The efficient frontier is a fundamental visualization in Modern Portfolio Theory (MPT) that displays a curve of optimal portfolios offering the highest expected return for each level of risk (standard deviation). Portfolios on the frontier are "efficient" because no other portfolio exists with higher return for the same risk, or lower risk for the same return. This plot is essential for asset allocation decisions and understanding the risk-return tradeoff in investment portfolios.
""" anyplot.ai
frontier-efficient: Efficient Frontier for Portfolio Optimization
Library: letsplot 4.9.0 | Python 3.13.13
Quality: 85/100 | Updated: 2026-05-17
"""
import os
import numpy as np
import pandas as pd
from lets_plot import (
LetsPlot,
aes,
element_blank,
element_line,
element_rect,
element_text,
geom_line,
geom_point,
ggplot,
ggsave,
ggsize,
labs,
scale_color_viridis,
theme,
theme_minimal,
)
LetsPlot.setup_html()
# Theme tokens
THEME = os.getenv("ANYPLOT_THEME", "light")
PAGE_BG = "#FAF8F1" if THEME == "light" else "#1A1A17"
ELEVATED_BG = "#FFFDF6" if THEME == "light" else "#242420"
INK = "#1A1A17" if THEME == "light" else "#F0EFE8"
INK_SOFT = "#4A4A44" if THEME == "light" else "#B8B7B0"
RULE = "rgba(26,26,23,0.10)" if THEME == "light" else "rgba(240,239,232,0.10)"
# Okabe-Ito palette
IMPRINT = ["#009E73", "#C475FD", "#4467A3", "#BD8233", "#AE3030", "#2ABCCD", "#954477"]
# Generate simulated asset data
np.random.seed(42)
n_assets = 6
n_portfolios = 300
risk_free_rate = 0.03
# Asset expected returns and volatilities
asset_returns = np.array([0.08, 0.10, 0.12, 0.15, 0.07, 0.18])
asset_volatility = np.array([0.12, 0.15, 0.18, 0.25, 0.10, 0.30])
# Correlation matrix
corr_matrix = np.array(
[
[1.00, 0.30, 0.25, 0.20, 0.50, 0.15],
[0.30, 1.00, 0.40, 0.35, 0.25, 0.30],
[0.25, 0.40, 1.00, 0.50, 0.20, 0.45],
[0.20, 0.35, 0.50, 1.00, 0.15, 0.60],
[0.50, 0.25, 0.20, 0.15, 1.00, 0.10],
[0.15, 0.30, 0.45, 0.60, 0.10, 1.00],
]
)
cov_matrix = np.outer(asset_volatility, asset_volatility) * corr_matrix
# Generate random portfolios
portfolio_returns = []
portfolio_risks = []
sharpe_ratios = []
for _ in range(n_portfolios):
weights = np.random.random(n_assets)
weights /= np.sum(weights)
port_return = np.sum(weights * asset_returns)
port_vol = np.sqrt(np.dot(weights.T, np.dot(cov_matrix, weights)))
portfolio_returns.append(port_return)
portfolio_risks.append(port_vol)
sharpe_ratios.append((port_return - risk_free_rate) / port_vol)
# Generate efficient frontier
target_returns = np.linspace(0.07, 0.18, 50)
frontier_risks = []
frontier_returns = []
for target in target_returns:
best_risk = float("inf")
for _ in range(2000):
weights = np.random.random(n_assets)
weights /= np.sum(weights)
port_return = np.sum(weights * asset_returns)
port_vol = np.sqrt(np.dot(weights.T, np.dot(cov_matrix, weights)))
if abs(port_return - target) < 0.003 and port_vol < best_risk:
best_risk = port_vol
if best_risk < float("inf"):
frontier_risks.append(best_risk)
frontier_returns.append(target)
# Find special portfolios
min_var_idx = np.argmin(frontier_risks)
min_var_risk = frontier_risks[min_var_idx]
min_var_return = frontier_returns[min_var_idx]
sharpe_frontier = [(r - risk_free_rate) / s for r, s in zip(frontier_returns, frontier_risks, strict=False)]
max_sharpe_idx = np.argmax(sharpe_frontier)
tangency_risk = frontier_risks[max_sharpe_idx]
tangency_return = frontier_returns[max_sharpe_idx]
# DataFrames
df_portfolios = pd.DataFrame({"risk": portfolio_risks, "return": portfolio_returns, "sharpe": sharpe_ratios})
df_frontier = pd.DataFrame({"risk": frontier_risks, "return": frontier_returns})
df_special = pd.DataFrame(
{
"risk": [min_var_risk, tangency_risk],
"return": [min_var_return, tangency_return],
"label": ["Min Variance", "Max Sharpe"],
}
)
# Capital Market Line
cml_risks = np.array([0, tangency_risk * 1.8])
cml_returns = risk_free_rate + (tangency_return - risk_free_rate) / tangency_risk * cml_risks
df_cml = pd.DataFrame({"risk": cml_risks, "return": cml_returns})
# Custom theme
anyplot_theme = theme(
plot_background=element_rect(fill=PAGE_BG),
panel_background=element_rect(fill=PAGE_BG),
panel_grid_major_x=element_line(color=RULE, size=0.3),
panel_grid_major_y=element_line(color=RULE, size=0.3),
panel_grid_minor=element_blank(),
axis_title=element_text(size=20, color=INK),
axis_text=element_text(size=16, color=INK_SOFT),
axis_line=element_line(color=INK_SOFT, size=0.5),
plot_title=element_text(size=24, color=INK),
legend_background=element_rect(fill=ELEVATED_BG, color=INK_SOFT),
legend_text=element_text(size=16, color=INK_SOFT),
legend_title=element_text(size=18, color=INK),
)
# Plot
plot = (
ggplot()
+ geom_point(data=df_portfolios, mapping=aes(x="risk", y="return", color="sharpe"), size=4, alpha=0.6)
+ geom_line(data=df_frontier, mapping=aes(x="risk", y="return"), color=IMPRINT[2], size=3)
+ geom_line(data=df_cml, mapping=aes(x="risk", y="return"), color=INK_SOFT, size=1.5, linetype="dashed")
+ geom_point(data=df_special, mapping=aes(x="risk", y="return"), color=IMPRINT[1], size=10, shape=18)
+ labs(
x="Risk (Standard Deviation)",
y="Expected Return",
title="frontier-efficient · letsplot · anyplot.ai",
color="Sharpe Ratio",
)
+ scale_color_viridis()
+ ggsize(1600, 900)
+ theme_minimal()
+ anyplot_theme
)
# Save
ggsave(plot, f"plot-{THEME}.png", path=".", scale=3)
ggsave(plot, f"plot-{THEME}.html", path=".")
Part of Efficient Frontier for Portfolio Optimization on anyplot.ai.