OBJECTEVES
· To understand the process of personal portfolio selection in theory and in practice.
·To build a quantitative model of the trade-off between risk and reward.
CONTENTS
12.1 The Process of Personal Portfolio Selection
12.2 The Trade-Off between Expected Return and Risk
12.3 Efficient Diversification with Many Risky Assets
This chapter is about how people should invest their wealth, a process called portfolio selection. A person’s wealth portfolio includes all of his or her assets (stocks, bonds, shares in unincorporated businesses, houses or apartments, pension benefits, insurance policies, etc.) and all of his or her liabilities (student loans, auto loans, home mortgages, etc.).
There is no single portfolio selection strategy that is best for all people. There are, however, some general principles, such as the principle of diversification that apply to all risk-averse people. In chapter 11 we discussed diversification as a method of managing risk. This chapter extends that discussion and analyzes the quantitative trade-off between risk and expected return.
Section 12.1 examines the role of portfolio selection in the context of a person’s life-cycle financial planning process and shows why there is no single strategy that is best for all people. It also examines how the investor’s time horizon and risk tolerance affect portfolio selection. Section 12.2 analyzes the choice between a single risky asset and a riskless asset, and Section 12.3 examines optimal portfolio selection with many risky assets.
12.1 THE PROCESS OF PERSONAL PORTFOLIO SELECTION
Portfolio selection is the study of how people should invest their wealth.It is a process of trading off risk and expected return to find the best portfolio of assets and liabilities.A narrow definition of portfolio selection includes only decisions about how much to invest in stocks, bonds,and other securities.A broader definition includes decisions about whether to buy or rent one's house,what types and amounts of insurance to purchase,and how to manage one’s liabilities.An even broader definition includes decisions about how much to invest m one’s human capital (e.g.,by furthering one’s professional education). The common element in all of these decisions is the trade-off between risk and expected return.
This chapter is devoted to exploring the concepts and techniques you need to know to evaluate risk-reward trade-offs and to manage your wealth portfolio efficiency. A major theme is that although there arc some general rules for portfolio selection that apply to virtually everyone,there is no single portfolio or portfolio strategy that is best for everyone.We begin by explaining why.
12.1.1 The Life Cycle
In portfolio selection the best strategy depends on an individual's personal circum- stances (age,family status,occupation, income,wealth,etc.).Forsome people holding a particular asset may add to their total risk exposure,but for others the same asset may be risk reducing. An asset that is risk reducing at an early stage in the life cycle may not be at a later stage.
For a young couple starting a family it may be optimal to buy a house and take out a mortgage loan.For an older couple about to retire it may be optimal to sell their house and invest the proceeds in some asset that will provide a steady stream of income for as long as they live.
Consider the purchase of life insurance.The optimal insurance policy for Miriam, a parent with dependent children,will differ from the policy appropriate for Sanjiv, a single person with no dependents, even if the two people are the same in all other respects (age, income, occupation, wealth, etc.). Miriam would be concerned about protecting her family in the event of her death and would,therefore, want a policy that provides cash benefits payable to her children upon her death. Sanjiv,on the other hand,would not be concerned about benefits payable if he dies; therefore, the purchase of life insurance would not be risk reducing for him.At a later stage in her life,Miriam too may find that her children can provide for them- selves and no longer need the protection afforded by life insurance.
Now consider the situation of Miriam and Sanjiv after they reach retirement age.Miriam has children and is happy to have them inherit any assets that are left after she dies-If she should live an extraordinarily long time and exhaust her own wealth,she is confident her children will provide financial support for her.
Sanjiv is a loner with no one to whom he cares to leave a bequest.He would like to consume all of his wealth during his own lifetime but is concerned that if he increases his spending he will exhaust his wealth if he happens to live an extraordinarily long time.For Sanjiv buying an insurance policy that guarantees him an income for as long as he lives would be risk reducing;for Miriam it would not be.Such an insurance policy is called a life annuity-
As these examples make clear,even people of the same age,with the same income and wealth,may have different perspectives on buying a house or buyinginsurance. The same is true of investing in stocks,bonds,and other securities. There is no single portfolio that is best for all people.
To see this,consider two different individuals of the same age and family status. Chang is 30 years old and works as a security analyst on Wall Street.His current and future earnings are very sensitive to the performance of the stock market. Obi is also 30years old and teaches English in the public school system.Her current and future earnings are not very sensitive to the stock market.For Chang investing a significant proportion ofhis investment portfolio in stocks would be z=t risky than itwould be for Obi.
12.1.2Time Horizons
In formulating a plan for portfolio selection you begin by determining you goalsand time horizons.The planning horizon is the total length of time for which one plans.
The longest time horizon would typically correspond to the retirement goal and would be the balance ofone’s lifetime. Thus, for a 25-year-old who expects live to age 85,the planning horizon would be60years. As one ages,the planning horizon typically gets shorter and shorter (see Box 12.1).
There are also shorter planning horizons that correspond to specific financialgoals,such as paying for a child's education. For example,if you have a child who is three years old and plan to pay for her college education when she reaches age 18, the planning horizon for this goal is 15years.
The decision horizon is the length of time between decisions to revisethe portfolio. The length of the decision horizon is controlled by the individual within certain limits.
Some people review their portfolios at regular intervals,for example, once a month (whentheypaytheirbills),oronce ayear(whentheyfileincome tax forms ). People of modest means with most of their wealth invested in bank accounts mightreview their portfolios very infrequently and at irregular intervals determined by some triggering event such as getting married or divorced,having a child, or receiving a bequest.A sudden rise or fall in the price of an asset a person ownsmight also trigger a review of the portfolio.
People with substantial investments in stocks and bonds might review theirportfolios every day or even more frequently.The shortest possible decision horizon is the trading horizon, defined as the minimum time interval over which investors can revise their portfolios.
The length of the trading horizon is not under the control of the individual. Whether the trading horizon is a week,a day,an hour,or a minute is determined bythe structure of the markets in the economy (e.g.,when the securities exchanges are open or whether organized off---exchange markets exist).
In today's global financial environment trading in many securities can be carried on somewhere on the globe around the clock.For these securities at least the trading horizon is very short.
Portfolio decisions you make today are influenced by what you think might happen tomorrow. A plan that takes account of future decisions in making current decisions is called a strategy.
How frequently investors can revise their portfolios by buying or selling securities is an important consideration in formulating investment strategies.If you know that you can adjust the composition ofyour portfolio frequently, you may invest differently than ifyou cannot adjust it.
For example,a person may adopt a strategy of investing "extra"wealth in stocks,meaning wealth in excess of the amount needed to insure a certain the old standard of living. If the stock market goes up over time,a person will increase the proportion of his or her portfolio invested in stocks.However,if the stock market goes down,a person will reduce the proportion invested in stocks.If thestock market fails to the point at which the person's threshold standard ofliving is threatened, he or she will get out of stocks altogether.An investor pursuing this particular strategy is more likely to have a higher threshold if stocks can only be infrequently.
12.1.3Risk Tolerance
Aperson's tolerance for bearing risk is a major determinant of portfolio choices. We expect risk tolerance to be influenced by such characteristics as age,family status,job status,wealth,and other attributes that affect people's ability to maintain their standard of living in the face of adverse movements in the market value of their investment portfolio.One's attitude towards risk also plays a role in determining a person's tolerance for bearing risk.Even among people with the same apparent personal, family, andjob characteristics,some may have a greater willingness to take risk than others.
When we refer to a person's risk tolerance in our analysis of optimal portfolio selection, we do not distinguish between capacity to bear risk and attitude toward risk. Thus,whether a person has a relatively high tolerance for risk because he is young or rich, because he handles stress we11,or because he was brought up to believe that taking chances is the morally right path,all that matters in the analysis to follow is that he is more willing than the average person to take on additional risk to achieve a higher expected return.
12.1.4The Roleof Professional Asset Managers
Most people have neither the knowledge nor the time to carry out portfolio optimization. Therefore they hire an investment advisor to do it for them or they buy a"finished product” from a financial intermediary.Such finished products include various investment accounts and mutual funds offered by banks,securities a vestment companies,and insurance companies.
When financial intermediaries decide what asset choices to offer to households,they are in a position analogous to a restaurant deciding on its menu.There are many ingredients available (the basic stocks,bonds,and other securities issued by firms and governments)and an infinite number of possible ways to combine them,but only a limited number of items will be offered to customers.The portfolio theory developed in the rest of this chapter offers some guidance in finding the least number of items to offer that still cover the full array of customer demands.
12.2 THE TRADE-OFF BETWEEN EXPECIED REIURN AND RISK
The next two sections present the analytical framework used by professional port- folio managers for examining the quantitative trade-off between risk and expected return. The objective is to find the portfolio that offers investors the highest expected rate of return for any degree of risk they arc willing to tolerate.Throughout the analysis we will refer to risky assets without specifically identifying them as bonds,stocks,options,insurance policies,and so on.This is because,as explained in the preceding sections of this chapter,the riskiness of a particular asset depends critically on the specific circumstances of the investor.
Portfolio optimization is often done as a two-step process:(1)Find the optimal combination of risky assets,and(2)mix this optimal risky-asset portfolio with the riskless asset.For simplicity,we start with the second step:mixing a single risky-asset portfolio and a riskless asset. (We discuss the identity of the riskless asset in the next section.)The single risky-asset portfolio is composed of many risky assets chosen in an optimal way.In section 12.3.4we investigate how the optimal composition of this risky-asset portfolio is found.
12.2.IWhat Is the Riskless Asset?
In chapter 4we discussed interest rates and showed that there is a different riskless asset that corresponds to each possible unit of account (dollars, yen,etc.)and toeachpossible maturity. Thus, a 10-year, dollar-denominated,zero-coupon bond that offers a default-free yield-to-maturity of6%per year is riskless only in terms of dollars and only ifheld to maturity.The dollar rate of return on that same bond is uncertain if it is sold before maturity because the price to be received is uncertain.And even if held to maturity,the bond's rate of return denominated in yen or in terms of consumer purchasing power is uncertain because future exchange rates and consumer prices are uncertain.
In the theory of portfolio selection the riskless asset is defined as a security that offers a perfectly predictable rate of return in terms of the unit of account selected for the analysis and the 1ength of the investor's decision horizon.When no specific investor is identified,the riskless asset refers to an asset that offers a predictable rate ofreturn over the tradinghorizon (i.e.,the shortest possible decision horizon).
Thus if the U.S.dollar is taken as the unit of account and the trading horizon is aday,the riskless rate is the interest rate on U.S-Treasury bills maturing the next day.
12.2.2 Combining the RiskIess Assetand a Single Risky Asset
Suppose that you have $100,000 to invest.You are choosing between a risklessset with an interest rate of .06per year and a risky asset with an expected rate of return of .14per year and standard deviation of .20.3 .How much of your$100,000 should you invest in the risky asset?
We examine all of the risk-return combinations open to you with the aid of Table 12.1and Figure 12.1.Start with the case in which you invest a11ofyour money in the riskless asset.This corresponds to the point labeled F in Figure 12.1 and the first row inTable12.1.Column 2in Tabie12.1gives the proportion of the portfolio invested in the risky asset (0)and column 3the proportion invested in the riskless asset (100%).The proportions always add to 100%.Columns 4and 5ofTable 12.1 give the expected return and standard deviation that correspond to portfolio F:E (r)of .06per year and σof 0.00.
The case in which you invest all of your money in the risky asset corresponds to the point labeled S in Figure 12.1and the last row in Table 12.1.Its expect return is .14and its standard deviation 20.
In Figure 12.1the portfolio expected rate of return, E (r),is measured L: the vertical axis and the standard deviation, σ,along the horizontal axis. The portfolio proportions are not explicitly shown in Figure 12.1; however,we know what they are from Table 121.
Figure 12.1graphically illustrates the trade-off between risk and reward. The line connecting points F, G, H, I, and S in Figure 12.1represents the set of alternatives open to you by choosing different combinations (portfolios)of the risky asset and riskless asset. Each point on the line corresponds to the mix of these two assets given in columns 2and 3ofTable12.1.
At point R which is on the vertical axis in Figure 12.1,with E(r)of .06per and σof zero,all of your money is invested in the riskless asset.You face EC and your expected return is .06per year.As you shift money out of the riskless asset and into the risky asset,you move to the right along the trade-off line and face both a higher expected rate of return and a greater risk. If you invest all of your money in the risky asset,you would be at point S with expected return,E(r),of .14 and standard deviation, σ,of .20.
Portfolio H (corresponding to the third row ofTable12.1)is half invested in the riskless asset and half in the risky asset.With $50,000invested in the risky asset and $50,000invested in the riskless asset,you would have an expected rate of return thatis halfwaybetweenthe expectedreturn onthe ail-stockportfolio (.14)and the riskless rate ofinterest(.06).The expected rate ofreturn of.10is shown in column 4and the standard deviation of .10in column 5.
Now let us show how we can find the portfolio composition for any point lying on the trade-off line in Figure 12.1,not only the points listed in Table 121.For example,suppose we want to identify the portfolio that has an expected rate of return of .09.We can tell from Figure 12.1that the point corresponding to such a portfolio lies on the trade-off line between points G and H.But what is the portfolio's composition and what is its standard deviation?In answering this question we shall also derive the formula for the trade-off line connecting ail of the points in Figure 12.1.
Step 1:Relate the portfolio's expected return to the proportion invested in the risky
Asset.
Let w denote the proportion of the $100,000investment to be allocated to the risky asset.The remaining proportion,1-w is to be invested in the riskless asset. The expected rate of return on any portfolio,E(r) ,is given by:
E(r)=wE(rs)+(1-w)rf
=rf+w[E(rs)-rf]
WhereE(rs) denotes the expected rate of return on the risky asset and rf is the z- less rate.Substituting .06for rf and .14for E (rs)we get:
E(r)=.06+w(.14 -.06)
=.06+.08w
Equation 12.1is interpreted as follows.The base rate of return form for any portfolio is the riskless rate (.06in our example).In addition,the portfolio is expected to earn a risk premium which depends on(1)the risk premium on the riskpremium on the risky asset, E(r) –r f (.08in our example)and (2)the proportion of the portfolio invested in the risky asset,denoted by w.