Thanks very much for posting this paper! I think there are some useful ideas here. I have worked with CVaR optimization (per the Uryasev paper you link to) as well as the L1 risk measure (i.e., MAD) per Konno. I was initially attracted to them because they are linear programming problems. However that was because, at the time, we did not have
cvxpy white-listed on Quantopian. Today with
cvxpy I don't think you need to be averse to quadratic terms. Anyhow, I think these methods have use as risk constraints not as objectives. You need to have some exogenous alpha vector to work in here. For example, I have an implementation of the L1 risk constraint here.
Here is an implementation of optimization with a limit on CVaR (thanks to Scott Sanderson for the insight that
avg_worst_returns = cvx.sum_smallest(portfolio_rets, nsamples) / nsamples captures the CVaR):
num_stocks = len(alpha_vector)
num_days = len(hist_returns)
A = hist_returns.fillna(0.0).as_matrix()[-(lookback_days-1):, :]
x = cvx.Variable(num_stocks)
nsamples = round(num_days * (1-cvar_ptile))
portfolio_rets = A*x
avg_worst_returns = cvx.sum_smallest(portfolio_rets, nsamples) / nsamples
objective = cvx.Maximize(alpha_vector*x)
constraints = [
avg_worst_returns > max_risk,
sum(x) == 0,
sum(cvx.abs(x)) <= 1,
x <= max_position,
x >= -max_position
prob = cvx.Problem(objective, constraints)
sol = np.asarray(x.value).flatten()
I hope to find some time to work through the Minimax risk constraint.
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