Question at position 15 Rather than tracing out the efficient frontier, you decide to calculate the Sharpe ratio-maximizing portfolio directly. You wrote the function to calculate the sharpe ratio of a portfolio given its weights as follows: def sharpe_ratio(w, μ, Σ, rf): r = w.transpose @ μ σ = np.sqrt(w.transpose @ Σ @ w) return (r - rf) / σ You then define an objective function objective(w): def objective(w): return a * sharpe_ratio(w, μ, Σ, rf) and minimize it subject to constraints using scipy.optimize.minimize. Which value of a will allow you to find the Sharpe ratio-maximizing portfolio? rf10You cannot find the maximum Sharpe ratio using the minimize function.-1

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We start by restating the scenario and listing the options to analyze them clearly.
Question: You define an objective function objective(w) = a * sharpe_ratio(w, μ, Σ, rf) and minimize it subject to constraints to locate the portfolio that maximizes the Sharpe ratio. The available choices for a are: rf, 1, 0, You cannot find the maximum Sharpe ratio using the minimize function., -1.
Option 1: a = rf. Here, a is a scalar multiplier for the Sharpe ratio, not the risk-free rate itself. Using rf as the multiplier would mix the constant rate into the optimization objective in a way that doesn’t systematically steer the optimizer toward maximizing Sharpe. This choice misinterprets the role of a as a scaling factor rather than a numeric constant tied to the risk-free rate.
Option 2: a = 1. If you set a to 1, the objective becomes the Sharpe ratio itself. When you minimize this, you are effectively seeking the portfolio with the smallest Sharpe ratio, not the largest. This is the opposite of what we want for maximizing Sharpe.
Option 3: a = 0. With a equal to zero, the objective is identically zero for all feasible weights. A constant objective provides no guidance to the optimizer about any relative performance of portfolios, so you cannot identify a Sharpe-maximizing portfolio from such an objective.
Option 4: You cannot find the maximum Sharpe ratio using the minimize function. This statement would be true only if the problem genuinely cannot be transformed into a minimization of any scalar objective that yields the maximum Sharpe under the given constraints. However, by choosing the appropriate sign for a, we can convert the maximization of Sharpe into a minimization problem. Specifically, minimizing the negative Sharpe yields a solution that maximizes Sharpe, provided the weight constraints are properly specified and the problem is well-posed.
Option 5: a = -1. This choice flips the objective to objective(w) = -1 * Sharpe(w). Minimizing the negative Sharpe is the standard mathematical trick to convert a maximization of Sharpe into a minimization. When the feasible set is defined (weights summing to 1, any bounds), this approach guides the optimizer toward the portfolio with the highest Sharpe ratio across the feasible region, assuming the Sharpe is finite and the problem is well-behaved.
Now, comparing these analyses, note the following misunderstandings and correct ones: using rf as a scaling factor (Option 1) confuses a with a fixed market parameter rather than a tunable sign/scale for the objective. Making a = 1 (Option 2) would minimize the Sharpe, not maximize it, so it does not help achieve the goal. Setting a = 0 (Option 3) erases the optimization signal entirely, yielding no information about the best Sharpe. The blanket statement in Option 4 is misleading: the minimization approach can find the maximum Sharpe if we choose the correct sign, i.e., a = -1, as in Option 5. Finally, Option 5 explicitly implements the standard transformation to convert a maximization problem into a minimization one, which is the correct strategy under typical assumptions about the feasible weight space.
In summary, the reasoning across options shows that the key idea is the sign flip via a, turning the Sharpe-maximization into Sharpe-minimization; the specific choice that implements this correctly is a = -1, while the other options fail for the reasons described. The conceptual pitfall is mistaking a scaling factor for a parameter that should be chosen to invert the optimization direction, and neglecting how the objective guides the optimizer within the feasible set.

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