Comp_Fin_HW3

Comp_Fin_HW3 - Jaime Frade Computational Finance: Dr....

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Jaime Frade Computational Finance: Dr. Kopriva Homework #3 0. Executive Summary In this assignment there is a need to determine the minimal interest rate, i , at which a person can invest $1000 per month for 15 years to ensure retirement nest egg of $1 million. In order to calculate i three root-finding algorithms, bisection method, Newton-Rapshon, and a combination, were programmed in C++. It will be shown that even though the bisection method does provide solutions, it suffers from a slow error linear convergence. On the other hand, using Newtons method will show fast quadratic convergence, however, highly depends on the choosing initial starting point appropriately. A combination of these methods will combine the speed and convergence stability needed to solve the problem. In order to reassure the person of my calculations and show a comparison of each of the three algorithms, I will test several functions with known roots. Results will be shown. This was done by advice of my collegue. From the calculations, it was found that an monthly interest rate, i , of approximately 1.5796708% under the IEEE standard of single precision. Therefore, to create this nest egg, the investor will need to find a fund offering an annual interest rate of 18.95% compounded monthly. Unfortunately, as of today, retirement will be impossible due to investments, such as CDs/savings, only offering rates 3-4%. 1. Statement of Problem Determine the interest rate which will satisfy the following equation: A = P i [(1 + i ) n- 1] , (1) where A amount in the account, P deposit amount at each month, and i interest rate per period. For the nest egg, n = 15 years (180 month since compounded monthly), P = 1 , 000 and A = 1 , 000 , 000. To determine i , will find a root using the bisection method, Newton-Rapshon, and a combination of the two. 2. Description of the Mathematics From lecture, the problem is to find a root for a function, f ( i ) : f ( i ) = A- P i [(1 + i ) n- 1] , i 6 = 0 (2) such that it solves f ( i ) = 0. The root exists by the Intermediate value theorem, if f ( i ) is a continuous function on an closed interval, [a,b], and f ( a ) f ( b ) < . Then there an i such that it satisfies, (2), f ( i ) = 0 for some i [ a, b ]. Bisection method will repeatedly check for an i , by making an initial guess such that i = a + b 2 . Newtons method will use functional values to determine the i which satisfies f ( i ) = 0 . If i is an approximation to i satisfying f ( i ) = 0, then by using simplification on the Taylor series expansion on the function (2) there exists a better approximation for i such that i = i n- f ( i n ) f ( i n ) , where (3) f ( i ) = Pni (1 + n ) n- 1- P (1 + i ) n + 1 i 2 (4) 1 3. Description of the Algorithm Bisection Method If trying to determine a root in a closed interval [a,b], denote AbsTol absolute tolerance and RelTol denote the relative tolerence....
View Full Document

This note was uploaded on 12/14/2011 for the course MAP 5611 taught by Professor Staff during the Fall '10 term at FSU.

Page1 / 15

Comp_Fin_HW3 - Jaime Frade Computational Finance: Dr....

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online