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This problem cannot be reduced to the solution of a set of linear equations It is termed a quadratic program, since the objective is quadratic and the constraints are linear equalities and inequalities. Special computer programs are available for solving such problems, but small to moderate-sized problems of this type can be solved readily with spreadsheet programs. In the financial industry there are a multitude of special-purpose programs designed to solve this problem for hundreds or even thousands of assets

A significant difference between the two formulations is that when short selling is allowed, most, if not all, of the optimal wfs have nonzero values (either positive or negative), so essentially all assets are used. By contrast, when short selling is not allowed, typically many weights are equal to zero

Example 6.10 (The three uncorrelated assets) Consider again the assets of Example 6.9, but with shorting not allowed Efficient points must solve problem (6 la) with the parameters of the earlier example. In this case the problem cannot be reduced to a system of equations, but by considering combinations of pairs of assets, the efficient frontier can be found The general solution is as follows:

6.7 THE TWO-FUND THEOREM*

The minimum-variance set has an important property that greatly simplifies its computation. Recall that points in this set satisfy the system of n + 2 linear equations [Eqs (6.5a~c)], which is repeated here:

CFijWj — krj ~ ¡i ~ 0 for i — 1,2,..,,« (6 8a)

Suppose that there are two known solutions, wl = (wj. . ..., w*), k{, /x] and w2 = (ujy, w;2, . , uj~), A.2, /x2, with expected rates of return F1 and F2, respectively, Let us form a combination by multiplying the first by a and the second by (1 — a). By direct substitution, we see that the result is also a solution to the n + 2 equations, corresponding to the expected value ar] + (I - a)72 To check this in detail, notice that aw1 +(1 — a)\v2 is a legitimate portfolio with weights that sum to 1; hence (6 8c) is satisfied. Next notice that the expected return is in fact oirj +(I — a)r2; hence (6.8/?) is satisfied for that value.. Finally, notice that since both solutions make the left side of (6.8a) equal to zero, their combination does also; hence (6 8«) is satisfied. This implies that the combination portfolio aw1 + (1 - a)w2 is also a solution; that is, it also represents a point in the minimum-variance set. This simple result is usually quite surprising to most people on their first exposure to the subject, but it highlights an important property of the minimum-variance set.

To use this result, suppose w1 and w2 are two different portfolios in the minimum-variance set. Then as a varies over -oo < a < oo, the portfolios defined by jf. (i ™ a;)w2 sweep out the entire minimum-variance set We can, of course, select the two original solutions to be efficient (that is, on the upper portion of the minimum-variance set), and these will generate all other efficient points (as well as all other points in the minimum-variance set). This result is often stated in a form that has operational significance for investors:

The two-fund theorem Two efficient funds (portfolios) can be established so that any efficient portfolio can be duplicated, in terms of mean and variance, as a combination of these two In other words, all investors seeking efficient portfolios need only invest in combinations of these two funds.

This result has dramatic implications According to the two-fund theorem, two mutual funds3 could provide a complete investment service for everyone There would be no need for anyone to purchase individual stocks separately; they could just purchase shares in the mutual funds. This conclusion, however, is based on the assumption that everyone cares only about mean and variance; that everyone has the same assessment of the means, variances, and covariances; and that a single-period framework is appropriate. All of these assumptions are quite tenuous Nevertheless, if you are an investor without the time or inclination to make careful assessments, you might choose to find two funds managed by people whose assessments you trust, and invest in those two funds.

The two-fund theorem also has implications for computation In order to solve (6.5fl-c) for all values of r it is only necessary to find two solutions and then form combinations of those two, A particularly simple way to specify two solutions is to specify values of X and p, Convenient choices are (a) X = 0, p = 1 and (b) X = 1, ¡1 = 0, In either of these solutions the constraint Yl'Ui w> = 1 may violated, but this can be remedied later by normalizing all s by a common scale factor The solution obtained by choice (a) ignores the constraint on the expected mean rate of return; hence this is the minimum-variance point The overall procedure is illustrated in the following example

Example 6.11 (A securities portfolio) The information concerning the 1-year co-variances and mean values of the rates of return on five securities is shown in the top part of Table 6.2. The mean values are expressed on a percentage basis, whereas the co variances are expressed in units of (percent)2/100 For example, the first security has an expected rate of return of 15 1% = .151 and a variance of return of ,023, which translates into a standard deviation of V 023 = 152 = 15.2% per year

3A mutual fund is an investment company that acccpts investment capital from individuals and reinvests that capital in a diversity of individual stocks Each individual is entitled to his or her proportionate share of the funds portfolio value, less certain operating fees and commissions

TABLE 6-2

A Securities Portfolio

TABLE 6-2

A Securities Portfolio

Security |
Covariance V |
r | ||||

1 |
2.30 |
9.3 |
.62 |
74 |
- 23 |
15 1 |

2 |
9.3 |
I 40 |
.22 |
56 |
26 |
125 |

3 |
62 |
22 |
1 80 |
78 |
27 |
34 7 |

4 |
74 |
56 |
78 |
3 40 |
- 56 |
9 02 |

5 |
-.23 |
26 |
- 27 |
- 56 |
2 60 |
17 68 |

141 |
3 652 |
088 |
158 | |

401 |
.3 583 |
251 |
155 | |

452 |
7 248 |
282 |
314 | |

166 |
874 |
104 |
0.38 | |

440 |
7 706 |
275 |
3.34 | |

Mean |
14413 |
15 202 | ||

Variance |
625 |
659 | ||

Std dev |
.791 |
812 |

The covariances and mean rales of return are shown for five securities The portfolio w1 is the minimum-variance point, and vr is another efficient portfolio made from these five securities

The covariances and mean rales of return are shown for five securities The portfolio w1 is the minimum-variance point, and vr is another efficient portfolio made from these five securities

We shall find two funds in the minimum-variance set. First we set X — 0 and [i — 1 in (6,5), We thus solve the system of equations for the vector v1 = (uj, . , uj). This solution can be found using a spreadsheet package that solves linear equations The coefficients of the equation are those of the covariance matrix, and the right-hand sides are ail l's. The resulting vj's are listed in the first column of the bottom part of Table 6.2 as components of the vector v! Next we normalize the u/'s so that they sum to 1, obtaining luj's as w =

The vector w1 = (uij, iv,, , w]) defines the minimum-variance point.

Second we set (i = 0 and X = I We thus solve the system of equations

6.8 inclusion of a risk-free asset 165

for a solution v2 = (u[, . , Again we normalize the resulting vector v2 so its components sum to I, to obtain w2. The vectors vl, v2, \v!, w2 are shown in the bottom part of Table 6 2 Also shown are the means, variances, and standard deviations corresponding to the portfolios defined by w1 and w2. All efficient portfolios are combinations of these two

In the previous few sections we have implicitly assumed that the n assets available are all risky; that is, they each have cr > 0, A risk-free asset has a return that is deterministic (that is, known with certainty) and therefore has a — 0 In other-words, a risk-free asset is a pure interest-bearing instrument; its inclusion in a portfolio corresponds to lending or borrowing cash at the risk-free rate Lending (such as the purchase of a bond) corresponds to the risk-free asset having a positive weight, whereas borrowing corresponds to its having a negative weight,

'The inclusion of a risk-free asset in the list of possible assets is necessary to obtain realism. Investors invariably have the opportunity to borrow or lend Fortunately, as we shall see shortly, inclusion of a risk-free asset introduces a mathematical degeneracy that greatly simplifies the shape of' the efficient frontier.

To explain the degeneracy condition, suppose that there is a risk-free asset with a (deterministic) rate of return rj. Consider any other risky asset with rate of return /, having mean 7 and variance cr Note that the covariance of these two returns must be zero. This is because the covariance is defined to be E[(/ — /T)(/y - ty)] — 0.

Now suppose that these two assets are combined to form a portfolio using a weight of a for the risk-free asset and 1 — a for the risky asset, with a < 1 The mean rate of return of this portfolio will be aif -f- (I — a)r. The standard deviation of the return will be >/(! — a)2cr2 — (1 — a)a. This is because the risk-free asset has no variance and no covariance with the risky asset. The only term left in the formula is that due to the risky asset

If we define, just for the moment, cry ~ 0, we see that the portfolio rate of return has mean = aiy 4- (1 — a)7

These equations show that both the mean and the standard deviation of the portfolio vary linearly with a. This means that as a varies, the point representing the portfolio traces out a straight line in the F-ct plane

Suppose now that there are n risky assets with known mean rates of return F, and known covariances cr,-,-. In addition, there is a risk-free asset with rate of return iy The inclusion of the risk-free asset in the list of available assets has a profound effect on the shape of the feasible region. The reason for this is shown in Figure 6, 13(a) First we construct the ordinary feasible region, defined by the n risky assets, (This region may be either the one constructed with shorting allowed or the one constructed without shorting.) This region is shown as the darkly shaded region in the figure. Next,

FIGURE 6,13 Effect of a risk-free asset. Inclusion of a risk-free asset adds lines to the feasible region (a) If both borrowing and lending are allowed, a complete infinite triangular region is obtained (b) If only lending is allowed, (he region will have a triangular front end, but will curve for larger a

FIGURE 6,13 Effect of a risk-free asset. Inclusion of a risk-free asset adds lines to the feasible region (a) If both borrowing and lending are allowed, a complete infinite triangular region is obtained (b) If only lending is allowed, (he region will have a triangular front end, but will curve for larger a for each asset (or portfolio) in this region we form combinations with the risk-free asset, In forming these combinations we allow borrowing or lending of the risk-free asset, but only purchase of the risky asset. These new combinations trace out the infinite straight line originating at the risk-free point, passing through the risky asset, and continuing indefinitely. There is a line of this type for every asset in the original feasible set. The totality of these lines forms a triangularly shaped feasible region, indicated by the light shading in the figure

This is a beautiful result. The feasible region is an infinite triangle whenever a risk-free asset is included in the universe of available assets,

If borrowing of the risk-free asset is not allowed (no shorting of this asset), we can adjoin only the finite line segments between the risk-free asset and points in the original feasible region. We cannot extend these lines further, since this would entail borrowing of the risk-free asset The inclusion of these finite line segments leads to a new feasible region with a straight-line front edge but a rounded top, as shown in Figure 6 13(b) .

When risk-free borrowing and lending are available, the efficient set consists of a single straight line, which is the top of the triangular feasible region. This line is tangent to the original feasible set of risky assets (See Figure 6.14.) There will be a point F in the original feasible set that is on the line segment defining the overall efficient set. It is clear that any efficient point (any point on the line) can be expressed as a combination of this asset and the risk-free asset. We obtain different efficient points by changing the weighting between these two (including negative weights of the risk-free asset to borrow money in order to leverage the buying of the risky asset) The portfolio

represented by the tangent point can be thought of as a fund made up of assets and sold as a unit. The role of this fund is summarized by the following statement:

The one-fund theorem There is a single fund F of risky assets such that any efficient portfolio can be constructed as a combination of the fund F and the risk-free asset

This is a final conclusion of mean-variance portfolio theory, and this conclusion is the launch point for the next chapter It is fine to stop reading here, and (after doing some exercises) to go on to the next chapter But if you want to see how to calculate the special efficient point F, read the specialized subsection that follows.

Solution Method*

How can we find the tangent point that represents the efficient fund? We just characterize that point in terms of an optimization problem. Given a point in the feasible region, we draw a line between the risk-free asset and that point We denote the angle between that line and the horizontal axis by 0 For any feasible (risky) portfolio p, we have tan 0 =

The tangent portfolio is the feasible point that maximizes 6 or, equivalently, maximizes tan6>. It turns out that this problem can be reduced to the solution of a system of linear equations

To develop the solution, suppose, as usual, that there are n risky assets We assign weights w\, u>2, w„ to the risky assets such that ( w-, = 1 There is zero weight on the risk-free asset in the tangent fund. (Note that we are allowing short selling among the risky assets ) For rp = Yl'l-i w>'»1 we have Tp — JZ/Li and if = EL I w>'f Thus>

anWiWt

It should be clear that multiplication of all w/s by a constant will not change the expression, since the constant will cancel Hence it is not necessary to impose the constraint Wj = 1 here

We then set the derivative of tant? with respect to each wk equal to zero. This leads (see Exercise 10) to the following equations:

where x is an (unknown) constant. Making the substitution = xwj for each /, (6 9) becomes n

We solve these linear'equations for the u/s and then normalize to determine the w^s; that is,

Example 6.12 (Three uncorrected assets) We consider again Example 6 9, where the three risky assets were uncorrelated and each had variance equal to 1. The three mean rates of return were n = 1, Ti = 2, and ô = 3. We assume in addition that there is a risk-free asset with rate ty = .5,

We apply (6,9), which is very simple in this case because the covariances are all zero, to find us = I — 5 — .5 V2 - 2 - .5 = 1.5 y3 = 3 - .5 = 2.5 We then normalize these values by dividing by their sum, 4 5, and find

Example 6.13 (A larger portfolio) Consider the five risky assets of Example 6.11 Assume also that there is a risk-free asset with iy = 10%. We can easily find the special fund F

We note that the system of equations (6.10) is identical to those used to find v1 and v2 in Example 6.11, but with a different right-hand side. Actually the right-hand side is a linear combination of those used for v! and v2; namely, 7k—iy = 1 xrk~~iyx I. Therefore the solution to (6.10) is v = v2 — tyv1 Thus (using ly = lOto be consistent with the units used in the earlier example), v = (2.242, -,427,2.728, - 786, .3..306) We normalize this to obtain the final result w = (..317, - 060, -386, -.111, 468)

Basically, we have used the fact that portfolio F is a combination of two known efficient points.

The study of one-period investment situations is based on asset and portfolio returns Both total returns and rates of return are used The return of an asset may be uncertain, in which case it is useful to consider it formally as a random variable. The probabilistic properties of such random returns can be summarized by their expected values, their variances, and their covariances with each other,

A portfolio is defined by allocating fractions of initial wealth to individual assets, The fractions (or weights) must sum to I; but some of these weights may be negative if short selling is allowed The return of a portfolio is the weighted sum of the returns of its individual assets, with the weights being those that define the portfolio, The expected return of the portfolio is, likewise, equal to the weighted average of the expected returns of the individual assets. The variance of the portfolio is determined by a more complicated formula: a2 = Yl'l j= i W/W/O);, where the w/'s are the weights and the cr,/s are the covariances

From a given collection of n risky assets, there results a set of possible portfolios made from all possible weights of the /; individual assets If the mean and the standard deviation of these portfolios are plotted on a diagram with vertical axis F (the mean) and horizontal axis a (the standard deviation), the region so obtained is called the feasible region Two alternative feasible regions are defined: one allowing shorting of assets and one not allowing shotting.

It can be argued that investors who measure the value of a portfolio in terms of its mean and its standard deviation, who are risk averse, and who have the nonsatiation property will select portfolios on the upper left-hand portion of the feasible region—the efficient frontier.

Points on the efficient frontier can be characterized by an optimization problem originally formulated by Markowitz. This problem seeks the portfolio weights that minimize variance for a given value of mean return. Mathematically, this is a problem with a quadratic objective and two linear constraints If shorting is allowed (so that the weights may be negative as well as positive), the optimal weights can be found by solving a system of /i-f-2 linear equations and n-\-2 unknowns. Otherwise if shorting is not allowed, the Markowitz problem can be solved by special quadratic programming packages

An important property of the Markowitz problem, when shorting is allowed, is that if two solutions are known, then any weighted combination of these two solutions is also a solution This leads to the fundamental two-fund theorem: investors seeking efficient portfolios need only invest in two master efficient funds

Usually it is appropriate to assume that, in addition to n risky assets, there is available a risk-free asset with fixed rate of return //. The inclusion of such an asset greatly simplifies the shape of the feasible region, transforming the upper boundary into a straight line. This line is the efficient frontier. The straight-line frontier touches the original feasible region (the region defined by the risky assets only) at a single point F This leads to the important one-fund theorem: investors seeking efficient portfolios need only invest in one master fund of risky assets and in the risk-free asset Different investors may prefer different combinations of these two.

The single efficient fund of risky assets F can be found by solving a system of n linear equations and n unknowns When the solution to this system is normalized so that its components sum to 1, the resulting components are the weights of the risky assets in the master fund.

1. (Shorting with margin) Suppose thai lo short a slock you are required to deposit an amount equal lo the initial price Xo of the stock At the end of I year the stock price is X\ and you liquidate your position You receive your profit from shorting equal to X0 — Yj and you recover your original deposit If R is the total return of the stock, what is the total return on your short?

2. (Dice product) Two dice are rolled and the two resulting values are multiplied together to form the quantity z- What are the expected value and the variance of the random variable zl [Hint. Use the independence of the two separate dice..]

3. (Two correlated assets) The correlation p between assets A and B is 1, and other data are given in Table 63 [Note: p = crAU/(crAai3).]

TABLE 6.3

Two Correlated Cases

TABLE 6.3

Two Correlated Cases

Asset |
j |
a |

A |
10 0% |
15% |

B |
18.0% |
30% |

(a) Find the proportions a of A and (1 -a) of B that define a portfolio of A and B having minimum standard deviation

(b) What is the value of this minimum standard deviation?

(c) What is the expected return of this portfolio?

4, (Two stocks) Two stocks are available The corresponding expected rates of return are F| and r2; the corresponding variances and covariances are of, and a{1 What percentages of total investment should be invested in each of the two stocks lo minimize the total variance of the rate of return of the resulting portfolio? What is the mean rate of return of this portfolio?

5. (Rain insurance) Gavin Jones's friend is planning to invest $1 million in a rock concert to be held 1 year from now The friend figures that he will obtain $3 million revenue from his $1 million investment—unless, my goodness, it rains If it rains, he will lose his entire investment. There is a 50% chance that it will rain the day of the concert Gavin suggests that he buy rain insurance He can buy one unit of insurance for $.50, and this unit pays $1 if it rains and nothing if it does not He may purchase as many units as he wishes, up lo $3 million

(a) What is the expected rate of return on his investment if he buys it units of insurance? (The cost of insurance is in addition to his $1 million investment.)

(b) What number of units will minimize the variance of his return? What is this minimum value? And what is the corresponding expected rate of return? [Hint-. Before calculating a general expression for variance, think about a simple answer ]

6. (Wild cats) Suppose there are n assets which are uncorrected (They might be n different "wild cat" oil well prospects) You may invest in any one, or in any combination of them The mean rate of return r is the same for each asset, but the variances are different The return on asset i has a variance of af for i = 1,2, ,n.

(a) Show the situation on an T-a diagram Describe the efficient set

(b) Find the minimum-variance point Express your result in terms of j

7. (Markowitz fun) There are just three assets with rates of return / / 2, and r^, respectively The covariance matrix and the expected rates of return are

"2 |
1 0" |
" 4" | ||

v = |
1 |
2 1 |
r = |
8 |

_0 |
1 2 |
8 |

{a) Find the minimum-variance portfolio [Hint: By symmetry w{ = wj ]

(b) Find another efficient portfolio by setting X = I, /i = 0

(c) If the risk-free rate is /y =s 2, find the efficient portfolio of risky assets

8. (Tracking) Suppose that it is impractical to use all the assets that are incorporated into a specified portfolio (such as a given efficient portfolio). One alternative is to find the portfolio, made up of a given set of n stocks, that tracks the specified portfolio most closely—in the sense of minimizing the variance of the difference in returns

Specifically, suppose that the target portfolio has (random) rate of return ist Suppose that there are n assets with (random) rates of return /¡.r^, . tr„. We wish to find the portfolio rate of return

(a) Find a set of equations for the or,\s

(b) Although this portfolio tracks the desired portfolio most closely in terms of variance, it may sacrifice the mean Hence a logical approach is to minimize the variance of the tracking error subject to achieving a given mean return As the mean is varied, this results in a family of portfolios that are efficient in a new sense—say, tracking efficient. Find the equation for the or, 's that are tracking efficient

9„ (Betting wheel) Consider a genera! betting wheel with n segments The payoff for a $1 bet on a segment i is A-, Suppose you bet an amount Bj = \/Ai on segment i for each / Show that the amount you win is independent of the outcome of the wheel. What is the risk-free rate of return for the wheel? Apply this to the wheel in Example 6 7.

10. (Efficient portfolio o) Derive (6 9). [Hint Note that aijw.w,

Ijj iv, Wj

Ijj iv, Wj

Mean-variance portfolio theory was initially devised by Markowitz [1-4]. Other important developments were presented in [5-8] The one-fund argument is due to Tobin [9] For comprehensive textbook presentations, see [S0-1I] and the other general investment textbooks listed as references for Chapter 2

1 Markowitz, H. M (1952), "Portfolio Selection," Journal of Finance, 7, no. 1, 77-91,

2 Markowitz, H M (1956), "The Optimization of a Quadratic Function Subject to Linear

Constraints," Naval Research L ogistics Quarterly, 3, nos 1-2, 111 —133

3 Markowitz, H M. (1987), Portfolio Selection, Wiley, New York

4 Markowitz, H M. (1987), Mean-Variance Analysis in Portfolio Choice and Capital Markets,

Basil Blackwell, New York 5. Hester, D D , and J Tobin (1967), Risk Aversion and Portfolio Choice, Wiley, New York

6 Fama, E F (1976), Foundations of Finance, Basic Books, New York

7 Sharpe, W F (1967), "Portfolio Analysis," Journal of Financial and Quantitative Analysis,

8 Levy, H {1979), "Does Diversification Always Pay?" TIMS Studies in Management Science

9 Tobin, I (1958), "Liquidity Preference as Behavior Toward Risk," Review of Economic

Studies, 26, February, 65-86. ¡0 Francis, J C, and G Alexander (1986), Portfolio Analysis, 3rd ed., Prentice Hall, Engle-wood Cliffs, NT

11 Elton, E .1., and M J Gruber (1991), Portfolio Theory and Investment Analysis, 4th ed , Wiley. New York

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