Sortino Ratios ... and Sharpe
motivated by e-mail from Dan M

First, we talk about the Sharpe Ratio which is an attemt to measure "Risk" and "Reward".

1. We look at a the average annual return of some asset (say, R = 9%) and compare it to some risk-free return (say Rf = 4%, so R - Rf = 5%).
Our "Reward" is the excess return: R - Rf.
2. That excess return may look pretty good, but the asset has volatility (or Standard Deviation, say SD = 25%).
That's our "Risk". It measures how far our annual returns deviate from their average.
In our example, we might expect most returns to vary between R - SD = 9 - 25 = -16% and R + SD = 9 + 25 = 34%.
3. We consider the Sharpe Ratio: (R - Rf) / SD ( In our example, it's: 5/25 = 0.20
>Big reward, small risk ... that's good, eh?
Yes, so we might look for assets with large Sharpe Ratios.

However, if the returns get really large compared to Rf, we should be happy.
But that means the deviations from the average can be large, and that'd make SD larger and that'd make the Sharpe Ratio smaller.

>Then don't buy the asset!
But you'd want lots of returns larger than Rf, wouldn't you?
Ay, there's the rub. Enter Frank Sortino.

If we really wanted to use deviations from the average as a measure of "risk", then why include returns that are larger than Rf?
We could just measure the deviation of those returns that are smaller than Rf, eh?
That means we could take the standard deviation of those less-than-Rf  returns, like so:

[1]       SDd2 = (1/n) Σ (r - Rf)2
where SDd is the "downside volatility" and the sum includes only returns for which r < Rf.

If we take SDd as our "risk", then we change Sharpe into Sortino:
 Sortino Ratio = (R - Rf) / SDd where R is the Mean annual return, Rf  is some risk-free return (or Minimum Acceptable Return: MAR) and SDd is the downside risk: SDd2 = (1/n) Σ (r - Rf)2 where the sum (or average) includes only returns for which r < Rf.
>And Sortino is better than Sharpe?
Define "better".
One thing is for sure: Sortino doesn't penalize an asset for "upside" volatility ... which is a "good" volatility.
However, there's a problem with Sortino. It may happen that no returns are less than Rf  in which case SDd = 0 and Sortino = ∞
 >But how do they compare, Sortino and Sharpe? First off, notice that the denominator in the Sortino Ratio is smaller than Sharpe's denominator ... since SDd contains fewer terms. That makes the Sortino Ratio larger, as in Figure 1, and ... >I'll bet fund managers like that! Smaller "risk", eh? Yes, and I understand that Sortino is (usually) used by hedge funds. Figure 1
>In Figure 1 you're using annual returns?
Well ... I used ten years worth of monthly returns and used (as Rf) 1/12 of the annual risk-free rate (or MAR).
>Ain't that cheating?
Define "cheating".
>Where's the spreadsheet and what did you do when Sortino = ∞ and ... ?
The spreadsheet will look like this:

but it isn't finished ... yet. In fact, I keep playing with it and it keeps getting bigger and bigger.

When it's finished, you can download three assets (from Yahoo) and compare Sortino and Sharpe and ...
>Wake me up when it's ready ... zzzZZZ
 Remember when I said that Sortino's "downside" volatility is smaller than the regular, garden-variety standard deviation used by Sharpe? You get a glimpse of that in Figure 2. See? >zzzZZZ The places where the downside volatility (that's SDd) is 0 is evident. See that? >zzzZZZ Fig. 2 is actually the XOM example we considered above. When it's 0 we just set the Sortino ratio = 0 ... instead of infinity. >zzzZZZ ... huh? Is that legal? Did you see the in progress sign? Figure 2

 calculating Sortino Ratios

Okay, so here's what I did with the spreadsheet:

I used monthly returns instead of annual returns ... since there are so many more of those available
I multiplied the monthly returns by 12 to simulate annual returns.
Then I calculated the Sortino and Sharpe ratios, using these "multiplied-by-12" returns.
Then I noticed that the ratios didn't change by this "multiply-by-12" ritual.

>Huh?
Suppose we multiply all our returns by some positive parameter λ (for example, λ = 12).
Note that SDd2, which is (1/n) Σ (r - Rf)2, becomes (1/n) Σ (λr -λRf)2 = λ2(1/n) Σ (r -Rf)2.
Hence the downside volatility SDd gets multiplied by λ as well.
(Indeed, any standard deviation gets multiplied by that parameter ... which is one reason I don't like standard deviation as a measure of "risk".)

That'd change (R - Rf) / SDd   into   (λR - λRf) / λSDd = (R - Rf) / SDd.
In other words, the Sortino Ratio don't change at all !

Okay, so I just forgot about any effort to change monthly to annual, and just used monthly returns.
In fact, I wanted to see the variations in ratios in a moving 2-year window ... that's 24 returns
 Note: The charts in Fig. 1 and 2 actually use a 1-year window ... but I've changed that. That makes it less likely that SDd = 0 as in Fig. 2. In fact, Figure 2 now looks like Figure 2a Figure 2a

One other thing.
In my moving 2-year window I need an average for 24 monthly returns. That means I use the first two of the 10 years just to get a single 2-year average.
That explains why, in the latest spreadsheet, the Sortino and Sharpe ratios (in the 2-year moving window) start two years later ... as in Figure 2a.

>And what about years when all returns were greater than Rf? You'd have SDd = 0 and ...
Yeah, and Sortino = infinity. I know! I know!
In fact, if you have just a single return less than Rf, then SDd = 0. That's a problem, eh?
However, in order that SDd = 0 in a 2-year window, you'd need to have 24 consecutive returns all less than Rf.
Since that's unlikely ...
>But I could choose Rf = 50% then they'd all be less ...
Yeah, sure ... so pick a more reasonable Rf.

You can try it out by clicking here, but there are no guarantees.
>But don't you offer a money-back guarantee?
Uh, yeah ... I forgot. The spreadsheet is guaranteed to work admirably.

Almost forgot this, too.
If you actually downloaded a bunch of monthly returns and wanted to estimate the annual standard deviation, it's more complicated than simply multipliying all returns by 12 and getting a standard deviation multiplied by 12. You'd multiply the standard deviation by SQRT(12) !!
That means (R - Rf) / SD will have the numerator multiplied by 12, but the denominator multiplied by SQRT(12).
That means the Sortino and Sharpe ratios get multiplied by SQRT(12) ... since 12/SQRT(12) = SQRT(12).
The spreadsheet will do that if'n you ask nice. Just say y:
It will give ratios which are more like the ones you might see in the literature or on the internet (even tho' the increased values don't change the look of the charts).

>Didn't you say you did not like standard deviation as a measure of risk?
Yes, because I can add 10% to all returns and the standard deviation doesn't change which makes it a lousy measure ...

>Then why don't you invent your own Sortino ratio, with your own measure of risk? You could call it ... uh, the Ponzorino Ratio.
Hmmm ... not a bad idea:

for Part II