# 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 $𝐸[𝑅]$ 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. $$𝑈=𝐸[𝑅]−\frac{1}{2}A\sigma^{2}$$ 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 ...

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