Sortino Ratios and Downside Risk ... a continuation of Part II

We talked about Sortino Ratios and noted that it was a ratio of Reward to Downside Risk.
The Reward was the excess of the Mean return M = (1/n) (r1+ r2+ ...+ rn) over some risk-free rate
... which we'll also call the Minimum Acceptable Return (or MAR): rf.
The problem is to define a reasonable Downside Risk.

Given a set of returns r1, r2, ... rn and a MAR rf, we let R[r1, ... rk+h, ... rn, rf] be our measure of Risk. (Downside Risk will come later.)

Risk should have the following properties:

  1. If any return is increased by an additive amount, h > 0, "risk" should decrease ... since increased returns should proved less risk.
          R[r1, ... rk+h, ... rn, rf] < R[r1, ... rk, ... rn, rf] ... if h > 0.
    And, of course, R should increase if h < 0 ... returns decrease => risk increases.
  2. If any return is increased by an multiplicative amount λ, the risk should decrease.
          R[r1, ... λrk, ... rn, rf] < R[r1, ... rk, ... rn, rf] ... if λrk > rk.
  3. Further, R should increase if rf is increased (since it's harder to achieve that larger minimum acceptable return).
>I suggest Standard Deviation of the returns!
Okay, remember that the Mean is M = (1/n) Σrk, so we'd have:
      R2[r1, ... rk, ... rn, rf] = SD2 = (1/n) Σ (rk - M)2 = (1/n) Σ rk2 - {(1/n) Σ rk}2 ... see SD-stuff
Alas, Standard Deviation fails condition #1 since adding a constant to a return doesn't change SD at all.
Further, multiplying all returns by λ > 0 will multiply SD by λ, so it fails #2 as well.
Finally, the SD of returns doesn't even involve our Minimum Acceptable Return rf, so #3 fails.
>So why is SD such a popular measure of risk?
I have no idea.

Anyway, in Part II, we used Value At Risk (or VAR) so let's look at that.

We calculate the SD of the returns and their Mean, M, and generate a cumulative normal distribution as illustrated in Figure 1. The probability of getting a return less than rf is then R.

If any r is increased, the location of the Mean increases and the curve is shifted to the right (see the magenta curve?) so the probability of being to the left of rf, hence R, increases ... so #1 and #2 are satisfied.

Further, if we increase (or decrease) rf, that probability increases (or decreases) ... so #3 is satisfied.


Figure 1
>Wait! If we change one of those r's, won't the shape of the distribution change as well?
Uh ... yeah, I guess it would.

If we multiplied all returns by some λ > 0 not only would the Mean increase by a factor λ but so would the Standard Deviation ... so the location of the Mean would move right and the distribution would spread out, like Figure 2.

>And would the VAR, our latest "risk" candidate, increase or decrease?
I have no idea.
>You don't have many ideas, do you?
Well, if all the returns that we're using to calculate the Mean and SD were positive, then multiplying by λ > 0 should decrease our risk, and we'd expect the shifted, expanded distribution to have a smaller value at rf.
That's the red value. See it, in Figure 2?


Figure 2
>So you like VAR as your "risk"?
Well, we've already noticed that Ponzorinos ain't so good, haven't we?

>What's next?
I'm thinking ...

Other Risk Measures

Gotta leave this tutorial for a while.
We're heading off for a 2-week vacation