CRRA Log Utility Maximization

Problem

Compute the optimal fraction of wealth invested in a binary gamble under log utility.

The gamble gains r_gain with probability p and loses r_loss otherwise. Maximizing p * ln(1 + alpha * r_gain) + (1 - p) * ln(1 + alpha * r_loss) gives alpha = -(p * r_gain + (1 - p) * r_loss) / (r_gain * r_loss). Return that fraction, or 0.0 when the denominator is zero.

Hint

The first-order condition is linear in the fraction for log utility.

Test cases

case 1

in: [0.6,0.2,-0.1]

expected: 3.9999999999999987

case 2

in: [0.5,0.1,-0.1]

expected: 0

case 3

in: [0.8,0.05,-0.2]

expected: 1.3877787807814455e-15

Starter codePython

def crra_log_optimal_fraction(p, r_gain, r_loss):
    # Your code here
    pass

3 test cases · run your code against them in the editor