| Coherent Risk: Part II ... a continuation of Part I |
We're trying to understand the theory of "coherent risk" as proposed by Artzner, Delbaen, Eber and Heath (ADEH).
The four axioms associated with "coherent risk" (as commonly stated) are:
If X and Y are representative of two portfolios and λ and c are constants, then the risk measure R
is coherent if:
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We want to consider each axiom in turn and (hopefully) understand what they're saying.
We give them the names used by Artzner et al (which we're calling ADEH):
| Axiom T: R[X + r c] = R[X] - c where "r" is the total gain of a risk-free investment. Translation Invariance |
We note that, since X and rc are added
(on the left side), then X and rc must be in the same units.
Further, since c is subtracted from R[X ] (on the right side),
then R[X ] and c must be in the same units.
We conclude that all of R[X ], X and c must be expressed in the same units.
>Huh?
No sensible argument would involve the addition of variables with different units.
Have you ever seen a formula that involves adding acres to hours or kilometres to kilograms?
In our case, it means that if X is in U.S. dollars, then R[X ] must not be expressed in deutschmarks.
>Or in hectares or miles or degrees centigrade or ... ?
Don't be silly!
In fact, one of the four authors, Heath, says that axiom T is there to ensure that all are in the same currency.
However, we'd like to understand what that first axiom means.
>In plain English, right?
Right.
Suppose that we interpret $X ≥ 0 as some portfolio goal that we wish to achieve ... at some time in the future, say in T months.
Suppose, further that we need an extra cash amount invested now to achieve that goal.
Further, we take R[X] to mean that required extra cash amount.
>If we wish to reach $100K we'd need R[100K] right now.
No, R[100K] stands for the EXTRA cash invested in order to reach $100K.
If R[100K] ≤ 0 then we don't need any extra cash invested now, so that future goal is a so-called "acceptable" goal.
>Acceptable?
Yes, that's the word used by ADEH.
Indeed, the set of goals for which R[X] ≤ 0 they call the "acceptance set".
In general, if R[X] ≤ 0, then X ≥ 0 is an acceptable goal because it won't require any extra cash investment.
>Isn't that one of those funny axioms in Part I? The one called [d]?
Yes.
Note that R[X] ≥ 0 means we're talking about a "risky" investment.
Now we consider having as our goal, not X but X+rc, where r is the total gain (over T months)
of a risk-free investment.
In other words, if we invest c now, we're guaranteed an amount rc in T months. (It's "risk-free, see?)
I should point out that if our risk-free investment has a total return of, say 4%, we'd take r = 1.04.
In general, r would be a number somewhat greater than (or maybe equal to) "1".
So, the big question:
How much extra cash is required to achieve that new goal of X+rc?
Whatever it is, we'd call it R[X+rc] ... according to our definition of R.
Remember, we need R[X] extra cash invested now in order to achieve X.
To achieve a portfolio of X+rc we'd need an extra R[X]
just to achieve X.
But now we'd need less to achieve X+rc since we're investing $c in a risk-free asset.
... since c grows to rc in T months.
How much less? $c less!
Hence R[X+rc] = R[X] - c
>That's confusing!
Don't worry about it. We don't have to agree. We just have to understand what ADEH are doing.
>And you're talking from the horse's mouth?
Yes.
Notice, however, that if our "risk-free" investment is to put the cash under our pillow, then r = 1 and the first axiom becomes simply:
R[X + c] =
R[X] - c
... and that's one of the (many!) ways the first axiom is (often!) stated 
| Axiom S: R[X+Y] ≤ R[X] + R[Y] Subadditivity |
The plain English interpretation is that, if we combine two portfolios, the risk is not greater than the sum of the risks associated with each.
In other words, we have the idea of "Risk Diversification" as being a financially healthy thing to do.
In other words ...
>What about that "extra" cash thing?
Okay, it means that the amount of extra cash required to achieve a total portfolio of X+Y is no greater than:
(the extra cash need to achieve X) + (the extra cash need to achieve Y).
This axiom is generally regarded as the most important of the four axioms.
>Then why isn't it axiom numero uno?
You asking me?
Anyway, ADEH state this axiom in the "brisk form": A merger does not create extra risk and give the following example:
A large company has two autonomous departments with risks R[X] and R[Y].
If Head Office has a cash amount C to cover the risks associated with both departments,
then they may decentralize their cash reserves, assigning cash C1 and C2 to each department ... so long as C1 + C2 = C.
>Huh?
These cash reserves C, C1 and C2 necessary to cover the risks are ... uh, Risks.
So the Risk C is no greater than the sum of the
Risks C1 + C2.
>It's still confusing!
Okay, here's another of their examples:
Suppose that a Risk measure did NOT satisfy axiom S.
That is, suppose that R[X+Y] > R[X] + R[Y].
Than an individual wishing to assume the risk R[X+Y] would open two accounts.
That would incur the smaller margin requirement of R[X] + R[Y].
>And that'd worry the stock exchange, right?
Apparently 
| Axiom PH: If λ > 0 , then R[λX] = λ R[X] Positive Homogeneity |
We understand this to mean ...
> The cash needed for λX is λ times greater than the cash needed for just X, right?
Yes. Since it seems more common sense than mathematical machination, let's go on to the next axiom.
>Wait! Doesn't this axiom mean that
R[2X] = 2 R[X] ?
Yes, so?
>And doesn't that second axiom, axiom S, say that R[X+X] ≤ R[X] + R[X] ?
Hmmm ... so we should have R[2X] ≤ 2 R[X], according to axiom S,
yet R[2X] = 2 R[X], according to axiom PH.
Interesting, eh?
I guess the "apparent" inconsistency is meant to imply that, for a set of identical portfolios, the equality holds in axiom S.
>Identical portfolios?
Yes. Axiom PH says that R[nX] = n R[X] for any integer n = 1, 2, 3 ...
Further, the equality holds in Axiom S if the n portfolios are identical.
After all, if you consider your own portfolio of $100K to be made up of 5 identical portfolios of $20K, then that should not reduce the risk, eh?
| Axiom M: If X ≥ Y, then R[X] ≤ R[Y] Monotonicity |
We interpret this to mean that if the portfolio (which is our eventual goal) is decreased from X to Y, then our Risk won't increase.
That's it for an explanation of the four axioms. Now let's see where "coherent risk" takes us ...
for Part III