Chapter 5, Problem 1EA

Lorri Morgan received $50 on her 16th birthday, and $70 on her 17th birthday, both of which she immediately invested in the bank, with interest compounded annually. On her 18th birthday, she had $127.40 in her account. Draw a time line, set up a polynomial equation, and calculate the YTM.

To determine

The timeline, a polynomial equation, and the YTM for the investments of $50 and $70 in two successive years.

Answer to Problem 1EA

Solution: $\underset{\_}{50{\left(1+i\right)}^2+70\left(1+i\right)=127.40}$, $0.043$

Explanation of Solution

Given: $50 is invested in the first year and $70 is invested in the next year. The total amount after 2 years is $127.40.

Explanation:

Lorri Morgan invested $50 on her 16^th birthday and $70 on her 17^th birthday. On her 18^th birthday, she had $127.40 in her account that pays interest compounded annually. The timeline for the given situation is as follows:

Assume that the interest is compounded annually at a rate $i$.

Use the compound interest formula:

$A=P{\left(1+i\right)}^n$

The compounded amount of first investment of $50 in two years will be

$A_1=50{\left(1+i\right)}^2$

The compounded amount of second investment of $70 in one year will be

$A_2=70\left(1+i\right)$

The sum of two amounts is $127.40. So, the polynomial equation in $i$ is as follows:

$50{\left(1+i\right)}^2+70\left(1+i\right)=127.40$

Assumingthat $1+i=x$, the polynomial equation will be

$\begin{array}{c}50x^2+70x=127.40\\ 50x^2+70x-127.40=0\end{array}$

Comparingthe above equation with quadratic equation $a{x}^2+bx+c=0$, we get

$a=50,b=70,\text{ and }c=-127.40$

Use quadratic formula:

$\begin{array}{c}x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ =\frac{-70\pm \sqrt{{70}^2-4\times 50\times \left(-127.40\right)}}{2\times 50}\\ =\frac{-70\pm \sqrt{30380}}{100}\\ =\frac{-70\pm 174.29859\cdots }{100}\end{array}$

Take positive sign:

$\begin{array}{c}x=\frac{-70+174.29859\cdots }{100}\\ =\frac{104.29859\cdots }{100}\\ =1.0429859\cdots \end{array}$

So, the value of $i$ will be

$\begin{array}{c}i=x-1\\ =1.0429859\cdots -1\\ =0.0429859\cdots \\ \approx 0.043\end{array}$

Conclusion: The polynomial equation for the calculation of YTM is $\underset{\_}{50{\left(1+i\right)}^2+70\left(1+i\right)=127.40}$ and the value of YTM is $0.043$.