Running Newton Method for multiple different inputs

I am currently trying to code for using Newtons Method.

My goal is that I have 30 prices (PV), each corresponds to a time in years. Such that the first PV of 98828.9817 is for the time t=1, and the next one is t=2, until t=30.

Each price has a YTM that needs to be calculated for it using newtons method.

For each price we have a set of payments C, for t=1 C = [104000], t =2 is [4000 104000], for up to t=30, as what Ive calculate in a for loop. The issue I am having is that it appears that I am not getting the correct output when i try to run Newtons method while attempting to loop through the prices. But im unsure where I have gone wrong. Any help is appreciated. Thanks

clear all;
format long;
global PV C t; %global variable
F = 100000; %Face value 
it_max = 100; %Number of iterations
y0 = 0.10; %Starting estimate for YTM
tol = 10e-6; %Convergence tolerance
for i = 1:1:30
    C = F * 0.04.*ones(i,1);
    C(i) = C(i)+F;
    for t = 1:1:length(C);
    %YTM
        [y, it] = newton_method(@func_bond, it_max, y0, tol);
        fprintf('Price: $%0.4f     YTM: %0.4f\n', PV(length(C)), y);

    end
end

function [fy, fpy] = func_bond(y)
    %Input: (y) 
    %Output: [fy, fpy] which are f(y) and f'(y)
    
    global PV C t;
    PV = [98828.9817; 97812.0511; 96937.8969; 96159.9962; 95269.2339;... 
        94353.5669; 93276.0334; 92237.8837; 91261.0455; 90214.0597;...
        89083.9301; 87944.5962; 86976.3584; 85928.4188; 84982.5065;...
        84248.2589; 83540.8304; 82911.5228; 82417.0923; 82009.0742;...
        81633.1317; 81287.3700; 81087.7608; 80919.5136; 80780.7515;...
        80669.7282; 80584.8196; 80524.5143; 80487.4060; 80472.1856];

    fy = sum(C.*exp(-t.*y))-PV(length(C));
    fpy = sum(-t.*C.*exp(-t.*y));
   
end


function [x, it] = newton_method(f, it_max, x0, tol)
    %Input: f: function
    %       it_max: maximum number of iterations
    %       x0: initial guess
    %       tol: user defined tolerance
    pre_x = x0;
    disp(sprintf('n \t  x_n \t |(x_n - x_{n-1})|'));
    disp(sprintf('0 \t %2.10g', pre_x));
    for it = 1:it_max
        [fx, fpx] = f(pre_x);
        new_x = pre_x - fx/fpx;
        disp(sprintf('%d \t %2.10g \t %2.8g', it, new_x, abs(new_x - pre_x)));
        if(abs(new_x - pre_x)<= tol)
            break;
        end
        pre_x = new_x;
        
    end
    x = new_x;
end

[Jan]

"it appears that I am not getting the correct output" - this is an important detail. It is useful to share it with the readers: Why do you think so?

[Jd45ter]

From what I’ve tested it I’m not getting the correct output as it looks like when I run the func_bond function I don’t think it it iterating over the time period of t=1:1length(C), I think instead it is just calculating for t=length(C)

Answer 1

[Shreshth]

Hello Jd45ter,

I understand that you are facing issue while looping through each bond price and implementing Newton’s method to find the YTM. While going through the code provided, I could spot some mistakes that can be avoided to make the function run efficiently.

1. The t variable in the func_bond function should not be a global variable that gets overwritten in the loop. Instead, it should be an array that corresponds to the time periods for each cash flow.
2. The loop structure is incorrect. You need to calculate the YTM for each price, and the inner loop should not be there. The t array should be constructed outside of the func_bond function
3. The PV array should also not be a global variable inside func_bond. Instead, it should be passed as a parameter to the function.
4. The fprintf statement prints the YTM for each price, but it's inside the loop that should not exist. It should be outside and after the Newton's method call.

By implementing the above corrections, you will be able to get the desired outputs.

To get a better understanding of global and local variables, please refer to the documentation below:

https://in.mathworks.com/help/matlab/ref/global.html

Thank you,

Shubham Shreshth.