## Short thoughts on Efficiency

Eugene Fama, Nobel laurate in 2013 for his empirical analysis of asset prices Source: http://www.nobelprize.org

The Efficient Market Hypothesis (EMH): The holy grail of the theory of finance, establishes that asset prices should reflect all information about future cash flows. But why should they? Who are we to decide how prices must behave? It reminds me of the Einstein-Bohr debate on the nature of quantum mechanics:

Einstein: God does not play with dice

Bohr: Stop telling God what to do

Should we be like Einstein, and impose a normative explanation on asset prices? Or should we be like Bohr, and just accept that if theories do not fit reality, we need a better explanation?

A not so formal Introduction to the EMH

Let me analyse how we think assets behave in a frictionless environment using a simple model .  Consider a representative agent that lives for two periods (present and future) $t=0,1$. The agent has some endowment (wages, dividends, etc) in each period that he/she uses to consume or invest. To invest, there is an asset that pays a random amount $X$ in the future. The budget constraint in period $0$ is:

$c_0=e_0-\eta p$

Where $c_0$ is present consumption, $e_0$ is present endowment, $p$ is the unitary price of the investment, and $\eta$ is the units of the asset bought (if $\eta<0$ the investor shorts the asset). Also, the constraint in period $t=1$ is:

$c_1=e_1+\eta X$

How much units to buy (sell)? The representative agent solves the following two period maximization problem:

$\max_\eta u(c_0)+\beta \mathbb{E}(u(c_1))$

Subject to the above constraints on $c_0$ and $c_1$. We impose the standard assumptions of risk aversion and non satiation on the utility function $u()$. The factor $\beta$ represents the impatience of the consumer and the way he/she values future consumption. A high $\beta$ means that the investor values present and future consumption almost equally, while a low $\beta$ means that the investor values more present consumption. Replacing the constraints in the objective function leads us to an unconstrained optimization problem with first order condition equal to:

$u'(c_0)p=\mathbb{E}(\beta u'(c_1) X)$ The marginal utility of consuming one extra unit in the future equals the marginal desutility of buying one marginal unit in the present. Rearranging terms give us the following expression for prices:

$p=\mathbb{E}(\frac{\beta u'(c_1)}{u'(c_0)}X)$

Defining $M=\beta \frac{u'(c_1)}{u'(c_0)}$ as the Stochastic Discount Factor (for short SDF which is a random variable as well, since consumption in the future is random)  give us one of the most important equations in finance

$p=\mathbb{E}(MX)$

According to this model, the price of any asset is the expected value of its cash flows times the stochastic discount factor. We can rearrange the equation to define a formula not for prices but for returns. Defining $R=\frac{X}{p}$ as the gross return, and using the formula for covariance we get the following expression:

$1=\mathbb{E}(MR)$

$1=Cov(M,R)+\mathbb{E}(M) \mathbb{E}(R)$

We can define the return on a risk-free asset (An asset thay pays a return $R_f$ independent on the state of nature) as

$1=\mathbb{E}(M)R_F \rightarrow R_F=\frac{1}{\mathbb{E}(M)}$

Using the above definition:

$R_f=\frac{Cov(M,R)}{\mathbb{E}(M)}+\mathbb{E}(R)$

Finally:

$\mathbb{E}(R)-R_f=\frac{Cov(M,R)}{Var(M)}(\frac{-Var(M)}{\mathbb{E}(M)})$

Defining $\beta=\frac{Cov(M,R)}{Var(M)}$ as the “sensitivity” to the stochastic discount factor, and $\lambda=\frac{-Var(M)}{\mathbb{E}(M)}$ as the market price of ris we get the following expression:

$\mathbb{E}(R)-R_f=\beta \lambda$

Which implies that the excess return on any asset (the return of borrowing at the risk free rate and investing in the asset) is equal to the product of the sensitivity of the asset to the stochastic discount factor and the so called market price of risk. If the asset pays more, when the stochastic discount factor is high (Low consumption) the asset will have a lower return. Why? Well for a risk averse agent having an extra income when things are bad, is more valuable than having it when things are good.

Testing Market Efficiency

What are the implications if markets are efficient? Well the above equation give us the answer. If markets are efficient, running a regression of the form

$R-R_f=\alpha+\beta \lambda$ must lead to an $\alpha$ that is undistinguishable from zero. Technically, the distribution of all alphas must be centred near zero. However there is one problem with this methodology. We have no idea about what really is the Stochastic discount factor (Measure of bad times). Every market hypothesis test is a joint test of the EMH and a specific asset pricing model.

An asset pricing model tries to model the stochastic discount factor using a measure of bad times. The CAPM, Fama and French 3-5 factor model, Momentum strategy, etc are ways to model the stochastic discount factor to explain the cross section of expected returns.

Off course just running an OLS regression just wont do the job. Why? well there are several reasons, there is a lot of cross sectional variation between assets, and we are using as an independent variable something that we can not even observe (the $\beta$ in the regression). There are some ways to deal with this, the most common ways are the so called Fama-Macbeth regression, and the GRS test (Gibbons, Ross, Shanken, 1989) that provides a joint hypothesis test on those $\alpha$‘s.

My thoughts

In my opinion, this methodology does not work properly. As explained in Garleanu and Pedersen (2011) and Imbett (2016) (A working paper I promise to upload shortly :p), in the presence of market frictions the methodology fails. In the presence of capital or liquidity requirements (requirements on the composition of assets or liabilities) assets exhibit $\alpha$‘s different from zero in equilibrium even if the market is efficient.

Suppose there is a liquidity requirement for investing in the asset. For every euro borrowed at the risk free rate investors must borrow an extra amount $\epsilon$ just to satisfy the liquidity requirement. What do you need to invest in the risky asset? Borrow $\latex (1+\epsilon)$ to invest. What is the payoff of this strategy?

$R-R_f(1+\epsilon)$

In equilibrium ( I skip the derivation)

$R-R_f=\epsilon R_f+\beta \lambda$

Which clearly give us an alpha different from zero. This is also the case when investors have margin requirements, or default risk. In the presence of margin requirements, investors are not able to leverage their positions to achieve higher expected returns. This increases the demand for risky assets (lowering their alphas below their natural levels) and decreases the demand for safer assets.

Conclusions

Well, we do not have yet a robust methodology to test if markets are efficient. Our tests are based on frictionless models. However as we discussed before, including real frictions to our models (something that ironically was not done until some years ago) has important implications on the distribution of the $\alpha$‘s.

References

N. Garleanu, L.H. Pedersen (2011) Margin-based asset pricing and deviations from the law of one price . Rev. Financial Stud., 24 (6), pp. 1980–2022

Imbett, J.F. (2016) Asset pricing and reserve requirements, Working Paper

# What is the relation between the Black Scholes equation and the Schrodinger equation?

### Parental Advisor: Explicit Math derivations

1) The Schrodinger Equation (Skip to section 2 if you are not into physics)

Let me derive step by step the Schrodinger equation. Schrodinger based most of his work on the special relativity theory of Albert Einstein, and the Maxwell’s Equations. In order to derive the equation we must revise some electromagnetics and relativity.

i) The four Maxwell Equations explain how electricity and magnetism behave. These equations are:

$\nabla\cdot E=\frac{\rho}{\epsilon_0}$ (The electric field that leaves a volume is proportional to the density of the charge inside)

$\nabla \cdot B=0$ (There are no magnetic monopoles)

$\nabla \times E=-\frac{\partial B}{\partial t}$ (The voltage accumulated around a closed circuit is proportional to the time rate of change of the magnetic flux it encloses.)

$\nabla \times B=\mu_0 (J+\epsilon_0 \frac{\partial E}{\partial t})$ (Electric currents are proportional to the magnetic field circulating about the are they pierce)

Now, let us see how these equations behave in the vacuum of space, where there is no charge permeability or diffusion medium.

$\nabla\cdot E=0$

$\nabla \cdot B=0$

$\nabla \times E=-\frac{\partial B}{\partial t}$

$\nabla \times B=\mu_0 \epsilon_0 \frac{\partial E}{\partial t}$

Applying the curl operator $\nabla \times$ to the third equation:

$\nabla \times (\nabla \times E)=-\frac{\partial \nabla \times B}{\partial t}=-\mu_0 \epsilon_0 \frac{\partial ^2}{\partial t^2}E$

Using  the following theorem:

$\nabla \times \nabla \times E =-\nabla^2 E+\nabla \cdot \nabla \cdot E$

Since $\nabla \cdot E=0$

$\nabla^2 E=\mu_0 \epsilon_0 \frac{\partial^2 E}{\partial t^2}E$

This is a three dimensional wave equation, assuming the wave only moves in one dimension gives us:

$\frac{\partial^2 E}{\partial x^2}=\mu_0 \epsilon_0 \frac{\partial^2 E}{\partial t^2}E$

Wave equations usually can be written in the form

$\frac{\partial^2 E}{\partial x^2}-\frac{1}{v^2} \frac{\partial^2 E}{\partial t^2}=0$

Where $v=\frac{1}{\sqrt{\mu_0 \epsilon_0}}=c$ is the speed of light in the vacuum.

Let us move now to some of the work of Albert Einstein, and derive the famous $E=mc^2$ formula formally.

The starting point is to usethe Lorentz transformation to understand the relativistic mass of an object. This means, the mass of the object when its speed approaches the speed of light. The relation between the relativistic mass $m$ and its rest or invariant mass $m_0$ is given by:

$m=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}m_0=\gamma m_0$ where $\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ is sometimes called the Lorentz factor. Let’s compute now the total energy of an object with rest mass $m_0$ and travelling at a speed $v$:

The kinetic energy of an object can be seen as the integral of the force with respect to distance:

$K =\int_0^s F ds$

Using Newton’s second law of motion $F=\frac{d mv}{dt}$

$K =\int_0^s \frac{d mv}{dt}ds$

Solving the integral

$K=\int_0^{mv}vd(mv)=\int_0^{mv}vd(\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}v)$

Integrating by parts

$K=\frac{m_0 v^2}{\sqrt{1-\frac{v^2}{c^2}}}-m_0 \int_0^{v}\frac{v dv}{\sqrt{1-\frac{v^2}{c^2}}}$

$K=\frac{m_0 v^2}{\sqrt{1-\frac{v^2}{c^2}}}+[m_0c^2\frac{m_0 c}{\sqrt{1-\frac{v^2}{c^2}}}]_0^v$

$K=\frac{m_0 c^2}{\sqrt{1-\frac{v^2}{c^2}}}-m_0c^2$

$K=mc^2-m_0c^2$

Since the total energy of the object is its rest and kinetic energy, we have that total energy is equal to:

$R+K=E=m_0c^2+mc^2-m_0c^2=mc^2$

Now, before continuing we need a relation between energy and momentum. Momentum squared equals:

$p^2=(mv)^2=m^2 v^2=m_0^2 \frac{v^2}{1-\frac{v^2}{c^2}}$

Which means that $\frac{v^2}{c^2}=1-\frac{m_0^2 v^2}{p^2}$

Replacing in our $E=mc^2$ formula leads us to:

$E=m_0 c^2 \sqrt{1+(\frac{p}{m_0 c})^2}$

$E^2=m_0^2 c^4 +\frac{m_0^2 c^4 p^2 }{m_0^2 c^2}=m_0^2 c^4 +p^2 c^2$

This is one of the most bad ass equations in physics. Look that the term $E^2$ actually has a positive and a negative root. This means that the equation also holds for particles with negative energies. Starting with this observation Paul Dirac predicted the existence of antiparticles decades before their discovery.

Now that we have derived both an equation for electromagnetic waves, the formula for the total energy of a particle, and the relationship between energy and momentum, we can  continue with the Schrodinger equation.

Recall the wave equation

$\frac{\partial^2 E}{\partial x^2}-\frac{1}{c^2} \frac{\partial^2 E}{\partial t^2}=0$

The general solution of this equation has the following form:

$E(x,t)=E_0 e^{i(kx-\omega t)}$

Where $k=\frac{2\pi}{\lambda}$ is the spatial frequency and $\omega=2\pi \upsilon$ is the temporal frequency of the  wave. Replacing this equation into the differential equation gives us:

$(\frac{\partial^2}{\partial x^2}-\frac{1}{c^2}\frac{\partial^2}{\partial t^2})E_0 e^{i(kx-\omega t)}=0$

$[-k^2+\frac{1}{c^2} \frac{\partial \omega^2}{\partial t^2}]E_0e^{i(kx-\omega t)}=0$

This means that:

$-k^2+\frac{\omega^2}{c^2}=0$

$k=\frac{\omega}{c}$ which relates both the spatial and temporal frequencies of the wave and the speed of light.

An important result from Max Planck was that the energy and the frequency of photons are related. Specially

$\mathbb{E}=\hbar \omega$ is the energy of the particle and $p=\hbar k$ the momentum, where $\hbar$ is the normalized Planck’s constant. Here I will use $\mathbb{E}$ to denote the energy of the particle and $E$ the electromagnetic field. Substituting into the equations

$-\frac{1}{\hbar^2}(p^2-\frac{\mathbb{E}^2}{c^2})E_0e^{\frac{i}{\hbar}(px-\mathbb{E}t)}=0$

Which leads to $\mathbb{E}^2=\rho^2 c^2$ which is offcourse the relativistic total energy $\mathbb{E}^2=m^2 c^4+\rho^2 c^2$ for massless particles like the photon.

Let’s try now to understand how particles with mass (e.g. electrons, positrons, neutrinos) move with out working with electromagnetic waves any more. Instead let’s just define a wave function $\Psi(x,t)$ that will lead us to the following differential equation:

$-\frac{1}{\hbar^2}(p^2 -\frac{\mathbb{E}^2}{c^2}+m^2 c^2)\Psi e^{\frac{i}{\hbar}(px-\mathbb{E}t)}=0$

With general wave equation

$\Psi(x,t)=\Psi_0e^{\frac{i}{\hbar}(px-\mathbb{E}t)}$

Normalized to unit probability

$\int \Psi^*\Psi dx=1$

The equation satisfying these properties is known as the Klein-Gordon equation (extending it to more than one dimensions)

$\nabla^2 \psi-\frac{m^2 c^2}{\hbar^2}\Psi=\frac{1}{c^2} \frac{\partial^2 \Psi}{\partial t^2}$

This a relativistic equation, Schrodinger’s equation is not. In order to make it non relativistic let’s approximate $\mathbb{E}^2$ as follows:

$\mathbb{E}=mc^2 \sqrt{1+\frac{p^2}{m^2 c^2}}$

$\mathbb{E}\approx mc^2 (1+\frac{1}{2}\frac{p^2}{m^2 c^2})$

$\mathbb{E} \approx mc^2 +\frac{p^2}{2m}$

Recall that the last term is nothing but the Kinetic energy of the particle. We can rewrite the wave equation as:

$\Psi(x,t)=e^{\frac{i}{\hbar}(mc^2 t)}\Psi_0e^{\frac{i}{\hbar}(px - Kt)}$

The term $\Psi_0e^{\frac{i}{\hbar}(px - Kt)}$  does not oscillate as fast as the first one since the speed of the particle is not close to $c$. We can call this term $\phi$ and see that:

$\Psi(x,t)=e^{\frac{i}{\hbar}(mc^2 t)} \phi$

The first and second derivatives with respect to time are:

$\frac{\partial \Psi}{\partial t}=-\frac{i}{\hbar}mc^2e^{-\frac{i}{\hbar}(mc^2 t)}\phi+e^{-\frac{i}{\hbar}(mc^2 t)}\frac{\partial \phi}{\partial t}$

$\frac{\partial^2 \Psi}{\partial t^2}=(-\frac{m^2 c^4}{\hbar^2}e^{-\frac{i}{\hbar}(mc^2 t)}\phi-\frac{2i}{\hbar}mc^2 e^{-\frac{i}{\hbar}(mc^2 t)}\frac{\partial \phi}{\partial t})+e^{-\frac{i}{\hbar}(mc^2 t)}\frac{\partial^2 \phi}{\partial t^2}$

The first term in brackets is large and the last term is small. We keep the large terms and discard the small one. The Klein Gordon equation is now:

$e^{-\frac{i}{\hbar}(mc^2 t)}[\frac{\partial^2}{\partial x^2}+\frac{2im}{\hbar}\frac{\partial}{\partial t}]\phi=0$

Which leads to the well known (in this case one-dimensional) Schrodinger equation

$-\frac{\hbar^2}{2m}\frac{\partial^2\phi}{\partial x^2}=i\hbar \frac{\partial \phi}{\partial t}$

Which can be written also as:

$\frac{\partial}{\partial t}V=HV$

Where $H$ is the Hamiltonian operator.

2) The Black Scholes Equation

The Black Scholes Partial Differential Equation (BSPDE) is an equation that explains how the value of any derivative must behave under no arbitrage opportunities. In their “seminal” paper (The Pricing of Options and Corporate Liabilities (1973) Journal of Political Economy) Black and Scholes derive the following equation for a derivative with value $V(S,t)$ that depends on a underlying asset $S$ and time $t$.

$\frac{\partial V}{\partial t}+rS\frac{\partial V}{\partial S}+\frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}-rV=0$

Im going to skip the derivation of this equation since it is widely known. However let’s see what is its relation with the Schrodinger’s equation. Organizing the time derivative on one side:

$\frac{\partial V}{\partial t}=-\frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}-rS\frac{\partial V}{\partial S}+rV$

Now, we can complete the square on the right hand side of the equation

$\frac{\partial V}{\partial t}=-\frac{\sigma^2}{2}[S\frac{\partial}{\partial S}-\frac{1}{2}(1-\frac{2r}{\sigma^2})]^2V+\frac{\sigma^2}{8}(1+\frac{2r}{\sigma^2})^2V$

Now let’s make some transformations. Define $p=-i \sigma [\frac{\partial}{\partial x}-\frac{1}{2}(1-\frac{2r}{\sigma^2})]$ where $x=log(S)$ and $U=\frac{\sigma^2}{8}(1+\frac{2r}{\sigma^2})^2$ the Black Scholes equation can be written as:

$\frac{\partial}{\partial t}V=\frac{p^2}{2}V+UV=HV$

The equation is a particular case of the Schrodinger equation with Hamiltonian equal to $\frac{p^2}{2}+U$. Analyzing in more detail the equation we see that the term $\sigma^2$ works as a Planck’s constant relating frequency with energy. Also the Black Scholes equation is like a Schrodinger equation for imaginary time since every solution lies on the Real Line.

We can understand prices as particles in which their position can not be known in advance. Instead, we assign to them a probability function that give us the probability of finding the particle in some interval tomorrow.

Some references:

http://arxiv.org/pdf/physics/0610121.pdf

https://phorgyphynance.wordpress.com/2008/06/03/black-scholes-and-schrodinger/

http://physics.stackexchange.com/questions/60242/relation-between-black-scholes-equation-and-quantum-mechanics

## When Physics meets Finance (Part I)

For better or for worse, Mathematics has always been a tool of modern Economics and Finance. It is not surprising that an important fraction of those who pursue an academic career in Economics come from highly mathematical disciplines. A few days ago I stumbled with a paper that describes arbitrage opportunities as gauge fields. For some of you the analogy may seem senseless, but for me it is beautiful. In this post I develop a small example of how physics ideas can be extended to Finance.

Before any Finance, let’s talk about  fields (not the one  with grass in the classic Windows  wallpaper).   A vector field consists in assigning a vector to every point in a space. Gravity can be seen as a field. Wherever you are, you always have a large arrow pulling you downwards to the centre of the earth. The length of this arrow (the magnitude) decreases as you move away at a ratio proportional to the inverse of the distance squared.

In a more modern framework, gravity can be seen as a “curvature” in space-time. This curvature induces a potential (the ability to move) which increases as objects approach to massive bodies.

At this point  you might be thinking, Where is the Economics behind all this Physics? The idea of curvature or force fields has a beautiful and simple correspondence in Finance.

Money across time and currencies

We all know 100 USD are not equivalent to 100 EUR, and 100 USD today are not equivalent to 100 USD in a year from now. Exchange rates allow us to convert between currencies, and discount factors (interest rates) between time. Consider an economy in which time is continuous, and there exists a measure of currencies of size $2\pi$. We can describe money (in some currency) at different points in time as all points in a ray starting at the origin $t=0$. Monetary values in this economy are points inside a disc of radius equal to the horizon time. The inclination of each ray corresponds to the currency analysed.

If exchange rates and interest rates (in each currency) are chosen appropriately, every point in the disk is equivalent to each other. In economic terms it means that there are no arbitrage opportunities. As an example, suppose the interest rate in USD is 1% and the interest rate in EUR is 2%. Let’s also suppose that the exchange rate at time $t=0$ is 1.1 EUR/USD and at $t=T$ is 1 EUR/USD.

You can obtain an arbitrage profit as follows:

Borrow $X$ USD at a 1% interest rate at $t=0$. Exchange those USD into euros at a rate of 1.1 EUR/USD, to get 1.1 $X$ EUR. Invest those euros at a 2% interest rate which will give you $1.1 X e^{0.02T}$ EUR in the future. Now is time to pay your credit, exchange your euros into dollars to obtain $1.1 Xe^{0.02T}$ USD. The amount you owe to the bank is $X e^{0.01T}$

Your profit will be: $1.1Xe^{0.02T}-X e^{0.01T}>0$

If there are no arbitrage opportunities, every point in the disk is equivalent to each other. In mathematical words any point in the disk is invariant to rotations and translations, in physics words every point is gauge invariant.

Arbitrage can be seen as a vector field or curvature in which we assign to every point a vector. Each point has a vector that points to another point in the disc. This pair of points (different interest rates and exchange rates) will give a potential arbitrage opportunity. Arbitrage can be seen as a curvature in the 2D disk. The degree of curvature measures the strength of the arbitrage opportunity and therefore the size of the potential investor’s profit. As time goes by, investors’ pressure will correct any price anomalies.

In conclusion, this is an interesting and peculiar way to visualise arbitrage opportunities. Not only we can visualize the direction of the potential arbitrage opportunity, but also predict the speed at which investors exploit this potential. If the curvature is small, the incentives to profit from arbitrage opportunities decrease due to transaction costs. If the curvature is small, profits will compensate transaction costs and more investors are willing to profit from these opportunities.

I believe in the cooperation among disciplines. If a tool works, why don’t borrow it?

Juan Imbett

## ¿Podemos usar el CAPM en Colombia?

El Capital Asset Pricing Model de Sharpe (1964) y Lintner (1969) es quizás la fórmula mas usada por profesionales en finanzas a la hora de estimar el retorno esperado de un activo. En este post realizo un análisis superficial de las ventajas y desventajas de usar este modelo en la práctica, de sus extensiones y distorsiones. Hago énfasis en los malos hábitos de los profesionales y como solucionarlos.

Empecemos donde empezó Sharpe y Lintner. En un mundo en el cual inversionistas toman decisiones a la Markowitz (1952), maximizando retornos esperados y minimizando varianzas. En este mundo la frontera eficiente se genera a partir de combinaciones convexas entre un activo libre de riesgo y el portafolio que genera el mayor Sharpe ratio  $\frac{R_i-r_f}{\sigma}$. Si suponemos que así se comportan los inversionistas,  la demanda de activos será proporcional a su importancia (weights) en el portafolio tangente. En equilibrio la oferta agregada de activos en el mercado tendrá la misma importancia (weights). El portafolio tangente será el portafolio de mercado. Un portafolio en el cual la idiosincrasia de cualquier activo no es significativa.

Omito la derivación por brevedad, en equilibrio el retorno esperado de todo activo será:

$E[R_i]=r_f + \beta (E[R_m]-r_f)$

Donde $r_f$ es la tasa libre de riesgo, $\beta=\frac{Cov(R_i, R_m)}{Var(R_m)}$ es la sensibilidad del activo al portafolio de mercado, y $E[R_m]-r_f$ se conoce como la Equity Risk Premium (ERP). Supongamos por ahora que el modelo funciona en la práctica, ¿Qué información usamos para estimar cada parámetro? ¿Qué suelen hacer los profesionales en inversión?

Tasa libre de riesgo

Forma correcta: Usar una tasa libre de riesgo en la moneda y el horizonte en el que se analizan los flujos de caja. Esta tasa se puede calcular a partir de bonos con calificación AAA emitidos por el país que emite la moneda. La deuda en moneda local es menos riesgosa, ya que el país puede recolectar ingresos en esa moneda e incluso imprimir dinero para pagar la deuda.

En caso que no existan bonos con calificación AAA en la misma moneda, la forma correcta de determinar la tasa de riesgo consiste en restarle un proxy del riesgo de default del pais a la tasa calculada. Un proxy comúnmente usado es el spread en los Credit Default Swaps en los bonos.

$r_f=r^{Local}-S$ donde $S$ es el spread del CDS, $r^{Local}$  es la tasa de retorno en los bonos en moneda local.

Forma Incorrecta:

Usar los TES para calcular la tasa libre de riesgo. El gobierno colombiano NO tiene calificación AAA ni en bonos emitidos en dólares o en pesos.

Forma correcta:  Usar un portafolio diversificado que represente el universo de inversiones para todo inversionista. En Economías desarrolladas equivale a índices diversificados y representativos (e.g. S&P 500). Cuando tratamos con economías emergentes usamos como base la ERP de una economía madura y sumamos una prima que refleja el riesgo extra que un inversionista afronta al invertir en activos en ese país. El riesgo país suele asociarse con el spread que tienen los bonos de ese país sobre bonos con calificación AAA. De esta forma el ERP en una economía emergente equivale a:

$ERP=E[R_m]-r_f+C$ donde $C$ refleja el riesgo de invertir marginalmente en la economía emergente. Ver detalles aquí

Forma incorrecta: Nunca se debe usar el COLCAP, no es un índice ni diversificado ni representativo. La idiosincrasia en las decisiones de Ecopetrol o ISAGEN afecta significativamente el índice.

Betas

Este es quizás el paso que mejor realizan los profesionales. Si hay información histórica el$\beta$ se puede estimar a partir de una regresión de la siguiente forma:

$R_t=a+bR_{mt}+\epsilon_t$

Y si no hay información pública, apalancar y desapalancar betas de compañías similares de la forma habitual.

Mean-Variance Analysis

Supongamos que podemos estimar correctamente los parámetros del modelo. ¿Vale la pena usarlo? Volvamos a las bases del modelo, los inversionistas miden el riesgo a partir de la varianza de los retornos tratando de maximizar su valor esperado. Ahora bien, ¿Es esto cierto teóricamente? Solo hay dos casos en los que solo los dos momentos de la distribución importan a la hora de tomar decisiones.

1. Los retornos siguen una distribución normal

¿Qué pasa en la realidad? Ninguna de las dos es cierta. Los retornos de las acciones por ejemplo muestran asimetría (The Leverage effect) y exceso de kurtosis. Lo primero sucede dado que al disminuir el valor de los activos, la deuda permanece constante y por lo tanto el equity se vuelve mas riesgoso. El segundo indica que la probabilidad de eventos extremos es mayor de lo que una distribución normal sugiere. (e.g. Las crisis financieras son mas comunes de lo que se piensa)

Por otro lado una función de utilidad cuadrática sugiere que la pendiente es negativa después de cierto nivel de consumo. Algo que no se observa en la realidad, las preferencias suelen ser monotónicas.

Queda por hacer un análisis estadístico a lo Fama and Mcbeth (1973) para ver que tal se comporta el CAPM valorando acciones en la BVC y América Latina. Hay mucho por trabajar en esta area, los mercados emergentes funcionan de manera distinta. Y el portafolio de mercado no es el único riesgo sistemático que afecta el comportamiento de los retornos.

Juan Felipe Imbett

Referencias:

Fama, Eugene F.; MacBeth, James D. (1973). “Risk, Return, and Equilibrium: Empirical Tests”. Journal of Political Economy 81 (3): 607–636

Lintner, J. (1969). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets: A reply. The Review of Economics and Statistics, 51(2):222–24.

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