At the beginning of each year, Al rose invested $1,200 semiannually at 12 percent for 8 years. The cash value of the annuity due at the end of the eighth year is?
careful, the payments are made at the beginning of each period, where the formulas we use assume that payments are made at the end of the period
i = .12/2 = .06
n = 8(2) = 16 but we will use only 15
and I will assume that the first payment is $1200 is made NOW.
PV = 1200 + 1200(1 - 1.06^-15)/.06
= .....
31654.70
To determine the cash value of the annuity due, we can use the formula for the future value of an annuity due:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = Future Value of the annuity due
P = Periodic payment (amount invested each period)
r = Interest rate per compounding period
n = Number of compounding periods
In this case:
P = $1,200 (amount invested semiannually)
r = 12% = 0.12 per compounding period
n = 8 years (since Al invested for 8 years)
First, we need to adjust the interest rate to match the semiannual compounding period. Since the interest rate stated is an annual rate, we divide it by 2 to get the semiannual rate.
r = 0.12 / 2 = 0.06 per compounding period
Now we can substitute the values into the formula and calculate the future value of the annuity due:
FV = $1,200 * [(1 + 0.06)^8 - 1] / 0.06
FV ≈ $1,200 * [1.485946 - 1] / 0.06
FV ≈ $1,200 * 0.485946 / 0.06
FV ≈ $9,239.64
Therefore, the cash value of the annuity due at the end of the eighth year is approximately $9,239.64.