Asset Allocation

Utility

Investors usually care about maximizing utility.

  • Suppose all investors have utility function
\[\begin{split}U(\mu, \sigma) & = \mu - \gamma \sigma^2.\end{split}\]
  • \(\mu\) and \(\sigma\) are the mean and standard deviation of asset returns.
  • What is the utility of holding a risk-free asset \(U(\mu_f, \sigma_f)\)?
\[U(\mu_f ,\sigma_f) = r_f.\]

Certainty Equivalent

For a risky portfolio, \(U(\mu_f, \sigma_f)\) can be thought of as a certainty equivalent return.

  • The return that a risk-free asset would have to offer to provide the same utility level as a risky asset.

Risk Aversion

The parameter \(\gamma\) is a measure of risk preference.

  • If \(\gamma > 0\) individuals are risk averse - volatility detracts from utility.
  • If \(\gamma = 0\) individuals are risk neutral - volatility doesn’t enter into the utility function.
    • In this case, investors rank portfolios by their expected return and don’t care about portfolio riskiness.

Risk Aversion

  • If \(\gamma < 0\) individuals are risk lovers - volatility is rewarded in the utility function.
    • In this case, investors enjoy and get utility by taking on risk.
  • We will generally assume investors are risk averse, with the magnitude of \(\gamma\) dictating the amount of risk aversion.

Mean-Variance Criterion

Under this utility model, investors prefer higher expected returns and lower volatility.

  • Let portfolio \(A\) have mean and volatility \(\mu_A\) and \(\sigma_A\).
  • Let portfolio \(B\) have mean and volatility \(\mu_B\) and \(\sigma_B\).
  • If \(\mu_A \geq \mu_B\) and \(\sigma_A \leq \sigma_B\), then \(A\) is preferred to \(B\).

Mean-Variance Criterion

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Indifference Curves

Portfolios in quadrant I are preferred to \(P\), which is preferred to portfolios in quadrant IV.

  • What about quadrants II and III?
  • If a portfolio \(Q\) has a mean and volatility that differ from \(P\) but yields the same utility level, then either
\[\begin{split}\mu_Q > \mu_p \text{ and } \sigma_Q > \sigma_p\end{split}\]

or

\[\begin{split}\mu_Q < \mu_p \text{ and } \sigma_Q < \sigma_p.\end{split}\]
  • That is, Q must be in quadrants II or III.

Indifference Curves

  • The portfolios that yield the same utility as \(P\) constitute an indifference curve.
  • We conclude that the indifference curve must cut through quadrants II and III.

Indifference Curve Plot

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Portfolios of Assets

Suppose an individual can invest in two assets: a risky portfolio \(P\) and a risk-free asset \(F\).

  • \(\omega\) will be the fraction wealth invested in \(P\).
  • \(1-\omega\) will be the fraction wealth invested in \(F\).
  • We will typically assume that portfolio weights sum to 1.
  • \(r_p\) will denote the return on asset \(P\), with \(\mu_p = E[r_p]\) and \(\sigma_p^2 = Var(r_p)\).
  • \(r_f\) will denote the return on asset \(F\), with \(\mu_f = r_f\) and \(\sigma_f^2 = 0\).

Portfolio Return

Let \(C\) denote the portfolio that combines the two assets.

  • \(C\) is a weighted average of \(P\) and \(F\):
\[C = \omega P + (1-\omega) F.\]
  • The return to \(C\), is
\[r_c = \omega r_p + (1-\omega) r_f.\]

Portfolio Return

By the linearity of expectations:

\[\mu_c = E[r_c] \qquad \qquad \qquad\]
\[\quad \enspace \, = \omega E[r_p] + (1 - \omega)E[r_f]\]
\[= \omega \mu_p + (1-\omega) r_f \enspace \,\]
\[= r_f + \omega(\mu_p - r_f). \enspace \,\]
  • The term in the parentheses is the risk premium of \(P\).

Portfolio Volatility

According to the properties of variance,

\[\sigma^2_c = Var(r_c) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \enspace\]
\[\quad \enspace \; \, = \omega^2 Var(r_p) + (1 - \omega)^2 Var(r_f) + 2 \omega (1 - \omega) Cov(r_p, r_f)\]
\[= \omega^2 Var(r_p) \qquad \qquad \qquad \qquad \qquad \qquad \quad \enspace \; \: \,\]
\[= \omega^2 \sigma^2_p. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \enspace \; \,\]
  • The third equality follows because \(r_f\) is a constant.
  • Thus,
\[\sigma_c = \omega \sigma_p.\]

Portfolios and Risk Aversion

Portfolio \(C\) earns a base return of \(r_f\) plus the risk premium associated with \(P\), weighted by the amount of wealth the investor allocates to \(P\).

  • More risk averse investors (small \(\omega\)) expect a rate of return closer to \(r_f\).
  • Less risk averse investors (high \(\omega\)) expect a rate closer to \(\mu_p - r_f\).
  • More risk averse investors have smaller portfolio volatilities.
  • Less risk averse investors have higher portfolio volatilities.

Portfolio Weight

Since \(\sigma_c = \omega \sigma_p\),

\[\omega =\frac{\sigma_c}{\sigma_p}.\]

Sharpe Ratio

Thus

\[\mu_c = r_f + \omega (\mu_p - r_f) \qquad\]
\[= r_f + \frac{\sigma_c}{\sigma_p} (\mu_p - r_f) \:\]
\[= r_f + \frac{\mu_p - r_f}{\sigma_p} \sigma_c \enspace \; \,\]
\[= r_f + \text{SR}_p \sigma_c. \qquad \enspace\]
  • \(\text{SR}_p\) is the Sharpe Ratio of portfolio \(P\).

Capital Allocation Line

The Capital Allocation Line (CAL) depicts the set of portfolios available to an investor (a budget constraint).

  • It plots pairs of \(\sigma_c\) and \(\mu_c\) that the investor can choose by selecting \(\omega\).
  • This is simply a plot of the equation
\[\mu_c = r_f + \text{SR}_p \sigma_c.\]
  • Clearly, the intercept will be \(r_f\) and the slope will be \(\text{SR}_p\).

Capital Allocation Line Example

Suppose

  • \(r_f = 0.07\).
  • \(\mu_p = 0.15\).
  • \(\sigma_p = 0.22\).

What is the CAL?

Capital Allocation Line Plot

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Simple Portfolio Choice

Individuals seek to maximize utility subject to the available choice set.

\[\max_{\mu_c, \sigma_c} U(\mu_c, \sigma_c) = \max_{\mu_c, \sigma_c} \mu_c - \frac{1}{2} \gamma \sigma^2_c,\]

subject to

\[\begin{split}\mu_c & = r_f + \omega (\mu_p - r_f) \\ \sigma_c & = \omega \sigma_p.\end{split}\]

Simple Portfolio Choice (Cont.)

Substituting the constraints, the maximization problem becomes

\[\max_{\omega} \left\{r_f + \omega (\mu_p - r_f) - \frac{1}{2} \gamma \omega^2 \sigma^2_p \right\}.\]

Taking the derivative of this equation w.r.t. \(\omega\),

\[\mu_p - r_f = \gamma \omega^* \sigma^2_p\]
\[\Rightarrow \omega^* = \frac{\mu_p - r_f}{\gamma \sigma^2_p} = \frac{\text{SR}_p}{\gamma \sigma_p}.\]

Simple Portfolio Choice

Since the CAL is a choice set (budget constraint), we can find the optimal portfolio by observing where an indifference curve is tangent to the CAL.

  • Fix the utility value at \(\bar{U} = r_f\).
  • Use the relation \(\bar{U} = \mu_c - \frac{1}{2} \gamma \sigma^2_c\) to solve for \(\mu_c\):
\[\mu_c = \bar{U} + \frac{1}{2} \gamma \sigma^2_c.\]

Simple Portfolio Choice

  • Using this equation, we find the pairs of \(\mu_c\) and \(\sigma_c\) that corresponds to utility \(\bar{U}\), which we plot.
  • Repeat this process, increasing the values of \(\bar{U}\) until a tangent indifference curve is found. The tangency corresponds to the optimal portfolio.

Spreadsheet Optimization

Given \(\gamma=2\), \(\mu=0.15\), \(\sigma=0.22\) and \(r_f=0.07\).

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Graphical Optimization

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Capital Market Line

Consider a value-weighted portfolio of all assets in the market.

  • We will call this the market portfolio and denote it by \(M\).
  • The CAL which connects \(r_f\) with \(M\) is called the Capital Market Line (CML).
  • Because the true market portfolio is unobserved, we use a proxy - a well diversified portfolio that provides a good representation of the entire market.
  • Typically we use the S&P 500.

Passive vs. Active Strategies

Holding the market portfolio or a market proxy is known as a passive strategy.

  • It requires no security analysis.
  • An active strategy is one that requires individual security analysis.