finance_solution/tvm/

rate.rs

//! **Periodic rate calculations.** Given an initial investment amount, a final amount, and a number
//! of periods what does the rate per period need to be?
//!
//! For most common usages, we recommend the [`rate_solution`](./fn.rate_solution.html) function to provide the best experience with debugging and additional features.
//!
// ! If you need to calculate the future value given a starting value, a number of periods, and one
// ! or more rates use [`future_value`] or related functions.
// !
// ! If you need to calculate the present value given a future value, a number of periods, and one
// ! or more rates use [`present_value`] or related functions.
// !
// ! If you need to calculate the number of periods given a fixed rate and a present and future value
// ! use [`periods`] or related functions.
//! # Formulas
//!
//! ## Simple Compounding
//!
//! With simple compound interest the rate is calculated with:
//!
//! > <img src="http://i.upmath.me/svg/rate%20%3D%20%5Csqrt%5Bperiods%5D%7B%5Cfrac%7Bfuture%5C_value%7D%7Bpresent%5C_value%7D%7D%20-%201" />
//!
//! Or using a few more common variable names:
//!
//! > <img src="http:i.upmath.me/svg/r%20%3D%20%5Csqrt%5Bn%5D%7B%5Cfrac%7Bfv%7D%7Bpv%7D%7D%20-%201" />
//!
//! `r` is the periodic rate, though this may appear as `i` for interest. `n` is often used for the
//! number of periods, though it may be `t` for time if each period is assumed to be one year as in
//! continuous compounding.
//!
//! Throughout this crate we use `pv` for present value and `fv` for future value. You may see these
//! values called `P` for principal in some references.
//!
//! ## Continuous Compounding
//!
//! With continuous compounding the formula is:
//!
//! > <img src="http://i.upmath.me/svg/rate%20%3D%20%5Cfrac%7B%5Cln%5Cleft(%5Cfrac%7Bfuture%5C_value%7D%7Bpresent%5C_value%7D%5Cright)%7D%7Bperiods%7D" />
//!
//! or:
//!
//! > <img src="http://i.upmath.me/svg/r%20%3D%20%5Cfrac%7B%5Cln%5Cleft(%5Cfrac%7Bfv%7D%7Bpv%7D%5Cright)%7Dn" />
//!
//! With continuous compounding the period is assumed to be years and `t` (time) is often used as
//! the variable name. Within this crate we stick with `n` for the number of periods so that all of
//! the functions use the same variables.
use crate::*;

/// Returns the periodic rate of an investment given the number of periods along with the present
/// and future values.
///
/// See the [rate](./index.html) module page for the formulas.
///
/// Related functions:
/// * To calculate a periodic rate and return a struct that shows the formula and optionally
/// produces the the period-by-period values use [`rate_solution`].
///
/// # Arguments
/// * `periods` - The number of periods such as quarters or periods. Often appears as `n` or `t`.
/// * `present_value` - The starting value of the investment. May appear as `pv` in formulas, or `C`
/// for cash flow or `P` for principal.
/// * `future_value` - The final value of the investment.
/// * `continuous_compounding` - True for continuous compounding, false for simple compounding.
///
/// If present_value and future_value are both zero then any rate will work so the function returns
/// zero.
///
/// # Errors
/// The call returns [`FinanceError`] if the present value is zero and the future value is nonzero or vice versa.
/// It also returns an error if the number of periods is zero and the present value is not equal to the
/// future value. In both cases this is because there's no periodic rate that could make that work.
///
/// # Examples
/// ```
/// use finance_solution::*;
///
/// // The interest will compound for 365 days.
/// let periods = 365;
///
/// // The starting value is $10,000.
/// let present_value = -10_000.00;
///
/// // The ending value is $11,000.
/// let future_value = 11_000.00;
///
/// let continuous_compounding = false;
///
/// // Calculate the periodic rate needed.
/// let rate = rate(periods, present_value, future_value, continuous_compounding).unwrap();
/// dbg!(&rate);
/// // The rate is 0.0261% per day.
/// assert_rounded_6(0.000261, rate);
/// ```
/// # Errors
/// Returns [`FinanceError::SameSignValues`], [`FinanceError::Unsolvable`], or
/// [`FinanceError::NonFinite`] when inputs cannot determine a rate.
///
/// # Examples
/// ```
/// use finance_solution::{rate, FinanceError};
///
/// let r = rate(365, -10_000.0, 11_000.0, false).unwrap();
/// assert!(r > 0.0);
///
/// match rate(12, 1_000.0, 1_100.0, false) {
///     Err(FinanceError::SameSignValues { .. }) => {}
///     other => panic!("expected SameSignValues, got {other:?}"),
/// }
/// ```
pub fn rate<P, F, C>(
    periods: u32,
    present_value: P,
    future_value: F,
    compounding: C,
) -> crate::FinanceResult<f64>
where
    P: Into<f64> + Copy,
    F: Into<f64> + Copy,
    C: Into<crate::Compounding>,
{
    rate_internal(
        periods,
        present_value.into(),
        future_value.into(),
        compounding.into().is_continuous(),
    )
}

/// Returns the periodic rate of an investment given the number of periods along with the present
/// and future values.
///
/// See the [rate](./index.html) module page for the formulas.
///
/// Related functions:
/// * To calculate a periodic rate returning an f64 value instead of solution object, use [`rate`](./fn.rate.html).
///
/// # Arguments
/// * `periods` - The number of periods such as quarters or periods. Often appears as `n` or `t`.
/// * `present_value` - The starting value of the investment. May appear as `pv` in formulas, or `C`
/// for cash flow or `P` for principal.
/// * `future_value` - The final value of the investment.
/// * `continuous_compounding` - True for continuous compounding, false for simple compounding.
///
/// If present_value and future_value are both zero then any rate will work so the function returns
/// zero.
///
/// # Errors
/// The call returns [`FinanceError`] if the present value is zero and the future value is nonzero or vice versa.
/// It also returns an error if the number of periods is zero and the present value is not equal to the
/// future value. In both cases this is because there's no periodic rate that could make that work.
///
/// # Examples
/// Calculate a periodic rate and examine the period-by-period values.
/// ```
/// use finance_solution::*;
/// // The interest will compound for ten periods.
/// // The starting value is $10,000.
/// // The ending value is $15,000.
/// let periods = 10;
/// let present_value = -10_000.00;
/// let future_value = 15_000.00;
/// let continuous_compounding = false;
/// /// // Calculate the periodic rate and create a struct with a record of the
/// // inputs, a description of the formula, and an option to calculate the
/// // period-by-period values.
/// let solution = rate_solution(periods, present_value, future_value, continuous_compounding).unwrap();
/// dbg!(&solution);
///
/// let rate = solution.rate();
/// dbg!(&rate);
/// // The rate is 4.138% per period.
/// assert_rounded_6(0.041380, rate);
///
/// // Examine the formulas.
/// let formula = solution.formula();
/// dbg!(&formula);
/// assert_eq!("0.041380 = ((-15000.0000 / -10000.0000) ^ (1 / 10)) - 1", formula);
/// let symbolic_formula = solution.symbolic_formula();
/// dbg!(&symbolic_formula);
/// assert_eq!("r = ((-fv / pv) ^ (1 / n)) - 1", symbolic_formula);
///
/// // Calculate the period-by-period values.
/// let series = solution.series();
/// dbg!(&series);
/// ```
pub fn rate_solution<P, F, C>(
    periods: u32,
    present_value: P,
    future_value: F,
    compounding: C,
) -> crate::FinanceResult<TvmSolution>
where
    P: Into<f64> + Copy,
    F: Into<f64> + Copy,
    C: Into<crate::Compounding>,
{
    rate_solution_internal(
        periods,
        present_value.into(),
        future_value.into(),
        compounding.into().is_continuous(),
    )
}

fn rate_internal(
    periods: u32,
    present_value: f64,
    future_value: f64,
    continuous_compounding: bool,
) -> crate::FinanceResult<f64> {
    crate::util::error::require_finite("present_value", present_value)?;
    crate::util::error::require_finite("future_value", future_value)?;
    if present_value + future_value == 0.0 {
        return Ok(0.0);
    }
    if future_value == 0.0 {
        return Ok(-1.0);
    }
    if (present_value < 0.0 && future_value < 0.0) || (present_value > 0.0 && future_value > 0.0) {
        return Err(crate::FinanceError::SameSignValues {
            present_value,
            future_value,
        });
    }
    if present_value == 0.0 && future_value != 0.0 {
        return Err(crate::FinanceError::Unsolvable {
            message: "present value is zero and future value is nonzero; cannot solve for rate",
        });
    }
    if periods == 0 && present_value + future_value != 0.0 {
        return Err(crate::FinanceError::Unsolvable {
            message: "periods is zero and present + future value is nonzero; cannot solve for rate",
        });
    }
    let rate = if continuous_compounding {
        (-future_value / present_value).ln() / periods as f64
    } else {
        (-future_value / present_value).powf(1.0 / periods as f64) - 1.0
    };
    if rate.is_finite() {
        Ok(rate)
    } else {
        Err(crate::FinanceError::NonFinite {
            field: "rate",
            value: rate,
        })
    }
}

pub(crate) fn rate_solution_internal(
    periods: u32,
    present_value: f64,
    future_value: f64,
    continuous_compounding: bool,
) -> crate::FinanceResult<TvmSolution> {
    if present_value == 0.0 && future_value == 0.0 {
        let formula = "{special case}";
        let symbolic_formula = "***";
        let rate = 0.0;
        return Ok(TvmSolution::new(
            TvmVariable::Rate,
            continuous_compounding,
            rate,
            periods,
            present_value,
            future_value,
            formula,
            symbolic_formula,
        ));
    }

    let rate = rate_internal(periods, present_value, future_value, continuous_compounding)?;
    let (formula, symbolic_formula) = if continuous_compounding {
        let formula = format!(
            "{:.6} = ln({:.4} / {:.4}) / {}",
            rate, -future_value, present_value, periods
        );
        let symbolic_formula = "r = ln(-fv / pv) / t";
        (formula, symbolic_formula)
    } else {
        let formula = format!(
            "{:.6} = (({:.4} / {:.4}) ^ (1 / {})) - 1",
            rate, -future_value, present_value, periods
        );
        let symbolic_formula = "r = ((-fv / pv) ^ (1 / n)) - 1";
        (formula, symbolic_formula)
    };
    Ok(TvmSolution::new(
        TvmVariable::Rate,
        continuous_compounding,
        rate,
        periods,
        present_value,
        future_value,
        &formula,
        symbolic_formula,
    ))
}

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn test_rate_edge() {
        // Zero periods, values add up to zero.
        assert_rounded_6(0.0, rate(0, 10_000.0, -10_000.0, false).unwrap());

        // Nonzero periods, values add up to zero.
        assert_rounded_6(0.0, rate(12, -10_000.0, 10_000.0, false).unwrap());
    }

    #[test]
    fn test_rate_err_present_value_nan() {
        // The present value is not a number.
        assert!(rate(12, std::f64::NAN, 1_000.0, false).is_err());
    }

    #[test]
    fn test_rate_err_present_value_inf() {
        // The present value is infinite.
        assert!(rate(12, std::f64::INFINITY, 1_000.0, false).is_err());
    }

    #[test]
    fn test_rate_err_future_value_nan() {
        // The future value is not a number.
        assert!(rate(12, 1_000.0, std::f64::NAN, false).is_err());
    }

    #[test]
    fn test_rate_err_future_value_inf() {
        // The future value is infinite.
        assert!(rate(12, 1_000.0, std::f64::NEG_INFINITY, false).is_err());
    }

    #[test]
    fn test_rate_err_zero_periods() {
        // Zero periods, values don't add up to zero.
        assert!(rate(0, 10_000.0, 10_000.0, false).is_err());
    }

    /*
    macro_rules! compare_to_excel {
        ( $n:expr, $pv:expr, $fv:expr, $r_excel:expr, $r_manual_simple:expr, $r_manual_cont:expr ) => {
            println!("$n = {}, $pv = {}, $fv = {}, $r_excel: {}, $r_manual_simple = {}, $r_manual_cont = {}", $n, $pv, $fv, $r_excel, $r_manual_simple, $r_manual_cont);
            assert_approx_equal!($r_excel, $r_manual_simple);

            let r_calc_simple = rate($n, $pv, $fv, false).unwrap();
            println!("r_calc_simple = {}", r_calc_simple);
            assert_approx_equal!($r_excel, r_calc_simple);

            let r_calc_cont = rate($n, $pv, $fv, true).unwrap();
            println!("r_calc_cont = {}", r_calc_cont);
            assert_approx_equal!($r_manual_cont, r_calc_cont);

            if is_approx_equal!(0.0, r_calc_simple) {
                assert_approx_equal!(0.0, r_calc_cont);
            } else {
                let ratio = r_calc_cont / r_calc_simple;
                println!("ratio = {}", ratio);
                if $r_excel < 0.0 {
                    assert!(ratio >= 1.0);
                    assert!(ratio <= 2.0);
                } else {
                    assert!(ratio >= 0.0);
                    assert!(ratio <= 1.0);
                }
            }
        }
    }
    */

    fn compare_to_excel(
        test_case: usize,
        n: u32,
        pv: f64,
        fv: f64,
        r_excel: f64,
        r_manual_simple: f64,
        r_manual_cont: f64,
    ) {
        let display = false;

        if display {
            println!("test_case = {}, n = {}, pv = {}, fv = {}, r_excel: {}, r_manual_simple = {}, r_manual_cont = {}", test_case, n, pv, fv, r_excel, r_manual_simple, r_manual_cont)
        }
        assert_approx_equal!(r_excel, r_manual_simple);

        let r_calc_simple = rate(n, pv, fv, false).unwrap();
        if display {
            println!("r_calc_simple = {}", r_calc_simple)
        }
        assert_approx_equal!(r_excel, r_calc_simple);

        let r_calc_cont = rate(n, pv, fv, true).unwrap();
        if display {
            println!("r_calc_cont = {}", r_calc_cont);
        }
        assert_approx_equal!(r_manual_cont, r_calc_cont);

        if is_approx_equal!(0.0, r_calc_simple) {
            assert_approx_equal!(0.0, r_calc_cont);
        } else {
            let ratio = r_calc_cont / r_calc_simple;
            if display {
                println!("ratio = {}", ratio)
            };
            if r_excel < 0.0 {
                assert!(ratio >= 1.0);
                assert!(ratio <= 2.0);
            } else {
                assert!(ratio >= 0.0);
                assert!(ratio <= 1.0);
            }
        }

        // Solution with simple compounding.
        let solution = rate_solution(n, pv, fv, false).unwrap();
        if display {
            dbg!(&solution);
        }
        solution.invariant();
        assert!(solution.calculated_field().is_rate());
        assert_eq!(false, solution.continuous_compounding());
        assert_approx_equal!(r_excel, solution.rate());
        assert_eq!(n, solution.periods());
        assert_approx_equal!(n as f64, solution.fractional_periods());
        assert_approx_equal!(pv, solution.present_value());
        assert_approx_equal!(fv, solution.future_value());

        // Solution with continuous compounding.
        let solution = rate_solution(n, pv, fv, true).unwrap();
        if display {
            dbg!(&solution);
        }
        solution.invariant();
        assert!(solution.calculated_field().is_rate());
        assert!(solution.continuous_compounding());
        assert_approx_equal!(r_manual_cont, solution.rate());
        assert_eq!(n, solution.periods());
        assert_approx_equal!(n as f64, solution.fractional_periods());
        assert_approx_equal!(pv, solution.present_value());
        assert_approx_equal!(fv, solution.future_value());
    }

    #[test]
    fn test_rate_against_excel() {
        compare_to_excel(
            1,
            90,
            -0.1f64,
            1f64,
            0.0259143654700119f64,
            0.0259143654700098f64,
            0.025584278811045f64,
        );
        compare_to_excel(
            2,
            85,
            1.05f64,
            -1.5f64,
            0.00420499208399443f64,
            0.00420499208399305f64,
            0.00419617581104391f64,
        );
        compare_to_excel(
            3,
            80,
            -2.25f64,
            2.25f64,
            8.10490132135311E-16f64,
            0f64,
            0f64,
        );
        compare_to_excel(
            4,
            75,
            4.3875f64,
            -3.375f64,
            -0.00349207865283533f64,
            -0.00349207865410572f64,
            -0.00349819019289988f64,
        );
        compare_to_excel(
            5,
            70,
            -10.125f64,
            5.0625f64,
            -0.00985323817917932f64,
            -0.00985323818144335f64,
            -0.00990210257942779f64,
        );
        compare_to_excel(
            6,
            65,
            0.759375f64,
            -7.59375f64,
            0.0360593046264088f64,
            0.0360593046256343f64,
            0.0354243860460622f64,
        );
        compare_to_excel(
            7,
            60,
            -7.9734375f64,
            11.390625f64,
            0.00596228650143506f64,
            0.00596228649269048f64,
            0.00594458239897887f64,
        );
        compare_to_excel(
            8,
            55,
            17.0859375f64,
            -17.0859375f64,
            4.31995919490311E-13f64,
            0f64,
            0f64,
        );
        compare_to_excel(
            9,
            50,
            -33.317578125f64,
            25.62890625f64,
            -0.00523354233611077f64,
            -0.00523354233613538f64,
            -0.00524728528934982f64,
        );
        compare_to_excel(
            10,
            45,
            76.88671875f64,
            -38.443359375f64,
            -0.0152852470655657f64,
            -0.0152852470655688f64,
            -0.0154032706791099f64,
        );
        compare_to_excel(
            11,
            40,
            -5.76650390625f64,
            57.6650390625f64,
            0.0592537251772898f64,
            0.0592537251772889f64,
            0.0575646273248511f64,
        );
        compare_to_excel(
            12,
            35,
            60.548291015625f64,
            -86.49755859375f64,
            0.010242814832087f64,
            0.0102428148320715f64,
            0.0101907126839638f64,
        );
        compare_to_excel(
            13,
            30,
            -129.746337890625f64,
            129.746337890625f64,
            2.53808542775445E-15f64,
            0f64,
            0f64,
        );
        compare_to_excel(
            14,
            25,
            253.005358886719f64,
            -194.619506835937f64,
            -0.0104396946981842f64,
            -0.0104396947068867f64,
            -0.0104945705786996f64,
        );
        compare_to_excel(
            15,
            20,
            -583.858520507812f64,
            291.929260253906f64,
            -0.0340636710696604f64,
            -0.0340636710751544f64,
            -0.0346573590279973f64,
        );
        compare_to_excel(
            16,
            15,
            43.7893890380859f64,
            -437.893890380859f64,
            0.165914401180033f64,
            0.165914401179832f64,
            0.15350567286627f64,
        );
        compare_to_excel(
            17,
            12,
            -459.788584899902f64,
            656.840835571289f64,
            0.0301690469250706f64,
            0.0301690469166949f64,
            0.0297229119948944f64,
        );
        compare_to_excel(
            18,
            10,
            985.261253356933f64,
            -985.261253356933f64,
            3.26110008113999E-16f64,
            0f64,
            0f64,
        );
        compare_to_excel(
            19,
            7,
            -1921.25944404602f64,
            1477.8918800354f64,
            -0.0367869049970667f64,
            -0.0367869049970667f64,
            -0.0374806092096416f64,
        );
        compare_to_excel(
            20,
            5,
            4433.6756401062f64,
            -2216.8378200531f64,
            -0.129449436703859f64,
            -0.129449436703876f64,
            -0.138629436111989f64,
        );
        compare_to_excel(
            21,
            4,
            -332.525673007965f64,
            3325.25673007965f64,
            0.778279410038923f64,
            0.778279410038923f64,
            0.575646273248511f64,
        );
        compare_to_excel(
            22,
            3,
            3491.51956658363f64,
            -4987.88509511947f64,
            0.126247880443697f64,
            0.126247880443606f64,
            0.118891647979577f64,
        );
        compare_to_excel(
            23,
            2,
            -7481.82764267921f64,
            7481.82764267921f64,
            -9.19973496824152E-17f64,
            0f64,
            0f64,
        );
        compare_to_excel(
            24,
            1,
            14589.5639032245f64,
            -11222.7414640188f64,
            -0.230769230769231f64,
            -0.230769230769231f64,
            -0.262364264467491f64,
        );
    }
}