There are a jillion definitions of "Risk" and I don't hardly like any of them. >Don't tell me you're talking about risk If stock A is "riskier" than stock B, then you'd expect: - The probability of getting large returns is higher for stock A
- The probability of getting large losses is also higher for stock A
You'd like to be rewarded for taking a larger "risk", knowing full well that there's also a greater possibility of getting large losses. >Yeah, so why call it "another definition of risk"? Why not stick with existing definitions like ...?
>And they are ...?
>You mean g-riskier, right? In that case, I'd say red ... that's Microsoft. But I coulda told you that without any g-risk mumbo-jumbo.
>Besides, you're talking history, not future. I suggest you take a look at For example,
Indeed, we're interested in probabilities of getting less than something (since both integrals in numerator and denominator are animals of this ilk).
That poses the following problem:
In other words, we need a mathematically defined distribution that mimics quantiles for some actual, historical distribution.
I'm thinking.
We look carefully at historical returns of some asset and extract the Mean Suppose we begin with the Normal distribution as a first approximation to the historical return distribution.
To be a distribution, we'll need: That gives:
Then we'll want Now on to the "second moment":
The above equation then reads: μ_{2} = m2 + s2 + _{∫} r ^{2} k(r) dr Then we have: [2] _{∫} r ^{2} k(r) dr = μ_{2} - m2 - s2
μ_{3} = _{∫} r ^{3} D(r) dr
= _{∫} r ^{3} N(r) dr
+ _{∫} r ^{3} k(r) dr
= m^{3} + 3ms^{2} + _{∫} r ^{3} k(r) dr
... since the 3rd moment for the Normal distribution is m^{3} + 3ms^{2}.
know the values of P_{z},
so we can determine the moments of k(r) .
That way we can modify the assumption that the actual return distribution is Normal. We just add our k(r) .
>Huh? We know P_{z}?
Sure. Just look 'em up in a book. In fact, we already know that:
P_{0} = 1 ... true of all distributions
P_{1} = 0 ... the Mean = 0 for a "standard" normal distribution
P_{2} = 1 ... since the Standard Deviation = 1
P_{3} = 0 ... since all odd powers give a zero integral
N, the Normal component of our modified distribution, D, will take care of them. It's them higher moments that we want to mimic by introducing our k-modification and ...
A good question. >Okay, suppose we know those moments. Now how do we construct k from its moments? Another good question. |