Question

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?

Answer 1

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

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