Jie is investing $15000 and is choosing between two options.
With Option A, the investment is compounded semi-annually and will grow to $20000 in 5 years.
With Option B, the investment is compounded monthly for 5 years.
The annual rate of interest is the same in both options. Which option should Jie choose?
option A:
semiannual rate --- i
15000(1+i)^10 = 20000
(1+i)^10 = 4/3
1+i = 1.029186
semi-annual rate is 2.9186 %
annual rate is 5.8372 %
option B, monthly rate is .004864..
15000(1 + .004864..)^60
= 20069.53
so option B
Since the annual rate is the same, clearly the more frequent compounding will give you a higher yield.
We knew that.
thank you so much! :)
To determine which option Jie should choose, we need to compare the interest rates of both options.
Let's denote the annual interest rate as r.
For option A (compounded semi-annually), we need to find the effective interest rate per compounding period:
The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (initial investment)
r = annual interest rate (in decimal form)
n = number of times the interest is compounded per year
t = number of years the money is invested for
In this case, the future value (A) is $20000, the principal (P) is $15000, and the investment period (t) is 5 years. The number of times interest is compounded per year (n) is 2 (semi-annually).
20000 = 15000(1 + r/2)^(2*5)
Simplifying the equation:
(1 + r/2)^(10) = 4/3
Taking the 10th root of both sides:
1 + r/2 = (4/3)^(1/10)
Subtracting 1 from both sides:
r/2 = (4/3)^(1/10) - 1
Multiplying both sides by 2:
r = 2[(4/3)^(1/10) - 1]
Now let's calculate the interest rate for option B (compounded monthly):
The formula for compound interest remains the same, but the number of times interest is compounded per year (n) is 12 (monthly).
Using a similar approach as before, we find:
r = 12[(20000/15000)^(1/(12*5)) - 1]
Now that we have both interest rates, we can compare them to determine which option offers a higher rate of return. The option with the higher interest rate should be chosen:
If the interest rate for option A is greater than that of option B (r_A > r_B), then Jie should choose Option A.
If the interest rate for option B is greater than that of option A (r_B > r_A), then Jie should choose Option B.
If the interest rates for both options are the same (r_A = r_B), then there is no preference between the two options.
To determine which option Jie should choose, we need to compare the compound interest earned in each option.
Option A: The investment is compounded semi-annually and will grow to $20,000 in 5 years.
Option B: The investment is compounded monthly for 5 years.
To find the rate of interest, we can use the compound interest formula:
A = P * (1 + r/n)^(nt)
Where:
A = the future value of the investment
P = the principal amount (initial investment)
r = the annual interest rate (unknown)
n = the number of times interest is compounded per year
t = the number of years
For Option A:
A = $20,000
P = $15,000
t = 5 years
n = 2 (since it is compounded semi-annually)
Using the formula, we can solve for r:
$20,000 = $15,000 * (1 + r/2)^(2 * 5)
Simplifying the equation:
(1 + r/2)^10 = 20,000 / 15,000
Taking the 10th root of both sides:
1 + r/2 = (20,000 / 15,000)^(1/10)
1 + r/2 = 1.0801
r/2 = 0.0801
r = 2 * 0.0801
r = 0.1602 (or 16.02%)
Now, let's calculate Option B:
A = $20,000
P = $15,000
t = 5 years
n = 12 (since it is compounded monthly)
Using the formula, we can solve for r:
$20,000 = $15,000 * (1 + r/12)^(12 * 5)
Simplifying the equation:
(1 + r/12)^60 = 20,000 / 15,000
Taking the 60th root of both sides:
1 + r/12 = (20,000 / 15,000)^(1/60)
1 + r/12 = 1.0315
r/12 = 0.0315
r = 12 * 0.0315
r = 0.378 (or 3.78%)
Comparing the two options, Option A has an annual interest rate of 16.02% and Option B has an annual interest rate of 3.78%.
Therefore, Jie should choose Option A as it offers a higher annual interest rate and will lead to a larger growth of the investment over 5 years.