (0) (4%)]4 2%

2

AB

AB

AC

AC

s

s

s

s

− + − + −√=−

+ + √√= = =

− + − + −√= −

+ +√= = =

(f) Portfolio AB is preferred: it provides the same return (14%) as Portfolio AC, but with less risk, as

measured by the standard deviation (s AB = 0%; s AC = 2%).

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7.

2006 2007 2008

Asset A 12 14 16Asset B 16 14 12Asset C 12 14 16Portfolio Return 13.33 14 14.67

2 2 2

Mean Return 1/3 (13.33) 1/3(14) 1/3(14.67) 14%

Standard Deviation [(13.33 14) (14 14) (14.67 14) ]/ 2

[0.4489 0 0.4489] / 2

0.4489 0.67

+ + =

= √ − + − + −

= √ + +

= √ =

The return would be the same with slight higher risk. This is because the assets are no longerperfectly negatively correlated. Two thirds of the portfolio has one characteristic return pattern andone third of the portfolio is constant over time.

8. (a) 1. Range of expected return: between 8% and 13%

2. Range of the risk: between 5% and 10%.(b) 1. Range of expected return: between 8% and 13%

2. Range of the risk: between 0 < risk < 10%.

(c) 1. Range of expected return: between 8% and 13%

2. Range of the risk: between 0 = risk < 10%.

9. (a) The figure showing the characteristic lines for investments A and B can be found on the book’sWeb site at www.awl.com/gitman_joehnk.

(b) Estimate beta by looking at the slope (angle) of the characteristic line for each investment. Asmarked on the graph, the beta for Investment A is about 0.75, and the beta for Investment B isabout 1.33. (A financial calculator with statistical functions can be used to perform linear

regression analysis. The beta (slope) of the Line A is 0.79; of Line B, 1.379.)(c) With a higher beta of 1.33, Asset B is more risky. Its return will increase or decrease 1.33 times

for each one point the market moves. Asset A’s return will increase or decrease at a lower rate, asindicated by its beta coefficient of 0.75.

10. You would buy the stock of Buyme Co., because it has the same expected return as Getit Corp. butwith less risk (a lower beta).

11. You may decide to purchase the stock of Getit Corp., since it should rise more than Buyme Co. in amarket rally.

12. The effect of change in market return on required return of an asset with beta of 1.20:

(a) 1.20 × (15%) = 18% increase(b) 1.20 × (–8%) = 9.6% decrease(c) 1.20 × (0) = no changeThe asset is more risky than the market portfolio, which has a beta of 1

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13. Use of beta: Change in security return = Beta × change in market return

(a) Security A return = 1.4 × 13.2% = 18.48%Security B return = 0.8 × 13.2% = 10.56%Security C return = –0.9 × 13.2% = –11.88%

(b) Security A return = 1.4 × –10.8% = –15.12%

Security B return = 0.8 × –10.8% = –8.64%Security C return = –0.9 × –10.8% = 9.72%

(c) Security A is the most risky. It has the highest relevant risk, as determined by the beta values andthe greater changes in security A’s return for a given change in the market return. Security Ccould be called defensive since it moves in the opposite direction from the market (its returnincreased when the market return fell and vice versa). Security B is the least risky since its returnis least responsive (regardless of direction) to changes in the market return.

During an economic downturn, it can probably be assumed that the market return woulddecrease. If this occurred, security C would perform best. Otherwise, security B would be bestsince it would be least responsive to a change in the market return.

14.

Security Beta Weight Weighted

A 1.4 0.333 0.447

B 0.8 0.333 0.27

C –0.9 0.333 (0 .30)Portfolio Beta 0.43

15. If the market rallied 207, the portfolio should increase by 8.6 (0.43 × 0.2) percent. The portfolio’svalue would be ($20,000 × 3) × 1.086 = $65,160.

If the market declines by 20%, the portfolio’s value will drop by $5,160 to $54,840 ($60,000 – (0.086 ×$60,000) = $60,000 − $5,160).

16. Capital Asset Pricing Model: r i = RF + [bi × (r m – RF )]

Investment ri R F [ bi × ( r m – R F)]

A 8.9% = 5% + [1.30 × (8% – 5%)]

B 12.5% = 8% + [0.90 × (13% – 8%)]

C 8.4% = 9% + [– 0.20 × (12% – 9%)]

D 15.0% = 10% + [1.00 × (15% – 10%)] E 8.4% = 6% + [0.60 × (10% – 6%)]

17. Using the CAPM, Bob’s required rate of return on the stock should be:

Required rate of return − risk free rate + [beta * (market rate – risk free rate)]

r j = 3 + 1.25(13 – 3) = 15.5%.Since Bob’s required rate of return exceeds the expected return, he should not buy the stock.

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18. If the risk-free rate is 7% and the market return is 12%.

(a) Vehicle E is the most risky because it has the highest beta, 2.00. Vehicle D, with a beta of 0, isthe least risky.

(b) Capital Asset Pricing Model: r i = RF + [bi × (r m – RF )]

Investment ri R F [ bi × ( r m – R F)]

A 14.5% = 7% + [1.50 × (12% – 7%)] B 12% = 7% + [1.00 × (12% – 7%)]

C 10.75% = 7% + [0.75 × (12% – 7%)]

D 7% = 7% + [0 × (12% − 7%)]

E 17% = 7% + [2.00 × (12% – 7%)]

(c) The figure showing the security market line (SML) can be found on the book’s Web site atwww.awl.com/gitman_joehnk.

(d) Based on the above graph and the calculations, there is a linear relationship between risk andreturn.

19. (a) and (b)

(c) Portfolios B, J, F, C, and H lie on the efficient frontier. These portfolios are the efficientportfolios, those that provide the best tradeoff between risk and return (the highest return for aparticular risk level or the lowest risk for the specified level of return). These portfolios dominatebecause all those to the left of the frontier are unattainable and all those to the right of the frontierare not desirable because they are not efficient.

(d) By plotting an investor’s utility function or risk-indifference curves, which show those risk-return combinations for which an investor would be indifferent, on the efficient frontier graph,the investor can determine the optimal portfolio. This portfolio would be the one that occurswhere an indifference curve meets the efficient frontier and represents the highest level ofsatisfaction for that investor for this set of portfolios.

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20. (a) and (b)

(c) Only nondiversifiable risk is relevant because, as shown by the graph, diversifiable risk can bevirtually eliminated through holding a portfolio of at least 20 securities which are not positivelycorrelated. David Finney’s portfolio, assuming diversifiable risk could no longer be reduced byadditions to the portfolio, has 6.47% relevant risk.

21. With a beta value of +1.5, portfolio A would be 1.5 times as responsive to changes in the market asthe market itself, while portfolio Z with a beta of –1.5 would also be 1.5 times as responsive, but inthe opposite direction, to these changes in the market. If the market portfolio return rises by20 percent, the expected return on portfolio A would go up by 30 percent. For A, the change is found

by multiplying 20% × 1.5 where 1.5 is the beta for A (20% × 1.5 = 30%). For B, the change is

–30% (20% × –1.5), since B has a beta of –1.5.

22. (a)

Stock Beta

Most risky B 1.40A 0.80

Least risky C –0.30

(b) and (c).

Stock Beta

Increase in

Market Return

Impact on

Asset Return

Decrease in

Market Return

Impact on

Asset Return

A 0.80 0.12 0.096 –0.05 –0.04 B 1.40 0.12 0.168 –0.05 –0.07

C –0.30 0.12 –0.036 –0.05 0.015

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(d) In a declining market, an investor would choose the defensive stock, Stock C. While the marketdeclines, the return on C increases.

(e) In a rising market, an investor would choose Stock B, the aggressive stock. As the market risesone point, Stock B rises 1.40 points.

23.1

Portfolio Betas :n

p j j

j

b w b=

= ×

∑(a)

Asset Beta w A w A × b A w B w B × b B

A 1.30 0.10 0.130 0.30 0.39

B 0.70 0.30 0.210 0.10 0.07 C 1.25 0.10 0.125 0.20 0.25

D 1.10 0.10 0.110 0.20 0.22

E 0.90 0.40 0.360 0.20 0 .18b A = 0.935 b B = 1.11

(b) Portfolio A is slightly less than the market (average risk), while Portfolio B is more risky than the

market. Portfolio B’s return will move more than Portfolio A’s for a given increase or decrease inmarket risk. Portfolio B is the more risky.

24. Required ReturnA = 2 + [0.935 × (12 – 2)] = 2 + 9.35 = 11.35

Required ReturnB = 2 + [1.11 × (12 – 2)] = 2 + 11.10 = 13.10

25.

[1] [2] [3] [2 × 3] [4] [2 × 4]

Asset Ra

% of

Portfolio A Pr

% of

Portfolio B Pr

1 16.5% 0.1 0.0165 0.3 0.0495

2 12.0% 0.3 0.036 0.1 0.012 3 15.0% 0.1 0.015 0.2 0.03

4 13.0% 0.1 0.013 0.2 0.026

5 7.0% 0.4 0 .028 0.2 0 .0140.1085 0.1315

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Portfolio B provides a return in excess (slightly) of the required rate of return, while Portfolio A doesnot. Portfolio B represents a better risk/reward tradeoff.

.8 Solutions to Case Problems

Case 5.1 Traditional Versus Modern Portfolio Theory: Who’s Right?

This case provides a basis for discussion of traditional and modern portfolio theory with emphasis on thereconciliation of the two.

(a) Walt’s arguments rely on the traditional approach to portfolio management. He believes that bybuilding a large portfolio, the maximum benefits of diversification can be achieved. For this reasonWalt insisted that an investor should buy mutual fund shares. However, according to modern portfoliotheory, all that is required to become adequately diversified is investment in about 8 to 20 differentstocks. It is not necessary for an investor to have hundreds of thousands of dollars in order todiversify. Evidently, Walt has not heard about the latest developments in modern portfolio theory.

(b) Shane is incorrect in assuming that the stock with a beta of 1.2 is equivalent to a mutual fund with abeta of 1.2. The error in logic occurs because a stock with a beta of 1.2 also has a certain amount ofdiversifiable (unsystematic) risk. On the other hand, a mutual fund with a beta of 1.2 has nodiversifiable risk. Therefore, the only risk in a mutual fund is its nondiversifiable risk. Simply stated, astock with a beta of 1.2 is riskier than a mutual fund with a beta of 1.2. This is because a mutual fundis a diversified investment, whereas an individual stock is not. It is important to remember that whenone uses beta to measure risk, he or she is assuming that all diversifiable risk will be eliminatedthrough diversification.

(c) The traditional approach to portfolio management simply involves forming a portfolio with a largevariety of stocks from different industries to obtain the benefits of diversification. These are usuallyhigh-quality, well-known stocks. Walt’s arguments basically revolve around the use of this traditionalapproach. On the other hand, Shane (though not completely correct) is trying to use modern portfoliotheory to state his belief. The MPT approach relies on statistical measures, such as beta, for assessing

risk and deciding whether to include a given security in the portfolio.

(d) Modern portfolio theory relies on statistical concepts. The use of the correlation coefficient and thebeta value are the most popular. Two securities that are negatively correlated tend to provide a greaterdegree of diversification than two securities that are positively correlated, and two securities that areperfectly positively correlated provide no risk diversification at all. Negative correlation is not anabsolute prerequisite for diversification. Any stocks that are less than perfectly positively correlatedwill provide some diversification benefit. A portfolio that is efficiently diversified has no diversifiablerisk. It only has nondiversifiable risk, and this can be measured using beta. While total risk is the sumof nondiversifiable and diversifiable risk, the only relevant risk is the nondiversifiable risk. Anyintelligent investor should be able to eliminate the diversifiable risk. Walt’s argument is, in a sense,related to this approach because he believes that a good strategy is to build a portfolio similar to a

mutual fund. Walt does understand the benefits of diversification. However, he is incorrect in hisbelief that the only way a small investor can adequately diversify is by purchasing mutual fund shares.Shane, on the other hand, recognizes that the relevant risk of a portfolio is the nondiversifiable risk.However, Shane is mistaken in assuming that a stock with a beta of 1.2 and a mutual fund with a betaof 1.2 are identical in terms of risk. A mutual fund is diversified; an individual stock is not.

(e) To reconcile the traditional portfolio approach and modern portfolio theory:

(1) Determine how much risk the investor is willing to bear.

(2) Seek diversification across different types of securities and across industry lines, paying attentionto the way the return from one security is related to another.

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(3) Using beta (from MPT), assemble a diversified portfolio consistent with the level of acceptablerisk.

(4) Consider different portfolios at the same level of risk and choose the one that provides the highestexpected return for the given level of acceptable risk.

In using this four-step procedure, we are in effect reconciling the approaches suggested by Walt andShane. We are forming a portfolio (though not as large as a mutual fund) to get the benefits of

diversification. We are using the concept of beta to measure the amount of risk in the portfolio. Thus,both Walt and Shane should find this procedure acceptable.

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Case 5.2 Susan Lussier’s Inherited Portfolio: Does It Meet Her Needs?

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This case demonstrates that a portfolio designed for one person is not likely to be appropriate for another.In particular, it emphasizes some of the considerations in designing a portfolio.

(a) Susan’s financial position is quite strong: she has a regular $125,000 per year job and also hasinherited a portfolio worth nearly $350,000 and $10,000 in cash. Susan has a good job and does nothave to rely on earnings from her portfolio to fulfill current income needs, at least for the present.(However, the oil and gas industry is rather volatile, and she should not be too complacent about her

job security in the future!) Her primary concerns are capital preservation and capital growth. Sheprobably prefers to receive capital gains in the future rather than current income. Although she doesnot want to talk about retirement. Susan should develop a financial/investment plan to address thisgoal. At her age and with her current level of income and wealth, she could deal with retirement at alater time, but she cannot ignore it forever.

(b) Reviewing Susan’s inherited portfolio indicates that current income was her father’s chief objectives;the portfolio’s current yield is nearly 10%. The asset allocation based on the total cost data is heavilyweighted toward bonds: 46% bonds ($158,100/$338,800), 29% common stock ($96,900/$338,800),and 25% mutual funds ($83,800/$338,800). Susan’s father apparently was not a risk-taker, so hisportfolio consisted primarily of bonds, low-risk (low-beta) stocks, and mutual funds. The portfolio’sreturn is not too high, but that is understandable given its low risk.

However, the same portfolio will result in an unduly high tax liability for Susan because of theportfolio’s current-income-orientation. This portfolio will not satisfy her financial objectives; she musttherefore restructure the portfolio to meet her objectives of capital appreciation and tax shelter.

(c) Since current yield is not an important consideration for Susan, she should revise the portfolio toinclude securities with low current yields and high capital appreciation potential. This will enable herto lower her annual tax liability. Her asset allocation should be shifted to more stocks and fewer bondsand mutual funds. Given the relatively large dollar size ($338,800) of the portfolio, she should be ableto achieve adequate diversification on her own (assuming she is willing to invest the time or hire aprofessional investment manager/advisor). She therefore does not have to rely as heavily on mutualfunds. An asset allocation scheme of around 25 percent bonds, 60 percent common stock, and 15percent mutual funds is one possible recommendation.

Within each asset category she should hold higher-risk, capital-appreciation-oriented securities ratherthan the income-oriented securities currently held. Since Susan is single and has adequate currentincome, she appears to be in a position to justify a higher-risk portfolio. For example, the existingportfolio has stocks with betas of 0.97 and 0.85, respectively. She could very well substitute for thesestocks those with higher betas. She should sell some of the highly income-oriented mutual fundinvestments. The proceeds should be invested in stocks with good potential for capital appreciation.For the bond segment she might consider convertible bonds that have the potential for large capitalgains.

(d) As discussed earlier, the inherited portfolio focuses on current income and capital preservation, ratherthan Susan’s objectives of capital gains and tax shelter. She will want to adjust the portfolio to includemore capital appreciation securities, and she may also want to restructure the portfolio to meetprojected retirement needs and objectives. Her objective should be to minimize taxation of theportfolio’s returns while meeting future net worth and investment objectives. This strategy wouldprobably initially introduce greater risk but, of course, the expected future returns would be greater.Furthermore, Susan can afford an increase in risk at this point in time, and with an appropriate strategythe risks can be minimized. Also, any losses can be minimized through portfolio adjustments, and anyrealized losses can be used to decrease her tax liability.

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(e) The inherited portfolio is very low-risk portfolio. As mentioned in the response to question c, this isnot a good portfolio for Susan. What Susan really needs is a portfolio offering greater capitalappreciation and consequently, lower taxable income. Susan should reallocate the assets in theportfolio (as noted in the response to question c) and introduce greater risk into the revised portfolio.She should sell some bonds and mutual fund shares and invest the proceeds in other stock issues. Herbond holdings could include convertibles, and the stocks should have betas in excess of 1. However,she should also include some fairly liquid investments in case her job situation changes, monitor herholdings carefully, and review her objectives and the portfolio if her personal situation changes.

.9 Outside Project

Chapter 5 Your Dream Portfolio

Understanding your own attitude toward risk is very important when you are selecting investment vehiclesfor inclusion in your portfolio. This project should help you consider what you would do if you came intosome money.

You bought a state lottery ticket last month and yesterday you learned that you won one million dollarsafter taxes. What would you do with the money? While there are certainly bills to be paid and things to

buy, assume that you will have at least $800,000 to invest. What goals do you wish to achieve with thesefunds? State them clearly, develop an asset allocation scheme, and design a portfolio of stocks, bonds,tangibles, and/or limited partnership investments that you feel would be consistent with achieving yourgoals. Using current price quotations from The Wall Street Journal or local newspapers, create your owndream portfolio.

Briefly explain the rationale for your asset allocation scheme and the reasons you included each specificinvestment vehicle in the portfolio. Point out any concerns or reservations you might have relative to theportfolio you created.

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