.. slideconf:: :slide_classes: appear ============================================================================== Asset Allocation ============================================================================== Utility ============================================================================== Investors usually care about maximizing utility. .. rst-class:: to-build - Suppose all investors have utility function .. rst-class:: to-build .. math:: U(\mu, \sigma) & = \mu - \gamma \sigma^2. .. rst-class:: to-build - :math:`\mu` and :math:`\sigma` are the mean and standard deviation of asset returns. .. rst-class:: to-build - What is the utility of holding a risk-free asset :math:`U(\mu_f, \sigma_f)`? .. rst-class:: to-build .. math:: U(\mu_f ,\sigma_f) = r_f. Certainty Equivalent ============================================================================== For a risky portfolio, :math:`U(\mu_f, \sigma_f)` can be thought of as a certainty equivalent return. .. rst-class:: to-build - 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 :math:`\gamma` is a measure of risk preference. .. rst-class:: to-build - If :math:`\gamma > 0` individuals are risk averse - volatility detracts from utility. .. rst-class:: to-build - If :math:`\gamma = 0` individuals are risk neutral - volatility doesn't enter into the utility function. .. rst-class:: to-build - In this case, investors rank portfolios by their expected return and don't care about portfolio riskiness. Risk Aversion ============================================================================== - If :math:`\gamma < 0` individuals are risk lovers - volatility is rewarded in the utility function. .. rst-class:: to-build - In this case, investors enjoy and get utility by taking on risk. .. rst-class:: to-build - We will generally assume investors are risk averse, with the magnitude of :math:`\gamma` dictating the amount of risk aversion. Mean-Variance Criterion ============================================================================== Under this utility model, investors prefer higher expected returns and lower volatility. .. rst-class:: to-build - Let portfolio :math:`A` have mean and volatility :math:`\mu_A` and :math:`\sigma_A`. .. rst-class:: to-build - Let portfolio :math:`B` have mean and volatility :math:`\mu_B` and :math:`\sigma_B`. .. rst-class:: to-build - If :math:`\mu_A \geq \mu_B` *and* :math:`\sigma_A \leq \sigma_B`, then :math:`A` is preferred to :math:`B`. Mean-Variance Criterion ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/pg164_1.jpg :width: 8.5in :align: center .. ifnotslides:: .. image:: /_static/pg164_1.jpg :width: 6in Indifference Curves ============================================================================== Portfolios in quadrant I are preferred to :math:`P`, which is preferred to portfolios in quadrant IV. .. rst-class:: to-build - What about quadrants II and III? .. rst-class:: to-build - If a portfolio :math:`Q` has a mean and volatility that differ from :math:`P` but yields the same utility level, then either .. rst-class:: to-build .. math:: \mu_Q > \mu_p \text{ and } \sigma_Q > \sigma_p .. rst-class:: to-build or .. rst-class:: to-build .. math:: \mu_Q < \mu_p \text{ and } \sigma_Q < \sigma_p. .. rst-class:: to-build - That is, Q must be in quadrants II or III. Indifference Curves ============================================================================== - The portfolios that yield the same utility as :math:`P` constitute an indifference curve. .. rst-class:: to-build - We conclude that the indifference curve must cut through quadrants II and III. Indifference Curve Plot ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/pg165_2.jpg :width: 7in :align: center .. ifnotslides:: .. image:: /_static/pg165_2.jpg :width: 6in Portfolios of Assets ============================================================================== Suppose an individual can invest in two assets: a risky portfolio :math:`P` and a risk-free asset :math:`F`. .. rst-class:: to-build - :math:`\omega` will be the fraction wealth invested in :math:`P`. .. rst-class:: to-build - :math:`1-\omega` will be the fraction wealth invested in :math:`F`. .. rst-class:: to-build - We will typically assume that portfolio weights sum to 1. .. rst-class:: to-build - :math:`r_p` will denote the return on asset :math:`P`, with :math:`\mu_p = E[r_p]` and :math:`\sigma_p^2 = Var(r_p)`. .. rst-class:: to-build - :math:`r_f` will denote the return on asset :math:`F`, with :math:`\mu_f = r_f` and :math:`\sigma_f^2 = 0`. Portfolio Return ============================================================================== Let :math:`C` denote the portfolio that *combines* the two assets. .. rst-class:: to-build - :math:`C` is a weighted average of :math:`P` and :math:`F`: .. rst-class:: to-build .. math:: C = \omega P + (1-\omega) F. .. rst-class:: to-build - The return to :math:`C`, is .. rst-class:: to-build .. math:: r_c = \omega r_p + (1-\omega) r_f. Portfolio Return ============================================================================== By the linearity of expectations: .. rst-class:: to-build .. math:: \mu_c = E[r_c] \qquad \qquad \qquad .. rst-class:: to-build .. math:: \quad \enspace \, = \omega E[r_p] + (1 - \omega)E[r_f] .. rst-class:: to-build .. math:: = \omega \mu_p + (1-\omega) r_f \enspace \, .. rst-class:: to-build .. math:: = r_f + \omega(\mu_p - r_f). \enspace \, .. rst-class:: to-build - The term in the parentheses is the risk premium of :math:`P`. Portfolio Volatility ============================================================================== According to the properties of variance, .. rst-class:: to-build .. math:: \sigma^2_c = Var(r_c) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \enspace .. rst-class:: to-build .. math:: \quad \enspace \; \, = \omega^2 Var(r_p) + (1 - \omega)^2 Var(r_f) + 2 \omega (1 - \omega) Cov(r_p, r_f) .. rst-class:: to-build .. math:: = \omega^2 Var(r_p) \qquad \qquad \qquad \qquad \qquad \qquad \quad \enspace \; \: \, .. rst-class:: to-build .. math:: = \omega^2 \sigma^2_p. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \enspace \; \, .. rst-class:: to-build - The third equality follows because :math:`r_f` is a constant. .. rst-class:: to-build - Thus, .. rst-class:: to-build .. math:: \sigma_c = \omega \sigma_p. Portfolios and Risk Aversion ============================================================================== Portfolio :math:`C` earns a base return of :math:`r_f` plus the risk premium associated with :math:`P`, weighted by the amount of wealth the investor allocates to :math:`P`. .. rst-class:: to-build - More risk averse investors (small :math:`\omega`) expect a rate of return closer to :math:`r_f`. .. rst-class:: to-build - Less risk averse investors (high :math:`\omega`) expect a rate closer to :math:`\mu_p - r_f`. .. rst-class:: to-build - More risk averse investors have smaller portfolio volatilities. .. rst-class:: to-build - Less risk averse investors have higher portfolio volatilities. Portfolio Weight ============================================================================== Since :math:`\sigma_c = \omega \sigma_p`, .. rst-class:: to-build .. math:: \omega =\frac{\sigma_c}{\sigma_p}. Sharpe Ratio ============================================================================== Thus .. rst-class:: to-build .. math:: \mu_c = r_f + \omega (\mu_p - r_f) \qquad .. rst-class:: to-build .. math:: = r_f + \frac{\sigma_c}{\sigma_p} (\mu_p - r_f) \: .. rst-class:: to-build .. math:: = r_f + \frac{\mu_p - r_f}{\sigma_p} \sigma_c \enspace \; \, .. rst-class:: to-build .. math:: = r_f + \text{SR}_p \sigma_c. \qquad \enspace .. rst-class:: to-build - :math:`\text{SR}_p` is the Sharpe Ratio of portfolio :math:`P`. Capital Allocation Line ============================================================================== The Capital Allocation Line (CAL) depicts the set of portfolios available to an investor (a budget constraint). .. rst-class:: to-build - It plots pairs of :math:`\sigma_c` and :math:`\mu_c` that the investor can choose by selecting :math:`\omega`. .. rst-class:: to-build - This is simply a plot of the equation .. rst-class:: to-build .. math:: \mu_c = r_f + \text{SR}_p \sigma_c. .. rst-class:: to-build - Clearly, the intercept will be :math:`r_f` and the slope will be :math:`\text{SR}_p`. Capital Allocation Line Example ============================================================================== Suppose .. rst-class:: to-build - :math:`r_f = 0.07`. .. rst-class:: to-build - :math:`\mu_p = 0.15`. .. rst-class:: to-build - :math:`\sigma_p = 0.22`. .. rst-class:: to-build What is the CAL? Capital Allocation Line Plot ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/pg172_1.jpg :width: 7.5in :align: center .. ifnotslides:: .. image:: /_static/pg172_1.jpg :width: 6in Simple Portfolio Choice ============================================================================== Individuals seek to maximize utility subject to the available choice set. .. rst-class:: to-build .. math:: \max_{\mu_c, \sigma_c} U(\mu_c, \sigma_c) = \max_{\mu_c, \sigma_c} \mu_c - \frac{1}{2} \gamma \sigma^2_c, .. rst-class:: to-build subject to .. rst-class:: to-build .. math:: \mu_c & = r_f + \omega (\mu_p - r_f) \\ \sigma_c & = \omega \sigma_p. Simple Portfolio Choice (Cont.) ============================================================================== Substituting the constraints, the maximization problem becomes .. rst-class:: to-build .. math:: \max_{\omega} \left\{r_f + \omega (\mu_p - r_f) - \frac{1}{2} \gamma \omega^2 \sigma^2_p \right\}. .. rst-class:: to-build Taking the derivative of this equation w.r.t. :math:`\omega`, .. rst-class:: to-build .. math:: \mu_p - r_f = \gamma \omega^* \sigma^2_p .. rst-class:: to-build .. math:: \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. .. rst-class:: to-build - Fix the utility value at :math:`\bar{U} = r_f`. .. rst-class:: to-build - Use the relation :math:`\bar{U} = \mu_c - \frac{1}{2} \gamma \sigma^2_c` to solve for :math:`\mu_c`: .. rst-class:: to-build .. math:: \mu_c = \bar{U} + \frac{1}{2} \gamma \sigma^2_c. Simple Portfolio Choice ============================================================================== - Using this equation, we find the pairs of :math:`\mu_c` and :math:`\sigma_c` that corresponds to utility :math:`\bar{U}`, which we plot. .. rst-class:: to-build - Repeat this process, increasing the values of :math:`\bar{U}` until a tangent indifference curve is found. The tangency corresponds to the optimal portfolio. Spreadsheet Optimization ============================================================================== Given :math:`\gamma=2`, :math:`\mu=0.15`, :math:`\sigma=0.22` and :math:`r_f=0.07`. :math:`\qquad` .. ifslides:: .. image:: /_static/calSS.png :width: 8in :align: center .. ifnotslides:: .. image:: /_static/calSS.png :width: 6in Graphical Optimization ============================================================================== :math:`\qquad` .. ifslides:: .. image:: /_static/pg177_2.jpg :width: 6.5in :align: center .. ifnotslides:: .. image:: /_static/pg177_2.jpg :width: 6in Capital Market Line ============================================================================== Consider a value-weighted portfolio of all assets in the market. .. rst-class:: to-build - We will call this the market portfolio and denote it by :math:`M`. .. rst-class:: to-build - The CAL which connects :math:`r_f` with :math:`M` is called the *Capital Market Line* (CML). .. rst-class:: to-build - Because the true market portfolio is unobserved, we use a proxy - a well diversified portfolio that provides a good representation of the entire market. .. rst-class:: to-build - 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*. .. rst-class:: to-build - It requires no security analysis. .. rst-class:: to-build - An *active strategy* is one that requires individual security analysis.