In this section of the course, I review the static portfolio choice problem. The investor chooses a portfolio structure which is then left alone. The investment criterion is the expected utility of wealth at a terminal date. I briefly review specifications for the utility function together with risk aversion concepts. I look at the case of constant absolute risk aversion and normal returns, with or without labour income. I then introduce mean variance preferences, linking them to expected utility. Mean variance with and without a risk free asset is studied. The link between mean variance preferences and the expected returns/beta relationship is explained (the key ingredient of the CAPM). I then touch on the implementation problem.


Timing

  • Two periods:

    • portfolio decisions in t=0

    • outcome observed in t=1

  • Outcomes in date \(1\) are uncertain as of date \(0\); they are described by random variables which we will identify in the notation using tildas

    • \(x\): particular outcome; \(\tilde{x}\): random variable

Instruments

  • Instrument \(i\) with price \(p_{i}\) in period \(0\) gives right to pay-off \(\tilde{x}_{i}\) in period \(1\)

  • A cash instrument is an instrument with known date \(1\) pay-off as of date \(0\)

  • For risky assets, \(\tilde{x}\) is uncertain as of date \(0\)

  • I'll assume there are \(N\) risky assets (\(i=1,\cdots,N\)) and potentially cash (the riskless asset), which will then have index \(0\)

  • The set of assets will be denoted by \(\cal{I}\), with either \({\cal I}=(1,\cdots,N)\) (no riskless asset) or \({\cal I}=(0,\cdots,N)\) (with a riskless asset)

Returns

  • The return of an instrument with price \(p\) and pay-off \(\tilde{x}\) is: \[\tilde{R}=\frac{\tilde{x}}{p}\]

  • The rate of return is \(\tilde{r}=\tilde{R}-1\)

  • The rate of return of cash is usually denoted \(r^{f}\); it is known as of date \(0\)

Investment and returns

  • From investment to pay-off

  • From \(t=0\) to \(t=1\):

    • \(\phi \longrightarrow \tilde{R}\phi\)
    • \(\phi \longrightarrow (1+\tilde{r})\phi\)

Portfolios

  • Wealth in period \(0\) is \(w_{0}\)

  • The portfolio is invested in period \(0\); quantities \((\theta_{i})_{i \in \cal{I}}\) are purchased

  • They need to satisfy: \[\sum_{i \in \cal{I}}\theta_{i}p_{i}=w_{0}\]

  • One can choose as control variables:

    • quantities \((\theta_{i})_{i \in \cal{I}}\)

    • dollar amounts invested on instruments \((\phi_{i})_{i \in \cal{I}}\) with \(\phi_{i}=\theta_{i}p_{i}\)

    • wealth shares \((\pi_{i})_{i \in \cal{I}}\), with \(\pi_{i}=\phi_{i}/w_{0}\)

Budget constraints

  • Quantities: \[\sum_{i \in \cal{I}}\theta_{i}p_{i}=w_{0}\]

  • Dollar amounts: \[\sum_{i \in \cal{I}}\phi_{i}=w_{0}\]

  • Wealth shares: \[\sum_{i \in \cal{I}}\pi_{i}=1\]

Borrowing

  • Borrowing is best understood as a negative position in cash:

    • from \(t=0\) to \(t=1\)

    • \(\phi=-d \longrightarrow -d(1+r^{f})\)

Accounting for future wealth

  • for a given initial wealth \(w_{0}\), a portfolio allocation leads to a random final wealth \(\tilde{w}\) with:

    • quantities: \(\tilde{w}=\sum_{i \in \cal{I}}\theta_{i}\tilde{x}_{i}\)

    • invested amounts: \(\tilde{w}=\sum_{i \in \cal{I}}\phi_{i}\tilde{R}_{i}\)

    • wealth shares: \(\tilde{w}=w_{0}\sum_{i \in \cal{I}}\pi_{i}\tilde{R}_{i}\)

    • It is sometimes useful to introduce at date \(1\) an exogenous income (amount to be received) or liability (amount to be paid) \(\tilde{y}\)

    • \(\tilde{w}=\tilde{y}+\sum_{i \in \cal{I}}\theta_{i}\tilde{x}_{i}\)

    • \(\tilde{w}=\tilde{y}+\sum_{i \in \cal{I}}\phi_{i}\tilde{R}_{i}\)

    • \(\tilde{w}=\tilde{y}+w_{0}\sum_{i \in \cal{I}}\pi_{i}\tilde{R}_{i}\)

Some return arithmetic

  • Without liability, we have:

    • portfolio return: \[\tilde{R_{p}}=\frac{\tilde{w}}{w_{0}}=\sum_{i \in \cal{I}}\pi_{i}\tilde{R}_{i}\]

    • portfolio rate of return: \[\tilde{r_{p}}=\frac{\tilde{w}}{w_{0}}=\sum_{i \in \cal{I}}\pi_{i}\tilde{r}_{i}\] (since \(\sum_{i \in \cal{I}}\pi_{i}=1\))

The space of excess returns

  • In the presence of a riskless asset, it is convenient to introduce excess returns versus the riskless rate: \[\tilde{r_{p}} = \sum_{i \in \cal{I}}\pi_{i}\tilde{r}_{i}\] \[=r^{f}+\sum_{i=1}^{N}\pi_{i}(\tilde{r}_{i}-r^{f}).\]

  • The choice variables are initially \((\pi_{i})_{i \in \cal{I}}\), under the constraint \(\sum_{i \in \cal{I}}\pi_{i}=1\).

  • In the excess return space, the choice variables are \((\pi_{i})_{i=1}^{N}\) to which no budget constraint applies since it is enforced by \(\pi_{0}=1-\sum_{i=1}^{N}\pi_{i}\).

The portfolio problem

  • Future wealth is a random variable, with a specific distribution

  • The portfolio problem:

    • choose quantities (amounts, wealth shares) so as to obtain the best wealth distribution possible
  • How do we compare random outcomes?

    • expected utility (Von Neumann Morgenstern - VNM) of outcome: \(E[u(\tilde{w})]\)

    • the utility function embodies attitudes towards risk of the decision maker

Some remarks

  • The optimization problem cannot have a solution if there are arbitrage opportunities

  • Reminder: an arbitrage is a way to generate a strictly positive pay-off without committing any funds

  • The existence of a solution to a portfolio optimization problem thus guarantees the existence of a strictly positive stochastic discount factor (see below). We will see this principle in action

Arbitrage, the law of one price and SDFs

  • A stochastic discount factor is a random variable \(\tilde{m}\) such that for any pay-off \(\tilde{x}\), the market price can be recovered: \[p=E[\tilde{m}\tilde{x}].\]

  • The law of one price is equivalent to the existence of a stochastic discount factor. The absence of arbitrage is equivalent to the existence of an almost everywhere strictly positive discount factor. Broadly speaking, strict positivity ensures that a (possibly synthetic) asset with strictly positive payoff cannot have a strictly negative price (this would be an arbitrage).

  • In the return space, the above relationship reads: \[E[\tilde{m}\tilde{R}]=1.\]

  • The expectation of the discount factor is linked to the risk free rate: \[E[\tilde{m}](1+r^{f})=1.\]

  • In the excess return space, this reads: \[E[\tilde{m}(\tilde{r}-r^{f})]=0.\]

  • We thus have, in the presence of a risk free asset1: \[E[\tilde{r}]-r^{f}=-R^{f}\text{cov}(\tilde{m},\tilde{R}),\] which describes the structure of risk premia across assets as a result of the covariances with the SDF.

Reminder on utility functions (1)

  • VNM utility functions are determined up to a linear transformation

  • Absolute risk aversion: \(\alpha(w)=-u''(w)/u'(w)\)

  • Relative risk aversion: \(\rho(w)=w\alpha(w)\)

  • Risk tolerance: \(\tau(w)=1/\alpha(w)\)

  • Additive certainty equivalent: for a centered distribution \(\tilde{\varepsilon}_{a}\) and an initial level of wealth \(w\), find \(\pi_{a}(w,\tilde{\varepsilon}_{a})\) such that: \[u(w-\pi_{a})=E[u(w+\tilde{\varepsilon}_{a})].\]

  • Multiplicative certainty equivalent: for a centered distribution \(\tilde{\varepsilon}_{m}\) and an initial level of wealth \(w\), find \(\pi_{m}(w,\tilde{\varepsilon}_{m})\) such that: \[u(w(1-\pi_{m}))=E[u(w(1+\tilde{\varepsilon}_{m}))].\]

Reminder on utility functions (2)

  • For small (centered) additive risks of variance \(\sigma^{2}\): \(\pi_{a} \approx \frac{1}{2}\sigma_{a}^{2}\alpha(w)\)

  • For small (centered) multiplicative risks of variance \(\sigma^{2}\): \(\pi_{m} \approx \frac{1}{2}\sigma_{m}^{2}\rho(w)\)

Some important utility functions

  • CARA: \(u(w)=-\exp(-\alpha w)\)

    • range: \(\mathbb{R}\)
    • absolute risk aversion: \(\alpha(w)=\alpha\)
  • CRRA: \[u_{\rho}(w)= \frac{c^{1-\rho}}{1-\rho},\, \rho \geq 0,\, \rho\neq 1,\] \[u_{\rho}(w)=\log(w),\, \rho=1,\]
    • range \(\mathbb{R}_{+}^{*}\)
    • relative risk aversion: \(\rho(w)=\rho\)

CRRA utility functions - fig 1

Figure 1: CRRA utility functions
Figure 1: CRRA utility functions

Utility functions and return distributions

  • Utility functions often have a restricted domain (frequently: positive consumption)

  • Assumptions on return distributions have to be consistent

  • For example, CRRA models require \(\tilde{R}\geq 0\) i.e. \(\tilde{r} \geq -1\). This assumption is sometimes called 'limited liability': the owner of an asset cannot end up having to transfer cash to the issuer.

  • This is a problem mainly for discrete time models (or continuous times models where prices can jump)

Absolute or relative?

  • The key consideration is the dependence of risk attitudes vis-à-vis the level of wealth

    • intuition suggests people accept greater dollar risk as their wealth rises

An important benchmark: CARA & normally distributed returns

  • Note that with normal returns, returns can be arbitrarily negative (no limited liability). Accordingly, the range of the utility function is \(\mathbb{R}\).

  • I assume that there is no labor income

  • \(\pmb{\pi}=(\pi_{i})_{i \in \cal{I}}'\) \[\underset{\pmb{\pi}}{\text{max}} \; E[-\exp(-\alpha \tilde{w})]\] \[\text{s.t.}:\] \[\tilde{w}=w_{0}\sum_{i \in \cal{I}}\pi_{i}\tilde{R}_{i}\] \[\sum_{i \in \cal{I}}\pi_{i}=1.\]

CARA normal case (1)

  • The random variable \(\tilde{w}\) is normally distributed. In this case, we know that: \[E[-\exp(-\alpha\tilde{w})] =\; -\exp(-\alpha E[\tilde{w}]+(\alpha^{2}/2)V[\tilde{w}])]\] \[=u(E[\tilde{w}]-(\alpha/2)V[\tilde{w}]).\]

  • Given that the function \(u(\cdot)\) is increasing, the program consists in maximizing the certainty equivalent \(E[\tilde{w}]-(\alpha/2)V[\tilde{w}]\), which reads, mean wealth minus the variance of wealth weighted by one half absolute risk aversion.

CARA normal case (2)

  • Preferences over the distribution of final wealth are thus entirely determined by the mean and the variance of the wealth distribution. This is an example of mean variance preferences.

  • We have: \[\tilde{w}=w_{0}\sum_{i \in \cal{I}}\pi_{i}\tilde{R}_{i}=w_{0}+w_{0}\sum_{i \in \cal{I}}\pi_{i}\tilde{r}_{i}\]

  • The maximized criterion is thus (dividing by \(w_{0}>0\)): \[E[\sum_{i \in \cal{I}}\pi_{i}\tilde{r}_{i}]-(\alpha w_{0}/2)V[\sum_{i \in \cal{I}}\pi_{i}\tilde{r}_{i}].\]

CARA normal case (3)

  • This is a standard mean-variance criterion, up to the fact that the risk aversion parameter depends on the level of wealth.

    • if this was not the case, optimal portfolio composition would be independent of the wealth level; this would imply that the investor take more dollar risk at higher wealth levels; in the CARA case, the appetite for dollar risk is independent of the level of wealth; thus the correction.

When do we get mean variance preferences?

  • How general is mean variance ?

    • preferences induced by utility functions will not, in general, correspond to mean-variance; additional assumptions are needed.

    • when the distribution of portfolio returns is characterized by mean and variance, all utility functions naturally lead to mean variance preferences (see elliptic distributions).

    • in the presence of stochastic labour income, mean variance needs to be amended

CARA normal case (4)

  • In the presence of normally distributed stochastic labor income, the optimal programme is: \[\underset{\pmb{\pi}}{\text{max}} \; E[-\exp(-\alpha \tilde{w})]\] \[\text{s.t.}\] \[\tilde{w}=\tilde{y}+\sum_{i \in \cal{I}}\theta_{i}\tilde{x}_{i}\] \[\sum_{i \in \cal{I}}\theta_{i}p_{i}=w_{0}.\]

  • It is this time more convenient to take as control variables the quantities: \((\theta_{i})_{i \in {\cal I}}\).

CARA normal case (5)

  • As before, we need to maximize the certainty equivalent: \(E[\tilde{w}]-(\alpha/2)V[\tilde{w}]\). This is equivalent to maximizing: \[E\left[\sum_{i \in {\cal I}}\theta_{i}\tilde{x}_{i}\right]-(\alpha/2)V\left[\tilde{y}+\sum_{i \in {\cal I}}\theta_{i}\tilde{x}_{i}\right].\]

  • We can decompose the variance term as: \[V\left[\tilde{y}\right]+V\left[\sum_{i \in \cal{I}}\theta_{i}\tilde{x}_{i}\right] +2\text{Cov}\left(\sum_{i \in \cal{I}}\theta_{i}\tilde{x}_{i},\tilde{y}\right).\]

CARA normal case (6)

  • I give the result assuming there is a riskless asset.

  • We assume the price of cash is \(p_{0}=1\), and the payoff \(\tilde{x}_{0}=1+r^{f}\).

  • Using the budget constraint \(\theta_{0}=w_{0}-\sum_{i=1}^{N}\theta_{i}p_{i}\), we can rewrite the criterion as: \[E\left[w_{0}(1+r^{f})+\sum_{i=1}^{N}\theta_{i}(\tilde{x}_{i}-p_{i}(1+r^{f}))\right]-(\alpha/2)V\left[\tilde{y}+\sum_{i=1}^{N}\theta_{i}\tilde{x}_{i}\right].\]

  • Notation:

    • \(\theta\) is the \(N\times 1\) vector of quantities invested on each risky asset
    • \(V[\tilde{x}]\) is the \(N\times N\) matrix where each \((i,j)\) is the covariance of the pay-offs of asset \(i\) and \(j\). It is assumed to have full rank, so that no financial asset is riskless or redundant.
    • \(\text{Cov}(\tilde{x},\tilde{y})\) is the \(N\times 1\) vector where each entry measures the covariance of a financial instrument with labour income
    • \(E[\tilde{\tilde{x}}]\) is the \(N\times 1\) vector of the expected excess pay-offs \((\tilde{x}_{i}-p_{i}(1+r^{f}))\) of the risky instruments instruments.

CARA normal case (7)

  • The first order condition leads to, in matrix notation: \[\theta=V[\tilde{x}]^{-1}\left(-\text{Cov}(\tilde{x},\tilde{y})+\frac{1}{\alpha}E[\tilde{\tilde{x}}]\right).\]

  • Remember that \(1/\alpha\) is risk tolerance.

  • The structure of the solution is as follows: the optimal porfolio consists of a hedging portfolio (which tries to replicate income variability using financial assets) and a speculative portfolio which has the same structure as in the case without labour income. The latter portfolio receives a weight equal to risk tolerance.

Optimization and SDF

  • I assume there is a solution \(\pmb{\pi}_{*}\) to the following problem: \[\underset{\pmb{\pi}}{\text{max}} \; E[u(\pmb{\pi}'\pmb{\tilde{R}})]\] \[\text{s.t.}\] \[\pmb{\pi}'\pmb{e}=1,\] where \(\pmb{e}\) is a vector where all components are equal to \(1\), and \(\pmb{\pi}\) is the vector of asset proportions.

  • The Lagrangian reads: \[{\cal L}=E[u(\pmb{\pi}'\pmb{\tilde{R}})]-\gamma \pmb{\pi}'\pmb{e},\] and the first order condition reads: \[E[u'(\pmb{\pi}'\pmb{\tilde{R}})\pmb{\tilde{R}}]=\gamma \pmb{e}.\]

  • Let: \[\tilde{m}=\frac{u'(\pmb{\pi}_{*}'\pmb{\tilde{R}})}{\gamma}.\] We then have: \[E[\tilde{m}\pmb{\tilde{R}}]=\pmb{e},\] i.e. for any asset \(i\): \[E[\tilde{m}\tilde{R}_{i}]=1.\] In other words, we have built an SDF from the solution of the optimization problem.

Mean variance efficiency

  • A portfolio \(p\) with mean and variance \((\mu_{p},\sigma_{p})\) is dominated by a portfolio \(q\) with mean and variance \((\mu_{q},\sigma_{q})\) if \(\mu_{q} \ge \mu_{p}\) and \(\sigma_{q} \le \sigma_{p}\) with at least one inequality being strict.

  • A portfolio is efficient in the mean variance sense if it is not dominated by any other portfolio.

  • Domination is a preorder. An efficient portfolio is a maximal element for the preorder. In particular, it is not a total order (all portfolio pairs cannot necessarily be ordered).

Mean variance without a riskfree asset (1)

  • The program: it consists in minimizing portfolio variance for a given level of expected returns \[\underset{\pmb{\pi}}{\text{min}} \; V\left[\sum_{i=1}^{N}\pi_{i}\tilde{r}_{i}\right]=\pmb{\pi}' \Sigma \pmb{\pi}\] \[\text{s.t.}\] \[\sum_{i=1}^{N}\pi_{i}=\pmb{\pi}'\pmb{e}=1\] \[E\left[\sum_{i=1}^{N}\pi_{i}\tilde{r}_{i}\right]=\pmb{\pi}'\pmb{\mu}=\mu_{p}.\]

Mean variance without a riskfree asset (2)

  • Bold notations denote vectors

    • \(\Sigma\) is the covariance matrix of returns, which we assume invertible
    • \(\pmb{e}\) is a vector of ones
    • \(\pmb{\tilde{r}}\) is the vector of returns
    • \(\pmb{\mu}\) is the vector of expected returns
  • We assume \(\pmb{\mu}\neq \pmb{e}\) to avoid degeneracy

Mean variance without a riskfree asset (3)

  • Lagrangian for the optimization problem (a factor \(1/2\) is convenient): \[\frac{1}{2}\pmb{\pi}' \Sigma \pmb{\pi}-\delta (\pmb{\pi}'\pmb{\mu}-\mu_{p})-\gamma (\pmb{\pi}'\pmb{e}-1)\] where I have introduced the Lagrange multipliers \(\delta\) and \(\gamma\).

  • The necessary and sufficient first order condition (positive definite quadratic problem) is: \[\Sigma \pmb{\pi}=\delta \pmb{\mu}+\gamma \pmb{e},\] or, assuming the covariance matrix is invertible: \[\pmb{\pi}=\delta\Sigma^{-1}\pmb{\mu}+\gamma \Sigma^{-1}\pmb{e}.\]

Mean variance without a riskfree asset (4)

  • Injecting this into the constraints leads to a system for the Lagrange multipliers: \[\delta \pmb{\mu}' \Sigma^{-1}\pmb{\mu}+\gamma \pmb{\mu}' \Sigma^{-1}\pmb{e}=\mu_{p},\] \[\delta \pmb{e}'\Sigma^{-1}\pmb{\mu}+\gamma \pmb{e}'\Sigma^{-1}\pmb{e}=1.\]

  • Reminder: \[\begin{pmatrix} a&b\\ c&d \end{pmatrix}^{-1}=\frac{1}{ad-bc}\begin{pmatrix} d&-b\\ -c&a \end{pmatrix}\]

  • It is useful to introduce two specific portfolios: \[\pmb{\pi}_{1}=\frac{1}{\pmb{e}'\Sigma^{-1}\pmb{e}}\Sigma^{-1}\pmb{e},\] \[\pmb{\pi}_{\mu}=\frac{1}{\pmb{e}'\Sigma^{-1}\pmb{\mu}}\Sigma^{-1}\pmb{\mu}.\]

Mean variance without a riskfree asset (5)

  • We can write : \[\pmb{\pi} = (\delta\pmb{e}'\Sigma^{-1}\pmb{\mu})\pmb{\pi}_{\mu}+(\gamma\pmb{e}'\Sigma^{-1}\pmb{e})\pmb{\pi}_{1}=\] \[\lambda \pmb{\pi}_{\mu}+(1-\lambda) \pmb{\pi}_{1}.\]

  • Thus, any optimal portfolio is a combination of the two portfolios we singled out:

    • \(\pmb{\pi}_{1}\) is the minimum variance portfolio
    • \(\pmb{\pi}_{\mu}\) is another portfolio as soon as \(\pmb{\mu} \neq \pmb{e}\)

Mean variance without a riskfree asset (6)

  • \(A=\pmb{\mu}'\Sigma^{-1}\pmb{\mu}\), \(B=\pmb{\mu}'\Sigma^{-1}\pmb{e}\), \(C=\pmb{e}'\Sigma^{-1}\pmb{e}\). \[\lambda=\frac{BC\mu_{p}-B^{2}}{AC-B^{2}},\] \[\sigma_{p}^{2}=\frac{A-2B\mu_{p}+C\mu_{p}^{2}}{AC-B^2{}}.\]

  • Check this.

  • The efficient frontier (in the standard deviation mean space) is the subset of non dominated portfolios in the set: \[\{(\sigma_{p},\mu_{p}),\; \mu_{p}\geq \mu_{1}\}\] where \(\mu_{1}=\pmb{\pi}_{1}'\pmb{\mu}\).

Mean variance without a riskfree asset (7) - fig 2

Figure 2: Efficient Frontier (without a risk free asset)
Figure 2: Efficient Frontier (without a risk free asset)

Mean variance without a riskfree asset (8)

  • I list the technical conditions below:

    • we assume that \(\pmb{\mu}\) and \(\pmb{e}\) are not colinear

    • we assume \(\pmb{e}'\Sigma^{-1}\pmb{\mu>0}\)

    • we have \(\pmb{e}'\Sigma^{-1}\pmb{e}>0\) as \(\Sigma^{-1}\) defines a positive definite quadratic form

    • we have \(\left(\pmb{\mu}'\Sigma^{-1}\pmb{e}\right)^2<\left(\pmb{e}'\Sigma^{-1}\pmb{e}\right)\left(\pmb{\mu}'\Sigma^{-1}\pmb{\mu}\right)\) from the Cauchy-Schwartz inequality and \(\pmb{e}'\Sigma^{-1}\pmb{e}>0\).

Mean variance with a riskfree asset (1)

  • It is convenient in this case to use the notation \(\pmb{\pi}\) to denote the vector of positions on the risky assets (see the slide on the space of excess returns). The cash position is thus: \[\pi_{0}=1-\pmb{e}'\pmb{\pi}.\]

  • The vector \(\pmb{\pi}\) is unconstrained. The optimization problem can be written: \[\underset{\pmb{\pi}}{\text{min}} \; \pmb{\pi}' \Sigma \pmb{\pi}\] \[\text{s.t.}\] \[\pmb{\pi}'(\pmb{\mu}-r^{f}\pmb{e})=\mu_{p}-r^{f}.\]

  • For reasons that will be clear below, I assume \(\pmb{e}'\Sigma^{-1}(\pmb{\mu}-r^{f}\pmb{e})>0\).

Mean variance with a riskfree asset (2)

  • First order condition for the Lagrangian: \(\pmb{\pi}=\delta \Sigma^{-1}(\pmb{\mu}-r^{f}\pmb{e})\)

  • From \((\pmb{\mu}-r^{f}\pmb{e})'\pmb{\pi}=\mu_{p}-r^{f}\), we get the value of \(\delta\) and then the value of \(\pmb{\pi}\): \[\pmb{\pi}=\frac{\mu_{p}-r^{f}}{(\pmb{\mu}-r^{f}\pmb{e})'\Sigma^{-1}(\pmb{\mu}-r^{f}\pmb{e})}\Sigma^{-1}(\pmb{\mu}-r^{f}\pmb{e}).\]

  • The standard deviation of the portfolio is: \[\frac{|\mu_{p}-r^{f}|}{\sqrt{(\pmb{\mu}-r^{f}\pmb{e})'\Sigma^{-1}(\pmb{\mu}-r^{f}\pmb{e})}}.\]

Mean variance with a riskfree asset (3)

  • The tangency portfolio is: \[\pmb{\pi_{*}}=\frac{1}{\pmb{e}'\Sigma^{-1}(\pmb{\mu}-r^{f}\pmb{e})}\Sigma^{-1}(\pmb{\mu}-r^{f}\pmb{e}).\]

  • It is a portfolio fully invested in risky assets which is on the overall efficient frontier. It is thus also on the risky asset efficient frontier.

Mean variance with a riskfree asset (4) - fig 3

Figure 3: Efficient Frontier (with risk free asset)
Figure 3: Efficient Frontier (with risk free asset)

Mean variance with a riskfree asset (4) - fig4

Figure 4: Efficient Frontier (with risk free asset): iso-utility curves
Figure 4: Efficient Frontier (with risk free asset): iso-utility curves

Data for the graphs (1)

  • Two risky assets:
    • \(\mu_{1}=0.05,\, \sigma_{1}=0.12\)
    • \(\mu_{2}=0.07,\, \sigma_{2}=0.16\)
    • \(\rho=0.7\)
    • \((\mu_{1}-r)/\sigma_{1}=0.33\)
    • \((\mu_{2}-r)/\sigma_{2}=0.375\)
    • \(\pmb{\pi_{1}}=(0.93,0.07)\)
    • \(\text{vol}(\pmb{\pi_{1}})=0.12\)
    • \(\pmb{\pi_{*}}=(0.4,0.6)\)
    • \(\text{vol}(\pmb{\pi_{*}})=0.13\)
    • \(\text{sharpe}(\pmb{\pi_{*}})=0.39\)

Data for the graphs (2)

  • The graphs shown assume positive Sharpe ratios for the underlying assets. This is the 'normal' situation. It ensures that the efficient frontier (with a riskfree asset!) is upward sloping.

A different description of the efficient frontier (1)

  • Maximize the expected return penalized for portfolio variance (\(\rho>0\)): \[\underset{\pmb{\pi}}{\text{max}} \; r^{f}+\pmb{\pi}'(\pmb{\mu}-r^{f}\pmb{e})-\frac{\rho}{2}\pmb{\pi}' \Sigma \pmb{\pi}.\]

  • Exercise: recover the lagrange multiplier of the traditional approach

  • The criteria are given by quadratic utility functions, indexed by \(\rho\)

A different description of the efficient frontier (2)

  • The first order condition reads: \[(\pmb{\mu}-r^{f}\pmb{e})=\rho\Sigma \pmb{\pi},\] and this implies that the optimal portfolio is proportional to the tangency portfolio.

  • How much of the tangency portfolio \(\pmb{\pi_{*}}\) does an investor with the above preferences and beliefs buy?

  • From the first order condition of the utility maximization problem2, we get that the weight \(\hat{\pi}=1-\pi_{0}\) invested in the tangency portfolio is: \[\hat{\pi}=\frac{1}{\rho}\frac{\mu_{*}-r^{f}}{\text{var}(\tilde{r}_{*})}.\]

  • We will remember that: \[\rho \hat{\pi}=\frac{\mu_{*}-r^{f}}{\text{var}(\tilde{r}_{*})},\] which is therefore independent of the risk aversion level of the investor. This will play a role in the derivation of the CAPM.

Interpretation of the first order condition (1)

  • Consider that the optimal portfolio of a mean-variance investor (\(p\) with weights \(\pmb{\pi}\)) is tilted by adding a long-short portfolio \(\pmb{\pi}_{\delta}\). How does that affect quadratic utility?

  • The utility level changes by (first order approximation): \[\mu_{\delta}-\rho\text{cov}(\tilde{r}_{\delta},\tilde{r}_{p})\] \[=\mu_{\delta}-\rho\frac{\text{cov}(\tilde{r}_{\delta},\tilde{r}_{p})}{\text{var}(\tilde{r}_{p})}\text{var}(\tilde{r}_{p}),\] \[=\mu_{\delta}-\rho \beta (\tilde{r}_{\delta},\tilde{r}_{p}) \text{var}(\tilde{r}_{p}),\] \[=\mu_{\delta}-\rho \hat{\pi}\beta (\tilde{r}_{\delta},\tilde{r}_{*}) \text{var}(\tilde{r}_{*}).\]

Interpretation of the first order condition (2)

  • Because the quantity \(\rho \hat{\pi}\) is independent of \(\rho\), the trade off between return and beta is a well defined consequence of the mean and variance assumptions.

  • Injecting the value of \(\rho \hat{\pi}\) into the first order condition delivers the quantity: \[\mu_{\delta}-(\mu_{*}-r^{f})\beta (\tilde{r}_{\delta},\tilde{r}_{*}).\]

  • Given the optimality of the tangency portfolio, the above quantity should be zero for all long short deviations to the tangency portfolio: \[\mu_{\delta}=(\mu_{*}-r^{f})\beta (\tilde{r}_{\delta},\tilde{r}_{*}).\]

  • For long short portfolios which borrow to buy a stock, the condition reads: \[(\mu_{i}-r^{f})=(\mu_{*}-r^{f})\beta (\tilde{r}_{i},\tilde{r}_{*}).\]

Interpretation of the first order condition (3)

  • The above relationship embodies the return beta trade off embedded in the mean variance assumptions.

  • At this stage, no equilibrium assumption has been made. We are looking at the implications of a portfolio being mean-variance optimal.

  • Note that the tangency portfolio can be replaced by any other efficient portfolio in the relationship.

The excess return-beta relationship (1) - fig 5

Figure 5: Return/beta relationship
Figure 5: Return/beta relationship

The excess return-beta relationship (2) - fig 6

Figure 6: Imperfectly priced portfolios
Figure 6: Imperfectly priced portfolios

The two fund theorem and the CAPM

  • We now move to equilibrium considerations. We assume all investors share the same beliefs on expected returns and risk, and all choose mean variance efficient portfolios.

  • As a result, they all hold a mixture of the risk free asset and a unique portfolio of risky asset, the tangency portfolio.

  • This is an instance of the two fund theorem, which also holds in more general contexts

  • The risky asset portfolio should be equal to the market portfolio of risky asset, with return \(r_{m}\). This gives: \[(\mu_{i}-r^{f})=(\mu_{m}-r^{f})\beta (\tilde{r}_{i},\tilde{r}_{m}).\]

Illustration of the CAPM - fig 7

Figure 7: CAPM
Figure 7: CAPM

The low beta anomaly - fig 8

Figure 8: The low beta anomaly
Figure 8: The low beta anomaly

Equity pricing anomalies

  • Take an investment universe (stocks) and an equity index

  • Follow the steps:

    • build equity portfolios by sorting stocks according to a financial characteristic
    • compute the beta of the portfolios and graph realized returns against betas
    • is the pricing error significant?
  • Examples of characteristics: size, book value, momentum, beta, vol

  • This procedure asks whether the index is mean variance efficient in sample

  • The pricing errors should be statistically significant

Mean variance in practice: the challenges

  • First, there is a question of interpretation: what is the investment horizon?

    • in particular, this conditions the nature of the risky asset (bonds or cash).
  • One also needs to be clear on whether real returns or nominal returns are considered

  • Once this has been clarified, input data needs to be estimated:

    • getting hold of expected returns
    • getting hold of the covariance matrix
  • The optimal portfolio is very sensitive to inputs
    • garbage in, garbage out

Examples of implementations

  • Give the same return to all risky assets

    • this delivers the minimum variance portfolio, which is not optimal unless the expected returns are truly equal across assets
  • Link the return assumptions to the risk estimates

    • this leads to various solutions...
  • For returns: estimate the payoffs of the asset and derive the implies return from the current asset price

  • Example: ERC ?

Links

  • The beamer presentation is here.

  1. Write the discount factor condition as: \[E[\tilde{m}\tilde{R}]=E[(\tilde{m}-E[\tilde{m}]+E[\tilde{m}])\tilde{R}]=1,\] and use the fact: \[E[(\tilde{m}-E[\tilde{m}])\tilde{R}]=\text{Cov}(\tilde{m},\tilde{R}).\]

  2. The first order condition reads: \[(\pmb{\mu}-r^{f}\pmb{e})=\rho\Sigma \pmb{\pi}.\] Multiply both sides on the left by \(\pmb{\pi}'\). Then use \(\pmb{\pi}=\hat{\pi}\pmb{\pi}_{\star}\).