Dynamic Portfolio Strategies

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Dynamic Portfolio Strategies

• Part 4
• Portfolio Choice Strategic Asset Allocation

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Strategic Asset Allocation

• Well now consider the problem of an investor who
  would like to save some portion of their current
  wealth.
• Well limit our attention to two asset classes
• A riskless investment (bonds) with relatively low
  returns.
• A risky investment (stocks) with relatively high
  returns.
• The strategic asset allocation problem is to
  decide what portion of current wealth to invest
  in each class.

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Portfolio Selection

• Investors will form a portfolio from these two
  assets.
• The return on the portfolio will be a weighted
  average of the individual assets simple returns

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Portfolio Returns

• Note
• There is a technical problem here. Weve assumed
  that the stock return is lognormal but the
  portfolio return is not.
• The saved wealth is directly proportional to the
  return, and it is not lognormal either.
• The portfolio weights must add to 1, but are not
  necessarily positive (borrowing and lending of
  both the stock and bond are allowed).

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Portfolio Theory - The Markowitz Approach

• Assumptions
• Investors want to buy and hold a portfolio of
  risky stocks.
• The investors like high expected returns, dont
  like high variance (equivalently standard
  deviation) and dont care about other aspects of
  portfolio return distributions.

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The Mean-Variance Tradeoff

• In the absence of arbitrage opportunities, the
  set of returns from all securities gives rise to
  an efficient frontier.
• The efficient frontier bounds the tradeoffs that
  are available between mean and variance by
  trading in all securities.

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Mean-Variance Analysis
Efficient mean/ std. dev. tradeoffs
All possible mean/std. dev. tradeoffs
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Mean-Variance Analysis
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Utility

• Investors are typically thought of as being risk
  averse
• When given the choice between
• a) a riskless payment of 10 and
• b) a 50/50 chance of 20 or 0
• most people choose a).
• This is a consequence of diminishing marginal
  utility of wealth.

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Diminishing Marginal Utility
Utility
Low Marginal Utility
High Marginal Utility
Wealth
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Indifference Curves

• If investors have utility functions of a special
  type, they only care about mean and variance.
  (More on this later.)
• Indifference curves define sets of mean-standard
  deviation pairs that make an investor equally
  well off.

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Indifference Curves
Mean
Higher Utility
Indifference Curves
Standard Deviation
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Combining Utility Theory and Mean-Variance Theory
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Mean-Variance Analysis

• Properties of MV frontier
• Higher mean returns can only be achieved by
  increasing portfolio variance.
• More risk-tolerant investors will choose higher
  variance portfolios but receive higher expected
  returns.
• These efficient portfolios can be calculated if
  we know the covariance matrix.
• There are companies who will calculate return
  covariances for you (e.g. BARRA)

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Portfolio Separation

• An important result
• all efficient portfolios are a combination of the
  same two efficient portfolios!
• Implication
• If there were no information issues and investors
  cared only about mean and variance, only two
  mutual funds would be required to satisfy all
  investors stock market demands.

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Mean-Variance Analysis
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Risk-free Borrowing and Lending

• You can expand the efficient set if risk-free
  borrowing and lending are possible.
• If you invest (1-?) in t-bills and ? in an
  efficient portfolio the mean and variance of the
  portfolio are

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Efficient Set
Tangency portfolio
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Portfolio Separation

• If there were no information issues and investors
  cared only about mean and variance, only t-bills
  and one mutual fund (the tangency portfolio)
  would be required to satisfy all investors stock
  market demands.
• Investments in this tangency portfolio and
  t-bills dominate all other investments.

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Commonly-used Utility Functions

• The class of power utility functions are commonly
  employed for the purposes of determining
  portfolio allocations.

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Maximizing Utility

• A formal definition of an investors asset
  allocation problem can now be given
• In words, the problem is to select the asset mix
  that maximizes utility.
• The investor will make a tradeoff between risk
  and expected return.

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Mean-Variance Preferences

• A useful benchmark is the solution to the problem
  of an investor who likes mean but dislikes
  variance (note that this is not a utility
  function as described above)

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Mean-Variance Preferences

• This investors optimal portfolio is given by
• Note
• S is the Sharpe ratio which is constant for all
  portfolios.
• The more the investor dislikes variance, the less
  is invested in the risky asset.

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Maximizing Utility

• The solution to the portfolio problem in the
  power utility case has no convenient form.
• If the forecasting horizon is very short, some
  convenient approximations may be used.
• For longer horizons, numerical solutions can be
  constructed.

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Maximizing Utility Numerical Solutions by
Simulation

• The solution to the portfolio choice problem in
  the power utility case must solve
• This is a non-linear equation in ? to which we
  can find an approximate solution using SOLVER in
  excel.

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Solution Strategy

• To solve the equation we need to find a
  convenient way to form the expectation.
• Using a large number of simulated excess returns
  (the Zt) we can represent this expectation
  arbitrarily accurately with a sum

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Solution Strategy

• The solution to the non-linear equation then
  gives an arbitrarily accurate appropriate asset
  allocation for the investor
• The benefit of this approach is that the effect
  of horizon on asset allocation can be examined in
  a quantitative way.

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Strategic Asset Allocation Example 1

• Suppose an index has IID return dynamics.
  Expected returns are 1 per month with a standard
  deviation of 4 per month. The risk-free rate is
  a constant .5 per month. Determine an optimal
  asset allocation for an investor with power
  utility and ? each of (2, 5, 10) if the investing
  horizon is
• 1 year.
• 5 years.

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Strategic Asset Allocation Example 2

• Suppose an index has return dynamics as given by
  the VAR model we studied earlier. Determine an
  optimal asset allocation for an investor with
  power utility and ? each of (2, 5, 10) if the
  investing horizon is
• 1 year.
• 5 years.