| Coherent Risk: Part III ... a continuation of Part II |
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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These axioms have been described like so:
- Translation Invariance: There is no additional capital requirement for an additional risk for which there is no uncertainty.
- Subadditivity: The capital requirement for two risks combined will not be greater than for the risks treated separately.
This is necessary, since otherwise companies would have an advantage (i.e. less required capital) to disaggregate into smaller companies.
OR: Diversification never hurts. Dissaggregation doesn't help. - Positive Homogeneity: The capital requirement is independent of currency changes in the unit in which the risk is measured.
- Monotonicity: If one risk always has greater losses than another risk, the capital requirement should be greater.
In Part II we understood R[X] to mean the "extra" cash investment required now in order to achieve some portfolio goal of $X at some time T months in the future.
Suppose we invest our $X in a risk-free investment that gives us a fixed annual return of, say 4%.
Then there is no risk associated with this return.
We don't need any "extra cash" or "risk capital" in order to cover the possibility of not reaching our goal.
We know what our initial investment should be. The $X goal is guaranteed.
We conclude that the Risk is 0.
If, however, our investment isn't risk-free, then we'd need an extra investment now in order to cover the possibility of not achieving our $X goal.
>Extra ... to cover the possibility of not achieving our goal? How come you didn't say that before?
I'm learning as we go along 
Consider the first case of a risk-free investment.
The standard deviation of returns is 0 and so is the Risk. (Remember that we're guaranteed a fixed 4% return.)
In the second case, the standard deviation is positive and so is the Risk. (That's because there's some risk associated with the second case.)
We ask: "Is Standard Deviation a coherent risk measure?"
If we consider the square of the standard deviation S (the VARiance), then the following is true:
S2[x+y] = S2[x] + S2[y] + 2 r S[x] S[y]
where r is the correlation between portfolios X and Y.
Since r ≤ 1, then S2[x+y] ≤ S2[x] + S2[y] + 2 S[x] S[y] =
(S[x] + S[y] )2
so that: S[x+y] ≤ S[x] + S[y] and the subadditivity axiom #2 is satisfied.
Further, S[λx] = λ S[x] is a characteristic of the standard deviation (for λ > 0)... so the positive homogeneity axiom #3 is satisfied.
>How come you use a lower case x and y whereas, when you're talking about the portfolios ...?
I use uppercase? Well ... uh, by "x" I mean the set of returns associated with the portfolio X and by "y" I mean the returns ...
>Yeah, I get it. Make it as confusing as possible, eh? But what about VAR?
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VAR? Uh ... yes, I remember VAR.
>What's the probability there?
84% of portfolios will be less than B and
>69% doesn't sound too promising to me!
>Do you believe this stuff?
|  Figure 1
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For example, Figure 3 is a sample cumulative distribution of portfolios, M months in the future.
>That's a pretty small portfolio.
|  Figure 3
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Since there's only a 5% chance that our portfolio will be less than V = 4965, there's only a 5% chance that our loss will be more than 1035.
On the other hand, if we started out with a portfolio worth V0 = 4,000 then V0 - V = 4000 - 4965 = -965 and we conclude that there's a 5% chance we'll lose -965, which is, after all, a gain of 965, eh?
>A negative value?
Why not? The greater the VAR value, the greater the risk. The smaller the VAR value, the smaller the risk.
Remember, VAR measure losses so negative VARs are to be preferred. 
However, since we expect an eventual portfolio of greater than V = 4965 (95% of the time),
then that 965 gain may be greater.
>Why VAR[95% ]?
Well, we could choose any confidence level and call it VAR[95%] or a VAR[99%] or ...
>Yeah, I get it. So VAR is a dollar amount and ...
Uh ... yes, but one could express VAR in terms of a percentage return, like so:
If anything less than V0 is considered a loss ... where V0 may (or may not!) be the initial portfolio value ... or maybe some benchmark,See? We'd want R to be positive so that, with 95% probability, the VAR-risk would be negative ... and that's good!
and there's a 95% probability of getting less than V ... with V determined from the probability distribution of future portfolio values,
and if V = (1+R)V0 ... so that going from V0 to V implies a change of R, or a percentage change of 100R,
then VAR-95% = V0 - V = -R V0
On the other hand, if R were negative, the VAR-risk would be positive... and that's bad!!
Let's summarize:
To determine VAR at the confidence level c: (Example: c = 0.95 or 95%)
The larger the VAR, the larger the "risk". VAR measures losses, not gains. The probability that x < G is 1 - c Example: the probability that x < G is 5% The probability that x > G is c Example: the probability that x > G is 95% |
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>So, if VAR[99%] is $1,000 I could say there's a 99% chance of losing $1,000?
You could, but it'd be wrong. If VAR = $1,000 then G = -$1,000.
The chances are 1% that your gain, x, is less than -$1,000 ... hence a 1% chance of losing more than $1,000.
>And if VAR[95%] is -$500 I could say ... uh ... I could say ...
If VAR = -$500, then G = $500.
The chances are 5% that your gain, x, is less than $500 ... hence a 95% chance of making more than $500.
>Or more?
Look again at Figure 3. For a 95% VAR, there's a 95% probability that you'll
end up with more than V.
That means that 95% of the time you'd make $500 or more
... and 5% of the time you'd make less.
>I assume, VAR is ... uh, coherent?
Well, it's actually ... may I call it incoherent? It doesn't satify that subadditivity axiom.
In other words, it's possible to consider two portfolios, X and Y, where the risk associated with X is greater than the risk associated with Y, yet
the VAR is smaller for the X-portfolio (implying a smaller risk).
>Huh? Wouldn't that depend upon how you define "risk"?
Good point, however we'll rely on a common sense understanding of "risk" as meaning the chances of losing money.
| VAR charts |
Look again at Figure 3. Since we'd like a small risk of losing money we'd be looking at the portfolio distribution near the 5% point, or even the 1% point. That means we're talking about the tail of the distribution.
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Let's shift the curve to the left by V0
(which is our original portfolio or some benchmark we'd like to achieve).
Now we're talking about our portfolio gains rather than our portfolio values. We'd get something like Figure 4a where there's a 5% chance of losing money ... meaning a negative gain. >You're talking about that 95% VAR, right?
Since we're interested in that small region near 0 (the border between making money and losing money) let's blow that part up ... as in Figure 4b. Suppose we have two portfolios, X and Y, that have the same gain distribution to the right of 0, but different to the left ... as in Figure 4c:  Figure 4c |  Figure 4a
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>X has smaller gains so I'd say X.
Wrong again!
Remember that we're plotting the probability of gains vs the gains themselves. When the gains are negative we're talking about losses. You'll notice that the probability of a given loss (say -6) is greater for Y than for X ... so Y is riskier.
>Isn't a loss of -6 really a gain?
Uh ... yes. I should have said a loss of 6.
>And if Y is riskier, why didn't you colour it red?
To see if you were paying attention 
Okay, since VAR is a "risk" measure, which has the larger VAR?
>Y does !
Wrong again!
They have the same VAR.
>Huh?
Remember that the VAR depends ONLY upon where that point occurs ... the point where our portfolio value
(after M months, or weeks or days) equals our initial value (or more) with a 95% probability
... and that's the same for both portfolios.
The interesting thing is that if the distribution of possible gains were Normal, then it'd be determined exactly by its Mean
and Standard Variation ... and that's how VAR got started (historically) and that's also what's wrong with it
In fact, all the pictures above, in Figures 1, 2, 3 and 4a,b are normal distributions ('cause they're easy to generate with Excel).
The red guy in Figure 4c ain't normal.
>Would you say it's abnormal?
Why not?
Notes:
If the gains (both positive and negative) were expressed as percentages of V0 which, as we've said, is some
reference value (maybe the initial value or a benchmark value that you'd like to achieve),
then VAR could be expressed as a percentage ... and it often is!
Remember that a positive VAR means there's a "risk".
Further, if you assume a Normal distribution and some Mean and Standard Deviation then you have eveything you need to know about the distribution and the calculation of VAR can be performed like so:
- Suppose we know the Mean annual Return = R and the annual Standard Deviation = SD ... perhaps from historical data.
- Suppose we want to estimate our losses M months into the future.
- We calculate the M-month return and standard deviation as: r = M*R/12 and s = S*SQRT(M/12)
- We generate a normal cumulative distribution with Mean and Standard Deviation r and s
... so it pertains to M months from now.
It might look like this which shows the case where the annual numbers are
R = 8% and S = 25% and we scaled it to an 8 month distribution:
The 8-month return is r = (8/12)*8% = 5.3% (shown in green).
(It's at the 50% probability level.) - If we want a 95% confidence level, we look at the return associated with a 5% probability.
- We can expect an M-month return less than this number, 5% of the time.
In the above example, there's a 5% probability of losing 28% ... or more.
>Or more?
Yes. There's a 5% chance of getting an 8-month return which is even less than -28%.
That's what the cumulative distribution does. It tells us the probability of getting less than something.
So we could announce, with 95% confidence: "There's a 5% probability of making -28% or less."
(Hence the 95% VAR is +28%)
For example, if you'd like a 40% return, look at the 95% level on the above chart. See the wee grey dot?
It says that there's a 95% probability of getting less than 40%.
>And what about a 99% confidence level, or maybe I can get a 10% return or ...?
Play with this:
For a much sexier calculator due to Peter Urbani, Click here!
>Hey! That's copyrighted!
Uh ... yes, but Peter has given me permission to use it
>Didn't you say that VAR didn't satisfy the subadditivity axiom?
I forgot.
| VAR and subadditivity |
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Consider owning a bond which has a 4% probability of defaulting
The cumulative distribution of gains would look like Figure 5. Note that, for this bond, VAR = 0. |  Figure 5 |
What's the VAR for this 2-bond portfolio?
We note the following:
Note that, for this portfolio, VAR = 100. >Huh? So it's riskier? Does that make sense ... for a more diversified portfolio?
|  Figure 5a |
We have two independent bonds, X and Y, where each happens to have the same distribution as shown in Figure 5
Hence VAR[X] = 0 and VAR[Y] = 0.
However, a portfolio with BOTH bonds has VAR[X+Y] = 100 as shown in Figure 5a
>And that's bigger than VAR[X] + VAR[Y], right?
Yes, so the subadditivity axiom, namely VAR[X+Y] ≤ VAR[X] + VAR[Y], is violated.
Note:
VAR answers the question: "What is the minimum loss I can expect in the worst case scenario?"
and completely ignores the size of losses worse than this minimum
(since the losses may be larger than VAR!) ... and it's this feature that makes it subadditive.
Indeed, in order to generate the examples above (showing that VAR can assign a greater "risk" to a portfolio which is clearly less risky), we investigated the features of the distribution where the losses were worse than VAR. Since VAR ignores any features involved with these "worse losses" (associated with the "tail" of the distribution), it's not surprising that Value at Risk is not a particularly "reasonable" risk measure.
