Maximize Sharpe ratio

[etowle]

Why this Nup?

Finance applications are probably plentiful enough to warrant their own nup category. If there is a dedicated Finance category, the maximal Sharpe ratio problem is popular and focused enough to be included.

Note that this is closely related to #9.

Does it fall under an existing category?

Other (Finance)

What will the API be?

asset_allocs = nupstup.max_sharpe_ratio(
    cov_matrix=Q  # variance-covariance matrix
    mu=mu  # estimated first moments of return function
    rf_rate=rf  # risk-free rate
)

where Q and mu are scipy/numpy data structures. We could alternatively require the factor matrix H as an argument instead of the covariance matrix Q (where Q = H.T @ H), though I think Q is more natural considering how the problem statement and model are written.

Additional context

One complication: to be solved as a QP, the model needs to be reformulated in a different variable space. Additional constraints must be added in the variable space of the reformulated model. The mapping between the two models is straightforward, but this does create a barrier for people interested in modifying the underlying model.

[rluce]

I like it. I think it won't harm much if the nup function allows for either the variance-covariance matrix, or its factor.

[rluce]

Maybe we should complement this Nup with a minimum variance portfolio selection (you'd minimize "just" x @ Q @ X), or can this be covered by some paramters in this Nup already?

[rluce]

I think we (@etowle, @silkehorn, and myself) should team up and create a small variety of base models in the financial space. This one and #9 are the first two on the list, but they would share the interface (and implementation) of the extra gadgets (min. buy-in, max. allocations, etc.). We may be even able to integrate the online-problem (you start with an existing allocation) into each formulation. A third variation may be needed for minimizing a tracking error.

[ChinSquared]

I would love to see the best way to include these in a multi-objective problem. Sharpe, TE, MV, etc...How would you link the y world and x in an efficient and convex way using the built in multi-objective method?

[simonbowly]

Hi @ChinSquared, could you please elaborate on what you mean here? Gurobi allows you to combine objectives either as a weighted sum, or in a hierarchical manner using priorities. Which are you looking for, and for which metrics?

[ChinSquared]

Sure, for example, if you specify a hierarchical objective, first is to max Sharpe with a budget of 20 bps. Then minimize tracking error as the second objective. Other constraints in the portfolio are active like max trade, step size, max positions. Thanks!

[etowle]

To incorporate other performance metrics into the Sharpe ratio model and retain convexity, these other metrics would have to be modeled in the space of the y variables (the reformulated variable space used in the Sharpe ratio model). This is because:

• The Sharpe ratio model is only convex in the space of the y variables
• We can't simultaneously model some metrics in the x-variable space and others in the y- variable space, unless we link together the x and y variables with the non-convex constraints x = y / (e^T y)

At any rate, I fear a multi-objective, multiple-metric model with side constraints is outside the scope of what this Mod is trying to accomplish. That said, you could always use the Sharpe ratio model-building code as a starting point for building a more complex model:

gurobi-optimods/src/gurobi_optimods/sharpe_ratio.py

Lines 87 to 100 in d1723e8

	def _max_sharpe_ratio_numpy(cov_matrix, mu, rf_rate, create_env):
	with create_env() as env, gp.Model("sharpe_ratio", env=env) as model:
	y = model.addMVar(mu.size, name="y")
	model.addConstr((mu - rf_rate) @ y == 1)
	model.setObjective(y @ cov_matrix @ y, sense=GRB.MINIMIZE)
	model.optimize()
	# Translate solution to original variable space
	x = y.X / y.X.sum()
	ret = mu @ x
	risk = x @ cov_matrix @ x
	sharpe_ratio = (ret - rf_rate) / math.sqrt(risk)
	return SharpeRatioResult(x, sharpe_ratio, ret, risk)