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13

Short of having a 'reasonable' predictive model for expected returns and the covariance matrix, there are a couple lines of attack. Shrinkage estimators (via Bayesian inference or Stein-class of estimators) Robust portfolio optimization Michaud's Resampled Efficient Frontier Imposing norm constraints on portfolio weights Naively, shrinkage methods ...


10

Both answers from Shane and Vishal Belsare make sense and detail different models. In my experience, I have never been satisfied by a unique model since the majority of papers out there can be split in two categories: Those that predict the mean component of the problem. Those that predict the variance component of the problem. The ideal (to read ...


8

Perhaps you may want to consider article by D. Levine - Modeling Tail Behavior with Extreme Value Theory who gives practicale example on how EVT can be used to calculate probabilities on returns in tails with use of the Pickands-Balkema-de Haan Theorem and generalized Pareto distribution. It also contains some criterias and points on other methods that can ...


8

One approach is Conditional Value at Risk (CVaR) a.k.a. Expected Shortfall (ES). It does, as you suggest, take into account the whole set of returns. However, instead of traditional VaR which asks "what is the worst 1% or 5% loss I can expect" in a given time frame, conditional VaR asks "assuming I sustain losses of at least 95% or 99% (and perhaps am ...


7

You raise a very important point, which unfortunately doesn't have a simple answer. Black-Litterman addresses the allocation problem by allowing you to provide a prior within a bayesian framework. It doesn't really tell you how to produce the prior itself. But more importantly, it doesn't address the fundamental problem: it's difficult to accurately ...


6

Here's a partial answer: This partly depends on the return characteristics. One way to look at this is to analyze the skewness and kurtosis of the returns. Most strategies have a negative skewness, which roughly means that they have mostly consistent small positive returns, with the occasional large negative return. Alternatively, some strategies have ...


5

If you don’t have any specific model which describes the behavior of the asset being traded, you can estimate the empirical distribution of returns by backtesting your momentum strategy. Then you can adjust this estimate during your strategy’s lifetime from your trading results. Additionally you can enhance this by accounting for different market regimes ...


5

You are trying to apply the Kelly Criterion, supposedly to maximize how aggressively to bet, and you are having trouble when the Kelly Value turns negative. The naive answer to your question is that when your kelly value turns negative, then $f=\frac{bp-q}{b}$ turning negative means the instantaneous expected return is negative, which means you should not ...


4

If the equation satisfies all the assumptions of OLS, particularly homoscedasticity and no autocorrelation in the errors, then the expected return for the equation you laid out is $E[r_{future}|r_{history},x_{news}]=\alpha+\beta_1r_{history}+\beta_2x_{news}+\beta_3r_{history}*x_{news}$ If the unconditional expected return is zero (as is likely to be ...


3

In effect, you are wondering whether to price this option on risk-free probability distributions (B-S drift $r_f$), or real-world ones (B-S drift $\mu$, however calibrated) One cannot short the mutual fund, so the argument for using risk-free is weakened. But, there are various economic equilibrium arguments why using it may still be OK. If you use the ...


3

The answer to the original question is simple: the Chopra-Ziemba paper is highly flawed and unreliable. Note that the framework is in-sample and based on a utility function. It has nothing to do with out-of-sample behavior of the mean vs. the covariance in an optimization. Estimation error grows linearly in the mean but quadratically in the covariance. At ...


3

Tail risk represents the probability that the magnitude of returns on an asset/portfolio will exceed some threshold (usually three standard deviations) on the normal curve. If you visualize a normal curve on standard axes, the tail on the left side corresponds to an extreme low return and the tail on the right side corresponds to an extreme high return. In ...


3

What a great question -- it touches on many issues at the core of quantitative finance. This answer might be a lot more than you bargained for, but it's too interesting to pass up. References Mostly, this subject falls somewhere at the intersection of these three highly-interrelated topics: risk-neutral valuation, rational pricing and the fundamental ...


3

Another possible approach is taking views a la Black-Litterman. There is a 2006 paper "Incorporating Trading Strategies in the Black-Litterman Framework" that discusses the methodology in more detail. There are several practical issues that one should consider when implementing a momentum strategy with optimization. I would pay careful attention to the ...


2

Returns-based analysis cannot calculate the expected return of a trading system. It yields nonsensical results and is not suited to this particular calculation. Consider a game where every time you play, you win 25% twice and lose 40% once. There are basically three permutations of this game. Represented in R vectors: first <- c(.25, .25, -.4) second ...


2

You can't add returns. You must multiply them. In your example above where daily returns are 25%, 25%, and -40% To compute expected return from a return series, simply use this formula: return = product( 1+return); in the case of you example this yields: return = (1.25 * 1.25 * .6) = .9375 To get the expected daily return use the geometric mean: ...


2

It seems that your real question is: is the PFP (Price Formation Process) diffusive from intraday to weekly sampling rate? It is a very good question since on intraday, some academics found some multifractal features into intraday returns, meaning that the PFP is not a Geometric Brownian Motion at small scales (even considering stochastic volatility). You ...


1

This may or may not be helpful, since I don't have anything to point you to that specifically addresses the high skewness of the distribution you mention. However, this sounds like it is probably an idiosyncratic risk, and that certainly has bearing on whether or not it would be priced. In the standard capital asset pricing model, the marginal investor ...


1

Isn't this a simple mathematical rule? $$\Delta r_{t}=r_{t} - r_{t-1} = ln(p_{t}) - ln(p_{0}) - ln(p_{t-1}) + ln(p_{0})=ln(\frac{p_{t}}{p_{t-1}})$$ i.e. logarithmic or continuously compounded return. As a result: $$E(\Delta r_{t})=\frac{1}{T}\sum_{t=1}^{T}\Delta r_{t} = ...


1

One approach which I've encountered in practice is Optimal risk budgeting (ORB). This method is similar to Black Litterman in the sense that it uses active investor views as a starting point. The mean variance optimization is then restricted to those assets for which an active investor view is available, and the allocation is calculated with the constraint ...



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