# Coherent Risk

*This tutorial written and reproduced with permission from Peter Ponzo*

Recently, a theory of coherent risk measures was proposed by Artzner, Delbaen, Eber and Heath (Thinking Coherently, 1997 and Coherent Measures of Risk, 1999)

#### Huh?

Patience. There are a jillion ways to measure “risk”, perhaps the most common being Volatility or Standard Deviation. That is, you might look at historical returns of a portfolio, determine the Mean annual return and calculate some average deviation of the returns from this Mean. If that deviation is large, we’re tempted to say that that particular portfolio is “risky”.

#### That’s volatility, eh? Just about everybody uses that as a risk measure, right?

Wrong, however …

#### What’s that coherent stuff?

Okay, let’s illustrate with an example:

I have a portfolio and so does my wife … and we trade independently. Suppose there is some measure of “risk”, we’ll call it R, and we apply it to both portfolios. The two “risks” are then: R[Me] and R[Her]. Now suppose we consider our total, family portfolio as being Me + Her and we apply this risk measure to the combined portfolio.

We’d get R[Me+Her].

Okay, here’s the question:** Is R[Me+Her] less than R[Me] + R[Her]?**

#### You mean the combined portfolio is less risky?

I’m just asking. What do you think? Would it be less … or more risky?

#### Wouldn’t it depend upon what’s R?

Exactly! But could our combined portfolio be more risky?

#### Uh … well, I really don’t understand that R-guy!

Okay, given a particular portfolio, the R function assigns some number which depends upon the makeup of the portfolio and …

#### Like percentage stocks or bonds or large cap or …

Sure, but that’s not so important as the “coherent” conditions that R should satisfy. I’ve found several “conditions”, but here’s the one we’ll consider:

If X and Y are representative of two portfolios and λ and c are constants, then the risk measure R is coherent if:

1. R[X + rc] = R[X] – c

where “r” is the total gain of a risk-free investment.

Translation Invariance

2. R[X + Y] ≤ R[X] + R[Y]

Subadditivity

3. If λ > 0 , then R[λX] = λ R[X]

Positive Homogeneity

4. If X ≥ Y, then R[X] ≤ R[Y]

Monotonicity

#### Aha! I recognize number 2!

Yes, so if we’re talking about a “coherent” risk measure, then the combined portfolio, mine and my wife’s, would NOT be more risky than my risk + her risk … and may, in fact, be less risky.

#### That’s something like “diversification decreases risk”, right?

Sort of, but …

#### And you understand all that homogeneity, monotonicity stuff?

Give me a minute or two. The literature is so confusing that I’ve seen:

*[a] the landmark paper by Artzner et al mentioned as appearing in 1998 and in 1999.
[b] the first of the four criteria for “coherent risk” written as: R[X + c] = R[X] + c.
[c] the first of the four written as: R[X + c] = R[X] – c.
[d] the first of the four replaced by: If X ≥ 0, then R[X] ≤ 0.
[e] the last of the criteria written: If X ≤ Y, then R[X] ≤ R[Y] *

Mamma mia! Who and what should we believe? Before we start this arduous journey (to understand “coherent risk”), we need to answer some questions, like:

*What does it mean to “combine” or “add” two portfolios?
What are the variables denoted by X and Y, above:
the dollar value of portfolios or their returns or expected dollar losses or the probability of losing x% … or what?
If X represents a portolio, what on earth is X + c or λX?
What is the meaning (in plain language) of the four criteria that define a “coherent risk measure”?*

#### Them’s the questions, but what’s the answers?

Give me a minute or two. I have to read this paper by Artzner, Delbaen, Eber and Heath.

#### Right from the horse’s mouth, eh?

Yes.