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.

Portfolio Type

Time Period

Start Year

First Month

End Year

Last Month

Optimization Goal

Targeted Annual Return

Targeted Annual Volatility

Targeted Tracking Error

Targeted Monthly CVaR

Robust Optimization

Use Historical Returns

Use Historical Volatility

Use Historical Correlations

Correlation Matrix

Browse…

Asset Constraints

Group Constraints

Compared Allocation

Benchmark

Benchmark Ticker

Portfolio Assets

Allocation

Expected Return

Volatility

Min. Weight

Max. Weight

Group

Select asset 1

Enter percentage allocation for asset 1 %

Expected return for asset 1 %

Expected volatility for asset 1 %

Minimum allocation for asset 1 %

Maximum allocation for asset 1 %

Group assignment for asset 1

Select asset 2

Enter percentage allocation for asset 2 %

Expected return for asset 2 %

Expected volatility for asset 2 %

Minimum allocation for asset 2 %

Maximum allocation for asset 2 %

Group assignment for asset 2

Select asset 3

Enter percentage allocation for asset 3 %

Expected return for asset 3 %

Expected volatility for asset 3 %

Minimum allocation for asset 3 %

Maximum allocation for asset 3 %

Group assignment for asset 3

Select asset 4

Enter percentage allocation for asset 4 %

Expected return for asset 4 %

Expected volatility for asset 4 %

Minimum allocation for asset 4 %

Maximum allocation for asset 4 %

Group assignment for asset 4

Select asset 5

Enter percentage allocation for asset 5 %

Expected return for asset 5 %

Expected volatility for asset 5 %

Minimum allocation for asset 5 %

Maximum allocation for asset 5 %

Group assignment for asset 5

Select asset 6

Enter percentage allocation for asset 6 %

Expected return for asset 6 %

Expected volatility for asset 6 %

Minimum allocation for asset 6 %

Maximum allocation for asset 6 %

Group assignment for asset 6

Select asset 7

Enter percentage allocation for asset 7 %

Expected return for asset 7 %

Expected volatility for asset 7 %

Minimum allocation for asset 7 %

Maximum allocation for asset 7 %

Group assignment for asset 7

Select asset 8

Enter percentage allocation for asset 8 %

Expected return for asset 8 %

Expected volatility for asset 8 %

Minimum allocation for asset 8 %

Maximum allocation for asset 8 %

Group assignment for asset 8

Select asset 9

Enter percentage allocation for asset 9 %

Expected return for asset 9 %

Expected volatility for asset 9 %

Minimum allocation for asset 9 %

Maximum allocation for asset 9 %

Group assignment for asset 9

Select asset 10

Enter percentage allocation for asset 10 %

Expected return for asset 10 %

Expected volatility for asset 10 %

Minimum allocation for asset 10 %

Maximum allocation for asset 10 %

Group assignment for asset 10

Total

Total allocation for portfolio %

Asset Groups

Min. Weight

Max. Weight

Full name for group A at index 1

Minimum allocation for group A at index 1 %

Maximum allocation for group A at index 1 %

Full name for group B at index 2

Minimum allocation for group B at index 2 %

Maximum allocation for group B at index 2 %

Full name for group C at index 3

Minimum allocation for group C at index 3 %

Maximum allocation for group C at index 3 %

Full name for group D at index 4

Minimum allocation for group D at index 4 %

Maximum allocation for group D at index 4 %

Full name for group E at index 5

Minimum allocation for group E at index 5 %

Maximum allocation for group E at index 5 %

Full name for group F at index 6

Minimum allocation for group F at index 6 %

Maximum allocation for group F at index 6 %

Portfolio Optimization Results (Mar 2010 - Mar 2021) Save

Note: The time period was constrained by the available data for ProShares UltraPro MidCap400 (UMDD) [Mar 2010 - Mar 2021].

Portfolio optimization results with goal to maximize kelly criterion. The possible range of expected annual portfolio returns for the given period is 18.61% to 46.60%. Refer to the efficient frontier section for additional details.

Portfolio Allocations

Maximum Kelly Criterion
Ticker	Name	Allocation
UPRO	ProShares UltraPro S&P500	85.61%
TMF	Direxion Daily 20+ Yr Trsy Bull 3X ETF	14.39%
Save portfolio »

Portfolio Performance Summary

Portfolio performance statistics
Metric	Maximum Kelly Criterion
Start Balance	$10,000
End Balance	$245,044
CAGR	33.46%
Expected Return	41.98%
Stdev	35.39%
Best Year	97.23%
Worst Year	-21.32%
Max. Drawdown	-49.86%
Sharpe Ratio (ex-ante)	1.17
Sharpe Ratio (ex-post)	0.99
Sortino Ratio	1.63
US Stock Market Correlation	0.98

Trailing Returns

Trailing Returns
Name	Annualized Return							Annualized Volatility
Name	3 Month	YTD	1 year	3 year	5 year	10 year	Full	3 year	5 year
Maximum Kelly Criterion	8.41%	8.41%	160.45%	30.68%	32.84%	31.17%	33.46%	48.19%	39.20%
Trailing annualized return and volatility are for full months ending in March 2021 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 2010 are based on full calendar months from March to December
The annual results for 2021 are based on full calendar months from January to March
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
#	Asset	CAGR	Expected Return	Standard Deviation	Sharpe Ratio	Min. Weight	Max. Weight
1	ProShares UltraPro S&P500 (UPRO)	32.64%	46.19%	43.91%	1.040	0.00%	100.00%
2	ProShares UltraPro MidCap400 (UMDD)	25.53%	45.35%	52.38%	0.855	0.00%	100.00%
3	ProShares UltraPro Russell2000 (URTY)	22.07%	46.60%	58.84%	0.783	0.00%	100.00%
4	Direxion Daily 20+ Yr Trsy Bull 3X ETF (TMF)	10.65%	19.12%	39.88%	0.466	0.00%	100.00%
5	Direxion Daily 7-10 Yr Trs Bull 3X ETF (TYD)	9.52%	11.30%	18.30%	0.588	0.00%	100.00%
6	ProShares Ultra Gold (UGL)	1.46%	6.83%	32.39%	0.194	0.00%	100.00%
Results based on historical returns. Ex ante Sharpe Ratio calculated using 3-month treasury bill returns as the risk-free rate (0.55% annualized).

Asset Correlations

Efficient Frontier Asset Correlations
Name	Ticker	UPRO	UMDD	URTY	TMF	TYD	UGL
ProShares UltraPro S&P500	UPRO	1.00	0.94	0.90	-0.45	-0.47	0.03
ProShares UltraPro MidCap400	UMDD	0.94	1.00	0.97	-0.46	-0.46	0.04
ProShares UltraPro Russell2000	URTY	0.90	0.97	1.00	-0.49	-0.48	0.00
Direxion Daily 20+ Yr Trsy Bull 3X ETF	TMF	-0.45	-0.46	-0.49	1.00	0.86	0.25
Direxion Daily 7-10 Yr Trs Bull 3X ETF	TYD	-0.47	-0.46	-0.48	0.86	1.00	0.34
ProShares Ultra Gold	UGL	0.03	0.04	0.00	0.25	0.34	1.00
Based on monthly returns from Mar 2010 to Mar 2021

Note: You can hover over any of the data points on the chart for more information.

Efficient Frontier Assets
#	UPRO	UMDD	URTY	TMF	TYD	UGL	Expected Return^*	Standard Deviation^*	Sharpe Ratio^*
1	20.71%	0.00%	2.41%	0.00%	76.74%	0.14%	18.61%	12.94%	1.397
2	21.97%	0.00%	1.92%	0.00%	76.11%	0.00%	18.87%	12.94%	1.416
3	23.28%	0.00%	1.40%	0.00%	75.32%	0.00%	19.13%	12.96%	1.434
4	24.60%	0.00%	0.88%	0.00%	74.53%	0.00%	19.38%	12.99%	1.450
5	25.91%	0.00%	0.36%	0.00%	73.74%	0.00%	19.64%	13.04%	1.465
6	27.06%	0.00%	0.00%	0.00%	72.94%	0.00%	19.90%	13.09%	1.478
7	27.84%	0.00%	0.00%	0.00%	72.16%	0.00%	20.16%	13.16%	1.490
8	28.63%	0.00%	0.00%	0.00%	71.37%	0.00%	20.42%	13.25%	1.500
9	29.42%	0.00%	0.00%	0.00%	70.58%	0.00%	20.68%	13.34%	1.509
10	30.20%	0.00%	0.00%	0.00%	69.80%	0.00%	20.94%	13.45%	1.516
11	30.99%	0.00%	0.00%	0.00%	69.01%	0.00%	21.20%	13.58%	1.521
12	31.78%	0.00%	0.00%	0.00%	68.22%	0.00%	21.46%	13.71%	1.525
13	32.52%	0.00%	0.00%	0.17%	67.30%	0.00%	21.72%	13.86%	1.528
14	33.08%	0.00%	0.00%	1.12%	65.80%	0.00%	21.98%	14.02%	1.530
15	33.63%	0.00%	0.00%	2.07%	64.30%	0.00%	22.25%	14.17%	1.531
16	34.18%	0.00%	0.00%	3.02%	62.80%	0.00%	22.51%	14.34%	1.532
17	34.73%	0.00%	0.00%	3.97%	61.29%	0.00%	22.78%	14.50%	1.533
18	35.29%	0.00%	0.00%	4.92%	59.79%	0.00%	23.04%	14.67%	1.533
19	35.84%	0.00%	0.00%	5.87%	58.29%	0.00%	23.31%	14.84%	1.533
20	36.39%	0.00%	0.00%	6.82%	56.78%	0.00%	23.57%	15.02%	1.533
21	36.94%	0.00%	0.00%	7.77%	55.28%	0.00%	23.84%	15.20%	1.532
22	37.50%	0.00%	0.00%	8.72%	53.78%	0.00%	24.10%	15.38%	1.532
23	38.05%	0.00%	0.00%	9.67%	52.28%	0.00%	24.37%	15.57%	1.531
24	38.60%	0.00%	0.00%	10.63%	50.77%	0.00%	24.64%	15.75%	1.529
25	39.15%	0.00%	0.00%	11.58%	49.27%	0.00%	24.91%	15.94%	1.528
26	39.71%	0.00%	0.00%	12.53%	47.77%	0.00%	25.18%	16.14%	1.526
27	40.26%	0.00%	0.00%	13.48%	46.26%	0.00%	25.44%	16.33%	1.525
28	40.81%	0.00%	0.00%	14.43%	44.76%	0.00%	25.71%	16.53%	1.523
29	41.36%	0.00%	0.00%	15.38%	43.26%	0.00%	25.99%	16.73%	1.521
30	41.92%	0.00%	0.00%	16.33%	41.76%	0.00%	26.26%	16.93%	1.518
31	42.47%	0.00%	0.00%	17.28%	40.25%	0.00%	26.53%	17.14%	1.516
32	43.02%	0.00%	0.00%	18.23%	38.75%	0.00%	26.80%	17.34%	1.514
33	43.58%	0.00%	0.00%	19.18%	37.25%	0.00%	27.07%	17.55%	1.511
34	44.13%	0.00%	0.00%	20.13%	35.74%	0.00%	27.35%	17.76%	1.509
35	44.68%	0.00%	0.00%	21.08%	34.24%	0.00%	27.62%	17.98%	1.506
36	45.23%	0.00%	0.00%	22.03%	32.74%	0.00%	27.89%	18.19%	1.503
37	45.79%	0.00%	0.00%	22.98%	31.24%	0.00%	28.17%	18.41%	1.501
38	46.34%	0.00%	0.00%	23.93%	29.73%	0.00%	28.44%	18.63%	1.498
39	46.89%	0.00%	0.00%	24.88%	28.23%	0.00%	28.72%	18.85%	1.495
40	47.44%	0.00%	0.00%	25.83%	26.73%	0.00%	28.99%	19.07%	1.492
41	48.00%	0.00%	0.00%	26.78%	25.23%	0.00%	29.27%	19.29%	1.489
42	48.55%	0.00%	0.00%	27.73%	23.72%	0.00%	29.55%	19.51%	1.486
43	49.10%	0.00%	0.00%	28.68%	22.22%	0.00%	29.83%	19.74%	1.483
44	49.65%	0.00%	0.00%	29.63%	20.72%	0.00%	30.11%	19.97%	1.480
45	50.21%	0.00%	0.00%	30.58%	19.21%	0.00%	30.38%	20.19%	1.478
46	50.76%	0.00%	0.00%	31.53%	17.71%	0.00%	30.66%	20.42%	1.475
47	51.31%	0.00%	0.00%	32.48%	16.21%	0.00%	30.94%	20.66%	1.472
48	51.87%	0.00%	0.00%	33.43%	14.71%	0.00%	31.22%	20.89%	1.469
49	52.42%	0.00%	0.00%	34.38%	13.20%	0.00%	31.51%	21.12%	1.466
50	52.97%	0.00%	0.00%	35.33%	11.70%	0.00%	31.79%	21.36%	1.463
51	53.52%	0.00%	0.00%	36.28%	10.20%	0.00%	32.07%	21.59%	1.460
52	54.08%	0.00%	0.00%	37.23%	8.69%	0.00%	32.35%	21.83%	1.457
53	54.63%	0.00%	0.00%	38.18%	7.19%	0.00%	32.64%	22.07%	1.454
54	55.18%	0.00%	0.00%	39.13%	5.69%	0.00%	32.92%	22.30%	1.452
55	55.73%	0.00%	0.00%	40.08%	4.19%	0.00%	33.21%	22.54%	1.449
56	56.29%	0.00%	0.00%	41.03%	2.68%	0.00%	33.49%	22.78%	1.446
57	56.84%	0.00%	0.00%	41.98%	1.18%	0.00%	33.78%	23.02%	1.443
58	57.50%	0.00%	0.00%	42.50%	0.00%	0.00%	34.06%	23.27%	1.440
59	58.54%	0.00%	0.00%	41.46%	0.00%	0.00%	34.35%	23.53%	1.437
60	59.59%	0.00%	0.00%	40.41%	0.00%	0.00%	34.64%	23.80%	1.432
61	60.63%	0.00%	0.00%	39.37%	0.00%	0.00%	34.92%	24.10%	1.426
62	61.68%	0.00%	0.00%	38.32%	0.00%	0.00%	35.21%	24.42%	1.420
63	62.72%	0.00%	0.00%	37.28%	0.00%	0.00%	35.50%	24.75%	1.412
64	63.77%	0.00%	0.00%	36.23%	0.00%	0.00%	35.79%	25.11%	1.404
65	64.81%	0.00%	0.00%	35.19%	0.00%	0.00%	36.08%	25.48%	1.395
66	65.86%	0.00%	0.00%	34.14%	0.00%	0.00%	36.37%	25.86%	1.385
67	66.90%	0.00%	0.00%	33.10%	0.00%	0.00%	36.66%	26.26%	1.375
68	67.95%	0.00%	0.00%	32.05%	0.00%	0.00%	36.96%	26.68%	1.365
69	68.99%	0.00%	0.00%	31.01%	0.00%	0.00%	37.25%	27.11%	1.354
70	70.04%	0.00%	0.00%	29.96%	0.00%	0.00%	37.54%	27.56%	1.343
71	71.08%	0.00%	0.00%	28.92%	0.00%	0.00%	37.83%	28.01%	1.331
72	72.13%	0.00%	0.00%	27.87%	0.00%	0.00%	38.13%	28.48%	1.320
73	73.17%	0.00%	0.00%	26.83%	0.00%	0.00%	38.42%	28.96%	1.308
74	74.22%	0.00%	0.00%	25.78%	0.00%	0.00%	38.72%	29.45%	1.296
75	75.26%	0.00%	0.00%	24.74%	0.00%	0.00%	39.01%	29.95%	1.284
76	76.31%	0.00%	0.00%	23.69%	0.00%	0.00%	39.31%	30.47%	1.272
77	77.35%	0.00%	0.00%	22.65%	0.00%	0.00%	39.61%	30.99%	1.261
78	78.40%	0.00%	0.00%	21.60%	0.00%	0.00%	39.91%	31.52%	1.249
79	79.44%	0.00%	0.00%	20.56%	0.00%	0.00%	40.20%	32.06%	1.237
80	80.48%	0.00%	0.00%	19.52%	0.00%	0.00%	40.50%	32.60%	1.226
81	81.53%	0.00%	0.00%	18.47%	0.00%	0.00%	40.80%	33.16%	1.214
82	82.57%	0.00%	0.00%	17.43%	0.00%	0.00%	41.10%	33.72%	1.203
83	83.62%	0.00%	0.00%	16.38%	0.00%	0.00%	41.40%	34.29%	1.191
84	84.66%	0.00%	0.00%	15.34%	0.00%	0.00%	41.70%	34.87%	1.180
85	85.71%	0.00%	0.00%	14.29%	0.00%	0.00%	42.01%	35.45%	1.170
86	86.75%	0.00%	0.00%	13.25%	0.00%	0.00%	42.31%	36.04%	1.159
87	87.80%	0.00%	0.00%	12.20%	0.00%	0.00%	42.61%	36.63%	1.148
88	88.84%	0.00%	0.00%	11.16%	0.00%	0.00%	42.91%	37.23%	1.138
89	89.89%	0.00%	0.00%	10.11%	0.00%	0.00%	43.22%	37.83%	1.128
90	90.93%	0.00%	0.00%	9.07%	0.00%	0.00%	43.52%	38.44%	1.118
91	91.98%	0.00%	0.00%	8.02%	0.00%	0.00%	43.83%	39.06%	1.108
92	93.02%	0.00%	0.00%	6.98%	0.00%	0.00%	44.13%	39.68%	1.099
93	94.07%	0.00%	0.00%	5.93%	0.00%	0.00%	44.44%	40.30%	1.089
94	95.11%	0.00%	0.00%	4.89%	0.00%	0.00%	44.75%	40.93%	1.080
95	96.16%	0.00%	0.00%	3.84%	0.00%	0.00%	45.06%	41.56%	1.071
96	97.20%	0.00%	0.00%	2.80%	0.00%	0.00%	45.36%	42.19%	1.062
97	98.25%	0.00%	0.00%	1.75%	0.00%	0.00%	45.67%	42.83%	1.054
98	99.29%	0.00%	0.00%	0.71%	0.00%	0.00%	45.98%	43.47%	1.045
99	75.60%	0.00%	24.40%	0.00%	0.00%	0.00%	46.29%	46.50%	0.984
100	0.00%	0.00%	100.00%	0.00%	0.00%	0.00%	46.60%	58.84%	0.783
^*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 (0.55% annualized).

Annual returns for the configured portfolios
Year	Inflation	Maximum Kelly Criterion Return	Maximum Kelly Criterion Balance	ProShares UltraPro S&P500 (UPRO)	Direxion Daily 20+ Yr Trsy Bull 3X ETF (TMF)
2010	1.12%	43.38%	$14,338	40.92%	10.27%
2011	2.96%	5.45%	$15,120	-11.88%	109.14%
2012	1.74%	42.82%	$21,594	46.80%	0.60%
2013	1.50%	84.86%	$39,919	118.49%	-39.01%
2014	0.76%	46.43%	$58,452	38.00%	97.34%
2015	0.73%	-4.00%	$56,115	-5.24%	-13.73%
2016	2.07%	28.15%	$71,910	30.79%	-2.52%
2017	2.11%	63.83%	$117,810	71.37%	22.73%
2018	1.91%	-21.32%	$92,698	-25.15%	-11.01%
2019	2.29%	97.23%	$182,827	102.31%	34.74%
2020	1.36%	23.63%	$226,037	10.08%	41.60%
2021	0.98%	8.41%	$245,044	17.58%	-37.24%
Annual return for 2010 is from 03/01/2010 to 12/31/2010 and annual return for 2021 is from 01/01/2021 to 03/31/2021

Monthly returns for the configured portfolios
Year	Month	Maximum Kelly Criterion Return	Maximum Kelly Criterion Balance	ProShares UltraPro S&P500 (UPRO)	Direxion Daily 20+ Yr Trsy Bull 3X ETF (TMF)
2010	3	15.21%	$11,521	18.90%	-6.73%
2010	4	4.93%	$12,089	4.08%	9.96%
2010	5	-18.30%	$9,877	-24.15%	16.54%
2010	6	-11.79%	$8,713	-16.58%	16.73%
2010	7	17.57%	$10,244	21.17%	-3.90%
2010	8	-7.79%	$9,446	-13.46%	25.94%
2010	9	22.62%	$11,583	27.94%	-9.02%
2010	10	7.84%	$12,491	11.44%	-13.57%
2010	11	-1.57%	$12,295	-0.82%	-6.03%
2010	12	16.62%	$14,338	21.36%	-11.63%
2011	1	4.27%	$14,949	6.56%	-9.40%
2011	2	9.54%	$16,376	10.39%	4.49%
2011	3	-0.76%	$16,251	-0.80%	-0.55%
2011	4	8.42%	$17,620	8.73%	6.57%
2011	5	-1.88%	$17,289	-3.93%	10.36%
2011	6	-6.09%	$16,236	-5.87%	-7.39%
2011	7	-3.56%	$15,657	-6.39%	13.28%
2011	8	-14.12%	$13,447	-21.09%	27.34%
2011	9	-11.76%	$11,865	-20.69%	41.35%
2011	10	26.06%	$14,957	32.61%	-12.87%
2011	11	-2.17%	$14,633	-3.35%	4.89%
2011	12	3.33%	$15,120	2.30%	9.46%
2012	1	11.68%	$16,886	13.90%	-1.50%
2012	2	10.19%	$18,607	13.24%	-7.95%
2012	3	6.08%	$19,739	9.29%	-13.02%
2012	4	0.03%	$19,744	-2.38%	14.37%
2012	5	-10.69%	$17,633	-17.35%	28.90%
2012	6	8.99%	$19,218	11.47%	-5.78%
2012	7	4.65%	$20,112	3.52%	11.43%
2012	8	4.92%	$21,103	6.53%	-4.61%
2012	9	5.30%	$22,222	7.52%	-7.87%
2012	10	-5.41%	$21,019	-5.96%	-2.13%
2012	11	1.43%	$21,319	1.01%	3.90%
2012	12	1.29%	$21,594	2.82%	-7.80%
2013	1	11.80%	$24,141	15.39%	-9.58%
2013	2	3.54%	$24,996	3.56%	3.39%
2013	3	9.19%	$27,293	11.03%	-1.77%
2013	4	6.72%	$29,126	5.45%	14.25%
2013	5	3.00%	$29,999	6.80%	-19.65%
2013	6	-5.96%	$28,211	-5.23%	-10.33%
2013	7	12.98%	$31,874	16.41%	-7.41%
2013	8	-8.53%	$29,154	-9.21%	-4.47%
2013	9	8.44%	$31,614	9.60%	1.53%
2013	10	12.37%	$35,526	13.76%	4.12%
2013	11	6.42%	$37,806	8.87%	-8.19%
2013	12	5.59%	$39,919	7.54%	-6.01%
2014	1	-6.37%	$37,375	-10.72%	19.47%
2014	2	11.92%	$41,831	13.65%	1.63%
2014	3	2.09%	$42,707	2.15%	1.76%
2014	4	2.27%	$43,677	1.65%	5.96%
2014	5	6.93%	$46,706	6.62%	8.79%
2014	6	5.19%	$49,128	6.22%	-1.00%
2014	7	-3.57%	$47,374	-4.44%	1.61%
2014	8	12.31%	$53,205	11.95%	14.42%
2014	9	-4.75%	$50,676	-4.44%	-6.65%
2014	10	6.32%	$53,881	6.03%	8.06%
2014	11	8.45%	$58,433	8.34%	9.08%
2014	12	0.03%	$58,452	-1.54%	9.38%
2015	1	-3.65%	$56,321	-9.47%	31.01%
2015	2	12.40%	$63,303	17.48%	-17.87%
2015	3	-4.15%	$60,676	-5.28%	2.57%
2015	4	0.74%	$61,123	2.59%	-10.30%
2015	5	1.80%	$62,224	3.43%	-7.91%
2015	6	-7.11%	$57,798	-6.21%	-12.49%
2015	7	7.21%	$61,968	6.17%	13.40%
2015	8	-16.55%	$51,715	-18.87%	-2.75%
2015	9	-6.66%	$48,271	-8.64%	5.10%
2015	10	22.68%	$59,221	26.79%	-1.73%
2015	11	0.31%	$59,403	0.81%	-2.71%
2015	12	-5.53%	$56,115	-6.06%	-2.41%
2016	1	-10.61%	$50,160	-15.38%	17.74%
2016	2	-0.19%	$50,067	-1.68%	8.71%
2016	3	18.31%	$59,235	21.50%	-0.64%
2016	4	0.35%	$59,439	0.81%	-2.39%
2016	5	4.25%	$61,967	4.65%	1.86%
2016	6	2.88%	$63,751	-0.20%	21.23%
2016	7	10.40%	$70,384	11.13%	6.07%
2016	8	-0.42%	$70,087	0.11%	-3.58%
2016	9	-1.15%	$69,283	-0.54%	-4.78%
2016	10	-6.73%	$64,621	-5.62%	-13.32%
2016	11	6.02%	$68,511	11.05%	-23.91%
2016	12	4.96%	$71,910	5.96%	-0.99%
2017	1	4.61%	$75,226	5.14%	1.49%
2017	2	10.83%	$83,373	11.91%	4.41%
2017	3	-0.52%	$82,936	-0.22%	-2.35%
2017	4	3.01%	$85,432	2.77%	4.43%
2017	5	3.91%	$88,773	3.70%	5.17%
2017	6	1.58%	$90,177	1.51%	1.98%
2017	7	4.68%	$94,395	5.87%	-2.43%
2017	8	1.64%	$95,940	0.21%	10.13%
2017	9	3.82%	$99,607	5.69%	-7.29%
2017	10	5.78%	$105,359	6.83%	-0.53%
2017	11	7.85%	$113,625	8.87%	1.73%
2017	12	3.68%	$117,810	3.45%	5.04%
2018	1	14.19%	$134,526	18.22%	-9.77%
2018	2	-12.96%	$117,088	-13.53%	-9.57%
2018	3	-6.22%	$109,802	-8.69%	8.44%
2018	4	-1.08%	$108,612	-0.14%	-6.71%
2018	5	6.29%	$115,447	6.46%	5.27%
2018	6	1.31%	$116,965	1.29%	1.49%
2018	7	8.51%	$126,918	10.73%	-4.70%
2018	8	8.27%	$137,411	9.09%	3.36%
2018	9	-0.24%	$137,086	1.20%	-8.77%
2018	10	-19.10%	$110,903	-20.78%	-9.11%
2018	11	4.35%	$115,722	4.28%	4.76%
2018	12	-19.90%	$92,698	-26.25%	17.93%
2019	1	20.51%	$111,714	23.89%	0.41%
2019	2	7.30%	$119,865	9.28%	-4.52%
2019	3	6.60%	$127,776	4.85%	17.00%
2019	4	9.17%	$139,489	11.81%	-6.56%
2019	5	-13.14%	$121,164	-18.89%	21.11%
2019	6	18.57%	$143,664	21.37%	1.90%
2019	7	3.30%	$148,410	3.80%	0.32%
2019	8	-0.61%	$147,498	-6.62%	35.11%
2019	9	3.23%	$152,261	5.30%	-9.07%
2019	10	4.29%	$158,789	5.70%	-4.11%
2019	11	8.90%	$172,920	10.73%	-2.01%
2019	12	5.73%	$182,827	8.38%	-10.05%
2020	1	2.64%	$187,657	-0.91%	23.80%
2020	2	-17.14%	$155,486	-23.04%	17.91%
2020	3	-39.49%	$94,084	-48.13%	11.91%
2020	4	32.21%	$124,386	37.17%	2.68%
2020	5	10.52%	$137,472	13.38%	-6.48%
2020	6	2.85%	$141,387	3.23%	0.54%
2020	7	17.04%	$165,478	17.70%	13.13%
2020	8	16.73%	$193,164	22.05%	-14.89%
2020	9	-10.18%	$173,492	-12.26%	2.15%
2020	10	-8.75%	$158,315	-8.48%	-10.35%
2020	11	30.74%	$206,984	35.20%	4.24%
2020	12	9.20%	$226,037	11.14%	-2.32%
2021	1	-4.73%	$215,335	-3.71%	-10.84%
2021	2	4.21%	$224,405	7.74%	-16.79%
2021	3	9.20%	$245,044	13.33%	-15.40%

Portfolio return and risk metrics
Metric	Maximum Kelly Criterion
Arithmetic Mean (monthly)	2.96%
Arithmetic Mean (annualized)	41.98%
Geometric Mean (monthly)	2.43%
Geometric Mean (annualized)	33.46%
Volatility (monthly)	10.22%
Volatility (annualized)	35.39%
Downside Deviation (monthly)	6.16%
Max. Drawdown	-49.86%
US Market Correlation	0.98
Beta^(*)	2.39
Alpha (annualized)	0.49%
R²	96.31%
Sharpe Ratio	0.99
Sortino Ratio	1.63
Treynor Ratio (%)	14.66
Calmar Ratio	0.62
Active Return	18.94%
Tracking Error	21.31%
Information Ratio	0.89
Skewness	-0.39
Excess Kurtosis	2.18
Historical Value-at-Risk (5%)	-13.53%
Analytical Value-at-Risk (5%)	-13.78%
Conditional Value-at-Risk (5%)	-20.66%
Upside Capture Ratio (%)	278.59
Downside Capture Ratio (%)	227.03
Safe Withdrawal Rate	33.02%
Perpetual Withdrawal Rate	23.90%
Positive Periods	90 out of 133 (67.67%)
Gain/Loss Ratio	1.03
^* US stock market is used as the benchmark for calculations. Value-at-risk metrics are based on monthly values.

Drawdowns for Maximum Kelly Criterion

Drawdowns for Maximum Kelly Criterion (worst 10)
Rank	Start	End	Length	Recovery By	Recovery Time	Underwater Period	Drawdown
1	Feb 2020	Mar 2020	2 months	Aug 2020	5 months	7 months	-49.86%
2	May 2011	Sep 2011	5 months	Feb 2012	5 months	10 months	-32.66%
3	Sep 2018	Dec 2018	4 months	Apr 2019	4 months	8 months	-32.54%
4	May 2010	Jun 2010	2 months	Oct 2010	4 months	6 months	-27.92%
5	Mar 2015	Sep 2015	7 months	Jun 2016	9 months	1 year 4 months	-23.75%
6	Feb 2018	Apr 2018	3 months	Aug 2018	4 months	7 months	-19.26%
7	Sep 2020	Oct 2020	2 months	Nov 2020	1 month	3 months	-18.04%
8	May 2019	May 2019	1 month	Jun 2019	1 month	2 months	-13.14%
9	May 2012	May 2012	1 month	Jul 2012	2 months	3 months	-10.69%
10	Aug 2013	Aug 2013	1 month	Oct 2013	2 months	3 months	-8.53%
Worst 10 drawdowns included above

Portfolio Assets

Performance statistics for portfolio components
Ticker	Name	CAGR	Stdev	Best Year	Worst Year	Max. Drawdown	Sharpe Ratio	Sortino Ratio	US Mkt Correlation
UPRO	ProShares UltraPro S&P500	32.64%	43.91%	118.49%	-25.15%	-60.44%	0.87	1.38	0.99
TMF	Direxion Daily 20+ Yr Trsy Bull 3X ETF	10.65%	39.88%	109.14%	-39.01%	-50.20%	0.43	0.78	-0.46

Monthly Correlations

Correlations for the portfolio assets
Ticker	Name	UPRO	TMF	Maximum Kelly Criterion
UPRO	ProShares UltraPro S&P500	1.00	-0.45	0.99
TMF	Direxion Daily 20+ Yr Trsy Bull 3X ETF	-0.45	1.00	-0.31

Portfolio Risk Decomposition

Portfolio risk decomposition
Ticker	Name	Maximum Kelly Criterion
UPRO	ProShares UltraPro S&P500	105.08%
TMF	Direxion Daily 20+ Yr Trsy Bull 3X ETF	-5.08%

Annual Asset Returns

Rolling returns summary
Roll Period	Average	High	Low
1 year	34.25%	160.45%	-26.37%
3 years	32.33%	62.25%	4.29%
5 years	31.11%	46.00%	9.17%
7 years	31.99%	41.85%	19.34%
10 years	30.49%	35.23%	23.37%