Question:
Assets: BTC-USD, BNO
BTC-USD: annualized_mean_return=1.0836, daily_std=0.0368
BNO: annualized_mean_return=0.5544, daily_std=0.0113
Minimum required portfolio return (annualized): 0.5974
Market regime: sideways

Compute portfolio weights (w_BTC-USD, w_BNO) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_BTC-USD=X.XXXX, w_BNO=X.XXXX

Answer: w_BTC-USD=0.1072, w_BNO=0.8928

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000127 - -0.000040) / (0.001352 + 0.000127 - -0.000080)
  Unconstrained: w_BTC-USD=0.1071
  After long-only clamp: w_BTC-USD=0.1071, w_BNO=0.8929.


MATIC-USD σ=0.0240, ICSH σ=0.0002, ρ=-0.005. Min-variance weights: MATIC-USD=0.000, ICSH=1.000.

Question:
Assets: MATIC-USD, ICSH
MATIC-USD: annualized_mean_return=-1.1088, daily_std=0.0240
ICSH: annualized_mean_return=0.0504, daily_std=0.0002
Minimum required portfolio return (annualized): 0.0503
Market regime: sideways

Compute portfolio weights (w_MATIC-USD, w_ICSH) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_MATIC-USD=X.XXXX, w_ICSH=X.XXXX

Answer: w_MATIC-USD=0.0001, w_ICSH=0.9999

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000000 - -0.000000) / (0.000577 + 0.000000 - -0.000000)
  Unconstrained: w_MATIC-USD=0.0001
  After long-only clamp: w_MATIC-USD=0.0001, w_ICSH=0.9999.


Question:
Assets: XLF, BIL
XLF: annualized_mean_return=0.0756, daily_std=0.0076
BIL: annualized_mean_return=0.0504, daily_std=0.0001
Minimum required portfolio return (annualized): 0.0595
Market regime: sideways

Compute portfolio weights (w_XLF, w_BIL) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_XLF=X.XXXX, w_BIL=X.XXXX

Answer: w_XLF=0.3611, w_BIL=0.6389

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000000 - 0.000000) / (0.000058 + 0.000000 - 0.000000)
  Unconstrained: w_XLF=-0.0030
  After long-only clamp: w_XLF=0.0000, w_BIL=1.0000.


DOT-USD σ=0.0175, HAUZ σ=0.0080, ρ=0.104. Min-variance weights: DOT-USD=0.145, HAUZ=0.855.

Question:
Assets: DOT-USD, HAUZ
DOT-USD: annualized_mean_return=-1.0080, daily_std=0.0175
HAUZ: annualized_mean_return=0.0504, daily_std=0.0080
Minimum required portfolio return (annualized): -0.6142
Market regime: sideways

Compute portfolio weights (w_DOT-USD, w_HAUZ) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_DOT-USD=X.XXXX, w_HAUZ=X.XXXX

Answer: w_DOT-USD=0.1445, w_HAUZ=0.8555

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000064 - 0.000015) / (0.000305 + 0.000064 - 0.000029)
  Unconstrained: w_DOT-USD=0.1455
  After long-only clamp: w_DOT-USD=0.1455, w_HAUZ=0.8545.


Question:
Assets: BNB-USD, SGOV
BNB-USD: annualized_mean_return=-0.8316, daily_std=0.0255
SGOV: annualized_mean_return=0.0504, daily_std=0.0002
Minimum required portfolio return (annualized): -0.4499
Market regime: sideways

Compute portfolio weights (w_BNB-USD, w_SGOV) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_BNB-USD=X.XXXX, w_SGOV=X.XXXX

Answer: w_BNB-USD=0.0000, w_SGOV=1.0000

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000000 - -0.000000) / (0.000652 + 0.000000 - -0.000001)
  Unconstrained: w_BNB-USD=0.0007
  After long-only clamp: w_BNB-USD=0.0007, w_SGOV=0.9993.


Question:
Assets: MTUM, CSHI
MTUM: annualized_mean_return=0.7560, daily_std=0.0105
CSHI: annualized_mean_return=0.0252, daily_std=0.0003
Minimum required portfolio return (annualized): 0.0257
Market regime: sideways

Compute portfolio weights (w_MTUM, w_CSHI) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_MTUM=X.XXXX, w_CSHI=X.XXXX

Answer: w_MTUM=0.0011, w_CSHI=0.9989

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000000 - -0.000000) / (0.000111 + 0.000000 - -0.000000)
  Unconstrained: w_MTUM=0.0020
  After long-only clamp: w_MTUM=0.0020, w_CSHI=0.9980.


Question:
Assets: XLI, SGOV
XLI: annualized_mean_return=0.4536, daily_std=0.0078
SGOV: annualized_mean_return=0.0756, daily_std=0.0002
Minimum required portfolio return (annualized): 0.0758
Market regime: sideways

Compute portfolio weights (w_XLI, w_SGOV) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_XLI=X.XXXX, w_SGOV=X.XXXX

Answer: w_XLI=0.0005, w_SGOV=0.9995

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000000 - 0.000000) / (0.000061 + 0.000000 - 0.000000)
  Unconstrained: w_XLI=-0.0006
  After long-only clamp: w_XLI=0.0000, w_SGOV=1.0000.


Question:
Assets: XLI, BIL
XLI: annualized_mean_return=0.3024, daily_std=0.0083
BIL: annualized_mean_return=0.0504, daily_std=0.0001
Minimum required portfolio return (annualized): 0.0504
Market regime: sideways

Compute portfolio weights (w_XLI, w_BIL) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_XLI=X.XXXX, w_BIL=X.XXXX

Answer: w_XLI=0.0003, w_BIL=0.9997

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000000 - 0.000000) / (0.000068 + 0.000000 - 0.000000)
  Unconstrained: w_XLI=-0.0014
  After long-only clamp: w_XLI=0.0000, w_BIL=1.0000.


Question:
Assets: SPY, BNO
SPY: annualized_mean_return=0.4284, daily_std=0.0058
BNO: annualized_mean_return=0.1512, daily_std=0.0124
Minimum required portfolio return (annualized): 0.2617
Market regime: sideways

Compute portfolio weights (w_SPY, w_BNO) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_SPY=X.XXXX, w_BNO=X.XXXX

Answer: w_SPY=0.8155, w_BNO=0.1845

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000153 - -0.000001) / (0.000034 + 0.000153 - -0.000002)
  Unconstrained: w_SPY=0.8155
  After long-only clamp: w_SPY=0.8155, w_BNO=0.1845.


Question:
Assets: XLI, CSHI
XLI: annualized_mean_return=0.0756, daily_std=0.0106
CSHI: annualized_mean_return=0.0756, daily_std=0.0007
Minimum required portfolio return (annualized): 0.0755
Market regime: sideways

Compute portfolio weights (w_XLI, w_CSHI) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_XLI=X.XXXX, w_CSHI=X.XXXX

Answer: w_XLI=0.0209, w_CSHI=0.9791

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000000 - -0.000002) / (0.000112 + 0.000000 - -0.000003)
  Unconstrained: w_XLI=0.0170
  After long-only clamp: w_XLI=0.0170, w_CSHI=0.9830.


Question:
Assets: VLUE, BIL
VLUE: annualized_mean_return=0.1764, daily_std=0.0096
BIL: annualized_mean_return=0.0756, daily_std=0.0002
Minimum required portfolio return (annualized): 0.0756
Market regime: bull

Compute portfolio weights (w_VLUE, w_BIL) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_VLUE=X.XXXX, w_BIL=X.XXXX

Answer: w_VLUE=0.0003, w_BIL=0.9997

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000000 - 0.000000) / (0.000092 + 0.000000 - 0.000000)
  Unconstrained: w_VLUE=-0.0002
  After long-only clamp: w_VLUE=0.0000, w_BIL=1.0000.


Question:
Assets: ADA-USD, ^VIX
ADA-USD: annualized_mean_return=-0.0252, daily_std=0.0391
^VIX: annualized_mean_return=-0.7812, daily_std=0.0614
Minimum required portfolio return (annualized): -0.2697
Market regime: sideways

Compute portfolio weights (w_ADA-USD, w_^VIX) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_ADA-USD=X.XXXX, w_^VIX=X.XXXX

Answer: w_ADA-USD=0.7242, w_^VIX=0.2758

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.003765 - 0.000153) / (0.001529 + 0.003765 - 0.000305)
  Unconstrained: w_ADA-USD=0.7241
  After long-only clamp: w_ADA-USD=0.7241, w_^VIX=0.2759.


Question:
Assets: VEA, ITB
VEA: annualized_mean_return=0.0504, daily_std=0.0091
ITB: annualized_mean_return=0.3024, daily_std=0.0147
Minimum required portfolio return (annualized): 0.0636
Market regime: sideways

Compute portfolio weights (w_VEA, w_ITB) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_VEA=X.XXXX, w_ITB=X.XXXX

Answer: w_VEA=0.9369, w_ITB=0.0631

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000215 - 0.000073) / (0.000083 + 0.000215 - 0.000146)
  Unconstrained: w_VEA=0.9381
  After long-only clamp: w_VEA=0.9381, w_ITB=0.0619.


Question:
Assets: ETH-USD, SHV
ETH-USD: annualized_mean_return=1.2096, daily_std=0.0234
SHV: annualized_mean_return=0.0504, daily_std=0.0002
Minimum required portfolio return (annualized): 0.0504
Market regime: sideways

Compute portfolio weights (w_ETH-USD, w_SHV) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_ETH-USD=X.XXXX, w_SHV=X.XXXX

Answer: w_ETH-USD=0.0001, w_SHV=0.9999

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000000 - 0.000000) / (0.000546 + 0.000000 - 0.000001)
  Unconstrained: w_ETH-USD=-0.0007
  After long-only clamp: w_ETH-USD=0.0000, w_SHV=1.0000.


Question:
Assets: DOT-USD, HYG
DOT-USD: annualized_mean_return=-1.1592, daily_std=0.0430
HYG: annualized_mean_return=0.0756, daily_std=0.0031
Minimum required portfolio return (annualized): -0.2730
Market regime: bull

Compute portfolio weights (w_DOT-USD, w_HYG) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_DOT-USD=X.XXXX, w_HYG=X.XXXX

Answer: w_DOT-USD=0.0000, w_HYG=1.0000

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000010 - 0.000024) / (0.001850 + 0.000010 - 0.000049)
  Unconstrained: w_DOT-USD=-0.0082
  After long-only clamp: w_DOT-USD=0.0000, w_HYG=1.0000.


Question:
Assets: XLV, BNO
XLV: annualized_mean_return=0.0252, daily_std=0.0065
BNO: annualized_mean_return=0.4032, daily_std=0.0119
Minimum required portfolio return (annualized): 0.1022
Market regime: sideways

Compute portfolio weights (w_XLV, w_BNO) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_XLV=X.XXXX, w_BNO=X.XXXX

Answer: w_XLV=0.7588, w_BNO=0.2412

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000142 - -0.000005) / (0.000042 + 0.000142 - -0.000009)
  Unconstrained: w_XLV=0.7599
  After long-only clamp: w_XLV=0.7599, w_BNO=0.2401.


Question:
Assets: XLP, PDBC
XLP: annualized_mean_return=0.3276, daily_std=0.0072
PDBC: annualized_mean_return=-0.4284, daily_std=0.0101
Minimum required portfolio return (annualized): -0.0805
Market regime: sideways

Compute portfolio weights (w_XLP, w_PDBC) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_XLP=X.XXXX, w_PDBC=X.XXXX

Answer: w_XLP=0.6702, w_PDBC=0.3298

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000103 - 0.000003) / (0.000052 + 0.000103 - 0.000006)
  Unconstrained: w_XLP=0.6698
  After long-only clamp: w_XLP=0.6698, w_PDBC=0.3302.


Question:
Assets: LINK-USD, GLD
LINK-USD: annualized_mean_return=-0.5544, daily_std=0.0419
GLD: annualized_mean_return=0.4788, daily_std=0.0094
Minimum required portfolio return (annualized): 0.0984
Market regime: sideways

Compute portfolio weights (w_LINK-USD, w_GLD) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_LINK-USD=X.XXXX, w_GLD=X.XXXX

Answer: w_LINK-USD=0.0504, w_GLD=0.9496

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000088 - -0.000005) / (0.001752 + 0.000088 - -0.000010)
  Unconstrained: w_LINK-USD=0.0502
  After long-only clamp: w_LINK-USD=0.0502, w_GLD=0.9498.


BNB-USD σ=0.0187, VNQI σ=0.0090, ρ=0.075. Min-variance weights: BNB-USD=0.168, VNQI=0.832.

Question:
Assets: BNB-USD, VNQI
BNB-USD: annualized_mean_return=-0.3276, daily_std=0.0187
VNQI: annualized_mean_return=0.0252, daily_std=0.0090
Minimum required portfolio return (annualized): -0.0010
Market regime: sideways

Compute portfolio weights (w_BNB-USD, w_VNQI) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_BNB-USD=X.XXXX, w_VNQI=X.XXXX

Answer: w_BNB-USD=0.0743, w_VNQI=0.9257

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000080 - 0.000012) / (0.000348 + 0.000080 - 0.000025)
  Unconstrained: w_BNB-USD=0.1680
  After long-only clamp: w_BNB-USD=0.1680, w_VNQI=0.8320.


Question:
Assets: SPY, IYR
SPY: annualized_mean_return=0.1512, daily_std=0.0070
IYR: annualized_mean_return=-0.0756, daily_std=0.0108
Minimum required portfolio return (annualized): 0.1489
Market regime: sideways

Compute portfolio weights (w_SPY, w_IYR) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_SPY=X.XXXX, w_IYR=X.XXXX

Answer: w_SPY=0.9899, w_IYR=0.0101

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000116 - 0.000047) / (0.000049 + 0.000116 - 0.000093)
  Unconstrained: w_SPY=0.9731
  After long-only clamp: w_SPY=0.9731, w_IYR=0.0269.


Question:
Assets: ETH-USD, TIP
ETH-USD: annualized_mean_return=1.9404, daily_std=0.0351
TIP: annualized_mean_return=0.0252, daily_std=0.0013
Minimum required portfolio return (annualized): 0.0252
Market regime: sideways

Compute portfolio weights (w_ETH-USD, w_TIP) that minimize portfolio variance while satisfying the minimum return constraint. Constraints: all weights >= 0, weights sum to 1. Report as: w_ETH-USD=X.XXXX, w_TIP=X.XXXX

Answer: w_ETH-USD=0.0000, w_TIP=1.0000

Explanation:
Analytic min-variance formula:
  w1* = (σ2² - σ12) / (σ1² + σ2² - 2σ12)
  = (0.000002 - 0.000004) / (0.001235 + 0.000002 - 0.000007)
  Unconstrained: w_ETH-USD=-0.0017
  After long-only clamp: w_ETH-USD=0.0000, w_TIP=1.0000.


4-asset optimization. Max-Sharpe: 1.544. Portfolio: return=4.54%, vol=0.35%. Weights: w_VLUE=0.0000, w_XRP-USD=0.0013, w_DBC=0.0000, w_BIL=0.9987.

Question:
Assets: VLUE, XRP-USD, DBC, BIL
Annualized mean returns: VLUE:-0.2972, XRP-USD:0.0846, DBC:-0.4397, BIL:0.0453
Covariance matrix (annualized):
[[0.021207, 0.017628, 0.015836, 7.9e-05], [0.017628, 0.179242, 0.002871, -0.000129], [0.015836, 0.002871, 0.028205, -1.9e-05], [7.9e-05, -0.000129, -1.9e-05, 1.2e-05]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_VLUE=0.0000, w_XRP-USD=0.0013, w_DBC=0.0000, w_BIL=0.9987

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_VLUE=0.0000, w_XRP-USD=0.0013, w_DBC=0.0000, w_BIL=0.9987
Portfolio annualized return: 4.54%, volatility: 0.35%
Sharpe ratio: (0.0454 - 0.0400) / 0.0035 = 1.5444


4-asset optimization. Max-Sharpe: 4.944. Portfolio: return=37.27%, vol=6.73%. Weights: w_XLB=0.3475, w_LINK-USD=0.0380, w_DBA=0.0000, w_TIP=0.6144.

Question:
Assets: XLB, LINK-USD, DBA, TIP
Annualized mean returns: XLB:0.6919, LINK-USD:1.5813, DBA:0.0106, TIP:0.1174
Covariance matrix (annualized):
[[0.023476, -0.008387, 0.00371, 0.001682], [-0.008387, 0.630498, 0.005426, -0.00014], [0.00371, 0.005426, 0.015518, 0.000623], [0.001682, -0.00014, 0.000623, 0.000772]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLB=0.3475, w_LINK-USD=0.0380, w_DBA=0.0000, w_TIP=0.6144

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLB=0.3475, w_LINK-USD=0.0380, w_DBA=0.0000, w_TIP=0.6144
Portfolio annualized return: 37.27%, volatility: 6.73%
Sharpe ratio: (0.3727 - 0.0400) / 0.0673 = 4.9444


4-asset optimization. Max-Sharpe: 1.992. Portfolio: return=4.52%, vol=0.26%. Weights: w_XLV=0.0005, w_SOL-USD=0.0000, w_BIL=0.9995, w_JNK=0.0000.

Question:
Assets: XLV, SOL-USD, BIL, JNK
Annualized mean returns: XLV:0.0594, SOL-USD:-2.1331, BIL:0.0452, JNK:-0.0370
Covariance matrix (annualized):
[[0.019542, 0.016427, 1.5e-05, 0.006159], [0.016427, 0.540059, -0.000412, 0.021847], [1.5e-05, -0.000412, 7e-06, -5e-05], [0.006159, 0.021847, -5e-05, 0.009941]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLV=0.0005, w_SOL-USD=0.0000, w_BIL=0.9995, w_JNK=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLV=0.0005, w_SOL-USD=0.0000, w_BIL=0.9995, w_JNK=0.0000
Portfolio annualized return: 4.52%, volatility: 0.26%
Sharpe ratio: (0.0452 - 0.0400) / 0.0026 = 1.9920


4-asset optimization. Max-Sharpe: 2.082. Portfolio: return=38.35%, vol=16.50%. Weights: w_XLK=1.0000, w_MATIC-USD=0.0000, w_VNQ=0.0000, w_CPER=0.0000.

Question:
Assets: XLK, MATIC-USD, VNQ, CPER
Annualized mean returns: XLK:0.3835, MATIC-USD:-1.4395, VNQ:0.0494, CPER:-0.3140
Covariance matrix (annualized):
[[0.027219, 0.018986, 0.018216, 0.012045], [0.018986, 0.242286, 0.005151, -0.000519], [0.018216, 0.005151, 0.041624, 0.008861], [0.012045, -0.000519, 0.008861, 0.036803]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLK=1.0000, w_MATIC-USD=0.0000, w_VNQ=0.0000, w_CPER=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLK=1.0000, w_MATIC-USD=0.0000, w_VNQ=0.0000, w_CPER=0.0000
Portfolio annualized return: 38.35%, volatility: 16.50%
Sharpe ratio: (0.3835 - 0.0400) / 0.1650 = 2.0821


4-asset optimization. Max-Sharpe: 3.855. Portfolio: return=5.03%, vol=0.27%. Weights: w_XLU=0.0000, w_LINK-USD=0.0000, w_ITB=0.0000, w_BIL=1.0000.

Question:
Assets: XLU, LINK-USD, ITB, BIL
Annualized mean returns: XLU:-0.2377, LINK-USD:-0.5109, ITB:-0.3178, BIL:0.0503
Covariance matrix (annualized):
[[0.018733, 0.007285, 0.007694, -0.000115], [0.007285, 0.242623, 0.022476, -0.000144], [0.007694, 0.022476, 0.058992, -0.000236], [-0.000115, -0.000144, -0.000236, 7e-06]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLU=0.0000, w_LINK-USD=0.0000, w_ITB=0.0000, w_BIL=1.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLU=0.0000, w_LINK-USD=0.0000, w_ITB=0.0000, w_BIL=1.0000
Portfolio annualized return: 5.03%, volatility: 0.27%
Sharpe ratio: (0.0503 - 0.0400) / 0.0027 = 3.8552


4-asset optimization. Max-Sharpe: 6.036. Portfolio: return=5.82%, vol=0.30%. Weights: w_VLUE=0.0000, w_BTC-USD=0.0000, w_REZ=0.0131, w_SGOV=0.9869.

Question:
Assets: VLUE, BTC-USD, REZ, SGOV
Annualized mean returns: VLUE:0.0943, BTC-USD:-0.2190, REZ:0.5300, SGOV:0.0519
Covariance matrix (annualized):
[[0.012158, 0.016911, 0.011514, -3.2e-05], [0.016911, 0.215594, 0.01475, -0.00015], [0.011514, 0.01475, 0.019974, -1.8e-05], [-3.2e-05, -0.00015, -1.8e-05, 6e-06]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: bull

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_VLUE=0.0000, w_BTC-USD=0.0000, w_REZ=0.0131, w_SGOV=0.9869

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_VLUE=0.0000, w_BTC-USD=0.0000, w_REZ=0.0131, w_SGOV=0.9869
Portfolio annualized return: 5.82%, volatility: 0.30%
Sharpe ratio: (0.0582 - 0.0400) / 0.0030 = 6.0360


4-asset optimization. Max-Sharpe: 2.512. Portfolio: return=50.66%, vol=18.58%. Weights: w_EEM=0.0000, w_AVAX-USD=0.1782, w_WEAT=0.0000, w_JNK=0.8218.

Question:
Assets: EEM, AVAX-USD, WEAT, JNK
Annualized mean returns: EEM:0.1737, AVAX-USD:2.2243, WEAT:-0.1687, JNK:0.1340
Covariance matrix (annualized):
[[0.025354, 0.02737, -0.007474, 0.007899], [0.02737, 0.854757, -0.04558, 0.011221], [-0.007474, -0.04558, 0.053962, -0.001572], [0.007899, 0.011221, -0.001572, 0.006029]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_EEM=0.0000, w_AVAX-USD=0.1782, w_WEAT=0.0000, w_JNK=0.8218

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_EEM=0.0000, w_AVAX-USD=0.1782, w_WEAT=0.0000, w_JNK=0.8218
Portfolio annualized return: 50.66%, volatility: 18.58%
Sharpe ratio: (0.5066 - 0.0400) / 0.1858 = 2.5116


4-asset optimization. Max-Sharpe: 2.098. Portfolio: return=12.07%, vol=3.84%. Weights: w_EWJ=0.1636, w_ADA-USD=0.0301, w_BNO=0.0000, w_TIP=0.8063.

Question:
Assets: EWJ, ADA-USD, BNO, TIP
Annualized mean returns: EWJ:0.1373, ADA-USD:0.8307, BNO:-0.8924, TIP:0.0907
Covariance matrix (annualized):
[[0.018059, 0.009057, 0.018051, -0.001792], [0.009057, 0.348933, 0.013469, 0.003089], [0.018051, 0.013469, 0.13288, -0.001454], [-0.001792, 0.003089, -0.001454, 0.001401]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_EWJ=0.1636, w_ADA-USD=0.0301, w_BNO=0.0000, w_TIP=0.8063

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_EWJ=0.1636, w_ADA-USD=0.0301, w_BNO=0.0000, w_TIP=0.8063
Portfolio annualized return: 12.07%, volatility: 3.84%
Sharpe ratio: (0.1207 - 0.0400) / 0.0384 = 2.0979


4-asset optimization. Max-Sharpe: 7.306. Portfolio: return=5.62%, vol=0.22%. Weights: w_XLV=0.0000, w_ETH-USD=0.0000, w_SGOV=0.9888, w_PDBC=0.0112.

Question:
Assets: XLV, ETH-USD, SGOV, PDBC
Annualized mean returns: XLV:-0.3073, ETH-USD:-0.2695, SGOV:0.0526, PDBC:0.3769
Covariance matrix (annualized):
[[0.010609, 0.008905, 1.3e-05, -0.001532], [0.008905, 0.425563, -0.000131, 0.009116], [1.3e-05, -0.000131, 4e-06, -1.9e-05], [-0.001532, 0.009116, -1.9e-05, 0.010901]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLV=0.0000, w_ETH-USD=0.0000, w_SGOV=0.9888, w_PDBC=0.0112

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLV=0.0000, w_ETH-USD=0.0000, w_SGOV=0.9888, w_PDBC=0.0112
Portfolio annualized return: 5.62%, volatility: 0.22%
Sharpe ratio: (0.0562 - 0.0400) / 0.0022 = 7.3056


4-asset optimization. Max-Sharpe: 1.473. Portfolio: return=6.65%, vol=1.80%. Weights: w_IVV=0.0000, w_SOL-USD=0.0174, w_SGOV=0.9826, w_WEAT=0.0000.

Question:
Assets: IVV, SOL-USD, SGOV, WEAT
Annualized mean returns: IVV:-0.0279, SOL-USD:1.5294, SGOV:0.0406, WEAT:-0.7051
Covariance matrix (annualized):
[[0.035625, 0.068326, -0.000105, 0.009821], [0.068326, 1.052109, -7.2e-05, -0.003283], [-0.000105, -7.2e-05, 9e-06, -7.4e-05], [0.009821, -0.003283, -7.4e-05, 0.059749]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_IVV=0.0000, w_SOL-USD=0.0174, w_SGOV=0.9826, w_WEAT=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_IVV=0.0000, w_SOL-USD=0.0174, w_SGOV=0.9826, w_WEAT=0.0000
Portfolio annualized return: 6.65%, volatility: 1.80%
Sharpe ratio: (0.0665 - 0.0400) / 0.0180 = 1.4727


4-asset optimization. Max-Sharpe: 4.498. Portfolio: return=80.64%, vol=17.04%. Weights: w_XLU=0.1861, w_ADA-USD=0.0000, w_CPER=0.0928, w_MORT=0.7211.

Question:
Assets: XLU, ADA-USD, CPER, MORT
Annualized mean returns: XLU:0.2918, ADA-USD:-0.3255, CPER:0.3194, MORT:1.0019
Covariance matrix (annualized):
[[0.022708, 0.007636, 0.010089, 0.006071], [0.007636, 0.638421, 0.042997, 0.037087], [0.010089, 0.042997, 0.029456, 0.008286], [0.006071, 0.037087, 0.008286, 0.047899]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLU=0.1861, w_ADA-USD=0.0000, w_CPER=0.0928, w_MORT=0.7211

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLU=0.1861, w_ADA-USD=0.0000, w_CPER=0.0928, w_MORT=0.7211
Portfolio annualized return: 80.64%, volatility: 17.04%
Sharpe ratio: (0.8064 - 0.0400) / 0.1704 = 4.4978


4-asset optimization. Max-Sharpe: 4.171. Portfolio: return=110.04%, vol=25.43%. Weights: w_EFA=0.0000, w_BNB-USD=0.3851, w_XHB=0.6149, w_DBB=0.0000.

Question:
Assets: EFA, BNB-USD, XHB, DBB
Annualized mean returns: EFA:0.2888, BNB-USD:1.7520, XHB:0.6923, DBB:0.0099
Covariance matrix (annualized):
[[0.01314, 0.015769, 0.016113, 0.005181], [0.015769, 0.238289, 0.020501, 0.010078], [0.016113, 0.020501, 0.051838, -0.000539], [0.005181, 0.010078, -0.000539, 0.026429]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_EFA=0.0000, w_BNB-USD=0.3851, w_XHB=0.6149, w_DBB=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_EFA=0.0000, w_BNB-USD=0.3851, w_XHB=0.6149, w_DBB=0.0000
Portfolio annualized return: 110.04%, volatility: 25.43%
Sharpe ratio: (1.1004 - 0.0400) / 0.2543 = 4.1707


4-asset optimization. Max-Sharpe: 5.774. Portfolio: return=5.36%, vol=0.24%. Weights: w_VLUE=0.0000, w_XRP-USD=0.0015, w_SGOV=0.9985, w_MORT=0.0000.

Question:
Assets: VLUE, XRP-USD, SGOV, MORT
Annualized mean returns: VLUE:-0.2960, XRP-USD:0.5660, SGOV:0.0528, MORT:-0.7379
Covariance matrix (annualized):
[[0.013969, -0.003946, -1.2e-05, 0.023363], [-0.003946, 0.134741, 1.1e-05, -0.015322], [-1.2e-05, 1.1e-05, 5e-06, 5.1e-05], [0.023363, -0.015322, 5.1e-05, 0.062528]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_VLUE=0.0000, w_XRP-USD=0.0015, w_SGOV=0.9985, w_MORT=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_VLUE=0.0000, w_XRP-USD=0.0015, w_SGOV=0.9985, w_MORT=0.0000
Portfolio annualized return: 5.36%, volatility: 0.24%
Sharpe ratio: (0.0536 - 0.0400) / 0.0024 = 5.7735


4-asset optimization. Max-Sharpe: 4.585. Portfolio: return=41.05%, vol=8.08%. Weights: w_EFA=0.0000, w_BNB-USD=0.0000, w_IGOV=0.6229, w_VNQ=0.3771.

Question:
Assets: EFA, BNB-USD, IGOV, VNQ
Annualized mean returns: EFA:0.0863, BNB-USD:-0.0522, IGOV:0.2891, VNQ:0.6110
Covariance matrix (annualized):
[[0.024045, 0.015257, 0.001795, 0.009686], [0.015257, 0.230954, -0.012396, 0.026409], [0.001795, -0.012396, 0.006327, 0.00119], [0.009686, 0.026409, 0.00119, 0.02472]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_EFA=0.0000, w_BNB-USD=0.0000, w_IGOV=0.6229, w_VNQ=0.3771

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_EFA=0.0000, w_BNB-USD=0.0000, w_IGOV=0.6229, w_VNQ=0.3771
Portfolio annualized return: 41.05%, volatility: 8.08%
Sharpe ratio: (0.4105 - 0.0400) / 0.0808 = 4.5849


4-asset optimization. Max-Sharpe: 0.035. Portfolio: return=4.61%, vol=17.48%. Weights: w_VEA=0.0000, w_ADA-USD=0.0000, w_REZ=1.0000, w_IGOV=0.0000.

Question:
Assets: VEA, ADA-USD, REZ, IGOV
Annualized mean returns: VEA:-0.0238, ADA-USD:-1.0110, REZ:0.0461, IGOV:-0.0895
Covariance matrix (annualized):
[[0.019123, 0.010543, 0.01429, 0.002011], [0.010543, 0.227738, -0.002848, 0.004517], [0.01429, -0.002848, 0.03057, 0.000454], [0.002011, 0.004517, 0.000454, 0.010489]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_VEA=0.0000, w_ADA-USD=0.0000, w_REZ=1.0000, w_IGOV=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_VEA=0.0000, w_ADA-USD=0.0000, w_REZ=1.0000, w_IGOV=0.0000
Portfolio annualized return: 4.61%, volatility: 17.48%
Sharpe ratio: (0.0461 - 0.0400) / 0.1748 = 0.0348


4-asset optimization. Max-Sharpe: 5.855. Portfolio: return=5.16%, vol=0.20%. Weights: w_QUAL=0.0000, w_ADA-USD=0.0000, w_MORT=0.0000, w_SGOV=1.0000.

Question:
Assets: QUAL, ADA-USD, MORT, SGOV
Annualized mean returns: QUAL:-0.3690, ADA-USD:-1.3348, MORT:-0.5714, SGOV:0.0516
Covariance matrix (annualized):
[[0.015192, 0.012688, 0.012485, -2.9e-05], [0.012688, 0.111582, 0.02687, -9.1e-05], [0.012485, 0.02687, 0.037248, 5e-06], [-2.9e-05, -9.1e-05, 5e-06, 4e-06]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_QUAL=0.0000, w_ADA-USD=0.0000, w_MORT=0.0000, w_SGOV=1.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_QUAL=0.0000, w_ADA-USD=0.0000, w_MORT=0.0000, w_SGOV=1.0000
Portfolio annualized return: 5.16%, volatility: 0.20%
Sharpe ratio: (0.0516 - 0.0400) / 0.0020 = 5.8551


4-asset optimization. Max-Sharpe: 2.879. Portfolio: return=4.80%, vol=0.28%. Weights: w_ACWI=0.0000, w_DOT-USD=0.0000, w_DBC=0.0000, w_SGOV=1.0000.

Question:
Assets: ACWI, DOT-USD, DBC, SGOV
Annualized mean returns: ACWI:0.0769, DOT-USD:-0.7643, DBC:-0.3801, SGOV:0.0480
Covariance matrix (annualized):
[[0.013214, 0.011926, 0.010528, 3.7e-05], [0.011926, 0.164188, 0.023027, -0.000106], [0.010528, 0.023027, 0.028335, -0.000117], [3.7e-05, -0.000106, -0.000117, 8e-06]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_ACWI=0.0000, w_DOT-USD=0.0000, w_DBC=0.0000, w_SGOV=1.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_ACWI=0.0000, w_DOT-USD=0.0000, w_DBC=0.0000, w_SGOV=1.0000
Portfolio annualized return: 4.80%, volatility: 0.28%
Sharpe ratio: (0.0480 - 0.0400) / 0.0028 = 2.8791


4-asset optimization. Max-Sharpe: 5.526. Portfolio: return=614.52%, vol=110.48%. Weights: w_VLUE=0.0000, w_XRP-USD=1.0000, w_XHB=0.0000, w_SOYB=0.0000.

Question:
Assets: VLUE, XRP-USD, XHB, SOYB
Annualized mean returns: VLUE:0.0066, XRP-USD:6.1452, XHB:-0.1787, SOYB:-0.1976
Covariance matrix (annualized):
[[0.016611, 0.028993, 0.016854, -0.000881], [0.028993, 1.220607, 0.011497, -0.015341], [0.016854, 0.011497, 0.052113, -0.003476], [-0.000881, -0.015341, -0.003476, 0.014772]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: bull

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_VLUE=0.0000, w_XRP-USD=1.0000, w_XHB=0.0000, w_SOYB=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_VLUE=0.0000, w_XRP-USD=1.0000, w_XHB=0.0000, w_SOYB=0.0000
Portfolio annualized return: 614.52%, volatility: 110.48%
Sharpe ratio: (6.1452 - 0.0400) / 1.1048 = 5.5260


4-asset optimization. Max-Sharpe: 3.663. Portfolio: return=11.07%, vol=1.93%. Weights: w_XLF=0.0000, w_BTC-USD=0.0471, w_SGOV=0.9079, w_JNK=0.0450.

Question:
Assets: XLF, BTC-USD, SGOV, JNK
Annualized mean returns: XLF:0.2352, BTC-USD:1.2395, SGOV:0.0424, JNK:0.3075
Covariance matrix (annualized):
[[0.026941, 0.025649, -0.000111, 0.014105], [0.025649, 0.113895, 0.000134, 0.018557], [-0.000111, 0.000134, 9e-06, -4.6e-05], [0.014105, 0.018557, -4.6e-05, 0.012828]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLF=0.0000, w_BTC-USD=0.0471, w_SGOV=0.9079, w_JNK=0.0450

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLF=0.0000, w_BTC-USD=0.0471, w_SGOV=0.9079, w_JNK=0.0450
Portfolio annualized return: 11.07%, volatility: 1.93%
Sharpe ratio: (0.1107 - 0.0400) / 0.0193 = 3.6627


4-asset optimization. Max-Sharpe: 0.627. Portfolio: return=12.36%, vol=13.33%. Weights: w_MTUM=0.4977, w_LINK-USD=0.0000, w_PALL=0.0000, w_SCHH=0.5023.

Question:
Assets: MTUM, LINK-USD, PALL, SCHH
Annualized mean returns: MTUM:0.1353, LINK-USD:-0.5799, PALL:-0.8427, SCHH:0.1120
Covariance matrix (annualized):
[[0.033323, 0.041693, 0.021057, 0.00728], [0.041693, 0.447598, 0.034074, 0.034727], [0.021057, 0.034074, 0.083793, 0.01203], [0.00728, 0.034727, 0.01203, 0.023253]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_MTUM=0.4977, w_LINK-USD=0.0000, w_PALL=0.0000, w_SCHH=0.5023

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_MTUM=0.4977, w_LINK-USD=0.0000, w_PALL=0.0000, w_SCHH=0.5023
Portfolio annualized return: 12.36%, volatility: 13.33%
Sharpe ratio: (0.1236 - 0.0400) / 0.1333 = 0.6272


4-asset optimization. Max-Sharpe: 2.808. Portfolio: return=67.03%, vol=22.45%. Weights: w_XLB=0.0000, w_SOL-USD=0.2058, w_IYR=0.0000, w_TLT=0.7942.

Question:
Assets: XLB, SOL-USD, IYR, TLT
Annualized mean returns: XLB:-0.0429, SOL-USD:2.2444, IYR:0.0374, TLT:0.2624
Covariance matrix (annualized):
[[0.01587, 0.008619, 0.015673, 0.004445], [0.008619, 0.903805, 0.014567, -0.012382], [0.015673, 0.014567, 0.030762, 0.010405], [0.004445, -0.012382, 0.010405, 0.025592]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLB=0.0000, w_SOL-USD=0.2058, w_IYR=0.0000, w_TLT=0.7942

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLB=0.0000, w_SOL-USD=0.2058, w_IYR=0.0000, w_TLT=0.7942
Portfolio annualized return: 67.03%, volatility: 22.45%
Sharpe ratio: (0.6703 - 0.0400) / 0.2245 = 2.8083


4-asset optimization. Max-Sharpe: 2.755. Portfolio: return=4.82%, vol=0.30%. Weights: w_XLU=0.0000, w_DOT-USD=0.0014, w_SGOV=0.9986, w_EMB=0.0000.

Question:
Assets: XLU, DOT-USD, SGOV, EMB
Annualized mean returns: XLU:-0.2550, DOT-USD:1.4808, SGOV:0.0462, EMB:-0.0178
Covariance matrix (annualized):
[[0.031143, 0.042026, 5e-06, 0.004389], [0.042026, 0.94802, 0.000264, 0.019685], [5e-06, 0.000264, 6e-06, 1e-06], [0.004389, 0.019685, 1e-06, 0.005317]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLU=0.0000, w_DOT-USD=0.0014, w_SGOV=0.9986, w_EMB=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLU=0.0000, w_DOT-USD=0.0014, w_SGOV=0.9986, w_EMB=0.0000
Portfolio annualized return: 4.82%, volatility: 0.30%
Sharpe ratio: (0.0482 - 0.0400) / 0.0030 = 2.7548


4-asset optimization. Max-Sharpe: -5.079. Portfolio: return=-52.13%, vol=11.05%. Weights: w_USMV=0.2500, w_AVAX-USD=0.2500, w_HAUZ=0.2500, w_BIL=0.2500.

Question:
Assets: USMV, AVAX-USD, HAUZ, BIL
Annualized mean returns: USMV:-0.0652, AVAX-USD:-1.8088, HAUZ:-0.2619, BIL:0.0509
Covariance matrix (annualized):
[[0.005446, 0.011043, 0.0039, -7.7e-05], [0.011043, 0.127681, 0.009535, -0.000204], [0.0039, 0.009535, 0.013842, 3e-06], [-7.7e-05, -0.000204, 3e-06, 7e-06]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_USMV=0.2500, w_AVAX-USD=0.2500, w_HAUZ=0.2500, w_BIL=0.2500

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_USMV=0.2500, w_AVAX-USD=0.2500, w_HAUZ=0.2500, w_BIL=0.2500
Portfolio annualized return: -52.13%, volatility: 11.05%
Sharpe ratio: (-0.5213 - 0.0400) / 0.1105 = -5.0792


4-asset optimization. Max-Sharpe: 4.996. Portfolio: return=10.91%, vol=1.38%. Weights: w_QQQ=0.0778, w_AVAX-USD=0.0000, w_STIP=0.2432, w_BIL=0.6790.

Question:
Assets: QQQ, AVAX-USD, STIP, BIL
Annualized mean returns: QQQ:0.7278, AVAX-USD:0.2103, STIP:0.0897, BIL:0.0451
Covariance matrix (annualized):
[[0.029387, 0.02024, -0.00178, 7.4e-05], [0.02024, 0.426145, 0.001149, -0.000243], [-0.00178, 0.001149, 0.00113, 2e-06], [7.4e-05, -0.000243, 2e-06, 1.2e-05]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_QQQ=0.0778, w_AVAX-USD=0.0000, w_STIP=0.2432, w_BIL=0.6790

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_QQQ=0.0778, w_AVAX-USD=0.0000, w_STIP=0.2432, w_BIL=0.6790
Portfolio annualized return: 10.91%, volatility: 1.38%
Sharpe ratio: (0.1091 - 0.0400) / 0.0138 = 4.9962


4-asset optimization. Max-Sharpe: 7.798. Portfolio: return=13.11%, vol=1.17%. Weights: w_QUAL=0.1085, w_AVAX-USD=0.0000, w_INDS=0.0000, w_CSHI=0.8915.

Question:
Assets: QUAL, AVAX-USD, INDS, CSHI
Annualized mean returns: QUAL:0.7121, AVAX-USD:-1.4722, INDS:0.4296, CSHI:0.0604
Covariance matrix (annualized):
[[0.009313, 0.022991, 0.007449, -4e-06], [0.022991, 0.508932, 0.051705, -5.4e-05], [0.007449, 0.051705, 0.027709, 1.7e-05], [-4e-06, -5.4e-05, 1.7e-05, 3.5e-05]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: bull

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_QUAL=0.1085, w_AVAX-USD=0.0000, w_INDS=0.0000, w_CSHI=0.8915

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_QUAL=0.1085, w_AVAX-USD=0.0000, w_INDS=0.0000, w_CSHI=0.8915
Portfolio annualized return: 13.11%, volatility: 1.17%
Sharpe ratio: (0.1311 - 0.0400) / 0.0117 = 7.7980


4-asset optimization. Max-Sharpe: 2.084. Portfolio: return=51.94%, vol=23.00%. Weights: w_XLF=0.0000, w_ADA-USD=0.0000, w_XHB=0.0000, w_WEAT=1.0000.

Question:
Assets: XLF, ADA-USD, XHB, WEAT
Annualized mean returns: XLF:0.0467, ADA-USD:-2.3709, XHB:-0.0172, WEAT:0.5194
Covariance matrix (annualized):
[[0.012944, 0.029405, 0.015934, 0.008132], [0.029405, 0.5242, 0.024877, 0.002513], [0.015934, 0.024877, 0.044784, 0.008474], [0.008132, 0.002513, 0.008474, 0.052917]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLF=0.0000, w_ADA-USD=0.0000, w_XHB=0.0000, w_WEAT=1.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLF=0.0000, w_ADA-USD=0.0000, w_XHB=0.0000, w_WEAT=1.0000
Portfolio annualized return: 51.94%, volatility: 23.00%
Sharpe ratio: (0.5194 - 0.0400) / 0.2300 = 2.0838


4-asset optimization. Max-Sharpe: -1.611. Portfolio: return=-20.54%, vol=15.23%. Weights: w_FXI=0.2500, w_BNB-USD=0.2500, w_BIL=0.2500, w_TLT=0.2500.

Question:
Assets: FXI, BNB-USD, BIL, TLT
Annualized mean returns: FXI:-0.2447, BNB-USD:-0.7822, BIL:0.0473, TLT:0.1578
Covariance matrix (annualized):
[[0.076759, 0.014324, 0.00014, -0.002328], [0.014324, 0.231056, 8.9e-05, 0.003159], [0.00014, 8.9e-05, 7e-06, 0.000107], [-0.002328, 0.003159, 0.000107, 0.032389]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_FXI=0.2500, w_BNB-USD=0.2500, w_BIL=0.2500, w_TLT=0.2500

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_FXI=0.2500, w_BNB-USD=0.2500, w_BIL=0.2500, w_TLT=0.2500
Portfolio annualized return: -20.54%, volatility: 15.23%
Sharpe ratio: (-0.2054 - 0.0400) / 0.1523 = -1.6115


4-asset optimization. Max-Sharpe: 6.592. Portfolio: return=5.34%, vol=0.20%. Weights: w_XLI=0.0000, w_BTC-USD=0.0000, w_GLD=0.0000, w_SGOV=1.0000.

Question:
Assets: XLI, BTC-USD, GLD, SGOV
Annualized mean returns: XLI:-0.1812, BTC-USD:-0.9494, GLD:-0.1262, SGOV:0.0534
Covariance matrix (annualized):
[[0.011129, 0.011529, 0.004462, -3.9e-05], [0.011529, 0.110936, 0.006769, -5.4e-05], [0.004462, 0.006769, 0.007178, -3.4e-05], [-3.9e-05, -5.4e-05, -3.4e-05, 4e-06]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLI=0.0000, w_BTC-USD=0.0000, w_GLD=0.0000, w_SGOV=1.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLI=0.0000, w_BTC-USD=0.0000, w_GLD=0.0000, w_SGOV=1.0000
Portfolio annualized return: 5.34%, volatility: 0.20%
Sharpe ratio: (0.0534 - 0.0400) / 0.0020 = 6.5924


4-asset optimization. Max-Sharpe: 1.991. Portfolio: return=4.96%, vol=0.48%. Weights: w_USMV=0.0158, w_ADA-USD=0.0006, w_HAUZ=0.0000, w_ICSH=0.9836.

Question:
Assets: USMV, ADA-USD, HAUZ, ICSH
Annualized mean returns: USMV:0.1148, ADA-USD:0.2557, HAUZ:0.1376, ICSH:0.0484
Covariance matrix (annualized):
[[0.005712, 0.010973, 0.007551, 8.5e-05], [0.010973, 0.536349, 0.015868, 4.2e-05], [0.007551, 0.015868, 0.020915, 0.000307], [8.5e-05, 4.2e-05, 0.000307, 1.9e-05]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_USMV=0.0158, w_ADA-USD=0.0006, w_HAUZ=0.0000, w_ICSH=0.9836

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_USMV=0.0158, w_ADA-USD=0.0006, w_HAUZ=0.0000, w_ICSH=0.9836
Portfolio annualized return: 4.96%, volatility: 0.48%
Sharpe ratio: (0.0496 - 0.0400) / 0.0048 = 1.9912


4-asset optimization. Max-Sharpe: 1.335. Portfolio: return=72.50%, vol=51.31%. Weights: w_IVV=0.0000, w_LINK-USD=1.0000, w_SOYB=0.0000, w_TLH=0.0000.

Question:
Assets: IVV, LINK-USD, SOYB, TLH
Annualized mean returns: IVV:-0.3535, LINK-USD:0.7250, SOYB:-0.1683, TLH:-0.4794
Covariance matrix (annualized):
[[0.013755, 0.021702, 0.003269, 0.003444], [0.021702, 0.263237, -0.007383, 0.014209], [0.003269, -0.007383, 0.030126, 0.001146], [0.003444, 0.014209, 0.001146, 0.016747]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_IVV=0.0000, w_LINK-USD=1.0000, w_SOYB=0.0000, w_TLH=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_IVV=0.0000, w_LINK-USD=1.0000, w_SOYB=0.0000, w_TLH=0.0000
Portfolio annualized return: 72.50%, volatility: 51.31%
Sharpe ratio: (0.7250 - 0.0400) / 0.5131 = 1.3352


4-asset optimization. Max-Sharpe: 1.930. Portfolio: return=26.48%, vol=11.65%. Weights: w_XLB=0.0000, w_ETH-USD=0.2803, w_HAUZ=0.0000, w_GLD=0.7197.

Question:
Assets: XLB, ETH-USD, HAUZ, GLD
Annualized mean returns: XLB:-0.2173, ETH-USD:0.4528, HAUZ:-0.1625, GLD:0.1916
Covariance matrix (annualized):
[[0.028103, 0.005677, 0.022589, -0.001336], [0.005677, 0.099436, 0.006381, -0.004119], [0.022589, 0.006381, 0.028293, -0.002159], [-0.001336, -0.004119, -0.002159, 0.014313]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLB=0.0000, w_ETH-USD=0.2803, w_HAUZ=0.0000, w_GLD=0.7197

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLB=0.0000, w_ETH-USD=0.2803, w_HAUZ=0.0000, w_GLD=0.7197
Portfolio annualized return: 26.48%, volatility: 11.65%
Sharpe ratio: (0.2648 - 0.0400) / 0.1165 = 1.9300


4-asset optimization. Max-Sharpe: 0.769. Portfolio: return=12.53%, vol=11.08%. Weights: w_VLUE=0.7243, w_ADA-USD=0.0000, w_VNQI=0.2757, w_USO=0.0000.

Question:
Assets: VLUE, ADA-USD, VNQI, USO
Annualized mean returns: VLUE:0.1219, ADA-USD:-1.1196, VNQI:0.1340, USO:0.0124
Covariance matrix (annualized):
[[0.012408, 0.011784, 0.010221, 0.006431], [0.011784, 0.2162, 0.030814, -0.006921], [0.010221, 0.030814, 0.022291, 0.003017], [0.006431, -0.006921, 0.003017, 0.054089]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: bull

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_VLUE=0.7243, w_ADA-USD=0.0000, w_VNQI=0.2757, w_USO=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_VLUE=0.7243, w_ADA-USD=0.0000, w_VNQI=0.2757, w_USO=0.0000
Portfolio annualized return: 12.53%, volatility: 11.08%
Sharpe ratio: (0.1253 - 0.0400) / 0.1108 = 0.7694


4-asset optimization. Max-Sharpe: 3.734. Portfolio: return=5.50%, vol=0.40%. Weights: w_XLK=0.0000, w_BNB-USD=0.0027, w_ICSH=0.9973, w_TIP=0.0000.

Question:
Assets: XLK, BNB-USD, ICSH, TIP
Annualized mean returns: XLK:-0.3025, BNB-USD:0.1034, ICSH:0.0549, TIP:0.0318
Covariance matrix (annualized):
[[0.030093, -0.001248, -9.1e-05, 0.000484], [-0.001248, 0.050409, -6.7e-05, 9.7e-05], [-9.1e-05, -6.7e-05, 1.6e-05, 3.4e-05], [0.000484, 9.7e-05, 3.4e-05, 0.000603]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLK=0.0000, w_BNB-USD=0.0027, w_ICSH=0.9973, w_TIP=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLK=0.0000, w_BNB-USD=0.0027, w_ICSH=0.9973, w_TIP=0.0000
Portfolio annualized return: 5.50%, volatility: 0.40%
Sharpe ratio: (0.0550 - 0.0400) / 0.0040 = 3.7336


4-asset optimization. Max-Sharpe: 3.361. Portfolio: return=6.95%, vol=0.88%. Weights: w_VEA=0.0169, w_ETH-USD=0.0106, w_CSHI=0.3195, w_SHV=0.6530.

Question:
Assets: VEA, ETH-USD, CSHI, SHV
Annualized mean returns: VEA:0.4461, ETH-USD:1.3721, CSHI:0.0581, SHV:0.0443
Covariance matrix (annualized):
[[0.025847, 0.043907, 0.000659, 4.9e-05], [0.043907, 0.220751, 0.001312, 0.000104], [0.000659, 0.001312, 7.6e-05, -1e-06], [4.9e-05, 0.000104, -1e-06, 9e-06]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_VEA=0.0169, w_ETH-USD=0.0106, w_CSHI=0.3195, w_SHV=0.6530

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_VEA=0.0169, w_ETH-USD=0.0106, w_CSHI=0.3195, w_SHV=0.6530
Portfolio annualized return: 6.95%, volatility: 0.88%
Sharpe ratio: (0.0695 - 0.0400) / 0.0088 = 3.3608


4-asset optimization. Max-Sharpe: 1.948. Portfolio: return=4.52%, vol=0.27%. Weights: w_IVV=0.0031, w_DOT-USD=0.0000, w_SGOV=0.9969, w_MORT=0.0000.

Question:
Assets: IVV, DOT-USD, SGOV, MORT
Annualized mean returns: IVV:0.1249, DOT-USD:-0.3430, SGOV:0.0449, MORT:-0.7438
Covariance matrix (annualized):
[[0.023542, 0.016235, 4.4e-05, 0.035513], [0.016235, 0.331105, 0.000133, -0.013081], [4.4e-05, 0.000133, 7e-06, -5e-05], [0.035513, -0.013081, -5e-05, 0.089507]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_IVV=0.0031, w_DOT-USD=0.0000, w_SGOV=0.9969, w_MORT=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_IVV=0.0031, w_DOT-USD=0.0000, w_SGOV=0.9969, w_MORT=0.0000
Portfolio annualized return: 4.52%, volatility: 0.27%
Sharpe ratio: (0.0452 - 0.0400) / 0.0027 = 1.9477


4-asset optimization. Max-Sharpe: 0.516. Portfolio: return=14.96%, vol=21.25%. Weights: w_XLF=0.0000, w_BNB-USD=0.0000, w_BNDX=0.0000, w_REZ=1.0000.

Question:
Assets: XLF, BNB-USD, BNDX, REZ
Annualized mean returns: XLF:0.0297, BNB-USD:-0.5806, BNDX:-0.0293, REZ:0.1496
Covariance matrix (annualized):
[[0.027607, 0.033142, 0.001589, 0.023308], [0.033142, 0.197402, 0.000319, 0.050454], [0.001589, 0.000319, 0.003453, 0.004328], [0.023308, 0.050454, 0.004328, 0.045171]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLF=0.0000, w_BNB-USD=0.0000, w_BNDX=0.0000, w_REZ=1.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLF=0.0000, w_BNB-USD=0.0000, w_BNDX=0.0000, w_REZ=1.0000
Portfolio annualized return: 14.96%, volatility: 21.25%
Sharpe ratio: (0.1496 - 0.0400) / 0.2125 = 0.5158


4-asset optimization. Max-Sharpe: 4.906. Portfolio: return=5.21%, vol=0.25%. Weights: w_VLUE=0.0093, w_ADA-USD=0.0000, w_BIL=0.9907, w_GLD=0.0000.

Question:
Assets: VLUE, ADA-USD, BIL, GLD
Annualized mean returns: VLUE:0.2624, ADA-USD:0.9292, BIL:0.0501, GLD:-0.1407
Covariance matrix (annualized):
[[0.012541, 0.018811, -5e-06, 0.003925], [0.018811, 0.517135, 0.000418, 0.002443], [-5e-06, 0.000418, 5e-06, -6.5e-05], [0.003925, 0.002443, -6.5e-05, 0.00908]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_VLUE=0.0093, w_ADA-USD=0.0000, w_BIL=0.9907, w_GLD=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_VLUE=0.0093, w_ADA-USD=0.0000, w_BIL=0.9907, w_GLD=0.0000
Portfolio annualized return: 5.21%, volatility: 0.25%
Sharpe ratio: (0.0521 - 0.0400) / 0.0025 = 4.9060


4-asset optimization. Max-Sharpe: 3.865. Portfolio: return=7.90%, vol=1.01%. Weights: w_^VIX=0.0098, w_MATIC-USD=0.0000, w_CORN=0.0000, w_CSHI=0.9902.

Question:
Assets: ^VIX, MATIC-USD, CORN, CSHI
Annualized mean returns: ^VIX:2.6236, MATIC-USD:-0.4830, CORN:0.0417, CSHI:0.0539
Covariance matrix (annualized):
[[0.980035, -0.045551, 0.003681, -0.00285], [-0.045551, 0.161832, -0.009345, 9.7e-05], [0.003681, -0.009345, 0.023144, -6e-06], [-0.00285, 9.7e-05, -6e-06, 6.5e-05]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_^VIX=0.0098, w_MATIC-USD=0.0000, w_CORN=0.0000, w_CSHI=0.9902

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_^VIX=0.0098, w_MATIC-USD=0.0000, w_CORN=0.0000, w_CSHI=0.9902
Portfolio annualized return: 7.90%, volatility: 1.01%
Sharpe ratio: (0.0790 - 0.0400) / 0.0101 = 3.8654


4-asset optimization. Max-Sharpe: 4.595. Portfolio: return=273.04%, vol=58.55%. Weights: w_XLU=0.3458, w_XRP-USD=0.6542, w_SOYB=0.0000, w_SCHH=0.0000.

Question:
Assets: XLU, XRP-USD, SOYB, SCHH
Annualized mean returns: XLU:0.1780, XRP-USD:4.0795, SOYB:-0.5095, SCHH:0.0740
Covariance matrix (annualized):
[[0.029305, 0.011389, 0.000166, 0.013748], [0.011389, 0.780779, -0.007016, -0.000223], [0.000166, -0.007016, 0.014838, 0.002125], [0.013748, -0.000223, 0.002125, 0.02177]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: bull

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLU=0.3458, w_XRP-USD=0.6542, w_SOYB=0.0000, w_SCHH=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLU=0.3458, w_XRP-USD=0.6542, w_SOYB=0.0000, w_SCHH=0.0000
Portfolio annualized return: 273.04%, volatility: 58.55%
Sharpe ratio: (2.7304 - 0.0400) / 0.5855 = 4.5950


4-asset optimization. Max-Sharpe: 3.115. Portfolio: return=4.77%, vol=0.25%. Weights: w_XLI=0.0000, w_BNB-USD=0.0000, w_SGOV=1.0000, w_IYR=0.0000.

Question:
Assets: XLI, BNB-USD, SGOV, IYR
Annualized mean returns: XLI:-0.1480, BNB-USD:-0.2464, SGOV:0.0477, IYR:-0.3986
Covariance matrix (annualized):
[[0.033423, 0.006346, -6.1e-05, 0.028414], [0.006346, 0.205907, 0.000216, 0.008639], [-6.1e-05, 0.000216, 6e-06, 8.7e-05], [0.028414, 0.008639, 8.7e-05, 0.05157]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLI=0.0000, w_BNB-USD=0.0000, w_SGOV=1.0000, w_IYR=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLI=0.0000, w_BNB-USD=0.0000, w_SGOV=1.0000, w_IYR=0.0000
Portfolio annualized return: 4.77%, volatility: 0.25%
Sharpe ratio: (0.0477 - 0.0400) / 0.0025 = 3.1152


4-asset optimization. Max-Sharpe: 5.648. Portfolio: return=12.98%, vol=1.59%. Weights: w_QQQ=0.0000, w_LINK-USD=0.0251, w_CSHI=0.9749, w_SCHH=0.0000.

Question:
Assets: QQQ, LINK-USD, CSHI, SCHH
Annualized mean returns: QQQ:-0.0375, LINK-USD:2.5433, CSHI:0.0676, SCHH:-0.2853
Covariance matrix (annualized):
[[0.032657, -0.002587, 0.000503, 0.022231], [-0.002587, 0.283753, -8e-05, -0.004915], [0.000503, -8e-05, 8.2e-05, 0.000513], [0.022231, -0.004915, 0.000513, 0.051602]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_QQQ=0.0000, w_LINK-USD=0.0251, w_CSHI=0.9749, w_SCHH=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_QQQ=0.0000, w_LINK-USD=0.0251, w_CSHI=0.9749, w_SCHH=0.0000
Portfolio annualized return: 12.98%, volatility: 1.59%
Sharpe ratio: (0.1298 - 0.0400) / 0.0159 = 5.6480


4-asset optimization. Max-Sharpe: 3.091. Portfolio: return=24.27%, vol=6.56%. Weights: w_EWJ=0.0000, w_XRP-USD=0.0000, w_DBC=0.0000, w_LQD=1.0000.

Question:
Assets: EWJ, XRP-USD, DBC, LQD
Annualized mean returns: EWJ:0.0489, XRP-USD:0.0489, DBC:-0.0631, LQD:0.2427
Covariance matrix (annualized):
[[0.022647, 0.028615, 0.005733, 0.004813], [0.028615, 0.136919, 0.011481, 0.010896], [0.005733, 0.011481, 0.017252, -0.000125], [0.004813, 0.010896, -0.000125, 0.004302]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_EWJ=0.0000, w_XRP-USD=0.0000, w_DBC=0.0000, w_LQD=1.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_EWJ=0.0000, w_XRP-USD=0.0000, w_DBC=0.0000, w_LQD=1.0000
Portfolio annualized return: 24.27%, volatility: 6.56%
Sharpe ratio: (0.2427 - 0.0400) / 0.0656 = 3.0908


4-asset optimization. Max-Sharpe: 5.188. Portfolio: return=5.51%, vol=0.29%. Weights: w_XLU=0.0033, w_DOT-USD=0.0004, w_ITB=0.0031, w_SGOV=0.9933.

Question:
Assets: XLU, DOT-USD, ITB, SGOV
Annualized mean returns: XLU:0.3786, DOT-USD:0.1402, ITB:0.5541, SGOV:0.0524
Covariance matrix (annualized):
[[0.035541, -0.000482, 0.027381, -1e-05], [-0.000482, 0.331274, 0.013717, -0.000109], [0.027381, 0.013717, 0.075579, -4e-05], [-1e-05, -0.000109, -4e-05, 7e-06]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLU=0.0033, w_DOT-USD=0.0004, w_ITB=0.0031, w_SGOV=0.9933

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLU=0.0033, w_DOT-USD=0.0004, w_ITB=0.0031, w_SGOV=0.9933
Portfolio annualized return: 5.51%, volatility: 0.29%
Sharpe ratio: (0.0551 - 0.0400) / 0.0029 = 5.1879


4-asset optimization. Max-Sharpe: 6.273. Portfolio: return=542.93%, vol=85.92%. Weights: w_XLRE=0.0000, w_SOL-USD=1.0000, w_ITB=0.0000, w_TIP=0.0000.

Question:
Assets: XLRE, SOL-USD, ITB, TIP
Annualized mean returns: XLRE:-0.2717, SOL-USD:5.4293, ITB:0.0148, TIP:0.0090
Covariance matrix (annualized):
[[0.055514, 0.015853, 0.049961, 0.002127], [0.015853, 0.738162, 0.063867, -0.001001], [0.049961, 0.063867, 0.078277, 0.003121], [0.002127, -0.001001, 0.003121, 0.000675]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLRE=0.0000, w_SOL-USD=1.0000, w_ITB=0.0000, w_TIP=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLRE=0.0000, w_SOL-USD=1.0000, w_ITB=0.0000, w_TIP=0.0000
Portfolio annualized return: 542.93%, volatility: 85.92%
Sharpe ratio: (5.4293 - 0.0400) / 0.8592 = 6.2728


4-asset optimization. Max-Sharpe: 2.205. Portfolio: return=73.53%, vol=31.53%. Weights: w_FXI=0.0000, w_ETH-USD=0.7118, w_XHB=0.2882, w_TLH=0.0000.

Question:
Assets: FXI, ETH-USD, XHB, TLH
Annualized mean returns: FXI:0.0065, ETH-USD:0.9689, XHB:0.1582, TLH:-0.1480
Covariance matrix (annualized):
[[0.058248, 0.004138, 0.033578, 0.016474], [0.004138, 0.188532, -0.004847, -0.011428], [0.033578, -0.004847, 0.070598, 0.029756], [0.016474, -0.011428, 0.029756, 0.032776]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_FXI=0.0000, w_ETH-USD=0.7118, w_XHB=0.2882, w_TLH=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_FXI=0.0000, w_ETH-USD=0.7118, w_XHB=0.2882, w_TLH=0.0000
Portfolio annualized return: 73.53%, volatility: 31.53%
Sharpe ratio: (0.7353 - 0.0400) / 0.3153 = 2.2053


4-asset optimization. Max-Sharpe: 5.601. Portfolio: return=6.18%, vol=0.39%. Weights: w_XLV=0.0000, w_BTC-USD=0.0071, w_PPLT=0.0000, w_SGOV=0.9929.

Question:
Assets: XLV, BTC-USD, PPLT, SGOV
Annualized mean returns: XLV:-0.1780, BTC-USD:1.7158, PPLT:0.1112, SGOV:0.0499
Covariance matrix (annualized):
[[0.009107, 0.007764, 0.004361, 5.2e-05], [0.007764, 0.166359, 0.036517, -2.6e-05], [0.004361, 0.036517, 0.066069, 1.5e-05], [5.2e-05, -2.6e-05, 1.5e-05, 7e-06]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLV=0.0000, w_BTC-USD=0.0071, w_PPLT=0.0000, w_SGOV=0.9929

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLV=0.0000, w_BTC-USD=0.0071, w_PPLT=0.0000, w_SGOV=0.9929
Portfolio annualized return: 6.18%, volatility: 0.39%
Sharpe ratio: (0.0618 - 0.0400) / 0.0039 = 5.6013


4-asset optimization. Max-Sharpe: 4.790. Portfolio: return=58.69%, vol=11.42%. Weights: w_SPY=0.9615, w_XRP-USD=0.0385, w_PALL=0.0000, w_MORT=0.0000.

Question:
Assets: SPY, XRP-USD, PALL, MORT
Annualized mean returns: SPY:0.5795, XRP-USD:0.7715, PALL:-0.5399, MORT:-0.1998
Covariance matrix (annualized):
[[0.012936, 0.010977, 0.005813, 0.016536], [0.010977, 0.178919, 0.030686, 0.038414], [0.005813, 0.030686, 0.137307, 0.038892], [0.016536, 0.038414, 0.038892, 0.061381]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_SPY=0.9615, w_XRP-USD=0.0385, w_PALL=0.0000, w_MORT=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_SPY=0.9615, w_XRP-USD=0.0385, w_PALL=0.0000, w_MORT=0.0000
Portfolio annualized return: 58.69%, volatility: 11.42%
Sharpe ratio: (0.5869 - 0.0400) / 0.1142 = 4.7896


4-asset optimization. Max-Sharpe: 3.630. Portfolio: return=9.54%, vol=1.52%. Weights: w_XLV=0.0023, w_LINK-USD=0.0065, w_CSHI=0.6815, w_HYG=0.3097.

Question:
Assets: XLV, LINK-USD, CSHI, HYG
Annualized mean returns: XLV:0.1040, LINK-USD:1.1080, CSHI:0.0656, HYG:0.1393
Covariance matrix (annualized):
[[0.007178, 0.009584, -3e-05, 0.000679], [0.009584, 0.321783, 0.000736, 0.005999], [-3e-05, 0.000736, 0.000126, 5.5e-05], [0.000679, 0.005999, 5.5e-05, 0.001094]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLV=0.0023, w_LINK-USD=0.0065, w_CSHI=0.6815, w_HYG=0.3097

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLV=0.0023, w_LINK-USD=0.0065, w_CSHI=0.6815, w_HYG=0.3097
Portfolio annualized return: 9.54%, volatility: 1.52%
Sharpe ratio: (0.0954 - 0.0400) / 0.0152 = 3.6298


4-asset optimization. Max-Sharpe: 3.269. Portfolio: return=9.72%, vol=1.75%. Weights: w_EEM=0.0233, w_AVAX-USD=0.0000, w_ITB=0.0000, w_STIP=0.9767.

Question:
Assets: EEM, AVAX-USD, ITB, STIP
Annualized mean returns: EEM:0.2784, AVAX-USD:-0.1277, ITB:-0.1493, STIP:0.0928
Covariance matrix (annualized):
[[0.015692, 0.01472, 0.01375, 0.000932], [0.01472, 0.481991, 0.065261, 0.003717], [0.01375, 0.065261, 0.061188, 0.00173], [0.000932, 0.003717, 0.00173, 0.000267]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_EEM=0.0233, w_AVAX-USD=0.0000, w_ITB=0.0000, w_STIP=0.9767

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_EEM=0.0233, w_AVAX-USD=0.0000, w_ITB=0.0000, w_STIP=0.9767
Portfolio annualized return: 9.72%, volatility: 1.75%
Sharpe ratio: (0.0972 - 0.0400) / 0.0175 = 3.2691


4-asset optimization. Max-Sharpe: 1.845. Portfolio: return=39.90%, vol=19.45%. Weights: w_XLY=0.7990, w_MATIC-USD=0.2010, w_VNQ=0.0000, w_SHY=0.0000.

Question:
Assets: XLY, MATIC-USD, VNQ, SHY
Annualized mean returns: XLY:0.2628, MATIC-USD:0.9404, VNQ:0.0176, SHY:-0.0065
Covariance matrix (annualized):
[[0.02487, 0.017978, 0.012412, 0.000447], [0.017978, 0.4008, 0.016605, 0.001428], [0.012412, 0.016605, 0.028751, 0.002043], [0.000447, 0.001428, 0.002043, 0.000345]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLY=0.7990, w_MATIC-USD=0.2010, w_VNQ=0.0000, w_SHY=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLY=0.7990, w_MATIC-USD=0.2010, w_VNQ=0.0000, w_SHY=0.0000
Portfolio annualized return: 39.90%, volatility: 19.45%
Sharpe ratio: (0.3990 - 0.0400) / 0.1945 = 1.8453


4-asset optimization. Max-Sharpe: 2.745. Portfolio: return=54.19%, vol=18.28%. Weights: w_XLK=0.8904, w_ADA-USD=0.1096, w_CPER=0.0000, w_TLH=0.0000.

Question:
Assets: XLK, ADA-USD, CPER, TLH
Annualized mean returns: XLK:0.5444, ADA-USD:0.5220, CPER:-0.6488, TLH:-0.2604
Covariance matrix (annualized):
[[0.036171, 0.012579, 0.015288, -0.006196], [0.012579, 0.190638, 0.006923, 0.01162], [0.015288, 0.006923, 0.039359, -0.011226], [-0.006196, 0.01162, -0.011226, 0.01699]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLK=0.8904, w_ADA-USD=0.1096, w_CPER=0.0000, w_TLH=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLK=0.8904, w_ADA-USD=0.1096, w_CPER=0.0000, w_TLH=0.0000
Portfolio annualized return: 54.19%, volatility: 18.28%
Sharpe ratio: (0.5419 - 0.0400) / 0.1828 = 2.7454


4-asset optimization. Max-Sharpe: 2.586. Portfolio: return=39.56%, vol=13.75%. Weights: w_XLU=0.8986, w_BNB-USD=0.0000, w_DBB=0.1014, w_IEF=0.0000.

Question:
Assets: XLU, BNB-USD, DBB, IEF
Annualized mean returns: XLU:0.4115, BNB-USD:-0.6732, DBB:0.2546, IEF:-0.0038
Covariance matrix (annualized):
[[0.020897, -0.004099, 0.009638, -0.000507], [-0.004099, 0.205701, 0.011164, 0.007082], [0.009638, 0.011164, 0.027101, 0.000896], [-0.000507, 0.007082, 0.000896, 0.006634]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLU=0.8986, w_BNB-USD=0.0000, w_DBB=0.1014, w_IEF=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLU=0.8986, w_BNB-USD=0.0000, w_DBB=0.1014, w_IEF=0.0000
Portfolio annualized return: 39.56%, volatility: 13.75%
Sharpe ratio: (0.3956 - 0.0400) / 0.1375 = 2.5862


4-asset optimization. Max-Sharpe: 2.067. Portfolio: return=6.35%, vol=1.14%. Weights: w_XLE=0.0000, w_ADA-USD=0.0000, w_DBA=0.0450, w_CSHI=0.9550.

Question:
Assets: XLE, ADA-USD, DBA, CSHI
Annualized mean returns: XLE:-0.3386, ADA-USD:-1.3999, DBA:0.1664, CSHI:0.0587
Covariance matrix (annualized):
[[0.059069, -0.005993, 0.01299, 0.00026], [-0.005993, 0.231966, 0.013012, -0.000951], [0.01299, 0.013012, 0.018852, -0.00016], [0.00026, -0.000951, -0.00016, 0.000115]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLE=0.0000, w_ADA-USD=0.0000, w_DBA=0.0450, w_CSHI=0.9550

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLE=0.0000, w_ADA-USD=0.0000, w_DBA=0.0450, w_CSHI=0.9550
Portfolio annualized return: 6.35%, volatility: 1.14%
Sharpe ratio: (0.0635 - 0.0400) / 0.0114 = 2.0674


4-asset optimization. Max-Sharpe: 7.302. Portfolio: return=6.29%, vol=0.31%. Weights: w_QUAL=0.0007, w_ADA-USD=0.0008, w_ICSH=0.5444, w_SHV=0.4541.

Question:
Assets: QUAL, ADA-USD, ICSH, SHV
Annualized mean returns: QUAL:0.0509, ADA-USD:0.3730, ICSH:0.0701, SHV:0.0537
Covariance matrix (annualized):
[[0.029937, 0.042472, -9.3e-05, 0.0], [0.042472, 0.34603, -0.0005, 0.000215], [-9.3e-05, -0.0005, 2.1e-05, 5e-06], [0.0, 0.000215, 5e-06, 7e-06]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: bull

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_QUAL=0.0007, w_ADA-USD=0.0008, w_ICSH=0.5444, w_SHV=0.4541

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_QUAL=0.0007, w_ADA-USD=0.0008, w_ICSH=0.5444, w_SHV=0.4541
Portfolio annualized return: 6.29%, volatility: 0.31%
Sharpe ratio: (0.0629 - 0.0400) / 0.0031 = 7.3023


4-asset optimization. Max-Sharpe: 2.294. Portfolio: return=78.10%, vol=32.31%. Weights: w_XLB=0.0000, w_BNB-USD=0.0000, w_MORT=0.0000, w_PPLT=1.0000.

Question:
Assets: XLB, BNB-USD, MORT, PPLT
Annualized mean returns: XLB:0.3470, BNB-USD:-1.0684, MORT:0.0316, PPLT:0.7810
Covariance matrix (annualized):
[[0.049703, 0.099525, 0.045141, 0.043375], [0.099525, 0.651454, 0.081185, 0.112356], [0.045141, 0.081185, 0.074932, 0.053374], [0.043375, 0.112356, 0.053374, 0.104383]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLB=0.0000, w_BNB-USD=0.0000, w_MORT=0.0000, w_PPLT=1.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLB=0.0000, w_BNB-USD=0.0000, w_MORT=0.0000, w_PPLT=1.0000
Portfolio annualized return: 78.10%, volatility: 32.31%
Sharpe ratio: (0.7810 - 0.0400) / 0.3231 = 2.2936


4-asset optimization. Max-Sharpe: 2.903. Portfolio: return=26.20%, vol=7.65%. Weights: w_USMV=1.0000, w_XRP-USD=0.0000, w_BNDX=0.0000, w_MORT=0.0000.

Question:
Assets: USMV, XRP-USD, BNDX, MORT
Annualized mean returns: USMV:0.2620, XRP-USD:-0.3199, BNDX:0.0137, MORT:0.0739
Covariance matrix (annualized):
[[0.005848, 0.003881, 0.001139, 0.006409], [0.003881, 0.106369, 0.00336, 0.009996], [0.001139, 0.00336, 0.001717, 0.002615], [0.006409, 0.009996, 0.002615, 0.026223]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_USMV=1.0000, w_XRP-USD=0.0000, w_BNDX=0.0000, w_MORT=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_USMV=1.0000, w_XRP-USD=0.0000, w_BNDX=0.0000, w_MORT=0.0000
Portfolio annualized return: 26.20%, volatility: 7.65%
Sharpe ratio: (0.2620 - 0.0400) / 0.0765 = 2.9026


4-asset optimization. Max-Sharpe: 5.231. Portfolio: return=5.19%, vol=0.23%. Weights: w_XLB=0.0007, w_ADA-USD=0.0000, w_BIL=0.9993, w_DBB=0.0000.

Question:
Assets: XLB, ADA-USD, BIL, DBB
Annualized mean returns: XLB:0.0349, ADA-USD:0.0536, BIL:0.0519, DBB:-0.3772
Covariance matrix (annualized):
[[0.024145, 0.032212, -1.9e-05, 0.00773], [0.032212, 0.367974, 0.00018, 0.009842], [-1.9e-05, 0.00018, 5e-06, 7.7e-05], [0.00773, 0.009842, 7.7e-05, 0.024241]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLB=0.0007, w_ADA-USD=0.0000, w_BIL=0.9993, w_DBB=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLB=0.0007, w_ADA-USD=0.0000, w_BIL=0.9993, w_DBB=0.0000
Portfolio annualized return: 5.19%, volatility: 0.23%
Sharpe ratio: (0.0519 - 0.0400) / 0.0023 = 5.2308


4-asset optimization. Max-Sharpe: 5.868. Portfolio: return=6.96%, vol=0.50%. Weights: w_EWJ=0.0000, w_AVAX-USD=0.0041, w_HYG=0.0000, w_ICSH=0.9959.

Question:
Assets: EWJ, AVAX-USD, HYG, ICSH
Annualized mean returns: EWJ:0.0707, AVAX-USD:2.7716, HYG:0.1456, ICSH:0.0585
Covariance matrix (annualized):
[[0.029284, -0.003584, 0.007626, 0.000116], [-0.003584, 0.797132, 0.010387, -0.000926], [0.007626, 0.010387, 0.00589, 0.0001], [0.000116, -0.000926, 0.0001, 2e-05]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_EWJ=0.0000, w_AVAX-USD=0.0041, w_HYG=0.0000, w_ICSH=0.9959

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_EWJ=0.0000, w_AVAX-USD=0.0041, w_HYG=0.0000, w_ICSH=0.9959
Portfolio annualized return: 6.96%, volatility: 0.50%
Sharpe ratio: (0.0696 - 0.0400) / 0.0050 = 5.8682


4-asset optimization. Max-Sharpe: 0.714. Portfolio: return=73.58%, vol=97.48%. Weights: w_XLF=0.0000, w_SOL-USD=1.0000, w_WEAT=0.0000, w_TIP=0.0000.

Question:
Assets: XLF, SOL-USD, WEAT, TIP
Annualized mean returns: XLF:-0.0198, SOL-USD:0.7358, WEAT:-0.6257, TIP:0.0079
Covariance matrix (annualized):
[[0.027447, 0.019603, 0.008688, 0.002604], [0.019603, 0.950143, -0.020593, 0.0047], [0.008688, -0.020593, 0.052942, 0.001703], [0.002604, 0.0047, 0.001703, 0.000923]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLF=0.0000, w_SOL-USD=1.0000, w_WEAT=0.0000, w_TIP=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLF=0.0000, w_SOL-USD=1.0000, w_WEAT=0.0000, w_TIP=0.0000
Portfolio annualized return: 73.58%, volatility: 97.48%
Sharpe ratio: (0.7358 - 0.0400) / 0.9748 = 0.7138


4-asset optimization. Max-Sharpe: -0.626. Portfolio: return=-40.31%, vol=70.73%. Weights: w_XLV=0.0000, w_XRP-USD=1.0000, w_EMB=0.0000, w_VNQ=0.0000.

Question:
Assets: XLV, XRP-USD, EMB, VNQ
Annualized mean returns: XLV:-0.1055, XRP-USD:-0.4031, EMB:0.0309, VNQ:-0.2047
Covariance matrix (annualized):
[[0.010592, 0.005022, 0.003671, 0.01227], [0.005022, 0.500311, 0.008684, 0.022424], [0.003671, 0.008684, 0.006648, 0.010034], [0.01227, 0.022424, 0.010034, 0.032141]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: bull

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLV=0.0000, w_XRP-USD=1.0000, w_EMB=0.0000, w_VNQ=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLV=0.0000, w_XRP-USD=1.0000, w_EMB=0.0000, w_VNQ=0.0000
Portfolio annualized return: -40.31%, volatility: 70.73%
Sharpe ratio: (-0.4031 - 0.0400) / 0.7073 = -0.6264


4-asset optimization. Max-Sharpe: 3.807. Portfolio: return=130.70%, vol=33.28%. Weights: w_XLRE=0.0000, w_DOT-USD=0.3427, w_PDBC=0.0000, w_ITB=0.6573.

Question:
Assets: XLRE, DOT-USD, PDBC, ITB
Annualized mean returns: XLRE:0.2614, DOT-USD:2.3264, PDBC:-0.2071, ITB:0.7755
Covariance matrix (annualized):
[[0.049235, 0.040547, 0.008244, 0.047899], [0.040547, 0.511693, 0.014027, 0.037384], [0.008244, 0.014027, 0.035244, 0.005078], [0.047899, 0.037384, 0.005078, 0.078347]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLRE=0.0000, w_DOT-USD=0.3427, w_PDBC=0.0000, w_ITB=0.6573

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLRE=0.0000, w_DOT-USD=0.3427, w_PDBC=0.0000, w_ITB=0.6573
Portfolio annualized return: 130.70%, volatility: 33.28%
Sharpe ratio: (1.3070 - 0.0400) / 0.3328 = 3.8066


4-asset optimization. Max-Sharpe: 5.744. Portfolio: return=10.34%, vol=1.10%. Weights: w_VEA=0.0590, w_AVAX-USD=0.0056, w_CSHI=0.9354, w_HAUZ=0.0000.

Question:
Assets: VEA, AVAX-USD, CSHI, HAUZ
Annualized mean returns: VEA:0.4662, AVAX-USD:3.6243, CSHI:0.0594, HAUZ:0.3946
Covariance matrix (annualized):
[[0.015042, 0.01445, -0.00016, 0.01706], [0.01445, 1.220481, -0.000875, 0.009321], [-0.00016, -0.000875, 5.5e-05, -0.000139], [0.01706, 0.009321, -0.000139, 0.02727]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_VEA=0.0590, w_AVAX-USD=0.0056, w_CSHI=0.9354, w_HAUZ=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_VEA=0.0590, w_AVAX-USD=0.0056, w_CSHI=0.9354, w_HAUZ=0.0000
Portfolio annualized return: 10.34%, volatility: 1.10%
Sharpe ratio: (0.1034 - 0.0400) / 0.0110 = 5.7442


4-asset optimization. Max-Sharpe: 5.682. Portfolio: return=5.39%, vol=0.24%. Weights: w_EEM=0.0022, w_AVAX-USD=0.0005, w_BIL=0.9973, w_SHY=0.0000.

Question:
Assets: EEM, AVAX-USD, BIL, SHY
Annualized mean returns: EEM:0.1797, AVAX-USD:0.4150, BIL:0.0534, SHY:0.0156
Covariance matrix (annualized):
[[0.029579, 0.002083, -6e-06, 0.001024], [0.002083, 0.225124, 5.5e-05, 0.000385], [-6e-06, 5.5e-05, 6e-06, 1e-06], [0.001024, 0.000385, 1e-06, 0.000451]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_EEM=0.0022, w_AVAX-USD=0.0005, w_BIL=0.9973, w_SHY=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_EEM=0.0022, w_AVAX-USD=0.0005, w_BIL=0.9973, w_SHY=0.0000
Portfolio annualized return: 5.39%, volatility: 0.24%
Sharpe ratio: (0.0539 - 0.0400) / 0.0024 = 5.6819


4-asset optimization. Max-Sharpe: 5.894. Portfolio: return=9.55%, vol=0.94%. Weights: w_XLE=0.0478, w_ADA-USD=0.0050, w_SHV=0.9472, w_CPER=0.0000.

Question:
Assets: XLE, ADA-USD, SHV, CPER
Annualized mean returns: XLE:0.7992, ADA-USD:1.9725, SHV:0.0500, CPER:0.2338
Covariance matrix (annualized):
[[0.022265, 0.003467, 0.000139, 0.007702], [0.003467, 0.530278, 0.000269, 0.029803], [0.000139, 0.000269, 9e-06, 1.1e-05], [0.007702, 0.029803, 1.1e-05, 0.027374]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: bull

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLE=0.0478, w_ADA-USD=0.0050, w_SHV=0.9472, w_CPER=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLE=0.0478, w_ADA-USD=0.0050, w_SHV=0.9472, w_CPER=0.0000
Portfolio annualized return: 9.55%, volatility: 0.94%
Sharpe ratio: (0.0955 - 0.0400) / 0.0094 = 5.8945


4-asset optimization. Max-Sharpe: 8.793. Portfolio: return=7.35%, vol=0.38%. Weights: w_USMV=0.0277, w_BTC-USD=0.0052, w_SGOV=0.9671, w_PALL=0.0000.

Question:
Assets: USMV, BTC-USD, SGOV, PALL
Annualized mean returns: USMV:0.5241, BTC-USD:1.0813, SGOV:0.0551, PALL:-0.2332
Covariance matrix (annualized):
[[0.008525, -0.007703, 1.5e-05, 0.005298], [-0.007703, 0.180488, -0.000292, -0.062367], [1.5e-05, -0.000292, 8e-06, 0.000205], [0.005298, -0.062367, 0.000205, 0.180194]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_USMV=0.0277, w_BTC-USD=0.0052, w_SGOV=0.9671, w_PALL=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_USMV=0.0277, w_BTC-USD=0.0052, w_SGOV=0.9671, w_PALL=0.0000
Portfolio annualized return: 7.35%, volatility: 0.38%
Sharpe ratio: (0.0735 - 0.0400) / 0.0038 = 8.7932


4-asset optimization. Max-Sharpe: 2.957. Portfolio: return=56.09%, vol=17.62%. Weights: w_VTI=0.1764, w_ETH-USD=0.0364, w_PPLT=0.0000, w_ITB=0.7872.

Question:
Assets: VTI, ETH-USD, PPLT, ITB
Annualized mean returns: VTI:0.2453, ETH-USD:0.4590, PPLT:-0.5986, ITB:0.6364
Covariance matrix (annualized):
[[0.010373, 0.007264, 0.009549, 0.012877], [0.007264, 0.128177, 0.0101, 0.024156], [0.009549, 0.0101, 0.048904, 0.016477], [0.012877, 0.024156, 0.016477, 0.041137]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_VTI=0.1764, w_ETH-USD=0.0364, w_PPLT=0.0000, w_ITB=0.7872

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_VTI=0.1764, w_ETH-USD=0.0364, w_PPLT=0.0000, w_ITB=0.7872
Portfolio annualized return: 56.09%, volatility: 17.62%
Sharpe ratio: (0.5609 - 0.0400) / 0.1762 = 2.9570


4-asset optimization. Max-Sharpe: 5.534. Portfolio: return=431.25%, vol=77.21%. Weights: w_XLE=0.0000, w_XRP-USD=0.6955, w_IYR=0.3045, w_IEF=0.0000.

Question:
Assets: XLE, XRP-USD, IYR, IEF
Annualized mean returns: XLE:-0.1719, XRP-USD:6.2149, IYR:-0.0318, IEF:-0.0304
Covariance matrix (annualized):
[[0.033335, 0.025274, -0.004437, -0.004677], [0.025274, 1.250292, -0.026038, 0.002578], [-0.004437, -0.026038, 0.02656, 0.006919], [-0.004677, 0.002578, 0.006919, 0.004063]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: bull

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLE=0.0000, w_XRP-USD=0.6955, w_IYR=0.3045, w_IEF=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLE=0.0000, w_XRP-USD=0.6955, w_IYR=0.3045, w_IEF=0.0000
Portfolio annualized return: 431.25%, volatility: 77.21%
Sharpe ratio: (4.3125 - 0.0400) / 0.7721 = 5.5335


4-asset optimization. Max-Sharpe: 5.969. Portfolio: return=5.89%, vol=0.32%. Weights: w_XLU=0.0000, w_ETH-USD=0.0038, w_BIL=0.9962, w_PPLT=0.0000.

Question:
Assets: XLU, ETH-USD, BIL, PPLT
Annualized mean returns: XLU:-0.0643, ETH-USD:1.6932, BIL:0.0527, PPLT:-0.5987
Covariance matrix (annualized):
[[0.027743, 0.00975, 0.000103, 0.005172], [0.00975, 0.268827, -0.000149, 0.013938], [0.000103, -0.000149, 7e-06, 0.000104], [0.005172, 0.013938, 0.000104, 0.04531]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLU=0.0000, w_ETH-USD=0.0038, w_BIL=0.9962, w_PPLT=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLU=0.0000, w_ETH-USD=0.0038, w_BIL=0.9962, w_PPLT=0.0000
Portfolio annualized return: 5.89%, volatility: 0.32%
Sharpe ratio: (0.0589 - 0.0400) / 0.0032 = 5.9692


4-asset optimization. Max-Sharpe: 3.338. Portfolio: return=257.58%, vol=75.98%. Weights: w_USMV=0.0000, w_BNB-USD=1.0000, w_INDS=0.0000, w_TLH=0.0000.

Question:
Assets: USMV, BNB-USD, INDS, TLH
Annualized mean returns: USMV:-0.0063, BNB-USD:2.5758, INDS:-0.4750, TLH:-0.2040
Covariance matrix (annualized):
[[0.005473, 0.005187, 0.009505, 0.00183], [0.005187, 0.577238, 0.021511, 0.000741], [0.009505, 0.021511, 0.033581, 0.009673], [0.00183, 0.000741, 0.009673, 0.011619]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_USMV=0.0000, w_BNB-USD=1.0000, w_INDS=0.0000, w_TLH=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_USMV=0.0000, w_BNB-USD=1.0000, w_INDS=0.0000, w_TLH=0.0000
Portfolio annualized return: 257.58%, volatility: 75.98%
Sharpe ratio: (2.5758 - 0.0400) / 0.7598 = 3.3377


4-asset optimization. Max-Sharpe: 4.602. Portfolio: return=67.25%, vol=13.74%. Weights: w_EFA=0.1579, w_SOL-USD=0.0000, w_REZ=0.0854, w_DBA=0.7567.

Question:
Assets: EFA, SOL-USD, REZ, DBA
Annualized mean returns: EFA:0.0798, SOL-USD:-0.6816, REZ:0.1798, DBA:0.8518
Covariance matrix (annualized):
[[0.011551, 0.033666, 0.012217, -0.002217], [0.033666, 0.748161, 0.020898, 0.023086], [0.012217, 0.020898, 0.034557, -0.000934], [-0.002217, 0.023086, -0.000934, 0.032603]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_EFA=0.1579, w_SOL-USD=0.0000, w_REZ=0.0854, w_DBA=0.7567

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_EFA=0.1579, w_SOL-USD=0.0000, w_REZ=0.0854, w_DBA=0.7567
Portfolio annualized return: 67.25%, volatility: 13.74%
Sharpe ratio: (0.6725 - 0.0400) / 0.1374 = 4.6023


4-asset optimization. Max-Sharpe: -0.988. Portfolio: return=-14.70%, vol=18.93%. Weights: w_XLI=0.0000, w_BNB-USD=0.0000, w_VNQI=0.0000, w_CPER=1.0000.

Question:
Assets: XLI, BNB-USD, VNQI, CPER
Annualized mean returns: XLI:-0.1642, BNB-USD:-0.7629, VNQI:-0.1444, CPER:-0.1470
Covariance matrix (annualized):
[[0.011477, 0.007466, 0.006875, 0.007267], [0.007466, 0.074698, 0.008884, -0.007539], [0.006875, 0.008884, 0.01353, 0.010604], [0.007267, -0.007539, 0.010604, 0.035833]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLI=0.0000, w_BNB-USD=0.0000, w_VNQI=0.0000, w_CPER=1.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLI=0.0000, w_BNB-USD=0.0000, w_VNQI=0.0000, w_CPER=1.0000
Portfolio annualized return: -14.70%, volatility: 18.93%
Sharpe ratio: (-0.1470 - 0.0400) / 0.1893 = -0.9879


4-asset optimization. Max-Sharpe: -3.150. Portfolio: return=-47.12%, vol=16.23%. Weights: w_XLF=0.2500, w_SOL-USD=0.2500, w_MORT=0.2500, w_SHV=0.2500.

Question:
Assets: XLF, SOL-USD, MORT, SHV
Annualized mean returns: XLF:-0.0036, SOL-USD:-1.8939, MORT:-0.0380, SHV:0.0506
Covariance matrix (annualized):
[[0.012985, 0.002578, 0.012341, -3.6e-05], [0.002578, 0.292831, 0.024437, -0.000112], [0.012341, 0.024437, 0.037444, -7.8e-05], [-3.6e-05, -0.000112, -7.8e-05, 5e-06]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLF=0.2500, w_SOL-USD=0.2500, w_MORT=0.2500, w_SHV=0.2500

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLF=0.2500, w_SOL-USD=0.2500, w_MORT=0.2500, w_SHV=0.2500
Portfolio annualized return: -47.12%, volatility: 16.23%
Sharpe ratio: (-0.4712 - 0.0400) / 0.1623 = -3.1495


4-asset optimization. Max-Sharpe: 4.565. Portfolio: return=58.18%, vol=11.87%. Weights: w_USMV=0.6352, w_MATIC-USD=0.0000, w_INDS=0.3648, w_PALL=0.0000.

Question:
Assets: USMV, MATIC-USD, INDS, PALL
Annualized mean returns: USMV:0.4422, MATIC-USD:0.3426, INDS:0.8248, PALL:0.2213
Covariance matrix (annualized):
[[0.00906, 0.028663, 0.012893, 0.015667], [0.028663, 0.52484, 0.037668, 0.071688], [0.012893, 0.037668, 0.033484, 0.022639], [0.015667, 0.071688, 0.022639, 0.120339]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: bull

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_USMV=0.6352, w_MATIC-USD=0.0000, w_INDS=0.3648, w_PALL=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_USMV=0.6352, w_MATIC-USD=0.0000, w_INDS=0.3648, w_PALL=0.0000
Portfolio annualized return: 58.18%, volatility: 11.87%
Sharpe ratio: (0.5818 - 0.0400) / 0.1187 = 4.5647


4-asset optimization. Max-Sharpe: 3.046. Portfolio: return=32.40%, vol=9.32%. Weights: w_XLRE=0.5657, w_SOL-USD=0.0000, w_DBB=0.3185, w_LQD=0.1158.

Question:
Assets: XLRE, SOL-USD, DBB, LQD
Annualized mean returns: XLRE:0.2589, SOL-USD:-0.1809, DBB:0.5162, LQD:0.1127
Covariance matrix (annualized):
[[0.011196, 0.008953, 0.000323, 0.00229], [0.008953, 0.275058, 0.052279, 0.006049], [0.000323, 0.052279, 0.044537, 0.001814], [0.00229, 0.006049, 0.001814, 0.003043]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLRE=0.5657, w_SOL-USD=0.0000, w_DBB=0.3185, w_LQD=0.1158

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLRE=0.5657, w_SOL-USD=0.0000, w_DBB=0.3185, w_LQD=0.1158
Portfolio annualized return: 32.40%, volatility: 9.32%
Sharpe ratio: (0.3240 - 0.0400) / 0.0932 = 3.0458


4-asset optimization. Max-Sharpe: 3.419. Portfolio: return=8.77%, vol=1.40%. Weights: w_VEA=0.0000, w_MATIC-USD=0.0149, w_IYR=0.0000, w_ICSH=0.9851.

Question:
Assets: VEA, MATIC-USD, IYR, ICSH
Annualized mean returns: VEA:-0.1307, MATIC-USD:2.8203, IYR:0.0679, ICSH:0.0463
Covariance matrix (annualized):
[[0.01282, 0.020709, 0.00635, 8.2e-05], [0.020709, 0.745556, 0.019562, 0.000227], [0.00635, 0.019562, 0.026119, 0.00017], [8.2e-05, 0.000227, 0.00017, 2.3e-05]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: bull

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_VEA=0.0000, w_MATIC-USD=0.0149, w_IYR=0.0000, w_ICSH=0.9851

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_VEA=0.0000, w_MATIC-USD=0.0149, w_IYR=0.0000, w_ICSH=0.9851
Portfolio annualized return: 8.77%, volatility: 1.40%
Sharpe ratio: (0.0877 - 0.0400) / 0.0140 = 3.4191


4-asset optimization. Max-Sharpe: 5.307. Portfolio: return=6.23%, vol=0.42%. Weights: w_MTUM=0.0015, w_AVAX-USD=0.0000, w_REZ=0.0154, w_BIL=0.9831.

Question:
Assets: MTUM, AVAX-USD, REZ, BIL
Annualized mean returns: MTUM:0.2337, AVAX-USD:0.1106, REZ:0.6800, BIL:0.0523
Covariance matrix (annualized):
[[0.068397, 0.097871, 0.010719, -0.00012], [0.097871, 0.596003, 0.047186, 0.000239], [0.010719, 0.047186, 0.022697, 0.000143], [-0.00012, 0.000239, 0.000143, 8e-06]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_MTUM=0.0015, w_AVAX-USD=0.0000, w_REZ=0.0154, w_BIL=0.9831

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_MTUM=0.0015, w_AVAX-USD=0.0000, w_REZ=0.0154, w_BIL=0.9831
Portfolio annualized return: 6.23%, volatility: 0.42%
Sharpe ratio: (0.0623 - 0.0400) / 0.0042 = 5.3071


4-asset optimization. Max-Sharpe: 2.455. Portfolio: return=43.31%, vol=16.01%. Weights: w_XLRE=0.0000, w_LINK-USD=0.0161, w_TLH=0.9839, w_VNQ=0.0000.

Question:
Assets: XLRE, LINK-USD, TLH, VNQ
Annualized mean returns: XLRE:0.1620, LINK-USD:0.2841, TLH:0.4355, VNQ:0.1762
Covariance matrix (annualized):
[[0.055907, 0.061308, 0.012237, 0.05346], [0.061308, 0.289396, 0.011467, 0.058569], [0.012237, 0.011467, 0.02604, 0.011693], [0.05346, 0.058569, 0.011693, 0.051626]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLRE=0.0000, w_LINK-USD=0.0161, w_TLH=0.9839, w_VNQ=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLRE=0.0000, w_LINK-USD=0.0161, w_TLH=0.9839, w_VNQ=0.0000
Portfolio annualized return: 43.31%, volatility: 16.01%
Sharpe ratio: (0.4331 - 0.0400) / 0.1601 = 2.4546


4-asset optimization. Max-Sharpe: -0.980. Portfolio: return=-15.92%, vol=20.33%. Weights: w_XLY=1.0000, w_BNB-USD=0.0000, w_MORT=0.0000, w_JNK=0.0000.

Question:
Assets: XLY, BNB-USD, MORT, JNK
Annualized mean returns: XLY:-0.1592, BNB-USD:-0.2329, MORT:-0.2945, JNK:-0.1036
Covariance matrix (annualized):
[[0.041346, 0.005234, 0.028466, 0.008519], [0.005234, 0.058804, 0.009638, 0.00328], [0.028466, 0.009638, 0.05851, 0.011392], [0.008519, 0.00328, 0.011392, 0.003938]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLY=1.0000, w_BNB-USD=0.0000, w_MORT=0.0000, w_JNK=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLY=1.0000, w_BNB-USD=0.0000, w_MORT=0.0000, w_JNK=0.0000
Portfolio annualized return: -15.92%, volatility: 20.33%
Sharpe ratio: (-0.1592 - 0.0400) / 0.2033 = -0.9796


4-asset optimization. Max-Sharpe: 2.112. Portfolio: return=65.43%, vol=29.09%. Weights: w_XLF=0.0000, w_AVAX-USD=0.0031, w_XHB=0.0000, w_PPLT=0.9969.

Question:
Assets: XLF, AVAX-USD, XHB, PPLT
Annualized mean returns: XLF:-0.6101, AVAX-USD:0.0798, XHB:-0.1164, PPLT:0.6561
Covariance matrix (annualized):
[[0.056816, 0.016439, 0.039247, -0.00507], [0.016439, 0.482495, 0.003791, 0.004013], [0.039247, 0.003791, 0.053138, -0.000882], [-0.00507, 0.004013, -0.000882, 0.085134]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLF=0.0000, w_AVAX-USD=0.0031, w_XHB=0.0000, w_PPLT=0.9969

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLF=0.0000, w_AVAX-USD=0.0031, w_XHB=0.0000, w_PPLT=0.9969
Portfolio annualized return: 65.43%, volatility: 29.09%
Sharpe ratio: (0.6543 - 0.0400) / 0.2909 = 2.1115


4-asset optimization. Max-Sharpe: 0.383. Portfolio: return=4.25%, vol=0.65%. Weights: w_XLY=0.0000, w_DOT-USD=0.0000, w_CSHI=1.0000, w_STIP=0.0000.

Question:
Assets: XLY, DOT-USD, CSHI, STIP
Annualized mean returns: XLY:-0.6001, DOT-USD:-1.9337, CSHI:0.0425, STIP:0.0132
Covariance matrix (annualized):
[[0.061046, 0.078443, 0.000645, 0.004646], [0.078443, 0.471721, 0.000637, 0.00587], [0.000645, 0.000637, 4.3e-05, 6.6e-05], [0.004646, 0.00587, 6.6e-05, 0.000951]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLY=0.0000, w_DOT-USD=0.0000, w_CSHI=1.0000, w_STIP=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLY=0.0000, w_DOT-USD=0.0000, w_CSHI=1.0000, w_STIP=0.0000
Portfolio annualized return: 4.25%, volatility: 0.65%
Sharpe ratio: (0.0425 - 0.0400) / 0.0065 = 0.3827


4-asset optimization. Max-Sharpe: 3.616. Portfolio: return=426.43%, vol=116.83%. Weights: w_FXI=0.0000, w_LINK-USD=1.0000, w_XHB=0.0000, w_IAU=0.0000.

Question:
Assets: FXI, LINK-USD, XHB, IAU
Annualized mean returns: FXI:-0.2225, LINK-USD:4.2643, XHB:-0.6236, IAU:-0.2021
Covariance matrix (annualized):
[[0.096722, -0.019845, 0.025735, 0.014364], [-0.019845, 1.364993, 0.051606, 0.005959], [0.025735, 0.051606, 0.055348, 0.003735], [0.014364, 0.005959, 0.003735, 0.029992]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_FXI=0.0000, w_LINK-USD=1.0000, w_XHB=0.0000, w_IAU=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_FXI=0.0000, w_LINK-USD=1.0000, w_XHB=0.0000, w_IAU=0.0000
Portfolio annualized return: 426.43%, volatility: 116.83%
Sharpe ratio: (4.2643 - 0.0400) / 1.1683 = 3.6156


4-asset optimization. Max-Sharpe: 3.271. Portfolio: return=7.07%, vol=0.94%. Weights: w_VTI=0.0642, w_BTC-USD=0.0000, w_CSHI=0.9358, w_IYR=0.0000.

Question:
Assets: VTI, BTC-USD, CSHI, IYR
Annualized mean returns: VTI:0.3132, BTC-USD:-0.0669, CSHI:0.0541, IYR:0.2071
Covariance matrix (annualized):
[[0.01178, 0.023189, 3e-05, 0.010827], [0.023189, 0.217508, -2.9e-05, 0.027886], [3e-05, -2.9e-05, 4.1e-05, 2.4e-05], [0.010827, 0.027886, 2.4e-05, 0.022761]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_VTI=0.0642, w_BTC-USD=0.0000, w_CSHI=0.9358, w_IYR=0.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_VTI=0.0642, w_BTC-USD=0.0000, w_CSHI=0.9358, w_IYR=0.0000
Portfolio annualized return: 7.07%, volatility: 0.94%
Sharpe ratio: (0.0707 - 0.0400) / 0.0094 = 3.2711


4-asset optimization. Max-Sharpe: 2.600. Portfolio: return=5.56%, vol=0.60%. Weights: w_FXI=0.0000, w_DOT-USD=0.0000, w_DBB=0.0000, w_CSHI=1.0000.

Question:
Assets: FXI, DOT-USD, DBB, CSHI
Annualized mean returns: FXI:-0.4247, DOT-USD:-1.8393, DBB:-0.7568, CSHI:0.0556
Covariance matrix (annualized):
[[0.039219, -0.017165, 0.012485, -0.000459], [-0.017165, 0.287923, 0.010495, 5e-06], [0.012485, 0.010495, 0.030289, -0.000284], [-0.000459, 5e-06, -0.000284, 3.6e-05]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_FXI=0.0000, w_DOT-USD=0.0000, w_DBB=0.0000, w_CSHI=1.0000

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_FXI=0.0000, w_DOT-USD=0.0000, w_DBB=0.0000, w_CSHI=1.0000
Portfolio annualized return: 5.56%, volatility: 0.60%
Sharpe ratio: (0.0556 - 0.0400) / 0.0060 = 2.5999


4-asset optimization. Max-Sharpe: 6.220. Portfolio: return=7.08%, vol=0.50%. Weights: w_ACWI=0.0000, w_AVAX-USD=0.0020, w_ICSH=0.9637, w_DBA=0.0343.

Question:
Assets: ACWI, AVAX-USD, ICSH, DBA
Annualized mean returns: ACWI:0.3287, AVAX-USD:-0.1804, ICSH:0.0590, DBA:0.4179
Covariance matrix (annualized):
[[0.010715, 0.014431, 0.000137, 0.002653], [0.014431, 0.717351, -0.001363, -0.008363], [0.000137, -0.001363, 2e-05, -3.2e-05], [0.002653, -0.008363, -3.2e-05, 0.01016]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_ACWI=0.0000, w_AVAX-USD=0.0020, w_ICSH=0.9637, w_DBA=0.0343

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_ACWI=0.0000, w_AVAX-USD=0.0020, w_ICSH=0.9637, w_DBA=0.0343
Portfolio annualized return: 7.08%, volatility: 0.50%
Sharpe ratio: (0.0708 - 0.0400) / 0.0050 = 6.2203


4-asset optimization. Max-Sharpe: 4.539. Portfolio: return=5.36%, vol=0.30%. Weights: w_EEM=0.0022, w_ETH-USD=0.0000, w_SGOV=0.9938, w_ITB=0.0040.

Question:
Assets: EEM, ETH-USD, SGOV, ITB
Annualized mean returns: EEM:0.2734, ETH-USD:0.1512, SGOV:0.0510, ITB:0.5583
Covariance matrix (annualized):
[[0.040756, 0.029614, 1.4e-05, 0.012254], [0.029614, 0.246264, 0.000116, 0.052448], [1.4e-05, 0.000116, 7e-06, 0.000117], [0.012254, 0.052448, 0.000117, 0.049021]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_EEM=0.0022, w_ETH-USD=0.0000, w_SGOV=0.9938, w_ITB=0.0040

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_EEM=0.0022, w_ETH-USD=0.0000, w_SGOV=0.9938, w_ITB=0.0040
Portfolio annualized return: 5.36%, volatility: 0.30%
Sharpe ratio: (0.0536 - 0.0400) / 0.0030 = 4.5388


4-asset optimization. Max-Sharpe: -2.108. Portfolio: return=-35.61%, vol=18.79%. Weights: w_XLI=0.2500, w_DOT-USD=0.2500, w_BIL=0.2500, w_DBB=0.2500.

Question:
Assets: XLI, DOT-USD, BIL, DBB
Annualized mean returns: XLI:0.0227, DOT-USD:-1.2470, BIL:0.0461, DBB:-0.2464
Covariance matrix (annualized):
[[0.028057, 0.011707, -0.000138, 0.010325], [0.011707, 0.369939, -5.8e-05, 0.044595], [-0.000138, -5.8e-05, 9e-06, -5.1e-05], [0.010325, 0.044595, -5.1e-05, 0.034115]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLI=0.2500, w_DOT-USD=0.2500, w_BIL=0.2500, w_DBB=0.2500

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLI=0.2500, w_DOT-USD=0.2500, w_BIL=0.2500, w_DBB=0.2500
Portfolio annualized return: -35.61%, volatility: 18.79%
Sharpe ratio: (-0.3561 - 0.0400) / 0.1879 = -2.1082


4-asset optimization. Max-Sharpe: 8.503. Portfolio: return=52.83%, vol=5.74%. Weights: w_XLV=0.4785, w_MATIC-USD=0.0000, w_VNQ=0.0000, w_VCIT=0.5215.

Question:
Assets: XLV, MATIC-USD, VNQ, VCIT
Annualized mean returns: XLV:0.7219, MATIC-USD:-0.0324, VNQ:0.8285, VCIT:0.3507
Covariance matrix (annualized):
[[0.009888, 0.007085, 0.008188, -0.000241], [0.007085, 0.704117, 0.055573, 0.006998], [0.008188, 0.055573, 0.033951, 0.006452], [-0.000241, 0.006998, 0.006452, 0.004244]]
Risk-free rate: 4.00%
Constraints: weights sum to 1, all weights ≥ 0
Market regime: sideways

Compute the portfolio weights that maximize the Sharpe Ratio for the above assets.

Answer: w_XLV=0.4785, w_MATIC-USD=0.0000, w_VNQ=0.0000, w_VCIT=0.5215

Explanation:
Solved max-Sharpe via SLSQP numerical optimization.
Optimal weights: w_XLV=0.4785, w_MATIC-USD=0.0000, w_VNQ=0.0000, w_VCIT=0.5215
Portfolio annualized return: 52.83%, volatility: 5.74%
Sharpe ratio: (0.5283 - 0.0400) / 0.0574 = 8.5033