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?

K = $10 , 000 an d S u m = K ⋅ ( 1 + 100 p ​ ) n − − − − − − − − − − − − − − − − − − − − − − − 1 ) 7% q u a r t er l y ⇒ 4 1 ​ ⋅ 7% = 1.75% ann u a ll y ⇒ p = 1.75 q u a r t er l y ⇒ 4 t im es ann u a ll y ⇒ 16 t im es in 4 ye a rs ⇒ n = 16 S u m ( 7% ) = $10 , 000 ⋅ ( 1 + 100 1.75 ​ ) 16 = . = $10 , 000 ⋅ ( 1 + 0 , 0175 ) 16 ≈ $13199.29 − − − − − − − − − − − − − − − − − − − − − − − −
\$13,096.69\\\\Ans.\ the\ larger\ amount\ gives\ the\ compounded\ quarterly."> 2 ) 6.94% d ai l y ⇒ 365 1 ​ ⋅ 6.94% ≈ 0.019% ann u a ll y ⇒ p = 0.019 d ai l y ⇒ 365 t im es ann u a ll y ⇒ 1420 t im es in 4 ye a rs ⇒ n = 1420 S u m ( 6.94% ) = $10 , 000 ⋅ ( 1 + 100 0.019 ​ ) 1420 = . = $10 , 000 ⋅ ( 1 + 0 , 00019 ) 1420 ≈ $13096.69 − − − − − − − − − − − − − − − − − − − − − − − − − − − $13 , 199.29 > $13 , 096.69 A n s . t h e l a r g er am o u n t g i v es t h e co m p o u n d e d q u a r t er l y .

Answered by kate200468 | 2024-06-10

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?

**7% compounded quarterly **

4 7 ​ %= 1.75% annually
quarterly for 4 years => 4*4 = 16 times
Accumulated value = present value * (1+\frac{r}{100})}^n
where present value = $10,000 , r = 1.75, n = 16 times
substituting values:
AV = 10000*{(1+\frac{1.75}{100})}^{16} = $13199.295
Thus, the value of $10,000 after 4 years at 7% compounded quarterly is $13199.295

**6.94% compounded continuously **

365 6.94 ​ % = 0.01904% per annum
365 days for 4 years => 365*4 = 1460 times
Accumulated value = present value * (1+\frac{r}{100})}^n
Where present value = $10,000, r = 0.019014 , n = 1460 times
Substituting values:
10000*{(1+\frac{0.019014}{100})}^{1460} = $ 13199.29085
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 * (1+\frac{r}{100})}^n
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Answered by rajagrewal768 | 2024-06-16

7% compounded quarterly yields approximately $13,199.29, while 6.94% compounded continuously yields about $13,194.31. Therefore, the better investment option is 7% compounded quarterly. The difference in yield is minimal but significant over time.
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Answered by rajagrewal768 | 2024-10-01