Get the answers you need at Westonci.ca, where our expert community is always ready to help with accurate information. Join our platform to connect with experts ready to provide detailed answers to your questions in various areas. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.

Suppose you have $10,000 to invest. Which of the two rates would yield the larger amount in 4 years. 7% compounded quarterly or 6.94% compounded continuously?

Sagot :

[tex]K=\$10,000\ \ \ and\ \ \ Sum=K\cdot(1+ \frac{p}{100})^n\\----------------------- \\1)\ \ \ 7\%\ quarterly\\\\ \ \Rightarrow\ \ \frac{1}{4} \cdot7\% =1.75\%\ \ annually\ \ \Rightarrow\ \ p=1.75\\\\ quarterly\ \Rightarrow\ \ 4\ times\ annually\ \Rightarrow\ \ 16\ times\ in\ 4\ years\ \Rightarrow\ \ n=16\\\\Sum(7\%)=\$10,000\cdot(1+ \frac{1.75}{100})^{16}=\\ \\.\ \ \ \ \ \ \ \ \ \ \ \ =\$10,000\cdot(1+0,0175)^{16}\approx\$13199.29\\------------------------\\[/tex]

[tex]2)\ \ \ 6.94\%\ daily\\\\ \ \Rightarrow\ \ \frac{1}{365} \cdot6.94\% \approx0.019\%\ \ annually\ \ \Rightarrow\ \ p=0.019\\\\ daily\ \Rightarrow\ 365\ times\ annually\ \Rightarrow\ 1420\ times\ in\ 4\ years\ \Rightarrow\ n=1420\\\\Sum(6.94\%)=\$10,000\cdot(1+ \frac{0.019}{100})^{1420}=\\ \\.\ \ \ \ \ \ \ \ \ \ \ \ =\$10,000\cdot(1+0,00019)^{1420}\approx\$13096.69\\---------------------------\\\$13,199.29 > \$13,096.69\\\\Ans.\ the\ larger\ amount\ gives\ the\ compounded\ quarterly.[/tex]

7% compounded quarterly > 6.94% compounded continuously.

What is compound interest?

Interest earned on the principal amount and the interest itself is known as compound interest. These increases exponentially.

How to solve?

1)   7% compounded quarterly

[tex]\frac{7}{4}% = 1.75%[/tex] %= 1.75% annually

quarterly for 4 years => 4*4 = 16 times

Accumulated value = present value * [tex](1+\frac{r}{100})}^n[/tex]

where present value = $10,000 , r = 1.75, n = 16 times

substituting values:

AV  = [tex]10000*{(1+\frac{1.75}{100})}^{16} = $13199.295[/tex]

Thus, the value of $10,000 after 4 years at 7% compounded quarterly is $13199.295

2)   6.94% compounded continuously

[tex]\frac{6.94}{365}[/tex]% = 0.01904%  per annum

365 days for 4 years => 365*4 = 1460 times

Accumulated value = present value * [tex](1+\frac{r}{100})}^n[/tex]

Where present value = $10,000, r = 0.019014 , n = 1460 times

Substituting values:

[tex]10000*{(1+\frac{0.019014}{100})}^{1460} = $ 13199.29085[/tex]

Thus, the value of $10,000 after 4 years at % compounded quarterly is $13199.29085

since $13199.295 > $13199.29085

Both values are approximately the same but the value of 7% compounded quarterly is comparatively more than  6.94% compounded continuously.

Formula used:

Accumulated value = present value * [tex](1+\frac{r}{100})}^n[/tex]

TO learn more about Interest rates visit:

https://brainly.com/question/27118582

#SPJ2