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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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a lower bound for the optimal solution of $VaR$ problem?

I need to find a lower bound for $VaR_\alpha$ in the problem $CVaR_\alpha(X)$. I think the optimal solution of the Exerted value problem it's work. but I'm not sure. Is the optimal solution of the ...
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27 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 ...
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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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$\text{ES}_{X^2}(p)$ where $X\sim \mathcal{N}(0,1)$

I am studying for my final exam and solving practice problems. One of the problems I solved was to calculate $\text{VaR}_{X^2}(p)$ and I managed to get; $$\text{Var}_{X^2}(p)=\left[\Phi^{-1}\left(1-\...
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1answer
65 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 ...
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52 views

Optimal classification function for two-classes non-overlapping

Given a known joint distribution p(x|y) for two classes and the two classes do not overlay $𝑚𝑖𝑛(𝑝(𝑥|𝑦=1),𝑝(𝑥|𝑦=−1))=0$ calculate the optimal classification function and the generalization ...
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8 views

Average risk and probability of error corresponding to feature vector

May you please help me how can I show the average risk as shown on picture? I don't know how to align the risk matrix L with the problem. Full body of the problem
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18 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 ...
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35 views

Bayes Rule and Bayesian Risk

Good day, When attempting this problem I came across some difficulties. A humanitarian charity wishes to classify a village as being at either high or low risk of flooding. The following ...
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2answers
59 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 ...
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67 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\...
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1answer
51 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 ...
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70 views

Moment generation function - compound Poisson

Let $S_1$ be distributed with compound Poisson $\lambda_1=2$ and discrete indeminizations: $f_1 (x), x \geq 0$. And let $S_2$ be compound Poisson distributed with $\lambda_2=4$, and discrete ...
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48 views

Exponentially distributed claims using Cramer-Lundberg approximation

I'm a bit stuck with my homework in a subject called "Risk Theory" and therefore I'm longing for your help! I know that the hints has been given etc, but I'm still confused and any kind of help would ...
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21 views

Value at Risk boundaries

For my bachelor thesis i need to proof the following proposition: $X_1, X_2$ random variables with $F_1=F_2=F$ continuous distribution concentrated on [0,$\infty$) with an ultimately decreasing ...
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29 views

Filtered Historical Simulation FHS for VAR

If I wish to run a FHS for VaR model, first I estimate the GARCH model on the historical returns $r_{t}$, then I obtain the historical innovation time series as $z_{t}=\frac{r_{t}}{\sigma_{t}}$, where ...
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2answers
29 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 ...
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68 views

Expected value or MAP to summarize a posterior distribution

Suppose we have a posterior distribution over a parameter, what would be the right way to summarize it? On one hand, we have the MAP (maximum a posterior), where we would get the mode (maximum) of the ...
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19 views

Adjustment coefficient does not exist, but MGF exists

The question is about Ruin theory and the Cramer-Lundberg model. I am wondering if there is an example of distribution where the MGF is finite, but the adjustment coefficient does not exist. Can you ...
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55 views

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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34 views

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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127 views

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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54 views

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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340 views

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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60 views

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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387 views

Calculating Risk of Ruin for known profit/loss and probabilities

In a scenario where I have an binary event where the outcome is uncertain and in one outcome, I profit and in the other I make a loss (and the sizes of the profit and loss are known), but overall I ...
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39 views

Stochastic stability

In this book (p. 37, 43, 44) the notion of stochastic stability of a stochastic process is defined by the condition $$ \mathbb{E} \left[\sum_{j=0}^{\infty} \|x_k\|^2\right] < \infty.\tag{1} $$ It ...
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1answer
33 views

How to find risk contingency reserve?

Attach, is one of the MCQ from exam-engine, I want to know how the answer was calculated. The subject is related to `risk assessment and controls selection.
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1answer
126 views

Risk adjusted premium principle

I had to do : Show that the risk-adjusted premium principle with $g(x) = x^{1/\rho}$ is consistent, scale-invariant and satisfies the no-ripoff property. I know from my course that if $g(x) = x^{1/\...
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701 views

Value at Risk (VAR), TVAR, ES for uniform distribution

The following is a problem for Kaas "Modern acturial risk theory". I am trying to understand my concepts for TVAR, VAR, and CTE Question: Give expressions from VAR, TVAR, and CTE in case S ~ Uniform[...
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1answer
39 views

Decision making based on epistemic uncertainty

You have a yellow button and a blue button. The yellow button saves 10 lives. The blue button saves 1 life. However, both buttons have an independent, unknown probability of working. Your objective is ...
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Function for Risk Score based on Decision Tree

I want to build a model that generates a risk score based on a decision tree of six levels. All these levels and options have the same weight (1). The first level has three options. The second level ...
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1answer
56 views

A question about probabilities and risk

Consider a game where you gamble with money. Several outcome may exist with different probabilities of occuring where you win or lose varying amounts. Let $X$ equal the amount of money retaind when ...
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1answer
83 views

Value-at-risk problems

I have this data Loss L1, L2, L3... Ln Probability P1, P2, P3... Pn And I'm trying to calculate value-at-risk using this model https://en.wikipedia.org/wiki/Value_at_risk#Mathematical_definition ...
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1answer
149 views

Directional derivative of risk measure

Let $S,Z \in L^\infty(\Omega,\mathcal{F},\mathbb{P})$ and $\rho \colon L^\infty \rightarrow \mathbb{R}$ be a risk measure. When is the directional derivative (Gâteaux derivative) $$\lim_{\varepsilon \...
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308 views

Proving function is convex argmin

How can I show that the following function is convex in which Z is a random variable? $$\rho(Z)=\frac{2}{3}\mathrm{argmin}_{t}\{t+10\mathbf{E}[Z-t]_{+}\}+\frac{1}{3}\mathrm{argmin}_{t}\{t+5\mathbf{E}[...
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1answer
424 views

Mean residual life function limits

In my paper about mean residual life functions and heavy tails I've come across the examples listed in the picture below. http://i.stack.imgur.com/OnklD.png source (page 124/125): http://www.mi.uni-...
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358 views

A level continuously paying annuity pay 1500 each month. Find the present value

Here is the full question: A level continuously paying annuty pays $\$ 1500$ each month for eight year. The force of interest is $\delta(t)=\frac{2t}{t^2+5}$ where time is measured in years. Find the ...
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1answer
89 views

$X<Y$ a.s. but $\mathrm{AV@R}_\alpha[X] = \mathrm{AV@R}_\alpha[Y]$

$\newcommand{\avar}{\mathrm{AV@R}_{\alpha}} \renewcommand{\Re}{\mathbb{R}}$Let $(\Omega, \mathscr{F}, \mathrm{P})$ be a probability space and define $\mathcal{Z}:=\mathcal{L}_p(\Omega, \mathscr{F}, \...
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3answers
113 views

Basic math, Confusion working out chance of failure.

While I am quite technical, I am not a mathematician. So please be forgiving if this question seems overly simple. My probability abilities left me long ago. I have a problem which I was arrogant ...
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1answer
330 views

Calculate the VaR at level alpha of the given CDF

I have to compute the mean value, the variance, the value-at-risk $\mathrm{V@R}_{\alpha}$ and the expected shortfall $\mathrm{ES}$ of a random variable with CDF $$ F(x) = \begin{cases} 1- (\frac{3}{...
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2answers
80 views

Convex risk measures

What is the intuitive explanation for convex risk measures represented as: $$\rho(X)=\sup_{P\in Q}\{E_{P}(-X)+\alpha(P)\}$$ where $\alpha(P)$ is a penalty function depending on the plausibility of P. ...
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23 views

Formulating simple games as Bayesian problems

Say you are in a game where every win doubles your money and every lose halves. You can walk away any time with the money you have by giving up? The rules of game is every step a problem is posed ...
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1answer
149 views

Using risk aversion

I'm trying to figure out what the non-stochastic equivalent payment is for someone who is risk-averse. Suppose we have a lottery that pays out\$100 with probability one half and \$0 with probability ...
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37 views

X% per turn over Y turns? Is my working out correct? risks and probabilities.

4% per turn * (over the course of) 25 turns. e.g. 1 Person $\alpha$ let's call him bob. Bob a 4% chance of winning something every day for 25 days. Whats the chance of him winning? (about 100%?) e....
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2k views

Tangent portfolio weights without short sales?

Consider a mean-variance investor in a world with a risk-free asset. Let $R_f>0$ be the return of the risk-free asset, $\mathbb{E}(R_i)>R_f$ the expected return of the risky asset $i$ and $SD(...
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1answer
360 views

Portfolio Theory - Finance Riskless Assets Return

A market consists of two risky assets and one riskless asset. Asset 1 has a return of 8% and a risk of 10%. Asset 2 has a return of 16% and a risk of 30% The correlation between the returns of the two ...
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287 views

Heavy-tailed distributions

I have encountered the following two definitions of heavy-tailedness (right tail) for a $[0,\infty)$-valued random variable $X$ satisfying $\mathbb{E}[X]<\infty$: (i) $\limsup_{x\to\infty}\frac{\...
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1answer
76 views

Algorithm for risky investments in banks

I made the following programming question on stack overflow but the users said it was more of math question. Here it is. Situation You start with a fixed amount of money, take it as $\$1000$. You ...
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1answer
505 views

Calculate single “battle” outcome odds for RISK

I am trying to reproduce the values in this odds ratio table from Wikipedia. For all those unfamiliar with RISK, this is a game where units fight against each other via the roll of the dice: The ...