Answer :
Answer:
(a) $10,072.94
Step-by-step explanation:
You want the interest earned on an investment of $500 per quarter for 10 years if interest of 8% is compounded semi-annually.
Effective rate
The multiplier (j) for figuring account value in each period when interest at rate r is compounded n times per year is ...
[tex]j=1 +\dfrac{r}{n}[/tex]
When there are m payments per year, the effective multiplier of each payment is ...
[tex]j_m=\left(1+\dfrac{r}{n}\right)^{n/m}[/tex]
And the effective interest rate per payment period is ...
[tex]r_m=j_m-1[/tex]
Account value
The value of the account after N payments will be ...
[tex]A=P\cdot\dfrac{(j_m)^N-1}{r_m}\\\\\\A=500\cdot\dfrac{\left(1+\dfrac{0.08}{2}\right)^{(2/4)\cdot40}-1}{\left(1+\dfrac{0.08}{2}\right)^{2/4}-1}=\dfrac{500(1.04^{20}-1)}{\sqrt{1.04}-1}\\\\\\A=30\,072.94[/tex]
Interest earned
The interest earned by the investment is the difference between the account value and the total value of the payments made:
I = A -NP
I = 30072.94 -40(500) = 10072.94
You earn $10,072.94 in interest during the entire term, choice A.
