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6

Risk-free rate is that you get for letting someone else use your money in a riskless manner. Suppose we live in a world where there is no risk whatsoever. In particular, if you lend someone \$100 there is 100% certainty that he will pay you back in a year. Before the pay date, he can do whatever he wants with your $100, while you have no access to it. Even ...


5

This the "Joint Hypothesis Problem". Basically, any test for abnormal returns is also implicitly a test of the model you use to define "abnormal". If you see a significant and positive $\alpha$, that could either mean that you actually are generating excess risk-adjusted returns, or it could mean that your risk model is incomplete. This is basically what ...


3

Fitting Fama-French or Carhart is as simple as learning how to perform Bayesian regression. Pretty much every introductory book Bayesian estimation will cover this. There are analytic formulae under certain assumptions, but I would definitely try to learn the basics of MCMC and Gibbs sampling before trying this out in practice. Here are two papers. The book ...


3

In my experience HFT has to balance the reward of any strategy with risk. In the case of a news-based trading strategy, the risk can be enormous, which means the algo will need a very high expected profit in order to trade the news. After important news events, volatility skyrockets and persists for some time (sometimes even days). If the market were able ...


3

The risk free rate is important and the reason for the inclusion and consideration of the risk free rate is that investors do not get compensated for not taking on risk. Now, we can argue whether the risk free rate truly provides risk free returns (we all should know that it does not, but ...) but it is important in the context of pricing risky assets that ...


2

Pretty good explanation is in Schweser CFA Study Notes for CFA level III. Books 3 and 5, at least from 2009, if I remember right. See also Tsay R.S. Analysis of Financial Time Series (Wiley Series in Probability and Statistics). // 2010. - good example with implementation in R.


2

In my opinion, risk free rate is not necessarily positive and not so important to pricing theory. It happened to be positive in most cases, but imagine a planet using Uranium-235 instead of gold as the money and unknowingly suffers from a shrinking population, likely the risk free rate is negative. Below are what I regard as important in pricing theory, ...


2

My non-rigorous answer: The future is uncertain. Even if there is no financial risk to investing in the "risk free" asset there is personal risk. For example, I could get hit by a car and die. Even if I survive till the moment that I liquidate my investment I will have less time left in my life to enjoy it. I need to be compensated for giving up this ...


2

Let $X$ be endowed with the following partial order: $y \geq x $ means that $\Bbb P(y\geq x) = 1$. The AOA condition in your case states that the pricing law $p$ is strictly inctreasing with respect to $\geq$, whereas LOP says that $p$ is a linear function. Neither if the two implies another one in general. For example, if $X = \Bbb R$ then $p(x) = x^3$ is a ...


2

The coefficients assuming they are statistically significant can be interpreted whether or not the underlying portfolio is efficient. The CAPM or FF4 simply tries to decompose a portfolio into a series of linear exposures + an intercept (alpha) which can be viewed as constant added value. In mathematical terms the regression is explaining how much of ...


2

They are actually exactly the same thing. CAPM say that expected risk premia are “explained" by the risk premium on the mean variance efficient (MVE) portfolio $$ R^i_{t+1} - R^f = \delta (R^{MVE}_{t+1}-R^f) + \varepsilon_{t+1} $$ De facto, you are saying that the systematic risk is just the projection of risk premia on MVE risk premium, and OLS are exactly ...


2

Regarding (1). Assume for some time $k$, $\mathcal{V}_k(\Phi_1) > \mathcal{V}_k(\Phi_2)$ (w.l.o.g.) with full knowledge that these strategies have equal value at $T$ ,($\mathcal{V}_T(\Phi_1)=\mathcal{V}_T(\Phi_2)$). I claim that this situation admits arbitrage. I can sell $\mathcal{V}_k(\Phi_1)$ and buy $\mathcal{V}_k(\Phi_2)$ and pocket the ...


2

Unfortunately, I do not know the model you talk about. However, the law of one price is a direct implication of the no-arbitrage assumption, which is assumed in many models (if not all). I do agree that the law of one price should be stated as a theorem rather than a definition. Anyway. Consider the case in which two portfolios A and B have the same value ...


2

if you have $p=0.5$ For example: $U(w)=ln(2w)$ why is that? relative risk aversion is given by $$RRA=\frac{-wU''(w)}{U('w)}=\frac{-w*(-1/4w^2)}{1/2w}=0.5$$ Now you can apply your formula. take for example: $x= 10000$ and $\pi=0.5=1-\pi.$ then expected utility is equal to $EU(x,w)=0.5*ln(2*(w+x))+0.5*ln(2*(w-x))=0.5ln(220000)+0.5ln(180000)$ you want to ...


2

(1) To get an arbitrage, buy low and sell high. Consider the following strategy: at $k$, in the event $V_k(\Phi_1) < V_k(\Phi_2)$, buy $\Phi_1$ and sell $\Phi_2$ invest the difference at the risk free rate. At maturity, your portfolio is worth what the you put in the bank plus interest. Formally, if $V_t(\Phi_\alpha) = \sum_{i=0}^d \Phi^i_{\alpha,t} ...


2

For non-normal asset price models you could look at the theory of Lévy-processes. If we assume that you work in the physical probability measure $P$ and that the random numbers that you have generated are daily log-returns, then you can do the following: Asset $i$ has starting price $S_0^i$ and for the future prices you can put $$ S_t^i = S_0^i ...


1

The answer depends somewhat on the type of bond. An odd lot of corporate bonds may be different than municipal bonds or US Government Agencies. Generally speaking, an odd lot would be any trade under $100,000 in face value.


1

A unique state price vector does not have to exist for there to be no arbitrage. It sounds like the state price vector in question has infinitely many solutions. Try to reduce the price matrix to row echelon form and show that at least one state price vector exists.


1

I just comment your second point, because in the definition i now of the LOP the state price vector (martingale measure) is involved. assuming the LOP holds then: state price vector is unique <=> the market is complete. to proof "=>" look at the trinomial model, show that the model is not complete by trying to find a hedge for a call, afterwards ...


1

The direct answer to your question on the choice of m is, "It depends." Your choice of m is dependent on the convention used by the source of your discount rate. Either may be appropriate. If you are actually looking to estimate a "fair" value, then the following will be relevant: A market yield(-to-maturity) approach assumes coupon reinvestment at that ...


1

In small sample, there is no reason why $x' \epsilon$ will be 0. In fact, there is no real reason why $\epsilon$ should be independent. The fact that you are assuming a linear specification for the returns means you are to some extent making assumptions of linear regression. Justifying the errors being uncorrelated with the independent variables is justified ...


1

Supply and demand... If you want an event that produce a change in the value of a currency, just look at the ruble. As Russia, gets more and more isolated and inflation spins out of control the ruble lose its value against other currencies.


1

I suggest to distinguish three things: Why risk-free rate? What defines its magnitude and sign? And what is its role in specific economic models? "Why risk-free rate" is very simple to answer: Because you want to compare cash at different points in time. One assumes that the difference between cash at time $t_1$ i.e. $C(t_1)$ and at time $t_2$ is $C(t_1) - ...


1

Edited: The risk free rate is positive because the factors of production, and the perception of time (from an individual's perspective) are limited. Limited supply of desirable goods such as houses (or limited capacity to make them from land, labour and capital), gives them a positive value in society (versus say air, which is in almost unlimited supply, ...


1

A risk free rate is the return rate from investing in an asset that has the lowest risk found in the market. It is a naming convention. The least risky of all returns is labelled as 'risk free' for the purpose of various models and resulting discussions. Another parallel answer is that you must understand what financial risk is in the first place. It is, ...


1

Ashwath Damodaran explains risk free rate in his 2008 paper, "What is risk free rate? A Search for the Basic Building Block". He has explained risk free rate from the perspective of investment. If we invest in an risk free asset then we expect guaranteed return. "An investment that delivers the same return, no matter what the scenario, should be ...


1

It is a decision problem, it is always a decision problem... The most basic decision problem is a lottery that gives you X (X > 0) for a chance of %Y and nothing for %(100-Y). Someone comes and offers you to buy your ticket for Z (0 < Z < X, otherwise it is trivial). What would you do? If Z is close to X, you need to be more risk-willing (or seeking) ...


1

That is true. Utility would not be concave anymore under prospect theory (only for gains), but convex for losses, which is evidence against CAPM. CAPM is valid either : -if the utility function is quadratic (which is nonsense in terms of economic interpretation, and in general, Von Neumann- Morgenstern utility describes poorly reality and should be ...


1

The general problem of the investor is: $$ \max_{w\in[0,1]^n} U(\mu_p(w),\sigma_p(w))\quad s.t. \sum_{i=1}^n w_i=1$$ where $w$ being the portfolio weights, and $U$ utility function of portfolio risk $\sigma_p$ and return $\mu_p$. CAPM assumes investors with concave utility function $U=\mu_p-\frac{1}{2}\sigma_p^2$, from which then follows that all ...


1

You can use the "Merton Jump Diffusion Model" to price European Options with jumps. The other points of your question are rather of practical relevance only. The negative drift of the underlying is usually not important, because the pricing goes under the riskneutral measure $Q$.



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