PORTFOLIO OPTIMIZATION: ANALYTICAL TECHNIQUES

Transcription

1 PORTFOLIO OPTIMIZATION: ANALYTICAL TECHNIQUES Keith Brown, Ph.D., CFA November 22 nd, 2007

2 Overview of the Portfolio Optimization Process The preceding analysis demonstrates that it is possible for investors to reduce their risk exposure simply by holding in their portfolios a sufficiently large number of assets (or asset classes). This is the notion of naïve diversification, but as we have seen there is a limit to how much risk this process can remove. Efficient diversification is the process of selecting portfolio holdings so as to: (i) minimize portfolio risk while (ii) achieving expected return objectives and, possibly, satisfying other constraints (e.g., no short sales allowed). Thus, efficient diversification is ultimately a constrained optimization problem. We will return to this topic in the next session. Notice that simply minimizing portfolio risk without a specific return objective in mind (i.e., an unconstrained optimization problem) is seldom interesting to an investor. After all, in an efficient market, any riskless portfolio should just earn the risk-free rate, which the investor could obtain more cost-effectively with a T-bill purchase. 1

3 The Portfolio Optimization Process As established by Nobel laureate Harry Markowitz in the 1950s, the efficient diversification approach to establishing an optimal set of portfolio investment weights (i.e., {w i }) can be seen as the solution to the following non-linear, constrained optimization problem: Select {w i } so as to minimize: σ 2 p = [w1σ wnσ n ] + [2w1w 2σ1σ 2ρ1, wn-1w nσ n 1σ nρn 1, n ] subject to: (i) E(R p ) = R* (ii) Σ w i = 1 The first constraint is the investor s return goal (i.e., R*). The second constraint simply states that the total investment across all 'n' asset classes must equal 100%. (Notice that this constraint allows any of the w i to be negative; that is, short selling is permissible.) Other constraints that are often added to this problem include: (i) All w i > 0 (i.e., no short selling), or (ii) All w i < P, where P is a fixed percentage 2

4 Solving the Portfolio Optimization Problem In general, there are two approaches to solving for the optimal set of investment weights (i.e., {w i }) depending on the inputs the user chooses to specify: 1. Underlying Risk and Return Parameters: Asset class expected returns, standard deviations, correlations) a. Analytical (i.e., closed-form) solution: True solution but sometimes difficult to implement and relatively inflexible at handling multiple portfolio constraints b. Optimal search: Flexible design and easiest to implement, but does not always achieve true solution 2. Observed Portfolio Returns: Underlying asset class risk and return parameters estimated implicitly 3

5 The Analytical Solution to Efficient Portfolio Optimization For any particular collection of assets, the efficient frontier refers to the set of portfolios that offers the lowest level of risk for a pre-specified level of expected return. Given information about the expected returns, standard deviations, and correlations amongst the securities, we have seen that efficient portfolio weights can be determined analytically by solving the following problem: Select {w i } so as to minimize: σ 2 p = [w 1 σ w n σ n ] + [2w 1 w 2 σ 1σ 2 ρ 1, w n -1 w n σ n 1σ n ρ n 1, n ] subject to: (i) E(Rp) = R* (ii) Σw i = 1. To make the solution somewhat more transparent, we can first rewrite the problem in matrix notation. Assuming there are "n" securities available, define: V = (n x n) covariance matrix (i.e., the diagonal elements are the n variances and the off-diagonal elements are the correlation coefficients amongst the securities); w = (n x 1) vector of portfolio weights; R = (n x 1) vector of expected security returns; i = (n x 1) "unit" vector (i.e., a vector of ones). With this notation, the efficient frontier problem can be recast as: subject to and w Min [(0.5) w' V w] R* = w' R 1 = w' i 4

6 The Analytical Solution to Efficient Portfolio Optimization (cont.) Note here that the variance of the portfolio (i.e., w' V w) has been divided by two; this is merely an algebraic convenience and does not change the values of the optimal weights. One approach to solving this constrained optimization problem is the Lagrange-multiplier method. That is, to convert the constrained problem into an unconstrained one, select the vector of portfolio weights so as to: Min L = Min [(0.5)w ' Vw - λ 1 (R * - w ' R ) - λ 2 (1 - w ' i )] where λ 1 and λ 2 are the Lagrangean multipliers. Notice that this method incorporates the constraints directly into the orginal function by creating two new variables to be solved. Consequently, the solution can proceed by differentiating L with respect to w, λ 1 and λ 2. The first order conditions of the minimum are: δ L δ w = Vw - λ 1 R - λ 2 i = 0 (1) δ L δλ 1 =R * - w ' R = 0 = R * - R ' w δ L =1 - w ' i = 0 = 1 - i ' w δλ 2 (2) (3) Solving (1) for w yields: w = λ 1 V -1 R + λ 2 V -1 i = V -1 [R i] λ (4) 5

7 The Analytical Solution to Efficient Portfolio Optimization (cont.) For a particular value of R*, solve for λ 1 * and λ 2 * by combining the constraint equations in (3a) and (3b) with an expanded form of (4): and R * = R ' w = λ 1 * R ' V -1 R + λ 2 * R ' V -1 i (5a) 1 = i ' w = λ 1 * i ' V -1 R + λ 2 * i ' V -1 i (5b) Also, define the following efficient set constants: A = R ' V -1 i = i ' V -1 R (6a) B = R ' V -1 R (6b) C = i ' V -1 i (6c) Substituting (6) into (5) yields: B A A C λ 1 λ 2 = M λ = R 1 (7) or: λ = M -1 R 1 (8) Substituting (8) into (4) leaves: w * = V -1 [ R i ] M -1 R * 1 (9) where w* is the vector of weights for the minimum variance portfolio having an expected return of R* and a variance of σ 2 = w*'vw*. 6

8 Example of Mean-Variance Optimization: Analytical Solution (Three Asset Classes, Short Sales Allowed) 7

9 Example of Mean-Variance Optimization: Analytical Solution (Three Asset Classes, Short Sales Allowed) 8

10 Example of Mean-Variance Optimization: Optimal Search Procedure (Three Asset Classes, Short Sales Allowed) 9

11 Example of Mean-Variance Optimization: Optimal Search Procedure (Three Asset Classes, No Short Sales) 10

12 Measuring the Cost of Constraint: Incremental Portfolio Risk Main Idea: Any constraint on the optimization process imposes a cost to the investor in terms of incremental portfolio volatility, but only if that constraint is binding (i.e., keeps you from investing in an otherwise optimal manner). 11

13 Mean-Variance Efficient Frontier With and Without Short-Selling 12

14 Optimal Search Efficient Frontier Example: Five Asset Classes 13

15 Example of Mean-Variance Optimization: Optimal Search Procedure (Five Asset Classes, No Short Sales) 14

16 Mean-Variance Optimization with Black-Litterman Inputs One of the criticisms that is sometimes made about the meanvariance optimization process that we have just seen is that the inputs (e.g., asset class expected returns, standard deviations, and correlations) must be estimated, which can effect the quality of the resulting strategic allocations. Typically, these inputs are estimated from historical return data. However, it has been observed that inputs estimated with historical data the expected returns, in particular lead to extreme portfolio allocations that do not appear to be realistic. Black-Litterman expected returns are often preferred in practice for the use in mean-variance optimizations because the equilibriumconsistent forecasts lead to smoother, more realistic allocations. 15

17 BL Mean-Variance Optimization Example Recall the implied expected returns and other inputs from the earlier example: Asset Allocation Analysis Analysis Inputs Case: Allocation Case Zephyr AllocationADVISOR: LBJ Asset Management Partners Analysis Inputs Forecast Date Constraint Return Risk Start End Min Max Assets US Bonds 4.6% 3.8% Mar 2002 Mar % 100% Global Bonds xusd 5.9% 8.3% Mar 2002 Mar % 100% US Equity 10.1% 12.3% Mar 2002 Mar % 100% Global Equity xus 11.2% 13.2% Mar 2002 Mar % 100% Emerging Equity 12.4% 17.5% Mar 2002 Mar % 100% Benchmark S&P % 12.3% Mar 2002 Mar 2007 Projection Inputs Target Return: 10.0% Time Horizon: 10 Years Initial Value: $1,000,000 Correlations US Bonds Global Bonds xusd US Equity Global Equity xus Emerging Equity Black-Litterman Model Inputs Palette Risk Premium 4.14% Risk-free Rate 4.63% Market Cap (millions) Date Weight US Bonds $9,901,772 Mar % Global Bonds xusd $13,926,593 Mar % US Equity $16,200,000 Mar % Global Equity xus $20,601,896 Mar % Emerging Equity $2,760,728 Mar % 16

18 BL Mean-Variance Optimization Example (cont.) These inputs can then be used in a standard mean-variance optimizer: 17

19 BL Mean-Variance Optimization Example (cont.) This leads to the following optimal allocations (i.e., efficient frontier): Asset Allocation Analysis Efficient Frontier Case: Allocation Case Return vs. Risk (Standard Deviation) Zephyr AllocationADVISOR: LBJ Asset Management Partners Emerging Equity 11 Global Equity xus 10 US Equity Return Asset Allocations 6 Global Bonds xusd US Bonds Global Bonds xusd US Equity Global Equity xus Emerging Equity 5 US Bonds Risk (Standard Deviation) 18

20 BL Mean-Variance Optimization Example (cont.) Asset Allocation Analysis Portfolio Statistics Case: Allocation Case Target Return: 10.00% - 10 Year Time Horizon - 95% of Projected Return Distribution Zephyr AllocationADVISOR: LBJ Asset Management Partners Portfolio Allocations E(R)=6% E(R)=7% E(R)=8% E(R)=9% E(R)=10% Asset Allocations US Bonds 73.9% 52.9% 31.9% 10.8% 0.0% Global Bonds xusd 2.2% 8.9% 16.1% 23.7% 19.9% US Equity 20.6% 22.1% 23.9% 26.0% 20.8% Global Equity xus 3.3% 16.0% 26.3% 34.4% 51.2% Emerging Equity 0.0% 0.0% 1.8% 5.1% 8.0% Portfolio Statistics Expected Return (Annualized) One Year 6.0% 7.0% 8.0% 9.0% 10.0% Time Horizon 5.9% 6.9% 7.8% 8.7% 9.5% Expected Risk One Year 3.6% 4.9% 6.6% 8.5% 10.6% Time Horizon 1.1% 1.5% 2.1% 2.7% 3.3% Best Case Return (Annualized) One Year 13.2% 16.8% 21.5% 26.7% 32.1% Time Horizon 8.2% 9.9% 12.0% 14.1% 16.2% Worst Case Return (Annualized) One Year -0.8% -2.2% -4.4% -6.8% -9.3% Time Horizon 3.8% 3.9% 3.8% 3.5% 3.2% Probability of Target Return One Year 13.2% 26.4% 37.1% 43.8% 48.1% Time Horizon 0.0% 2.3% 14.8% 31.1% 44.0% Probability of Negative Return One Year 4.3% 7.1% 11.0% 14.4% 17.2% Time Horizon 0.0% 0.0% 0.0% 0.0% 0.1% Tracking to Market Benchmark Benchmark Tracking R-Squared 48% 69% 74% 74% 73% Tracking Error 11.64% 9.92% 8.48% 7.51% 7.25% 19

21 BL Mean-Variance Optimization Example (cont.) Another advantage of the BL Optimization model is that it provides a way for the user to incorporate his own views about asset class expected returns into the estimation of the efficient frontier. Said differently, if you do not agree with the implied returns, the BL model allows you to make tactical adjustments to the inputs and still achieve well-diversified portfolios that reflect your view. Two components of a tactical view: Asset Class Performance - Absolute (e.g., Asset Class #1 will have a return of X%) - Relative (e.g., Asset Class #1 will outperform Asset Class #2 by Y%) User Confidence Level - 0% to 100%, indicating certainty of return view (See the article A Step-by-Step Guide to the Black-Litterman Model by T. Idzorek of Zephyr Associates for more details on the computational process involved with incorporating userspecified tactical views) 20

22 BL Mean-Variance Optimization Example (cont.) Suppose we adjust the inputs in the process to include two tactical views: - US Equity will outperform Global Equity by 50 basis points (70% confidence) - Emerging Market Equity will outperform US Equity by 150 basis points (50% confidence) Asset Allocation Analysis Analysis Inputs Case: Allocation Case with Relative Views Zephyr AllocationADVISOR: LBJ Asset Management Partners Analysis Inputs Forecast Date Constraint Return Risk Start End Min Max Assets US Bonds 4.5% 3.8% Mar 2002 Mar % 100% Global Bonds xusd 5.2% 8.3% Mar 2002 Mar % 100% US Equity 10.4% 12.3% Mar 2002 Mar % 100% Global Equity xus 10.4% 13.2% Mar 2002 Mar % 100% Emerging Equity 11.7% 17.5% Mar 2002 Mar % 100% Benchmark S&P % 12.3% Mar 2002 Mar 2007 Projection Inputs Target Return: 10.0% Time Horizon: 10 Years Initial Value: $1,000,000 Correlations US Bonds Global Bonds xusd US Equity Global Equity xus Emerging Equity Palette Risk Premium 4.14% Risk-free Rate 4.63% Black-Litterman Model Inputs Market Cap (millions) Date Weight US Bonds $9,901,772 Mar % Global Bonds xusd $13,926,593 Mar % US Equity $16,200,000 Mar % Global Equity xus $20,601,896 Mar % Emerging Equity $2,760,728 Mar % Relative Views US Equity will outperform Global Equity xus 0.5% with 70% confidence Emerging Equity will outperform US Equity 1.5% with 50% confidence 21

23 BL Mean-Variance Optimization Example (cont.) The new optimal allocations reflect these tactical views (i.e., more Emerging Market Equity and less Global Equity): Asset Allocation Analysis Portfolio Statistics Case: Allocation Case with Relative Views Target Return: 10.00% - 10 Year Time Horizon - 95% of Projected Return Distribution Zephyr AllocationADVISOR: LBJ Asset Management Partners Portfolio Allocations Active Portfolio E(R)=6% E(R)=7% E(R)=8% E(R)=9% E(R)=10% Asset Allocations US Bonds 85.5% 74.1% 49.9% 27.3% 4.8% 0.0% Global Bonds xusd 0.0% 0.7% 8.9% 16.1% 23.2% 11.3% US Equity 14.5% 25.3% 41.2% 53.7% 65.8% 75.0% Global Equity xus 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% Emerging Equity 0.0% 0.0% 0.0% 3.0% 6.2% 13.7% Portfolio Statistics Expected Return (Annualized) One Year 5.4% 6.0% 7.0% 8.0% 9.0% 10.0% Time Horizon 5.3% 5.9% 6.9% 7.8% 8.7% 9.5% Expected Risk One Year 3.3% 3.6% 5.0% 6.9% 8.9% 11.1% Time Horizon 1.0% 1.1% 1.6% 2.2% 2.8% 3.5% Best Case Return (Annualized) One Year 11.9% 13.2% 17.1% 22.1% 27.5% 33.4% Time Horizon 7.4% 8.2% 10.0% 12.1% 14.3% 16.5% Worst Case Return (Annualized) One Year -0.9% -0.9% -2.5% -4.9% -7.5% -10.2% Time Horizon 3.3% 3.7% 3.8% 3.6% 3.3% 2.8% Probability of Target Return One Year 8.0% 13.4% 26.9% 37.4% 43.9% 48.0% Time Horizon 0.0% 0.0% 2.6% 15.6% 31.5% 43.7% Probability of Negative Return One Year 4.8% 4.5% 7.7% 12.0% 15.6% 18.6% Time Horizon 0.0% 0.0% 0.0% 0.0% 0.1% 0.2% Tracking to Market Benchmark Benchmark Tracking R-Squared 17% 54% 81% 88% 90% 91% Tracking Error 12.98% 11.36% 8.99% 6.85% 5.14% 4.13% 22

24 BL Mean-Variance Optimization Example (cont.) This leads to the following new efficient frontier: 12 Asset Allocation Analysis Efficient Frontier Case: Allocation Case with Relative Views Return vs. Risk (Standard Deviation) Zephyr AllocationADVISOR: LBJ Asset Management Partners Emerging Equity 11 US Equity Global Equity xus 10 9 Return 8 7 Asset Allocations 6 5 Global Bonds xusd US Bonds Global Bonds xusd US Equity Global Equity xus Emerging Equity US Bonds Risk (Standard Deviation) 23

25 Optimal Portfolio Formation With Historical Returns: Examples Suppose we have monthly return data for the three recent years on the following six asset classes: Chilean Stocks (IPSA Index) Chilean Bonds (LVAG & LVAC Indexes) Chilean Cash (LVAM Index) U.S. Stocks (S&P 500 Index) U.S. Bonds (SBBIG Index) Multi-Strategy Hedge Funds (CSFB/Tremont Index) Assume also that the non-clp denominated asset classes can be perfectly and costlessly hedged in full if the investor so desires 24

26 Optimal Portfolio Formation With Historical Returns: Examples (cont.) Consider the formation of optimal strategic asset allocations under a wide variety of conditions: With and without hedging non-clp exposure With and Without Investment in Hedge Funds With and Without 30% Constraint on non-clp Assets With different definitions of the optimization problem: 1. Mean-Variance Optimization 2. Mean-Lower Partial Moment (i.e., downside risk) Optimization 3. Alpha -Tracking Error Optimization Each of these optimization examples will: Use the set of historical returns directly rather than the underlying set of asset class risk and return parameters Be based on historical return data from the period October 2002 September 2005 Restrict against short selling (except those short sales embedded in the hedge fund asset class) 25

27 1. Mean-Variance Optimization: Non-CLP Assets 100% Unhedged 26

28 Unconstrained Efficient Frontier: 100% Unhedged E(R) σp Relative σp W cs W cb W cc W uss W usb W hf 5.00% 1.13% % 6.18% 85.87% 0.00% 0.00% 0.00% 6.00% 1.65% % 9.15% 79.05% 0.00% 0.00% 0.00% 7.00% 2.18% % 12.12% 72.23% 0.00% 0.00% 0.00% 8.00% 2.71% % 15.09% 65.42% 0.00% 0.00% 0.00% 9.00% 3.24% % 18.06% 58.60% 0.00% 0.00% 0.00% 10.00% 3.77% % 21.03% 51.78% 0.00% 0.00% 0.00% 11.00% 4.30% % 24.00% 44.96% 0.00% 0.00% 0.00% 12.00% 4.83% % 26.97% 38.15% 0.00% 0.00% 0.00% 13.00% 5.36% % 29.94% 31.33% 0.00% 0.00% 0.00% 14.00% 5.90% % 32.92% 24.51% 0.00% 0.00% 0.00% 27

29 One Consequence of the Unhedged M-V Efficient Frontier Notice that because of the strengthening CLP/USD exchange rate over the October 2002 September 2005 period, the optimal allocation for any expected return goal did not include any exposure to non-clp asset classes This unhedged foreign investment efficient frontier is equivalent to the efficient frontier that would have resulted from a domestic investment only constraint. 28

30 Mean-Variance Optimization: Non-CLP Assets 100% Hedged 29

31 Unconstrained M-V Efficient Frontier: 100% Hedged E(R) σp Relative σp W cs W cb W cc W uss W usb W hf 5.00% 0.71% % 9.45% 68.45% 0.59% 0.00% 19.31% 6.00% 1.02% % 13.82% 53.57% 0.74% 0.00% 28.51% 7.00% 1.34% % 18.19% 38.68% 0.89% 0.00% 37.72% 8.00% 1.67% % 22.56% 23.80% 1.04% 0.00% 46.93% 9.00% 2.00% % 26.93% 8.92% 1.19% 0.00% 56.13% 10.00% 2.33% % 26.81% 0.00% 0.68% 0.00% 63.93% 11.00% 2.72% % 20.28% 0.00% 0.00% 0.00% 68.54% 12.00% 3.16% % 13.98% 0.00% 0.00% 0.00% 72.28% 13.00% 3.63% % 7.67% 0.00% 0.00% 0.00% 76.01% 14.00% 4.11% % 1.37% 0.00% 0.00% 0.00% 79.75% 30

32 Comparison of Unhedged (i.e. Domestic Only ) and Hedged (i.e., Unconstrained Foreign ) Efficient Frontiers Expected Return Unhedged M-V σp Hedged M-V σp Relative σp 5.00% 1.13% 0.71% % 1.65% 1.02% % 2.18% 1.34% % 2.71% 1.67% % 3.24% 2.00% % 3.77% 2.33% % 4.30% 2.72% % 4.83% 3.16% % 5.36% 3.63% % 5.90% 4.11%

33 A Related Question About Foreign Diversification What allocation to foreign assets in a domestic investment portfolio leads to a reduction in the overall level of risk? Van Harlow of Fidelity Investments performed the following analysis: Consider a benchmark portfolio containing a 100% allocation to U.S. equities Diversify the benchmark portfolio by adding a foreign equity allocation in successive 5% increments Calculate standard deviations for benchmark and diversified portfolios using monthly return data over rolling three-year holding periods during For each foreign allocation proportion, calculate the percentage of rolling three-year holding periods that resulted in a risk level for the diversified portfolio that was higher than the domestic benchmark 32

34 Portfolio Risk Reduction and Diversifying Into Foreign Assets United States, % 25% Frequency of Higher Risk (vs Domestic Only) Rolling 3 Year Periods % 15% 10% 5% 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% Foreign Stock Allocation 33

35 Foreign Diversification Potential (cont.) Ennis Knupp Associates (EKA) have provided an alternative way of quantifying the diversification benefits of adding international stocks to a U.S. stock portfolio: Impact of Diversification on Volatility Volatility Percentage in F oreign Stocks EKA concludes that international diversification adds an important element of risk control within an investment program; the optimal allocation from a statistical standpoint is approximately 30%-40% of total equities, although they generally favor a slightly lower allocation due to cost considerations. 34

36 Foreign Diversification Potential: One Caveat During recent periods, it appears as though the correlations between U.S. and non-u.s. markets are increasing, reducing the diversification benefits of non-u.s. markets. While this is true, the fact that these markets are less than perfectly correlated means that there is still a diversification benefit afforded to investors who allocate a portion of their assets overseas. Rolling 3-Year Correlations U.S. and Non-U.S. Stocks

37 More on Mean-Variance Optimization: The Cost of Adding Additional Constraints Start with the following base case: Six asset classes: Three Chilean, Three Foreign (Including Hedge Funds) No Short Sales 100% Hedged Foreign Investments No Constraint on Total Foreign Investment No Constraint on Hedge Fund Investment Consider the addition of two more constraints: 30% Limit on Foreign Asset Classes No Hedge Funds 36

38 Additional Constraints: 30% Foreign Investment E(R) σp Relative σp W cs W cb W cc W uss W usb W hf 5.00% 0.71% % 9.45% 68.45% 0.59% 0.00% 19.31% 6.00% 1.02% % 13.82% 53.57% 0.74% 0.00% 28.51% 7.00% 1.40% % 16.77% 46.22% 0.63% 0.00% 29.37% 8.00% 1.85% % 19.60% 39.53% 0.50% 0.00% 29.50% 9.00% 2.34% % 22.43% 32.84% 0.37% 0.00% 29.63% 10.00% 2.84% % 25.25% 26.15% 0.24% 0.00% 29.76% 11.00% 3.35% % 27.97% 19.58% 0.10% 0.00% 29.90% 12.00% 3.87% % 30.93% 12.76% 0.00% 0.00% 30.00% 13.00% 4.39% % 33.90% 5.94% 0.00% 0.00% 30.00% 14.00% 4.92% % 36.01% 0.00% 0.00% 0.00% 30.00% 37

39 Additional Constraints: 30% Foreign Investment & No Hedge Funds E(R) σp Relative σp W cs W cb W cc W uss W usb W hf 5.00% 1.03% % 11.38% 72.44% 4.63% 5.73% 0.00% 6.00% 1.51% % 16.68% 60.13% 6.66% 7.78% 0.00% 7.00% 1.99% % 21.98% 47.82% 8.69% 9.83% 0.00% 8.00% 2.48% % 27.28% 35.50% 10.72% 11.88% 0.00% 9.00% 2.97% % 32.57% 23.19% 12.76% 13.93% 0.00% 10.00% 3.46% % 37.75% 11.69% 14.64% 15.36% 0.00% 11.00% 3.96% % 42.31% 3.71% 15.85% 14.15% 0.00% 12.00% 4.46% % 42.55% 0.00% 16.67% 13.33% 0.00% 13.00% 4.98% % 39.03% 0.00% 17.13% 12.87% 0.00% 14.00% 5.51% % 36.16% 0.00% 17.52% 11.74% 0.00% 38

40 2. Mean-Downside Risk Optimization Scenario Start with Same Base Case as Before: Six Asset Classes: Three Domestic, Three Foreign Fully Hedged Foreign Investments; No Short Sales No Constraint on Foreign Investments No Constraint on Hedge Funds Downside Risk Conditions: Threshold Level = 2.93% (i.e., annualized return from Chilean cash market) Power Factor for Downside Deviations =

41 Mean-Downside Risk Optimization: Non-CLP Assets 100% Hedged 40

42 Unconstrained M-LPM Efficient Frontier: 100% Hedged E(R) LPMp Rel. LPMp W cs W cb W cc W uss W usb W hf 5.00% 0.19% % 12.91% 67.61% 0.65% 0.00% 15.47% 6.00% 0.26% % 16.98% 54.49% 0.61% 0.00% 22.79% 7.00% 0.33% % 20.98% 41.13% 0.57% 0.00% 30.53% 8.00% 0.41% % 24.98% 27.78% 0.52% 0.00% 38.27% 9.00% 0.48% % 28.99% 14.42% 0.48% 0.00% 46.01% 10.00% 0.56% % 32.99% 1.06% 0.44% 0.00% 53.75% 11.00% 0.65% % 26.91% 0.00% 0.00% 0.00% 59.01% 12.00% 0.79% % 20.05% 0.00% 0.00% 0.00% 63.55% 13.00% 0.94% % 13.64% 0.00% 0.00% 0.00% 67.44% 14.00% 1.11% % 7.76% 0.00% 0.00% 0.00% 70.57% 41

43 Additional Constraints: 30% Foreign Investment E(R) LPMp Rel. LPMp W cs W cb W cc W uss W usb W hf 5.00% 0.19% % 12.91% 67.61% 0.65% 0.00% 15.47% 6.00% 0.26% % 16.98% 54.49% 0.61% 0.00% 22.79% 7.00% 0.33% % 20.75% 42.13% 0.42% 0.00% 29.58% 8.00% 0.45% % 24.30% 34.66% 0.00% 0.00% 30.00% 9.00% 0.59% % 27.81% 27.29% 0.00% 0.00% 30.00% 10.00% 0.76% % 32.63% 18.58% 0.00% 0.00% 30.00% 11.00% 0.93% % 37.57% 9.75% 0.00% 0.00% 30.00% 12.00% 1.12% % 43.47% 0.07% 0.89% 0.00% 29.11% 13.00% 1.32% % 39.71% 0.00% 0.00% 0.00% 30.00% 14.00% 1.53% % 36.01% 0.00% 0.00% 0.00% 30.00% 42

44 Additional Constraints: 30% Foreign Investment & No Hedge Funds E(R) LPMp Rel. LPMp W cs W cb W cc W uss W usb W hf 5.00% 0.33% % 12.96% 75.04% 5.44% 0.84% 0.00% 6.00% 0.47% % 18.62% 62.72% 8.16% 2.11% 0.00% 7.00% 0.62% % 24.24% 50.41% 10.88% 3.40% 0.00% 8.00% 0.77% % 29.74% 38.10% 13.66% 4.79% 0.00% 9.00% 0.92% % 35.30% 25.76% 16.41% 6.15% 0.00% 10.00% 1.07% % 40.86% 13.43% 19.17% 7.52% 0.00% 11.00% 1.22% % 46.41% 1.81% 21.75% 8.25% 0.00% 12.00% 1.38% % 46.27% 0.00% 23.66% 5.20% 0.00% 13.00% 1.56% % 45.57% 0.00% 25.30% 1.02% 0.00% 14.00% 1.76% % 41.76% 0.00% 27.22% 0.00% 0.00% 43

45 3. Alpha-Tracking Error Optimization Scenario Start with Same Base Case as Before: Six Asset Classes: Three Domestic, Three Foreign Fully Hedged Foreign Investments; No Short Sales No Constraint on Foreign Investments or Hedge Funds Optimization Process Defined Relative to Benchmark Portfolio: Minimize Tracking Error Necessary to Achieve a Required Level of Excess Return (i.e., Alpha) Relative to Benchmark Return Benchmark Composition: Chilean Stock: 35%; Chilean Bonds: 30%, Chilean Cash: 5%; U.S. Stock: 15%; U.S. Bonds: 15%; Hedge Funds: 0% Notice that Benchmark Portfolio Could Be Defined as Average Peer Group Allocation 44

46 Alpha-Tracking Error Optimization: Non-CLP Assets 100% Hedged 45

47 Unconstrained α-te Efficient Frontier: 100% Hedged α TE Relative TE W cs W cb W cc W uss W usb W hf 0.20% 0.07% % 30.87% 2.16% 15.01% 14.83% 1.90% 0.40% 0.13% % 31.35% 0.00% 14.94% 14.46% 3.74% 0.60% 0.22% % 30.57% 0.00% 14.60% 13.46% 5.41% 0.80% 0.31% % 29.79% 0.00% 14.26% 12.45% 7.08% 1.00% 0.40% % 29.02% 0.00% 13.93% 11.45% 8.75% 1.20% 0.49% % 28.24% 0.00% 13.59% 10.44% 10.42% 1.40% 0.59% % 27.47% 0.00% 13.25% 9.44% 12.09% 1.60% 0.68% % 26.69% 0.00% 12.92% 8.43% 13.76% 1.80% 0.78% % 25.92% 0.00% 12.58% 7.43% 15.43% 2.00% 0.87% % 25.14% 0.00% 12.25% 6.42% 17.10% 46

48 Additional Constraints: 30% Foreign Investment α TE Relative TE W cs W cb W cc W uss W usb W hf 0.20% 0.09% % 30.68% 3.80% 14.86% 13.89% 1.25% 0.40% 0.18% % 31.37% 2.60% 14.72% 12.78% 2.50% 0.60% 0.27% % 32.05% 1.39% 14.58% 11.67% 3.75% 0.80% 0.35% % 32.76% 0.16% 14.45% 10.56% 5.00% 1.00% 0.44% % 32.42% 0.00% 14.15% 9.40% 6.45% 1.20% 0.54% % 31.91% 0.00% 13.83% 8.23% 7.94% 1.40% 0.63% % 31.41% 0.00% 13.52% 7.06% 9.42% 1.60% 0.72% % 30.90% 0.00% 13.20% 5.89% 10.91% 1.80% 0.82% % 30.40% 0.00% 12.88% 4.72% 12.40% 2.00% 0.91% % 29.89% 0.00% 12.56% 3.55% 13.89% 47

49 Additional Constraints: 30% Foreign Investment & No Hedge Funds α TE Relative TE W cs W cb W cc W uss W usb W hf 0.20% 0.10% % 30.92% 3.40% 15.24% 14.76% 0.00% 0.40% 0.21% % 31.83% 1.81% 15.49% 14.51% 0.00% 0.60% 0.31% % 32.77% 0.19% 15.73% 14.27% 0.00% 0.80% 0.42% % 32.25% 0.00% 15.84% 14.16% 0.00% 1.00% 0.53% % 31.55% 0.00% 15.94% 14.06% 0.00% 1.20% 0.64% % 30.84% 0.00% 16.03% 13.97% 0.00% 1.40% 0.75% % 30.14% 0.00% 16.12% 13.88% 0.00% 1.60% 0.87% % 29.70% 0.00% 16.19% 13.51% 0.00% 1.80% 0.98% % 29.29% 0.00% 16.25% 13.13% 0.00% 2.00% 1.10% % 28.88% 0.00% 16.31% 12.74% 0.00% 48

50 The Portfolio Optimization Process: Some Summary Comments The introduction of the portfolio optimization process was an important step in the development of what is now considered to be modern finance theory. These techniques have been widely used in practice for more than fifty years. Portfolio optimization is an effective tool for establishing the strategic asset allocation policy for a investment portfolio. It is most likely to be usefully employed at the asset class level rather than at the individual security level. There are two critical implementation decisions that the investor must make: The nature of the risk-return problem: Mean-Variance, Mean-Downside Risk, Excess Return-Tracking Error Estimates of the required inputs : Expected returns, asset class risk, correlations Portfolio optimization routines can be adapted to include a variety of restrictions on the investment process (e.g., no short sales, limits on foreign investing). The cost of such investment constraints can be viewed in terms of the incremental volatility that the investor is required to bear to obtain the same expected outcome 49