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8/8/2019 The Insights of Modern Portfolio Theory http://slidepdf.com/reader/full/the-insights-of-modern-portfolio-theory 1/17 CHAPTER 2 The Insights of Modern Portfolio Theory Bob Litterman I n order to be successful, an investor must understand and be comfortable with taking risks. Creating wealth is the object of making investments, and risk is the energy that in the long run drives investment returns. Investor tolerance for taking risk is limited, though. Risk quantifies the likeli- hood and size of potential losses, and losses are painful. When a loss occurs it im- plies consumption must be postponed or denied, and even though returns are largely determined by random events over which the investor has no control, when a loss occurs it is natural to feel that a mistake was made and to feel regret about taking the risk. If a loss has too great an impact on an investor’s net worth, then the loss itself may force a reduction in the investor’s risk appetite, which could create a significant limitation on the investor’s ability to generate future in- vestment returns. Thus, each investor can only tolerate losses up to a certain size. And even though risk is the energy that drives returns, since risk taking creates the opportunity for bad outcomes, it is something for which each investor has only a limited appetite. But risk itself is not something to be avoided. As we shall discuss, wealth cre- ation depends on taking risk, on allocating that risk across many assets (in order to minimize the potential pain), on being patient, and on being willing to accept short- term losses while focusing on long-term, real returns (after taking into account the effects of inflation and taxes). Thus, investment success depends on being prepared for and being willing to take risk. Because investors have a limited capacity for taking risk it should be viewed as a scarce resource that needs to be used wisely. Risk should be budgeted, just like any other resource in limited supply. Successful investing requires positioning the risk one takes in order to create as much return as possible. And while investors have in- tuitively understood the connection between risk and return for many centuries, only in the past 50 years have academics quantified these concepts mathematically and worked out the sometimes surprising implications of trying to maximize ex- pected return for a given level of risk. This body of work, known today as modern portfolio theory, provides some very useful insights for investors, which we will highlight in this chapter.

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CHAPTER 2

The Insights ofModern Portfolio Theory

Bob Litterman

In order to be successful, an investor must understand and be comfortable withtaking risks. Creating wealth is the object of making investments, and risk is the

energy that in the long run drives investment returns.Investor tolerance for taking risk is limited, though. Risk quantifies the likeli-

hood and size of potential losses, and losses are painful. When a loss occurs it im-plies consumption must be postponed or denied, and even though returns are

largely determined by random events over which the investor has no control,when a loss occurs it is natural to feel that a mistake was made and to feel regretabout taking the risk. If a loss has too great an impact on an investor’s net worth,then the loss itself may force a reduction in the investor’s risk appetite, whichcould create a significant limitation on the investor’s ability to generate future in-vestment returns. Thus, each investor can only tolerate losses up to a certain size.And even though risk is the energy that drives returns, since risk taking creates theopportunity for bad outcomes, it is something for which each investor has only alimited appetite.

But risk itself is not something to be avoided. As we shall discuss, wealth cre-ation depends on taking risk, on allocating that risk across many assets (in order tominimize the potential pain), on being patient, and on being willing to accept short-term losses while focusing on long-term, real returns (after taking into account theeffects of inflation and taxes). Thus, investment success depends on being preparedfor and being willing to take risk.

Because investors have a limited capacity for taking risk it should be viewed asa scarce resource that needs to be used wisely. Risk should be budgeted, just like anyother resource in limited supply. Successful investing requires positioning the risk

one takes in order to create as much return as possible. And while investors have in-tuitively understood the connection between risk and return for many centuries,only in the past 50 years have academics quantified these concepts mathematicallyand worked out the sometimes surprising implications of trying to maximize ex-pected return for a given level of risk. This body of work, known today as modernportfolio theory, provides some very useful insights for investors, which we willhighlight in this chapter.

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The interesting insights provided by modern portfolio theory arise from the in-terplay between the mathematics of return and risk. It is important at this junctureto review the different rules for adding risks or adding returns in a portfolio con-text. These issues are not particularly complex, but they are at the heart of modernportfolio theory. The mathematics on the return side of the investment equation isstraightforward. Monetary returns on different investments at a point in time areadditive. If one investment creates a $30,000 return and another creates a $40,000return, then the total return is $70,000. The additive nature of investment returnsat a point in time is illustrated in Figure 2.1.

Percentage returns compound over time. A 20 percent return one year followedby a 20 percent return the next year creates a 44 percent1 return on the original in-vestment over the two-year horizon.

The risk side of the investment equation, however, is not so straightforward.Even at a point in time, portfolio risk is not additive. If one investment creates avolatility2 of $30,000 per year and another investment creates a volatility of $40,000 per year, then the total annual portfolio volatility could be anywhere be-tween $10,000 and $70,000. How the risks of different investments combine de-pends on whether the returns they generate tend to move together, to moveindependently, or to offset. If the returns of the two investments in the precedingexample are roughly independent, then the combined volatility is approximately3

$50,000; if they move together, the combined risk is higher; if they offset, lower.This degree to which returns move together is measured by a statistical quantitycalled correlation, which ranges in value from +1 for returns that move perfectly to-gether to zero for independent returns, to –1 for returns that always move in oppo-

8 THEORY

FIGURE 2.1 Expected Return Sums Linearly

A B

C

C = A + B

A = Old Portfolio Expected Return

B = New Investment Expected Return

C = New Portfolio Expected Return

1The two-period return is z, where the first period return is x, the second period return is y,and (1 + z) = (1 + x)(1 + y).2Volatility is only one of many statistics that can be used to measure risk. Here “a volatility”refers to one standard deviation, which is a typical outcome in the distribution of returns.3In this calculation we rely on the fact that the variance (the square of volatility) of indepen-dent assets is additive.

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site directions. The fact that risks are not additive, but combine in a way that de-pends on how returns move together, leads to the primary insight of portfolio the-ory—that diversification, the spreading of investments across less correlated assets,tends to reduce overall portfolio risk.

This risk reduction benefit of diversification can be a free lunch for investors.Given the limited appetite each investor has for risk, the diversification benefit itself creates the opportunity to generate higher expected returns. An additional diversifi-cation benefit accrues over time. Due to the relatively high degree of independenceof returns during different intervals of time, risk generally compounds at a rateclose to the square root of time, a rate that is much less than the additive rate atwhich returns accrue.4 This difference between the rate at which return grows overtime and the rate at which risk grows over time leads to the second insight of port-folio theory—that patience in investments is rewarded and that total risk should be

spread relatively evenly over time.Consider a simple example. Taking one percentage point of risk per day createsonly about 16 percent5 of risk per year. If this one percentage point of risk per dayis expected to create two basis points6 of return per day, then over the course of 252business days in a year this amount of risk would generate an approximately 5 per-cent return. If, in contrast, the same total amount of risk, 16 percent, were concen-trated in one day rather than spread over the year, at the same rate of expectedreturn, two basis points per percentage point, it would generate only 2 · 16 = 32basis points of expected return, less than one-fifteenth as much. So time diversifica-

tion—that is, distributing risk evenly over a long time horizon—is another poten-tial free lunch for investors.All of us are familiar with the trade-offs between quality and cost in making

purchases. Higher-quality goods generally are more expensive; part of being a con-sumer is figuring out how much we can afford to spend on a given purchase. Simi-larly, optimal investing depends on balancing the quality of an investment (theamount of excess return an investment is expected to generate) against its cost (thecontribution of an investment to portfolio risk). In an optimal consumption plan, aconsumer should generate the same utility per dollar spent on every purchase. Oth-

erwise, dollars can be reallocated to increase utility. Similarly, in an optimal portfo-

The Insights of Modern Portfolio Theory  9

4In fact, as noted earlier, due to compounding, returns accrue at a rate greater than additive.To develop an intuition as to why risk does not increase linearly in time, suppose the risk ineach of two periods is of equal magnitude, but independent. The additive nature of the vari-ance of independent returns implies that the total volatility, the square root of total variance,sums according to the same Pythagorean formula that determines the hypotenuse of a righttriangle. Thus, in the case of equal risk in two periods, the total risk is not two units, but thesquare root of 2, as per the Pythagorean formula. More generally, if there are the square root

of t units of risk (after t periods), and we add one more unit of independent risk in period t +1, then using the same Pythagorean formula there will be the square root of  t + 1 units of risk after the t + 1st period. Thus, the total volatility of independent returns that have a con-stant volatility per unit of time grows with the square root of time. This will be a reasonablefirst-order approximation in many cases.5Note that 16 is just slightly larger than the square root of 252, the number of business daysin a year.6A basis point is one-hundredth of a percent.

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lio, the investor should generate the same expected return per unit of portfolio riskcreated in each investment activity. Otherwise, risk can be reallocated to achieve aportfolio with higher expected returns. The analogy between budgeting dollars inconsumption and budgeting risk in portfolio construction is powerful, but one has

to constantly keep in mind that in investing, risk is the scarce resource, not dollars.Unfortunately, many investors are not aware that such insights of modern port-folio theory have direct application to their decisions. Too often modern portfoliotheory is seen as a topic for academia, rather than for use in real-world decisions.For example, consider a common situation: When clients of our firm decide to sellor take public a business that they have built and in which they have a substantialequity stake, they receive very substantial sums of money. Almost always they willdeposit the newly liquid wealth in a money market account while they try to decidehow to start investing. In some cases, such deposits stay invested in cash for a sub-

stantial period of time. Often individuals do not understand and are not comfort-able taking investment risks with which they are not familiar. Portfolio theory isvery relevant in this situation and typically suggests that the investor should createa balanced portfolio with some exposure to public market securities (both domesticand global asset classes), especially the equity markets.

When asked to provide investment advice to such an individual, our first task isto determine the individual’s tolerance for risk. This is often a very interesting exer-cise in the type of situation described above. What is most striking is that in manysuch cases the individual we are having discussions with has just made or is contem-

plating an extreme shift in terms of risk and return—all the way from one end of therisk / return spectrum to the other. The individual has just moved from owning anilliquid, concentrated position that, when seen objectively, is extremely risky7 to amoney market fund holding that appears to have virtually no risk at all.8 Portfoliotheory suggests that for almost all investors neither situation is a particularly goodposition to be in for very long. And what makes such situations especially interestingis that if there ever happens to be a special individual, either a very aggressive risktaker or an extremely cautious investor, who ought to be comfortable with one of these polar situations, then that type of investor should be the least comfortable

with the other position. Yet we often see the same individual investor is comfortablein either situation, and even in moving directly from one to the other.The radically different potential for loss makes these two alternative situations

outermost ends of the risk spectrum in the context of modern portfolio theory. Andyet it is nonetheless difficult for many individuals to recognize the benefit of a morebalanced portfolio. Why is that? One reason is that people often have a very hardtime distinguishing between good outcomes and good decisions—and this is particu-larly true of good outcomes associated with risky investment decisions. The risk is of-

10 THEORY

7Of course, perceptions of risk can differ markedly from objective reality. This topic has beenrecently investigated by two academics, Tobias Moskowitz and Annette Vising-Jorgensen, ina paper entitled, “The Returns to Entrepreneurial Investment: The Private Equity PremiumPuzzle,” forthcoming in the American Economic Review.8We will come back to the important point that the short-term stability of the nominal pre-tax returns from a money market fund can actually create considerable real after-tax riskover longer periods of time.

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ten not recognized. Generally speaking, an investor who has just been successful in aninvestment wants to take credit for the good decisions that created this result and tothink of the result as being an almost inevitable consequence of the investor’s gooddecisions rather than to recognize that the outcomes of investment decisions, no mat-

ter how good, are, at least in the short run, usually very much a function of luck.Consider an investor in the situation just described. Such an individual is cer-tainly not typical. He or she has just joined the elite group of people who have expe-rienced the closest equivalent in the business world to winning the lottery. Thisindividual is among the lucky few with a concentrated risk position whose companieshave survived, grown profitably, and at an opportune time have been sold to the pub-lic. In retrospect, the actions taken by these individuals to create their wealth—thehard work, the business acumen, and in particular the holding of a concentrated po-sition—might seem unassailable. We might even suppose that other investors should

emulate their actions and enter into one or more such illiquid concentrated positions.However, there is a bigger picture. Many small business owners have businessesthat fail to create significant wealth. Just as in a lottery, the fact that there are a fewbig winners does not mean that a good outcome is always the result of a good invest-ment choice. Granting that there may be many psychic benefits of being a small busi-ness owner with a highly concentrated investment in one business, it is nonethelesstypically a very risky investment situation to be in. When a single business representsa significant fraction of one’s investment portfolio, there is an avoidable concentra-tion of risk. The simplest and most practical insight from modern portfolio theory is

that investors should avoid concentrated sources of risk.9

Concentrated risk positionsignore the significant potential risk reduction benefit derived from diversification.While it is true that to the extent that a particular investment looks very attractive itshould be given more of the overall risk budget, too much exposure can be detrimen-tal. Portfolio theory provides a context in which one can quantify exactly how muchof an overall risk budget any particular investment should consume.

Now consider the investors who put all of their wealth in money market funds.There is nothing wrong with money market funds; for most investors such fundsshould be an important, very liquid, and low-risk portion of the overall portfolio.

The problem is that some investors, uncomfortable with the potential losses fromrisky investments, put too much of their wealth in such funds and hold such posi-tions too long. Over short periods of time, money market funds almost always pro-duce steady, positive returns. The problem with such funds is that over longerperiods of time the real returns (that is, the purchasing power of the wealth createdafter taking into account the effects of inflation and taxes) can be quite risky andhistorically have been quite poor.

Modern portfolio theory has one, and really only one, central theme: In con-structing their portfolios investors need to look at the expected return of each in-

vestment in relation to the impact that it has on the risk of the overall portfolio. Wewill come back to analyze in more detail why this is the case, but because it is, thepractical message of portfolio theory is that sizing an investment is best understood

The Insights of Modern Portfolio Theory  11

9Unfortunately, in the years 2000 and 2001 many employees, entrepreneurs, and investors intechnology, telecommunications, and Internet companies rediscovered firsthand the risks as-sociated with portfolios lacking diversification.

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as an exercise in balancing its expected return against its contribution to portfoliorisk.10 This is the fundamental insight from portfolio theory. This insight was firstsuggested by Harry Markowitz (1952) and developed in his subsequent texts (1959and 1987). Upon first reflection, this insight seems intuitive and not particularly re-

markable. As we will see, however, getting it right in building portfolios is generallyneither easy nor intuitive.The first complication is perhaps obvious. It is hard to quantify either expected

returns or contributions to portfolio risk.11 Thus, balancing the two across differentinvestments is especially difficult. Coming up with reasonable assumptions for ex-pected returns is particularly problematic. Many investors focus on historical re-turns as a guide, but in this book we will emphasize an equilibrium approach toquantifying expected returns. We will return to this topic in Chapters 5 and 6.Here, we focus on measuring the contribution to portfolio risk, which, though still

complex, is nonetheless more easily quantified. For an investor the risk that eachinvestment adds to a portfolio depends on all of the investments in the portfolio, al-though in most cases in a way that is not obvious.

The primary determinant of an investment’s contribution to portfolio risk isnot the risk of the investment itself, but rather the degree to which the value of thatinvestment moves up and down with the values of the other investments in theportfolio. This degree to which these returns move together is measured by a statis-tical quantity called “covariance,” which is itself a function of their correlationalong with their volatilities. Covariance is simply the correlation times the volatili-

ties of each return. Thus, returns that are independent have a zero covariance,while those that are highly correlated have a covariance that lies between the vari-ances of the two returns. Very few investors have a good intuition about correla-tion, much less any practical way to measure or monitor the covariances in theirportfolios. And to make things even more opaque, correlations cannot be observeddirectly, but rather are themselves inferred from statistics that are difficult to esti-mate and which are notoriously unstable.12 In fact, until very recently, even profes-sional investment advisors did not have the tools or understanding to takecovariances into account in their investment recommendations. It is only within the

past few years that the wider availability of data and risk management technologyhas allowed the lessons of portfolio theory to be more widely applied.The key to optimal portfolio construction is to understand the sources of risk

in the portfolio and to deploy risk effectively. Let’s ignore for a moment the difficul-ties raised in the previous paragraph and suppose we could observe the correlationsand volatilities of investment returns. We can achieve an increased return by recog-nizing situations in which adjusting the sizes of risk allocations would improve the

12 THEORY

10In an optimal portfolio this ratio between expected return and the marginal contribution toportfolio risk of the next dollar invested should be the same for all assets in the portfolio.11Each of these topics will be the subject of later chapters. Equilibrium expected returns arediscussed in Chapters 5 and 6, and deviations from equilibrium in Chapter 7. Estimating co-variance is the topic of Chapter 16.12Whether the unobserved underlying correlations themselves are unstable is a subtle ques-tion. The statistics used to measure correlations over short periods of time, which have esti-mation error, clearly are unstable.

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expected return of the overall portfolio. A typical situation would be one in whichan asset is relatively independent of other investments in a portfolio and eventhough it may be risky by itself, it tends to add little to the overall risk of the port-folio. We refer to such investments as diversifiers, and we use them to increase re-

turn while living within an overall risk budget. Understanding and being able tomeasure and monitor the contribution to portfolio risk of every investment be-comes a key part of the decision about how much to invest in each asset or invest-ment activity. Assets that contribute less risk to a portfolio are less expensive interms of using up the risk budget, and, everything else being equal, we should in-vest more in them.

The intuition behind the mathematics that determines portfolio volatility canbe seen in the geometry of a simple diagram. An asset affects the risk of a portfolioin the same way that the addition of a side to a line segment changes the distance of 

the end point to the origin. This nonlinear nature of adding risks, and the depen-dence on correlation, is illustrated in Figure 2.2.The length of the original line segment represents the risk of the original port-

folio. We add a side to this segment; the length of the side represents the volatilityof the new asset. The distance from the end of this new side to the origin representsthe risk of the new portfolio. In the geometry of this illustration, it is clear how theangle between the new side and the original line segment is critical in determininghow the distance to the origin is changed. In the case of portfolio risk, the correla-tion of the new asset with the original portfolio plays the same role as the angle be-

tween the new side and the original line segment. Correlations range between –1and +1 and map into angles ranging from 0 to 180 degrees. The case of no correla-tion corresponds to a 90-degree angle. Positive correlations correspond to anglesbetween 90 and 180 degrees, and negative correlations correspond to angles be-tween 0 and 90 degrees.

Let us consider a relatively simple example of how to use measures of contribu-tion to portfolio risk to size investments and to increase expected returns. A keyquestion that faces both individual and institutional investors is how much to in-vest in domestic versus international equities. One school of thought is that as

The Insights of Modern Portfolio Theory  13

FIGURE 2.2 Summation of Risk Depends on Correlation

A

C

A = Old Portfolio Risk

B = New Investment Risk

C = New Portfolio Risk

Correlation Determines

the Angle between A and B

B

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global markets have become more correlated recently, the value of diversifying intointernational equities decreases. Let us see how modern portfolio theory addressesthis question. In this example we will initially treat domestic and international eq-uities as if they were the only two asset classes available for investment.

In the absence of other constraints (transactions costs, etc.), optimal allocationof the risk budget requires equities to be allocated from domestic to internationalmarkets up to the point where the ratio of expected excess return13 to the marginalcontribution to portfolio risk is the same for both assets. We focus on this marginalcondition because it can provide guidance toward improving portfolios. Although afull-blown portfolio optimization is straightforward in this context, we deliberatelyavoid approaching the problem in this way because it tends to obscure the intuitionand it does not conform to most investors’ behavior. Portfolio decisions are almostalways made at the margin. The investor is considering a purchase or a sale and

wants to know how large to scale a particular transaction. The marginal conditionfor portfolio optimization provides useful guidance to the investor whenever suchdecisions are being made.

This example is designed to provide intuition as to how this marginal conditionprovides assistance and why it is the condition that maximizes expected returns fora given level of risk. Notice that we assume that, at a point in time, the total risk of the portfolio must be limited. If this were not the case, then we could always in-crease expected return simply by increasing risk.

Whatever the initial portfolio allocation, consider what happens if we shift a

small amount of assets from domestic equities to international equities and adjustcash in order to hold the risk of the portfolio constant. In order to solve for theappropriate trades, we reallocate the amounts invested in domestic and interna-tional equities in proportion to their marginal contribution to portfolio risk. Forexample, if at the margin the contribution to portfolio risk of domestic equities istwice that of international equities, then in order to hold risk constant for eachdollar of domestic equities sold we have to use a combination of proceeds pluscash to purchase two dollars’ worth of international equities. In this context, if the ratio of expected excess returns on domestic equities to international equities

is less than this 2 to 1 ratio of marginal risk contribution, then the expected re-turn on the portfolio will increase with the additional allocation to internationalequities. As long as this is the case, we should continue to allocate to interna-tional equities in order to increase the expected return on the portfolio withoutincreasing risk.

Let us adopt some notation and look further into this example. Let ∆ be themarginal contribution to the risk of the portfolio on the last unit invested in an as-set. The value of ∆ can be found by calculating the risk of the portfolio for a givenasset allocation and then measuring what happens when we change that allocation.

That is, suppose we have a risk measurement function, Risk(d , f ), that we use tocompute the risk of the portfolio with an amount of domestic equities, d , and anamount of international equities, f .

We use the notation Risk(d , f ) to emphasize that different measures of risk

14 THEORY

13Throughout this book when we use the phrase “expected excess return,” we mean the ex-cess over the risk-free rate of interest.

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could be used. Many alternative functional forms have been proposed to measureinvestors’ utility as a function of return distributions. As noted earlier, while in-vestors are generally very sensitive to losses, they often seem much less cognizant of the risk that can lead to losses. We will explore some of these issues in the next

chapter. To be concrete, we will here use the statistical measure, volatility, to quan-tify risk. For example, suppose we have some relevant data that allows us to mea-sure the volatilities and correlation of the returns of domestic and internationalequities. Let these quantities be σ

d , σ

f , and ρ, respectively. Then one example of a

simple risk function would be the volatility of the portfolio, given by:

(2.1)

Let us use the notation ∆d 

to refer to the marginal contribution to portfoliorisk of domestic equities. This quantity is defined to be the derivative of the riskfunction with respect to the quantity of domestic equity, that is, the difference inthe risk of portfolios that have the same amount of international equities, but asmall difference, δ, in domestic equities, divided by δ. Thus, we can formalize thisas an equation:

(2.2)

and let ∆d be the limit of ∆d (δ) as δ goes to zero.Similarly, the marginal contribution to risk of international equities is given by

∆f , which is defined as the limit of ∆

f (δ) as δ goes to zero, where:

(2.3)

These marginal contributions to risk are the key to optimal portfolio alloca-tions. As we shall see, a condition for a portfolio to be optimal is that the ratio of 

expected excess return to marginal contribution to risk is the same for all assets inthe portfolio.Let us return to the question of whether we can improve the portfolio by selling

domestic equity and buying international equity. The ratio of marginal contribu-tions to risk is ∆

d  / ∆

f . Let the expected excess returns on domestic and international

equities be given by ed 

and ef , respectively. Now suppose e

d  / e

f is less than ∆

d  / ∆

f .

How much international equity must we purchase in order to keep risk constant if we sell a small amount of domestic equities? The rate of change in risk from thesale of domestic equity sales is –∆

d per unit sold. In order to bring risk back up to

its previous level, we need to purchase (∆d  / ∆f ) units of international equity. The ef-fect on expected return to the portfolio is –ed 

per unit sold of domestic equity and+(∆

d  / ∆

f )e

f from the purchase of an amount of international equity that leaves risk

unchanged. If, in this context, expected return is increased, then we should con-tinue to increase the allocation to international equity. If expected return is de-creased, then we should sell international equity and buy domestic equity. The onlycase in which the expected return of the portfolio cannot be increased while hold-ing risk constant is if the following condition is true:

∆f d f d f  

( )( , ) ( , )

δδδ

=+ −Risk Risk

∆d d f d f  

( )( , ) ( , )

δδ

δ=

+ −Risk Risk

Risk( , ) / 

d f d f d f  d f d f  = + +( )• • • • • • •2 2 2 2

1 22σ σ σ σ ρ

The Insights of Modern Portfolio Theory  15

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(2.4)

Rearranging terms, we have:

(2.5)

Thus, in this simple two-asset example we have derived a simple version of thegeneral condition that the expected return divided by the marginal contribution toportfolio risk should be the same for all assets in order for a portfolio to be opti-mal. If this condition is not met, then we can increase the expected return of theportfolio without affecting its risk.

More generally, we can consider sales and purchases of any pair of assets in amultiple asset portfolio. The above analysis must hold, where in this context let therisk function, Risk(w), give the risk for a vector w, which gives the weights for allassets. Let Risk

m(w, δ) give the risk of the portfolio with weights w and a small in-

crement, δ, to the weight for asset m. Define the marginal contribution to portfoliorisk for asset m as ∆

m, the limit as δ goes to zero of:

(2.6)

Then, as earlier, in an optimal portfolio it must be the case that for every pairof assets, m and n, in a portfolio the condition

(2.7)

is true. If not, the prescription for portfolio improvement is to buy the asset forwhich the ratio is higher and sell the asset for which the ratio is lower and to con-

tinue to do so until the ratios are equalized. Note, by the way, that if the expectedreturn of an asset is zero then the optimal portfolio position must be one in whichthe ∆ is also zero. Readers familiar with calculus will recognize that this condi-tion—that the derivative of the risk function is zero—implies that the risk functionis at a minimum with respect to changes in the asset weight.

Let us consider how this approach might lead us to the optimal allocation tointernational equities. To be specific, let us assume the values shown in Table 2.1for the volatilities and expected excess returns for domestic and international eq-uity, and for cash. Assume the correlation between domestic and international

equity is .65.We will use as the risk function the volatility of the portfolio:

(2.8)

In order to make the analysis simple, let us assume that the investor wants tomaximize expected return for a total portfolio volatility of 10 percent. Consider an

Risk( , ) / 

d f d f d f  d f d f  = + +( )• • • • • • •2 2 2 2

1 2

2σ σ σ σ ρ

e em

m

n

n∆ ∆=

∆mm w w

( )( , ) ( )

δδδ

=−Risk Risk

e ed 

f ∆ ∆=

− + 

 

 

   =e ed 

f f 

∆∆

0

16 THEORY

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investor starting with an equity allocation that is totally domestic. In order to gen-erate a volatility of 10 percent the investor must hold a combination of cash plusdomestic equity. In particular, given the assumed 15 percent volatility of domesticequity, the proportion allocated to equity is two-thirds of the total value and the al-location to cash is one-third of the total value.

What happens as the investor starts to sell domestic equity and buy interna-tional equity? The marginal contributions to risk are simply the derivatives of thisrisk function with respect to the two arguments and can easily be shown to begiven by the formulas:

(2.9)

(2.10)

In the special case when f = 0, these formulas simplify to:

Suppose the portfolio has a valuation, v, which is a large number, and an in-vestor sells one unit of domestic equity; that is, let δ = –1. Recalling equation (2.6),

(2.11)

The risk of the portfolio is decreased by approximately:

(2.12)

In order to keep risk unchanged, the investor must purchase

(2.13)∆∆

= =.

..

15

1041 442

Risk Risk( , ) ( , ) . .d f d f  + − = = −•δ δ15 15

∆d d f d f  

( )( , ) ( , )

δδ

δ=

+ −Risk Risk

d d 

d f 

=

( )= =

= ( ) = =

• • •

••

σ

σσ

σ σ ρ

σ σ ρ

2

2 21 2

2 21 2

150

104

 / 

 / 

.

.

∆f f d f f d 

d f =+• • • •σ σ σ ρ2

Risk( , )

∆d d d f d f 

d f =

+• • • •σ σ σ ρ2

Risk( , )

The Insights of Modern Portfolio Theory  17

TABLE 2.1 Values for Volatilities and Expected Excess Returns

Volatility Expected Excess Return Total Return

Domestic equity 15% 5.5% 10.5%

International equity 16 5.0 10.0Cash 0 0.0 5.0

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units of international equity. The sale of one unit of domestic equity reduces portfo-lio expected excess return by .055.

The purchase of 1.442 units of international equity increases expected excessreturn by:

(2.14)

Thus, at the margin, selling domestic equity and purchasing international equity ata rate that keeps risk constant raises expected excess by

(2.15)

per unit of domestic equity sold.The signal provided by this marginal analysis is clear and intuitive. The in-

vestor should continue to sell domestic equity as long as the effect on portfolio ex-pected excess returns is positive and the risk is unchanged. Unfortunately, of course, this increasing of expected return cannot go on indefinitely. As soon as theinvestor sells domestic equity and purchases international equity, the marginal con-tribution to risk of domestic equity begins to fall and that of international equitybegins to rise. This effect is why the marginal analysis is only an approximation,valid for small changes in portfolio weights.

Before we investigate what happens as the investor moves from domestic to in-ternational equities, however, we might consider what is the expected excess return

on international equities for which the investor would be indifferent to such atransaction. Clearly, from the preceding analysis this point of indifference is givenby the value, e

f , such that:

(2.16)

In other words, the hurdle rate, or point of indifference for expected return, suchthat expected returns beyond that level justify moving from domestic to foreign eq-uity, is e

f = 3.8%.

To put it differently, if the expected excess return on foreign equity is less than

this value, then we would not have any incentive to purchase international equities.If we were to look only at the risks and not expected excess returns, we might

suppose that because of the diversification benefit we would always want to holdsome international equity, at least at the margin. In fact, when assets are positivelycorrelated, as they are in this example, even the first marginal allocation creates mar-ginal risk and requires an expected excess return hurdle in order to justify a purchase.

Now suppose the investor has sold 10 percent of the domestic equity. In or-der to keep risk constant the investor can purchase 13.18 percent of interna-tional equity. Using the new values d  = .5667 and f  = .1318 in the above

formulas we can confirm that the volatility of the portfolio remains 10 percentand that ∆

d = .148 and ∆

f = .122.

The impact on expected excess return of the portfolio per unit sold at this pointis given by:

(2.17)∆∆

f f d e e

 

 

 

   − = − =•( . . . ) .1 212 05 055 01667

( . . )1 442 055 0• − =ef 

. . .07215 055 01715− =

1 442 05 07215. . .• =

18 THEORY

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The investor should continue to sell domestic equity since the value is positive,though at this level the value in terms of incremental expected excess return to theportfolio per unit sold has dropped slightly, from .17 to .1667.

Again we might consider what is the expected excess return on international

equities for which the investor would be indifferent to an additional purchase. Thepoint of indifference is the value, ef , such that:

(2.18)

That is, ef = 4.5.

The hurdle rate to justify continued purchase of international equities has in-creased from 3.8 to 4.5 because the marginal contribution of international equitiesto portfolio risk has increased relative to that of domestic equities.

Suppose the investor decides to keep only 10 percent of the portfolio value in

domestic equity. In order to keep risk constant, the investor must purchase 56 per-cent of international equity. Using the new values d = .10 and f = .56 in the earlierformulas we can confirm that the volatility of the portfolio remains 10 percent andthat ∆

d = .110 and ∆

f = .159.

The impact on expected excess return of the portfolio per unit sold at this pointis given by

(2.19)

which simplifies as

(2.20)

Now the investor has sold too much domestic equity. The value in terms of in-cremental expected excess return to the portfolio per unit sold has dropped so farthat it has become negative. The negative impact on the portfolio expected returnsignals that at the margin the investor has too much risk coming from international

equity and the expected excess return does not justify it.The hurdle rate to justify continued purchase of international equities is thevalue, e

f , such that:

(2.21)

That is, ef = 8.0%.

Clearly this hurdle rate has continued to increase as the marginal contributionof international equities to portfolio risk has continued to increase relative to that

of domestic equities.Throughout this example, we have assumed that the investor has a set of ex-pected excess returns for domestic and international equities. In practice, few in-vestors have such well-formulated views on all asset classes. Notice, however, thatgiven an expected excess return on any one asset class, in this case domestic equities,we can infer the hurdle rate, or point of indifference for purchases or sales of everyother asset. We refer to these hurdle rates as the implied views of the portfolio.Rather than following the traditional portfolio optimization strategy, which requires

(. . )691 055 0• − =ef 

(. . . ) .691 05 055 012• − = −

∆∆

f f d e e

 

 

 

 

  −

( . . )1 212 055 0• − =ef 

The Insights of Modern Portfolio Theory  19

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prior specification of expected excess returns for all assets, we can take an existingportfolio, make an assumption of excess return on one asset (or more generally onany one combination of assets such as a global equity index), and back out the im-plied views on all others. Purchases of an asset are warranted when the hurdle rate

given by the implied view appears to be lower than one’s view of what a reasonablevalue is. Conversely, sales are warranted when the implied view appears to be abovea reasonable value. Implied views provide insight for deciding how large to make in-vestments in an existing portfolio.

There is, however, an additional layer of complexity that we have not yet re-flected: the role of correlation in determining optimal positions. In the earlier analy-sis, the role correlation played, through its impact on portfolio risk and marginalcontribution to portfolio risk, was not highlighted.

In order to highlight the role of correlation, we extend the previous example by

considering a new asset, commodities, which we suppose has volatility of 25 per-cent, and correlations of –.25 with both domestic and international equities. Con-sider again the original portfolio invested two-thirds in domestic equities and therest in cash. If we consider adding commodities to this portfolio, the marginal con-tribution to portfolio risk of commodities, ∆

c, is –.066. Because domestic equity

risk is the only risk in the portfolio, a marginal investment in commodities, which isnegatively correlated with domestic equity, reduces risk.

This negative marginal contribution to portfolio risk for commodities leads toa new phenomenon. Commodities are a diversifier in the portfolio. The previous

type of analysis, where we sold domestic equity and bought enough internationalequity to hold risk constant, doesn’t work. If we sell domestic equities and try toadjust the commodity weight to keep risk constant, we have to sell commodities aswell. If instead we were to purchase commodities, then we would reduce risk onboth sides of the transaction.

Retain the assumption that the expected excess return on domestic equities is5.5 percent and consider the hurdle rate for purchases of commodities, which isgiven by the expected excess return, e

c, such that:

(2.22)

That is,

(2.23)

(2.24)

(2.25)

Here we see an interesting result. When there is no existing position in com-modities in this portfolio, the implied view for commodities is a negative ex-pected excess return.

ec = −2 42. %

− − =•2 27 055 0. .ec

.

..

150

066055 0

− 

  

   − =ec

∆∆

cce 

      − =.055 0

20 THEORY

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Now suppose we assume a 5 percent long position in commodities. Most in-vestors believe that a long position implies a positive expected excess return andthat the larger the position, the larger is the implied view. As we shall see here, thatis not necessarily the case; the implied view may not even have the same sign as the

position. With the commodity position at 5 percent and the domestic equity posi-tion unchanged, the portfolio volatility drops to 9.76 percent. The marginal contri-butions to portfolio risk, ∆

d and ∆

c, become .149 and –.032, respectively. The

marginal contribution of domestic equity has declined while the marginal contribu-tion for commodities remains negative, but has increased closer to zero. Considerthe new implied view for commodities, the value of e

csuch that:

(2.26)

–4.65 · ec– .055 = 0 (2.27)

ec= –1.18% (2.28)

Here we see a truly counterintuitive result. Despite our positive holding of a signifi-cant 5 percent of the portfolio weight in the volatile commodities asset class, theimplied view for commodities is a negative expected excess return.

Perhaps one might at this point jump to the conclusion that this counterintu-

itive sign reversal will always be the case when the correlations between two assetsare negative. However, that is not so. Let us see what happens when we further in-crease the size of the commodity position from 5 percent to 15 percent of the port-folio. The volatility of the portfolio remains unchanged at 9.76 percent. Theportfolio volatility is minimized at 9.68 when there is a 10 percent weight in com-modities. At 15 percent commodities the volatility is increasing as more commodi-ties are added. The marginal contributions to portfolio risk, ∆

d and ∆

c, are now

.139 and .032, respectively. The contribution of domestic equity continues to de-cline while the marginal contribution for commodities has increased from a nega-

tive value to a positive value.The new hurdle rate for commodities is given by the expected excess return, e

c,

such that:

(2.29)

4.35 · ec– .055 = 0 (2.30)

ec= 1.26% (2.31)

Clearly at 15 perceent of portfolio weight, the hurdle rate on commodities hasbecome positive. As the weight on commodities increased from 5 percent to 15 per-cent the impact on the portfolio changed from being a diversifier to being a sourceof risk. In fact, there is a weight in commodities for which the portfolio volatility isminimized. This risk-minimizing value for commodities, holding all other assets

.

..

139

032055 0

 

  

   − =ec

.

..

149

032055 0

− 

  

   − =ec

The Insights of Modern Portfolio Theory  21

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constant, is a special and interesting position. It has the property that this is thepoint where the marginal contribution to risk, and therefore the implied excess re-turn on commodities, is zero. We can solve for the risk-minimizing position by set-ting ∆

c= 0, or equivalently, solving for c such that (c · σ

c2 + d · σ

d  · σc · ρ

dc) = 0

where ρdc is the correlation between commodities and domestic equity. Holdingfixed the two-thirds weight in domestic equity, this risk-minimizing position incommodities is 10 percent.

Thus, an important intuition that helps make sense of implied views is as fol-lows: Holding fixed the weights in all other assets, there is a risk-minimizing posi-tion for each asset. Weights greater than that risk-minimizing position reflectpositive implied views; weights less than that risk-minimizing position reflect bear-ish views. In terms of implied views, there is nothing special about positions greaterthan or less than zero; the neutral point is the risk-minimizing position. In a single-

asset portfolio the risk-minimizing position is, of course, zero. More generally,however, the risk-minimizing position is a function of the positions, volatilities, andcorrelations of all assets in the portfolio. Moreover, in multiple-asset portfolios, therisk-minimizing position for each asset can be a positive or a negative value.

We can use the correlations among assets and the risk-minimizing position toidentify opportunities to improve allocations in portfolios. In multiple-asset portfo-lios, the risk-minimizing position will only be at zero for assets that are uncorre-lated with the rest of the portfolio. Such uncorrelated assets are likely to be veryvaluable. Any asset or investment activity that is uncorrelated with the portfolio,

but also has a positive expected excess return, should be added to the portfolio. Inaddition to commodities, such uncorrelated activities might include the active riskrelative to benchmark of traditional active asset managers, certain types of hedgefunds, active currency overlays, and global tactical asset allocation mandates.

More generally, in the case of assets or activities that do have correlations withthe existing portfolio and therefore that have nonzero risk-minimizing positions,any position that lies between zero and the risk-minimizing position is likely to rep-resent an opportunity for the investor. Such positions are counterintuitive in thesame sense that the 5 percent commodity position was. The implied view is oppo-

site to the sign of the position. Typically investors hold positive positions becausethey have positive views, and vice versa. Whenever this is the case and the actualposition is less than the risk-minimizing position, it makes sense to increase the sizeof the position. This situation is an opportunity because increasing the size of theposition will both increase expected return and decrease risk.

In terms of asset allocation, the counterintuitive positions described here arenot very common. Most positions in asset classes are long positions (very few in-vestors hold short positions in asset classes), most asset returns correlate positivelywith portfolio returns (commodities are an exception), and most assets are ex-

pected to have positive excess returns. More generally, though, we will see thatwhen portfolios of securities are constructed with risk measured relative to abenchmark, such counterintuitive positions arise quite often.

In this chapter we have taken the simple idea of modern portfolio theory—thatinvestors wish to maximize return for a given level of risk—and developed somevery interesting, and not particularly obvious, insights into the sizing of positions.We have tried to develop these ideas in a way that is intuitive and which can beused to help make portfolio decisions at the margin. We avoid the usual approach

22 THEORY

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