Ed Long promised to pay his son $400 semiannually for 12 years. Assume Ed can invest his money at 6% in an ordinary annunity. How much must Ed invest to pay his son $400 semiannually for 12 years? 24 periods, 3% (Table 13.2)
To determine how much Ed must invest to pay his son $400 semiannually for 12 years, we can use the formula for the present value of an ordinary annuity.
The formula for the present value of an ordinary annuity is:
PV = PMT * [(1 - (1 + r)^(-n)) / r]
Where:
PV = Present Value (the amount Ed must invest)
PMT = Payment per period ($400)
r = Interest rate per period (6% = 0.06 / 2 = 0.03)
n = Number of periods (12 years = 24 semiannual periods)
Now, let's plug in the values into the formula:
PV = $400 * [(1 - (1 + 0.03)^(-24)) / 0.03]
Next, let's calculate the portion within the brackets first:
(1 - (1 + 0.03)^(-24)) = (1 - (1.03)^(-24)) = (1 - 0.523654452) = 0.476345548
Now, let's plug in the calculated value:
PV = $400 * (0.476345548 / 0.03)
Next, let's simplify the equation:
PV = $400 * 15.87818493
Finally, let's calculate the final answer:
PV = $6,351.27
Therefore, Ed must invest $6,351.27 to pay his son $400 semiannually for 12 years at an interest rate of 6% in an ordinary annuity.
To calculate the amount Ed must invest to pay his son $400 semiannually for 12 years, we can use the formula for the present value of an ordinary annuity.
The formula for the present value of an ordinary annuity is:
PV = P * (1 - (1 + r)^-n) / r
Where:
PV = present value (the amount Ed must invest)
P = payment amount per period ($400 in this case)
r = interest rate per period (6% / 2 = 3% or 0.03 in this case)
n = number of periods (12 years * 2 periods per year = 24 periods)
Plugging the values into the formula, we get:
PV = 400 * (1 - (1 + 0.03)^-24) / 0.03
At this point, we can use Table 13.2 to find the factor (1 - (1 + 0.03)^-24) / 0.03 corresponding to 24 periods and 3% interest rate. From the table, we find that the factor is approximately 10.9927.
Substituting this value back into the equation, we get:
PV = 400 * 10.9927
Calculating this, we find:
PV ≈ $4,397.08
Therefore, Ed must invest approximately $4,397.08 in order to pay his son $400 semiannually for 12 years.