Questions tagged [risk-assessment]

For questions about risk assessment, a systematic process of evaluating the potential risks that may be involved in a projected activity or undertaking.

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Is "extreme CVaR" (CVaR from extreme value theory) elicitable or conditionally elicitable with some other statistical mapping (like VaR)?

I am recently using deep reinforcement learning (DRL) for solving risk-averse markov decision processes (RA-MDP) using risk measures which are either elicitable or conditionally elicitable (as in the ...
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Probability Risk Evaluation

I have a batch of 50 items. There is an inspection pass/fail criteria and 3 out of 50 failed inspection. How do I determine if I pick 10 random items from this batch of 50 the probability %age that I ...
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How Proof $ I(a)=\int ^{}_{s} {a}dv=\int ^{\infty }_{0} {v}\left \{ {s\in S|a(s)≥\alpha } \right \} d\alpha ,a\in {B}_{+}\left ( {S,\Sigma } \right )$

Let $ \Sigma $ denote a non-empty algebra of subsets of a set S, let B(S, $ \Sigma $) denote the set of bounded, real valued, $ \Sigma $-measurable functions on S, and let v denote a monotonic real ...
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How to find a bayes risk

Let $X$ follows a normal distribution with parameters $\theta$ and $\sigma^2$. Let a prior for $\theta$ be a Normal distribution with parameters $\mu$ and $b^2$. How can I find the Bayes risk of this ...
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Risk measures and Bellman's principle of optimality in the context of risk-sensitive MDPs

I'm learning about finite-horizon risk-sensitive Markov Decision Processes (MDP) in the context of stochastic shortest path problems. In a nutshell, instead of finding a policy that minimizes the ...
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I don't know where I'm going wrong in calculating the expected replacement cost

This is the question: Suppose the probability a plant is destroyed in year 1 is 2% and the probability a plant is destroyed in year 2 assuming it was not destroyed previously is 4%. When the plant is ...
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Distribution of $X(T)$ where $T$ is the time of ruin.

Given a ordinary renewal process $$ X\left(t\right)=u+ct-\sum^{N\left(t\right)}_{i=1}{Y_i},\ \ \ t\geq{0} $$ where $u\geq 0$ is the initial surplus, $c$ represents the insurer’s premium income per ...
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Need help to understand how an individual model can be transformed into a collective one

Can someone help me to understand the following section which is intended to show that the individual model is a special case of the collective model. (To be precise: From a given individual model we ...
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How to calculate Relative Risk Reduction and Absolute Risk Reduction?

In this guide they try to teach how to calculate relative and absolute risk reduction. Problem: My understanding of the formulas leads to different result than they say. Data and results given ...
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Correctly comparing the risk posed by a single task to the risk posed by a repetitive task

I have been told that I have come up with a comparison which statistically doesn't make sense. But no one has told me how to correct the comparison. Although I believe it is related to an increased ...
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IB Extended Essay Research Idea Inquiries

I am currently a senior in the IB program at my school and I have to write a full-length paper on mathematical research I have conducted. My idea for my paper was to create a gaussian function (normal ...
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Show translation invariance from axioms

In https://sites.math.washington.edu/~rtr/papers/rtr206-RiskTutorial_INFORMS2007.pdf, Rockafellar defines a coherent risk measure in the following way: $\rho : \mathcal{L}^2(\Omega, \mathcal{F}, P) \...
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Discrete random walk/risk process characterized by a step distribution with a heavy tail and a mean existing only as a Hadamard finite-part integral

This question is an attempt to make more precise two previous questions of mine on another SE (this and this). The random walk/risk process of interest Consider the following discerete process. Let $n=...
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Analyzing a Gambling Race Paradox

Suppose a number of players are given $100$ points each, and repeatedly engage in a gamble having positive expected value, with the goals of being the first player to reach $100000$ points. Solving ...
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Find the sample size $n$ for a stratified sampling method

Let’s say we have a finite population $N=5000$.We have divided the population into 5 subgroups according to a risk characteristic “very low risk”,”low risk”,”medium risk”,”high risk” and “very high ...
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Insurance statistics book

Am am looking for a book on insurance statistics. The problem is that there are a lot of googleable variants that it is difficult to decide which one is good and which ones are better in good ones. ...
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Question about the mean excess loss function.

I'm new to this risk thing. I am trying to obtain the mean excess loss function evaluated at a point for the following Pareto distribution: $F(x)=1-(\frac{\theta}{x})^{\alpha}$ The excess loss ...
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Prove that $E_θ(θ-\frac{X}{100})^2 = \frac {θ(1-θ)}{100}$

$X \text {~} Binomial(100,θ)$, $δ(X)=\frac {X}{100}$, $g(θ)=θ$ and the loss function is given by $L(θ,d)=(θ-d)^2$. The risk function for δ is $R(θ,δ) = E_θ(θ-\frac{X}{100})^2 = \frac {θ(1-θ)}{100}$ I ...
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Example of a risk measure that is not law-invariant

In some theorems about risk measures, the property of law invariance is required. Let $\mathcal{Z} = \mathcal{L}(\Omega, \mathcal{F}, P)$. A risk measure $\rho\colon \mathcal{Z}\to \mathbb{R}$ is law ...
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Optimization of stop-loss contract with respect to CVaR.

I'm trying to figure out where my mistake is. I'm trying to optimize a parameter for a stop-loss type contract with respect to conditional value-at-risk. I have attempted to differentiate with respect ...
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Find the m.g.f., the mean and the variance of the aggregate claims?

Assume that the number of losses $N$ has a geometric distribution with parameter $p$, while the claim amounts,${\{X_i}\}_{i≥1}$, is a sequence of i.i.d. Gamma distributed r.v. with parameters $a$ and $...
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How can I calculate win percentage needed based on risk/reward ratio?

Let's say I have calculated the probability of a win in two games to be $60\%$. I have $2$ bets I am looking at. For the first bet I need to risk $165$ to win $85$. For the second bet I need to risk $...
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Philosophically motivated risk metrics

During a random process over a sample space $\Omega$, an agent incurs a cost (say, in money) given by a random variable $Z:\Omega \rightarrow R$, which is determined by the agent's action. The agent ...
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Markov chain exercise 1

I have the transition matrix $$P= \begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 1\\ \frac14 & \frac14 & \frac12 \end{pmatrix}$$ and I have to determine the period of each state, compute $...
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optima of a random function / stochastic process

Consider the probability space $(\Omega, \mathcal{F},\mathbb{P})$, and let $X:\Omega \mapsto \mathbb{R}^T$, where $T$ is an index set, be a random function. What is the canonical definition of the ...
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The best way to calculate a risk score based on driving conditions

If I were to have a list of current weather conditions, including measurements like temperature, humidity, etc. What would the mathematical best way to calculate a risk score for bad driving ...
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Proof that conditional value at risk is coherant risk measure

Conditional value at risk, sometimes called expected shortfall, is defined as: $$CVaR_{\alpha}(X) = E[X|X \ge VaR_{\alpha}(X)]$$ For a risk measure $\left( \rho(x)\right)$, to be coherent, we require ...
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Translation equivariance of AV@R (average value at risk), proof

I am trying to prove that the average value at risk is translation equivariant: $$AV@R_\alpha[Z+\tau] = AV@R_\alpha[Z] + \tau$$ where $$AV@R_\alpha[Z] := \inf_{t\in \mathbb{R}} \{t+\alpha^{-1} \mathbb{...
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Total probability on a vector of Bernoulli random variables

My question concerns text following Equation 3.4.8 in Varshney (pg. 76): Varshney, P. K. (2012). Distributed detection and data fusion. Springer Science & Business Media. Seems trivial, but I don'...
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Find a joint distribution function with known Kendall's tau (Copula)

Assume we have two random variables $X_1\sim Exp(2)$, $X_2\sim Exp(1/2)$. I need to find a joint distribution function, so that Kendall's Tau will be $\rho_\tau(X_1,X_2)=-0.85$. I know that somehow I ...
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Test difference between two transition matrices (probabilities of default)

Recently, at work, I developed a project to estimate probabilities of default (PDs) using Markov Chains. We have a portafolio of loans classified in an internal rating (ratings from "Bucket" ...
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Expected shortfall

I have some diffculties with the following question; Prove that $ES_{\alpha}(L) = \frac{1}{1- \alpha}inf_{c \in {R}}\{E[(L-c)^{+}] + (1-\alpha)c\}$ Hint use $ES_{\alpha}(L) = \frac{1}{1-\alpha}E\big[...
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Steps to do a Monte Carlo simulation

I'm trying to do a Monte Carlo simulation but I'm lost in the process. The big question I want to answer is what's the probability I have to do a certain amount of work. In my solution I've already ...
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Risk adjusted premium principle (subadditivity)

Define the premium for a risk $X\geq0$ as: $$ π(X)=∫^∞_0(1−F_X(t))^{1/ρ}dt $$ With $\rho\geq1$. Let $X,Y\geq0$ two independent random variables. Prove or find a counterexample for: $$ \pi(X+Y)\leq\...
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Calculate risk of unrelated groups

Let's say we have a few variables (made up numbers). Each percentage is a risk of failure later in life. Gender Male 10% Woman 20% Age 25 years: 2% 50 years: 5% 70 years: 7% Interests Fotball: ...
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Reserve for Credit Loss

This is a regulatory way to calculate a reserve for credit loss. I want to interpret this formula. $$Lifetime\ Reserve=\frac{PD\times LGD\times EAD}{(1+i)}\left[\frac{1-(1-PD)^n}{PD}\right]-$$ $$\...
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Why must risk averse be correlated with a concave utility function?

Let my utility function $U: \mathbb{R}\to\mathbb{R}$ be arbitrary and suppose I am a risk averse person. Now suppose there was no risk involved with the attainment of money. Should should my utility ...
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Cramér-Lundberg model for on-demand insurance

I am looking for inspiration and perhaps guidance on the following as I’ve been stuck for a while now: Context: I am working on a practically oriented project to adjust the Cramér-Lundberg model ...
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Existence of functionals on $L^0$

Studying a paper about risk measures by F. Delbaen, I bumped into this statement: Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space: if $\mathbb{P}$ is atomless, then there exists no ...
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Utility premium

For the customer, the utility premium $H^{\star}$ is defined as the premium making the homeowner indifferent to insure or not, e.g. $H^{\star}$ solves $$u(x-H^{\star})=\mathbb{E}u(x-X).$$ How does it ...
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Toy problem: How much should I rationally be willing to pay for this hypothetical and simplified insurance?

Note: This is a sub-question to help answer a larger question I've posted in the personal finance stack exchange: Rational risk-assessment decision framework: Should I buy health insurance? Consider ...
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$ES_X(p)$ of a Lognormal

I have been trying to find the expected shortfall for $X \sim Lognormal(\mu, \sigma^2)$. I have calculated the following; $$ ES_X(p)=\frac{1}{p}\exp\left(\mu+\frac{\sigma^2}{2}\right)\left[1-\Phi\...
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Two-horse race probabilities

I have a rather basic question about probabilities, and this is a reference request. Assume you have 100 units of something, and you have to bet the 100 units into two events, A or B. You can ...
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Percent loss covered on average with a deductible

Suppose you have auto insurance with a deductible of \$200, and with no restriction on maximal payment. The probability of a loss is 0.10, and suppose that the distribution of the loss is exponential ...
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Compound Poisson models with completely monotone claim sizes

Assume that the free surplus of an insurance company at time t is modeled by \begin{equation} R(t)=u+ct-\sum_{k=1}^{N(t)}X_k \end{equation} where u is the initial surplus, the stochastic process N(t) ...
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Risk Management System

I am attempting to create a hypothetical risk management system for crypto trading based on the size of my portfolio. I have initial conditions of 200 dollars of initial capital and 200 dollars per ...
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What is a good introductory book on mathematical risk theory?

I'm looking for a book that introduces the following concepts and illustrates them with examples and exercises: Risk measures (Value at Risk, Expected Shortfall, distortion risk measures, and more) ...
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EVT: correlation standard normal distribution and Gumbel distribution

The standard normal distribution given here by $F$, satisfies the following limit: $$ n(1-F(a_n x+b_n)) \rightarrow e^{-x} \;\;\text{ as }\;\; n \rightarrow \infty$$ with $$b_n=(2\log(n)+\log(\...
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Two boxes with money - switch or not? [duplicate]

Someone gave me the following riddle and I am really not sure what the answer is. I have two boxes in front of me. One box contains $x$ Dollar, the other box contains $2x$ Dollar. The rules are as ...
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How to calculate the distortion function for EVaR?

As defined in Entropic Value-at-Risk: A New Coherent Risk Measure (reference) by A.Ahmadi-Javid, the entropic value-at-risk (EVaR) of $x\in L_{M^{+}}$ with confidence level $1-\alpha$ is: $$EVaR_{1-\...
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