More on Asset Allocation

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More on Asset Allocation

• Week 5

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Minimum Variance and Efficient Frontier

• When we only have two risky assets, as in our KO
  PEP example, it is easy to construct this graph
  by simply calculating the portfolio returns for
  all possible weights.
• When we have more than 2 assets, it becomes more
  difficult to represent all possible portfolios,
  and instead we will only graph only a subset of
  portfolios. Here, we will choose only those
  portfolios that have the minimum volatility for a
  given return. We will call this graph the minimum
  variance frontier.
• The upper half of the minimum variance frontier
  is called the efficient frontier. The optimal
  portfolio is one of the portfolios on the
  efficient frontier.
• The next few slides describes how this frontier
  can be created.
• Once the frontier is created, we can find the
  portfolio with the highest Sharpe ratio by

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Creating the minimum variance frontier

• How to use a spreadsheet to calculate the
  frontier when there are more than 2 assets

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The Minimum Variance Frontier

• With two assets, as we saw, we can construct the
  frontier by brute force - by listing almost all
  possible portfolios.
• When we have more than 2 assets, its gets
  difficult to consider all possible portfolio
  combinations. Instead, we will make the process
  simpler by considering only a subset of
  portfolios those portfolios that have the
  minimum volatility for a given return.
• When we plot the return and volatilities of these
  portfolios, the resultant graph will be known as
  the minimum variance (or volatility) frontier.
• We will use Excels Solver for these
  calculations (look under Tools. If it is not
  there, then add it into the menu through Add-in).

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The Steps

• We will implement the procedure in three steps
• 1. For each asset (and for the time period that
  you have chosen), calculate the mean return,
  volatility and the correlation matrix.
• 2. Set up the spreadsheet so that the Solver can
  be used. See the sample spreadsheet. Your
  objective here is to determine the weights of the
  portfolio that will allow you to achieve a
  specified required rate of return with the lowest
  possible volatility.
• 3. Repeat 2 for a range of returns, and plot the
  frontier (return vs. volatility).

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Step 1 Assembling the Data

• A. Fix the time period for the analysis. You want
  a sufficiently long period so that your estimates
  of the mean return, volatility and correlation
  are accurate. But you also dont want too long a
  period as very old data may not be valid.
• B. Estimate the mean return and volatility for
  each of your assets. Next, calculate the
  correlation between each pair of assets. If there
  are N assets, you will have to calculate N(N-1)/2
  correlations.

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Step 2 Setting up the spreadsheet to use the
Solver (1/4)

• The objective here is to set up the spreadsheet
  in a manner that is easy to use with the solver.
• The estimates of the return, volatility and the
  correlation matrix are used to set up a matrix
  for covariances, which is then used to calculate
  the portfolio volatility for a given set of
  weights.
• To create the frontier, you will ask the solver
  to find you the weights that gives you the
  minimum volatility for a required return.

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Step 2 Using the Solver (2/4)

• 1.Target Cell When you call the solver, it will
  ask you to specify the objective or the target
  cell. Your objective is to minimize the
  volatility - so in this case, you will specify
  the cell that calculates the portfolio variance
  B36. As you want to minimize the variance,
  you click the Min.
• 2. Constraints You will have to specify the
  constraints under which the optimization must
  work. There are two constraints that hold, and a
  third which will usually also apply.

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Using the Solver Constraints on the Optimization
(3/4)

• 1. First, the sum of the weights must add up to
  1.
• 2. Second, you have to specify the required rate
  of return for which you want the portfolio of
  least volatility. For each level of return, you
  will solve for the weights that give you the
  minimum volatility. To construct the frontier,
  you will vary this required return over a range.
  Thus, you will have to change this constraint
  every time you change the required return.
• Third, if you want to impose a short-selling
  constraint, you can specify that each portfolio
  weight is positive.

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Step 2 (4/4)

• Finally, you specify the arguments that need to
  be optimized. In this case, you are searching for
  the optimal weights, so you will have to specify
  the range in the spreadsheet where the portfolio
  weights used A29 to A34.

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Step 3

• The final step is to simply repeat step 2, until
  you have a sufficiently large data set so that
  the minimum variance frontier can be plotted.
• .

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The Optimal Allocation

• We can now use the graph of the minimum variance
  frontier to figure out the portfolio with the
  highest Sharpe Ratio. This portfolio will be the
  portfolio such that the CAL passing through it is
  tangent to the minimum variance frontier.
• The weights of this portfolio determines the
  optimal allocation within the assets that make up
  the risky portfolio. All investors should opt
  for this allocation.
• The portfolio will always be on the upper portion
  of the frontier, above the portfolio with the
  lowest volatility - this portion is called the
  efficient frontier.

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Example

• The spreadsheet MinimumVarianceFrontier
  provides an example of the computation of the
  minimum variance frontier for a portfolio of 6
  stocks KO, PEP, WMT, IBM, XOM, MSFT.
• The optimal portfolio turns out to have the
  following weights 13 in WMT, 3.50 in IBM, 68
  in XOM, and 15.50 in MSFT. This portfolio had an
  average return of 23, and a volatility of
  15.45.
• The return and volatility of SP 500 are also
  plotted on the graph for comparison.

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Minimum Variance Frontier
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The Sharpe Ratio for the Portfolios
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The Optimal Portfolio vs. SP 500

• In our example, the optimal portfolio provides a
  risk-return tradeoff far superior to investing in
  the SP 500.
• For example, if we invest 48.8 in the optimal
  portfolio and the remainder in the Treasury Bill,
  we would expect to earn a return equal to that of
  the SP 500 (of 12.14), but with a volatility of
  7.50, far lower than the SP 500 volatility of
  28.
• On the other hand, if we leverage up (borrowing
  at the risk-free rate) to make the volatility of
  the optimal portfolio equal to the volatility of
  SP 500, we would expect to earn a return of
  38.25, much higher than the SP 500 return of
  12.14.

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In Summary (1/2)

• 1. The optimal allocation is determined in two
  steps. First, we decide the allocation between
  the risky portfolio, and the riskless asset.
  Second, we determine the allocation between the
  assets that comprise the risky portfolio.
• 2. As every portfolio of the risky assets and the
  riskless asset has the same Sharpe ratio, there
  is not one optimal portfolio for all investors.
  Instead, the allocation will be determined by
  individual-specific factors like risk aversion
  and the objectives of the investor, taking into
  account factors like the investors horizon,
  wealth, etc.
• 3. When we are considering the allocation between
  different classes of risky assets, it is possible
  to create a portfolio that has the highest Sharpe
  Ratio. The weights of the risky assets in this
  portfolio will determine the optimal allocation
  between various risky assets. This portfolio can
  be determined graphically by drawing the capital
  allocation line (CAL) such that it is tangent to
  the minimum variance frontier. This portfolio
  will always lie on the upper part of the frontier
  (or on the efficient part of the frontier).

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In Summary (2/2)

• 4. The extent to which you can decrease the
  volatility of the portfolio depends also on the
  correlation. The lower the average correlation of
  the stocks in your portfolio, the lower you can
  decrease the volatility of your portfolio.
• 5. The homework provides you with an exercise to
  determine the optimal allocations.