________ is the amount of a $3,000.00 annuity due at 12 percent compounded semiannually for 3 years.
A. $334,321.00
B. $43,068.93
C. $22,180.50
D. $41,814.50
Annuity "due" implies that the payment is made at the end of the interest period instead of the usual end of the period.
so , assuming there are 6 payments of $3000, each made at the beginning of the period. (the wording of the question could be improved)
Amount = 3000( 1.06^6 - 1)/.06 * 1.06 = 22,181.51
Lay out a time-graph to understand why I am multiplying by the extra 1.06
(I don't understand why their answer is off by a whole dollar, none of the
other three answers come remotely close to make any sense)
Thank you! I was very confused.
To calculate the amount of an annuity due, we can use the formula:
A = P * (1 + r/n)^(nt) - 1 / (r/n)
where:
A = the amount of the annuity due
P = the principal amount
r = interest rate per period
n = number of compounding periods per year
t = number of years
Given:
P = $3,000.00
r = 12% (convert to decimal by dividing by 100, so r = 0.12)
n = 2 (compounded semiannually, so 2 times per year)
t = 3 years
Now, let's substitute the values into the formula:
A = 3000 * (1 + 0.12/2)^(2*3) - 1 / (0.12/2)
Simplifying,
A = 3000 * (1 + 0.06)^(6) - 1 / (0.06)
A = 3000 * (1.06)^6 - 1 / 0.06
A = 3000 * 1.418519 - 1 / 0.06
A = 4255.56 - 1 / 0.06
A = 4255.56 - 16.67
A ≈ 4238.89
Therefore, the amount of the $3,000.00 annuity due at 12 percent compounded semiannually for 3 years is approximately $4238.89.
None of the given answer choices match the calculated result, so the correct answer is not provided.
To calculate the future value of an annuity due, we can use the formula:
Future Value = P * (1 + r/n)^(nt)
Where:
P = the periodic payment
r = the interest rate (as a decimal)
n = the number of compounding periods per year
t = the number of years
In this case, the annuity is $3,000.00, the interest rate is 12% (0.12 as a decimal), and it compounds semiannually, so there are 2 compounding periods per year. The duration is 3 years, so t = 3.
Let's substitute these values into the formula:
Future Value = $3,000.00 * (1 + 0.12/2)^(2*3)
Future Value = $3,000.00 * (1 + 0.06)^6
Future Value = $3,000.00 * (1.06)^6
Future Value = $3,000.00 * 1.418519 = $4,255.56
Therefore, the amount of the $3,000.00 annuity due after 3 years is approximately $4,255.56.
Now we can compare the calculated value with the options given:
A. $334,321.00 - This option is significantly larger and not likely to be correct.
B. $43,068.93 - This option is larger, but still unlikely to be correct.
C. $22,180.50 - This option is smaller and seems reasonable.
D. $41,814.50 - This option is slightly smaller and also seems reasonable.
Based on the calculations, it seems that option C. $22,180.50 is the closest to the correct answer.