## Percentile Series

The Manager vs. Universe graph and table allow you to present the data as a percentile or ranked series. In both cases, every manager is ranked according to the statistic being shown on the graph or table. The universe is then separated into quartiles according to those ranks.

The percentile series plots the actual statistic (return, i.e.) on the vertical axis of the Manager vs. Universe Graph. The universe is represented by colored bands that represent the span of returns for each quartile of the universe.

Here, we are given:

• a universe of N manager series, each running over t periods:

M1 = M11, M12, ..., M1t
M2 = M21, M22, ..., M2t
...
MN = MN1, MN2, ..., MNt

• a percentage p

We are trying to construct a return series S = s1, s2, ..., st such that for each i with 1 <= i <= t, the return si is the p-th percentile of the returns m1i, , mNi. To calculate si for any i with 1 <= i <= t follow these steps:

1) Order the manager returns m1i, ..., mNi in ascending order, say r0, ..., rN-1.
2) Find the position between 0 and N - 1 that the percentage p corresponds to:
pos = (N - 1) * (100 - p)/100
3) Find the two indices that pos lies between, say k and k+1: k <= pos <= k + 1
4) Interpolate the return between rk and rk+1 according to pos:
si = rk + (pos - k) * (rk+1 - rk)

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