Asset Allocation

1
Asset Allocation

• Week 4

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Asset Allocation The Fundamental Question

• How do you allocate your assets amongst different
  assets?
• Traditionally, we divide the discussion here into
  two parts
• A. The allocation between riskfree and a
  portfolio of risky assets.
• B. The allocation between different risky asset
  within the portfolio of risky assets.

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The Decisions That an Investor Must Make

• Thus, there are two decisions that an investor
  must make
• 1. Which is the risky stock portfolio that
  results in the best risk-return tradeoff?
• 2. After making the choice of the risky stock
  portfolio, how should you allocate your assets
  between this risky portfolio and the riskfree
  asset?
• Typically, the first objective of a financial
  advisor is to determine for her clients the
  appropriate allocation between the risky and
  riskless assets, and then to choose how the risky
  portfolio should be constructed.

4
The Sharpe Ratio

• To compare one portfolio with another, we will
  use a metric called the Sharpe Ratio.
• The Sharpe ratio measures the tradeoff between
  risk and return for each portfolio.
• R Expected Portfolio Return.
• Rf Riskfree Rate.
• Vol Portfolio Vol.
• Portfolio Vol (w1)2 (vol of asset 1)2 (w2
  )2 (vol of asset 2)2 2 (correlation) (w1 )(w2
  ) (vol of asset 1)(vol of asset 2)
  .(additional terms of volatilities and
  correlations).
• Sharpe Ratio (R-Rf)/(Vol).
• We may use the Sharpe ratio as a criteria for
  determining the right portfolio.

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Asset Allocation Risky vs. Riskless Asset

• Consider the allocation between the risky and
  riskless asset.
• Rf expected return on riskfree asset.
• Rp expected return on risky asset portfolio.
• Volatility of riskfree asset 0.
• W1 proportion in riskfree asset.
• W2 proportion in risky asset.
• Is there an optimal w1, w2?
• We shall show that the choice of w1, w2 is
  individual-specific, and will depend on the
  individuals risk aversion and objectives. Thus,
  there is no one optimal portfolio.

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Portfolio of Risky Riskless Asset

• To calculate the portfolio return and portfolio
  variance when we combine the risky asset and
  riskless asset, we can use the usual formulas,
  noting that the volatility of the riskfree rate
  is zero.
• Portfolio Return w1 Rf w2 Rp.
• Portfolio Variance (w1)2 (0) (w2 )2 (vol of
  risky asset)2 2 (correlation) (w1 )(w2 )
  (0)(vol of risky asset).
• Portfolio Volatility w2 (vol of risky asset).
• This simplification in the formula for the
  portfolio volatility occurs because the vol of
  the riskfree asset is zero.
• To understand the tradeoff between risk and
  return, we can graph the portfolio return vs the
  the portfolio volatility.
• The following graph shows this graph for the case
  when the mean return for the riskfree asset is
  5, the mean return for the risky asset is 12,
  and the volatility of the risky asset is 15.

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Riskfree Return0.05, Risky Return0.12, Vol of
Risky Asset0.15
8
Portfolio Return vs. Portfolio Volatility
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Capital Allocation Line (CAL)

• The graph from the previous slide is called the
  capital allocation line (CAL).
• For the special case when one of the two assets
  is the riskfree asset, the CAL is a straight
  line, with a slope of (Rp-Rf)/(Vol of risky
  portfolio).
• This slope equals the increase in return of the
  portfolio for a unit increase in volatility.
  Therefore, it is also called the
  reward-to-variability ratio. We will also refer
  to this ratio as the Sharpe ratio.
• The greater the slope the greater the reward for
  taking risk. Ideally, you want to achieve the
  highest return per unit risk, so that you choose
  a risky portfolio that gives you the steepest
  slope.
• Note that this tradeoff will be essentially
  determined by the mean return and volatility of
  the risky portfolio.

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How to allocate between the riskfree asset and
the risky stock portfolio.

• The conclusion we draw from the straight-line
  graph is that when we combine a riskfree asset
  with the risky stock portfolio, all portfolios
  have the same Sharpe ratio.
• Therefore, it is not possible to make a decision
  on allocation between the riskfree asset and the
  risky stock portfolio based solely on the Sharpe
  ratio. Instead, we will have to take into account
  individual-specific considerations. There is no
  single allocation here that is best for all
  investors.
• Your decision to allocate between the risky asset
  and the riskfree asset will be determined by your
  level of risk aversion and your objectives,
  depending on factors like your age, wealth,
  horizon, etc. The more risk averse you are, the
  less you will invest in the risky asset.
• Although different investors may differ in the
  level of risk they take, they are also alike in
  that each investor faces exactly the same
  risk-return tradeoff.

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A Digression into Market Timing

• Why not actively manage the allocation between
  the riskfree asset and the risky stock portfolio?
• There are funds that actively manage the decision
  to allocate between the risky/riskless asset for
  the investor these funds are typically called
  market allocation funds.
• Typically, the funds actively manage a mix of
  stocks, bonds and money market securities, and
  they may change the fraction of their holding in
  each of these assets, depending on what they
  think is optimal at that time.
• Such a trading strategy is also called market
  timing. The objective of market timing is to be
  invested in stocks in a bull market, and to be
  invested in bonds/cash in a bear market.

12
Returns to Market Timing

• Here is an example that illustrates how you could
  do if you were a good/bad market timer. If you
  could time the market, using the SP 500, what
  would your returns be over the period Jan 1950-
  Dec 2002? We start with 1 on January 1, 1950,
  and ask how much we would have on December 31,
  2002.
• 1. Buy and hold strategy 51.60 (average
  return7.72).
• 2. Perfect timer 238,203 (26.31) (!!).
• 3. Occasional timer (miss the worst 10 months)
  200 (10.52).
• 4. Mis-timer (miss the best 10 months) 16.87
  (5.48).
• 5. Miss both best/worst 10 months 65.49
  (8.21).
• Moral of the story time the market only if you
  have a good crystal ball.
• But its tempting to keep trying even when one
  doesnt have a crystal ball.

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The Optimal Risky Stock Portfolio

• We discussed the allocation between the risky
  (stock) portfolio and the riskless (cash)
  portfolio.
• Now we will consider the other decision that an
  investor must make how should the risky stock
  portfolio be constructed?
• Once again we will assume that investors want to
  maximize the Sharpe ratio (so that investors want
  the best tradeoff between return and volatility).

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Determining the Optimal Portfolio

• If we can plot the portfolio return vs. Portfolio
  volatility for all possible allocations
  (weights), then we can easily locate the optimal
  portfolio with the highest Sharpe ratio of (Rp -
  Rf)/(Vol of risky portfolio).
• When we only have two risky assets, as in this
  case, 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
  variance-return frontier.
• Once we solve for this minimum variance frontier,
  we will show that there exists one portfolio on
  this frontier that has the highest Sharpe ratio,
  and thus is the optimal stock portfolio.
• Because there exists one specific portfolio with
  the highest Sharpe ratio, all investors will want
  to invest in that portfolio. Thus, the weights
  that make up this portfolio determines the
  optimal allocation between the risky assets for
  all investors.

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Frontier with KO and PEP

• As an example, consider a portfolio of KO and
  PEP. What should be the optimal combination of KO
  and PEP?
• Refer to excel file on web page.
• As we only have two assets here, we can easily
  tabulate the Sharpe ratio for a range of
  portfolio weights, and check which portfolio has
  the highest Sharpe ratio.
• The next slide shows the results. In the
  calculation of the Sharpe ratio, it is assumed
  that the riskfree rate is constant (which is not
  strictly true). The portfolio mean and portfolio
  return are calculated with the usual formulae
  over the 10-year sample period 1993-2002, with
  monthly data.
• As can be seen, the optimal weight for a
  portfolio (to get the maximum Sharpe ratio)
  appears to be in the range of 0.6 in KO. If the
  exact answer is required, we can easily solve for
  it using the excel solver.
• It can also be observed that, amongst these 11
  portfolios, the portfolio with the minimum
  volatility is one that invests 50 in each of the
  two stocks. This is the minimum variance
  portfolio. The minimum variance portfolio may be
  different from the portfolio with the highest
  Sharpe ratio.

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The Sharpe Ratio KO PEP
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Volatility-Return Frontier

• Consider the graph of the portfolio return vs.
  Portfolio volatility.
• Graphically, the optimal portfolio (with the
  highest Sharpe ratio) is the portfolio that lies
  on a tangent to the graph. This tangent is drawn
  so that it has the riskfree rate as its
  intercept.
• This is because the slope of the line that passes
  connects the riskfree asset and the risky
  portfolio is equal to the Sharpe ratio. Thus, the
  steeper the line, the higher the Sharpe ratio.
  The tangent to the graph has the steepest slope,
  and thus the portfolio that lies on this tangent
  is the optimal portfolio (having the highest
  Sharpe ratio).
• This tangent is now the capital allocation line.
  All investments represented on this line are
  optimal (and will comprise of combination of the
  riskfree asset and risky stock portfolio).

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Portfolio Return-Volatility Frontier
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Creating the mean 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 dont want a period too
  long, because the 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)
  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 minium
  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 volatility
  B25. As you want to minimize the volatility,
  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 there are constraints to short-selling,
  you will have to 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 A20, A21, A22.

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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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Diversification (1/6)

• We have observed that by combining stocks into
  portfolios, we can create an asset with a better
  risk-return tradeoff.
• The reduction of risk in a portfolio occurs
  because of diversification. By combining
  different assets into a portfolio, we can
  diversify risk and reduce the overall volatility
  of the portfolio.
• Let us review the factors that affect how risk
  can be diversified. Here we will ignore the issue
  of allocation (as we have already considered it),
  and instead assume that our portfolio is equally
  weighted.

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Factors that affect diversification in an equally
weighted portfolio (2/6)

• There are two main factors that affect the extent
  to which volatility can be reduced the number of
  assets in the portfolio, and the correlation
  between the assets.
• Increasing the number of assets reduces the
  volatility of the portfolio.
• Adding an asset with a low correlation with the
  existing assets of a portfolio also helps to
  reduce the volatility of the portfolio.

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(3/6)

• To examine the effect of correlation and the
  number of assets, lets assume, for simplicity,
  that each of the assets have the same volatility
  (say, 40) and the same average correlation with
  each other.
• The portfolio volatility can then be calculated
  by the usual formula, and we can examine the
  reduction in volatility of the portfolio as we
  change the number of assets, or the correlation.

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Sample spreadsheet (4/6)
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Some Conclusions (5/6)

• By changing Nnumber of stocks in portfolio, and
  the correlation, we can examine how the portfolio
  volatility decreases.
• We can make the following observations
• 1. For all positive correlation, there is a
  threshold beyond which we cannot reduce the
  portfolio volatility. This threshold depends on
  the magnitude of the correlation. If the
  correlation is zero or less than zero, then it is
  possible to bring down the portfolio volatility
  to zero by having a large number of assets. This
  threshold represents the undiversifiable or the
  systematic risk of the portfolio.

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Some Conclusions (6/6)

• 2. As the correlation decreases, the more we can
  reduce the portfolio volatility. However, it
  takes more assets to bring down the portfolio
  volatility to its theoretical minimum.
• Example if the correlation is 0.9 and the
  average volatility of each stock in the portfolio
  is 40, then the lowest portfolio volatility that
  is possible is about 37.95. We can reach within
  0.5 of this minimum volatility by creating a
  portfolio of only 4 assets. Suppose instead that
  the average correlation is 0.5. Then the lowest
  possible portfolio volatility is 28.28 however,
  to reach within 0.5 of this value, we need as
  many as 30 stocks.

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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.