# Tag Info

9

In the below figure, CDF $F(\cdot)$ is first-order stochastically dominated by $G(\cdot)$. But $X_1$ and $X_2$ fall within the support of both distributions. So it would be possible to draw $X_1$ from $F$ and $X_2$ from $G$, or to draw $X_2$ from $F$ and $X_1$ from $G$. More generally, if $X_G$ is a draw from $G$ and $X_F$ is a draw from $F$ then $X_F-X_G$ ...

8

The efficient market hypothesis does not imply that there are no patterns! As Eugene Fama pointed out decades ago, any test of market efficiency is a joint test of market efficiency and an asset pricing model. The EMH on its own is not a testable theory. If I understand your statement properly, you're claiming that forecasting variance would violate market ...

7

Quadratic utility is given by $$u(w) = w - b w^2$$ which has derivative $$u'(w) = 1- 2b w$$ such that for high levels of $w, u'(w)<0$. That is, the utility is not everywhere increasing. This may be weird because even people with high wealth should prefer more to less. The second derivative is $$u'(w) = -2b$$ such that absolute risk aversion is $$\frac{- u'... 7 Generally speaking no. You wouldn't be able to distinguish re-balancing for risk aversion reasons from re-balancing motivated by changes in expected returns or the co-variance of returns. Consider the simple case of a household periodically re-balancing their investments in across both a fixed index fund and an equity index fund. The econometrician sees ... 5 All you need for this particular question is the following. Let \mathbf{X} be a T \times K matrix, \mathbf{w} a K-dimensional vector and \mathbf{y} a T-dimensional vector, then$$ \begin{eqnarray*} \frac{\partial \mathbf{w}^{\prime}\mathbf{X}^{\prime}\mathbf{y}}{\partial \mathbf{w}}&=& \mathbf{X}^{\prime}\mathbf{y}\\ \frac{\partial \mathbf{w}...

4

In mean-variance optimisation I have typically seen the below quadratic utility function where $๐ธ[๐ ]$ is the expected return (or mean return) of a possible portfolio, $\sigma^{2}$ is the return variance of that portfolio and $A$ is a parameter that the determines the sensitivity to variance. $$๐=๐ธ[๐ ]โ\frac{1}{2}A\sigma^{2}$$ As an aside, maximising % ...

3

Say you have a portfolio with returns described by a random variable X. Call the lowest possible realization of X: xmin. If you take a levered position in that portfolio with leverage A and financing cost r your returns are r(A-1)+AX. There will exist a value of A no larger than 1/xmin where when you get the worst return of X and the levered portfolio ...

3

$\newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\F}{\mathbb{F}} \newcommand{\C}{\mathbb{C}} \newcommand{\E}{\mathbb{E}} %short command for inseting abbreviated "such that" in a math environment \newcommand{\st}{\text{ s.t. }} %text in a math environment "as" \newcommand{\as}{\text{ as }} %various referencing commands \newcommand{\rref}[1]... 2 Here's how you would account for risk aversion. First of all, I think this problem is usually set up using weights as the variables (see here http://www.math.ku.dk/~rolf/CT_FinOpt.pdf, page 141). If you want to use dollar weights, you will have to transform the result. The optimization problem is: $$\max_w \quad w'\mu โ\frac{\gamma}{2} w'\Sigma w$$ ... 2 Welcome to EC.SE! Hopefully this will help. I don't think this characterization is correct. These are not opposing theories. Modern portfolio theory is based on the idea that people choose portfolios to maximize their utility. After that, it depends on the specification of utility. As a simple example, a person with a quadratic utility function will choose ... 2 This seems to be specific to the CFA exam and is a badly formulated question. First, an indifference curve for some fixed utility level can be viewed as a function mapping$\sigma$to$\mu$. At any value of$\sigma$, this function has a slope (which is given by$\sigma A$). No economist would call this a "slope coefficient" in this context, as this ... 2 It would seem that both options are correct given the specific mean-variance utility function. Use$\mu$to denote the expected value. In the$(\sigma,\mu)$-plain, an indifference curve representing a particular utility level$\overline U$is given by $$\overline U=\mu-\frac12 A\sigma^2.$$ Applying implicit differentiation, we can ... 2 One reason why you might want to do this, and perhaps is the motivation for this example, is that it gives you a simple example for thinking about option pricing. Under some assumptions, the value of any option can be replicated by continuously trading in the underlying asset. Here, you can think of the$\Delta$shares that you have as your replicating ... 1 Using$\mathbb E$for the expected value symbol,$E_R$and$V_R$for the mean and the avriance of returns$R\$, for a utility function of the form $$U(R) = \ln(1+R)$$ the second-order Taylor expansion gives (ignoring the remainder) $$U(R) \approx \ln(1+E_R) + \frac{1}{1+E_R} (R- E_R) -\frac{1}{2(1+E_R)^2}(R-E_R)^2.$$ Then $$\mathbb E[U(R)] \approx E_R + 0 - \... 1 You should be fine to use negative interest rates for the risk-free rate when calculating a Sharpe ratio. If rates are negative, they are negative. That does not change the calculation of excess returns. One point, however: EURIBOR and EONIA are interbank rates and thus are not risk-free but have a small credit spread above risk-free rates. The latest ECB ... 1 The latest Freakonomics podcast topic Stupidest money may shed some light. I just quote part of the conversation from John Bogle: The markets are highly efficient โ although, importantly, not perfectly efficient. Sometimes theyโre very efficient and sometimes theyโre not. Itโs hard for we poor souls on Earth to know which is which It is a gamble ... 1 Let me begin with noting that models such as the CAPM have been extensively falsified, beginning with Mandelbrot in 1963 and ending with the Fama-MacBeth testing in 1973. Other falsifications followed, but really, it has been a zombie theory since the early seventies. The difficulty is that the falsification has been like the Michaelson-Morley experiments ... 1 You could say the return on Johnson & Johnson is a random variable R_J with expected value \mu^{\,}_J and variance E[(R_P-\mu^{\,}_P)^2] = \sigma^2_J, and on Ford is R_F with expected value \mu^{\,}_F and variance E[(R_F-\mu^{\,}_F)^2] = \sigma^2_F R_J and R_F perfectly correlated means \frac{\mathbb E[(R_J-\mu^{\,}_J)(R_F-\mu^{\,}_F)]}... 1 Why are log returns used in finance? It really is about compactness when devising models. The mathematical property of logarithms$$log(S_{t+n}/S_t)=log(S_{t+n})-log(S_t)$$makes log returns more convenient for modeling than the traditional$$\frac{ S_{t+1}-S_t}{S_t}$$because the former allow the modeler to think in terms of subtraction. The only "con" ... 1 A \lambda = 0 means that the objetive function's derivative with respect to the restriction is zero. In more intuitive terms, one cannot change the expected utility of consumption by relaxing or tightening the budget restriction. This is a weird case, for sure, I think you're missing something here. Maybe telling us what the variables mean can help? 1 Note: This answer is preliminary, as I am unsure about some components of the question. I will note that it is a very long time since I studied portfolio immunization, and so I am describing how this would be looked at in the industry. The simplest strategy for immunization in this case is to purchase five zero coupon bonds, each with a principal value of \... 1 An equlibrium is a property of an entire market, not of any given investor's portfolio. The CAPM is an equilibrium theory. In equilibrium, all (nontoxic!) assets must have non-negative prices. Hence, the weights of all assets in the market portfolio must be non-negative as well. The market portfolio weights will be strictly positive only if the assets have ... 1 By itself, no it does not, at least how I am understanding your post. While I am an opponent of the Efficient Market Hypothesis, what you would need to show is that there is a "free lunch," with your methodology. You would also have to show it persists out-of-sample and does so for decades. You would need to measure commissions as well. A \3 gain that ... 1 It appears you are looking for literature on Ambiguity Aversion and/or "Uncertainty Aversion". You can start by looking up the work of L.G Epstein, I. Gilboa, and D. Schmeidler. It is an attempt to formalize the behavior of agents exhibiting behavior consistent with the Ellsberg paradox. 1 The Sharpe ratio tells us the amount of excess return we get for taking on each additional unit of portfolio standard deviation.$$\frac{\mu_p - r_f}{\sigma_p }$$We are looking for the combination of the two risky assets with the highest Sharpe ratio (P^*). Once we do that, we can take linear combinations of that portfolio and the risk-less asset and ... 1 Thanks a lot for the reference. I think the result does hold. Here is what I found: The validity of such a law of large numbers what subject to some debate in the 1980s. See Judd (1985), Feldman and Gilles (1985) and Uhlig (1996) for representative papers. Luckily Feldman and Gilles show that for a probability space (Y; B(Y ); \Pi) there exists a ... 1 The problem is equivalent to maximizing$$ \max_\phi \left( \phi ' \mu - \frac{1}{2} \alpha \phi ' \Sigma \phi \right) $$subject to$$ \mathbf{1}' \phi = w_0. $$(boldface 1 is column vector of ones, ' stands for transposition). The lagrangian is$$ L(\phi, \lambda) = \phi ' \mu - \frac{1}{2} \alpha \phi ' \Sigma \phi - \lambda \left( \mathbf{1}' \phi ...

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