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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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Isoelastic utility functions, how to calculate risk boundaries.

I am trying to replicate a calculation I found in a paper that calculates the level of risk aversion for an individual to choose one option over the other, but I am struggling to solve it. Ideally, ...
anona's user avatar
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Find the trivial interval for fair prices

Suppose there is a risk-free asset with $ r^0 = 5 $ and $2$ risky assets with $E r^1 = 8, \sigma^1 = 5, E r^2 = 20, \sigma^2 = 20, \text{corr}(r^1, r^2) = 0 $. What is the maximum average profit and ...
WOL - THE WORLD OF LESSONS's user avatar
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Calculating the RRR

I want to calculate my Risk-Reward-Ratio to pipe it into an equity simulator. The risk is calculated by dividing the reward by the risk. I also need the number of iterations and the hit rate to get ...
Hadsga's user avatar
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Equality case in the subadditive property of Expected Shortfall

If $Z_1$ and $Z_2$ are two $L^1$ real-values random variables, it is well known that $ES_\alpha(Z_1+Z_2) \le ES_\alpha(Z_1) + ES_\alpha(Z_2)$ for any $\alpha \in (0,1)$ However, what about the ...
Aristodog's user avatar
2 votes
1 answer
96 views

How do I prove monotonicity of risk measures?

Assume I have a risk measure, that is a functional $R:L^\infty(\Omega, F,P)\to \mathbb{R}$, that applies to random variables $X:\Omega\to\mathbb{R}$. Assume the functional satisfies the following ...
mathCurious's user avatar
1 vote
1 answer
120 views

For what values of $\alpha <\frac 12$ occurs $CVaR_{\alpha}(X) \ge VaR_{\alpha} (X)$

The payoff function for an investment is modeled with a random variable $X$ with a density function $$f(x)=\frac 12 e^{-|x|}$$ For what values of $\alpha <\frac 12$ occurs $$CVaR_{\alpha}(X) \ge ...
dsk62's user avatar
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Why risk measures are defined on the space of bounded random variables?

I'm reading an article where risk measures are defined on the space of random variable (not necessarily bounded), so I want to understand why risk measures are defined at first (for example in Follmer ...
MathLover's user avatar
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The only way that $\max\{X_1, X_2\} >x$ in the subexponetial case, is if either $X_1$ or $X_2$ is large?

In my book about risk, there is a chapter where they show that \begin{align*} \mathbb{P}(\max\{X_1, X_2\} > x) = \mathbb{P}(X_1 > x) + \mathbb{P}(X_2 > x) - \mathbb{P}(X_1 > x, X_2 > x) ...
Mathias Nissen's user avatar
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1 answer
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Prove that $S^0=S^1$ so there is one bank account in the market without arbitrage

In a two-period market with four possible scenarios, the risk-free interest rate is $10$%. There is one stock in this market whose prices are described by process $S$: $$S_0 = 100, S_1(\omega_1) = S_1(...
john1235's user avatar
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Proof that $VaR_c(L)=(\Phi^{-1}(\frac{c+1}2))^2$

The loss $L$ has the $\lambda_1^2$ distribution, i.e. the distribution of the random variable $X^2$, where $X$ has a standard normal distribution. Proof that $VaR_c(L)=(\Phi^{-1}(\frac{c+1}2))^2$, ...
john1235's user avatar
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1 answer
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How are the two statements same mathematically? [closed]

I am doing a data course on statistics and in one of the questions, they're suggesting that the statements given below are mathematically the same. I disagree. Can they be mathematically proven to be ...
djumanji's user avatar
2 votes
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What is the occupation measure for constrained risk-averse markov decision processes with only expectation constraints (no risk constraints)?

I have formulated my optimization problem as a constrained risk averse markov decision process. In my solution methodology I want to use occupation measure of the formulated problem. I have searched ...
Engr. Moiz Ahmad's user avatar
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23 views

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 ...
j0nr's user avatar
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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 ...
crab13's user avatar
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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 ...
Manda Burnham's user avatar
1 vote
1 answer
895 views

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 ...
surfmuggle's user avatar
1 vote
1 answer
22 views

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 ...
Steffan's user avatar
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1 vote
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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 ...
Viraj Singh's user avatar
1 vote
1 answer
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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) \...
user2005142's user avatar
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108 views

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=...
linguisticturn's user avatar
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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 ...
user10478's user avatar
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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 ...
Homer Jay Simpson's user avatar
3 votes
1 answer
105 views

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. ...
Evgeny Kuznetsov's user avatar
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1 answer
357 views

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 ...
IchVerlore's user avatar
1 vote
1 answer
45 views

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 ...
user913386's user avatar
0 votes
1 answer
132 views

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 ...
MDescamps's user avatar
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2 answers
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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 ...
user2005142's user avatar
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2 answers
255 views

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 $...
user avatar
1 vote
1 answer
306 views

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 $...
CAN's user avatar
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5 votes
1 answer
111 views

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 ...
Frank Seidl's user avatar
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2 votes
1 answer
101 views

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 $...
Marina 's user avatar
1 vote
0 answers
57 views

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 ...
ChargeShivers's user avatar
0 votes
1 answer
23 views

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 ...
Zero's user avatar
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1 vote
1 answer
452 views

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 ...
ITOM97's user avatar
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1 vote
1 answer
60 views

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{...
MDescamps's user avatar
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1 vote
2 answers
86 views

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'...
Joshua D Carmichael's user avatar
1 vote
0 answers
56 views

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 ...
Fitzjerald's user avatar
1 vote
0 answers
628 views

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" ...
alejandrogrrl's user avatar
1 vote
1 answer
274 views

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 ...
Hugo Assanti's user avatar
0 votes
0 answers
103 views

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\...
Edgar Alarcón's user avatar
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0 answers
10 views

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: ...
Jens Törnell's user avatar
2 votes
0 answers
47 views

Reserve for Credit Loss

This is a regulatory way to calculate a reserve for credit loss. I want to interpret this formula. $$Lifetime\ Reserve=\frac{PD\times LGD\times EAD}{(1+i)}\left[\frac{1-(1-PD)^n}{PD}\right]-$$ $$\...
alejandrogrrl's user avatar
1 vote
1 answer
2k views

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 ...
Kohler Fryer's user avatar
1 vote
0 answers
92 views

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 ...
johncarver19's user avatar
6 votes
1 answer
166 views

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 ...
Riccardo Ceccon's user avatar
0 votes
0 answers
67 views

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 ...
Jennifer's user avatar
1 vote
2 answers
117 views

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 ...
cpotter's user avatar
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2 votes
1 answer
956 views

$ES_X(p)$ 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\...
Nelly's user avatar
  • 179
1 vote
1 answer
287 views

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 ...
Tal-Botvinnik's user avatar
0 votes
2 answers
80 views

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 ...
Marcel's user avatar
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