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Asset Allocation


Asset Allocation Week 4 Asset Allocation: The Fundamental Question How do you allocate your assets amongst different assets? There are literally thousands of assets ... – PowerPoint PPT presentation

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Title: Asset Allocation

Asset Allocation
  • Week 4

Asset Allocation The Fundamental Question
  • How do you allocate your assets amongst different
  • 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

The Two Decisions
  • How do you allocate your assets amongst different
    assets? There are two decisions that you have to
  • 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
  • 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.

  • Traditionally, when we decide asset allocation,
    we will assume that all the assets are fairly
  • 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.

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
  • We will measure our risk by the volatility of the
    return. The volatility is the standard deviation
    of the return.

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.

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
  • 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.

Notations and Useful Formulae
  • Let there be two assets, Asset 1 and Asset 2.
  • R1, R2 expected returns on Asset 1 and Asset 2,
  • Vol1, Vol2 volatilities of the returns on Asset
    1 and Asset 2, respectively.
  • The volatility is the standard deviation of the
  • 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

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.

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.

Riskfree Return5, Risky Return12, Vol of
Risky Asset0.15
Portfolio Return vs. Portfolio Volatility
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
  • 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.

B. Portfolio of Risky Assets
  • We discussed the allocation between the risky
    (stock) portfolio and the riskless (cash)
  • 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).

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.

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..

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 Return-Volatility Frontier KO PEP,