Asset Allocation

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Asset Allocation

• Week 4

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Asset Allocation The Fundamental Question

• How do you allocate your assets amongst different
  assets?
• There are literally thousands of assets available
  to you for investment purposes. Which ones will
  you invest in, and how much will you invest in
  each of these?
• In this class, we will limit our discussion only
  to the universe of stocks and one risk-free asset
  (the Treasury). However, the same ideas can be
  extended to any set of risky assets, like real
  estate, hedge fund investments, etc.
• We can divide this question into two separate
  decisions.

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The Two Decisions

• How do you allocate your assets amongst different
  assets? There are two decisions that you have to
  make
• A. How will you allocate between the risk-free
  asset and the portfolio of risky assets (stocks)?
  
• To figure this out, ask yourself this question
  Of all the money you have available to you for
  investment, how much do you want to keep in
  cash?
• We shall see that there is no best way to
  allocate your allocation will depend on your
  preferences and risk tolerance. Thus, your
  decision will depend on criteria like your
  current age, total wealth, current financial
  commitments, etc.
• B. How will you allocate between different risky
  assets within the portfolio of risky assets.
• We shall see that there is an optimal allocation
  between risky assets one best way to divide
  all your cash between the risky stocks.
• This is the primary question we shall deal with
  over here.

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Assumptions

• Traditionally, when we decide asset allocation,
  we will assume that all the assets are fairly
  priced.
• If, instead, one asset is not fairly priced (it
  is under- or over-valued), it may be optimal for
  you to simply allocate all your money into that
  one asset!
• Moreover, we will assume that we know, or can
  estimate from past history, all that we need to
  know about the expected returns, volatilities,
  and correlations of our stocks.

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The Objective of Allocation

• What should be our objective when we decide to
  allocate between different assets?
• For example, why should we not invest in only
  one asset? We may not wish to invest in one asset
  as one of our goals is to diversify (and, thus,
  reduce) risk.
• Specifically, we will set our objective as Earn
  the highest return per unit of risk.
• We will measure our return as excess returns R
  Rf
• We will measure our risk by the volatility of the
  return. The volatility is the standard deviation
  of the return.

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The Sharpe Ratio

• Given our objective of maximizing the return per
  unit of risk, we will use a metric based on the
  expected return and volatility of the asset that
  is commonly known as the Sharpe Ratio.
• The Sharpe ratio measures the tradeoff between
  risk and return for each portfolio.
• Sharpe Ratio (R-Rf)/(Vol).
• We will use the Sharpe ratio as our criteria for
  choosing between different allocations.

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Maximizing the Sharpe Ratio

• It is important to note that maximizing the
  Sharpe ratio, i.e., maximizing the excess return
  per unit of risk is not equivalent to either (a)
  maximizing the return, or (b) minimizing the
  risk.
• Example
• Between 1/1994 and 9/2004, the average return
  earned on a stock of KO was 11.85/year. Over the
  same period, the return earned on PEP was
  14.69/year. The volatility of KOs return was
  25.35/year, and the volatility of PEPs return
  was 24.28/year. Thus, PEP earned a higher return
  with a lower risk than KO over this period.
• Qt Suppose you expect KO and PEP to perform the
  same over the next 10 years. Does this mean that
  you should invest all your money in Pepsi, and
  nothing in Coke? Answer No.

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Notations and Useful Formulae

• Let there be two assets, Asset 1 and Asset 2.
• R1, R2 expected returns on Asset 1 and Asset 2,
  respectively.
• Vol1, Vol2 volatilities of the returns on Asset
  1 and Asset 2, respectively.
• The volatility is the standard deviation of the
  returns.
• Rho12 correlation between returns on Asset 1
  and Asset 2
• W1 proportion in Asset 1.
• W2 proportion in Asset 2.
• Rp expected return on portfolio of the two
  assets w1 R1 w2 R2
• Volp Volatility of portfolio of the two assets
  (w1)2 (Vol1)2 (w2)2 (Vol2)2 2 x Rho12 x
  w1 x w2 x Vol1 x Vol2

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Asset Allocation A. Risky vs. Riskless Asset

• First, consider the allocation between the risky
  and riskless asset.
• Rf expected return on riskfree asset.
• Rp expected return on risky 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. Thus, there is no one best
  portfolio allocation.

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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 Return5, Risky Return12, Vol of
Risky Asset0.15
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Portfolio Return vs. Portfolio Volatility
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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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B. Portfolio of Risky Assets

• 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 investor
  allocate between two or more risky stocks?
• 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 portfolio).
• When we only have two risky assets, 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.
• 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.
• 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 over the 10-year sample
  period 1994-2004, with monthly data.
• As can be seen, the optimal weight for a
  portfolio (to get the maximum Sharpe ratio) is
  about 28 for KO.
• If the exact answer is required, we can easily
  solve for it using the solver in Excel..

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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, drawn such that it has
  the risk-free rate as its intercept.
• This is because the slope of the line that passes
  connects the risk-free 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 also called 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 KO PEP,
1994-2004