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Question 3 (1 point)
You start to save for a major purchase. You can invest $500 every
three months for 10 years. You are able to earn 8% compounded
semi-annually. What is the amount of interest that you earn during the
entire term?
$10 072.94
a
b
$20 072.94
Ос
$30 072.94
$15 072.94
e
$25 072.94
​

Sagot :

sqdancefan

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.

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