Overview
Given that a series of cash flows is discounted at different rates depending on the date of the cash flow, it is natural to ask if there exists a single rate that when used to discount all cash flows, equals the present value as calculated with the appropriate rates.
This rate, if it existed, would represent an "average" of sorts of the rates of each cash flow, and would provide a single number by which to compare various fixed income instruments.
In general this rate exists, but can only be calculated using numeric methods. It is called the internal rate of return, or in the context of fixed income, yield to maturity.
Internal Rate of Return - Yield to Maturity
Given a series of cash flows, equally spaced in time. (that is, each separated by the same time period)
{% x_0,x_1,...x_n %}
The internal rate of return is defined to be the single rate, r, that solves
{% 0 = x_0 + \frac{x_1}{1+r} + \frac{x_2}{(1+r)^2} + ... + \frac{x_n}{(1+r)^n} %}
Note: at least one of these cash flows needs to be negative. The negative cash flow typically represents the cost of purchasing the bond
or instrument up front. However, when modeling a project or other series of cash flows, it could very well be that there are multiple
cash flows, representing the expenses, and positive cash flows representing the returns.
The IRR can be calculated using the Yield Library
let yd = await import('/lib/finance/yield/v1.0.0/yield.mjs');
let cashFlows = [{ time:0,value:100 }, { time:0.1,value:10 },{ time:0.5,value:5 },{ time:0.9,value:10 },{ time:1,value: -120 },];
let rate = yd.irr(cashFlows);
Annualizing Yields
When the income payments on a fixed income instrument occurs at a different period than yearly (say monthly or quarterly), it is often convenient to annualize the yield, in order put comparisons of yields on an apples to apples basis.
When {% m %} is the number or payment periods per year, the effective annual yield is
{% yield = (1 + rate)^m - 1 %}
A simple approximation to the above is
{% yield \approx rate \times m %}
Continuous Compounding
The internal rate of return on a continuous compounding basis is the single rate {% r %} that solves
{% 0 = x_0 + x_1e^{-rt_1} + ... + x_ne^{-rt_n} %}
where {% t_i %} is the time of the {% i^{th} %} cash flow.