Interest Rates & Compounding

Interest rate mathematics translates nominal rates into time value of money discount factors and compound factors.

Compounding Conventions

Given annual interest rate $r$ and year fraction $t$:

1. Compounded ($m$ times per year)

$C(r, t) = \left( 1 + \frac{r}{m} \right)^{m \cdot t}, \quad d(r, t) = \frac{1}{C(r, t)} = \left( 1 + \frac{r}{m} \right)^{-m \cdot t}$

2. Continuous Compounding

$C(r, t) = e^{r \cdot t}, \quad d(r, t) = e^{-r \cdot t}$

3. Simple Compounding

$C(r, t) = 1 + r \cdot t, \quad d(r, t) = \frac{1}{1 + r \cdot t}$

4. Simple Then Compounded

Used for money market discount instruments with stub periods: simple interest for $t \le 1/m$, compounded thereafter.

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Equivalent Rate Conversions

To convert a rate $r_1$ with frequency $m_1$ to an equivalent rate $r_2$ with frequency $m_2$ such that their discount factors match:

$\left( 1 + \frac{r_1}{m_1} \right)^{m_1} = \left( 1 + \frac{r_2}{m_2} \right)^{m_2}$

$r_2 = m_2 \left[ \left( 1 + \frac{r_1}{m_1} \right)^{\frac{m_1}{m_2}} - 1 \right]$

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Rust Code Example

use sabiquant::core::rates::{InterestRate, Compounding};
use sabiquant::core::Frequency;

fn main() -> Result<(), Box<dyn std::error::Error>> {
    // 5.0% semi-annually compounded rate
    let rate = InterestRate::new(0.05, Compounding::Compounded, Frequency::SemiAnnual);

    let time = 2.5; // 2.5 years

    // Compound and discount factors
    let compound_factor = rate.compound_factor(time);
    let discount_factor = rate.discount_factor(time);

    println!("Compound Factor (2.5Y): {:.6}", compound_factor);
    println!("Discount Factor (2.5Y): {:.6}", discount_factor);

    // Convert to equivalent continuously compounded rate
    let continuous_rate = rate.equivalent_rate(Compounding::Continuous, Frequency::Annual, time)?;
    println!("Continuous Equivalent:  {:.5}%", continuous_rate.rate() * 100.0);

    // Convert to quarterly compounded rate
    let quarterly_rate = rate.equivalent_rate(Compounding::Compounded, Frequency::Quarterly, time)?;
    println!("Quarterly Equivalent:   {:.5}%", quarterly_rate.rate() * 100.0);

    Ok(())
}