Lecture 2: Delineating efficient portfolios, the shape of the meanvariance frontier, techniques for calculating the efficient frontier Prof. Massimo Guidolin Portfolio Management Spring 2016
Overview The two-asset case o Perfectly correlated assets o Perfectly negatively correlated assets o 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 2
The two-asset case Call X A the fraction of a portfolio held in asset A and X B the fraction held in asset B o However these may as well be portfolio and not individual assets We require the investor to be fully invested, X A + X B = 1 and X B = 1 X A, 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 3
The two-asset case: = +1 When = +1, there is a linear relationship btw. expected ptf. returns and ptf. standard deviation o This derives from the presence of a perfect square inside brackets o C and S refer to an example (see above) o The expected return on the portfolio is o Thus with = + l, risk and return are linear combinations of the risk and return of each security and because X C = (E[R P ] - E[R S ])/(E[R C ] - E[R S ]), we have When = +1, there is a linear relationship btw. expected ptf. returns and std. dev. 4
The two-asset case: = -1 o E.g., with our inputs for Colonel Motors and Separated Edison, substituting this expression for X C into the equation for E[R P ] and rearranging yields: In the case of perfectly correlated assets, there is no reduction in risk from purchasing both assets o 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 >0 5
The two-asset case: = -1 If two securities are perfectly negatively correlated, it is always possible to find some combination that has zero risk 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 X C 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 combination that has zero risk By setting either (*) or (**) equal to 0, we find that a portfolio with will have zero risk 6
The two-asset case: -1 < < +1 o Because S + C > C this implies that 0 < X C < 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 X C the appropriate one is the one that guarantees P 0 Case 3: Uncorrelated assets ( = 0), in this case shows for any value for X C 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 7
The two-asset case: = 0 GMVP 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. 8
The two-asset case: -1 < < +1 In the case -1 < < +1, the global minimum variance ptf. is the set of weights that minimizes the resulting ptf.risk o This portfolio can be found by looking at the equation for risk and minimizing by taking the FOC w.r.t. X C and solving: (***) = 0 o Continuing with our example, the value of X C that minimizes risk is The correlation between any two actual stocks is almost always greater than 0 and considerably less than 1 o 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 X C = 0 or 100% in Separated Edison 9
The critical coefficient in the two-asset case 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 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 X C 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 10
The shape of the mean-variance frontier Note that the portion of the portfolio possibility curve (aka meanvariance 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. 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 11
The shape of the mean-variance frontier The (efficient) segment of the mean-variance frontier above the GMVP must have a concave shape In (c) all portfolios have less risk than the straight line connecting Colonel Motors and the MVP, but this shape is impossible o Examine the portfolios labeled U and V, combinations of the minimum variance portfolio and Colonel Motors o 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 12
The efficient frontier in the general N-asset case Mean-variance dominance criteria simplify the opportunity set 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 ptfs that (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 risk o Ptf. C would be preferable to portfolio A because it offers less risk for the same level of return 13
The efficient frontier in the general N-asset case 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 riskreturn space we want to move as far as possible in the direction of increasing mean return and as far as possible in the direction of decreasing risk o Therefore we can eliminate D since portfolio E exists, which has higher mean 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 GMVP o Ptf. F is on the outer shell, but E has less risk for the same return o As we move up the outer shell from point F, all ptfs are dominated 14
The efficient frontier in the general N-asset case The efficient frontier consists of the envelope curve of all portfolios that lie between the global MVP and the maximum return portfolio 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 portfolios that 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 15
The efficient frontier with short selling 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 selling o In practice, this amounts to borrowing an asset under the promise to the lender that 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 returns 16
Unrestricted Borrowing and Lending Up to this point we have dealt with portfolios of risky assets only The introduction of a riskless asset, that yields R F, into the investment opportunity set considerably simplifies the analysis o 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 17
Unrestricted Borrowing and Lending The risk on the combination is (C stands for combination) Since we have already argued that σ F is zero, Solving this expression for X yields X = σ C /σ A and substituting this expression into the expression for expected return and rearranging, yields This is the equation of a straight line with slope = Sharpe ratio of ptf. A: The line passes through point (σ A, E[R A ]) 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 18
The tangency portfolio All investors facing the same efficient frontier ABGH will select the same tangency portfolio G Combinations along the ray R F B are superior to combinations along R F A since they offer greater expected return for the same risk We would like to rotate the straight line passing through R F 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 R F Cannot rotate the ray further because by definition there are no ptfs lying above the line passing through R F 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 19
The tangency portfolio and the separation theorem 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 o Investors who are very risk-averse select a ptf along the segment R F G 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 20
The effects of frictions on the separation theorem Whe borrowing at the riskless rate is impossible, the efficient frontier becomes R F GH 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: different investors may select different risky ptfs A possibility is that investors can lend at one rate (R F ) but must pay a different and higher rate to borrow (R F ) The efficient frontier becomew R F GHI There is a small range of risky ptfs that is optional for investors to hold and two different tangency ptfs, G and H 21
One practical issue in portfolio choice 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 estimates o E.g., the formula for the standard error of the mean of a sequence of independent random variables is σ 2 /N where N is the sample size o This effect may be first-order: imagine an investor choosing between two investments, each with identical sample means and variances o The standard approach would view the two investments as equivalent o 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: 22
One practical issue in portfolio choice 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 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 23
One practical issue in portfolio choice 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 correlations are very different and will affect the frontier 24
The stock-bond choice again (shorting allowed) 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 continues to the right) (tangency) 25
The stock-bond choice again (no short sales) The tangency portfolio is T and we will see how it is calculated soon Assuming a 5% T-bill rate, we have E[R T ] = 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 26
Hints to techniques to calculate the efficient frontier We distinguish among 4 cases: o Short sales are allowed and riskless lending and borrowing is possible o Short sales are allowed but riskless lending or borrowing is not o Short sales are disallowed but riskless lending and borrowing exists o 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 R F and B An equivalent way of identifying the ray R F -B is to recognize that it is the ray with the greatest slope, θ The efficient set is determined by finding the ptf with the greatest θ (Sharpe) ratio that satisfies the weight sum constraint 27
Hints to techniques to calculate the efficient frontier o This is a constrained maximization problem for which there are standard solution techniques o 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 28
Hints to techniques to calculate the efficient frontier o 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 X k is multiplied by a constant, define a new variable Z k = X k and substituting Z k for the X k simplifies the formulation: We have one equation like this for each value of i Now solve the system of simultaneous equations: 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 Z k ) 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 modified o Assume a riskless lending and borrowing rate and find the optimum o 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 R F 30
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 R F 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 weights o 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 31
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 solution o 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 32
Summary and conclusions 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 33