A decile is a quantile at a multiple of 10%: the first decile is the 10th percentile, the second is the 20th, and so on through the ninth at 90%. These nine internal cutpoints describe ten ordered groups. In financial research, decile also means one of those groups: “the highest-loss decile” is a collection of observations, whereas “the ninth-decile loss” is a threshold in currency units. Statistics Canada's income-decile definition explicitly separates groups from their cutpoints.
For an increasing continuous distribution with cumulative distribution function , the cutpoints satisfy , for . Thus is the median. For distributions with probability atoms, a generalised inverse provides a convention, for example . The probability below a cutpoint need not then equal its nominal percentage exactly.
A loss report with twenty scenarios
Suppose a risk analyst has twenty equally weighted synthetic one-period loss scenarios, in USD:
The parentheses denote sorted positions, not dates. Positive values mean losses. No scenario probabilities for a real portfolio have been estimated here.
To report sample quantiles, the analyst must choose a calculation rule. Use R type 7 linear interpolation in this example. At probability , set
and interpolate as for these interior probabilities. For and , the position is , giving . This is a location between observations, not a recorded loss. Type 7 is a named choice among the nine algorithms documented by R.
| Cutpoint | Probability level | Type 7 result, USD | Interpretation in this sample |
|---|---|---|---|
| 10% | 2.9 | Losses 1 and 2 lie below it | |
| 50% | 10.5 | Ten losses lie below it | |
| 90% | 18.1 | Losses 19 and 20 lie above it |
Here the highest-loss group contains 19 and 20, not every loss above . The threshold is neither that group's average nor the worst loss. Its average is USD 19.5; the largest scenario loss is USD 20.
Original synthetic loss scenarios. Dashed lines are interpolated type 7 cutpoints; filled dots are observations. Groups are ordered from smaller to larger losses.
The algorithm and the grouping rule are separate choices
Another defensible convention takes the first observed value whose cumulative empirical proportion reaches . With these twenty equally weighted observations, that empirical inverse-CDF rule gives : , and , all in USD. Both sets of results use the same data. NIST describes several interpolation conventions and the absence of a universally accepted choice. Carry the method with the reported number.
To turn cutpoints into groups, state which group includes a boundary observation. One rule puts values at or below in group 1, values strictly above and at or below in group 2, and continues through values above in group 10. The figure has two observations per group because of its particular untied data and sample size.
Ten groups do not guarantee ten equal counts. If the sample size is not divisible by ten, equal integer group sizes may be impossible. Ties are a stronger obstacle: twenty identical losses give nine identical cutpoint values. Keeping equal losses together cannot produce ten nonempty value groups. A rank-based allocation may split tied records using an explicit secondary rule, but that rule adds no economic distinction between equal losses. Weighted data require cumulative weights rather than a silent assumption of equal counts.
Reading a decile portfolio table
A researcher might rank stocks by a signal and compare subsequent returns of the highest and lowest groups. First establish what “highest” means: high forecast return and high estimated loss point in different economic directions. Check whether decile 1 labels the lowest or highest ranks; tables use both conventions. Also check the eligible universe and whether ranking occurs within sectors or across all stocks.
A ranked basket becomes a strategy only after specifying information availability, formation and execution times, holdings, weights, rebalancing, and trading costs. A historical spread between group-average returns does not supply those choices or establish a future profit. The Portfolio sorts lesson develops sector rankings, weights and forward holding windows; the public page is a preview and the full lesson requires membership.
Sources
- Statistics Canada, Dictionary, Census of Population, 2016: Economic family after-tax income decile group — distinction between ranked groups and the income cutpoints defining them.
- R Core Team, Sample Quantiles, stats documentation — type 1 empirical inverse and type 7 interpolation.
- NIST/SEMATECH, e-Handbook of Statistical Methods, §7.2.6.2: Percentiles — ordered data, deciles and alternative quantile conventions.
- Larry Wasserman, All of Statistics: A Concise Course in Statistical Inference (Springer, 2004), Example 7.14 — empirical inverse-CDF sample quantile.