Linear interpolation and annuities problem

Question

For 10,000, Kelly purchases an annuity−immediate that pays 400 quarterly for the next 10 years. Calculate the annual nominal interest rate convertible monthly earned by Kelly’s investment. Hint: Use linear interpolation.

Correct solution: 10.01%, https://www.youtube.com/watch?v=HLW_E6N_rss&feature=youtu.be

In the above video solution, I do not understand how he got the accurate estimate for $j$ using linear interpolation. Given the rearranged equation of value (let $j$ be the effective quarterly interest rate) $$ 400\frac{1-\frac{1}{(1+j)^{40}}}{j}-10000 = f(j) $$ and our goal is to find value of $j$ s.t $f(j)=0$. By using bisection method several times, I found $x_1=0.0406, x_2=0.0559, f(x_1) = 1857.52, f(x_2)=-3656.7$, then used linear interpolation $$ f(j) = f(x_1) + \frac{f(x_2)-f(x_1)}{x_2-x_1}\cdot (j-x_1) $$ by setting $f(j)=0$, and solving for $j$, I got $j=0.0457$. However, the $j$ in the video solution is .025 ish.

(a) What am I doing wrong, does my $x_1, x_2$ need to be closer together? How would I know what is "close enough" on an exam setting? (b) Are there alternate methods to linear interpolation?

Answer 1

I don´t know what went wrong. Let a,b the lower bound and upper bound, respectively. And $m=\frac{a+b}2$. Then I have the following decision rule:

1. If $f(a)\cdot f(m)<0$, then the lower bound $a$ remains the same and the upper bound becomes $m$.

2. If $f(a)\cdot f(m)>0$, then the upper bound $b$ remains the same and the lower bound becomes $m$.

It results in the following table:

(a) What am I doing wrong, does my x1,x2 need to be closer together?

It is necessary that the root (j) is between $x_1$ and $x_2$. If it is not then the bisection method went wrong. In your case $j=0.025244$ is outside of the interval $(0.0406, 0.0559)$