# Obtaining Accurate Gold Prices

## Abstract

**:**

## 1. Introduction

_{t}and P

_{t−1}are the prices of two adjacent periods, µ and σ are the drift and diffusion terms, respectively, proxied by the averaged return and the standard deviation, Δt is the change in time, while ε is a random variable with a mean of zero and a standard deviation of 1. Refs. [27,30,31] provide a good understanding of the GBM process. Generally, it is assumed that the drift and diffusion terms would remain constant, but Ref. [30] pointed out that this assumption could be weakened. Accordingly, Ref. [18] estimated the drift and diffusion terms using a twenty-year-long rolling window, essentially obtaining a unique set of drift and diffusion terms, at the start of every forecasting period. Moreover, by randomly altering the numerical values of ε, many different numerical values of P

_{t}can be obtained. From the universe of simulated prices, associated probabilities for those values can be obtained, and expected prices can be estimated by summing up products of these simulated prices and their multiplication with the associated probabilities. The expected price obtained in such a way can be compared to the observed price at monthly, quarterly, and yearly frequencies. A similar approach to forecasting stock indices was used in Refs. [18,32,33], where results indicate these forecasts to be reliable. In the current manuscript, the methodology used in Refs. [18,32,33] is applied to gold prices, with similar results, in that the expected prices can provide reliable forecasts of the actual prices.

## 2. Data

## 3. Methodology

_{t,sim}is the simulated prices, while P

_{t−1}is the last period’s actual closing price, which also serves as the beginning price for the period in which the simulation is carried out. ${\overline{\mu}}_{t}$ and ${\sigma}_{t}$ are the average return and the standard deviation, respectively, used to proxy the drift and diffusion terms, calculated using the previous 20 years, 80 quarters, and 240 months for the yearly, quarterly, and monthly bases, respectively. While there are a number of ways in which the drift and diffusion terms can be estimated, using historical averages and standard deviation as the proxies in the GBM framework has been applied in a number of manuscripts [12,28,32,33,34,35,36].

_{t}are obtained by arbitrarily changing the values of ε in each period. For each simulated gold price, the return is estimated using Equation (9). The associated probability of each simulated return is estimated by calculating the difference between the cumulative density function of two adjacent simulated returns, assuming a normal distribution with the mean and standard deviation estimated using Equations (7) and (8), respectively.

_{t,actual}is the actual gold price at the end of a period, while P

_{t,exp}is the GBM-simulated expected price for that period. α is the intercept, while β is the slope coefficient. The regression error term is represented by $\u03f5.$ In Equation (11), the natural logarithms of both the actual and the expected prices are used in the regressions. This equation can be used to test whether the expected price is a good reliable forecast of the actual price. Alternatively, it could also be used as a predictive model for the actual price using the expected price as the independent variable. Values of 0 for α and one for β would indicate the usefulness of the expected price. A similar logic was used in Refs. [18,32,33] as well as by Ref. [37] who tested the persistence of mutual fund performance between two adjacent periods. The regression represented in Equation (11) was carried out at annual, quarterly, and monthly frequencies in our study, and the results are presented in Table 2.

_{1}= n

_{2}, and the difference between the two means, (µ

_{1}–µ

_{2}), is hypothesized to not be different from zero. Table 3 presents the results of these tests.

## 4. Results

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Expected and actual gold prices at an annual frequency. The figure plots the actual vs. the expected prices at an annual frequency. The prices on the y-axis are the actual and expected prices. The expected price is obtained using the manuscript’s methodological procedure.

**Figure 2.**Expected and actual gold prices at a quarterly frequency. The figure plots the actual vs. the expected prices at a quarterly frequency. The prices on the y-axis are the actual and expected prices. The expected price is obtained using the manuscript’s methodological procedure.

**Figure 3.**Expected and actual gold prices at a monthly frequency. The figure plots the actual vs. the expected prices at a monthly frequency. The prices on the y-axis are the actual and expected prices. The expected price is obtained using the manuscript’s methodological procedure.

**Figure 4.**Expected vs. actual gold prices at an annual frequency. This figure plots the natural logarithms of the observed prices (y-axis) against the natural logarithms of the expected prices (x-axis) estimated using the manuscript’s methodology at an annual frequency.

**Figure 5.**Expected vs. actual gold prices at a quarterly frequency. This figure plots the natural logarithms of the observed prices (y-axis) against the natural logarithms of the expected prices (x-axis) estimated using the manuscript’s methodology at a quarterly frequency.

**Figure 6.**Expected vs. actual gold prices at a monthly frequency. This figure plots the natural logarithms of the observed prices (y-axis) against the natural logarithms of the expected prices (x-axis) estimated using the manuscript’s methodology at a monthly frequency.

Date | Price | |||||
---|---|---|---|---|---|---|

First date | 29 December 1978 | 226.38 | ||||

Last date | 29 December 2023 | 2065.45 | ||||

Lowest price | 15 January 1979 | 216.63 | ||||

Highest price | 27 December 2023 | 2078.95 | ||||

Annual | Quarterly | Monthly | ||||

Mean | 4.91% | 1.23% | 0.41% | |||

Standard Deviation | 19.51% | 8.16% | 5.09% |

Frequency | |||
---|---|---|---|

Annual | Quarterly | Monthly | |

Intercept | 0.30 | 0.08 | 0.03 |

Standard Error | (0.0243) | (0.0024) | (0.0005) |

t-stat (for integer = 0) | 12.5 | 32.35 | 59.99 |

Expected Price | 0.95 | 0.99 | 0.99 |

Standard Error | (0.0036) | (0.00036) | (0.00007) |

t-stat (for coefficient = 0) | 266.08 | 2748.16 | 14,742.02 |

t-stat (for coefficient = 1) | 14.73 | 37.89 | 70.41 |

N | 25 | 100 | 300 |

R-square | 99.97% | 100.00% | 100.00% |

Adjusted R-Square | 99.97% | 100.00% | 100.00% |

Annual | Quarterly | Monthly | ||||
---|---|---|---|---|---|---|

Expected | Actual | Expected | Actual | Expected | Actual | |

Mean | 6.78 | 6.73 | 6.77 | 6.76 | 6.77 | 6.76 |

Variance | 0.51 | 0.46 | 0.46 | 0.45 | 0.46 | 0.45 |

Testing Differences in Variances | ||||||

F-stat | 1.11 | 1.03 | 1.01 | |||

F-critical (one-tail) | 1.98 | 1.39 | 1.21 | |||

F-critical (two-tail) | 2.90 | 1.68 | 1.35 | |||

p-value | 0.40 | 0.45 | 0.47 | |||

Testing Differences in Means | ||||||

Hypothesized Difference | 0.00 | 0.00 | 0.00 | |||

Pooled Variance | 0.48 | 0.45 | 0.45 | |||

t-stat | 0.27 | 0.14 | 0.08 | |||

t-critical | 2.01 | 1.97 | 1.96 | |||

p-value | 0.79 | 0.89 | 0.93 |

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**MDPI and ACS Style**

Sinha, A.K.
Obtaining Accurate Gold Prices. *Commodities* **2024**, *3*, 115-126.
https://doi.org/10.3390/commodities3010008

**AMA Style**

Sinha AK.
Obtaining Accurate Gold Prices. *Commodities*. 2024; 3(1):115-126.
https://doi.org/10.3390/commodities3010008

**Chicago/Turabian Style**

Sinha, Amit K.
2024. "Obtaining Accurate Gold Prices" *Commodities* 3, no. 1: 115-126.
https://doi.org/10.3390/commodities3010008