Efficient Frontier

This tool uses mean-variance optimization to calculate and plot the efficient frontier for the specified asset classes, mutual funds, ETFs or stocks for the specified time period. The efficient frontier shows the set of optimal portfolios that provide the best possible expected return for the level of risk in the portfolio. Monte Carlo method can be used for more robust optimization that resamples the optimization inputs in order to mitigate the impact of input estimation errors and improve diversification. You can also use the efficient frontier forecast tool to specify expected future returns for the assets. The required inputs for the efficient frontier include the portfolio assets. You can optionally specify the asset allocation and allocation constraints for the portfolio assets. If an asset allocation is specified, the provided portfolio will be rendered on the efficient frontier chart.

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Asset Groups
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Efficient Frontier Results (Jan 1987 - Dec 2018)

Portfolio Allocations

EF #1 Tangency Portfolio
Asset Class Allocation
US Stock Market 40.97%
Long Term Treasury 59.03%
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EF #2 Tangency Portfolio
Asset Class Allocation
US Stock Market 15.26%
Total US Bond Market 84.74%
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Efficient Frontier #1 Assets

Efficient Frontier #1 Assets
#AssetCAGRExpected Return*Standard DeviationSharpe Ratio*Min. WeightMax. Weight
1US Stock Market9.76%11.02%14.99%0.5280.00%100.00%
2Long Term Treasury7.26%7.75%9.64%0.4820.00%100.00%
*Expected return is annualized from historical monthly mean return. Ex ante Sharpe Ratio calculated using historical 1-month treasury bill returns as the risk-free rate (3.10% annualized).

Efficient Frontier #2 Assets

Efficient Frontier #2 Assets
#AssetCAGRExpected Return*Standard DeviationSharpe Ratio*Min. WeightMax. Weight
1US Stock Market9.76%11.02%14.99%0.5280.00%100.00%
2Total US Bond Market5.78%5.85%3.81%0.7200.00%100.00%
*Expected return is annualized from historical monthly mean return. Ex ante Sharpe Ratio calculated using historical 1-month treasury bill returns as the risk-free rate (3.10% annualized).

Asset Correlations

Asset Correlations
AssetUS Stock MarketTotal US Bond MarketLong Term Treasury
US Stock Market1.000.07-0.12
Total US Bond Market0.071.000.87
Long Term Treasury-0.120.871.00
Based on monthly returns from Jan 1987 to Dec 2018
Efficient Frontier #1 Points
#US Stock MarketLong Term TreasuryExpected Return*Standard Deviation*Sharpe Ratio*
131.25%68.75%8.76%7.66%0.739
231.95%68.05%8.79%7.66%0.741
332.64%67.36%8.81%7.67%0.744
433.33%66.67%8.83%7.67%0.746
534.03%65.97%8.85%7.68%0.749
634.72%65.28%8.88%7.69%0.751
735.42%64.58%8.90%7.70%0.752
836.11%63.89%8.92%7.72%0.754
936.81%63.19%8.94%7.73%0.755
1037.50%62.50%8.97%7.75%0.756
1138.20%61.80%8.99%7.77%0.757
1238.89%61.11%9.01%7.80%0.758
1339.58%60.42%9.04%7.82%0.758
1440.28%59.72%9.06%7.85%0.759
1540.97%59.03%9.08%7.88%0.759
1641.67%58.33%9.10%7.91%0.759
1742.36%57.64%9.13%7.94%0.758
1843.06%56.94%9.15%7.98%0.758
1943.75%56.25%9.17%8.01%0.757
2044.45%55.55%9.19%8.05%0.756
2145.14%54.86%9.22%8.09%0.755
2245.83%54.17%9.24%8.14%0.754
2346.53%53.47%9.26%8.18%0.753
2447.22%52.78%9.28%8.23%0.751
2547.92%52.08%9.31%8.27%0.749
2648.61%51.39%9.33%8.32%0.748
2749.31%50.69%9.35%8.38%0.746
2850.00%50.00%9.37%8.43%0.744
2950.69%49.31%9.40%8.48%0.742
3051.39%48.61%9.42%8.54%0.739
3152.08%47.92%9.44%8.60%0.737
3252.78%47.22%9.47%8.66%0.735
3353.47%46.53%9.49%8.72%0.732
3454.17%45.83%9.51%8.78%0.729
3554.86%45.14%9.53%8.85%0.727
3655.56%44.44%9.56%8.91%0.724
3756.25%43.75%9.58%8.98%0.721
3856.94%43.06%9.60%9.05%0.718
3957.64%42.36%9.62%9.12%0.715
4058.33%41.67%9.65%9.19%0.712
4159.03%40.97%9.67%9.26%0.709
4259.72%40.28%9.69%9.34%0.706
4360.42%39.58%9.71%9.41%0.702
4461.11%38.89%9.74%9.49%0.699
4561.81%38.19%9.76%9.56%0.696
4662.50%37.50%9.78%9.64%0.693
4763.19%36.81%9.81%9.72%0.689
4863.89%36.11%9.83%9.80%0.686
4964.58%35.42%9.85%9.88%0.682
5065.28%34.72%9.87%9.97%0.679
5165.97%34.03%9.90%10.05%0.676
5266.67%33.33%9.92%10.14%0.672
5367.36%32.64%9.94%10.22%0.669
5468.06%31.94%9.96%10.31%0.666
5568.75%31.25%9.99%10.40%0.662
5669.44%30.56%10.01%10.48%0.659
5770.14%29.86%10.03%10.57%0.655
5870.83%29.17%10.06%10.66%0.652
5971.53%28.47%10.08%10.75%0.649
6072.22%27.78%10.10%10.85%0.645
6172.92%27.08%10.12%10.94%0.642
6273.61%26.39%10.15%11.03%0.638
6374.31%25.69%10.17%11.12%0.635
6475.00%25.00%10.19%11.22%0.632
6575.69%24.31%10.22%11.31%0.628
6676.39%23.61%10.24%11.41%0.625
6777.08%22.92%10.26%11.51%0.622
6877.78%22.22%10.28%11.60%0.619
6978.47%21.53%10.31%11.70%0.615
7079.17%20.83%10.33%11.80%0.612
7179.86%20.14%10.35%11.90%0.609
7280.56%19.44%10.38%12.00%0.606
7381.25%18.75%10.40%12.10%0.603
7481.94%18.06%10.42%12.20%0.600
7582.64%17.36%10.44%12.30%0.596
7683.33%16.67%10.47%12.41%0.593
7784.03%15.97%10.49%12.51%0.590
7884.72%15.28%10.51%12.61%0.587
7985.42%14.58%10.54%12.71%0.584
8086.11%13.89%10.56%12.82%0.581
8186.81%13.19%10.58%12.92%0.579
8287.50%12.50%10.60%13.03%0.576
8388.19%11.81%10.63%13.13%0.573
8488.89%11.11%10.65%13.24%0.570
8589.58%10.42%10.67%13.35%0.567
8690.28%9.72%10.70%13.45%0.564
8790.97%9.03%10.72%13.56%0.561
8891.67%8.33%10.74%13.67%0.559
8992.36%7.64%10.76%13.77%0.556
9093.06%6.94%10.79%13.88%0.553
9193.75%6.25%10.81%13.99%0.551
9294.44%5.56%10.83%14.10%0.548
9395.14%4.86%10.86%14.21%0.545
9495.83%4.17%10.88%14.32%0.543
9596.53%3.47%10.90%14.43%0.540
9697.22%2.78%10.92%14.54%0.538
9797.92%2.08%10.95%14.65%0.535
9898.61%1.39%10.97%14.76%0.533
9999.31%0.69%10.99%14.87%0.530
100100.00%0.00%11.02%14.99%0.528
*Ex-ante values shown for portfolio return and volatility. Ex ante Sharpe Ratio calculated using historical 1-month treasury bill returns as the risk-free rate (3.10% annualized).
Efficient Frontier #2 Points
#US Stock MarketTotal US Bond MarketExpected Return*Standard Deviation*Sharpe Ratio*
14.67%95.33%6.09%3.75%0.796
25.63%94.37%6.14%3.75%0.808
36.59%93.41%6.19%3.76%0.820
47.56%92.44%6.23%3.77%0.829
58.52%91.48%6.28%3.79%0.838
69.48%90.52%6.33%3.82%0.845
710.45%89.55%6.38%3.85%0.851
811.41%88.59%6.43%3.88%0.856
912.37%87.63%6.48%3.93%0.859
1013.33%86.67%6.53%3.97%0.862
1114.30%85.70%6.58%4.02%0.863
1215.26%84.74%6.63%4.08%0.863
1316.22%83.78%6.67%4.14%0.862
1417.19%82.81%6.72%4.20%0.861
1518.15%81.85%6.77%4.27%0.858
1619.11%80.89%6.82%4.34%0.855
1720.08%79.92%6.87%4.42%0.852
1821.04%78.96%6.92%4.50%0.848
1922.00%78.00%6.97%4.58%0.843
2022.96%77.04%7.02%4.67%0.838
2123.93%76.07%7.07%4.76%0.833
2224.89%75.11%7.12%4.85%0.827
2325.85%74.15%7.17%4.94%0.821
2426.82%73.18%7.21%5.04%0.815
2527.78%72.22%7.26%5.14%0.809
2628.74%71.26%7.31%5.24%0.803
2729.70%70.30%7.36%5.34%0.797
2830.67%69.33%7.41%5.45%0.790
2931.63%68.37%7.46%5.56%0.784
3032.59%67.41%7.51%5.67%0.778
3133.56%66.44%7.56%5.78%0.771
3234.52%65.48%7.61%5.89%0.765
3335.48%64.52%7.66%6.00%0.759
3436.45%63.55%7.71%6.12%0.752
3537.41%62.59%7.76%6.23%0.746
3638.37%61.63%7.81%6.35%0.740
3739.33%60.67%7.86%6.47%0.734
3840.30%59.70%7.91%6.59%0.728
3941.26%58.74%7.96%6.71%0.723
4042.22%57.78%8.00%6.83%0.717
4143.19%56.81%8.05%6.96%0.711
4244.15%55.85%8.10%7.08%0.706
4345.11%54.89%8.15%7.21%0.701
4446.07%53.93%8.20%7.33%0.695
4547.04%52.96%8.25%7.46%0.690
4648.00%52.00%8.30%7.59%0.685
4748.96%51.04%8.35%7.71%0.680
4849.93%50.07%8.40%7.84%0.676
4950.89%49.11%8.45%7.97%0.671
5051.85%48.15%8.50%8.10%0.666
5152.82%47.18%8.55%8.23%0.662
5253.78%46.22%8.60%8.36%0.657
5354.74%45.26%8.65%8.49%0.653
5455.70%44.30%8.70%8.62%0.649
5556.67%43.33%8.75%8.76%0.645
5657.63%42.37%8.80%8.89%0.641
5758.59%41.41%8.85%9.02%0.637
5859.56%40.44%8.90%9.16%0.633
5960.52%39.48%8.95%9.29%0.629
6061.48%38.52%9.00%9.42%0.626
6162.44%37.56%9.05%9.56%0.622
6263.41%36.59%9.10%9.69%0.619
6364.37%35.63%9.15%9.83%0.615
6465.33%34.67%9.20%9.96%0.612
6566.30%33.70%9.25%10.10%0.608
6667.26%32.74%9.30%10.24%0.605
6768.22%31.78%9.35%10.37%0.602
6869.19%30.81%9.40%10.51%0.599
6970.15%29.85%9.45%10.65%0.596
7071.11%28.89%9.50%10.78%0.593
7172.07%27.93%9.55%10.92%0.590
7273.04%26.96%9.60%11.06%0.587
7374.00%26.00%9.65%11.20%0.585
7474.96%25.04%9.70%11.34%0.582
7575.93%24.07%9.75%11.47%0.579
7676.89%23.11%9.80%11.61%0.577
7777.85%22.15%9.85%11.75%0.574
7878.81%21.19%9.90%11.89%0.572
7979.78%20.22%9.95%12.03%0.569
8080.74%19.26%10.00%12.17%0.567
8181.70%18.30%10.05%12.31%0.565
8282.67%17.33%10.11%12.45%0.562
8383.63%16.37%10.16%12.59%0.560
8484.59%15.41%10.21%12.73%0.558
8585.56%14.44%10.26%12.87%0.556
8686.52%13.48%10.31%13.01%0.554
8787.48%12.52%10.36%13.15%0.552
8888.44%11.56%10.41%13.29%0.550
8989.41%10.59%10.46%13.43%0.548
9090.37%9.63%10.51%13.57%0.546
9191.33%8.67%10.56%13.71%0.544
9292.30%7.70%10.61%13.85%0.542
9393.26%6.74%10.66%13.99%0.540
9494.22%5.78%10.71%14.14%0.538
9595.19%4.81%10.76%14.28%0.536
9696.15%3.85%10.81%14.42%0.535
9797.11%2.89%10.86%14.56%0.533
9898.07%1.93%10.91%14.70%0.531
9999.04%0.96%10.97%14.84%0.530
100100.00%0.00%11.02%14.99%0.528
*Ex-ante values shown for portfolio return and volatility. Ex ante Sharpe Ratio calculated using historical 1-month treasury bill returns as the risk-free rate (3.10% annualized).
Notes on results:
  • Past performance is not a guarantee of future returns.
  • The efficient frontier is based on the monthly returns, volatilities, and correlations of the specified assets.
  • The displayed standard deviation and expected return in the efficient frontier are annualized from the monthly returns
  • Ex-ante Sharpe Ratio for the efficient frontier data points is calculated using historical 1-month treasury bill returns as the risk free rate