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In order to understand this problem, I will work through the generic case. Say that a user had generalized quadratic (Bernoulli) utility, similar to your problem: $$u(x) = \beta x^2 + \gamma x$$ and suppose that there is a distribution for the outcome of $x$, denoted $F(x)$. Thus, utility over this distribution is equal to \begin{align} \int u(x) \text{... 8 \begin{eqnarray*} \displaystyle U(L) & = &\sum_{s=1}^{S}\pi_s U(Y_s) = \sum_{s=1}^{S} \left(-\frac{1}{2}\pi_s(\alpha - Y_s)^2\right) = -\frac{1}{2}\sum_{s=1}^{S} \left(\pi_s(\alpha^2 + Y_s^2-2\alpha Y_s)\right) \\ &=& -\frac{1}{2}\left(\alpha^2\sum_{s=1}^{S} \pi_s + \sum_{s=1}^{S} \pi_sY_s^2-2\alpha \sum_{s=1}^{S} \pi_sY_s\right) = -\frac{1}{... 8 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 ... 7 Let P = \alpha A + (1-\alpha) B where A and B are returns (random) from the two assets, and P is their portfolio. Variance of portfolio P can therefore be written as \begin{eqnarray*} \sigma^2_P & = & \alpha^2 \sigma^2_A + (1-\alpha)^2 \sigma^2_B + 2\alpha (1-\alpha)\text{Cov}(A, B) \\ & \leq &\alpha^2 \sigma^2_A + (1-\alpha)^2 \... 6 1) Would lead to the return of general panic led bank runs, and introduce additional instability to the system. It´s not generally appreciated, that 19th century bank runs were not just a symptom of insolvency or illiquidity, but were also occasionally triggered by competitors (other banks) spreading rumours. Generally this is a bad idea that rests on the ... 6 This is an important result in financial economics (asset pricing) but not trivial to explain intuitively. I do my best to give you the big picture and get you started on your research. R is the return on an asset or portfolio. Any asset or portfolio. m is the stochastic discount factor or pricing kernel. The expectation here is over possible states ... 5 The Aerobie Sports company mostly makes odd-ball equipment for having a catch. But they made a coffee maker called the Aeropress as a side project and now it makes up more than half of their sales. The Woolworth Company (of five and dime fame) founded the Foot Locker shoe store as a side business and now it is the core-surviving brand of that corporation ... 5 The easiest way to model short-term but risky debt in continuous time is to have your \psi be the increment of a compound Poisson process. Jumps in this process correspond to events that might or might not cause default; the size of the jump can enter together with the state X(t) into a function like your g(X(t),\psi) to determine whether or not ... 5 Easiest fix: if you're worried about it you should value weight your results. This is suggest by, for instance, Kothari, Shanken and Sloan (1995). Firms that are delisted tend to have extremely small market cap, so value weighting gives them very little impact on summary statistics. Delisted returns should also be used, although I'm not sure how much impact ... 5 The first equation can be written as: r_E(Levered) = \frac{E+D}{E}r_E(Unlevered) - \frac{D}{E}r_D $$Then, isolating the unlevered return gives:$$ r_E(Unlevered) = \frac{E}{E+D}r_E(Levered) + \frac{D}{E+D}r_D$$And this is the WACC. 5 Let w denote the weight on A so that 1-w is the weight on B. Recall from the properties of variance that \sigma_p^2 = w^2\sigma_A^2 + 2w(1-w)\sigma_A\sigma_B \rho_{AB}+ (1-w)^2\sigma_B^2 Without loss of generality, assume \sigma_A \geq \sigma_B. We wish to show that w^2\sigma_A^2 + 2w(1-w)\sigma_A\sigma_B \rho_{AB}+ (1-w)^2\sigma_B^2\leq ... 5 With a European view: The CEPR VOX EU: an economics blog created by the Centre for Economic Policy Research, which promotes research excellence and policy relevance in European economics. It covers a diverse range of research based analyses on policy and economic issues. The Economist a global scope with an European perspective. EconomicsUK: personal ... 5 Net Present Value (NPV) as a soft concept existed probably even in antiquity but it was formalized and made popular by Irving Fisher in his book the Rate of Interest. Internal rate of return is basically a special application of NPV. It was also first formally introduced in Fisher's book although he called it 'rate of return over costs'. Duration of bonds ... 4 Fundamental theorem of asset pricing tells us that if there is no arbitrage, there must exist a positive random variable M (also called stochastic discount factor) such that for any return R, we must have 1 = E[M R]. Thus to understand asset prices, in cross-section as well as across time, we need to understand which factors affect M. Now, the most ... 4 Your question relates more broadly to modern portfolio theory, and can be illustrated via mean-variance analysis of a univariate time-series. The extension to the multi variate (normal) setting, is trivial. Below I use the term accuracy, in a non-formal way, relating to the variance of an estimator. In particular, when the variance of an estimator can be ... 4 The paper is assuming that some form of "law of large numbers" (LLN) applies for the continuum. The expected value of capital for an individual agent is$$\mathbb{E}[R\cdot 1_{R\geq R^*}]=\int_{R^*}^1 R\,dR\tag{1}$$The LLN assumption says that when we have a mass-1 continuum of agents, their actual total will equal their expectation in (1). What justifies ... 4 Microplane graters grew out of a company that was manufacturing dot-matrix printer parts, using a process that created extremely sharp metal bands— which was, at the time, more of a liability than a strength. As dot-matrix printers fell into obsolescence, the company decided to find a use for its unnaturally sharp strips of metal, and "pivoted" toward making ... 4 (Looking at the question and notation used more closely, the formulation seems to be problematic in couple places.) General Fact Let W be standard Brownian motion with respect to filtration ( \mathscr F_t )_{t \in [0,T]}. Consider (L_t)_{t \in [0,T]} defined by$$ \frac{dL_t}{L_t} = \psi_t dL_t, \; L_0 = 1.  In general, $L_t = e^{\int_0^t \psi_s ... 4 The recommended books are decent. From these two I'd go with Bailey first and if you're comfortable with that, then LeRoy & Werner. The latter requires some background in linear algebra and optimization theory. If you want to study some econometric applications for financial economics, you might try: Cuthbertson & Nitzsche: Quantitative Financial ... 4 Sure, because dollars are nominal goods. Perhaps a ticket to Star Wars Episode CXVIII will cost one trillion dollars. If you meant today, without huge changes in the price level: Yes, there is way more than one trillion dollars of wealth in the world. Depending on the laws of physics and psychology there is probably something you would be able to market for ... 4 We are given two CDFs$F$and$G$, such that$F$FOSD$G$i.e.$F(x) \leq G(x)\forall x$. Consider the random variables$X\sim F$and$Y\sim G$. Also, suppose$X$and$Y$take non-negative values. We want to show that$\mathbb{E}(X) \geq \mathbb{E}(Y)$. Here is the intuition:$F(x) \leq G(x)\forall x$means that the probability that the random ... 4 Looking at gross output (which includes using the outputs of other industries) and value-added (largely wages and profits) by industry you get numbers like this for 2017 in USD trillion. Adding up the value-added gives total GDP, and you can see that there is a lot more to the economy than manufacturing ... 3 NA, in my opinion, is the lack of a trading strategy to generate excess returns, that means returns greater than those of assets in the same risk category. Note that it doesn't mean no risk. A market is efficient, again in my own understanding, when all the contents of an information set are reflected in the market's assets prices. There are 3 forms of ... 3 "Short selling" is a method of trading that includes selling a security (in this case currency) that the trader doesn't own, by borrowing it from a broker. Prices on forex trading websites are quoted at the current market price, meaning that people around the world are buying and selling at that price at that time. Brokers buy for slightly under market price ... 3 When you look at many things in economics it is worth noting that everything is relative.$1 is only worth what it can purchase. The decisions you make are made in the contexts of your next best option (opportunity cost). In this case, the Australian dollar isn't worth any less domestically. The US has just gotten stronger. With the US outlook being quite ...

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The following are NOT my words. Found this online. Please check full article here: http://www.multifactorworld.com/Lists/Posts/Post.aspx?List=3b285530-ccfa-416a-ba3b-de55213abe17&ID=59 Hope this helps... Size Premium: Why Does it Exist? There seems to be general agreement that the size premium is a reward for bearing the risk that small companies may ...

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This is equation $(13)$ of Whited, T. M., & Wu, G. (2006). Financial constraints risk. Review of Financial Studies, 19(2), 531-559. It is empirically estimated as regards the specific coefficient values. The important question is, What is the left-hand-side? Looking at eq. $(12)$ of the paper the left-hand-side of $(13)$ is $\lambda_{i,t+1}$ which ...

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I guess you're asking three inter-relatated though conceptually separate questions: What is the latent or efficient price? What is market microstructure noise (as opposed to non-market-microstructure noise)? What does it mean to assume that the efficient price process and the microstructure noise process are independent? I won't attempt to provide a full ...

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Prices changing so frequently is simply the result of an adjustment between demand and supply in the market. Demand and supply change all the time, because market venues allow them to change very frequently, and so does the prices. This does not mean that the fundamentals of the underlying assets are also changing, this is just the way market process (what ...

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While the prices of equities do fluctuate in response to information about the underlying enterprises, this is not the first order or likely even second order reason why they do so. Discount-rate variation is the central organizing question of current asset-pricing research. I survey facts, theories, and applications. Previously, we thought returns ...

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