beta_parameters | Compute the parameters of the beta distribution and plot normalized data. |

FV_post_artan | Compute the final expected value of an n-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate X, using the tetraparametric function approach. |

FV_post_beta_kmom | Compute the final expected value of an n-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate X, using the estimated moments of the beta distribution. |

FV_post_mood | Compute the final expected value of an n-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate X, using the method of Mood _et al._ |

FV_post_norm_kmom | Compute the final expected value of an n-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate X, using the estimated moments of the normal distribution. |

FV_post_quad | Compute the final expected value of an n-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate X, using the quadratic discount method. |

FV_pre_artan | Compute the final expected value of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate X, using the tetraparametric function approach. |

FV_pre_beta_kmom | Compute the final expected value of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate X, using the estimated moments of the beta distribution. |

FV_pre_mood | Compute the final expected value of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate X, using the method of Mood _et al._ |

FV_pre_norm_kmom | Compute the final expected value of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate X, using the estimated moments of the normal distribution. |

FV_pre_quad | Compute the final expected value of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate X, using the quadratic discount method. |

moment | Compute the exact moments of a distribution. |

norm_mom | Fit the data to a normal curve and compute the moments of the normal distribution according to the definition (as integral). |

norm_test_jb | Compute the Jarque-Bera test for checking the assumption of normality of the interest rates distribution and returns the parameters of the fitted normal distribution. |

plot_FVs_post | Plot the final expected values of an n-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate X, using different approaches. |

plot_FVs_pre | Plot the final expected values of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate X, using different approaches. |

plot_FV_post_beta_kmom | Plot the final expected value of an n-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate X, using the estimated moments of the beta distribution. |

plot_FV_post_norm_kmom | Plot the final expected value of an n-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate X, using the estimated moments of the normal distribution. |

plot_FV_pre_beta_kmom | Plot the final expected value of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate X, using the estimated moments of the beta distribution. |

plot_FV_pre_norm_kmom | Plot the final expected value of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate X, using the estimated moments of the normal distribution. |

plot_PVs_post | Plot the present expected values of an n-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate X, using different approaches. |

plot_PVs_pre | Plot the present expected values of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate X, using different approaches. |

PV_post_artan | Compute present expected value of an n-payment annuity, with payments of 1 unit each, made at the end of every year (annuity-immediate), valued at the rate X, using the tetraparametric function approach. |

PV_post_cubic | Compute the present expected value of an n-payment annuity, with payments of 1 unit each made at the end of every year (annuity-due), valued at the rate X, using the cubic discount method. |

PV_post_exact | Computes the present value of an annuity-immediate considering only non-central moments of negative orders. |

PV_post_mood_nm | Compute the present expected value of an n-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate X, with the method of Mood _et al._ using some negative moments of the distribution. |

PV_post_mood_pm | Compute the present expected value of an n-payment annuity, with payments of 1 unit each made at the end of every year (annuity-immediate), valued at the rate X, with the method of Mood _et al._ using some positive moments of the distribution. |

PV_post_triang_3 | Compute the present value of an annuity-immediate considering only non-central moments of negative orders. The calculation is performed by using the function triangular\_moments\_3 for the moments greater than -2 (in absolute value). |

PV_post_triang_dis | Compute the present value of an annuity-immediate considering only non-central moments of negative orders. The calculation is performed by using the moments of the fitted triangular distribution of the random variable "capitalization factor" U (which are obtained from the definition of negative moment of a continuous random variable). |

PV_pre_artan | Compute the present expected value of an n-payment annuity, with payments of 1 unit each, made at the beginning of every year (annuity-due), valued at the rate X, using the tetraparametric function approach. |

PV_pre_cubic | Compute the present expected value of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate X, using the cubic discount method. |

PV_pre_exact | Compute the present value of an annuity-due considering only non-central moments of negative orders. |

PV_pre_mood_nm | Compute the present expected value of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate X, with the method of Mood _et al._ using some negative moments of the distribution. |

PV_pre_mood_pm | Compute the present expected value of an n-payment annuity, with payments of 1 unit each made at the beginning of every year (annuity-due), valued at the rate X, with the method of Mood _et al._ using some positive moments of the distribution. |

PV_pre_triang_3 | Compute the present value of an annuity-due considering only non-central moments of negative orders. The calculation is performed by using the function $triangular\_moments\_3$ for the moments greater than -2 (in absolute value). |

PV_pre_triang_dis | Compute the present value of an annuity-due considering only non-central moments of negative orders. The calculation is performed by using the moments of the fitted triangular distribution of the random variable "capitalization factor" U (which are obtained from the definition of negative moment of a continuous random variable) |

triangular_moments_3 | Compute the negatives moments (different from orders 1 and 2) of the fitted triangular distribution of the random variable X. |

triangular_moments_3_U | Compute the negatives moments (different from orders 1 and 2) of the fitted triangular distribution of the random variable "capitalization factor" U. |

triangular_moments_dis | Compute the negative moments of the fitted triangular distribution of the random variable X according to the definition (as integral). |

triangular_moments_dis_U | Compute the negative moments of the fitted triangular distribution of the random variable "capitalization factor" U according to the definition (as integral). |

triangular_parameters | Compute the parameters and plot the fitted triangular distribution of the random variable X. |

triangular_parameters_U | Return the parameters of the fitted triangular distribution of the random variable "capitalization factor" U. |

variance_drv | Compute the variance of the present value of an annuity using "discrete random variable" approach. |

variance_post_mood_nm | Compute the variance of the present value of an annuity-immediate using the Mood _et al._ approximation and some non-central moments of negative order. |

variance_post_mood_pm | Compute the variance of the present value of an annuity-immediate using the Mood _et al._ approximation and some non-central moments of positive order. |

variance_pre_mood_nm | Compute the variance of the present value of an annuity-due using the Mood _et al._ approximation and some non-central moments of negative order. |

variance_pre_mood_pm | Compute the variance of the present value of an annuity-due using the Mood _et al._ approximation and some non-central moments of positive order. |