# New Insights on Hedge Ratios in the Presence of Stochastic Transaction Costs

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## Abstract

**:**

## 1. Introduction

## 2. Previous Studies

- (a)
- Learning cost. This cost corresponds to the time and effort invested into learning how futures markets and hedging work.
- (b)
- Exchange and brokerage fees. This is the most frequently addressed cost in the literature, and refers to the fees charged by the futures exchanges and brokerage houses for the execution of trades in the futures market.
- (c)
- Liquidity cost. This one refers to the cost of entering and exiting the market, and is closely related to the trading volume in a given futures contract. The bid-ask spread is commonly used to measure this cost.
- (d)
- Position management. When hedgers trade futures contracts, they need to dedicate time to monitor and manage their position in the futures market or pay someone to do so. Either way, there is a cost associated with the management of their position.
- (e)
- Initial margin deposit. The margin system in the futures market requires that all traders make an initial margin deposit when they first trade a contract. Hence, there is an opportunity cost of having funds tied to the margin system.
- (f)
- Margin calls. Traders can receive margin calls if the positions in the futures market start losing money. Hence, traders need to set aside some funds in order to meet those margin calls, which also corresponds to an opportunity cost.
- (g)
- Taxation. If traders make a profit in the futures market, they have to pay taxes on their earnings. Although specific tax regulations can vary across different countries, this can be seen as another cost involved in futures trading.

## 3. Transaction Costs in Brazil

## 4. Research Method

_{1}as defined in Equation (1), where W

_{0}is initial wealth in period t = 0, S

_{1}is the spot price in period t = 1, CP is the cost of production, Q is the quantity produced and sold by the producer, tax is the income tax rate on the spot position, F

_{0}and F

_{1}are the respective futures prices in periods t = 0 and t = 1, h is the hedge ratio, and TC is the total transaction cost involved in trading futures contracts.

_{1}/W

_{0}), final wealth can be expressed as W

_{1}= W

_{0}R. Thus, return R can be used as the argument of the utility function as in Equation (3), where θ is the coefficient of relative risk aversion (θ = αW

_{0}).

_{R}and ${\sigma}_{R}^{2}$ are the mean and variance of the return distribution, respectively.

_{1}/W

_{0}, Equation (1) can be algebraically manipulated such that R is given by Equation (5), where ${r}_{spot}=\left({S}_{1}-CP\right)/CP$ and ${r}_{fut}=\left({F}_{0}-{F}_{1}\right)/{F}_{0}$ are the respective returns on the spot and futures positions, tax is the income tax rate on the spot position, h is the hedge ratio, tc is the total transaction cost of trading futures contracts as a proportion of the initial futures price $\left(tc=TC\xb7{F}_{0}\xb7h\xb7Q\right)$ and $z={F}_{0}/CP$. The mean and variance of R are then given by Equations (6) and (7), where µ

_{spot}and ${\sigma}_{spot}^{2}$ are the mean and variance of spot return distribution, µ

_{fut}and ${\sigma}_{fut}^{2}$ are the mean and variance of the futures return distribution, µ

_{tc}and ${\sigma}_{tc}^{2}$ are the mean and variance of the futures transaction cost distribution, σ

_{spot,fut}is the covariance between spot and futures returns, σ

_{spot,tc}is the covariance between spot returns and transaction costs, and σ

_{fut,tc}is the covariance between futures returns and transaction costs.

_{spot,tc}= 0), the optimal hedge ratio is given by Equation (8). This expression will be used to calculate hedge ratios in the simulations performed in this study. Equation (8) incorporates the magnitude of transaction costs (${\mu}_{tc}$) as well as their stochastic nature (variance of transaction costs ${\sigma}_{tc}^{2}$, and covariance between transaction costs and futures returns ${\sigma}_{fut,tc}$).

_{fut}= 0)2, three hedge ratios can be discussed. First, in order to provide a benchmark for comparison, no transaction costs are assumed $\left({\mu}_{tc}=0,\text{}{\sigma}_{tc}^{2}=0,{\sigma}_{fut,tc}=0\right)$. In this case, the optimal hedge ratio is simply the minimum-variance hedge ratio $h=-{\sigma}_{fut,spot}/z{\sigma}_{fut}^{2}$. Given how returns are calculated, spot and futures returns are negatively correlated. Hence, the covariance between spot and futures returns is negative and the optimal hedge ratio is a positive number.

_{tc}) will reduce the magnitude of the numerator and thus lead to lower hedge ratios.

_{fut,tc}).

## 5. Data

_{fut}= 0). In fact, based on our data set, mean returns are not statistically distinguishable from zero. Assuming returns follow a normal distribution $N\left({\mu}_{fut},{\sigma}_{fut}\right)$, three probability distributions of futures returns are generated: ${N}_{1}\left(0,0.1\right)$, ${N}_{2}\left(0,0.2\right)$, and ${N}_{3}\left(0,0.3\right)$, which are used to simulate futures price trajectories in each hedging horizon. For each probability distribution 5000 futures price trajectories are simulated for each hedging horizon. At this point, it was necessary to choose the same starting point for all price trajectories, which was set at the average price observed in the sample collected for this study (R$90.00 per 15 kg). Each trajectory starts with F

_{0}= 90 and the second daily price is generated by randomly picking a daily return from the probability distribution and multiplying it by the initial price. The subsequent prices are also generated by randomly picking daily returns from the probability distribution and multiplying them by the price in the previous day. This process continues for 42 days in the 2-month hedging horizon and 84 days in the 4-month hedging horizon, and is repeated 5000 times for each probability distribution. At the end, there will be six sets of 5000 simulated futures price trajectories: 2-month hedging horizon with ${N}_{1}\left(0,0.1\right)$, 2-month hedging horizon with ${N}_{2}\left(0,0.2\right)$, 2-month hedging horizon with ${N}_{3}\left(0,0.3\right)$, 4-month hedging horizon with ${N}_{1}\left(0,0.1\right)$, 4-month hedging horizon with ${N}_{2}\left(0,0.2\right)$, and 4-month hedging horizon with ${N}_{3}\left(0,0.3\right)$.

_{1}= F

_{1}) and spot and futures returns are calculated as ${r}_{spot}=\left({S}_{1}-CP\right)/CP$ and ${r}_{fut}=\left({F}_{0}-{F}_{1}\right)/{F}_{0}$, respectively. Cost of production (CP) used to calculate spot returns were obtained from the Center for Advanced Studies on Applied Economics (CEPEA). The set of spot and futures returns calculated from the 5000 trajectories are used to generate the covariance between spot and futures returns, which is then used to calculate the hedge ratios in (8)–(10).

_{tax}) are calculated from all tax payments during the hedge. Opportunity costs are considered based on margin deposits required during the hedge. An interest cost of 11% (average rate observed in Brazil during the period in which prices were collected) is applied to the funds deposited by the producer during the hedge. The “margin cost” (C

_{margin}) is the sum of these costs during the hedge. Lastly, fees charged by brokers and futures exchanges are also considered in the calculation of transaction costs. Consistent with values charged from Brazilian hedgers, a fixed rate of 0.3% of the contract value is used to represent expenses with brokerage and exchange fees (C

_{fees}). The total transaction cost TC

_{i}for each price trajectory i in each hedging horizon is calculated as TC

_{i}= C

_{tax,i}+ C

_{margin,i}+ C

_{fees}. Costs with income tax and margins will vary according to the price trajectory, while expenses with fees are the same for all trajectories.

_{tc}) and variances (${\sigma}_{tc}^{2}$) for transaction costs that are used to find the optimal hedge ratio in (8)–(10). Covariances between futures returns and transaction costs $\left({\sigma}_{fut,tc}\right)$ needed to calculate the hedge ratio in (8)–(10) are obtained from the simulated price trajectories.

## 6. Results

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1 | In fact, Benninga et al. (1983) show that utility-maximizing hedge ratios are consistent with minimum-variance hedge ratios. |

2 | This assumption is based on previous studies (e.g., Lence 1996; Mattos et al. 2008) and follows the notion that futures prices are unbiased (Garcia and Leuthold 2004). |

**Table 1.**Mean and standard deviation of transaction costs in futures trading based on simulated price trajectories with annual price volatility of 10%

^{(a)}.

Income Tax | Margins Cost | Exchange Fees | Total Cost | |
---|---|---|---|---|

2-month hedge | ||||

mean (%) | 0.278 | 0.019 | 0.300 | 0.597 |

std. dev. (%) | 0.345 | 0.023 | 0.000 | 0.333 |

4-month hedge | ||||

mean (%) | 0.453 | 0.055 | 0.300 | 0.808 |

std. dev. (%) | 0.478 | 0.068 | 0.000 | 0.441 |

^{(a)}transaction costs are expressed as a percentage of the initial futures price.

**Table 2.**Mean and standard deviation of transaction costs in futures trading based on simulated price trajectories with annual price volatility of 20%

^{(a)}.

Income Tax | Margins Cost | Exchange Fees | Total Cost | |
---|---|---|---|---|

2-month hedge | ||||

mean (%) | 0.551 | 0.038 | 0.300 | 0.889 |

std. dev. (%) | 0.662 | 0.048 | 0.000 | 0.636 |

4-month hedge | ||||

mean (%) | 0.872 | 0.111 | 0.300 | 1.284 |

std. dev. (%) | 0.896 | 0.140 | 0.000 | 0.819 |

^{(a)}transaction costs are expressed as a percentage of the initial futures price.

**Table 3.**Mean and standard deviation of transaction costs in futures trading based on simulated price trajectories with annual price volatility of 30%

^{(a)}.

Income Tax | Margins Cost | Exchange Fees | Total Cost | |
---|---|---|---|---|

2-month hedge | ||||

mean (%) | 0.809 | 0.059 | 0.300 | 1.168 |

std. dev. (%) | 0.942 | 0.076 | 0.000 | 0.904 |

4-month hedge | ||||

mean (%) | 1.261 | 0.175 | 0.300 | 1.737 |

std. dev. (%) | 1.298 | 0.220 | 0.000 | 1.178 |

^{(a)}transaction costs are expressed as a percentage of the initial futures price.

Volatility of Simulated Price Trajectories | |||
---|---|---|---|

10% | 20% | 30% | |

No costs (standard minimum-variance hedge) | |||

2-month hedge | 0.976 | 0.992 | 0.992 |

4-month hedge | 0.995 | 1.000 | 1.000 |

Income tax on spot position; no futures costs in futures trading | |||

2-month hedge | 0.721 | 0.727 | 0.770 |

4-month hedge | 0.710 | 0.759 | 0.791 |

Volatility of Simulated Price Trajectories | |||
---|---|---|---|

10% | 20% | 30% | |

Deterministic transaction costs in futures trading | |||

2-month hedge | 0.000 | 0.596 | 0.761 |

4-month hedge | 0.276 | 0.717 | 0.841 |

Deterministic transaction costs in futures trading and income tax on spot position | |||

2-month hedge | 0.000 | 0.332 | 0.539 |

4-month hedge | 0.000 | 0.473 | 0.619 |

Volatility of Simulated Price Trajectories | |||
---|---|---|---|

10% | 20% | 30% | |

Stochastic transaction costs in futures trading | |||

2-month hedge | 0.000 | 0.671 | 0.848 |

4-month hedge | 0.305 | 0.784 | 0.907 |

Stochastic transaction costs in futures trading and income tax on spot position | |||

2-month hedge | 0.000 | 0.373 | 0.601 |

4-month hedge | 0.000 | 0.518 | 0.668 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Andrade, E.; Mattos, F.; Lima, R.A.d.S.
New Insights on Hedge Ratios in the Presence of Stochastic Transaction Costs. *Risks* **2018**, *6*, 118.
https://doi.org/10.3390/risks6040118

**AMA Style**

Andrade E, Mattos F, Lima RAdS.
New Insights on Hedge Ratios in the Presence of Stochastic Transaction Costs. *Risks*. 2018; 6(4):118.
https://doi.org/10.3390/risks6040118

**Chicago/Turabian Style**

Andrade, Elisson, Fabio Mattos, and Roberto Arruda de Souza Lima.
2018. "New Insights on Hedge Ratios in the Presence of Stochastic Transaction Costs" *Risks* 6, no. 4: 118.
https://doi.org/10.3390/risks6040118