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: $$\rho(w)=\rho$$

## 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 (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.

## 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 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: $(\mu_{i}-r^{f})=(\mu_{m}-r^{f})\beta (\tilde{r}_{i},\tilde{r}_{m}).$

## Equity pricing anomalies

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

• 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

• 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 ?

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 $$\pmb{\pi}=\hat{\pi}\pmb{\pi}_{\star}$$.