$$ \newcommand{\E}{\mathbb{E}} \newcommand{\cond}{\, |\, } \newcommand{\var}{\text{var}} \newcommand{\cov}{\text{cov}} \newcommand{\corr}{\text{corr}} \newcommand{\std}{\text{std}} \newcommand{\covmat}{\boldsymbol{\Sigma}} \newcommand{\cdf}{\Phi} \newcommand{\Normal}{\mathcal{N}} \newcommand{\lp}{\mathbb{L}} \newcommand{\dlim}{\overset{D}{\to \;}} \newcommand{\plim}{\overset{P}{\to \;}} \newcommand{\iid}{i.i.d.} \newcommand{\free}{f} \newcommand{\ex}[1]{\tilde{#1}} \newcommand{\R}[1][]{R^{#1}} \newcommand{\Rf}{\R[\free]} \newcommand{\Rx}[1][]{\ex{R}^{#1}} \renewcommand{\r}[1][]{r^{\scriptscriptstyle {#1}}} \newcommand{\rf}{\r[\free]} \newcommand{\rx}[1][]{\ex{r}^{\scriptscriptstyle {#1}}} \newcommand{\rlog}[1][]{{\texttt{r}^{#1}}} \newcommand{\rflog}{\rlog[\free]} \newcommand{\rvec}[1][]{\boldsymbol{\r[#1]}} \newcommand{\rxvec}[1][]{\boldsymbol{\rx[#1]}} \newcommand{\pay}[1][]{\Gamma^{\scriptscriptstyle {#1}}} \renewcommand{\P}{\mathcal{P}} \newcommand{\ind}[1]{_{[#1]}} \newcommand{\notind}[1]{\ind{-#1}} \newcommand{\coord}{\boldsymbol{\iota}} \newcommand{\obs}{n} \newcommand{\Nobs}{N} \newcommand{\lag}{h} \newcommand{\Nlag}{H} \newcommand{\indx}{i} \newcommand{\indxalt}{j} \newcommand{\I}{\mathcal{I}} \newcommand{\one}{\textbf{1}} \newcommand{\zeros}{\textbf{0}} \newcommand{\x}{\textbf{x}} \newcommand{\z}{\textbf{z}} \newcommand{\y}{\textbf{y}} \newcommand{\w}{\textbf{w}} \newcommand{\X}{\textbf{X}} \newcommand{\Z}{\textbf{Z}} \newcommand{\Y}{\textbf{Y}} \newcommand{\W}{\textbf{W}} \newcommand{\alphavec}{\boldsymbol{\alpha}} \newcommand{\betavec}{\boldsymbol{\beta}} \newcommand{\epsilonvec}{\boldsymbol{\epsilon}} \newcommand{\sigmavec}{\boldsymbol{\sigma}} \newcommand{\mux}{\ex{\mu}} \newcommand{\muvec}{\boldsymbol{\mu}} \newcommand{\muxvec}{\boldsymbol{\ex{\mu}}} \newcommand{\muP}{\mu^p} \newcommand{\Sigmamat}{\boldsymbol{\Sigma}} \newcommand{\wt}{\boldsymbol{\omega}} \newcommand{\wtx}{\boldsymbol{w}} \newcommand{\wtxstar}{\wtx^*} \newcommand{\wtTan}{\wt^{\tan}} \newcommand{\wtxTan}{\wtx^{\tan}} \newcommand{\wtGMV}{\wt^{\gmv}} \newcommand{\mv}{\scriptscriptstyle {\subset}} \newcommand{\MV}{$\ex{\text{MV}}\ $} \newcommand{\MVscale}{\delta} \newcommand{\ifac}{k} \newcommand{\Nfacs}{k} \newcommand{\fbeta}[1][]{\beta^{\scriptscriptstyle {#1}}} \newcommand{\fbetavec}[1][]{\betavec^{\scriptscriptstyle {#1}}} \newcommand{\radon}{q} \newcommand{\mkt}{m} \newcommand{\act}{a} \renewcommand{\tan}{\texttt{t}} \newcommand{\gmv}{\texttt{v}} \newcommand{\size}{s} \newcommand{\val}{v} \newcommand{\up}{u} \newcommand{\mom}{\text{mom}} \newcommand{\orthog}{\alpha} \newcommand{\fac}{z} \newcommand{\facs}{\boldsymbol{z}} \newcommand{\facmac}{f} \newcommand{\facsmac}{\boldsymbol{f}} \newcommand{\prem}{\lambda} \newcommand{\premvec}{\boldsymbol{\prem}} \newcommand{\pos}{i} \newcommand{\hedge}{j} \newcommand{\etavec}{\boldsymbol{\eta}} \newcommand{\Q}{\textbf{Q}} \newcommand{\eig}{\psi} \newcommand{\Eig}{\Psi} \newcommand{\eigv}{\textbf{q}} \newcommand{\Eigv}{Q} \newcommand{\PCwt}{\textbf{q}} \newcommand{\PCfac}{x} \newcommand{\VaR}{\text{VaR}} \newcommand{\ES}{\text{ES}} \newcommand{\Rrate}{\,\r} \newcommand{\Rratevec}{\,\rvec} \newcommand{\RVaR}{\, \r[\text{VaR}]} \newcommand{\RES}{\,\r[\text{ES}]} \newcommand{\rVaR}{\, \r[\text{VaR}]} \newcommand{\rES}{\, \r[\text{ES}]} \newcommand{\thresh}{\pi} \newcommand{\quantile}{\texttt{z}_{\thresh}} \newcommand{\gain}{\Delta V} \newcommand{\loss}{L} \newcommand{\lossvec}{\textbf{L}} \newcommand{\cdfnorm}{\Phi} \newcommand{\cdflosst}{F^\ell_\tau} \newcommand{\cdfgaint}{F^g_\tau} \newcommand{\cdfretst}{F^{\r}_\tau} \newcommand{\invcdflosst}{F_\tau^{\ell(-1)}} \newcommand{\invcdfgaint}{F_\tau^{g[-1]}} \newcommand{\invcdfretst}{F^{\r(-1)}_\tau} \newcommand{\mawt}{\theta} \newcommand{\DP}{\text{DP}} \newcommand{\n}{{(n)}} \newcommand{\0}{(0)} \newcommand{\1}{{(1)}} \newcommand{\2}{{(2)}} \newcommand{\3}{{(3)}} \newcommand{\4}{{(4)}} \newcommand{\5}{{(5)}} \newcommand{\bonds}{B} \newcommand{\fx}{S} \newcommand{\fxlog}{\texttt{s}} \newcommand{\meuro}{\text{\euro}} \newcommand{\usd}{\$} \newcommand{\for}{F} \newcommand{\forlog}{\texttt{f}} $$ # Fundamental Theorem of Asset Pricing #### Notation | notation | description | | --- | --- | | $\rx$ | excess return rate over the period | | $\rx[i]$ | arbitrary asset $i$ | | $\rx[p]$ | arbitrary portfolio $p$ | | $\rx[\tan]$ | tangency portfolio | | $\rx[\mkt]$ | market portfolio | | $\fbeta[i,j]$ | regression beta of $\rx[i]$ on $\rx[j]$ | #### Fundamental Theorem The First Fundamental Theorem of Asset Pricing guarantees... - there exists a probability measure, $\mathbb{P}^*$, - under which the expected return, $\E^*$, - the expected return of any portfolio\footnote{Of course, this includes portfolios of a single asset, $\r[i]$.} $\r[p]$ equals the risk-free rate: $$ \begin{aligned} \rf=& \E^*\left[\r[p]\right]\\ 0=& \E^*\left[\rx[p]\right] \end{aligned} $$ FINM 33000 covers this in detail. #### Pricing kernel The so-called risk-neutral probability measure defining $\E^*$, depends on a change of measure, $\radon$.\footnote{From FINM 33000, you'll recall this is the Radon-Nikodym derivative.} $$ \begin{aligned} \E^*\left[x\right] \equiv\, & \E\left[x\radon\right] \end{aligned} $$ Thus we can write the Fundamental Theorem as $$ \begin{aligned} \rf=& \E\left[\radon\r[p]\right] \label{eq:sdfpricing} \end{aligned} $$ #### A function of $\r[\tan]$ We state without proof that $\radon$ can be written as a linear function of $\r[\tan]$: $$ \begin{aligned} \radon = a + b\,\r[\tan] \end{aligned} $$ and that this results in\footnote{ We can write $\radon$ as, $$ \begin{aligned} \radon =& 1-\left(\r[\tan]-\E\left[\r[\tan]\right]\right)\frac{\E\left[\r[\tan]-\rf\right]}{\var\left[\r[\tan]\right]}\\ \end{aligned} $$ Then, subbing this into equation \eqref{eq:sdfpricing}, we get the desired equation, \eqref{eq:tanbeta}. } $$ \begin{aligned} \E\left[\r[p]\right]=& \rf + \frac{\cov\left[\r[p],\r[\tan]\right]}{\var\left[\r[\tan]\right]}\E\left[\r[\tan]-\rf\right]\notag\\ \E\left[\rx[p]\right] =& \fbeta[p,\tan]\E\left[\rx[\tan]\right] \label{eq:tanbeta} \end{aligned} $$ #### Outline - Beta-Factor Representation - Fundamental Theorem #### Sketch of proof To see this from the MV mathematics, recall the following formula.\footnote{We could write this formula in terms of excess return space $\wtx$ or return space, $\wt$. Recall that the tangency portfolio is the one portfolio on both MV and \MV frontiers. That is, $\wtx^{\tan}=\wt^{\tan}$.} $$ \begin{aligned} \wtx^{\tan} =& \Sigmamat^{-1}\muxvec\frac{1}{\gamma} \label{eq:wtxtan}\\ \gamma \equiv& \one'\Sigmamat^{-1}\muxvec \notag \end{aligned} $$ where $\gamma$ is just a scaling constant to ensure $\wtx^{\tan}$ adds to one, $\left(\wtx^{\tan}\right)'\one=1$. #### Sketch continued Using this formula, note that the covariance of any portfolio return, $\r[p]$, with $\r[\tan]$ is, $$ \begin{aligned} \cov\left[\rx[p],\rx[\tan]\right] =& \left(\wtx^p\right)'\Sigmamat\wtx^{\tan}\\ =& \left(\wtx^p\right)'\Sigmamat\Sigmamat^{-1}\muxvec\frac{1}{\gamma}\\ =& \left(\wtx^p\right)'\muxvec\frac{1}{\gamma}\\ =& \mu^p\frac{1}{\gamma} \end{aligned} $$ Thus, $$ \begin{aligned} \mux^p = \cov\left[\rx[p],\rx[\tan]\right]\gamma \label{eq:tanmv} \end{aligned} $$ #### Sketch finished It is easy to show\footnote{ Using the formula for $\wtx^{\tan}$, $$ \begin{aligned} \E\left[\rx[\tan]\right] = \muxvec'\Sigmamat^{-1}\muxvec\gamma \end{aligned} $$ And $$ \begin{aligned} \var\left[\r[\tan]\right] =& \left(\wtx^{\tan}\right)'\Sigmamat\wtx^{\tan}\\ =& \gamma\muxvec\Sigmamat^{-1}\Sigmamat\Sigmamat^{-1}\muxvec\gamma\\ =& \gamma\muxvec\Sigmamat^{-1}\muxvec\gamma\\ =& \gamma\E\left[\rx[\tan]\right] \end{aligned} $$ } $$ \begin{aligned} \gamma = \frac{\E\left[\rx[\tan]\right]}{\var\left[\rx[\tan]\right]} \end{aligned} $$ Thus Equation \eqref{eq:tanmv} is equivalent to \eqref{eq:tanbeta}. #### Factor pricing We then conclude that all asset returns have a factor-beta notation, $$ \begin{aligned} \E\left[\rx[i]\right]=& \; \fbeta[i,\tan]\; \E\left[\rx[\tan]\right] \label{eq:MV_LFM} \\[7pt] \fbeta[i,\tan]\equiv& \; \frac{\cov\left(\rx[i],\rx[\tan]\right)}{\var\left(\rx[\tan]\right)} \nonumber \end{aligned} $$ This will hold in sample exactly, by mathematical identity. #### Multiple factors Suppose we have a set of (excess) factor returns, $\rxvec[\facs]$, such that the tangency portfolio is a linear combination of them: (no need for linear combination to sum to one.) $$ \begin{aligned} \rx[\tan] = \boldsymbol{\omega}'\rxvec[\facs] \end{aligned} $$ Then, $$ \begin{aligned} \E\left[\rx[p]\right] = \left(\fbetavec[p,\facs]\right)'\E\left[\rxvec[\facs]\right] \end{aligned} $$ where $\fbetavec[p,\facs]$ is the vector of betas from a multivariate regression of $\rx[p]$ on $\rxvec[\facs]$. #### Generalization This relationship is not special to the tangency portfolio; it holds for any arbitrary MV portfolio, $\wt^{\mv}$. $$ \begin{aligned} \E\left[\rx[p]\right] =& \fbeta[p,\mv]\E\left[\rx[\mv]\right] \end{aligned} $$ Though this relationship works for any MV portfolio, we focus on the tangency portfolio, without loss of generality. #### Optimization Conditions The First Fundamental Theorem of Asset Pricing holds if---and only if---the mean-variance efficient portfolios are well defined. In our MV optimization language, the theorem relies on... - Convexity of the control space. - Convexity of the constraint and objective. - Well-defined solution. #### Payoff conditions \textcolor{structure}{Portfolio Formation} For any two security payoffs $\pay[i]$ and $\pay[j]$, the payoff $a\pay[i]+b\pay[j]$ is also an available security. \tcolor{Law of One Price} (LOOP) The pricing function, $\P\left(\cdot\right)$ is linear: $$ \begin{aligned} \P(w^i\,\pay[i] + w^j\,\pay[j])= w^i\,\P(\pay[i])+ w^j\,\P(\pay[j]) \end{aligned} $$ \tcolor{No Arbitrage} For every payoff $\pay$, if $$ \begin{aligned} Pr(\pay\ge 0)=1 \text{ and } Pr(\pay>0)>0 \Longrightarrow \P(\pay)>0. \end{aligned} $$ \label{slide:arb} #### Portfolio implications Portfolio Formation \bit \item if $\wtx^i, \wtx^j$ are permissible, \item $\wtx^k = \ex{\delta}\wtx^i + (1-\ex{\delta})\wtx^j$ is permissible for any $\ex{\delta}\in(-\infty,\infty)$. \eit LOOP \bit \item $\ex{r}^k = \ex{\delta}\ex{r}^i + (1-\ex{\delta})\ex{r}^j$ \item More generally, *Suppose $\wtx$ is a vector of portfolio weights. If $\wtx'\boldsymbol{\pay*=0$ for every state, then $\wtx'\P\left(\boldsymbol{\pay}\right)=0$.} \eit No Arbitrage \bit \item Non-trivial, limited-liability, portfolios have well-defined returns. \eit #### Meaning Portfolio Formation \bit \item Short-selling and leverage are allowed. \eit LOOP \bit \item Prices and returns equal the sum of their parts. \eit No arbitrage \bit \item If a portfolio has cash-flow in any contingency, (without incurring liabilities in any contingencies,) then the portfolio must have a positive price. \eit #### Outline - Beta-Factor Representation - Fundamental Theorem #### Necessity of risk-free rate? We derived \eqref{eq:tanbeta} in terms of excess returns, which assumes the existence of a risk-free rate. - Without a risk-free rate, the same arguments apply, simply adjusting the equations to have 0 in place of $\rf$. - In deriving \eqref{eq:tanmv}, the argument holds simply changing $\left(\wtx^{\tan}, \muxvec,\rx\right)$ to $\left(\wt^{\tan}, \muvec, \r\right)$. #### Modeling vs Estimation This factor-beta notation seems to give us a model for all mean returns. But it depends on knowing $\rx[\tan]$ via - a theoretical model for $\rx[\tan]$. - direct empirical estimation of $\rx[\tan]$. One might consider using equation \eqref{eq:wtxtan}, for direct estimation, but this does not work in practice. #### Circularity in direct estimation Suppose we want to use the factor pricing model to estimate $\mu^i$. - The estimation of $\rx[\tan]$ via Equation \eqref{eq:wtxtan} requires $\muvec$. $$ \begin{aligned} \wtx^{\tan} = \covmat^{-1}\muvec\frac{1}{\phi} \end{aligned} $$ - But $\wtx^{\tan}$ should consider all available assets, meaning $\muvec$ should includes $\mu^i$ itself! - So this cannot be a way to estimate $\mu^i$. #### Imprecision in direct estimaiton Suppose we instead estimate $\rx[\tan]$ from one set of assets and then use it to estimate the mean return for some other asset, $i$. - Still, direct estimation does not work well. - We will be ignoring the weight of $\wtx^{\tan}$ that should be in $i$. - Worse, the poor conditioning of $\covmat$ means that inverting it will greatly magnify the (substantial!) estimation errors in $\muvec$. Thus, it is not practical to statistically extract an \MV portfolio to use in the linear pricing formula above. #### Linear Factor Pricing Models **\textcolor{structure**{Linear factor pricing models (LFPM)}} are assertions about the identity of the tangency portfolio. - This avoids the problems of direct estimation. - But it relies on the assumption about the identity of the tangency portfolio (or some other mean-variance portfolio.) #### Allocation vs Pricing The theory does not assume investors allocate to this assumed MV portfolio. - It assumes the portfolio is MV for the purposes of pricing expected returns. - If we additionally assume investors prefer MV portfolios, then this portfolio will both price securities and be the equilibrium allocation.