Correcting Annual Return and Downside Deviation in a Sortino Ratio
Summary
The document reviews a Sortino ratio calculation for S&P 500 total returns over a historical sample. It computes daily returns from adjusted closes, treats negative daily returns as downside observations against a zero target, averages their squared shortfalls across the full observation count, and annualizes the resulting downside deviation using the square root of the assumed trading days per year. It also compares a geometric annualized return from start and end values with the annualized average daily return.
The accepted answer identifies an arithmetic error in the start-to-end annual return calculation and says the average daily return’s annualization is the appropriate numerator for the ratio shown. With that correction, the reported ratio falls substantially. The answer also notes that changing the sample start date produces a very different ratio, underscoring sensitivity to the observation window. The treatment is tied to the specified data, zero target, and annualization convention; it does not settle all methodological choices in Sortino calculations.
Key ideas
- Daily returns are calculated from consecutive adjusted closing prices.
- Downside deviation is based on squared returns below the selected target, here zero.
- The sample downside deviation is annualized by multiplying by the square root of trading days per year.
- The original start-to-end annualized return calculation contains an arithmetic error.
- Sortino results depend on the return sample window and the chosen calculation conventions.
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Full text
# Did I correctly calculate all the elements of the Sortino ratio?
# 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.
## Answer by user17688 (score 2, accepted)
https://quant.stackexchange.com/a/54588
Having reviewed the documentation sent by Noob2 and rechecking everything, I came to the following conclusion:
- ((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%.
- 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.