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with a large positive correlation of returns, the portfolio standard deviation will increase monotonically from the low-risk asset to the high-risk asset. Even in this case, however, there is a positive (if small) benefit from diversification.

What is the minimum level to which portfolio standard deviation can be held? For the parameter values stipulated in Table 7.1, the portfolio weights that solve this minimization problem turn out to be4

wMin(D)!=!.82 wMin(E )!=!1!"!.82!=!.18

This minimum-variance portfolio has a standard deviation of #Min!=! [ (.822!$!122 )!+!(.182!$!202 )!+!(2!$!.82!$!.18!$!72 ) ] 1/2!=!11.45%

as indicated in the last line of Table 7.3 for the column % = .30. The solid colored line in Figure 7.4 plots the portfolio standard deviation when % = .30

as a function of the investment proportions. It passes through the two undiversified port- folios of wD = 1 and wE = 1. Note that the minimum-variance portfolio has a standard deviation smaller than that of either of the individual component assets. This illustrates the effect of diversification.

The other three lines in Figure 7.4 show how portfolio risk varies for other values of the correlation coefficient, holding the variances of each asset constant. These lines plot the values in the other three columns of Table 7.3.

The solid dark straight line connecting the undiversified portfolios of all bonds or all stocks, wD = 1 or wE = 1, shows portfolio standard deviation with perfect positive correlation, % = 1. In this case there is no advantage from diversification, and the port- folio standard deviation is the simple weighted average of the component asset standard deviations.

The dashed colored curve depicts portfolio risk for the case of uncorrelated assets, % = 0. With lower correlation between the two assets, diversification is more effective and portfolio risk is lower (at least when both assets are held in positive amounts). The mini- mum portfolio standard deviation when % = 0 is 10.29% (see Table 7.3), again lower than the standard deviation of either asset.

Finally, the triangular broken line illustrates the perfect hedge potential when the two assets are perfectly negatively correlated (% = "1). In this case the solution for the minimum-variance portfolio is, by Equation 7.12,

wMin(D; %!=!"1)!=! #E _______

#D!+!#E !=! 20 _______

12!+!20 !=!.625

wMin(E; %!=!"1)!=!1!"!.625!=!.375 and the portfolio variance (and standard deviation) is zero.

We can combine Figures 7.3 and 7.4 to demonstrate the relationship between portfolio risk (standard deviation) and expected return—given the parameters of the available assets.

4This solution uses the minimization techniques of calculus. Write out the expression for portfolio variance from Equation 7.3, substitute 1 " wD for wE, differentiate the result with respect to wD, set the derivative equal to zero, and solve for wD to obtain

wMin(D)!=! # E 2 " Cov(rD, rE) ____________________

# D 2 + # E 2 " 2 Cov(rD, rE)

Alternatively, with a spreadsheet program such as Excel, you can obtain an accurate solution by using the Solver to minimize the variance. See Appendix A for an example of a portfolio optimization spreadsheet.

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When we maximize the objective function, Sp, we have to satisfy the constraint that the portfolio weights sum to 1.0, that is, wD + wE = 1. Therefore, we solve an optimization problem formally written as

Max w i

S p !=! E(rp)!"!rf ________

#p

subject to !wi = 1. This is a maximization problem that can be solved using standard tools of calculus.

In the case of two risky assets, the solution for the weights of the optimal risky portfolio, P, is given by Equation 7.13. Notice that the solution employs excess returns (denoted R) rather than total returns (denoted r).6

wD!=! E(RD) # E 2 !"!E(RE) Cov(RD, RE) ___________________________________________

E(RD) # E 2 !+!E(RE) # D 2 !"![E(RD)!+!E(RE)] Cov(RD, RE) (7.13)

wE!=!1!"!wD

6The solution procedure for two risky assets is as follows. Substitute for E(rp) from Equation 7.2 and for #p from Equation 7.7. Substitute 1 " wD for wE. Differentiate the resulting expression for Sp with respect to wD, set the derivative equal to zero, and solve for wD.

Using our data, the solution for the optimal risky portfolio is:

wD!=! ("!"!#)$%%!"!(&'!"!#)()

___________________________________ ("!"!#)$%%!+!(&'!"!#)&$$!"!("!"!#!+!&'!"!#)() !=!.$%

wE!=!&!"!.$%!=!.*%

The expected return and standard deviation of this optimal risky portfolio are

E(rP)!=!(.$!$!")!+!(.*!$!&')!=!&&% #P!=! [ (.$)!$!&$$)!+!(.*)!$!$%%)!+!()!$!.$!$!.*!$!()) ] &/)!=!&$.)%

This asset allocation produces an optimal risky portfolio whose CAL has a slope of

SP!=! &&!"!#

_____ &$.)

!=!.$)

which is the Sharpe ratio of portfolio P. Notice that this slope exceeds the slope of any of the other feasible portfolios that we have considered, as it must if it is to be the slope of the best feasible CAL.

Example 7.2 Optimal Risky Portfolio

In Chapter 6 we found the optimal complete portfolio given an optimal risky portfolio and the CAL generated by a combination of this portfolio and T-bills. Now that we have constructed the optimal risky portfolio, P, we can use its expected return and volatility along with the individual investor’s degree of risk aversion, A, to calculate the optimal proportion of the complete portfolio to invest in the risky component.

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The points marked by squares are the result of a variance-minimization program. We first draw the constraints, that is, horizontal lines at the level of required expected returns. We then look for the portfolio with the lowest standard deviation that plots on each hori- zontal line—in other words, we look for the portfolio that will plot farthest to the left (smallest standard deviation) on that line. When we repeat this for many levels of required expected returns, the shape of the minimum-variance frontier emerges. We then discard the bottom (dashed) half of the frontier, because it is inefficient.

In the alternative approach, we draw a vertical line that represents the standard devia- tion constraint. We then consider all portfolios that plot on this line (have the same stan- dard deviation) and choose the one with the highest expected return, that is, the portfolio that plots highest on this vertical line. Repeating this procedure for many vertical lines (levels of standard deviation) gives us the points marked by circles that trace the upper por- tion of the minimum-variance frontier, the efficient frontier.

When this step is completed, we have a list of efficient portfolios because the solution to the optimization program includes the portfolio proportions, wi, the expected return, E(rp), and the standard deviation, !p.

Let us restate what our portfolio manager has done so far. The estimates generated by the security analysts were transformed into a set of expected rates of return and a covari- ance matrix. We call this group of estimates the input list. This input list is then fed into the optimization program.

Before we proceed to the second step of choosing the optimal risky portfolio from the frontier set, let us consider a practical point. Some clients may be subject to addi- tional constraints. For example, many institutions are prohibited from taking short positions in any asset. For these clients the portfolio manager will add to the opti- mization program constraints that rule out negative (short) positions in the search for efficient portfolios. In this special case it is possible that single assets may be, in and of themselves, efficient risky portfolios. For example, the asset with the highest expected return will be a frontier portfolio because, without the opportunity of short sales, the only way to obtain that rate of return is to hold the asset as one’s entire risky portfolio.

Short-sale restrictions are by no means the only such constraints. For example, some clients may want to ensure a minimal level of expected dividend yield from the opti- mal portfolio. In this case the input list will be expanded to include a set of expected dividend yields d1, . . . , dn and the optimization program will include an additional con- straint that ensures that the expected dividend yield of the portfolio will equal or exceed the desired level, d.

Another type of constraint is aimed at ruling out investments in industries or countries considered ethically or politically undesirable. This is referred to as socially responsible investing." Portfolio managers can tailor the efficient set to conform to any desire of the client. Of course, any constraint carries a price tag in the sense that an efficient fron- tier constructed subject to extra constraints will offer a Sharpe ratio inferior to that of a less constrained one. The client should be made aware of this cost and should carefully consider constraints that are not mandated by law.

Capital Allocation and the Separation Property Now that we have the efficient frontier, we proceed to step two and introduce the risk-free asset. Figure 7.13 shows the efficient frontier plus three CALs representing various portfo- lios from the efficient set. As before, we ratchet up the CAL by selecting different portfo- lios until we reach portfolio P, which is the tangency point of a line from F to the efficient

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If we define the average variance and average covariance of the securities as

̄ ! 2"=" 1 __ n ! i = 1

n ! i 2 (7.18)

̄ Cov "=" 1 _______ n(n"#"1)

" ! j = 1 ____ j $ i

n !

i=1

n Cov(ri , rj ) (7.19)

we can express portfolio variance as

! p 2 "=" 1 __ n

̄ ! 2"+" n"#"1 _____ n ̄ Cov (7.20)

Now examine the effect of diversification. When the average covariance among security returns is zero, as it would be if all risk were firm-specific, portfolio variance can be driven to zero. We see this from Equation 7.20. The second term on the right-hand side will be zero in this scenario, while the first term approaches zero as n becomes larger. Hence when security returns are uncorrelated, the power of diversification to reduce portfolio risk is unlimited.

However, the more important case is the one in which economywide risk factors impart positive correlation among stock returns. In this case, as the portfolio becomes more highly diversified (n increases), portfolio variance remains positive. Although firm-specific risk, represented by the first term in Equation 7.20, approaches zero as diversification increases (n gets ever larger), the second term simply approaches ̄ Cov . [Note that (n # 1)/n = 1 # 1/n, which approaches 1 for large n.] Thus, the irreducible risk of a diversified portfolio depends on the covariance of the returns of the component securities, which in turn is a function of the importance of systematic factors in the economy.

To see further the fundamental relationship between systematic risk and security corre- lations, suppose for simplicity that all securities have a common standard deviation, !, and all security pairs have a common correlation coefficient, %. Then the covariance between any pair of securities is %!2, and Equation 7.20 becomes

! p 2 "=" 1 __ n ! 2 "+" n"#"1 ____

n % ! 2 (7.21)

The effect of correlation is now explicit. When % = 0, we again obtain the insurance principle, where portfolio variance approaches zero as n becomes greater. For % > 0, how- ever, portfolio variance remains positive. In fact, for % = 1, portfolio variance equals !2 regardless of n, demonstrating that diversification is of no benefit: In the case of perfect correlation, all risk is systematic. More generally, as n becomes ever larger, Equation 7.21 shows that variance approaches %!2. We can think of this limit as the “systematic variance” of the security market.

Table 7.4 presents portfolio standard deviation as we include ever-greater numbers of securities in the portfolio for two cases, % = 0 and % = .40. The table takes ! to be 50%. As one would expect, portfolio risk is greater when % = .40. More surprising, perhaps, is that portfolio risk diminishes far less rapidly as n increases in the positive correlation case. The correlation among security returns limits the power of diversification.

Note that for a 100-security portfolio, the standard deviation is 5% in the uncorrelated case—still significant compared to the potential of zero standard deviation. For % = .40, the standard deviation is high, 31.86%, yet it is very close to undiversifiable systematic standard deviation in the infinite-sized security universe, "

___ % ! 2 "=" "

_______ .4"&" 50 2 "="31.62% .

At this point, further diversification is of little value.

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spectrum of financial markets and financial instruments has put sophisticated investment beyond the capacity of many amateur investors. Finally, there are strong economies of scale in investment analysis. The end result is that the size of a competitive investment company has grown with the industry, and efficiency in organization has become an important issue.

A large investment company is likely to invest both in domestic and international mar- kets and in a broad set of asset classes, each of which requires specialized expertise. Hence the management of each asset-class portfolio needs to be decentralized, and it becomes impossible to simultaneously optimize the entire organization’s risky portfolio in one stage, although this would be prescribed as optimal on theoretical grounds. In future chap- ters we will see how optimization of decentralized portfolios can be mindful as well of the entire portfolio of which they are a part.

The practice is, therefore, to optimize the security selection of each asset-class portfolio independently. At the same time, top management continually updates the asset allocation of the organization, adjusting the investment budget allotted to each asset-class portfolio.

Optimal Portfolios and Non-Normal Returns The portfolio optimization techniques we have used so far assume normal distributions of returns in that standard deviation is taken to be a fully adequate measure of risk. However, potential non-normality of returns requires us to pay attention as well to risk measures that focus on worst-case losses such as value at risk (VaR) or expected shortfall (ES).

In Chapter 6 we suggested that capital allocation to the risky portfolio ideally should account for fat-tailed distributions that can result in extreme values of VaR and ES. Spe- cifically, forecasts of greater than normal VaR and ES should encourage more moderate capital allocations to the risky portfolio. Accounting for the effect of diversification on VaR and ES would be useful as well. Unfortunately, the impact of diversification on tail risk cannot be easily estimated.

A practical way to estimate values of VaR and ES in the presence of fat tails is called bootstrapping. We start with a historical sample of returns of each asset in our prospective portfolio. We compute the portfolio return corresponding to a draw of one return from each asset’s history. We thus calculate hypothetical (but still empirically based) returns on as many of these random portfolio returns as we wish. Fifty thousand portfolio returns pro- duced in this way can provide a good estimate of VaR and ES values. The forecasted values for VaR and ES of the mean-variance optimal portfolio can then be compared to other candidate portfolios. If these other portfolios yield sufficiently better VaR and ES values, we may prefer one of those to the mean-variance efficient portfolio.

Diversification entails spreading the investment budget across a variety of assets in order to limit overall risk. It is common, as we have done here, to use the analogy between insur- ance companies spreading risk across policies and investors diversifying their portfolios to illustrate how diversification reduces risk. While that analogy is useful, we actually have to be a bit careful about the source of the risk reduction. We will see here that risk reduction actually requires both!risk pooling (spreading your exposures across multiple uncorrelated

7.5 Risk Pooling, Risk Sharing, and the Risk of Long-Term Investments*

*The material in this section is more challenging. It may be skipped without impairing the ability to understand later chapters.

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risky ventures) as well as risk sharing (allowing other investors to share in the risk of a portfolio of assets). The utility of capital markets in a developed economy rests in large part on this distinction.

The confusion between the roles of risk pooling and risk sharing leads to another, related confusion. A widespread, but incorrect, belief is that spreading investments across time, so that average performance reflects returns in several investment periods, offers a sort of “time diversification.” Many observers argue that time diversification can make long-term investing less risky. We will see that extending the horizon of a risky investment is analogous to risk pooling. But this application of “the insurance principle” to long-term investments ignores the crucial role of risk sharing in portfolio risk management, and can easily lead to poor investment decisions. Long-term investments are not necessarily safer.

In this section, therefore, we try to clarify these issues and explore the appropriate extension of the insurance principle to investment risk over different horizons. We start by reviewing the respective contributions of risk pooling and risk sharing to the benefits of portfolio diversification. With these insights in hand, we can better understand the risk of long-term investments.

Risk Pooling and the Insurance Principle Risk pooling means adding uncorrelated, risky projects to the investor’s portfolio. Applied to the insurance business, risk pooling entails selling many uncorrelated insurance policies. This application of risk pooling as a means to reduce risk has therefore come to be known as the insurance principle. Conventional wisdom holds that such pooling is the driving force behind risk management for the insurance industry.

But even brief reflection should convince you that risk pooling cannot be the entire story. How can adding new bets (selling additional insurance policies) that are independent of your other bets reduce your total exposure to risk? This would be little different from a gambler in Las Vegas arguing that a few more trips to the roulette table will reduce his total risk by diversifying his overall “portfolio” of wagers. You would immediately realize that the gambler now has more money at stake, and the overall uncertainty in his wealth is clearly greater: While his average gain or loss per bet may become more predictable as he repeatedly returns to the table, his total proceeds become less so.

Think about investing $1 in a single risky security, call it A, with risky rate of return rA and total payoff of 1 + rA dollars. The mean of rA is E(r), the standard deviation is !, and the variance is !2. Now think about risk pooling by taking on another investment of $1 in an uncorrelated investment, call it B, with return rB that has the same mean and variance as security A. The total payoff to your two-asset portfolio, which we will call portfolio"P, is (1 + rA) + (1 + rB).

The expected dollar profit on your $2 investment is 2"#!E(r), and, because the covari- ance between the two investments is zero, the variance of the dollar payoff is Var(rA + rB) = 2!2. It is clear that with double the variance, this position is riskier than one invested only in asset A. This is just like the gambler in Las Vegas who goes to the roulette table twice rather than once.

But this greater risk is not readily apparent when we compute only the rate of return statistics of the portfolio. With half the portfolio invested in A and half in B, the weights on each security are 1 ⁄ 2 , so the expected rate of return and the volatility of the rate of return on the two-asset portfolio are:

E(rP)"=" 1 ⁄ 2 E(r)"+" 1 ⁄ 2 E(r)"="E(r) Var(rP)"="( 1 ⁄ 2 )2 !2"+"( 1 ⁄ 2 )2 !2"+"2"#" 1 ⁄ 2 "#" 1 ⁄ 2 "#"Cov(rA,"rB)"=" 1 ⁄ 2 !2

SD(rP)"="!P"=" ! ___

1 ⁄ 2 "#"!

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It looks like the “diversified” two-asset portfolio is safer; its variance has fallen by a fac- tor of 1 ⁄ 2 . But this apparent safety is an illusion. While its rate of return is more predictable, we know the dollar variance of the “risk-pooled portfolio” is double the variance of the one-asset portfolio. Again, this is just like our Las Vegas gambler. With every trip to the roulette table, percentage gains (in this case, losses) are more predictable, but dollar gains or losses are less so.

Why do we obtain these seemingly conflicting signals of risk? Because we are compar- ing two portfolios of different sizes. The first investment, in asset A, is for only $1. The investment in the two-asset risk-pooled portfolio, P, is $2. With twice as much money at risk, it’s no wonder that the two-asset portfolio has riskier dollar profits even if its rate of return is more predictable. The standard deviation of the dollar profit of portfolio P is $2 ! "P = $2 ! [ !

___ 1 / 2 ! "] =# !

__ 2 ! ", and the variance is 2"2, just as we found above.

Risk Sharing Now think of a variation on our investor’s two-asset portfolio. Let him start as before, by putting a dollar into both assets A and B, but now imagine that he sells off half of his total investment to other investors. In so doing, he augments his risk-pooling strategy with a risk sharing strategy. Crucially, this strategy maintains the size of his total investment at $1, even as he adds the second security to his portfolio. Now that the investor’s total invest- ment is held fixed, we can compare portfolios using their expected rates of return together with the standard deviation and variance of those rates of return. This is because the rates of return are now applied to the same investment base, and we don’t have to worry about scaling up these risk and return measures by the different amounts put at risk.14#

We’ve already established that the rate of return on the two-asset portfolio is 1 ⁄ 2 (rA + rB), with expected return E(r) and standard deviation# !

___ 1 / 2 ! ". This reduction in standard

deviation on a fixed $1 investment is truly a reduction in risk. Risk pooling, together with risk sharing, results in portfolios with a superior risk–return trade-off, in this example, a portfolio with the same expected return but lower volatility.

What does this have to do with the insurance principle? Risk pooling is a large part of what makes the insurance industry tick. But by itself, risk pooling actually increases the volatility of profits as the company writes more and more policies. However, when risk sharing is part of the strategy, allowing more and more investors to share the risk, each investor’s personal investment does not have to grow as the insurance company sells more policies. Instead, the many thousands of investors who own shares in the insurance company can determine exactly how much of their investment budget to place in the company independently of how many policies the company has written.

Similarly, capital markets allow investors to gain the benefits of diversification by widely sharing firm-specific risks. When an investor adds more stocks to a risky portfolio but leaves the total size of the portfolio unchanged, she necessarily must own a smaller fraction of each firm included in her portfolio. In other words, she must be sharing progres- sively more of the risk of that firm with the rest of the capital market. Such sharing reduces her exposure to any particular stock and allows her to be less and less concerned about its firm-specific risk. By enabling investors to combine risk sharing with risk pooling, 14There is a good analogy to corporate finance here. You may remember that the IRR rule for evaluating a capital budgeting project expresses profitability on a rate of return basis, whereas the NPV rule evaluates projects in terms of dollar values. While the IRR rule usually will give us the correct accept/reject decision for projects considered in isolation, it does not allow you to compare projects of different sizes. In that case, you must use NPV to deter- mine the best choice from a set of mutually exclusive projects. We face a similar problem here. We can’t properly compare investments in A and P using the mean and standard deviation of their rates of return when portfolio P is double the size of A. But if we compare portfolios of the same size, we can compare them using rates of return.

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capital markets allow firms to engage in risky projects without imposing undue risk on their shareholders, and in this way facilitate the collective economy’s ability to undertake large, possibly risky, ventures.

Diversification and the Sharpe Ratio While it is clear that risk pooling by itself does not reduce risk, we do not mean to suggest that it is unimportant. Risk pooling improves the risk–return trade-off and, therefore, is one crucial part of a diversification strategy. Consider again the one-asset versus two-asset investments above. Security A has a Sharpe ratio of

SA!=! [ E(r)!"!rf ] # $ while the ratio for the two-asset risk-pooled portfolio P is

SP!=! Expected!dollar!risk!premium _________________________

SD(dollar!profit) !=!

2! [ E(r)!"!rf ] ___________ $ !

__ 2 !=! !

__ 2 SA

The Sharpe ratio for the risk-sharing portfolio (with a fixed $1 investment) also is ! __

2 times that of the one-asset portfolio:

Expected!dollar!risk!premium _________________________ SD(dollar!profit)

!=! E(r)!"!rf ________ $ # !

__ 2 !=! !

__ 2 S A

Thus, risk pooling improves the Sharpe ratio, regardless of whether risk sharing is part of the strategy.

To summarize, when risk was simply pooled by adding a second, uncorrelated, asset to the initial one-asset portfolio, the expected dollar risk premium doubled, dollar variance doubled, and standard deviation increased by !

__ 2 . Therefore, risk pooling increased the

Sharpe ratio by the factor 2 # ! __

2 = ! __

2 . ! So while risk increased, the risk–return trade-off was improved. But by adding risk-sharing to the strategy, we reap both the higher Sharpe ratio as well as lower total risk. In the risk-sharing strategy, the expected risk premium was unchanged, and standard deviation fell by !

__ 2 . The Sharpe ratio increased by !

__ 2 , just as for

the risk-pooling strategy. These results generalize to diversification beyond two assets. Suppose we consider

holding n identical, uncorrelated assets in the portfolio. The expected dollar risk premium scales up in proportion to n in the risk-pooling strategy, but the dollar standard deviation increases by the factor !

__ n . Therefore, the Sharpe ratio increases by a factor n # !

__ n = !

__ n .

Risk sharing combined with risk pooling (thus holding the size of the investment budget fixed) does not affect the expected risk premium, but it does reduce the standard deviation by the factor !

__ n and therefore also increases the Sharpe ratio by !

__ n .

Think back to our gambler at the roulette wheel one last time. He would be wrong to argue that diversification means that 100 bets are less risky than 1 bet. His intuition would be correct, however, if he shared those 100 bets with 100 of his friends. A 1/100 share of 100 bets is in fact less risky than one bet. Fixing the amount of his total money at risk as that money is spread across more independent bets is the way for him to reduce risk. True diversification entails spreading a portfolio of fixed size across many assets, not merely adding more risky bets to an ever-growing risky portfolio.

Time Diversification and the Investment Horizon Now we turn to the implications of risk pooling and risk sharing for long-term investing. Think of extending the investment horizon for another period (which means that we now incur the uncertainty of that period’s risky return) as analogous to adding another risky

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asset to our portfolio or adding a new insurance policy to a pool of existing ones. So it is already clear that we should not expect extending the investment horizon while maintain- ing a fixed asset allocation in each period to reduce risk.

Examining the impact of an extension of the investment horizon requires us to clarify what the alternative is. Suppose you consider an investment in a risky portfolio over the next two years, which we’ll call the “long-term investment.” How should you compare this decision to a “short-run investment”? We must compare these two strategies over the same period, that is, two years. The short-term investment therefore must be interpreted as an investment in the risky portfolio over the first year and in the risk-free asset over the second year.

Given this comparison, and assuming the risky return in the first year is uncorrelated with that in the second, it becomes clear that the “long-term” strategy is analogous to the multi-asset risk-pooled portfolio. This is because holding on to the risky investment in the second year (rather than withdrawing to the risk-free asset) piles up more risk, just as sell- ing another insurance policy or adding another risky asset to the portfolio would. While extending a risky investment to a longer horizon improves the Sharpe ratio (as does risk pooling), it also increases risk. Thus “time diversification” is not really diversification.

The more accurate analogy to risk sharing for a long-term horizon would be to spread the risky investment budget across each of the investment periods. Compare the following three strategies applied to the whole investment budget over a two-year horizon:

1. Invest the whole budget at risk for one period, and then withdraw the entire pro- ceeds, placing them in a risk-free asset in the other period. Because you are invested in the risky asset for only one year, the risk premium over the whole investment period is R = E(r)!!!rf , the SD over the two-year period is ", and the Sharpe ratio is S = R/".

2. Invest the whole budget in the risky asset for both periods. The two-year risk premium is 2R (assuming continuously compounded rates), the two-year variance is 2"2, the two-year SD is# !

__ 2 "#, and the Sharpe ratio is !

__ 2 R/" . This is analogous to

risk pooling, taking two “bets” on the risky portfolio instead of just one. 3. Invest half the investment budget in the risky position in each of two periods,

placing the remainder of funds in the risk-free asset. The risk premium in each year is 1 ⁄ 2 R, and the standard deviation in each year is 1 ⁄ 2 ". Therefore, the two-year cumulative risk premium is R, the two-year variance is 2 $ ( 1 ⁄ 2 ")2 = 1 ⁄ 2 "2, the SD is " !

___ 1 / 2 , and the Sharpe ratio is S = !

__ 2 R/" . This is analogous to risk sharing,

taking a fractional position in each year’s investment return.

Strategy 3 is less risky than either alternative. Its expected total return equals Strategy 1’s, yet its risk is lower and therefore its Sharpe ratio is higher. It achieves the same Sharpe ratio as Strategy 2 but with standard deviation reduced by a factor of !

__ 2 . In summary, its

Sharpe ratio is at least as good as either alternative and, more to the point, its total risk is less than either.

We conclude that risk does not fade in the long run. An investor who contemplates investing in an attractive portfolio, and considers investing a given amount in that portfolio for one period, would find it preferable to put money at risk in that portfolio in as many periods as is feasible; however, to limit cumulative risk, he must also correspondingly decrease the risky budget in each period.#If “time diversification” really were a way to limit risk, one would expect it to allow longer-term investors to increase the amount invested in the risky portfolio without increasing the risk of final wealth, but that is unfortunately not the case.

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1. The expected return of a portfolio is the weighted average of the component security expected returns with the investment proportions as weights.

2. The variance of a portfolio is the weighted sum of the elements of the covariance matrix with the product of the investment proportions as weights. Thus the variance of each asset is weighted by the square of its investment proportion. The covariance of each pair of assets appears twice in the covariance matrix; thus the portfolio variance includes twice each covariance weighted by the product of the investment proportions in each of the two assets.

3. Even if the covariances are positive, the portfolio standard deviation is less than the weighted average of the component standard deviations, as long as the assets are not perfectly positively correlated. Thus, portfolio diversification is of value as long as assets are less than perfectly correlated.

4. The greater an asset’s covariance with the other assets in the portfolio, the more it contributes to portfolio variance. An asset that is perfectly negatively correlated with a portfolio can serve as a perfect hedge. That perfect hedge asset can reduce the portfolio variance to zero.

5. The efficient frontier is the graphical representation of a set of portfolios that maximize expected return for each level of portfolio risk. Rational investors will choose a portfolio on the efficient frontier.

6. A portfolio manager identifies the efficient frontier by first establishing estimates for asset expected returns and the covariance matrix. This input list is then fed into an optimization pro- gram that produces as outputs the investment proportions, expected returns, and standard devia- tions of the portfolios on the efficient frontier.

7. In general, portfolio managers will arrive at different efficient portfolios because of differences in methods and quality of security analysis. Managers compete on the quality of their security analysis relative to their management fees.

8. If a risk-free asset is available and input lists are identical, all investors will choose the same portfolio on the efficient frontier of risky assets: the portfolio tangent to the CAL. All investors with identical input lists will hold an identical risky portfolio, differing only in how much each allocates to this optimal portfolio and to the risk-free asset. This result is characterized as the separation principle of portfolio construction.

9. Diversification is based on the allocation of a portfolio of fixed size across several assets, limiting the exposure to any one source of risk. Adding additional risky assets to a portfolio, thereby increasing the total amount invested, does not reduce dollar risk, even if it makes the rate of return more predictable. This is because that uncertainty is applied to a larger invest- ment base. Nor does investing over longer horizons reduce risk. Increasing the investment horizon is analogous to investing in more assets. It increases total risk. Analogously, the key to the insurance industry is risk sharing—the spreading of many independent sources of risk across many investors, each of whom takes on only a small exposure to any particular source of risk.!

SUMMARY

diversification insurance principle market risk systematic risk nondiversifiable risk unique risk firm-specific risk

KEY TERMS nonsystematic risk diversifiable risk minimum-variance portfolio portfolio opportunity set Sharpe ratio optimal risky portfolio

minimum-variance frontier efficient frontier of risky assets input list separation property risk pooling risk sharing

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The expected rate of return on a portfolio: E(rp)!=!wD E(rD)!+!wE E(rE)

The variance of the return on a portfolio: " p 2 !=!(wD "D)2!+!(wE "E)2!+!2(wD "D)(wE "E) #DE

The Sharpe ratio of a portfolio: Sp!=! E(rp)!$!rf ________

"p

Sharpe ratio maximizing portfolio weights with two risky assets (D and E) and a risk-free asset:

wD!=! [ E(rD)!$!rf ] " E 2 !$! [ E(rE)!$!rf ] "D "E #DE

________________________________________________________ [ E(rD)!$!rf ] " E 2 !+! [ E(rE)!$!rf ] " D 2 !$! [ E(rD)!$!rf!+!E(rE)!$!rf ] "D "E #DE

wE!=!1!$!wD

Optimal capital allocation to the risky asset, y: E(rp)!$!rf ________

A " p 2

KEY EQUATIONS

1. Which of the following factors reflect pure market risk for a given corporation? a. Increased short-term interest rates. b. Fire in the corporate warehouse. c. Increased insurance costs. d. Death of the CEO. e. Increased labor costs.

2. When adding real estate to an asset allocation program that currently includes only stocks, bonds, and cash, which of the properties of real estate returns affect portfolio risk? Explain. a. Standard deviation. b. Expected return. c. Correlation with returns of the other asset classes.

3. Which of the following statements about the minimum-variance portfolio of all risky securities is valid? (Assume short sales are allowed.) Explain. a. Its variance must be lower than those of all other securities or portfolios. b. Its expected return can be lower than the risk-free rate. c. It may be the optimal risky portfolio. d. It must include all individual securities.

The following data apply to Problems 4 through 10: A pension fund manager is considering three mutual funds. The first is a stock fund, the second is a long-term government and corporate bond fund, and the third is a T-bill money market fund that yields a rate of 8%. The probability distribution of the risky funds is as follows:

$ Expected Return Standard Deviation

Stock fund (S) !"% #"% Bond fund (B) $! $%

The correlation between the fund returns is .10. 4. What are the investment proportions in the minimum-variance portfolio of the two risky funds,

and what is the expected value and standard deviation of its rate of return? 5. Tabulate and draw the investment opportunity set of the two risky funds. Use investment propor-

tions for the stock fund of 0% to 100% in increments of 20%. 6. Draw a tangent from the risk-free rate to the opportunity set. What does your graph show for the

expected return and standard deviation of the optimal portfolio?

PROBLEM SETS

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7. Solve numerically for the proportions of each asset and for the expected return and standard deviation of the optimal risky portfolio.

8. What is the Sharpe ratio of the best feasible CAL? 9. You require that your portfolio yield an expected return of 14%, and that it be efficient, on the

best feasible CAL. a. What is the standard deviation of your portfolio? b. What is the proportion invested in the T-bill fund and each of the two risky funds?

10. If you were to use only the two risky funds, and still require an expected return of 14%, what would be the investment proportions of your portfolio? Compare its standard deviation to that of the optimized portfolio in Problem 9. What do you conclude?

11. Stocks offer an expected rate of return of 18%, with a standard deviation of 22%. Gold offers an expected return of 10% with a standard deviation of 30%. a. In light of the apparent inferiority of gold with respect to both mean return and volatility,

would anyone hold gold? If so, demonstrate graphically why one would do so. b. Given the data above, reanswer (a) with the additional assumption that the correlation

coefficient between gold and stocks equals 1. Draw a graph illustrating why one would or would not hold gold in one’s portfolio.!

c. Could the set of assumptions in part (b) for expected returns, standard deviations, and correlation represent an equilibrium for the security market?

12. Suppose that there are many stocks in the security market and that the characteristics of stocks A and B are given as follows:

Stock Expected Return Standard Deviation

A !"% #% B !# !"

Correlation = "!

Suppose that it is possible to borrow at the risk-free rate, rf. What must be the value of the risk- free rate? (Hint: Think about constructing a risk-free portfolio from stocks A and B.)

13. True or false: Assume that expected returns and standard deviations for all securities (including the risk-free rate for borrowing and lending) are known. In this case, all investors will have the same optimal risky portfolio.!

14. True or false: The standard deviation of the portfolio is always equal to the weighted average of the standard deviations of the assets in the portfolio.!

15. Suppose you have a project that has a .7 chance of doubling your investment in a year and a .3 chance of halving your investment in a year. What is the standard deviation of the rate of return on this investment?

16. Suppose that you have $1 million and the following two opportunities from which to construct a portfolio: a. Risk-free asset earning 12% per year. b. Risky asset with expected return of 30% per year and standard deviation of 40%.

If you construct a portfolio with a standard deviation of 30%, what is its expected rate of return? The following data are for Problems 17 through 19: The correlation coefficients between several pairs of stocks are as follows: Corr(A, B) = .85; Corr(A, C) = .60; Corr(A, D) = .45. Each stock has an expected return of 8% and a standard deviation of 20%. 17. If your entire portfolio is now composed of stock A and you can add some of only one stock to

your portfolio, would you choose (explain your choice): a. B b. C c. D d. Need more data

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18. Would the answer to Problem 17 change for more risk-averse or risk-tolerant investors? Explain. 19. Suppose that in addition to investing in one more stock you can invest in T-bills as well. Would

you change your answers to Problems 17 and 18 if the T-bill rate is 8%? The following table of compound annual returns by decade applies to Problems 20 and 21. ! $%"&s* $%'&s $%(&s $%#&s $%)&s $%!&s $%*&s $%%&s "&&&s

Small-company stocks !!."#% ".#$% #%.&!% '(.%'% '!."#% $.")% '#.*&% '!.$*% &."%% Large-company stocks '$.!& !'.#) (.'' '(.*' ".$* ).(% '".&% '$.#% !'.%% Long-term gov’t bonds !.($ *.&% !.)( %.#) '.'* &.&! ''.)% $.&% ).%% Treasury bills !.)& %.!% %.!" '.$" !.$( &.#( (.%% ).%# #."% Inflation !'.%% !#.%* ).!& #.## #.)# ".!& ).'% #.(! #.)% *Based on the period '(#&–'(#(.

20. Input the data from the table into a spreadsheet. Compute the serial correlation in decade returns for each asset class and for inflation. Also find the correlation between the returns of various asset classes. What do the data indicate?

21. Convert the asset returns by decade presented in the table into real rates. Repeat the analysis of Problem 20 for the real rates of return.

The following information applies to Problems 22 through 27: Greta, an elderly investor, has a degree of risk aversion of A = 3 when applied to return on wealth over a one-year horizon. She is pondering two portfolios, the S&P 500 and a hedge fund, as well as a number of one-year strategies. (All rates are annual and continuously compounded.) The S&P 500 risk premium is estimated at 5% per year, with a SD of 20%. The hedge fund risk premium is estimated at 10% with a SD of 35%. The returns on both of these portfolios in any particular year are uncorrelated with its own returns in other years. They are also uncorrelated with the returns of the other portfolio in other years. The hedge fund claims the correlation coefficient between the annual return on the S&P 500 and the hedge fund return in the same year is zero, but Greta is not fully convinced by this claim. 22. Compute the estimated annual risk premiums, SDs, and Sharpe ratios for the two portfolios. 23. Assuming the correlation between the annual returns on the two portfolios is indeed zero, what

would be the optimal asset allocation?" 24. What should be Greta’s capital allocation? 25. If the correlation coefficient between annual portfolio returns is actually .3, what is the covari-

ance between the returns? 26. Repeat Problem 23 using an annual correlation of .3." 27. Repeat Problem 24 using an annual correlation of .3.

The following data apply to CFA Problems 1 through 3: Hennessy & Associates manages a $30 million equity portfolio for the multimanager Wilstead Pension Fund. Jason Jones, financial vice president of Wilstead, noted that Hennessy had rather consistently achieved the best record among the Wilstead’s six equity managers. Performance of the Hennessy portfolio had been clearly superior to that of the S&P 500 in four of the past five years. In the one less-favorable year, the shortfall was trivial.

Hennessy is a “bottom-up” manager. The firm largely avoids any attempt to “time the market.” It also focuses on selection of individual stocks, rather than the weighting of favored industries.

There is no apparent conformity of style among Wilstead’s six equity managers. The five manag- ers, other than Hennessy, manage portfolios aggregating $250 million made up of more than 150 individual issues.

Jones is convinced that Hennessy is able to apply superior skill to stock selection, but the favorable returns are limited by the high degree of diversification in the portfolio. Over the years, the portfolio generally held 40–50 stocks, with about 2%–3% of total funds committed to each issue.

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The reason Hennessy seemed to do well most years was that the firm was able to identify each year 10 or 12 issues that registered particularly large gains.

On the basis of this overview, Jones outlined the following plan to the Wilstead pension committee:

Let’s tell Hennessy to limit the portfolio to no more than 20 stocks. Hennessy will double the commitments to the stocks that it really favors, and eliminate the remainder. Except for this one new restriction, Hennessy should be free to manage the portfolio exactly as before.

All the members of the pension committee generally supported Jones’s proposal because all agreed that Hennessy had seemed to demonstrate superior skill in selecting stocks. Yet the pro- posal was a considerable departure from previous practice, and several committee members raised questions. Respond to each of the following questions. 1. a. Will the limitation to 20 stocks likely increase or decrease the risk of the portfolio? Explain.

b. Is there any way Hennessy could reduce the number of issues from 40 to 20 without signifi- cantly affecting risk? Explain.

2. One committee member was particularly enthusiastic concerning Jones’s proposal. He suggested that Hennessy’s performance might benefit further from reduction in the number of issues to 10. If the reduction to 20 could be expected to be advantageous, explain why reduction to 10 might be less likely to be advantageous. (Assume that Wilstead will evaluate the Hennessy portfolio independently of the other portfolios in the fund.)

3. Another committee member suggested that, rather than evaluate each managed portfolio indepen- dently of other portfolios, it might be better to consider the effects of a change in the Hennessy portfolio on the total fund. Explain how this broader point of view could affect the committee decision to limit the holdings in the Hennessy portfolio to either 10 or 20 issues.

4. Which one of the following portfolios cannot lie on the efficient frontier as described by Markowitz?

! Portfolio Expected Return (%) Standard Deviation (%)

a. W !" #$ b. X !% !" c. Z " & d. Y ' %!

5. Which statement about portfolio diversification is correct? a. Proper diversification can reduce or eliminate systematic risk. b. Diversification reduces the portfolio’s expected return because it reduces a portfolio’s total

risk. c. As more securities are added to a portfolio, total risk typically can be expected to fall at a

decreasing rate. d. The risk-reducing benefits of diversification do not occur meaningfully until at least 30

individual securities are included in the portfolio. 6. The measure of risk for a security held in a diversified portfolio is:

a. Specific risk. b. Standard deviation of returns. c. Reinvestment risk. d. Covariance.

7. Portfolio theory as described by Markowitz is most concerned with: a. The elimination of systematic risk. b. The effect of diversification on portfolio risk. c. The identification of unsystematic risk. d. Active portfolio management to enhance return.

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8. Assume that a risk-averse investor owning stock in Miller Corporation decides to add the stock of either Mac or Green Corporation to her portfolio. All three stocks offer the same expected return and total variability. The correlation of return between Miller and Mac is !.05 and between Miller and Green is +.05. Portfolio risk is expected to: a. Decline more when the investor buys Mac. b. Decline more when the investor buys Green. c. Increase when either Mac or Green is bought. d. Decline or increase, depending on other factors.

9. Stocks A, B, and C have the same expected return and standard deviation. The following table shows the correlations between the returns on these stocks.

Stock A Stock B Stock C

Stock A +!." Stock B +".# +!." Stock C +".! !".$ +!."

Given these correlations, the portfolio constructed from these stocks having the lowest risk is a portfolio: a. Equally invested in stocks A and B. b. Equally invested in stocks A and C. c. Equally invested in stocks B and C. d. Totally invested in stock C.

10. Statistics for three stocks, A, B, and C, are shown in the following tables. Standard Deviations of Returns

Stock: A B C

Standard deviation (%): $" %" $"

Correlations of Returns

Stock A B C

A !."" ".#" ".&" B ' !."" ".!" C ' ' !.""

Using only"the information provided in the tables, and given a choice between a portfolio made up of equal amounts of stocks A and B or a portfolio made up of equal amounts of stocks B and C, which portfolio would you recommend? Justify your choice.

11. George Stephenson’s current portfolio of $2 million is invested as follows: ! Summary of Stephenson’s Current Portfolio

# Value Percent of

Total Expected Annual

Return Annual Standard

Deviation

Short-term bonds $ %"",""" !"% $.(% !.(% Domestic large-cap equities ("",""" )" !%.$ !#.& Domestic small-cap equities !,%"",""" ("''' !(." %#.#''' Total portfolio $%,""",""" !""% !).* %).!

Stephenson soon expects to receive an additional $2 million and plans to invest the entire amount in an index fund that best complements the current portfolio. Stephanie Coppa, CFA, is evaluating the four index funds shown in the following table for their ability to produce a portfo- lio that will meet two criteria relative to the current portfolio: (1) maintain or enhance expected return and (2) maintain or reduce volatility.

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Each fund is invested in an asset class that is not substantially represented in the current portfolio.

Index Fund Characteristics

Index Fund Expected Annual Return Expected Annual

Standard Deviation Correlation of Returns with Current Portfolio

Fund A !"% #"% +$.%$ Fund B !! ## +$.&$ Fund C !& #" +$.'$ Fund D !( ## +$.&"

Which fund should Coppa recommend to Stephenson? Justify your choice by describing how your chosen fund best meets both of Stephenson’s criteria. No calculations are required.

12. Abigail Grace has a $900,000 fully diversified portfolio. She subsequently inherits ABC Com- pany common stock worth $100,000. Her financial adviser provided her with the following forecast information:

Risk and Return Characteristics

# Expected Monthly

Returns Standard Deviation of

Monthly Returns

Original Portfolio $.&)% #.*)% ABC Company !.#" #.'"

The correlation coefficient of ABC stock returns with the original portfolio returns is .40. a. The inheritance changes Grace’s overall portfolio, and she is deciding whether to keep the

ABC stock. Assuming Grace keeps the ABC stock, calculate the: i. Expected return of her new portfolio, which includes the ABC stock. ii. Covariance of ABC stock returns with the original portfolio returns. iii. Standard deviation of her new portfolio, which includes the ABC stock.

b. If Grace sells the ABC stock, she will invest the proceeds in risk-free government securi- ties yielding .42% monthly. Assuming Grace sells the ABC stock and replaces it with the government securities, calculate the i. Expected return of her new portfolio, which includes the government securities. ii. Covariance of the government security returns with the original portfolio returns. iii. Standard deviation of her new portfolio, which includes the government securities.

c. Determine whether the systematic risk of her new portfolio, which includes the government securities, will be higher or lower than that of her original portfolio.

d. On the basis of conversations with her husband, Grace is considering selling the $100,000 of ABC stock and acquiring $100,000 of XYZ Company common stock instead. XYZ stock has the same expected return and standard deviation as ABC stock. Her husband comments, “It doesn’t matter whether you keep all of the ABC stock or replace it with $100,000 of XYZ stock.” State whether her husband’s comment is correct or incorrect. Justify your response.

e. In a recent discussion with her financial adviser, Grace commented, “If I just don’t lose money in my portfolio, I will be satisfied.” She went on to say, “I am more afraid of losing money than I am concerned about achieving high returns.” i. Describe one weakness of using standard deviation of returns as a risk measure

for Grace. ii. Identify an alternate risk measure that is more appropriate under the circumstances.

13. Dudley Trudy, CFA, recently met with one of his clients. Trudy typically invests in a master list of 30 equities drawn from several industries. As the meeting concluded, the client made the fol- lowing statement: “I trust your stock-picking ability and believe that you should invest my funds in your five best ideas. Why invest in 30 companies when you obviously have stronger opinions

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on a few of them?” Trudy plans to respond to his client within the context of modern portfolio theory. a. Contrast the concepts of systematic risk and firm-specific risk, and give an example of each

type of risk. b. Critique the client’s suggestion. Discuss how both systematic and firm-specific risk change

as the number of securities in a portfolio is increased.

E$INVESTMENTS EXERCISES Go to the www.investopedia.com/articles/basics/!"/!#!$!".asp Web site to learn more about diversification, the factors that influence investors’ risk preferences, and the types of invest- ments that fit into each of the risk categories. Then check out www.investopedia.com/articles/ pf/!#/!%&#!#.asp for asset allocation guidelines for various types of portfolios from conserva- tive to very aggressive. What do you conclude about your own risk preferences and the best portfolio type for you? What would you expect to happen to your attitude toward risk as you get older? How might your portfolio composition change?

SOLUTIONS TO CONCEPT CHECKS 1. a. The first term will be w D ! "! w D ! "! # D 2 because this is the element in the top corner of the

matrix ( # D 2 ) times the term on the column border (wD) times the term on the row border (wD). Applying this rule to each term of the covariance matrix results in the sum

w D 2 # D 2 !+! w D w E Cov ( r E , r D ) !+! w E w D Cov ( r D , r E ) !+! w E 2 # E 2 , which is the same as Equation 7.3, because Cov(rE, rD) = Cov(rD, rE).

b. The bordered covariance matrix is

' wX wY wZ wX # X ! Cov(rX, rY) Cov(rX, rZ)

wY Cov(rY, rX!) # Y ! Cov(rY, rZ)

wZ Cov(rZ, rX!) Cov(rZ, rY!) # Z !

There are nine terms in the covariance matrix. Portfolio variance is calculated from these nine terms:

# P 2 !=! w X 2 # X 2 !+! w Y 2 # Y 2 !+! w Z 2 # Z 2 +!wXwY!Cov(rX, rY)!+!wY wX Cov(rY, rX) +!wX wZ Cov(rX, rZ)!+!wZ wX Cov(rZ, rX) + wY wZ Cov(rY, rZ)!+!wZ wY Cov(rZ, rY)

=! w X 2 # X 2 !+! w Y 2 # Y 2 !+! w Z 2 # Z 2 +!2 wX wY Cov(rX, rY)!+!2 wX wZ Cov(rX, rZ)!+!2 wY wZ Cov(rY, rZ )

2. The parameters of the opportunity set are E(rD) = 8%, E(rE) = 13%, #D = 12%, #E = 20%, and $(D,E) = .25. From the standard deviations and the correlation coefficient, we generate the covariance matrix:

Fund D E

D "## $% E $% #%%

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The global minimum-variance portfolio is constructed so that

wD!=! " E 2 !#!Cov(rD, rE) ____________________

" D 2 !+! " E 2 !#!2 Cov(rD, rE)

=! 400!#!60 ___________________ (144!+!400)!#!(2!$!60)

!=!.8019

wE!=!1!#!wD!=!.1981

Its expected return and standard deviation are

E(rP)!=!(.8019!$!8)!+!(.1981!$!13)!=!8.99% "P!=! [ w D 2 " D 2 !+! w E 2 " E 2 !+!2 wD wE Cov(rD, rE) ] 1/2

=! [ (.80192!$!144)!+!(.19812!$!400)!+!(2!$!.8019!$!.1981!$!60) ] 1/2 =!11.29%

For the other points we simply increase wD from .10 to .90 in increments of .10; accordingly, wE ranges from .90 to .10 in the same increments. We substitute these portfolio proportions in the formulas for expected return and standard deviation. Note that when wE = 1.0, the portfolio parameters equal those of the stock fund; when wD = 1, the portfolio parameters equal those of the debt fund. We thus generate the following table:

wE wD E!(r!) "

!.! ".! #.! "$.!! !." !.% #.& "".'( !.$ !.# %.! "".$% !.) !.* %.& "".'# !.' !.( "!.! "$.!) !.& !.& "!.& "$.## !.( !.' "".! ").%% !.* !.) "".& "&.)! !.# !.$ "$.! "(.*( !.% !." "$.& "#.)' ".! !.! ").! $!.!! !."%#" !.#!"% #.%% "".$% minimum variance portfolio

You can now draw your graph. 3. a. The computations of the opportunity set of the stock and risky bond funds are like those of

Question 2 and will not be shown here. You should perform these computations, however, in order to give a graphical solution to part a. Note that the covariance between the funds is

Cov(rA, rB)!=!%(A, B)!$!"A!$!"B = #.2!$!20!$!60!=!#240

b. The proportions in the optimal risky portfolio are given by

wA!=! (10!#!5)602!#!(30!#!5)(#240) ________________________________

(10!#!5)602!+!(30!#!5)202!#!30(#240)

=!.6818 wB =!1!#!wA!=!.3182

The expected return and standard deviation of the optimal risky portfolio are

E(rP)!=!(.6818!$!10)!+!(.3182!$!30)!=!16.36% "P!=! {(.68182!$!202)!+!(.31822!$!602)!+! [ 2!$!.6818!$!.3182(!#!240) ] } 1/2

=!21.13%

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Note that portfolio P is not the global minimum-variance portfolio. The proportions of the latter are given by

wA!=! 602!"!("240) __________________

602!+!202!"!2("240) !=!.8571

wB!=!1!"!wA!=!.1429

With these proportions, the standard deviation of the minimum-variance portfolio is

#(min)!=!(.85712!$!202)!+!(.14292!$!602)!+! [ 2!$!.8571!$!.1429!$!("240) ] 1/2 =!17.75%

which is less than that of the optimal risky portfolio. c. The CAL is the line from the risk-free rate through the optimal risky portfolio. This line

represents all efficient portfolios that combine T-bills with the optimal risky portfolio. The slope of the CAL is

S!=! E(rP)!"!rf ________

#P !=! 16.36!"!5 _________

21.13 !=!.5376

d. Given a degree of risk aversion, A, an investor will choose the following proportion, y, in the optimal risky portfolio (remember to express returns as decimals when using A):

y!=! E(rP)!"!rf ________

A # P 2 !=! .1636!"!.05 __________

5!$!.21132 !=!.5089

This means that the optimal risky portfolio, with the given data, is attractive enough for an investor with A = 5 to invest 50.89% of his or her wealth in it. Because stock A makes up 68.18% of the risky portfolio and stock B makes up 31.82%, the investment proportions for this investor are

Stock A: .!"#$ $ %#.&# = '(.)"% Stock B: .!"#$ $ '&.#* = &%.&$% Total !".#$%

4. Efficient frontiers derived by portfolio managers depend on forecasts of the rates of return on various securities and estimates of risk, that is, the covariance matrix. The forecasts themselves do not control outcomes. Thus preferring managers with rosier forecasts (northwesterly frontiers) is tantamount to rewarding the bearers of good news and punishing the bearers of bad news. What we should do is reward bearers of accurate news. Thus, if you get a glimpse of the frontiers (forecasts) of portfolio managers on a regular basis, what you want to do is develop the track record of their forecasting accuracy and steer your advisees toward the more accurate forecaster. Their portfolio choices will, in the long run, outperform the field.

5. The parameters are E(r) = 15, # = 60, and the correlation between any pair of stocks is % = .5.

a. The portfolio expected return is invariant to the size of the portfolio because all stocks have identical expected returns. The standard deviation of a portfolio with n = 25 stocks is

#P!=! [ #2 / n!+!%!$!#2(n!"!1) / n ] 1/2 =![602 / 25!+!.5!$!602!$!24 / 25]1/2!=!43.27%

b. Because the stocks are identical, efficient portfolios are equally weighted. To obtain a standard deviation of 43%, we need to solve for n:

432!=! 60 2 ___

n !+!.5!$! 60

2(n!"!1) _________ n

1,849n!=!3,600!+!1,800n!"!1,800

n!=! 1,800 _____ 49

!=!36.73

Thus we need 37 stocks and will come in with volatility slightly under the target.

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c. As n gets very large, the variance of an efficient (equally weighted) portfolio diminishes, leaving only the variance that comes from the covariances among stocks. Therefore,

Systematic standard deviation!=! ! ______

"!#!$2 !=! ! _______

.5!#!602 !=!42.43%

Note that with 25 stocks we came within .84% of the systematic risk, that is, the standard deviation of a portfolio of 25 stocks is only .84% higher than 42.43%. With 37 stocks, the standard deviation is 43.01%, which!is only .58% greater than 42.43%.

d. If the risk-free is 10%, then the risk premium on any size portfolio is 15 % 10 = 5%. The standard deviation of a well-diversified portfolio is (practically) 42.43%; hence the slope of the CAL is

S!=!5 / 42.43!=!.1178

APPENDIX A:!A Spreadsheet Model for Efficient Diversification

Several software packages can be used to generate the efficient frontier. We will dem- onstrate the method using Microsoft Excel. Excel is far from the best program for this purpose and is limited in the number of assets it can handle, but working through a simple portfolio optimizer in Excel can illustrate concretely the nature of the calculations used in more sophisticated “black-box” programs. You will find that even in Excel, the computa- tion of the efficient frontier is not difficult.

We apply the Markowitz portfolio optimization program to the problem of interna- tional diversification. We take the perspective of a portfolio manager serving U.S. clients, who wishes to construct for the next year an optimal risky portfolio of large stocks in the U.S. and six developed capital markets (Japan, Germany, the U.K., France, Canada, and Australia). First we describe the input list: forecasts of risk premiums and the covariance matrix. Next, we describe Excel’s Solver, and finally we show the solution to the man- ager’s problem.

The Input List The manager needs to compile an input list of expected returns, variances, and covariances to compute an efficient frontier and the optimal risky portfolio.!Spreadsheet 7A.1 shows the calculations.

Panel A lays out the expected excess return for each country index.!While these esti- mates may be guided by historical experience, as we discussed in Chapter 5, using simple historical averages would yield extremely noisy estimates of expected risk premiums because returns are so variable over time. Average returns fluctuate enormously across subperiods, making historical averages highly unreliable estimators. Here, we simply assume the manager has arrived at some reasonable estimates of each country’s risk pre- mium through scenario analysis informed by historical experience. These values are pre- sented in column B.

Panel B is the bordered covariance matrix corresponding to Table 7.2 earlier in the chapter. The covariances in this table might reasonably be estimated from a sample of historical returns, as empirically based variance and covariance estimates are far more precise than corresponding estimates of mean returns. It would be common to estimate Panel B using perhaps five years of monthly returns. The Excel function COVARIANCE would compute the covariance between the time series of returns for any pair of coun- tries. We assume the manager has already collected historical returns on each index, plugged each pair of returns into the COVARIANCE function, and obtained the entries that appear in Panel B. The elements on the diagonal of the covariance matrix in Panel B

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A B C D E F G H I J K L

! "

Efficient Frontier Spreadsheet

Expected excess returns (risk premiums) of each country indexPanel A

Bordered Covariance Matrix

Various points along the efficient frontier.

Panel B

Panel C

Formulas used in key cells

# $

!.!"! !.!#$ !.!%! !.!&! !.!#& !.!'# !.!#(

% & ' ( ) !* !!

Portfolio weights *.&!!" *.(''( -*.%*)' *."*%%

!."))* !.&%%&

-!.*)'! -!.#!(% !.!"(# !.*!##

-!.!'!* ).!!!!

Cell A)% - A*$ These are portfolio weights. You can set initial value arbitrarily as long as sum = ) = A)%, and so on. The portfolio weights in column A are copied to row )%.

=SUM(A)%:A*$)

=C)#*SUMPRODUCT($A)%:$A*$,C)%:C*$) Copied from C*' (note the use of absolute addresses)

=SUMPRODUCT(A)%:A*$,$B#:$B)))

=SUM(C*':I*')^!.# =C*#/C*"

Cell C)#

Cell A*'

Cell C*'

Cell D*' through I*'

Cell C*#

Cell C*"

Cell C*%

!.!**' !.!)&' !.!*#! !.!*&& !.!)(# !.!)*) !.!*!# !.!!%& !.!$&$ !.))$* !.$$&"

!.!)&' !.!**$ !.!*%# !.!*(( !.!*!' !.!)*' !.!*!" !.!))$

!.!*&& !.!*(( !.!'$& !.!#)# !.!$!) !.!)&$ !.!$!#

-!.!!"#

!.!)*) !.!)*' !.!)%% !.!)&$ !.!)'% !.!$#$ !.!)#& !.!!*"

U.S. U.K. -*."!$*

!.!*#! !.!*%# !.!'!$ !.!'$& !.!*#( !.!)%% !.!*%$

-!.!!*%

France Germany *.*&)%

!.!)(# !.!*!' !.!*#( !.!$!) !.!*") !.!)'% !.!*$' !.!!!(

Australia Japan U.S. U.K. France Germany Australia Japan Canada

-*.*$*"

!.!*!# !.!*!" !.!*%$ !.!$!# !.!*$' !.!)#& !.!*(&

-!.!!!#

Canada

!# !$

!"

!%

!& !' !( !) "* "! "" "# "$ "% "& "' "( ")

#*

#!

#"

##

#$

#%

#&

#' #(

U.K.

Std Dev: Risk Prem:

Sharpe:

!.!$#! !.))') !.$!"" !.#('' ).!)%#

-!.*$"# -!."!%% !.!#&& !.*)(*

-!.!'#(

Min var portfolio

Optimal (tangency) portfolio

!.!$&$ !.))$* !.$$&" !."))* !.&%%&

-!.*)'! -!.#!(% !.!"(# !.*!##

-!.!'!* !.!'"#

!.!'!! !.))$# !.$#*# !.")(# !.&!&$

-!.*!*( -!.'")! !.!%'& !.)(&%

-!.!$%' !.!'""

!.!'#! !.))"& !.$&#$ !."''" !.#((*

-!.)"($ -!.$)'' !.!(!% !.)%&)

-!.!*&& !.!'%(

!.!#!! !.)*$& !.'!$% !.""(" !.$(!!

-!.)$#% -!.)"%( !.)!"% !.)#%#

-!.!*!$ !.!#!&

!.!##! !.)$'! !.')!' !."('% !.)&!(

-!.)!*) -!.!*)$ !.)**" !.)$"(

-!.!))& !.!##!

!.!#"' !.)$%' !.')!% !.%!)& !.)*)'

-!.!(*" !.!*!# !.)*%) !.)$))

-!.!!($ !.!#"'

!.!#%# !.)'!) !.')!" !.%!%$ !.!%#&

-!.!&#* !.!#*' !.)$!" !.)*""

-!.!!%# !.!#%#

!.!"!! !.)'"" !.'!(* !.%)(&

-!.!*&$ -!.!"&# !.)*#$ !.)$&# !.))"'

-!.!!$* !.!"!*

!.!%!! !.)%%) !.$(#$ !.%"((

-!.''"# -!.!!)' !.')&# !.)%!' !.!%#* !.!)$( !.!%*%

!.!&!! !.*))( !.$%%' !.&*!)

-!.&"'& !.!"#& !.%))% !.*!*$ !.!$') !.!$!( !.!&%!!.!'"(

U.S.

France Germany Australia Japan Canada CAL* *Risk premium along the CAL

= StdDev of portfolio times slope of optimal risky portfolio (Cell I'#)

$" $# $$ $% $& $' $( $) %* %! %"

$!

#)

$*

Risk Prem Std Dev Sharpe

U.S. U.K. France Germany Australia Japan Canada

Spreadsheet 7A.1 Spreadsheet models for international diversification

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are the variances of each country index. You can take square roots to find each country’s standard deviation.

The covariance matrix in Panel B is bordered along the top (row 15) and left (column!A) by a set of portfolio weights. You can start the procedure off using any arbi- trary set of weights as long as they sum to 1. You will be asking Excel to replace these initial weights with the weights that correspond to the portfolios on the efficient frontier.

Cells C25, C26, and C27 calculate some important characteristics of the portfolio defined by the set of portfolio weights. C25 is the expected risk premium on the port- folio, computed by taking a weighted average of country risk premiums (using Excel’s SUMPRODUCT function. The formula for cell C25 is presented in cell C35.) Cell C26 computes the standard deviation of any portfolio using the formula developed in Table 7.2. Portfolio variance is given by the sum of cells C24–I24 below the bordered covariance matrix. Cell C26 takes the square root of this sum to produce the standard deviation. Finally C27 is the portfolio’s Sharpe ratio, expected excess return divided by standard deviation. This is also the slope of the CAL (capital allocation line) that runs through the portfolio corresponding to the weights in column A. The optimal risky portfolio is the one that maxi- mizes the Sharpe ratio.

Panel C shows the properties of several portfolios along the efficient frontier. The high- lighted columns correspond to the minimum-variance and the tangency portfolios. Each column in the panel shows portfolio characteristics and the weighting in each country.

Using the Excel Solver Excel’s Solver is a user-friendly, but quite powerful, optimizer. It has three parts: (1) an objective function, (2) decision variables, and (3) constraints. Figure 7A.1 shows three pic- tures of the Solver. We will begin by finding the minimum-variance portfolio. The problem is set up in Panel A of the figure.

The top line of the Solver lets you choose a target cell for the “objective function,” that is, the variable you are trying to optimize. In Panel A of Figure 7A.1, the target cell is!C26, the portfolio standard deviation. Below the target cell, you can choose whether your objec- tive is to maximize, minimize, or set your objective function equal to a value that you specify. Here we choose to minimize the portfolio standard deviation.

The next part of Solver contains the decision variables. These are cells that the Solver can change in order to optimize the objective function in the target cell. Here, we input cells A17–A23, the portfolio weights that we select to minimize portfolio volatility.

The bottom panel of the Solver can include any number of constraints. One constraint that must always appear in portfolio optimization is the “feasibility constraint,” namely, that portfolio weights sum to 1.0. When we bring up the dialogue box for constraints, we specify that cell A24 (the sum of weights) must be set equal to 1.0.

When we!click “Solve,” the Solver finds the weights of the minimum-variance portfo- lio and places them in column A. The spreadsheet then calculates the risk premium and standard deviation of that portfolio. We save the portfolio weights and its key statistics by copying them to Panel C in Spreadsheet 7A.1. Column C in Panel C shows that the lowest standard deviation (SD) that can be achieved with our input list is 11.32%. This standard deviation is considerably lower than even the lowest SD of the individual indexes.!

Now we are ready to find other points along the efficient frontier. We will do this by finding the portfolio that has the lowest possible variance for any targeted risk premium. You can produce the entire efficient frontier in minutes following this procedure.

1. Input to the Solver a constraint that says: Cell C25 (the portfolio risk premium) must equal some desired value, for example .04. The Solver at this point is shown

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APPENDIX B:!Review of Portfolio Statistics

We base this review of scenario analysis on a two-asset portfolio. We denote the assets D and E (which you may think of as debt and equity), but the risk and return parameters we use in this appendix are not necessarily consistent with those used in Section 7.2.

Expected Returns We use “expected value” and “mean” interchangeably. For an analysis with n scenarios, where the rate of return in scenario i is r(i ) with probability p(i ), the expected return is

E(r)!=! ! i = 1

n p(i )r (i ) (7B.1)

If you were to increase the rate of return assumed for each scenario by some amount ", then the mean return will increase by ". If you multiply the rate in each scenario by a factor w, the new mean will be multiplied by that factor:

! i = 1

n p(i )!#! [ r(i )!+!" ] !=! !

i = 1

n p(i )!#!r (i )!+!" !

i = 1

n p(i )!=!E(r)!+!"

! i = 1

n p(i )!#! [ wr (i ) ] !=!w !

i = 1

n p(i )!#!r (i )!=!wE(r)

(7B.2)

Column C of Spreadsheet !B." shows scenario rates of return for debt, D. In column D we add #% to each scenario return and in column E we multiply each rate by .$. The spread- sheet shows how we compute the expected return for columns C, D, and E. It is evident that the mean increases by #% (from .%& to ."") in column D and is multiplied by .$ (from .%& to %.%#') in column E.

Example 7B.1 Expected Rates of Return

Spreadsheet 7B.1 Scenario analysis for bonds

A B C D E GF

!

"

#

$

%

&

'

(

)

!*

!!

!"

Scenario Rates of Return

rD(i ) rD(i ) + 0.03 0.4*rD(i )

Mean Cell C8

-!."!

!.!!

!."!

!.#$

!.!%! =SUMPRODUCT($B$4:$B$7,C4:C7)

-!.!&

!.!#

!."#

!.#'

!.""!

-!.!(!

!.!!!

!.!(!

!."$%

!.!#$

!."(

!.#)

!.#!

!.$!

Probability

"

$

#

(

Scenario

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Now let’s construct a portfolio that invests a fraction of the investment budget, w(D), in bonds and the fraction w(E ) in stocks. The portfolio’s rate of return in each scenario and its expected return are given by

rP(i )!=!wD rD(i )!+!wE rE(i ) E(rP)!=!! p(i ) [ wD rD(i )!+!wE rE(i ) ] !=!! p(i )wD rD(i )!+!! p(i )wE rE(i ) (7B.3)

=!wD E(rD)!+!wE E(rE) The rate of return on the portfolio in each scenario is the weighted average of the com- ponent rates. The weights are the fractions invested in these assets, that is, the portfolio weights. The expected return on the portfolio is the weighted average of the asset means.

Spreadsheet !B." lays out rates of return for both stocks and bonds. Using assumed weights of .# for debt and .$ for equity, the portfolio return in each scenario appears in column L. Cell L% shows the portfolio expected return as .&'#', obtained using the SUMPRODUCT func- tion, which multiplies each scenario return (column L) by the scenario probability (column I) and sums the results.

Example 7B.2 Portfolio Rate of Return

15Variance (here, of an asset rate of return) is not the only possible choice to quantify variability. An alternative would be to use the absolute deviation from the mean instead of the squared deviation. Thus, the mean absolute deviation (MAD) is sometimes used as a measure of variability. The variance is the preferred measure for several reasons. First, working with absolute deviations is mathematically more difficult. Second, squaring deviations gives more weight to larger deviations. In investments, giving more weight to large deviations (hence, losses) is compatible with risk aversion. Third, when returns are normally distributed, the variance is one of the two param- eters that fully characterizes the distribution.

Variance and Standard Deviation The variance and standard deviation of the rate of return on an asset from a scenario analysis are given by15

"2(r )!=! ! i=1

n p(i ) [ r(i )!#!E(r ) ] 2

"(r )!=! " _____

"2(r ) (7B.4)

Spreadsheet 7B.2 Scenario analysis for bonds and stocks

H I J K L

!

"

#

$

%

&

'

(

)

!*

!!

!"

Scenario Rates of Return Portfolio Return 0.4*rD (i ) + 0.6*rE (i )rD(i ) rE (i )

!."#

!.$%

!.$!

!.&!

-!."!

!.!!

!."!

!.$&

!.!'Mean

Cell L4 =0.4*J4+0.6*K4

Cell L8 =SUMPRODUCT($I$4:$I$7,L4:L7)

-!.$(

!.&!

!.#(

-!.")

!."&

-!.&(!!

!."&!!

!.$"!!

!.!"#!

!."!#!

"

&

$

#

ProbabilityScenario

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Notice that the unit of variance is percent squared. In contrast, standard deviation, the square root of variance, has the same dimension as the original returns, and therefore is easier to interpret as a measure of return variability.

When you add a fixed incremental return, !, to each scenario return, you increase the mean return by that same increment. Therefore, the deviation of the realized return in each scenario from the mean return is unaffected, and both variance and SD are unchanged. In contrast, when you multiply the return in each scenario by a factor w, the variance is multiplied by the square of that factor (and the SD is multiplied by w):

Var(wr)"=" ! i = 1

n p(i )"#" [ wr (i )"$"E(wr ) ] 2"="w2 !

i = 1

n p(i ) [ r (i )"$"E(r ) ] 2"="w2 %2

SD(wr)"=" " _____

w2 %2 "="w%(r) (7B.5)

Excel does not have a direct function to compute variance and standard deviation for a scenario analysis. Its STDEV and VAR functions are designed for time series. We need to calculate the probability-weighted squared deviations directly. To avoid having to first compute columns of squared deviations from the mean, however, we can simplify our problem by expressing the variance as a difference between two easily computable terms:

%2(r)"="E [ r"$"E(r) ] 2"="E {r2"+" [ E(r) ] 2"+"2rE(r)} ="E(r)2"+"[E(r)]2"$"2E(r)E(r) (7B.6)

="E(r 2)"$" [ E(r) ] 2"=" ! i = 1

n p(i )r (i )2"$" [ ! i = 1

n p(i )r(i )]

2

You can compute the first expression, E(r!!), in Equation "B.# using Excel’s SUMPRODUCT function. For example, in Spreadsheet "B.$, E(r!!) is first calculated in cell C!% by using SUM- PRODUCT to multiply the scenario probability times the asset return times the asset return again. Then [E(r!)]! is subtracted (notice the subtraction of C!&! in cell C!%), to arrive at variance.

Example 7B.3 Calculating the Variance of a Risky Asset in Excel

Spreadsheet 7B.3 Scenario analysis for bonds

A B C D E F G

!"

!#

!$

!%

!&

!'

!(

)*

)!

))

)"

)#

Scenario Rates of Return

rD(i ) rD(i ) + *.*" *.#*rD(i ) !."#

!.$%

!.$!

!.&!

Mean

Cell C21

Variance

SD

-!."!

!.!!

!."!

!.$&

!.!'!!

!.!"'(

!."$()

=SUMPRODUCT($B$16:$B$19,C16:C19,C16:C19)-C20^2

Cell C22 =C21^0.5

-!.!*

!.!$

!."$

!.$(

!.""!!

!.!"'(

!."$()

-!.!#!

!.!!!

!.!#!

!."&'

!.!&#!

!.!!$#

!.!('#

"

&

$

#

ProbabilityScenario

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The variance of a portfolio return is not as simple to compute as the mean. The portfolio variance is not the weighted average of the asset variances. The deviation of the portfolio rate of return in any scenario from its mean return is

rP!"!E(rP)!=!wD rD(i )!+!wE rE(i )!"! [ wD E(rD)!+!wE E(rE) ] =!wD [ rD(i )!"!E(rD) ] !+!wE [ rE(i )!"!E(rE) ] (7B.7) = wD d(i )!+!wE e(i )

where the lowercase variables denote deviations from the mean:

d(i )!=!rD(i )!"!E(rD) e(i )!=!rE (i )!"!E(rE)

We express the variance of the portfolio return in terms of these deviations from the mean in Equation 7B.8:

# P 2 !=! ! i = 1

n p(i ) [ rP!"!E(rP) ] 2!=! !

i = 1

n p(i ) [ wD d(i )!+!wE e(i ) ] 2

=! ! i = 1

n p(i ) [ w D 2 d (i )2!+! w E 2 e (i )2!+!2 wD wE d(i )e(i ) ]

=! w D 2 ! i = 1

n p(i )d (i )2+ w E 2 !

i = 1

n p(i )e (i )2!+!2 wD wE !

i = 1

n p(i )d(i )e(i )

(7B.8)

=! w D 2 # D 2 !+! w E 2 # E 2 !+!2 wD wE ! i = 1

n p(i )d(i )e(i )

The last line in Equation 7B.8 tells us that the variance of a portfolio is the weighted sum of portfolio variances (notice that the weights are the squares of the portfolio weights), plus an additional term that, as we will soon see, makes all the difference.

Notice also that d(i ) $ e(i ) is the product of the deviations of the scenario returns of the two assets from their respective means. The probability-weighted average of this product is its expected value, which is called covariance and is denoted Cov(rD, rE). The covariance between the two assets can have a big impact on the variance of a portfolio.

Covariance The covariance between two variables equals

Cov(rD, rE) = E(d!$!e)!=!E { [ rD!"!E(rD) ] [ rE!"!E(rE) ] } =!E(rD rE)!"!E(rD)E(rE)

(7B.9)

The covariance is an elegant way to quantify the covariation of two variables. This is easi- est seen through a numerical example.

Imagine a three-scenario analysis of stocks and bonds such as that given in Spreadsheet 7B.4. In scenario 1, bonds go down (negative deviation) while stocks go up (positive devi- ation). In scenario 3, bonds are up, but stocks are down. When the rates move in opposite directions, as in this case, the product of the deviations is negative; conversely, if the rates moved in the same direction, the sign of the product would be positive. The magnitude of the product shows the extent of the opposite or common movement in that scenario. The

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probability-weighted average of these products therefore summarizes the average tendency for the variables to co-vary across scenarios. In the last line of the spreadsheet, we see that the covariance is !80 (cell H6).

Suppose our scenario analysis had envisioned stocks generally moving in the same direction as bonds. To be concrete, let’s switch the forecast rates on stocks in the first and third scenarios, that is, let the stock return be !10% in the first scenario and 30% in the third. In this case, the absolute value of both products of these scenarios remains the same, but the signs are positive, and thus the covariance is positive, at +80, reflecting the tendency for both asset returns to vary in tandem. If the levels of the scenario returns change, the intensity of the covariation also may change, as reflected by the magnitude of the product of deviations. The change in the magnitude of the covariance quantifies the change in both direction and intensity of the covariation.

If there is no co-movement at all, because positive and negative products are equally likely, the covariance is zero. Also, if one of the assets is risk-free, its covariance with any risky asset is zero, because its deviations from its mean are identically zero.

The computation of covariance using Excel can be made easy by using the last line in Equation 7B.9. The first term, E(rD " rE), can be computed in one stroke using Excel’s SUMPRODUCT function. Specifically, in Spreadsheet 7B.4, SUMPRODUCT(A3:A5, B3:B5, C3:C5) multiplies the probability times the return on debt times the return on equity in each scenario and then sums those three products.

Notice that adding # to each rate would not change the covariance because deviations from the mean would remain unchanged. But if you multiply either of the variables by a fixed factor, the covariance will increase by that factor. Multiplying both variables results in a covariance multiplied by the products of the factors because

Cov(wD rD, wE rE) = E { [ wD rD$!$wD E(rD) ] [ wE rE$!$wE E(rE) ] } =$wD wE Cov(rD, rE)

(7B.10)

The covariance in Equation 7B.10 is actually the term that we add (twice) in the last line of the equation for portfolio variance, Equation 7B.8. So we find that portfolio variance is the weighted sum (not average) of the individual asset variances, plus twice their covariance weighted by the two portfolio weights (wD " wE).

Like variance, the dimension (unit) of covariance is percent squared. But here we can- not get to a more easily interpreted dimension by taking the square root, because the aver- age product of deviations can be negative, as it was in Spreadsheet 7B.4. The solution in

Spreadsheet 7B.4 Three-scenario analysis for stocks and bonds

A

Rates of Return Deviation from Mean

B C D E F G H

! " # !."#

!.#!

!."#

Mean:

-"

$

%&

$

'!

%!

-%!

%!

-(

!

(

!

"!

!

-"!

!

-%$!

!

-%$!

-(!

$ % &

Probability Bonds Stocks Bonds Stocks Product of Deviations

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this case is to scale the covariance by the standard deviations of the two variables, produc- ing the correlation coefficient.

Correlation Coefficient Dividing the covariance by the product of the standard deviations of the variables will generate a pure number called correlation. We define correlation as follows:

Corr(rD, rE)!=! Cov(rD, rE) __________

"D "E (7B.11)

The correlation coefficient must fall within the range [#1, 1]. This can be explained as follows. What two variables should have the highest degree co-movement? Logic says a variable with itself, so let’s check it out.

Cov(rD, rD)!=!E { [ rD!#!E(rD) ] !$! [ rD!#!E(rD) ] } =!E[rD!#!E(rD)]2!=! " D 2 (7B.12)

Corr(rD, rD)!=! Cov(rD, rD) __________

"D "D !=! " D

2 ___ " D 2

!=!1

Similarly, the lowest (most negative) value of the correlation coefficient is #1. (Check this for yourself by finding the correlation of a variable with its own negative.)

An important property of the correlation coefficient is that it is unaffected by both addition and multiplication. Suppose we start with a return on debt, rD, multi- ply it by a constant, wD, and then add a fixed amount %. The correlation with equity is unaffected:

Corr(%!+!wD rD, rE)!=! Cov(%!+!wD rD, rE) ___________________

! _________________

Var(%!+!wD rD)!$!"E

=! wD Cov(rD, rE) ____________ !

______ w D 2 " D 2 !$!"E

!=! wD Cov(rD, rE) _____________ wD "D!$!"E

(7B.13)

=!Corr(rD, rE)

Because the correlation coefficient gives more intuition about the relationship between rates of return, we sometimes express the covariance in terms of the correlation coeffi- cient. Rearranging Equation 7B.11, we can write covariance as

Cov(rD, rE)!=!"D "E Corr(rD, rE) (7B.14)

Spreadsheet !B." shows the covariance and correlation between stocks and bonds using the same scenario analysis as in the other examples in this appendix. Covariance is calculated using Equation !B.#. The SUMPRODUCT function used in cell J$$ gives us E(rD $ rE !), from which we subtract E(rD) $ E(rE !) (i.e., we subtract J$% $ K$%). Then we calculate correlation in cell J$& by dividing covariance by the product of the asset standard deviations.

Example 7B.4 Calculating Covariance and Correlation

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C H A P T E R ! Optimal Risky Portfolios "#$

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Spreadsheet 7B.5 Scenario analysis for bonds and stocks

H I J K L M

!" !# !$ !% !& !' !( )* )! )) )" )# )$

Scenario Rates of Return rD(i ) rE(i )

!."# !.$% !.$! !.&!

-!."! !.!! !."! !.$& !.!'

!."$() -!.!!$# -!.!'#*

Mean SD Covariance Correlation

Cell J22

Cell J23

=SUMPRODUCT(I16:I19,J16:J19,K16:K19)-J20*K20 =J22/(J21*K21)

-!.$( !.&! !.#(

-!.") !."&

!.&)"'

" & $ #

ProbabilityScenario

Spreadsheet 7B.6 Scenario analysis for bonds and stocks

A B C D E F G !" !# !$ !% !& '( ') '! '' '* '" '# '$ '% '&

Cell E35 =SUMPRODUCT(B30:B33,E30:E33,E30:E33)-E34^2)^0.5 Cell E36 =(0.4*C35)^2+(0.6*D35)^2+2*0.4*0.6*C36)^0.5

Scenario Rates of Return Portfolio Return rD(i ) rE(i ) 0.4*rD(i )+0.6rE(i )

!."# !.$% !.$! !.&!

Mean SD Covariance Correlation

-!."! !.!! !."! !.$& !.!'

!."$() -!.!!$# -!.!'#*

-!.$( !.&! !.#(

-!.") !."&

!.&)"' SD:

-!.&( !."& !.$"

!.!"# !."!#! !."*'' !."*''

" & $ #

ProbabilityScenario

We calculate portfolio variance in Spreadsheet !B.". Notice there that we calculate the portfolio standard deviation in two ways: once from the scenario portfolio returns (cell E#$) and again (in cell E#") using the first line of Equation !B.%$. The two approaches yield the same result. You should try to repeat the second calculation using the correlation coefficient from the second line in Equation !B.%$ instead of covariance in the formula for portfolio variance.

Example 7B.5 Calculating Portfolio Variance

Portfolio Variance We have seen in Equation 7B.8, with the help of Equation 7B.10, that the variance of a two-asset portfolio is the sum of the individual variances multiplied by the square of the portfolio weights, plus twice the covariance between the rates, multiplied by the product of the portfolio weights: ! P 2 "=" w D 2 ! D 2 "+" w E 2 ! E 2 "+"2 wD wE Cov(rD, rE) =" w D 2 ! D 2 "+" w E 2 ! E 2 "+"2 wD wE !D !E Corr(rD, rE)

(7B.15)

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!"" P A R T I I Portfolio Theory and Practice

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Suppose that one of the assets, say, E, is replaced with a money market instrument, that is, a risk-free asset. The variance of E is then zero, as is the covariance with D. In that case, as seen from Equation 7B.15, the portfolio standard deviation is just wD!D. In other words, when we mix a risky portfolio with the risk-free asset, portfolio standard deviation equals the risky asset’s standard deviation times the weight invested in that asset. This result was used extensively in Chapter 6.

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!"# P A R T I I Portfolio Theory and Practice

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To see this, notice that the covariance can be computed from the correlation coefficient, !DE, as Cov(rD, rE)"="!DE #D#E (7.6) Therefore, # p 2 "=" w D 2 # D 2 "+" w E 2 # E 2 "+"2 w D w E # D # E ! DE (7.7) Other things equal, portfolio variance is higher when !DE is higher. In the case of perfect positive correlation, !DE = 1, the right-hand side of Equation 7.7 is a perfect square and simplifies to # p 2 "=" ( w D # D "+" w E # E ) 2 (7.8) or # p "=" w D # D "+" w E # E (7.9) Therefore, the standard deviation of the portfolio with perfect positive correlation is just the weighted average of the component standard deviations. In all other cases, the correlation coefficient is less than 1, making the portfolio standard deviation less than the weighted average of the component standard deviations.

A hedge asset has negative correlation with the other assets in the portfolio. Equation"7.7 shows that such assets will be particularly effective in reducing total risk. Moreover, Equation 7.2 shows that expected return is unaffected by correlation between returns. Therefore, other things equal, we will always prefer to add to our portfolios assets with low or, even better, negative correlation with our existing position.

Because the portfolio’s expected return is the weighted average of its component expected returns, whereas its standard deviation is less than the weighted average of the component standard deviations, portfolios of less than perfectly correlated assets always offer some degree of diversification benefit. The lower the correlation between the assets, the greater the gain in efficiency.

How low can portfolio standard deviation be? The lowest possible value of the correla- tion coefficient is $1, representing perfect negative correlation. In this case, Equation 7.7 simplifies to # p 2 "=" ( w D # D "$" w E # E ) 2 (7.10) and the portfolio standard deviation is # p "="Absolute"value" ( w D # D "$" w E # E ) (7.11) When ! = $1, a perfectly hedged position can be obtained by choosing the portfolio pro- portions to solve

wD #D"$"wE #E"="0 The solution to this equation is

wD"=" #E _______

#D"+"#E

wE"=" #D _______

#D"+"#E "="1"$"wD

(7.12)

These weights drive the standard deviation of the portfolio to zero.

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