# Did I correctly calculate all the elements of the Sortino ratio?

I wish to verify my understanding and correctness of my methodology with this question, when calculating the sortino ratio for the SP500. My base data is the total returns SP500 index from Yahoo finance:

https://finance.yahoo.com/quote/%5ESP500TR/history?p=%5ESP500TR

I took the adj. closing price starting with the 3rd of January 2000, going all the way up to 22nd of May 2020. The first question I wanted to answer was, what the average yearly return was for the SP500 for that period. This was calculated the following way (all functions in question are excel functions):

Annualized return = $$(\frac{Endprice - Start price}{Start Price} + 1)^{\frac{1}{Years}} - 1$$

where Years = $$\frac{Days(end date, start date)}{365.25}$$

Plugging in the numbers gets me => $$(\frac{6044 - 2002}{2002})^{\frac{1}{20.38}} = 5.57$$%

Next thing I want to know is what the downside deviation (semi-deviation) is, where the benchmark return that needs to be achieved is at-least 0. Admittedly this is a relatively low benchmark, but all I care about are the days where the returns were negative (the following is done for all the 5130 data-points).

Daily return = $$\frac{Price_{t} - Price_{t-1}}{Price_{t-1}}$$

Daily Downside variance value $$(DDVR)$$ = $$\min{(Daily return, 0)^{2}}$$

Daily downside variance = $$\frac{1}{N}\sum^{N}_{i=1}DDVR_{i}$$ = $$\frac{0.44}{5129} = 0.0086\%$$

Daily downside deviation = $$\sqrt{0.0086} = 0.9272\%$$

Annualized downside deviation = $$0.9272\% \times \sqrt{252} = 14.72\%$$

Sortino ratio = $$\frac{5.57\% - 0}{14.72\%} = 0.3784$$

I am unsure if my result is correct at this point. Another consideration I have is, is whether I calculated the annualized returns correctly. If I take the average return of all the values I derived when I calculated the Daily return I get $$0.014\%$$. Annualizing this gets me $$(1 + 0.014)^{252}-1 = 3.49\%$$, which is some way away from $$5.57\%$$, that I have initially calculated. It seems to me I am doing something wrong here, but I cant see it.

• Sortino ratio is not a standard mathematical notion. So, unless you have a reference for that notion or you're lucky enough that someone in the know passes by, the question will remain unanswered. For all I know, you're just stringing a number of operations and pulling out a number. So what? May 24 '20 at 17:48
• Does the small example given here help you cmegroup.com/education/files/rr-sortino-a-sharper-ratio.pdf May 24 '20 at 20:01
• @Raskolnikov I would actually expect exactly that, that someone in the know would review my calculations. I mean this is the "quantitative finance" forum, I think it is not unreasonable to expect, right?
– Noir
May 24 '20 at 20:25
• @Noob2 thank you for the paper I will review it.
– Noir
May 24 '20 at 20:26
• If your "benchmark return that needs to be achieved is at-least 0" then maybe you should just put your money in the bank? That's the whole reason why they introduced MAR, so that every investor could calculate the risk specific to their goals. Jun 2 '20 at 7:19

Having reviewed the documentation sent by Noob2 and rechecking everything, I came to the following conclusion:

1. ((6044−2002)/2002)^1/20.38=5.57% is absolutely wrong. If one does the calculations for this you get 1.035 (I have no idea how I managed do come up with 5.57% in the first place). Thus this resolves the question where I am confused about the difference in returns of 3.49% to 5.57%.

2. Correcting for the number above the sortino ratio then is (3.49% - 0)/14.72% = 0.237, which is a very low ratio if we think about the fact that this is the SP500, but my data start point is just before a major market drop. If we for example take the starting point to be May 2010 (thus 10 years before now), the ratio increases to 1.3, with way higher returns on average.

• While you can use discrete form of downside risk, Sortino, Rom and Ferguson 1 themselves listed a few reasons NOT to use it. They advocate using only a continuous form. Which amongst other advantages allows one to use actual returns distribution and not just assume normal distribution. Jun 2 '20 at 6:53
• Thank you for the reference Rusi, I did not know. Using google I got the exact section where they discuss it with the math in question.
– Noir
Jun 2 '20 at 7:07
• Do you remember what page or Chapter that is? Jun 2 '20 at 8:47
• Hey Noob2 here is the link where I found the description: books.google.lu/… The page is 60-61.
– Noir
Jun 2 '20 at 9:02