Second Investment Course

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Second Investment Course November 2005

• Topic Four
• Portfolio Optimization Analytical Techniques

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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.

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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., wi) can be
  seen as the solution to the following non-linear,
  constrained optimization problem
• Select wi so as to minimize
• subject to (i) E(Rp) R
• (ii) S wi 1
• The first constraint is the investors 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 wi to be negative
  that is, short selling is permissible.)
• Other constraints that are often added to this
  problem include (i) All wi gt 0 (i.e., no short
  selling), or (ii) All wi lt P, where P is a fixed
  percentage

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Solving the Portfolio Optimization Problem

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

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The Analytical Solution to Efficient Portfolio
Optimization
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The Analytical Solution to Efficient Portfolio
Optimization (cont.)
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The Analytical Solution to Efficient Portfolio
Optimization (cont.)
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Example of Mean-Variance Optimization Analytical
Solution(Three Asset Classes, Short Sales
Allowed)
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Example of Mean-Variance Optimization Analytical
Solution (cont.) (Three Asset Classes, Short
Sales Allowed)
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Example of Mean-Variance Optimization Optimal
Search Procedure (Three Asset Classes, Short
Sales Allowed)
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Example of Mean-Variance Optimization Optimal
Search Procedure (Three Asset Classes, No Short
Sales)
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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).
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Mean-Variance Efficient Frontier With and Without
Short-Selling
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Optimal Search Efficient Frontier Example Five
Asset Classes
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Example of Mean-Variance Optimization Optimal
Search Procedure (Five Asset Classes, No Short
Sales)
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Mean-Variance Optimization with Black-Litterman
Inputs

• One of the criticisms that is sometimes made
  about the mean-variance 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
  datathe expected returns, in particularlead 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
  equilibrium-consistent forecasts lead to
  smoother, more realistic allocations.

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BL Mean-Variance Optimization Example

• Recall the implied expected returns and other
  inputs from the earlier example

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BL Mean-Variance Optimization Example (cont.)

• These inputs can then be used in a standard
  mean-variance optimizer

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BL Mean-Variance Optimization Example (cont.)

• This leads to the following optimal allocations
  (i.e., efficient frontier)

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BL Mean-Variance Optimization Example (cont.)
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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
  user-specified tactical views)

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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)

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BL Mean-Variance Optimization Example (cont.)

• The new optimal allocations reflect these
  tactical views (i.e., more Emerging Market Equity
  and less Global Equity

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BL Mean-Variance Optimization Example (cont.)

• This leads to the following new efficient
  frontier

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Optimal Portfolio Formation With Historical
Returns Examples

• Suppose we have monthly return data for the last
  three years on the following six asset classes
• Chilean Stocks (IPSA Index)
• Chilean Bonds (LVAG LVAC Indexes)
• Chilean Cash (LVAM Index)
• U.S. Stocks (SP 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

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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
• Mean-Variance Optimization
• Mean-Lower Partial Moment (i.e., downside risk)
  Optimization
• 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)

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1. Mean-Variance Optimization Non-CLP Assets
100 Unhedged
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Unconstrained Efficient Frontier 100 Unhedged
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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.
• The issue of foreign currency hedging will be
  considered in a separate topic

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Mean-Variance Optimization Non-CLP Assets 100
Hedged
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Unconstrained M-V Efficient Frontier 100 Hedged
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Comparison of Unhedged (i.e. Domestic Only) and
Hedged (i.e., Unconstrained Foreign) Efficient
Frontiers
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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
  1970-2005
• 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

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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

• 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.

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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.

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More on Mean-Variance OptimizationThe 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

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Additional Constraints 30 Foreign Investment
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Additional Constraints 30 Foreign Investment
No Hedge Funds
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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 2.0

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Mean-Downside Risk Optimization Non-CLP Assets
100 Hedged
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Unconstrained M-LPM Efficient Frontier 100
Hedged
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Additional Constraints 30 Foreign Investment
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Additional Constraints 30 Foreign Investment
No Hedge Funds
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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

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Alpha-Tracking Error Optimization Non-CLP Assets
100 Hedged
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Unconstrained a-TE Efficient Frontier 100 Hedged
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Additional Constraints 30 Foreign Investment
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Additional Constraints 30 Foreign Investment
No Hedge Funds
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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