# Expectation and Optimal Allocations in Existential Contests of Finite, Heavy-Tail-Distributed Outcomes

## Abstract

**:**

## 1. Introduction and Review of the Literature

#### 1.1. Fat-Tailed Distributions in Economic Time Series—History

#### 1.1.1. Subsequent Historical Milestones in Heavy-Tailed Distributions Applied to Financial Time Series

#### 1.1.2. The Generalized Hyperbolic Distribution (GHD)

- Substantial flexibility afforded by multiple shape parameters, enabling the GHD to accurately fit myriad empirically observed, non-normal behaviors in financial and economic data sets. Both leptokurtic and platykurtic distributions can be readily captured.
- Mathematical tractability, with the probability density function, cumulative distribution function, and characteristic function expressible in closed analytical form. This facilitates rigorous mathematical analysis and inference.
- Theoretical connections to fundamental economic concepts, such as utility maximization. Various special cases also share close relationships with other pivotal distributions, such as the variance-gamma distribution.
- Empirical studies across disparate samples and time horizons consistently demonstrate a superior goodness-of-fit compared to normal and stable models when applied to asset returns, market indices, and other economic variables.
- Ability to more accurately model tail risks and extreme events compared to normal models, enabling robust quantification of value-at-risk, expected shortfall, and other vital risk metrics.
- Despite the lack of a simple analytical formula, the distribution function can be reliably evaluated through straightforward numerical methods, aiding practical implementation.

#### 1.1.3. Extensions of the Normal Distribution for Greater Flexibility (GHD)

#### 1.2. Mathematical Expectation—History

#### 1.3. Optimal Allocations—History

## 2. Characteristics of the Generalized Hyperbolic Distribution (GHD)—Distributional Form and Corresponding Parameters

- A more general distribution class that includes Student’s t, Laplace, hyperbolic, normal-inverse Gaussian, and variance-gamma as special cases. Its mathematical tractability provides the ability to derive other distribution properties.
- Semi-heavy tail behavior, which allows modeling data with extreme events and fat-tailed probabilities.
- A density function, involving modified Bessel functions of the second kind (BesselK functions). Thus, although there is a lack of a closed-form density function, it can still be numerically evaluated in a straightforward manner.

#### Parameter Description

- The concentration parameter (α) modulates tail weight, with larger values engendering heavier tails and hence elevating the probability mass attributable to extreme deviations. α typically assumes values on the order of 0.1 to 10 for modeling economic phenomena.
- The scale parameter (δ) controls the spread about the central tendency, with larger values yielding expanded variance and wider dispersion. Applied economic analysis often utilizes δ ranging from 0.1 to 10.
- The location parameter (μ) shifts the distribution along the abscissa, with positive values effecting rightward translation. In economic contexts, μ commonly falls between −10 and 10.
- The shape parameter (λ) influences peakedness and flatness, with larger values precipitating more acute modes and sharper central tendencies. λ on the order of 0.1 to 10 frequently appears in economic applications.

## 3. Parameter Estimation of the Symmetrical GHD

_{0}/d

_{1}− 1, where d

_{0}represents the most recent data and d

_{1}the data from the previous period. In the case of outputs of a trading approach, the divisor is problematic—the results must be expressed in log terms, in terms of a percentage change. Such results must be expressed, therefore, in terms of an amount put at risk to assume such results, which are generalized to the largest potential loss. The notion of “potential loss” is another Pandora’s box worthy of a separate discussion), or any other time series data that comport to an assumed probability density function whose parameters we wish to fit to this sample data.

## 4. Discussion

#### 4.1. The Goal of Median-Sorted Outcome after N Trials—The Process of Determining the Representative “Expected” Set of Oucomes

- Determine what these finite-length expectations are, given a probability distribution we believe the outcomes comport to, so as to base on our sample data and the fitted parameters of that distribution.
- Determine a likely N-length sequence of outcomes for what we “expect”; that is, a likely set of N outcomes representative of the outcome stream that would see half of the outcome streams of N length be better than, and half less than, such a stream of expected outcomes. It is this “expected, representative stream” that we will use as input to other functions, such as determining growth-optimal allocations.

#### 4.2. Determining Median-Sorted Outcome after N Trials

#### 4.2.1. Calculating the Inverse CDF

#### 4.2.2. Worst-Case Expected Outcome

^{N}

#### 4.2.3. Calculating the Representative Stream of Outcomes from the Median-Sorted Outcome

- For each of N outcomes, we determined the corresponding probability by taking the value returned by the inverse CDF as input to the CDF. When using the inverse CDF function, the random number generated is not the probability of the value returned by the inverse CDF. Instead, the random number generated is used as the input to the inverse CDF function to obtain a value from the distribution. This value is a random variable that follows the distribution specified by the CDF. To determine the probability of this value, one would need to use the CDF function and evaluate it at the value returned by the inverse CDF function. The CDF function gives the probability that a random variable is less than or equal to a specific value. Therefore, if the CDF function is evaluated at the value returned by the inverse CDF function, the probability of that value will be obtained.
- For our purposes, however, we took any of the probabilities that were >0.5 and took the absolute value of their difference to 0.5. We called this p`:
- p` = p if p ≤ 0.5
- p` = |1 − p| if p > 0.5

- We took M sets of N outputs, corresponding to outputs drawn from our fitted distribution as well as the corresponding p` probabilities.
- For each of these M sets of N outcome—p` combinations, we summed the outcomes, and took the product of all of the p` values.
- We sorted these M sets based on their summed outcomes.
- We then took the sum of these M product of p` values. We called this the “SumOfProbabilities”.
- For each M set, we proceeded sequentially through them, where each M has a cumulative probability, specified as the cumulative probability of the previous M in the (outcome-sorted) sequence, plus the probability of that M divided by the SumOfProbabilities.
- It is the final calculation, we sought that M whose final calculation was closest to 0.5. This is our median-sorted outcome, and the values for N that comprise it are our typical, median-sorted outcome, our expected, finite-sequence set of outcomes.

#### 4.2.4. Example—Calculating the Representative Stream of Outcomes from the Median-Sorted Outcome

#### 4.3. Optimal Allocations

#### 4.3.1. Growth-Optimal Fraction

#### 4.3.2. Return/Risk Optimal Points

#### 4.3.3. Risk of Ruin/Drawdown Calculations

#### 4.3.4. Growth Diminishment in Existential Contests

## 5. Further Research

## 6. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Date | S&P500 | % Change | Date | S&P500 | % Change |
---|---|---|---|---|---|

20221230 | 3839.5 | 20230403 | 4124.51 | 0.003699 | |

20230103 | 3824.14 | −0.004 | 20230404 | 4100.6 | −0.0058 |

20230104 | 3852.97 | 0.007539 | 20230405 | 4090.38 | −0.00249 |

20230105 | 3808.1 | −0.01165 | 20230406 | 4105.02 | 0.003579 |

20230106 | 3895.08 | 0.022841 | 20230410 | 4109.11 | 0.000996 |

20230109 | 3892.09 | −0.00077 | 20230411 | 4108.94 | −4.1 × 10^{−5} |

20230110 | 3919.25 | 0.006978 | 20230412 | 4091.95 | −0.00413 |

20230111 | 3969.61 | 0.012849 | 20230413 | 4146.22 | 0.013263 |

20230112 | 3983.17 | 0.003416 | 20230414 | 4137.64 | −0.00207 |

20230113 | 3999.09 | 0.003997 | 20230417 | 4151.32 | 0.003306 |

20230117 | 3990.97 | −0.00203 | 20230418 | 4154.87 | 0.000855 |

20230118 | 3928.86 | −0.01556 | 20230419 | 4154.52 | −8.4 × 10^{−5} |

20230119 | 3898.85 | −0.00764 | 20230420 | 4129.79 | −0.00595 |

20230120 | 3972.61 | 0.018918 | 20230421 | 4133.52 | 0.000903 |

20230123 | 4019.81 | 0.011881 | 20230424 | 4137.04 | 0.000852 |

20230124 | 4016.95 | −0.00071 | 20230425 | 4071.63 | −0.01581 |

20230125 | 4016.22 | −0.00018 | 20230426 | 4055.99 | −0.00384 |

20230126 | 4060.43 | 0.011008 | 20230427 | 4135.35 | 0.019566 |

20230127 | 4070.56 | 0.002495 | 20230428 | 4169.48 | 0.008253 |

20230130 | 4017.77 | −0.01297 | 20230501 | 4167.87 | −0.00039 |

20230131 | 4076.6 | 0.014642 | 20230502 | 4119.58 | −0.01159 |

20230201 | 4119.21 | 0.010452 | 20230503 | 4090.75 | −0.007 |

20230202 | 4179.76 | 0.014699 | 20230504 | 4061.22 | −0.00722 |

20230203 | 4136.48 | −0.01035 | 20230505 | 4136.25 | 0.018475 |

20230206 | 4111.08 | −0.00614 | 20230508 | 4138.12 | 0.000452 |

20230207 | 4164 | 0.012873 | 20230509 | 4119.17 | −0.00458 |

20230208 | 4117.86 | −0.01108 | 20230510 | 4137.64 | 0.004484 |

20230209 | 4081.5 | −0.00883 | 20230511 | 4130.62 | −0.0017 |

20230210 | 4090.46 | 0.002195 | 20230512 | 4124.08 | −0.00158 |

20230213 | 4137.29 | 0.011449 | 20230515 | 4136.28 | 0.002958 |

20230214 | 4136.13 | −0.00028 | 20230516 | 4109.9 | −0.00638 |

20230215 | 4147.6 | 0.002773 | 20230517 | 4158.77 | 0.011891 |

20230216 | 4090.41 | −0.01379 | 20230518 | 4198.05 | 0.009445 |

20230217 | 4079.09 | −0.00277 | 20230519 | 4191.98 | −0.00145 |

20230221 | 3997.34 | −0.02004 | 20230522 | 4192.63 | 0.000155 |

20230222 | 3991.05 | −0.00157 | 20230523 | 4145.58 | −0.01122 |

20230223 | 4012.32 | 0.005329 | 20230524 | 4115.24 | −0.00732 |

20230224 | 3970.04 | −0.01054 | 20230525 | 4151.28 | 0.008758 |

20230227 | 3982.24 | 0.003073 | 20230526 | 4205.45 | 0.013049 |

20230228 | 3970.15 | −0.00304 | 20230530 | 4205.52 | 1.66 × 10^{−5} |

20230301 | 3951.39 | −0.00473 | 20230531 | 4179.83 | −0.00611 |

20230302 | 3981.35 | 0.007582 | 20230601 | 4221.02 | 0.009854 |

20230303 | 4045.64 | 0.016148 | 20230602 | 4282.37 | 0.014534 |

20230306 | 4048.42 | 0.000687 | 20230605 | 4273.79 | −0.002 |

20230307 | 3986.37 | −0.01533 | 20230606 | 4283.85 | 0.002354 |

20230308 | 3992.01 | 0.001415 | 20230607 | 4267.52 | −0.00381 |

20230309 | 3918.32 | −0.01846 | 20230608 | 4293.93 | 0.006189 |

20230310 | 3861.59 | −0.01448 | 20230609 | 4298.86 | 0.001148 |

20230313 | 3855.76 | −0.00151 | 20230612 | 4338.93 | 0.009321 |

20230314 | 3919.29 | 0.016477 | 20230613 | 4369.01 | 0.006932 |

20230315 | 3891.93 | −0.00698 | 20230614 | 4372.59 | 0.000819 |

20230316 | 3960.28 | 0.017562 | 20230615 | 4425.84 | 0.012178 |

20230317 | 3916.64 | −0.01102 | 20230616 | 4409.59 | −0.00367 |

20230320 | 3951.57 | 0.008918 | 20230620 | 4388.71 | −0.00474 |

20230321 | 4002.87 | 0.012982 | 20230621 | 4365.69 | −0.00525 |

20230322 | 3936.97 | −0.01646 | 20230622 | 4381.89 | 0.003711 |

20230323 | 3948.72 | 0.002985 | 20230623 | 4348.33 | −0.00766 |

20230324 | 3970.99 | 0.00564 | 20230626 | 4328.82 | −0.00449 |

20230327 | 3977.53 | 0.001647 | 20230627 | 4378.41 | 0.011456 |

20230328 | 3971.27 | −0.00157 | 20230628 | 4376.86 | −0.00035 |

20230329 | 4027.81 | 0.014237 | 20230629 | 4396.44 | 0.004474 |

20230330 | 4050.83 | 0.005715 | 20230630 | 4450.38 | 0.012269 |

20230331 | 4109.31 | 0.014437 |

**Table 2.**Best-fit parameter set to Table 1, only optimizing three parameters to the first six months in 2023 of daily S&P500 closing price changes. Location was set to the mean (and median, being symmetrical) of the sample data of 0.0029048 and the scale was kept at the sample standard deviation of 0.0105381.

Name | Values | |
---|---|---|

α | Concentration | 2.1 |

δ | Scale | 0.01053813 |

μ | Location | 0.0029048 |

λ | Shape | 1.64940218 |

**Table 3.**Sorted outcomes. Outcomes are labeled as columns o1…o7, and each row is one of 1000 of the M sets of N = 7 outcomes.

o1 | o2 | o3 | o4 | o5 | o6 | o7 | ∑o | ∏p | Cum Sum Probs | |
---|---|---|---|---|---|---|---|---|---|---|

0.01898 | −0.01143 | −0.02245 | −0.00848 | −0.00245 | 0.00988 | −0.04847 | −0.064417766 | 0.000000000791 | 0.000367729532 | |

−0.01148 | −0.01351 | −0.01114 | −0.00849 | −0.00428 | −0.0135 | −0.00353 | −0.065923437 | 0.000000015425 | 0.000361021142 | |

−0.023 | −0.00209 | −0.00094 | −0.01424 | −0.0132 | −0.01449 | −0.00144 | −0.069414377 | 0.000000020100 | 0.000230227464 | |

−0.00535 | −0.01323 | −0.00791 | −0.01091 | 0.00768 | −0.02873 | −0.01846 | −0.076917579 | 0.000000002948 | 0.000059794713 | |

−0.00817 | −0.0109 | −0.00531 | −0.01709 | −0.01115 | −0.0179 | −0.00953 | −0.080049157 | 0.000000003151 | 0.000034795240 | |

0.01658 | −0.0653 | −0.00424 | −0.01429 | −0.01636 | −0.01301 | 0.01241 | −0.084203152 | 0.000000000228 | 0.000008072898 | |

−0.00544 | −0.11487 | 0.01205 | −0.01222 | −0.00734 | 0.01483 | −0.00471 | −0.117685371 | 0.000000000724 | 0.000006137397 | <-- Previous Cum Sum Probs + current ∏p/SumOfProbs |

0 | ||||||||||

0.000117933 | <-- Sum of ∏p’s (SumOfProbs) |

o1 | o2 | o3 | o4 | o5 | o6 | o7 | ∑o | ∏p | Cum Sum Probs |
---|---|---|---|---|---|---|---|---|---|

−0.01076 | −0.0096 | 0.01428 | 0.03664 | −0.00401 | −0.01816 | 0.00873 | 0.017127894 | 0.000000003902 | 0.503768567507 |

−0.00529 | −0.00806 | 0.01028 | 0.01246 | −0.00586 | −0.00624 | 0.0196 | 0.016892532 | 0.000000096482 | 0.503735481158 |

−0.01023 | −0.00529 | −0.00461 | 0.01046 | 0.00852 | 0.01987 | −0.00196 | 0.01676511 | 0.000000268421 | 0.502917368488 |

−0.00667 | −0.00638 | −0.00662 | −0.00515 | 0.01318 | 0.01693 | 0.01147 | 0.016764245 | 0.000000106581 | 0.500641320047 |

0.00863 | 0.02348 | −0.00931 | −0.00549 | 0.01595 | −0.00801 | −0.00849 | 0.01676081 | 0.000000026072 | 0.499737576467 |

−0.00721 | −0.01002 | 0.02994 | −0.01062 | −0.00915 | −0.01151 | 0.03524 | 0.016677462 | 0.000000000506 | 0.499516499541 |

−0.00352 | 0.01047 | 0.02715 | −0.00517 | −0.01393 | −0.01018 | 0.01182 | 0.016631721 | 0.000000023517 | 0.499512210603 |

−0.0099 | 0.01316 | −0.00314 | 0.0184 | −0.00324 | 0.00952 | −0.00819 | 0.01661394 | 0.000000135827 | 0.499312798 |

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**MDPI and ACS Style**

Vince, R.
Expectation and Optimal Allocations in Existential Contests of Finite, Heavy-Tail-Distributed Outcomes. *Mathematics* **2024**, *12*, 11.
https://doi.org/10.3390/math12010011

**AMA Style**

Vince R.
Expectation and Optimal Allocations in Existential Contests of Finite, Heavy-Tail-Distributed Outcomes. *Mathematics*. 2024; 12(1):11.
https://doi.org/10.3390/math12010011

**Chicago/Turabian Style**

Vince, Ralph.
2024. "Expectation and Optimal Allocations in Existential Contests of Finite, Heavy-Tail-Distributed Outcomes" *Mathematics* 12, no. 1: 11.
https://doi.org/10.3390/math12010011