Market Psychology

Five stocks, with the funds buying and selling each in one quarter. Across all of them 57% of trades were buys. Lakonishok, Shleifer and Vishny measure herding as how far each stock’s buy share sits from 57%, minus how far it would sit by chance. Stock A (18 buyers, 2 sellers) scores 0.24, stock D (5 and 15) 0.23; E scores −0.07, no more lopsided than chance. The average is 0.09.

• Trading together is not proof of copying: funds reacting to the same news will look like a herd.
• Using the same measure, LSV (1992) found only a little herding among U.S. pension fund managers — far less than the stories suggest.
• Herding can be rational for each person (see Information Cascades); it is the collective result that goes wrong.

from math import comb
q = pd.DataFrame({'buyers': [18, 9, 14, 5, 11], 'sellers': [2, 11, 6, 15, 9]},
                 index=list('ABCDE'))            # funds buying and selling five stocks in one quarter, made up
n = q.buyers + q.sellers
p = q.buyers / n
p_all = q.buyers.sum() / n.sum()                 # share of all trades that are buys

def af(n, p):                                    # |p_i - p_all| expected by chance alone
    return sum(comb(n, k) * p**k * (1 - p)**(n - k) * abs(k / n - p) for k in range(n + 1))

H = (p - p_all).abs() - n.apply(af, p=p_all)     # Lakonishok, Shleifer & Vishny's herding measure

Assumes import numpy as np and import pandas as pd. Each snippet carries its own example numbers; the comments say which are made up.