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//! **Present value calculations.** Given a final amount, a number of periods such as years, and fixed //! or varying interest rates, what is the current value? //! //! For most common usages, we recommend the [`present_value_solution`](./fn.present_value_solution.html) function to provide a better debugging experience and additional features. //! //! If you have a more complicated use case which has varying rates per period, use the [`present_value_schedule_solution`](./fn.present_value_schedule_solution.html) function. //! // ! If you need to calculate the future value given a present value, a number of periods, and one // ! or more rates use [`future_value`] or related functions. // ! // ! If you need to calculate a fixed rate given a present value, future value, and number of periods // ! use [`rate`] 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 present value is calculated with: //! //! > <img src="http://i.upmath.me/svg/present%5C_value%20%3D%20%7Bfuture%5C_value%20%5Cover%20(1%2Brate)%5E%7Bperiods%7D%7D" /> //! //! Or using some more usual variable names: //! //! > <img src="http://i.upmath.me/svg/pv%20%3D%20%7Bfv%20%5Cover%20(1%2Br)%5En%7D" /> //! //! `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. `r` is the periodic rate, though this may //! appear as `i` for interest. //! //! 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. //! //! Within the [TvmSolution](././tvm_simple/struct.TvmSolution.html) struct we record the formula used for the particular calculation //! using both concrete values and symbols. For example if we calculated the present value of an //! investment that grows by 1.5% per month for 48 months using simple compounding and reaches a //! future value of $50,000 the solution struct would contain these fields: //! ```text //! formula: "24468.0848 = 50000.0000 / (1.015000 ^ 48)", //! symbolic_formula: "pv = fv / (1 + r)^n", //! ``` //! //! ## Continuous Compounding //! //! With continuous compounding the formula is: //! //! > <img src="http://i.upmath.me/svg/present%5C_value%20%3D%20%7Bfuture%5C_value%20%5Cover%20e%5E%7Brate%20%5Ctimes%20periods%7D%7D" /> //! //! or: //! //! > <img src="http:i.upmath.me/svg/pv%20%3D%20%7Bfv%20%5Cover%20e%5E%7Br%20%5Ctimes%20n%7D%7D" /> //! //! 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 it's //! easier to compare formulas when they're printed as simple text as part of the [TvmSolution](./struct.TvmSolution.html) //! struct. Taking the example above but switching to continuous compounding the struct would //! contain these fields: //! ```text //! formula: "24337.6128 = 50000.0000 / 2.718282^(0.015000 * 48)", //! symbolic_formula: "pv = fv / e^(rt)", //! ``` use log::warn; use super::tvm::*; /// Returns the current value of a future amount using a fixed rate. /// /// Related functions: /// * To calculate a present value with a fixed rate and return a struct that shows the formula and /// optionally produces the the period-by-period values use [`present_value_solution`](./fn.present_value_solution.html). /// * To calculate the present value if the rates vary by period use [`present_value_schedule`](./fn.present_value_schedule.html) /// or [`present_value_schedule_solution`](./fn.present_value_schedule_solution.html). /// /// See the [present_value](./index.html) module page for the formulas. /// /// # Arguments /// * `rate` - The rate at which the investment grows or shrinks per period, /// expressed as a floating point number. For instance 0.05 would mean 5% growth. Often appears as /// `r` or `i` in formulas. /// * `periods` - The number of periods such as quarters or years. Often appears as `n` or `t`. /// * `future_value` - The final value of the investment. /// * `continuous_compounding` - True for continuous compounding, false for simple compounding. /// /// # Panics /// The call will fail if `rate` is less than -1.0 as this would mean the investment is /// losing more than its full value every period. It will fail also if the future value is zero as /// in this case there's no way to determine the present value. /// /// # Examples /// Investment that grows month by month. /// ``` /// use finance_solution::*; /// /// // The investment will grow by 1.1% per month. /// let rate = 0.011; /// /// // The investment will grow for 12 months. /// let periods = 12; /// /// // The final value will be $50,000. /// let future_value = 50_000; /// /// let continuous_compounding = false; /// /// // Find the current value. /// let present_value = present_value(rate, periods, future_value, continuous_compounding); /// dbg!(&present_value); /// /// // Confirm that the present value is correct to four decimal places (one hundredth of a cent). /// assert_rounded_4(-43_848.6409, present_value); /// ``` /// Error case: The investment loses 105% per year. There's no way to work out what this means so /// the call to present_value() will panic. /// ```should_panic /// let rate = -1.05; /// let periods = 6; /// let present_value = -10_000.75; /// let continuous_compounding = false; /// let present_value = finance_solution::present_value(rate, periods, present_value, continuous_compounding); /// ``` pub fn present_value<T>(rate: f64, periods: u32, future_value: T, continuous_compounding: bool) -> f64 where T: Into<f64> + Copy { present_value_internal(rate, periods as f64, future_value.into(), continuous_compounding) } /// Calculates the current value of a future amount using a fixed rate and returns a struct /// with the inputs and the calculated value. This is used for keeping track of a collection of /// financial scenarios so that they can be examined later. /// /// See the [present_value](./index.html) module page for the formulas. /// /// Related functions: /// * For simply calculating a single present value using a fixed rate use [`present_value`](./fn.present_value.html). /// * To calculate the present value if the rates vary by period use [`present_value_schedule`](./fn.present_value_schedule.html) /// or [`present_value_schedule_solution`](./fn.present_value_schedule_solution.html). /// /// # Arguments /// * `rate` - The rate at which the investment grows or shrinks per period, /// expressed as a floating point number. For instance 0.05 would mean 5% growth. Often appears as /// `r` or `i` in formulas. /// * `periods` - The number of periods such as quarters or years. Often appears as `n` or `t`. /// * `future_value` - The final value of the investment. /// * `continuous_compounding` - True for continuous compounding, false for simple compounding. /// /// # Panics /// The call will fail if `rate` is less than -1.0 as this would mean the investment is /// losing more than its full value every period. It will fail also if the future value is zero as /// in this case there's no way to determine the present value. /// /// # Examples /// Calculate a present value and examine the period-by-period values. /// ``` /// use finance_solution::*; /// /// // The rate is 8.45% per year. /// let rate = 0.0845; /// /// // The investment will grow for six years. /// let periods = 6; /// /// // The final value is $50,000. /// let future_value = 50_000; /// /// let continuous_compounding = false; /// /// // Calculate the present value and create a struct with the input values and /// // the formula used. /// let solution = present_value_solution(rate, periods, future_value, continuous_compounding); /// dbg!(&solution); /// /// let present_value = solution.present_value(); /// assert_rounded_4(present_value, -30_732.1303); /// /// // Examine the formulas. /// let formula = solution.formula(); /// dbg!(&formula); /// assert_eq!(formula, "-30732.1303 = -50000.0000 / (1.084500 ^ 6)"); /// let symbolic_formula = solution.symbolic_formula(); /// dbg!(&symbolic_formula); /// assert_eq!("pv = -fv / (1 + r)^n", symbolic_formula); /// /// // Calculate the amount at the end of each period. /// let series = solution.series(); /// dbg!(&series); /// ``` /// Build a collection of present value calculations where the future value and periodic rate are /// fixed but the number of periods varies, then filter the results. /// ``` /// // The rate is 0.9% per month. /// # use finance_solution::*; /// let rate = 0.009; /// /// // The final value is $100,000. /// let future_value = 100_000; /// /// let continuous_compounding = false; /// /// // We'll keep a collection of the calculated present values along with their inputs. /// let mut scenarios = vec![]; /// /// // Calculate the present value for terms ranging from 1 to 36 months. /// for periods in 1..=36 { /// // Calculate the future value for this number of months and add the details to the /// // collection. /// scenarios.push(present_value_solution(rate, periods, future_value, continuous_compounding)); /// } /// dbg!(&scenarios); /// assert_eq!(36, scenarios.len()); /// /// // Keep only the scenarios where the present value (which is negative) is greater than or /// // than or equal to -$80,000. /// scenarios.retain(|x| x.present_value() >= -80_000.00); /// dbg!(&scenarios); /// assert_eq!(12, scenarios.len()); /// /// // Find the range of months for the remaining scenarios. /// let min_months = scenarios.iter().map(|x| x.periods()).min().unwrap(); /// let max_months = scenarios.iter().map(|x| x.periods()).max().unwrap(); /// dbg!(min_months, max_months); /// assert_eq!(25, min_months); /// assert_eq!(36, max_months); /// /// // Check the formulas for the first of the remaining scenarios. /// let formula = scenarios[0].formula(); /// dbg!(&formula); /// assert_eq!("-79932.0303 = -100000.0000 / (1.009000 ^ 25)", formula); /// let symbolic_formula = scenarios[0].symbolic_formula(); /// dbg!(&symbolic_formula); /// assert_eq!("pv = -fv / (1 + r)^n", symbolic_formula); /// /// ``` /// Error case: The investment loses 111% per year. There's no way to work out what this means so /// the call to present_value() will panic. /// ```should_panic /// # use finance_solution::*; /// let rate = -1.11; /// let periods = 12; /// let present_value = 100_000.85; /// let continuous_compounding = false; /// let present_value = present_value_solution(rate, periods, present_value, continuous_compounding); /// ``` pub fn present_value_solution<T>(rate: f64, periods: u32, future_value: T, continuous_compounding: bool) -> TvmSolution where T: Into<f64> + Copy { present_value_solution_internal(rate, periods as f64, future_value.into(), continuous_compounding) } /// Calculates a present value based on rates that change for each period. /// /// Related functions: /// * To calculate the present value with varying rates and return a struct that can produce the /// period-by-period values use [`present_value_schedule_solution`](./fn.present_value_schedule_solution.html). /// * If there is a single fixed rate use [present_value](./fn.present_value.html) or [present_value_solution](./fn.present_value_solution.html). /// /// # Arguments /// * `rates` - A collection of rates, one for each period. /// * `future_value` - The ending value of the investment. /// /// # Panics /// The call will fail if any of the rates is less than -1.0 as this would mean the investment is /// losing more than its full value every period. It will fail also if the future value is zero as /// in this case there's no way to determine the present value. /// /// # Examples /// Calculate the present value of an investment whose rates vary by year. /// ``` /// // The annual rate varies from -3.4% to 12.9%. /// let rates = [0.04, -0.034, 0.0122, 0.129, 8.5]; /// /// // The value of the investment after applying all of these periodic rates /// // will be $30_000. /// let future_value = 30_000.00; /// /// // Calculate the present value. /// let present_value = finance_solution::present_value_schedule(&rates, future_value); /// dbg!(&present_value); /// ``` pub fn present_value_schedule<T>(rates: &[f64], future_value: T) -> f64 where T: Into<f64> + Copy { let periods = rates.len(); let future_value = future_value.into(); // Check the parameters including all of the provided rates. for rate in rates { check_present_value_parameters(*rate, periods as f64, future_value); } let mut present_value = -future_value; for i in (0..periods).rev() { present_value /= 1.0 + rates[i]; } present_value } /// Calculates a present value based on rates that change for each period and returns a struct /// with the inputs and the calculated value. /// /// Related functions: /// * To calculate the present value as a single number if the rates vary by period use /// [present_value_schedule](./fn.present_value_schedule.html). /// * If there is a single fixed rate use [present_value](./fn.present_value.html) or /// [present_value_solution](./fn.present_value_solution.html). /// /// # Arguments /// * `rates` - A collection of rates, one for each period. /// * `future_value` - The ending value of the investment. /// /// # Panics /// The call will fail if any of the rates is less than -1.0 as this would mean the investment is /// losing more than its full value every period. It will fail also if the future value is zero as /// in this case there's no way to determine the present value. /// /// # Examples /// Calculate the value of an investment whose rates vary by year, then view only those periods /// where the rate is negative. /// ``` /// use finance_solution::*; /// /// // The quarterly rate varies from -0.5% to 4%. /// let rates = [0.04, 0.008, 0.0122, -0.005]; /// /// // The value of the investment after applying all of these periodic rates /// // will be $25_000. /// let future_value = 25_000.00; /// /// // Calculate the present value and keep track of the inputs and the formula /// // in a struct. /// let solution = present_value_schedule_solution(&rates, future_value); /// dbg!(&solution); /// /// let present_value = solution.present_value(); /// assert_rounded_4(present_value, -23_678.6383); /// /// // Calculate the value for each period. /// let series = solution.series(); /// dbg!(&series); /// ``` pub fn present_value_schedule_solution<T>(rates: &[f64], future_value: T) -> TvmScheduleSolution where T: Into<f64> + Copy { let present_value = present_value_schedule(rates, future_value); TvmScheduleSolution::new(TvmVariable::PresentValue, rates, present_value, future_value.into()) } pub(crate) fn present_value_internal(rate: f64, periods: f64, future_value: f64, continuous_compounding: bool) -> f64 { check_present_value_parameters(rate, periods, future_value); let present_value = if continuous_compounding { -future_value / std::f64::consts::E.powf(rate * periods as f64) } else { -future_value / (1. + rate).powf(periods) }; assert!(present_value.is_finite()); present_value } pub(crate) fn present_value_solution_internal(rate: f64, periods: f64, future_value: f64, continuous_compounding: bool) -> TvmSolution { let present_value = present_value_internal(rate, periods, future_value, continuous_compounding); let rate_multiplier = 1.0 + rate; assert!(rate_multiplier >= 0.0); let (formula, symbolic_formula) = if continuous_compounding { let formula = format!("{:.4} = {:.4} / {:.6}^({:.6} * {})", present_value, -future_value, std::f64::consts::E, rate, periods); let symbolic_formula = "pv = -fv / e^(rt)"; (formula, symbolic_formula) } else { let formula = format!("{:.4} = {:.4} / ({:.6} ^ {})", present_value, -future_value, rate_multiplier, periods); let symbolic_formula = "pv = -fv / (1 + r)^n"; (formula, symbolic_formula) }; TvmSolution::new_fractional_periods(TvmVariable::PresentValue, continuous_compounding, rate, periods, present_value, future_value, &formula, symbolic_formula) } fn check_present_value_parameters(rate: f64, _periods: f64, future_value: f64) { assert!(rate.is_finite(), "The rate must be finite (not NaN or infinity)"); assert!(rate >= -1.0, "The rate must be greater than or equal to -1.0 because a rate lower than -100% would mean the investment loses more than its full value in a period."); if rate.abs() > 1. { warn!("You provided a periodic rate ({}) greater than 1. Are you sure you expect a {}% return?", rate, rate * 100.0); } assert!(future_value.is_finite(), "The future value must be finite (not NaN or infinity)"); assert!(future_value.is_normal(), "The future value is zero (or subnormal) so there is no way to calculate the present value."); } #[cfg(test)] mod tests { use super::*; use crate::*; #[test] fn test_present_value_schedule() { let rates = [0.04, 0.07, -0.12, -0.03, 0.11]; let future_value = 100_000.25; let present_value = present_value_schedule(&rates, future_value); assert_rounded_4(-94843.2841, present_value); let solution = present_value_schedule_solution(&rates, future_value); assert_rounded_4(100000.2500, solution.future_value()); assert_rounded_4(-94843.2841, solution.present_value()); let series = solution.series(); assert_eq!(6, series.len()); let period = &series[0]; assert_eq!(0, period.period()); assert_rounded_6(0.0, period.rate()); assert_rounded_4(-present_value,period.value()); let period = &series[1]; assert_eq!(1, period.period()); assert_rounded_6(0.04, period.rate()); assert_rounded_4(98_637.0154,period.value()); let period = &series[2]; assert_eq!(2, period.period()); assert_rounded_6(0.07, period.rate()); assert_rounded_4(105_541.6065,period.value()); let period = &series[3]; assert_eq!(3, period.period()); assert_rounded_6(-0.12, period.rate()); assert_rounded_4(92_876.6137,period.value()); let period = &series[4]; assert_eq!(4, period.period()); assert_rounded_6(-0.03, period.rate()); assert_rounded_4(90_090.3153, period.value()); let period = &series[5]; assert_eq!(5, period.period()); assert_rounded_6(0.11, period.rate()); assert_rounded_4(100_000.2500, period.value()); } /* macro_rules! compare_to_excel { ( $r:expr, $n:expr, $fv:expr, $pv_excel:expr, $pv_manual_simple:expr, $pv_manual_cont:expr ) => { println!("$r = {}, $n = {}, $fv = {}, $pv_excel: {}, $pv_manual_simple = {}, $pv_manual_cont = {}", $r, $n, $fv, $pv_excel, $pv_manual_simple, $pv_manual_cont); assert_approx_equal!($pv_excel, $pv_manual_simple); let pv_calc_simple = present_value($r, $n, $fv, false); println!("pv_calc_simple = {}", pv_calc_simple); assert_approx_equal!($pv_excel, pv_calc_simple); let pv_calc_cont = present_value($r, $n, $fv, true); println!("pv_calc_cont = {}", pv_calc_cont); assert_approx_equal!($pv_manual_cont, pv_calc_cont); let ratio = pv_calc_cont / pv_calc_simple; println!("ratio = {}", ratio); assert!(ratio > 0.0); assert!(ratio <= 1.0); } } */ fn compare_to_excel (test_case: usize, r: f64, n: u32, fv: f64, pv_excel: f64, pv_manual_simple: f64, pv_manual_cont: f64) { let display = false; if display { println!("test_case = {}, r = {}, n = {}, fv = {}, pv_excel: {}, pv_manual_simple = {}, pv_manual_cont = {}", test_case, r, n, fv, pv_excel, pv_manual_simple, pv_manual_cont) }; assert_approx_equal!(pv_excel, pv_manual_simple); let pv_calc_simple = present_value(r, n, fv, false); if display { println!("pv_calc_simple = {}", pv_calc_simple) }; assert_approx_equal!(pv_excel, pv_calc_simple); let pv_calc_cont = present_value(r, n, fv, true); if display { println!("pv_calc_cont = {}", pv_calc_cont) }; assert_approx_equal!(pv_manual_cont, pv_calc_cont); let ratio = pv_calc_cont / pv_calc_simple; if display { println!("ratio = {}", ratio) }; assert!(ratio >= 0.0); assert!(ratio <= 1.0); // Solution with simple compounding. let solution = present_value_solution(r, n, fv, false); if display { dbg!(&solution); } solution.invariant(); assert!(solution.calculated_field().is_present_value()); assert_eq!(false, solution.continuous_compounding()); assert_approx_equal!(r, solution.rate()); assert_eq!(n, solution.periods()); assert_approx_equal!(n as f64, solution.fractional_periods()); assert_approx_equal!(pv_excel, solution.present_value()); assert_approx_equal!(fv, solution.future_value()); // Solution with continuous compounding. let solution = present_value_solution(r, n, fv, true); if display { dbg!(&solution); } solution.invariant(); assert!(solution.calculated_field().is_present_value()); assert!(solution.continuous_compounding()); assert_approx_equal!(r, solution.rate()); assert_eq!(n, solution.periods()); assert_approx_equal!(n as f64, solution.fractional_periods()); assert_approx_equal!(pv_manual_cont, solution.present_value()); assert_approx_equal!(fv, solution.future_value()); let rates = initialized_vector(n as usize, r); // Schedule solution. let solution = present_value_schedule_solution(&rates, fv); if display { dbg!(&solution); } solution.invariant(); assert!(solution.calculated_field().is_present_value()); assert_eq!(n, solution.periods()); assert_approx_equal!(pv_excel, solution.present_value()); assert_approx_equal!(fv, solution.future_value()); } #[test] fn test_present_value_against_excel() { compare_to_excel(1, 0.01f64, 90, 1f64, -0.408391185151344f64, -0.408391185151344f64, -0.406569659740599f64); compare_to_excel(2, -0.01f64, 85, -1.5f64, 3.52451788132823f64, 3.52451788132823f64, 3.50947027788899f64); compare_to_excel(3, 0f64, 80, 2.25f64, -2.25f64, -2.25f64, -2.25f64); compare_to_excel(4, 0.05f64, 75, -3.375f64, 0.0869113201859512f64, 0.0869113201859512f64, 0.0793723922640307f64); compare_to_excel(5, -0.05f64, 70, 5.0625f64, -183.53236712846f64, -183.53236712846f64, -167.64697554088f64); compare_to_excel(6, 0.01f64, 65, -7.59375f64, 3.97710447262579f64, 3.97710447262579f64, 3.96428511727897f64); compare_to_excel(7, -0.01f64, 60, 11.390625f64, -20.8178501685176f64, -20.8178501685176f64, -20.7550719606981f64); compare_to_excel(8, 0f64, 55, -17.0859375f64, 17.0859375f64, 17.0859375f64, 17.0859375f64); compare_to_excel(9, 0.05f64, 50, 25.62890625f64, -2.23493614322574f64, -2.23493614322574f64, -2.10374873426328f64); compare_to_excel(10, -0.05f64, 45, -38.443359375f64, 386.597546504632f64, 386.597546504632f64, 364.740438412197f64); compare_to_excel(11, 0.01f64, 40, 57.6650390625f64, -38.7309044888379f64, -38.7309044888379f64, -38.6540316390219f64); compare_to_excel(12, -0.01f64, 35, -86.49755859375f64, 122.962317182378f64, 122.962317182378f64, 122.745878432934f64); compare_to_excel(13, 0f64, 30, 129.746337890625f64, -129.746337890625f64, -129.746337890625f64, -129.746337890625f64); compare_to_excel(14, 0.05f64, 25, -194.619506835937f64, 57.4716797951039f64, 57.4716797951039f64, 55.7594222710607f64); compare_to_excel(15, -0.05f64, 20, 291.929260253906f64, -814.33953749853f64, -814.33953749853f64, -793.546003343685f64); compare_to_excel(16, 0.01f64, 15, -437.893890380859f64, 377.179672510109f64, 377.179672510109f64, 376.898764278606f64); compare_to_excel(17, -0.01f64, 12, 656.840835571289f64, -741.033445550103f64, -741.033445550103f64, -740.585974095395f64); compare_to_excel(18, 0f64, 10, -985.261253356933f64, 985.261253356933f64, 985.261253356933f64, 985.261253356933f64); compare_to_excel(19, 0.05f64, 7, 1477.8918800354f64, -1050.31016709206f64, -1050.31016709206f64, -1041.45280575294f64); compare_to_excel(20, -0.05f64, 5, -2216.8378200531f64, 2864.94240503709f64, 2864.94240503709f64, 2846.47610562283f64); compare_to_excel(21, 0.01f64, 4, 3325.25673007965f64, -3195.50635796575f64, -3195.50635796575f64, -3194.87154873072f64); compare_to_excel(22, -0.01f64, 3, -4987.88509511947f64, 5140.56501667989f64, 5140.56501667989f64, 5139.78881110503f64); compare_to_excel(23, 0f64, 2, 7481.82764267921f64, -7481.82764267921f64, -7481.82764267921f64, -7481.82764267921f64); compare_to_excel(24, 0.05f64, 1, -11222.7414640188f64, 10688.3252038274f64, 10688.3252038274f64, 10675.4019041389f64); compare_to_excel(25, -0.05f64, 0, 16834.1121960282f64, -16834.1121960282f64, -16834.1121960282f64, -16834.1121960282f64); } }