fundamentals of mean-variance portfolio selection

Lecture 2: Delineating efficient portfolios, the shape of the mean-variance frontier, techniques for calculating the efficient frontier

Prof. Massimo Guidolin

Portfolio Management

Spring 2016

Overview

2Lecture 2: The Efficient Mean-Variance Frontier – Prof. Guidolin

The two-asset caseo Perfectly correlated assetso Perfectly negatively correlated assetso The case of -1 < < +1 (and = 0, uncorrelated assets)

The shape of the mean-variance frontier

The efficient frontier in the general N-asset case

The efficient frontier with unrestricted borrowing and lending and the riskless rate

The tangency portfolio and the separation theorem

One practical issue in the construction of the efficient frontier: parameter uncertainty

The two-asset case

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Call XA the fraction of a portfolio held in asset A and XB the fraction held in asset Bo However these may as well be portfolio and not individual assets

We require the investor to be fully invested, XA + XB = 1 and XB = 1 –XA, so that:

Such a simple, weighted way of combining is not necessarily true of the risk (standard deviation of the return) of the portfolio:

Using the equation for means returns and , we obtain

In order to learn more about this relationship, we study specific cases involving different degrees of co-movement btw. securities

Case 1: Perfect Positive Correlation ( = +1), the securities move in unison

Lecture 2: The Efficient Mean-Variance Frontier – Prof. Guidolin

The two-asset case: = +1

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o This derives from the presence of a perfect square inside bracketso C and S refer to an example (see above) o The expected return on the portfolio iso Thus with = + l, risk and return are linear combinations of the risk

and return of each security and because XC = (E[RP] - E[RS])/(E[RC] - E[RS]), we have

When = +1, there is a linear relationship btw. expected ptf. returns and std. dev.

When = +1, there is a linear relationship btw. expected ptf. returns and ptf. standard deviation


The two-asset case: = -1

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o E.g., with our inputs for Colonel Motors and Separated Edison, substituting this expression for XC into the equation for E[RP] and rearranging yields:

In the case of perfectly correlated assets, there is no reduction in risk from purchasing both assetso In the plot, any combinations of the two assets lie along a straight line

connecting the two assets Case 2: Perfect Negative Correlation ( = -1), the securities move

perfectly together but in exactly opposite directions In this case the standard deviation of the portfolio is:

The term in the brackets is equivalent to either of the following:or

Ptf. volatility is (*)or (**)o Since we took the square root and the square root of a negative

number is imaginary, the equations hold only when right-hand side >0Lecture 2: The Efficient Mean-Variance Frontier – Prof. Guidolin

The two-asset case: = -1

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o Since one is always positive when the other is negative (except when both equations equal zero), each also plots as a straight line when expected return is plotted against volatility

o In fact, the value of P in this case is always smaller than the value of P for the case where = + 1, for all values of XC between 0 and 1

We can go one step further: If two securities are perfectly negatively correlated, it should always be possible to find some combinationthat has zero risk

By setting either (*) or (**) equal to 0, we find that a portfolio with

will have zero risk

If two securities are perfectly negatively correlated, it is alwayspossible to find some combination that has zero risk


The two-asset case: -1 < < +1

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o Because S + C > C this implies that 0 < XC < 1 the zero risk ptf. will always involve positive investment in both securities

o In our example, zero risk obtains for a simple 1/3-2/3 portfolio because 3/(3 + 6) = 1/3

o Two equations relating mean and standard deviation, and for each selection of XC the appropriate one is the one that guarantees P 0

Case 3: Uncorrelated assets ( = 0), in this case

shows for any value for XC between 0 and 1 the lower the correlation the lower is the standard deviation of the ptf.

Ptf. standard deviation reaches its lowest value for = -1 (curve SBC) and its highest value for = + 1 (curve SAC)

These two curves represent the limits within which all portfolios of these two securities must lie for intermediate values of


The two-asset case: = 0

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o E.g., when = 0, noting that the covariance term drops out, the expression for standard deviation becomes

There is one point on this figure that is worth special attention: the portfolio that has minimum risk, the global minimum variance ptf.


GMVP

The two-asset case: -1 < < +1

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o This portfolio can be found by looking at the equation for risk

and minimizing by taking the FOC w.r.t. XC and solving:

o Continuing with our example, the value of XC that minimizes risk is

The correlation between any two actual stocks is almost always greater than 0 and considerably less than 1o E.g., in the case of = 0.5, the ptf. risk equation becomes

o In our example, minimum risk is obtained at a value of XC = 0 or 100% in Separated Edison

In the case -1 < < +1, the global minimum variance ptf. is the set of weights that minimizes the resulting ptf.risk


= 0

(***)

The critical coefficient in the two-asset case

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o Analytically,

In this example (i.e., = 0.5) there is no combination of the two securities that is less risky than the least risky asset by itself, though combinations are still less risky than under = +1

The particular value of the correlation coefficient for which no combination of two securities is less risky than the least risky security depends on the characteristics of the assets in question

For all assets there is some critical value of such that the risk on the portfolio can no longer be made less than the risk of the least risky asset in the portfolio without selling short

Setting XC equal to zero in (***) above and solving for * gives * = S/C so that when is equal to or higher than *, the least risky combination involves short selling the most risky security and may be impossible

There is some critical value of such that ptf. risk cannot be made less than the risk of the least risky asset without selling short



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Note that the portion of the portfolio possibility curve (aka mean-variance frontier) that lies above the MVP is concave while that which lies below the minimum variance portfolio is convex

This is a general characteristic of all portfolio problems

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o The three figures represent three hypothesized shapes for combinations of Colonel Motors and the MV portfolio

Shape (b) is impossible because combinations of assets cannot have more risk than on a straight line connecting two assets



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In (c) all portfolios have less risk than the straight line connecting Colonel Motors and the MVP, but this shape is impossibleo Examine the portfolios labeled U and V, combinations of the minimum

variance portfolio and Colonel Motorso Since U and V are portfolios, all combinations of U and V must lie

either on a straight line connecting U and V or above such a line The only legitimate shape is that shown in (a), which is concave

o Analogous reasoning can be used to show that if we consider combinations of the MVP and a security or portfolio with higher variance and lower return, the curve must be convex

What if the number of assets is some general N >> 2? In theory we could plot all conceivable risky assets and their

combinations in a diagram in return standard deviation space “In theory," not because there is a problem, but because there are

an infinite number of possibilities that must be considered

The (efficient) segment of the mean-variance frontier above the GMVP must have a concave shape



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If we were to plot all possibilities in risk-return space, we would get Examine the diagram and see if

we can eliminate any part of it from consideration by the investor

A rational investor would prefer a higher mean return to less and would prefer less risk to more

Thus, if we can find a set of ptfsthat (i) offered a bigger mean return for the same risk, or (ii) offered a lower risk for the same mean return, we have the choice set o E.g., ptf. B would be preferred by all investors to ptf. A because it

offers a higher return with the same level of risko Ptf. C would be preferable to portfolio A because it offers less risk for

the same level of return

Mean-variance dominance criteria simplify the opportunity set



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However, we can find no portfolio that dominates portfolio C or portfolio B

For this reason, an efficient set of ptfs. cannot include interior ptfs.

Moreover, for any point in risk-return space we want to move as far as possible in the direction of increasing mean return and as far as possible in the direction ofdecreasing risko Therefore we can eliminate D since

portfolio E exists, which has highermean return for the same risk

o This is true for every other portfolio as we move up the outer shell from D to point C

o Point C cannot be eliminated because it is the GMVPo Ptf. F is on the outer shell, but E has less risk for the same returno As we move up the outer shell from point F, all ptfs are dominated



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o This until we come to B that cannot be eliminated for there is no ptf. that has same return and less risk or the same risk and more return

o Point B represents that ptf. (usually a single security) that offers the highest expected return of all ptfs.

The efficient set consists of the envelope curve of all portfoliosthat lie between the global minimum variance portfolio and the maximum return portfolio

See a graph of the efficient frontier Based on our earlier proof, it is

a concave function Only linear segments may exist

The efficient frontier consists of the envelope curve of all portfoliosthat lie between the global MVP and the maximum return portfolio


The efficient frontier with short selling

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The portfolio problem, then, is to find all portfolios along this frontier, which we shall examine later

So far, one could only combine long positions in existing assets In many capital markets, an investor can often sell a security that he

or she does not own, a process called short sellingo In practice, this amounts to borrowing

an asset under the promise to the lenderthat she will be no worse off lending it

and with a commitment to return it at same date (say, end-of day)

o This requires re-funding any cash flows (e.g., dividends or coupons that the asset may pay out over time)

Short selling allows us to leverage up the return of best performing securities but also increases risks

With short sales, ptfs. exist that give infinite expected returnsLecture 2: The Efficient Mean-Variance Frontier – Prof. Guidolin

Unrestricted Borrowing and Lending

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Up to this point we have dealt with portfolios of risky assets only The introduction of a riskless asset, that yields RF, into the

investment opportunity set considerably simplifies the analysiso Because the return is certain, the standard deviation of the return on

the riskless asset must be zero Of course such a step requires assuming that a risk-free asset exists Borrowing can be considered as selling such a security short, so

that also borrowing can take place at the riskless rate The investor is interested in placing part of the funds in some

portfolio A and either lending or borrowing Call X the fraction of original funds that the investor places in ptf A

o X may exceed 1 because we are assuming that investors can borrow at the riskless rate and invest more than his initial funds in ptf. A

The expected return on the combination of riskless asset and risky portfolio is given by


Unrestricted Borrowing and Lending

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The risk on the combination is (C stands for combination)

Since we have already argued that σF is zero, Solving this expression for X yields X = σC/σA and substituting this

expression into the expression for expected return and re-arranging, yields

This is the equation of a straight line withslope = Sharpe ratio of ptf. A:

The line passes through point (σA, E[RA]) To the left of point A we have

combinations of lending and portfolio A, whereas to the right of point A we have combinations of borrowing and ptf. A

Problem: ptf. A we selected has no special properties; we could have combined portfolio B with riskless lending and borrowing


The tangency portfolio

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Combinations along the ray RFB are superior to combinations along RFA since they offer greater expected return for the same risk

We would like to rotate the straight line passing through RF as far as we can in a counterclockwise direction

The furthest we can rotate it is through G Point G is the tangency point between the

efficient frontier and a ray through RF

Cannot rotate the ray further because by definition there are no ptfs lying above the line passing through RF and G

All investors who believed they faced the efficient frontier and riskless lending and borrowing rates shown in the figure would hold the same ptf. of risky asset: G

All investors facing the same efficient frontier ABGH will select the same tangency portfolio G


The tangency portfolio and the separation theorem

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o Investors who are very risk-averse select a ptf along the segment RFG and place some money in a riskless asset and some in risky ptf G

o Others who were much more tolerant of risk would hold portfolios along the segment G-H, borrowing funds and placing their original capital plus the borrowed funds in portfolio G

Yet all of these investors would hold the tangency portfolio G. Thus, for the case of riskless lending and borrowing, identification

of portfolio G constitutes a solution to the problem The ability to determine the optimal ptf. of risky assets without

knowing anything about an investor is the separation theorem Our our assumptions realistic? While there is no question about the

ability of investors to lend at the risk-free rate (buy government securities), they could possibly not borrow at this rate

According to the separation theorem, all investors facing the same efficient frontier select the tangency ptf. of risky assets regardless of their preferences towards risk


The effects of frictions on the separation theorem

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Whe borrowing at the riskless rate is impossible, the efficient frontier becomes RFGH

Certain investors will hold portfolios of risky assets located between G and H

However, any investor who held some riskless asset would place all remaining funds in the tangency portfolio G

The separation theorem fails: differentinvestors may select different risky ptfs

A possibility is that investors can lend at one rate (RF) but must pay a different and higher rate to borrow (R’F)

The efficient frontier becomew RFGHI There is a small range of risky ptfs that

is optional for investors to hold and two different tangency ptfs, G and H


One practical issue in portfolio choice

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Also in this case the separation theorem fails Reliable inputs on means, variances, and covariances are crucial to

the proper use of mean-variance optimization Common to use historical risk, return, and correlation as a starting

point in obtaining inputs for calculating the efficient frontier If return characteristics do not change through time, then the

longer the data are available the more accurate are the estimateso E.g., the formula for the standard error of the mean of a sequence of

independent random variables is σ2/N where N is the sample sizeo This effect may be first-order: imagine an investor choosing between

two investments, each with identical sample means and varianceso The standard approach would view the two investments as equivalento If you consider the additional information that the first sample mean

was based on 1 year of data and the second on 10 years of data, common sense would suggest that the second alternative is less risky:



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o The first part of the expression captures the inherent risk in the return; the second term captures the uncertainty that comes from lack of knowledge about the true mean return

Characteristics of asset returns usually change over time There is a trade-off between using a long time frame to improve the

estimates and having potentially inaccurate estimates from the longer time period because the characteristics have changed

Because of this, most analysts modify historical estimates to reflect beliefs about how current conditions differ from past conditions

The choice of the time period is more complicated when a relatively new asset class is added to the mix, and the available data for the new asset is much less than for other assets

For example, consider the case of CDS or CDOs as asset classes

There is a trade-off between using a long time frame to improve the estimates and having potentially inaccurate estimates from the longer time period because the characteristics have changed



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o An analyst who wishes to use historical data could use all available data or use shorter data only from the common period of observation

Consider the IFC emerging markets index example in the table

o Differences may be substantial: e.g., statistics over the longer term are consistent with an equilibrium in which a higher investor risk is compensated by higher investor expected return

o Statistics over the period of common observation, beginning in 1985, are inconsistent with this argument

Also correlationsare very differentand will affect thefrontier


The stock-bond choice again (shorting allowed)

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Consider again the allocation between equity and debt The estimated historical inputs are:

The minimum variance portfolio is given by

Unsurprisingly, it implies selling the index short (write futures?) The associated st. dev. is 4.75%,

which is slightly less than the one associated with 100% in bonds, so were slightly below the critical ρ

This is the efficient frontier with short sales allowed (it continuesto the right)


(tangency)

The stock-bond choice again (no short sales)

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The tangency portfolio is T and we will see how it is calculated soon Assuming a 5% T-bill rate, we have E[RT] = 13.54% and T =

16.95% so that the slope of the line connecting the tangency portfolio and the efficient frontier is (13.54% - 5%)/16.95% = 0.5

The equation of the efficient frontier with riskless lending and borrowing is

Knowing the expected return of T we can easily determine its composition:

The efficient frontier with no short sales is to the right

In this case the GMVP is 100% in bonds


Hints to techniques to calculate the efficient frontier

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We distinguish among 4 cases:o Short sales are allowed and riskless lending and borrowing is possibleo Short sales are allowed but riskless lending or borrowing is noto Short sales are disallowed but riskless lending and borrowing existso Neither short sales nor riskless lending and borrowing are allowed

The derivation of the efficient set when short sales are allowed and there is a riskless lending and borrowing rate is the simplest case

In this case, the efficient frontier is the entire length of the ray extending through RF and B

An equivalent way of identifying the ray RF-B is to recognize that it is the ray with thegreatest slope,

The efficient set is determined by findingthe ptf with the greatest θ (Sharpe) ratio that satisfies the weight sum constraint


θ

Hints to techniques to calculate the efficient frontiero This is a constrained maximization problem for which there are

standard solution techniqueso For example, it can be solved by the method of Lagrangian multipliers

There is an alternative: the constraint could be substituted into the objective function and the objective function maximized as in an unconstrained problem

Making this substitution in the objective function and stating the expected return and standard deviation of return in the general form, one maximizes

This problem can be solved in standard ways by imposing and solving first order conditions

28Lecture 2: The Efficient Mean-Variance Frontier – Prof. Guidolin

Hints to techniques to calculate the efficient frontiero In this case the FOCs are also sufficient because the objective function

is concave (we are dividing by a positive quadratic form) We can prove that

Because each Xk is multiplied by a constant , define a new variable Zk = Xk and substituting Zk for the Xk simplifies the formulation:

We have one equation like this for each value of i Now solve the system of simultaneous equations:

Lecture 2: The Efficient Mean-Variance Frontier – Prof. Guidolin 29

Hints to techniques to calculate the efficient frontier To determine the optimum amount to invest, we first solve the

equations for the Zs and this is generally possible because there are N equations (one for each security) and N unknowns (the Zk)

Then the optimum proportions to invest in stock k is:

When short sales are allowed but there is no riskless lending and borrowing rate, the solution above must be modifiedo Assume a riskless lending and borrowing rate and find the optimumo Assume that a different riskless lending

and borrowing rate exists and find the optimum that corresponds to this rate

o Continue changing the riskless rate until the full efficient frontier is deterrnined

One can show that the optimal proportion to invest in any security is simply a linear function of RF


Hints to techniques to calculate the efficient frontier Furthermore, the entire efficient frontier can be constructed as a

combination of any two portfolios that lie along it Therefore the identification of the characteristics of the optimal

portfolio for any two arbitrary values of RF is sufficient to trace out the total efficient frontier

When short sales are not allowed but there is riskless lending and borrowing, the solution comes from solving

This is a nonlinear mathematical programming problem because of the inequality restriction on the weightso Equations involving squared terms and cross-product terms are called

quadratic equation There are computer packages for solving quadratic programming

problems subject to constraints In our case, the Excel solver will do


Hints to techniques to calculate the efficient frontier The imposition of short sales constraints has complicated the

solution technique, forcing us to use quadratic programming Once we resort to this technique, it is a simple matter to impose

other requirements on the solution Any set of requirements that can be formulated as linear functions

of the investment weights can be imposed on the solutiono For example, some managers wish to select optimum ptfs given that

the dividend yield on the portfolios is at least some number, D

o If one wants no short sale constraints, these can be imposed:

o Perhaps, the most frequent constraints are those that place an upper limit on the fraction of the portfolio that can be invested in any asset

o Upper limits on the amount that can invested in any one stock are often part of the charter of mutual funds

o It is possible to build in constraints on the amount of turnover in a portfolio and to allow the consideration of transaction costs


Summary and conclusions

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Provided that < 1 diversification offers a costless payoff: risk reduction without any costs in terms of lower expected return

Such a risk-reduction is maximal when = -1, when a special ptf. can be found that implies zero risk but positive expected return

When = 0, in the limit risk can also be removed by increasing the total number of assets (N) in the portfolio

When > 0, even though N →∞, the total amount of risk does not level off to zero, but converges to the average covariance across all pairs of assets in the economy

The efficient mean-variance set (frontier) is the subset of the opportunity set that lies above the global minimum variance portfolio and has a concave shape

The tangency portfolio is unique across all investors that perceive the same efficient set

The separation theorem states that all investors will demand the same risky portfolio irrespective of their risk aversion


fundamentals of mean-variance portfolio selection

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