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Coherent Risk Measure - Monotonicity

ami44

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Hallo,

I got stuck on practice question 4. in the study notes for Chapter 3 of Dowd.

4. Portfolios (X) and (Y) each have volatility of 20%, but portfolio (Y) has a higher return and therefore its absolute VaR is lower; i.e., Absolute VaR = - return * T + deviate * volatility * SQRT(T). Which coherence property does this illustrate?
a) Monotonicity
b) Subadditivity
c) Positive Homogeneity
d) Translational invariance

The given answer is a), but I don't understand why. Just because Y has a higher return than X doesn't mean Y > X a.s. which I thought is the condition for monotonicity.
I would appreciate it, if somebody can point me in the right direction here.
 

brian.field

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This post may help.

https://www.bionicturtle.com/forum/threads/coherent-risk-measure.4723/


. while Dowd (FRM assigned; he gives your expression above, that I copied of course) says "Let X and Y represent any two portfolios’ P/L (or future values, or more loosely, the two portfolios themselves)" and, in the footnote, "Monotonicity means that a random cash flow or future value Y that is always greater than X should have a lower risk: this makes sense, because it means that less has to be added to Y than to X to make it acceptable, and the amount to be added is the risk measure." (FRM assigned Dowd)
 

David Harper CFA FRM

David Harper CFA FRM
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Thank you @brianhfield I swear I was just going to include a link to that thread ....

Hi @ami44 Monotonicity has been a challenge over the years, because we've had different authors employ the formula differently; e.g., Wilmott's is different than Dowd's. I've come to view Dowd's formula as correct, if we are mindful of his context. He gives:

Monotonicity: Y ≥ X --> ρ(Y) < ρ(X)

Note this is the only coherence property which includes the "naked" X and Y, rather than operate entirely on the risk measure rho(.). So, as I read it, this is the only coherence property which involves return, as it relates to the risk measure. ρ(.) is the risk measure; it's worth groking Dowd's specific meaning: "We can then interpret the risk measure ρ(.) as the minimum extra cash that has to be added to the risky position and invested prudently in some reference asset to make the risky position acceptable. If ρ(.) is negative, its negativity can be interpreted as the maximum amount that can be safely withdrawn, and still leave the position acceptable" (footnote 11, page 33). But (Y) and (X) are future portfolio values (or they could be returns).

So this monotonicity (IMO) is saying: if the future value of portfolio (Y) is greater than the future value of portfolio (X), then ceteris paribus, a monotonic risk measure will be lower Y than X; i.e., it will say that X is riskier. Notice that absolute VaR does this: if absolute VaR = -μ + ασ, then for unchanged volatility (σ), higher return lowers the risk (absolute VaR).

I guess it's an opportunity, also, to grok Dowd's "definition" of relative ρ(.) in the deeper abstract. Imagine we have an (absolute) VaR budget and portfolio (Y), with the higher return (or higher future value or higher expected cash flow) and volatility = 20%, just happens to exactly match our VaR budget. If we instead invested in portfolio (X), it's VaR would be higher and we would exceed the VaR budget, such that we would need to allocate some portion to cash, in order to reduce the VaR until we hit the budget. By needing to add cash to (X), we "prove" that it is riskier.

Maybe to your point: in my view, relative VaR only satisfies monotonicity due to the "="! If volatilities are equal, then relative VaR is unchanged (unchanged mean variance). I hope that's helpful!
 
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ami44

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Brianhfield and David thank you for your answers.
If I understand that correctly, then Dowd uses X and Y as portfolio P/L while others define them as Losses. The two definitions differ in the sign. That seems to cause the confusion about the < and > signs in the monotonicity condition.
This also changes the sign in the translation invariance:
X is P/L (Dowd): rho(X+c) = rho(X) - c
X is Loss: rho(X+c) = rho(X) + c

@David: you are right, that the VaR for portfolio Y is higher than for X if the return is higher and the variance the same. But I still can't deduce that from the monotonicity condition, which assumes that Y > X, which is more than the returns of Y being higher then of X.
(I interpret return here as being the expected value of the portfolios, the μ in your formula above).
Do I miss something.
 

David Harper CFA FRM

David Harper CFA FRM
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Hi @ami44

Yes I agree with your interpretation of why sometimes the signs are different! For monotonicity, Wilmott gives: if Y ≤ X --> ρ(Y) < ρ(X) [intro quant financie, page 470] but presumably he means losses rather than profits. But, we know that it's okay to express either L/P or P/L such that I think your point is good: we don't need to get too caught up in the sign. The point of monotonicity is that, ceteris paribus, the portfolio with a higher expected future value is less risky according to a monotonic measure.

But that gets to your question ("But I still can't deduce that from the monotonicity condition"). I don't think you can deduce it. Dowd is exact, this is the only property that uses mathematical "implication:"

Y ≥ X ⇒ ρ(Y) < ρ(X); i.e.,
If Y ≥ X implies ρ(Y) < ρ(X) then the risk measure ρ(.) satisfies the monotonic condition

... so i don't think you deduce the monotonicity condition so much as you determine it to be true if and only if ρ(Y) < ρ(X) follows from Y ≥ X.

Re: This also changes the sign in the translation invariance: X is P/L (Dowd): rho(X+c) = rho(X) - c; if X is Loss: rho(X+c) = rho(X) + c
I don't think i agree because ρ(.) does not suffer the ambiguity of L/P versus P/L. So I could be wrong but i don't see how this works: X is Loss: rho(X+c) = rho(X) + c, as rho(X) + c looks to be unambigiously an increase in risk.

Invariance is ρ(X + n) = ρ(X) - n ... it says that adding cash (n) reduces the risk by (n). Thanks!
 

ami44

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Hello David,

thanks for the clarification, I will ignore the mentioned practice question then.

@translation invariance: if X is P/L than X+c is adding cash and the risk is reduced. On the other hand if X is L/P then X+c is adding loss and the risk is increased.
 

David Harper CFA FRM

David Harper CFA FRM
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Hi @ami44
  • Okay I see why you say that notionally (i.e., "On the other hand if X is L/P then X+c is adding loss") but translation invariance doesn't contemplate "adding loss" so I see how you are being consistent but "adding loss" forces you outside the condition. The (n) must be riskless.
  • Re: "I will ignore the mentioned practice question then." I am not sure what you mean? I think the original practice question survived our inquiry and looks good; i.e., absolute VaR looks to satisfy montonicity. Thanks!
 

ami44

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Hallo David,

I understood the practice question as stating the following:
If you have two portfolios X and Y with the same variance, but Y has higher return than X, than you can deduce from monotonicity of rho(.) that rho(X) >= rho(Y).

I interpret Y having higher returns in the context of this Question as E(Y) > E(X). But to use Monotonicity you need Y > X a.s. which is something different, as you confirmed.

So something is wrong either with the wording or with my understanding of the question.
 

David Harper CFA FRM

David Harper CFA FRM
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Hi @ami44

No, I don't agree, sorry.

The question you quoted says: "4. Portfolios (X) and (Y) each have volatility of 20%, but portfolio (Y) has a higher return and therefore its absolute VaR is lower; i.e., Absolute VaR = - return * T + deviate * volatility * SQRT(T)."
This appears, to me, to be consistent with Dowd's: Monotonicity: Y ≥ X --> ρ(Y) < ρ(X)

Dowd's variables are symbolic:
  • As he himself states, X and Y are portfolio P/Ls but can be future cash flows. They are already future expected values, E(.), such that it is equivalent to say: Monotonicity: if E(Y) ≥ E(X) --> ρ(Y) < ρ(X)
  • There isn't a deduction. Monotonicity is met if ρ(Y) < ρ(X) is observed given that E(Y) ≥ E(X), and the ρ(Y) is entirely symbolic, it means any risk measure. All monotonity really means, i think, is: ceritis peribus, a higher expected return does not by itself make the portfolio riskier, according to the risk metric. I hope that makes sense!
 

David Harper CFA FRM

David Harper CFA FRM
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Hey @ami44 I was just supporting a (lower board) source Q&A (same question as here: https://www.bionicturtle.com/forum/threads/l2-t5-71-coherent-risk-measures.3655/#post-33122) and @Aenny helpfully shared this paper, which I've never seen before: http://www.columbia.edu/~mh2078/RiskMeasures.pdf
This is interesting because it speaks exactly to our conversation. This author is using losses (L1 and L2) rather than expected future values (X and Y in Dowd), so as we confirmed ourselves, he's got an alternative definition for monotonicity that only looks different.

It's also interesting because his Translation Invariance matches yours above, your X is Loss: rho(X+c) = rho(X) + c.
Ergo, it looks like you are correct about that and I am incorrect ... although I don't understand how it can be, it still doesn't make sense to me that adding cash (or capital) can increase the risk. And, in this paper, greater rho is apparently greater risk. So I disagree with this paper, or probably I don't understand it. Because it agrees with you! :)
 

David Harper CFA FRM

David Harper CFA FRM
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About that paper @ http://www.columbia.edu/~mh2078/RiskMeasures.pdf

It has the same definition of the risk measure, ρ(.), as Dowd's:
  1. Paper: "ρ(L) may be interpreted as the riskiness of a portfolio or the amount of capital that should be added to a portfolio with a loss given by L, so that the portfolio can then be deemed acceptable from a risk point of view. Note that under this latter interpretation, portfolios with ρ(L) < 0 are already acceptable and do not require capital injections. In fact, if ρ(L) < 0 then capital could even be withdrawn while the portfolio would still remain acceptable."
  2. Dowd: "let ρ(.) be a measure of risk over a chosen horizon (footnote 11) ... footnote 11: "11 At a deeper level, we can also start with the notion of an acceptance set, the set of all positions acceptable to some stakeholder (e.g., a financial regulator). We can then interpret the risk measure ρ(.) as the minimum extra cash that has to be added to the risky position and invested prudently in some reference asset to make the risky position acceptable. Ifρ(.) is negative, its negativity can be interpreted as the maximum amount that can be safely withdrawn, and still leave the position acceptable"
Dowd's translation invariance, ρ(X + n ) --> ρ(X) - n , continues to make sense to me: adding (n) reduced the additional cash required by (n); or, if you like, adding cash reduces the risk. Note that, for example, volatility as a risk measure appears to satisfy translation invariance: if a portfolio has only a single risky asset with volatility of 20% (eg), then doubling the portfolio by adding cash reduces the portfolio volatility to 10%. But I don't quite understand the paper's function: I don't see how adding (a) can increase the risk by (a). Thanks,
 
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ami44

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Hello David,

Thanks for the interesting link, I skimmed over it.

You wrote:
------ begin citation -----
  • As he himself states, X and Y are portfolio P/Ls but can be future cash flows. They are already future expected values, E(.), such that it is equivalent to say: Monotonicity: if E(Y) ≥ E(X) --> ρ(Y) < ρ(X)
  • There isn't a deduction. Monotonicity is met if ρ(Y) < ρ(X) is observed given that E(Y) ≥ E(X), and the ρ(Y) is entirely symbolic, it means any risk measure. All monotonity really means, i think, is: ceritis peribus, a higher expected return does not by itself make the portfolio riskier, according to the risk metric.
------ end citation -----
I don't have Dowds book and maybe that's why I'm confused. I'm coming at it from a mathematical perspective. So I understand X and Y as random variables that represent future portfolio values and rho(.) as a function that assigns a risk measure (a number) to all such random variables. X and Y as random variables are different from their expected values E(X) and E(Y).

With that in mind, montonicity is Y ≥ X --> ρ(Y) < ρ(X). This means that if the value of Portfolio Y is always greater than the Value of Portfolio X, then the risk of X is higher than the risk of Y.
If Y ≥ X, it follows that E(Y) ≥ E(X), but not the other way around and so the monotonicity condition is different than E(Y) ≥ E(X) --> ρ(Y) < ρ(X).

Obviously, you and Dowd seem to have a different interpretation. I understand that now. I just wanted to tell you my point of view, so that you can understand where I was coming from and understand why I found the practice question confusing. Our discussion helped me to understand your interpretation better. That is a plus.

Referring to translation invariance, I understand it thusly:
If X is P/L and c is positive, then X+c is adding cash to the portfolio (decreasing risk). However, c can also be negative, then X+c would be taking cash out (increasing risk). I think c can have either negative or positive values (I think you mentioned that too).

If X is a loss, then c also changes sign, i.e. a positive c in X+c means adding loss (taking cash out, increasing risk), and vice versa. So rho(X+c) = rho(X) + c just means that, taking cash c out, increases the risk by c.
 

David Harper CFA FRM

David Harper CFA FRM
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Hi @ami44

re: translation invariance: thank you, I didn't see that: by interpreting(+c) as removing cash or capital, the paper's equation does make sense. Nice!

re: "you and Dowd seem to have a different interpretation." please don't get me wrong. I agree with Dowd on coherence, Dowd's coherence all makes perfect sense to me. But, yes, of course I agree with: If Y ≥ X, it follows that E(Y) ≥ E(X), but not the other way around and so the monotonicity condition is different than E(Y) ≥ E(X) --> ρ(Y) < ρ(X). Yes, that was lazy ... :oops:

This is all Dowd defines of X & Y: "Let X and Y represent any two portfolios’ P/L (or future values, or more loosely, the two portfolios themselves)" such that I still think my question is okay, but I think I see that you are being precise (accurate) about "expected", E(.), then I think I agree with that. Please correct me if I am wrong, but it sounds like you would be satisfied with (for example):

"4. Portfolios (X) and (Y) each have volatility of 20%, but portfolio (Y)'s return dominates portfolio (X)'s return--such that it's also true that E(Y) is greater than E(Y)-- and therefore its absolute VaR is lower; i.e., Absolute VaR = - return * T + deviate * volatility * SQRT(T). Which coherence property does this illustrate?
... i'm thinking some kind of dominance (http://en.wikipedia.org/wiki/Stochastic_dominance ) as i don't think monotonicity's Y > X requires, in non-overlapping fashion, the worst Y to be better than the best X
I think i finally see that you are getting at the imprecision of my implied expectation (although I'm not sure I am any less than Dowd's precision!). Please let me know? Because that's a great point, thanks!
 

ami44

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re: "re: "you and Dowd seem to have a different interpretation.""
I was unclear, I meant yours and Dowds interpretation differ from mine. Since I don't know Dowds book and you do, I just assumed you are in line with the book.

Indeed I like your knew wording of the question better than the old one, but reading the wikipedia article I realized that there are multiple definition of dominance and I think only statewise dominance would be appropriate here. May I suggest the following wording:
4. Portfolios (X) and (Y) each have volatility of 20%, but portfolio (Y)'s return dominates portfolio (X)'s return in a way, that the value of Y is always greater or equal than the value of X. .....

I agree, that non-overlapping dominance like you describe it is not appropriate here. Y > X means Y is greater than X for every outcome (mathematical technicality: except on null sets http://en.wikipedia.org/wiki/Almost_surely ).
 

David Harper CFA FRM

David Harper CFA FRM
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awesome @ami44 thank you for your patience in conveying your points. It was worth it to me, you pushed me to a deeper appreciation of these coherence properties that I had started to take for granted :)
 
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