# Questions tagged [distribution]

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### Can I apply the Kelly criterion directly, without fitting any distributions?

Problem I want to apply the Kelly criterion to asset returns, so that I know how much to hold of each, ideally (and how much I should keep as a cash reserve). As far as I understand the Kelly ...
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### What is the Accumulation/Distribution formula doing geometrically?

My understanding of the essence of the Accumulation/Distribution Index is that it tracks the closing price of a security during each period relative to its price range for that period, something like ...
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### Pareto comparison of return distributions

In making a choice among financial strategies, each of which has some estimated return distribution, some strategies will clearly be better than others. But many times, the choice is a question of ...
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### Integral of brownian motion wrt. time over [t;T]

From the post Integral of Brownian motion w.r.t. time we have an argument for $$\int_0^t W_sds \sim N\left(0,\frac{1}{3}t^3\right).$$ However, how does this generalise for the interval $[t;T]$? I.e. ...
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### How to compute the combined probability of loss for 2 time series (consisting of historical stock prices)?

May I please ask the community's support with the following problem? I have 2 time series, with approximately 1000 observations each (same number of observations for both). They represent the daily ...
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### Market Sentiment Concept Question

I came across an interesting concept question and was curious what other people thought: Let's say some commodity has a certain return distribution. Now, if one knows that over the next five days, ...
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### Terminal wealth distribution from dollar cost averaging

If monthly stock market returns follow an IID lognormal distribution, the terminal wealth distribution of investing a lump sum for many years is also lognormal. What is the terminal wealth ...
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1 vote
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### Assymetric Rate Distribution

The pandemic has disavowed any notion of nominal rate distributions to being truncated at 0%. However, if Central Banks at Debtor nations are conflicted in that they are incented to suppress interest ...
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### Probablity distributions of zero crossings in 1D random-walk

Consider a simple 1D random walk that starts at position zero, and each second changes position by either +1 or -1 with 50-50 probabalities. I know it is proven to cross zero infinitely many times, ...
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### Why stock prices changes don't follow Pareto Distribution?

I calculated the distribution of the stock price changes (diffs). The diffs are multiplicative, $d_t=p_{t} / p_{t-1}$. As far as I know the distribution should look like Power law distribution (Pareto ...
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### basic numerical integration question related to case of high positive volatility skew

is the below equation true irrespective of if that 2nd derivative turns out to be negative or >1 , (ie even if theres an arbitrage) ? the reason i ask is that i am writing a single asset montecarlo ...
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### Fat tailed can be estimated through a t-distributions?

I have a simple question that makes me doubt a bit. In a multiple choise exam I ecountered this question: "if the stocks returns are not normally distributed, the fat tail effect can be estimated ...
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### Calculating the Value-at-Risk when changing the confidence level

If I have a VaR estimate at a 95% confidence interval is 10, how do I calculate the approximate level of the VaR if the confidence level was raised to 99%, assuming a one-tailed normal distribution?
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### Should stock return series be modeled with a parametric distribution, or an autoregressive function? [closed]

If I have prior knowledg that a stock return series follows a parametric distribution, such as a Student t-distribution with 4 degrees of freedom, without actively looking for prior knowledge of ...
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### What does this absolute return distribution chart show?

I was reading some pages in Professional Automated Trading by Eugene Durenard when I came across this chart: The caption says: "S&P Absolute Return Distribution: Log-Log Scale". The ...
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### FX spot distribution with student-t returns

If I am modelling my returns as $\sim N(0, \sigma^2)$, then I can evolve my spot distribution as: $$S_{t} = S_{0}e^{(\mu - \frac{1}{2}\sigma^{2})t + \sigma dW_{t}}$$ where $S_{0}$ is the spot, $\mu$ ...
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### What should degrees of freedom $\nu$ be set to when modeling financial returns that follow the t-distribution?

The closer the t-distribution degrees of freedom ($\nu$) is to 0, the more heavy are the tails, whereas high degrees of freedom recovers the normal distribution. In finance, what value is usually used ...
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### Large deviations theory in finance

In probability theory, the theory of large deviations concerns the asymptotic behavior of remote tails of sequences of probability distributions. A related post says: Large deviations theory is ...
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1 vote
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### Do portfolio mean and portfolio variance have probability distributions?

If $X$ is a $T\times N$ matrix of multivariate asset returns, and $w$ is some optimal portfolio weight vector, then the portfolio return series is $r_p = X w \in\mathbb{R}^{T}$. This return series ...
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### law of absolute of max of brownian motion

What is the law of $\max\left(|B_t|\right)$ for $t$ in $[0,T]$ and $B_t$ is a Brownian motion? Any references for properties of this process?
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### Does Value-at-Risk have any mathematical equivalence to copulas?

Portfolio Value-at-Risk estimated using the copula approach often just means generating artificial data sampled from a parametric copula('s joint multivariate distribution) as a model fit over the ...
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### What is the distribution of the risk-free asset?

If the risk-free asset has a volatility of $0$, therefore making its mean equal to the risk-free rate, $r_f$, does this mean that it has no probability distribution, and therefore there is no reason ...
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### How important is the chronological ordering of historical returns?

The returns of asset $A$ in chronological order are 0.03 0.01 -0.04 0.02 0.05 -0.10 0.02 The expected return, or sample mean, is $-0.00143$ while its sample ...
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### Interpretation of a uniform asset return distribution

Typically asset return distributions are bell-shaped with most mass occurring in and around the center, 0% returns, and less so in the tails, with the left tail representing the probability of large ...
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### Monte Carlo approach and methods for generating random returns

Recently I found myself reading more about Monte Carlo approach in m.v. portfolio optimization framework. I already discuss the topic on this forum (if interested please consider the following links - ...
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### Minimizing variance vs. expected shortfall: distributions where the difference is salient

In portfolio theory in finance, given a set of $n$ assets to choose from, one often selects portfolio weights so as to maximize expected return and minimize some measure of risk, e.g. variance or ...
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### Are mean-variance efficient portfolio weights random variables with probability distributions?

The mean-variance model outputs a portfolio weight vector whose elements are individual asset weights that sum to 1. Regardless of which portfolio along the efficient frontier is being solved, the ...
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### Sampling from an empirical distribution

I want to sample from the empirical distribution of returns. To do so, I do not want to make the preliminary assumption of which distribution the returns follow, rather I would like to sample from the ...
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