# Portfolio Optimization

This portfolio optimizer tool supports the following portfolio optimization strategies:

• Mean Variance Optimization – Find the optimal risk adjusted portfolio that lies on the efficient frontier
• Conditional Value-at-Risk – Optimize the portfolio to minimize the expected tail loss
• Risk Parity – Find the portfolio that equalizes the risk contribution of portfolio assets
• Tracking Error – Find the portfolio that minimizes the tracking error against the selected benchmark
• Information Ratio – Find the portfolio that maximizes the information ratio against the selected benchmark
• Kelly Criterion – Finds the portfolio with the maximum expected geometric growth rate
• Sortino Ratio – Find the portfolio that maximizes the Sortino ratio for the given minimum acceptable return
• Omega Ratio – Find the portfolio that maximizes the Omega ratio for the given minimum acceptable return
• Maximum Drawdown – Find the portfolio with the minimum worst case drawdown with optional minimum acceptable return

The optimization is based on the monthly return statistics of the selected portfolio assets for the given time period. The optimization result does not predict what allocation would perform best outside the given time period, and the actual performance of portfolios constructed using the optimized asset weights may vary from the given performance goal.

The required inputs for the optimization include the time range and the portfolio assets. Portfolio asset weights and constraints are optional. You can also use the Black-Litterman model based portfolio optimization, which allows the benchmark portfolio asset weights to be optimized based on investor's views.

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Asset Groups
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## Portfolio Optimization Results (Sep 2011 - Dec 2020)

Portfolio optimization results with goal to minimize maximum drawdown subject to 20.00% targeted annual return. The possible range of expected annual portfolio returns for the given period is 19.39% to 48.12%. Refer to the efficient frontier section for additional details.

### Portfolio Allocations

Minimum Maximum Drawdown
Ticker Name Allocation
UPRO ProShares UltraPro S&P500 32.70%
TMF Direxion Daily 20+ Yr Trsy Bull 3X ETF 30.57%
XVZ iPath S&P 500 Dynamic VIX ETN 36.73%
Save portfolio »

### Portfolio Performance Summary

Portfolio performance statistics
MetricMinimum Maximum Drawdown
Start Balance\$10,000
End Balance\$49,938
CAGR18.80%
Expected Return 20.00%
Stdev14.50%
Best Year78.51%
Worst Year-12.06%
Max. Drawdown -14.62%
Sharpe Ratio (ex-ante)1.34
Sharpe Ratio (ex-post)1.22
Sortino Ratio2.44
US Stock Market Correlation0.25

### Trailing Returns

Trailing Returns
NameAnnualized ReturnAnnualized Volatility
3 Month1 year3 year5 yearFull3 year5 year
Minimum Maximum Drawdown6.10%78.51%30.51%25.02%18.80%19.53%16.64%
Trailing annualized return and volatility are for full months ending in December 2020 excluding portfolio cashflows.
Notes on results:
• Past performance is no guarantee of future results, which may vary. All use is subject to terms of service.
• Investing involves risk, including possible loss of principal. The value of the investments and the income derived from them may fluctuate over time.
• All portfolio returns presented are hypothetical and backtested. Hypothetical returns do not reflect trading costs, transaction fees, or taxes.
• Portfolio optimization determines target weights for portfolio assets based on mathematical models that can use either historical or forecasted data as inputs. Optimization results are not guarantees of future performance.
• The results are based on information from a variety of sources we consider reliable, but we do not represent that the information is accurate or complete.
• The results do not constitute investment advice or recommendation, are provided solely for informational purposes, and are not an offer to buy or sell any securities.
• The results are based on the total return of assets and assume that all received dividends and distributions are reinvested.
• The annual results for 2011 are based on full calendar months from September to December
• CAGR = Compound Annual Growth Rate
• Stdev = Annualized standard deviation of monthly returns
• Sharpe ratio is calculated and annualized from monthly excess returns over the risk free rate (3-month treasury bill)
• Drawdowns are calculated based on monthly returns.
• The backtested results assume monthly rebalancing of portfolio assets to match the specified allocation.
• The results from the evolutionary algorithm for non-convex optimization may vary slightly between optimization runs depending on convergence to the global optimum.
Efficient Frontier Assets
#AssetCAGRExpected ReturnStandard DeviationSharpe RatioMin. WeightMax. Weight
1ProShares UltraPro S&P500 (UPRO)34.85%48.12%42.99%1.1050.00%100.00%
2Direxion Daily 20+ Yr Trsy Bull 3X ETF (TMF)12.49%20.50%38.64%0.5140.00%100.00%
3iPath S&P 500 Dynamic VIX ETN (XVZ)-4.63%-1.17%31.39%-0.0570.00%100.00%
Results based on historical returns. Ex ante Sharpe Ratio calculated using 3-month treasury bill returns as the risk-free rate (0.63% annualized).

### Asset Correlations

Efficient Frontier Asset Correlations
NameTickerUPROTMFXVZ
ProShares UltraPro S&P500UPRO1.00-0.38-0.54
Direxion Daily 20+ Yr Trsy Bull 3X ETFTMF-0.381.000.15
iPath S&P 500 Dynamic VIX ETNXVZ-0.540.151.00
Based on monthly returns from Sep 2011 to Dec 2020
Efficient Frontier Assets
#UPROTMFXVZExpected Return*Standard Deviation*Sharpe Ratio*
135.03%23.21%41.77%19.39%14.12%1.329
235.43%23.49%41.08%19.66%14.13%1.347
335.84%23.77%40.39%19.92%14.14%1.364
436.24%24.05%39.70%20.18%14.16%1.381
536.65%24.34%39.02%20.45%14.19%1.397
637.05%24.62%38.33%20.71%14.22%1.413
737.46%24.90%37.64%20.98%14.26%1.427
837.86%25.18%36.95%21.24%14.31%1.441
938.27%25.47%36.26%21.51%14.37%1.454
1038.67%25.75%35.58%21.78%14.43%1.466
1139.08%26.03%34.89%22.04%14.50%1.477
1239.49%26.31%34.20%22.31%14.58%1.487
1339.89%26.60%33.51%22.58%14.66%1.497
1440.30%26.88%32.83%22.85%14.76%1.506
1540.70%27.16%32.14%23.12%14.85%1.514
1641.11%27.44%31.45%23.39%14.96%1.522
1741.51%27.73%30.76%23.66%15.07%1.528
1841.92%28.01%30.08%23.93%15.19%1.534
1942.32%28.29%29.39%24.20%15.31%1.540
2042.73%28.57%28.70%24.48%15.44%1.544
2143.13%28.85%28.01%24.75%15.58%1.549
2243.54%29.14%27.32%25.02%15.72%1.552
2343.94%29.42%26.64%25.30%15.87%1.555
2444.35%29.70%25.95%25.57%16.02%1.557
2544.75%29.98%25.26%25.85%16.18%1.559
2645.16%30.27%24.57%26.12%16.34%1.560
2745.56%30.55%23.89%26.40%16.51%1.561
2845.97%30.83%23.20%26.67%16.68%1.562
2946.38%31.11%22.51%26.95%16.85%1.562
3046.78%31.40%21.82%27.23%17.04%1.562
3147.19%31.68%21.14%27.51%17.22%1.561
3247.59%31.96%20.45%27.79%17.41%1.560
3348.00%32.24%19.76%28.07%17.61%1.559
3448.40%32.53%19.07%28.35%17.80%1.557
3548.81%32.81%18.39%28.63%18.01%1.555
3649.21%33.09%17.70%28.91%18.21%1.553
3749.62%33.37%17.01%29.19%18.42%1.551
3850.02%33.65%16.32%29.47%18.63%1.548
3950.43%33.94%15.63%29.76%18.85%1.545
4050.83%34.22%14.95%30.04%19.07%1.543
4151.24%34.50%14.26%30.32%19.29%1.539
4251.64%34.78%13.57%30.61%19.52%1.536
4352.05%35.07%12.88%30.89%19.74%1.533
4452.45%35.35%12.20%31.18%19.97%1.530
4552.86%35.63%11.51%31.46%20.21%1.526
4653.27%35.91%10.82%31.75%20.44%1.522
4753.67%36.20%10.13%32.04%20.68%1.519
4854.08%36.48%9.45%32.33%20.92%1.515
4954.48%36.76%8.76%32.62%21.17%1.511
5054.89%37.04%8.07%32.90%21.41%1.507
5155.29%37.33%7.38%33.19%21.66%1.504
5255.70%37.61%6.69%33.49%21.91%1.500
5356.10%37.89%6.01%33.78%22.16%1.496
5456.51%38.17%5.32%34.07%22.42%1.492
5556.91%38.45%4.63%34.36%22.67%1.488
5657.32%38.74%3.94%34.65%22.93%1.484
5757.72%39.02%3.26%34.95%23.19%1.480
5858.13%39.30%2.57%35.24%23.45%1.476
5958.53%39.58%1.88%35.53%23.72%1.472
6058.94%39.87%1.19%35.83%23.98%1.468
6159.35%40.15%0.51%36.12%24.25%1.464
6259.92%40.08%0.00%36.42%24.51%1.460
6360.98%39.02%0.00%36.72%24.80%1.456
6462.03%37.97%0.00%37.02%25.09%1.450
6563.09%36.91%0.00%37.31%25.41%1.444
6664.14%35.86%0.00%37.61%25.74%1.437
6765.20%34.80%0.00%37.91%26.09%1.429
6866.25%33.75%0.00%38.21%26.45%1.421
6967.30%32.70%0.00%38.51%26.83%1.412
7068.36%31.64%0.00%38.81%27.22%1.403
7169.41%30.59%0.00%39.11%27.62%1.394
7270.47%29.53%0.00%39.42%28.03%1.384
7371.52%28.48%0.00%39.72%28.46%1.374
7472.58%27.42%0.00%40.02%28.90%1.363
7573.63%26.37%0.00%40.33%29.35%1.353
7674.69%25.31%0.00%40.63%29.81%1.342
7775.74%24.26%0.00%40.94%30.28%1.331
7876.80%23.20%0.00%41.24%30.76%1.321
7977.85%22.15%0.00%41.55%31.24%1.310
8078.91%21.09%0.00%41.85%31.74%1.299
8179.96%20.04%0.00%42.16%32.25%1.288
8281.02%18.98%0.00%42.47%32.76%1.277
8382.07%17.93%0.00%42.78%33.28%1.267
8483.13%16.87%0.00%43.09%33.81%1.256
8584.18%15.82%0.00%43.40%34.34%1.246
8685.23%14.77%0.00%43.71%34.88%1.235
8786.29%13.71%0.00%44.02%35.43%1.225
8887.34%12.66%0.00%44.33%35.98%1.215
8988.40%11.60%0.00%44.65%36.54%1.205
9089.45%10.55%0.00%44.96%37.10%1.195
9190.51%9.49%0.00%45.27%37.67%1.185
9291.56%8.44%0.00%45.59%38.25%1.176
9392.62%7.38%0.00%45.90%38.82%1.166
9493.67%6.33%0.00%46.22%39.41%1.157
9594.73%5.27%0.00%46.53%39.99%1.148
9695.78%4.22%0.00%46.85%40.59%1.139
9796.84%3.16%0.00%47.17%41.18%1.130
9897.89%2.11%0.00%47.49%41.78%1.122
9998.95%1.05%0.00%47.81%42.38%1.113
100100.00%0.00%0.00%48.12%42.99%1.105
*Ex-ante values shown for portfolio return and volatility. Ex ante Sharpe Ratio calculated using historical 3-month treasury bill returns as the risk-free rate (0.63% annualized).
Annual returns for the configured portfolios
YearInflationMinimum Maximum Drawdown ReturnMinimum Maximum Drawdown BalanceProShares UltraPro S&P500 (UPRO)Direxion Daily 20+ Yr Trsy Bull 3X ETF (TMF)iPath S&P 500 Dynamic VIX ETN (XVZ)
2011-0.39%16.79%\$11,6793.97%41.41%-3.00%
20121.74%9.26%\$12,76046.80%0.60%-18.59%
20131.50%2.77%\$13,113118.49%-39.01%-26.74%
20140.76%30.95%\$17,17138.00%97.34%-15.57%
20150.73%-4.77%\$16,352-5.24%-13.73%-9.44%
20162.07%14.94%\$18,79630.79%-2.52%7.37%
20172.11%19.53%\$22,46671.37%22.73%-17.16%
20181.91%-12.06%\$19,757-25.15%-11.01%-9.62%
20192.29%41.59%\$27,974102.31%34.74%-6.79%
20201.36%78.51%\$49,93810.08%41.60%93.79%
Annual return for 2011 is from 09/01/2011 to 12/31/2011
Monthly returns for the configured portfolios
YearMonthMinimum Maximum Drawdown ReturnProShares UltraPro S&P500 (UPRO)Direxion Daily 20+ Yr Trsy Bull 3X ETF (TMF)iPath S&P 500 Dynamic VIX ETN (XVZ)
201198.88%-20.69%41.35%8.18%
2011102.19%32.61%-12.87%-12.35%
2011111.76%-3.35%4.89%3.71%
2011123.14%2.30%9.46%-1.36%
201214.38%13.90%-1.50%0.80%
201222.88%13.24%-7.95%2.68%
20123-1.36%9.29%-13.02%-1.13%
201242.59%-2.38%14.37%-2.80%
201253.80%-17.35%28.90%1.74%
201261.79%11.47%-5.78%-0.52%
201273.67%3.52%11.43%-2.66%
201281.11%6.53%-4.61%1.04%
20129-2.07%7.52%-7.87%-5.77%
201210-4.72%-5.96%-2.13%-5.77%
2012110.28%1.01%3.90%-3.40%
201212-2.99%2.82%-7.80%-4.14%
20131-1.23%15.39%-9.58%-9.08%
201321.25%3.56%3.39%-2.58%
201333.31%11.03%-1.77%0.66%
201344.19%5.45%14.25%-5.31%
20135-2.57%6.80%-19.65%3.29%
20136-2.49%-5.23%-10.33%6.48%
201371.00%16.41%-7.41%-5.73%
20138-4.32%-9.21%-4.47%0.15%
201392.20%9.60%1.53%-3.84%
2013103.06%13.76%4.12%-7.34%
2013110.30%8.87%-8.19%-0.28%
201312-1.56%7.54%-6.01%-5.94%
201412.85%-10.72%19.47%1.10%
201422.78%13.65%1.63%-5.95%
201430.37%2.15%1.76%-2.37%
201441.38%1.65%5.96%-2.68%
201455.26%6.62%8.79%1.12%
201460.58%6.22%-1.00%-3.14%
20147-1.23%-4.44%1.61%-0.75%
201488.70%11.95%14.42%1.05%
20149-2.43%-4.44%-6.65%2.87%
2014102.23%6.03%8.06%-6.01%
2014116.10%8.34%9.08%1.62%
2014121.20%-1.54%9.38%-3.16%
201516.41%-9.47%31.01%0.07%
20152-0.09%17.48%-17.87%-0.93%
20153-0.17%-5.28%2.57%2.11%
20154-1.82%2.59%-10.30%1.31%
20155-1.71%3.43%-7.91%-1.13%
20156-5.95%-6.21%-12.49%-0.28%
201573.38%6.17%13.40%-7.45%
20158-2.95%-18.87%-2.75%11.05%
20159-1.46%-8.64%5.10%-0.54%
2015105.44%26.79%-1.73%-7.61%
201511-0.80%0.81%-2.71%-0.64%
201512-4.40%-6.06%-2.41%-4.57%
201611.95%-15.38%17.74%4.23%
201622.43%-1.68%8.71%0.86%
201635.89%21.50%-0.64%-2.58%
201640.70%0.81%-2.39%3.18%
201653.01%4.65%1.86%2.50%
201666.25%-0.20%21.23%-0.47%
201676.42%11.13%6.07%2.52%
20168-0.05%0.11%-3.58%2.74%
20169-2.21%-0.54%-4.78%-1.57%
201610-6.66%-5.62%-13.32%-2.05%
201611-4.91%11.05%-23.91%-3.29%
2016122.17%5.96%-0.99%1.43%
201711.63%5.14%1.49%-1.37%
201725.40%11.91%4.41%0.44%
20173-1.45%-0.22%-2.35%-1.79%
20174-0.30%2.77%4.43%-6.98%
201752.34%3.70%5.17%-1.23%
20176-0.08%1.51%1.98%-3.22%
201770.61%5.87%-2.43%-1.53%
201781.81%0.21%10.13%-3.68%
201790.51%5.69%-7.29%2.40%
2017101.56%6.83%-0.53%-1.40%
2017114.41%8.87%1.73%2.66%
2017121.72%3.45%5.04%-2.59%
201811.98%18.22%-9.77%-2.70%
20182-11.81%-13.53%-9.57%-12.14%
201830.99%-8.69%8.44%3.42%
20184-4.14%-0.14%-6.71%-5.57%
201853.07%6.46%5.27%-1.78%
201861.15%1.29%1.49%0.74%
201871.24%10.73%-4.70%-2.27%
201885.15%9.09%3.36%3.12%
20189-2.34%1.20%-8.77%-0.13%
201810-7.74%-20.78%-9.11%5.02%
2018112.64%4.28%4.76%-0.57%
201812-1.60%-26.25%17.93%4.10%
201914.51%23.89%0.41%-9.34%
20192-0.21%9.28%-4.52%-5.07%
201937.22%4.85%17.00%1.18%
201942.62%11.81%-6.56%2.07%
201950.12%-18.89%21.11%-0.41%
201966.94%21.37%1.90%-1.72%
201971.51%3.80%0.32%0.46%
201988.30%-6.62%35.11%-0.74%
201990.03%5.30%-9.07%2.91%
2019101.50%5.70%-4.11%2.43%
2019114.20%10.73%-2.01%3.54%
201912-0.90%8.38%-10.05%-1.55%
202016.14%-0.91%23.80%-2.28%
202020.34%-23.04%17.91%6.53%
2020319.12%-48.13%11.91%85.01%
2020412.83%37.17%2.68%-0.40%
202052.78%13.38%-6.48%1.04%
202061.41%3.23%0.54%0.52%
2020711.12%17.70%13.13%3.58%
202083.69%22.05%-14.89%2.82%
20209-2.13%-12.26%2.15%3.32%
202010-5.13%-8.48%-10.35%2.20%
2020118.11%35.20%4.24%-12.78%
2020123.45%11.14%-2.32%1.41%
Portfolio return and risk metrics
MetricMinimum Maximum Drawdown
Arithmetic Mean (monthly)1.53%
Arithmetic Mean (annualized)20.00%
Geometric Mean (monthly)1.45%
Geometric Mean (annualized)18.80%
Volatility (monthly)4.19%
Volatility (annualized)14.50%
Downside Deviation (monthly)2.08%
Max. Drawdown-14.62%
US Market Correlation0.25
Beta(*)0.25
Alpha (annualized)14.58%
R26.14%
Sharpe Ratio1.22
Sortino Ratio2.44
Treynor Ratio (%)70.58
Calmar Ratio2.09
Active Return3.81%
Tracking Error17.66%
Information Ratio0.22
Skewness0.51
Excess Kurtosis2.95
Historical Value-at-Risk (5%)-4.80%
Analytical Value-at-Risk (5%)-5.32%
Conditional Value-at-Risk (5%)-7.03%
Upside Capture Ratio (%)63.99
Downside Capture Ratio (%)3.07
Safe Withdrawal Rate16.89%
Perpetual Withdrawal Rate13.66%
Positive Periods76 out of 112 (67.86%)
Gain/Loss Ratio1.30
* US stock market is used as the benchmark for calculations. Value-at-risk metrics are based on monthly values.

### Drawdowns for Minimum Maximum Drawdown

Drawdowns for Minimum Maximum Drawdown
RankStartEndLengthRecovery ByRecovery TimeUnderwater PeriodDrawdown
1Feb 2018Oct 20189 monthsJun 20198 months1 year 5 months-14.62%
2Aug 2016Nov 20164 monthsNov 20171 year1 year 4 months-13.25%
3Feb 2015Dec 201511 monthsMay 20165 months1 year 4 months-10.50%
4Sep 2012Jan 20135 monthsApr 20141 year 3 months1 year 8 months-10.34%
5Sep 2020Oct 20202 monthsNov 20201 month3 months-7.16%
6Sep 2014Sep 20141 monthNov 20142 months3 months-2.43%
7Mar 2012Mar 20121 monthApr 20121 month2 months-1.36%
8Jul 2014Jul 20141 monthAug 20141 month2 months-1.23%
9Dec 2019Dec 20191 monthJan 20201 month2 months-0.90%

### Portfolio Assets

Performance statistics for portfolio components
TickerNameCAGRStdevBest YearWorst YearMax. DrawdownSharpe RatioSortino RatioUS Mkt Correlation
UPROProShares UltraPro S&P50034.85%42.99%118.49%-25.15%-60.44%0.911.450.99
TMFDirexion Daily 20+ Yr Trsy Bull 3X ETF12.49%38.64%97.34%-39.01%-49.74%0.470.88-0.39
XVZiPath S&P 500 Dynamic VIX ETN-4.63%31.39%93.79%-26.74%-71.71%-0.06-0.15-0.50

### Monthly Correlations

Correlations for the portfolio assets
TickerNameUPROTMFXVZMinimum Maximum Drawdown
UPROProShares UltraPro S&P5001.00-0.38-0.540.24
TMFDirexion Daily 20+ Yr Trsy Bull 3X ETF-0.381.000.150.56
XVZiPath S&P 500 Dynamic VIX ETN-0.540.151.000.39

### Portfolio Risk Decomposition

Portfolio risk decomposition
TickerNameMinimum Maximum Drawdown
UPROProShares UltraPro S&P50022.79%
TMFDirexion Daily 20+ Yr Trsy Bull 3X ETF45.83%
XVZiPath S&P 500 Dynamic VIX ETN31.37%

### Annual Asset Returns

Rolling returns summary
Roll PeriodAverageHighLow
1 year17.71%88.22%-12.06%
3 years12.58%32.39%3.01%
5 years12.05%25.02%7.33%
7 years12.84%21.07%7.35%