Additivity of Value at Risk
$begingroup$
I'm struggling with the following exercise:
Let $(Omega,mathcal{F},mathbb{P})$ be a probability space and $X,Y$ be two real-valued random varaibles such that their corresponding cumulative distribution functions $F_X$ and $F_Y$ are continuous and strictly monotonically increasing. Further, we assume that $(X,Y)sim(q_X^-(U),q_Y^-(U))$, where the random variable $U$ is uniformly distributed on $(0,1)$. Show that in this case
$$
VaR_lambda(X+Y)=VaR_lambda(X)+VaR_lambda(Y)
$$
My idea: Since $F_X$ and $F_Y$ are continuous and strictly monotonically increasing, there exist continuous inverse functions $F_X^{-1}$ and $F_Y^{-1}$ of $F_X$ and $F_Y$ such that $VaR_lambda(X)=F^{-1}_X(lambda)$ and $VaR_lambda(Y)=F^{-1}_Y(lambda)$. My main problem is that I don't really know what this $(X,Y)sim(q_X^-(U),q_Y^-(U))$ actually means (where $q_X^-(lambda)$ is the lower $lambda$-quantile) or how to deal with this. One thing I saw is that for a uniformly distributed $U$ on $(0,1)$, we have that $F(X)=mathbb{P}(Uleq F(X))=mathbb{P}(q(U)leq x)$. I'm not sure if this is correct in this setting or helpful, but these are the only approaches I have so far.
Thank you for your help.
probability-distributions random-variables finance
edited Jan 22 at 19:14
gt6989b
34.5k22456
asked Jan 22 at 19:05
ToxxiqqToxxiqq
173
$endgroup$
add a comment |
$begingroup$
I'm struggling with the following exercise:
Let $(Omega,mathcal{F},mathbb{P})$ be a probability space and $X,Y$ be two real-valued random varaibles such that their corresponding cumulative distribution functions $F_X$ and $F_Y$ are continuous and strictly monotonically increasing. Further, we assume that $(X,Y)sim(q_X^-(U),q_Y^-(U))$, where the random variable $U$ is uniformly distributed on $(0,1)$. Show that in this case
$$
VaR_lambda(X+Y)=VaR_lambda(X)+VaR_lambda(Y)
$$
My idea: Since $F_X$ and $F_Y$ are continuous and strictly monotonically increasing, there exist continuous inverse functions $F_X^{-1}$ and $F_Y^{-1}$ of $F_X$ and $F_Y$ such that $VaR_lambda(X)=F^{-1}_X(lambda)$ and $VaR_lambda(Y)=F^{-1}_Y(lambda)$. My main problem is that I don't really know what this $(X,Y)sim(q_X^-(U),q_Y^-(U))$ actually means (where $q_X^-(lambda)$ is the lower $lambda$-quantile) or how to deal with this. One thing I saw is that for a uniformly distributed $U$ on $(0,1)$, we have that $F(X)=mathbb{P}(Uleq F(X))=mathbb{P}(q(U)leq x)$. I'm not sure if this is correct in this setting or helpful, but these are the only approaches I have so far.
Thank you for your help.
probability-distributions random-variables finance
edited Jan 22 at 19:14
gt6989b
34.5k22456
asked Jan 22 at 19:05
ToxxiqqToxxiqq
173
$endgroup$
$begingroup$
So, readers are supposed to guess what your notation means?
$endgroup$
– uniquesolution
Jan 22 at 19:17
1
$begingroup$
If the functions $F_X, F_Y$ are continuous and strictly monotonically increasing, $q^-_X=F^{-1}_X$ and likewise for $Y$.
$endgroup$
– Raskolnikov
Jan 22 at 23:10
add a comment |
$begingroup$
I'm struggling with the following exercise:
Let $(Omega,mathcal{F},mathbb{P})$ be a probability space and $X,Y$ be two real-valued random varaibles such that their corresponding cumulative distribution functions $F_X$ and $F_Y$ are continuous and strictly monotonically increasing. Further, we assume that $(X,Y)sim(q_X^-(U),q_Y^-(U))$, where the random variable $U$ is uniformly distributed on $(0,1)$. Show that in this case
$$
VaR_lambda(X+Y)=VaR_lambda(X)+VaR_lambda(Y)
$$
My idea: Since $F_X$ and $F_Y$ are continuous and strictly monotonically increasing, there exist continuous inverse functions $F_X^{-1}$ and $F_Y^{-1}$ of $F_X$ and $F_Y$ such that $VaR_lambda(X)=F^{-1}_X(lambda)$ and $VaR_lambda(Y)=F^{-1}_Y(lambda)$. My main problem is that I don't really know what this $(X,Y)sim(q_X^-(U),q_Y^-(U))$ actually means (where $q_X^-(lambda)$ is the lower $lambda$-quantile) or how to deal with this. One thing I saw is that for a uniformly distributed $U$ on $(0,1)$, we have that $F(X)=mathbb{P}(Uleq F(X))=mathbb{P}(q(U)leq x)$. I'm not sure if this is correct in this setting or helpful, but these are the only approaches I have so far.
Thank you for your help.
probability-distributions random-variables finance
edited Jan 22 at 19:14
gt6989b
34.5k22456
asked Jan 22 at 19:05
ToxxiqqToxxiqq
173
$endgroup$
I'm struggling with the following exercise:
Let $(Omega,mathcal{F},mathbb{P})$ be a probability space and $X,Y$ be two real-valued random varaibles such that their corresponding cumulative distribution functions $F_X$ and $F_Y$ are continuous and strictly monotonically increasing. Further, we assume that $(X,Y)sim(q_X^-(U),q_Y^-(U))$, where the random variable $U$ is uniformly distributed on $(0,1)$. Show that in this case
$$
VaR_lambda(X+Y)=VaR_lambda(X)+VaR_lambda(Y)
$$
My idea: Since $F_X$ and $F_Y$ are continuous and strictly monotonically increasing, there exist continuous inverse functions $F_X^{-1}$ and $F_Y^{-1}$ of $F_X$ and $F_Y$ such that $VaR_lambda(X)=F^{-1}_X(lambda)$ and $VaR_lambda(Y)=F^{-1}_Y(lambda)$. My main problem is that I don't really know what this $(X,Y)sim(q_X^-(U),q_Y^-(U))$ actually means (where $q_X^-(lambda)$ is the lower $lambda$-quantile) or how to deal with this. One thing I saw is that for a uniformly distributed $U$ on $(0,1)$, we have that $F(X)=mathbb{P}(Uleq F(X))=mathbb{P}(q(U)leq x)$. I'm not sure if this is correct in this setting or helpful, but these are the only approaches I have so far.
Thank you for your help.
probability-distributions random-variables finance
probability-distributions random-variables finance
edited Jan 22 at 19:14
gt6989b
34.5k22456
asked Jan 22 at 19:05
ToxxiqqToxxiqq
173
edited Jan 22 at 19:14
gt6989b
34.5k22456
asked Jan 22 at 19:05
ToxxiqqToxxiqq
173
edited Jan 22 at 19:14
gt6989b
34.5k22456
edited Jan 22 at 19:14
gt6989b
34.5k22456
edited Jan 22 at 19:14
gt6989b
34.5k22456
34.5k22456
asked Jan 22 at 19:05
ToxxiqqToxxiqq
173
asked Jan 22 at 19:05
ToxxiqqToxxiqq
173
asked Jan 22 at 19:05
ToxxiqqToxxiqq
173
173
$begingroup$
So, readers are supposed to guess what your notation means?
$endgroup$
– uniquesolution
Jan 22 at 19:17
1
$begingroup$
If the functions $F_X, F_Y$ are continuous and strictly monotonically increasing, $q^-_X=F^{-1}_X$ and likewise for $Y$.
$endgroup$
– Raskolnikov
Jan 22 at 23:10
add a comment |
$begingroup$
So, readers are supposed to guess what your notation means?
$endgroup$
– uniquesolution
Jan 22 at 19:17
1
$begingroup$
If the functions $F_X, F_Y$ are continuous and strictly monotonically increasing, $q^-_X=F^{-1}_X$ and likewise for $Y$.
$endgroup$
– Raskolnikov
Jan 22 at 23:10
$begingroup$
So, readers are supposed to guess what your notation means?
$endgroup$
– uniquesolution
Jan 22 at 19:17
$begingroup$
So, readers are supposed to guess what your notation means?
$endgroup$
– uniquesolution
Jan 22 at 19:17
1
1
$begingroup$
If the functions $F_X, F_Y$ are continuous and strictly monotonically increasing, $q^-_X=F^{-1}_X$ and likewise for $Y$.
$endgroup$
– Raskolnikov
Jan 22 at 23:10
$begingroup$
If the functions $F_X, F_Y$ are continuous and strictly monotonically increasing, $q^-_X=F^{-1}_X$ and likewise for $Y$.
$endgroup$
– Raskolnikov
Jan 22 at 23:10
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Here's a hint on top of my comment:
$$X+Y sim q_X^-(U) + q_Y^-(U) = (q_X^- + q_Y^-)(U) ; .$$
answered Jan 23 at 10:53
RaskolnikovRaskolnikov
12.6k23571
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3083566%2fadditivity-of-value-at-risk%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Here's a hint on top of my comment:
$$X+Y sim q_X^-(U) + q_Y^-(U) = (q_X^- + q_Y^-)(U) ; .$$
answered Jan 23 at 10:53
RaskolnikovRaskolnikov
12.6k23571
$endgroup$
add a comment |
$begingroup$
Here's a hint on top of my comment:
$$X+Y sim q_X^-(U) + q_Y^-(U) = (q_X^- + q_Y^-)(U) ; .$$
answered Jan 23 at 10:53
RaskolnikovRaskolnikov
12.6k23571
$endgroup$
add a comment |
$begingroup$
Here's a hint on top of my comment:
$$X+Y sim q_X^-(U) + q_Y^-(U) = (q_X^- + q_Y^-)(U) ; .$$
answered Jan 23 at 10:53
RaskolnikovRaskolnikov
12.6k23571
$endgroup$
Here's a hint on top of my comment:
$$X+Y sim q_X^-(U) + q_Y^-(U) = (q_X^- + q_Y^-)(U) ; .$$
answered Jan 23 at 10:53
RaskolnikovRaskolnikov
12.6k23571
answered Jan 23 at 10:53
RaskolnikovRaskolnikov
12.6k23571
answered Jan 23 at 10:53
RaskolnikovRaskolnikov
12.6k23571
answered Jan 23 at 10:53
RaskolnikovRaskolnikov
12.6k23571
12.6k23571
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3083566%2fadditivity-of-value-at-risk%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
So, readers are supposed to guess what your notation means?
$endgroup$
– uniquesolution
Jan 22 at 19:17
1
$begingroup$
If the functions $F_X, F_Y$ are continuous and strictly monotonically increasing, $q^-_X=F^{-1}_X$ and likewise for $Y$.
$endgroup$
– Raskolnikov
Jan 22 at 23:10