w = np.linalg.solve(δ * Σ_est, μ_est)

fig, ax = plt.subplots(figsize=(8, 5))

ax.set_title('Mean-variance portfolio weights recommendation \

and the market portfolio')

ax.set_title('Mean-variance portfolio weights recommendation and the market portfolio')

ax.plot(np.arange(N)+1, w, 'o', c='k', label='$w$ (mean-variance)')

ax.plot(np.arange(N)+1, w_m, 'o', c='r', label='$w_m$ (market portfolio)')

ax.vlines(np.arange(N)+1, 0, w, lw=1)

- They want to continue to allow the customer to express his or her

risk tolerance by setting $\delta$.

- Leaving $\Sigma$ at its maximum-likelihood value, they push

$\mu$ away from its maximum value in a way designed to make

$\mu$ away from its maximum-likelihood value in a way designed to make

portfolio choices that are more plausible in terms of conforming to

what most people actually do.

μ_m = (d_m * Σ_est @ w_m).reshape(N, 1)

fig, ax = plt.subplots(figsize=(8, 5))

ax.set_title(r'Difference between $\hat{\mu}$ (estimate) and \

$\mu_{BL}$ (market implied)')

ax.set_title(r'Difference between $\hat{\mu}$ (estimate) and $\mu_{BL}$ (market implied)')

ax.plot(np.arange(N)+1, μ_est, 'o', c='k', label='$\hat{\mu}$')

ax.plot(np.arange(N)+1, μ_m, 'o', c='r', label='$\mu_{BL}$')

ax.vlines(np.arange(N) + 1, μ_m, μ_est, lw=1)

ax[0].vlines(np.arange(N)+1, μ_m, μ_est, lw=1)

ax[0].axhline(0, c='k', ls='--')

ax[0].set(xlim=(0, N+1), xlabel='Assets',

title=r'Relationship between $\hat{\mu}$, \

$\mu_{BL}$and$\tilde{\mu}$')

title=r'Relationship between $\hat{\mu}$, $\mu_{BL}$, and $ \tilde{\mu}$')

ax[0].xaxis.set_ticks(np.arange(1, N+1, 1))

ax[0].legend(numpoints=1)