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Unformatted text preview: IOE 310 Fall 2010 Project Maximizing Omega David L. Kaufman [email protected] October 11, 2010 Keywords: Omega, portfolio optimization, efficient frontier. 1 Introduction In 1952 Harry M. Markowitz published one of the most famous (if not the most famous) papers in finance, “Portfolio Selection” [2]. In this paper, Markowitz presents a one period decision model for optimal portfolio allocation. After realizing that previous present value models ignored the impacts of risk, he incorporated risk of financial assets with uncertain outcomes through correlations. His model allowed investors to systematically study the benefits of investment diversification. Markowitz’s work was truly seminal, and in 1990 he was awarded a Nobel Prize in Economic Sciences. William F. Sharpe was also awarded a Nobel Prize in Economic Sciences the same year. Sharpe is perhaps most recognized for the so-called Sharpe Ratio. The Sharpe Ratio is a performance ratio of reward-to-risk that plays an important role in Modern Portfolio Theory. Similar to Markowitz’s model assumptions, the Sharpe Ratio assumes that reward is an expected (that is, average) value and risk is measured by the standard deviation of uncertain returns. The downside of using standard deviation as a risk measure, however, is that it weighs losses and gains equally. If the distribution of potential outcomes is skewed, standard deviation may be inappropriate. Standard deviation is especially inappropriate for portfolios that include derivative securities (like out-of-the-money call or put options), because their returns can be heavily skewed. Alternative measures of risk have been proposed. The 1992 RiskMetrics methodology of J.P. Morgan [?], for example, introduced value-at-risk, or VaR, for evaluating the risk of multi asset class portfolios. VaR considers the risk of losses in the tail of the portfolio’s return distribution, where the tail is defined relative to some quantile of the distribution. The advantage of VaR is that it does not penalize potentially large gains. Other popular risk measures, including expected shortfall (also known as conditional value-at-risk, or CVaR), have a similar advantage. More recently, a performance ratio known as Omega has been gaining traction as an alternative to the Sharpe Ratio. Omega might be well suited for portfolios with asymmetric return distributions. 1.1 Omega Let the random variable X denote the future return of a portfolio over a single period of time. We will assume that this is a percent return, although we could just as easily work in absolute terms. Omega is a 1 Copyright 2010. Do not distribute without permission. function, of a parameter t ∈ R, defined by the following relationship: Ω(t ) = E [(X − t )+ ] , E [(t − X )+ ] (1) where ‘E’ denotes expectation (the average value) and the superscript ‘+’ denotes the positive part: z+ = max{z, 0}. The input t is referred to as the target. If gains are considered to be those realizations of X that exceed t , and losses are realizations that fall below t , then Ω(t ) is “a measure of the quality of our investment ‘bet’” [3, p. 9]. The larger the value of Omega, the larger the gains relative to the losses. In selecting a portfolio, some portfolio managers’ objective is to maximize Omega. Omega has some nice properties. As a function of t over (−∞, ∞), it carries the same information as the distribution of X ; all moments of the distribution function can be recovered from Ω(t ). Omega might be value of Omega for a specific threshold, like Ω(.01), might carry some relevant information without having to share the entire distirubtion. Ω(t ) is a decreasing convex function that equals 1 when t equals E[X ]. The flatter the function, the riskier the portfolio. easier to convey to investors than the distribution function, at least in parts. For example, knowing a fund’s 2 Optimal Portfolio Allocation Suppose that the investable universe consists of N (possibly) correlated assets. Denote the period return for asset n by the random variable Yn . The return on the portfolio is a weighted combination of Y1 , . . . , YN . The allocation weights are given by an N -dimensional (column) vector w. For example, wn = .10 means that the nth asset makes up 10% of the portfolio. The total value of the portfolio at the end of the period is X (w) = ∑N=1 wnYn . The weights are chosen by the portfolio manager from a set of feasible weights, W , n satisfying ￿ ￿ W ⊆ w:w∈R , N n=1 ∑ wn = 1 N , where RN is the N -dimensional set of real numbers. The exact form of W will depend on the portfolio manager’s mandate. Typically, W is a convex set – for instance, a polyhedron formed by linear constraints. For threshold t , determined exogenously, let Pt (w) = E Ct (w) = E ￿￿ ￿￿ t − ∑ wnYn n=1 N N ￿+ ￿ −t + ∑ wnYn n=1 ￿+ ￿ 2 Developed for IOE 310 Fall 2010 Copyright 2010. Do not distribute without permission. so that, by (1), Omega equals Ωt (w) = Ct (w) . Pt (w) The functions Ct (w) and Pt (w) are referred to as the “call” and “put,” respectively, because of their similarities to pricing functions for standard European call and put options. The call has the interpretation of a reward and the put is the risk. In order to evaluate the call and put, we need to take expectations. Instead of trying to get closed-form solutions for these expressions, in practice the asset returns are simulated. For example, the RiskMetrics methodology uses Monte-Carlo simulation to draw random samples from a multivariate distribution for Y1 , . . . , YN that takes into account asset correlations. We will assume a total of K samples, with the kth sample given by a vector yk ∈ RN , where yk is the profit/loss of the nth asset. It is assumed that each of the n K draws are equally likely, and therefore equally weighted, by 1/K . In this case, Pt (w) = Ct (w) = ￿+ 1 K￿ t − wT yk ∑ K k=1 ￿+ 1 K￿ ∑ −t + wT yk . K k=1 We are interested in the following optimization problem: max Ωt (w). w∈W Unfortunately, Ωt (w) is not necessarily (jointly) concave in w and it may exhibit multiple local maxima. This presents a problem for optimization routines. As an alternative to maximizing Ωt (w) directly, a twostage process is proposed. First, an efficient frontier for Ct vs. Pt could be constructed. That is, for each possible put value, we could find a portfolio with the highest possible call value. Then, a portfolio that maximizes Omega could be found by searching over the efficient frontier for the highest call/put ratio. 3 Efficient Frontier Markowitz was the first to consider an efficient frontier of portfolio return vs. risk. A point on the efficient frontier corresponds to a portfolio that maximizes return subject to a given upper bound on risk. Likewise, an efficient portfolio minimizes risk subject to a lower bound on return. Portfolios that are efficient are not dominated in both risk and reward by other portfolios. Otherwise, portfolios off of the frontier are dominated by a portfolio with either a higher reward for the same level of risk or lower risk for the same level of reward. Figure 1 depicts an efficient frontier. The Markowitz frontier is concave and nondecreasing 3 Developed for IOE 310 Fall 2010 Copyright 2010. Do not distribute without permission. in risk. Assuming a multivariate Normal distribution for returns, standard deviation for risk, and the minimal constraints that weights sum to one, it can be shown that the efficient frontier is the top part of a parabola. The lower leftmost point represents the min risk portfolio, and the uppermost point on the efficient frontier is the max reward portfolio. M ax R ewa r d * * * * * * * * ** * * ** * * * * ** * * * * * **** * * ** *** * * ** ** * * ** * * * * * ** * * *** ** * * * * R ewa r d M i n R is k R is k Figure 1: An efficient frontier. Individual assets from the asset universe are represented with a ‘∗’. In general, there are two methods for finding portfolios on the efficient frontier. Method 1 solves the following optimization problem: min w Risk(w) Reward(w) ≥ ￿ w∈W, (2) subject to where ￿ is a lower bound on the reward. The efficient frontier can be traced by varying ￿ and repeatedly solving this optimization problem. The range of ￿ is first determined by solving the min risk problem: min Risk(w) w∈W. The resulting min risk portfolio gives a lower bound on ￿. An upper bound on ￿ is found by solving max Reward(w) w∈W. (Similarly, one might instead trace the frontier by maximizing reward subject to a lower bound on risk.) 4 Developed for IOE 310 Fall 2010 Copyright 2010. Do not distribute without permission. Method 2 solves the following optimization problem: min w subject to Risk(w) − λReward(w) w∈W, (3) where the parameter λ (which is a Lagrange multiplier), called the risk aversion parameter, is varied instead. If λ = 0 then the objective is to minimize risk. As λ → ∞ the problem becomes that of maximizing return. The two methods are equivalent, producing the same set of efficient portfolios, when Risk(w) is convex, Reward(w) is concave, and W is a convex set; see [1, Theorem 3]. Under these circumstances, both (2) and (3) are convex optimization problems – minimizing a convex function over a convex feasible region. In problem (2), when Reward(w) is concave, the set {w : Reward(w) ≥ ￿} is convex. Since W is also convex, problem (2) is a minimization over a convex feasible set. In (3), when Reward(w) in concave, −λReward(w) is convex. Since the sum of convex functions is convex, the objective function in (3) is convex when Risk(w) is convex and Reward(w) is concave. In general, convex optimization problems are desirable because they are relatively easy to solve. The problem faced when trying to optimize Omega is that, while Pt (w) is convex as desired, the reward function Ct (w) is, unfortunately, also convex and not concave. Neither (2) nor (3) are convex optimization problems. This presents an obstacle. The solution is to recast this optimization problem using a well-known financial relationship called put-call parity: call − put = expected value. In our setting, ￿ ￿ ￿ ￿ Ct (w) − Pt (w) = E (X (w) − t )+ − E (t − X (w))+ = E[X (w) − t ]. Ωt (w) = Ct (w) Ct (w) − Pt (w) E [X (w)] − t = +1 = + 1. Pt (w) Pt (w) Pt (w) It follows that We see that for a fixed put value, Ωt (w) is maximized if E[Xt (w)] is maximized. Likewise, for a fixed expected return, Ωt (w) is maximized if Pt (w) is minimized. This suggests instead constructing an efficient frontier in which Risk(w) = Pt (w) and Reward(w) = E[Xt (w)]. Now, the reward E[Xt (w)] is linear in w; it is both convex and concave. Since it is concave, the efficient frontier optimization problems (2) and (3) are convex optimization problems, and they should be relatively easy to solve. 5 Developed for IOE 310 Fall 2010 Copyright 2010. Do not distribute without permission. When the distribution of X is simulated, the optimization problem corresponding to (2) is: min w ￿+ 1 K￿ t − wT yk ∑ K k=1 1 K Tk ∑w y ≥￿ K k=1 w∈W (4) subject to Note that the left-hand-side of the reward constraint is linear in w. The objective function (4) is not linear, but is piecewise linear. 4 Case Study You are approached by a portfolio manager who wants to “optimize” a portfolio by maximizing Omega. The assets in the investable universe have these ticker symbols on the NYSEArca exchange: SHY (iShares Barclays 1-3 Year Treasury Bond), XLB (Materials Select Sector SPDR), XLE (Energy Select Sector SPDR), XLF (Financial Select Sector SPDR), XLI (Industrial Select Sector SPDR ), XLK (Technology Select Sector SPDR), XLP (Consumer Staples Select Sector SPDR), XLU (Utilities Select Sector SPDR), XLV (Health Care Select Sector SPDR), EWZ (iShares MSCI Brazil Index), GLD (SPDR Gold Shares), and SPY (SPDR S&P 500). You will perform an optimization study using historical data. Rather than using some analytical model for randomly generating forecasted returns, you will perform a “historical simulation.” That is, your “draws” are historical monthly returns, that are each equally likely; yk is a vector of returns for the kth historical month (either counting forwards or backwards in time). The return for asset n in month k is yk . The assets are n correlated via their historical observations. For example, you might see that gold (GLD) and the “Spider” (SPY – the S&P 500 market index) have a tendency to move in opposite directions over time. If this is the case, they might be negatively correlated. There are 4 parts to your assignment: 1. Assume that the portfolio is a long-only portfolio. This means that the percent allocation, wn , satisfies 0 ≤ wn ≤ 1. (This is opposed to a long/short portfolio in which the weights can be negative.) Formulate an LP (not necessarily in standard form) to solve (4). Throughout we will use Method 1 and not use Method 2. (a) As a function of K and N , how many decision variables are there? Include in the count any slack variables that might be missing in your formulation but would be needed to transform the LP to standard form. How many constraints are there? 6 Developed for IOE 310 Fall 2010 Copyright 2010. Do not distribute without permission. (b) If N = 100 and K = 10, 000, how many constraints and decision variables are there? (c) If N = 100 and K = 100, 000, how many constraints and decision variables are there? Also, state the percent increases for this 10-fold increase in K . (d) If N = 1, 000 and K = 10, 000, how many constraints and decision variables are there? Also, state the percent increases for this 10-fold increase in N . 2. Assume from now on that the max allocation to any one of the assets listed above is limited to at most 20%. We will consider a monthly horizon and a target monthly return of 1%; t = .01. Download 5 years worth of historical monthly data for the above symbols (available on CTools). Numerically solve this problem: (a) What are the optimal min risk portfolio weights? What is the min risk portfolio’s expected return? (b) What is the optimal max reward portfolio? Max expected return? (c) Plot the efficient frontier. For the plot, divide the distance between the min risk portfolio’s expected return and the max expected return into 20 equal segments. (d) For the efficient portfolios, plot Ω.01 vs. risk (Pt ). (e) Of the resulting 21 portfolios that make up the efficient frontier, what is the optimal Omega portfolio? (If there is more than one optimal Omega portfolio, choose the one with the higher expected return.) What is the maximum Ω.01 ? (f) For each asset n, plot (on the same chart) wn for the efficient portfolios vs. risk (Pt ). 3. Now suppose that the mandate of the portfolio allows for a positive allocation to a risk-free asset: one whose monthly return is, for sure, constant. Let’s assume that the monthly return is a constant .05% (0.0005). Assume that the max allocation to the risk-free asset is 10%. (After all, portfolio managers aren’t getting paid to invest in cash.) (a) Plot the new efficient frontier. Again, use 21 portfolios in a manner similar to before, but with new min risk and max return portfolios. Superimpose the first frontier that does not allocate to a risk-free asset. (b) For the efficient portfolios, plot the percent allocation to the risk-free asset vs risk (Pt ). Is it increasing or decreasing in risk? Is there some point (in terms of risk) at which the upper bound on the risk-free asset allocation becomes non-binding? If so, what is that point, and what is the associated expected return? (c) For the efficient portfolios, plot Ω.01 vs. risk (Pt ). (d) What is the new optimal Omega portfolio? What is the maximum Ω.01 ?. As compared to the first optimal Omega portfolio (without a risk-free asset), has risk gone down or up? What about expected return? 7 Developed for IOE 310 Fall 2010 Copyright 2010. Do not distribute without permission. 4. Take the risk-free asset allocation to the extreme: Assume that the max allocation to the risk-free asset is 100%. (a) What is the min risk portfolio? What is its expected return? (b) Plot the new efficient frontier. Superimpose the other two frontiers. (c) For the efficient portfolios, plot Ω.01 vs. risk (Pt ). (d) What is the new optimal Omega portfolio? What is the maximum Ω.01 ?. As compared to the first two optimal Omega portfolios, has risk gone down or up? What about expected return? (e) For the efficient portfolios, plot the percent allocation of the risk-free asset vs. risk (Pt ). (f) For each of risky assets, again plot (on one chart) a percent allocation vs. risk. This time, however, rather than plotting wn (the percent allocation to the entire portfolio), instead plot the percent allocation to the portion of the non-risk-free portfolio. That is, if λ ∈ [0, 1] is the proportion of the total portfolio that is allocated to the risk-free asset, plot wn /(1 − λ) vs. Pt . Of course, λ and wn both may vary with Pt . (g) Finally, can you justify your observations? References [1] P. Krokhmaland, J. Palmquist, and S. Uryasev. Portfolio optimization with conditional value-at-risk objective and constraints. The Journal of Risk, 4(2):11–27, 2002. [2] H. Markowitz. Portfolio selection. The Journal of Finance, 7(1):77–91, March 1952. [3] W. Shadwick and C. Keating. A universal performance measure. Journal of Performance Measurement, pages 59–84, Spring 2002. [4] P. Zangari. Riskmetrics technical document. RiskMetrics Group Technical Documents Series, 1996. 8 Developed for IOE 310 Fall 2010 ...
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This note was uploaded on 12/20/2010 for the course IOE 310 taught by Professor Saigal during the Fall '08 term at University of Michigan.

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