Computing Bond Yield with Newton’s Method

On By ElenaIn Numerical Analysis

The cat of Entrevaux

In fixed income analysis it is often required to calculate the yield of some coupon paying instrument, for example a bond. Very simply, an yield of a bond is defined as a single rate of return. Under continuous compounding, given a set of interest rates \{r_{T_1}, r_{T_2}, ..., r_{T_n}\}, the yield y equates the present value of the bond computed under a given set of rates. In other words:

\sum_{i=1}^{n}exp(-r_{T_i}T_i) C_{T_i}=\sum_{i=1}^{n}exp(-yT_i) C_{T_i}

where C_{T_i} is a coupon paid out at time T_i . Usually, one solves for y using some iterative guess-work, e.g. goal seek. In this post I will show you a well-know Newton’s recursive method that can be used to base such goal seek on.

The Newton’s Method

The Newton’s method is based on a recursive approximation formula:

x_{i+1}=x_i-\frac{f(x_i)}{f'(x_i)}

Let B=\sum_{i=1}^{n}exp(-yT_i) C_{T_i} be the price(or present value) of the bond expressed via its yield. Then, taking x to represent interest rate, we have:
f(x) = \sum_{i=1}^{n}exp(-x_{T_i}T_i) C_{T_i} - B = 0 and f'(x)=-\sum_{i=1}^{n}T_i C_{T_i} exp (-x_{T_i}T_i)

The recursive formula to solve for yield becomes:

x_{i+1}=x_i+\frac{\sum_{i=1}^{n}exp(-x_{T_i}T_i) C_{T_i} - B}{\sum_{i=1}^{n}T_i C_{T_i} exp (-x_{T_i}T_i)}

Goal Seek in C#

Now let’s take a look at an example implementation. The while loop in CalculateYield method implements the goal seek. This is a very simple example, and we pretend to have the bond price.

using System;
using System.Collections.Generic;
using System.Linq;

namespace YieldCalculator
{
    class EntryPoint
    {
        static void Main(string[] args)
        {
            // use some numbers to illustrate the point
            // maturities in months
            int[] maturities = { 6, 12, 18, 24, 30, 36 };
            double[] yearFrac = maturities.Select(i => i / 12.0).ToArray();
            // bond principle
            double P = 100;
            // bond price
            double B = 108;
            double couponRate = 0.1;
            // calculate future cashflows
            double[] cashflows = maturities.Select(i => (P * couponRate) / 2.0).ToArray();
            // add principle repayment
            cashflows[cashflows.Count() - 1] = P + cashflows[cashflows.Count() - 1];

            double initialGuess = 0.1;
            Console.WriteLine(CalculateYield(initialGuess, cashflows, yearFrac, B));
        }

        static double CalculateYield(double initialGuess, double[] cashflows, double[] yearFrac, double B)
        {
            double error = 0.000000001;
            double x_i = initialGuess-1.0;
            double x_i_next = initialGuess;

            double numerator;
            double denominator;
 
            while (Math.Abs(x_i_next - x_i) > error)
            {
                x_i = x_i_next;
                // linq's Zip is handy to perform a sum over expressions involving several arrays
                numerator = cashflows.Zip(yearFrac, (x,y) => (x * Math.Exp(y *-1*x_i))).Sum();
                denominator = yearFrac.Zip(cashflows, (x,y) => (x*y*Math.Exp(x*-1*x_i))).Sum();
                x_i_next = x_i + (numerator - B) / denominator;
            }
            return x_i_next;
        }
    }
}

Note how handy is LINQ’s Zip function to calculate a sum over an expression involving two arrays.

And now to the burning question – why the picture of a cat? The answer is even simpler than this post: It is a cute cat, so, why not?