The terms of a single parent's will indicate that a child will receive an ordinary annuity of $20,000 per year from age 18 to age 24 (so the child can attend college) and that the balance of the estate goes to a niece. If the parent dies on the child's 17th birthday, how much money must be removed from the estate to purchase the annuity? (Assume an interest rate of 9%, compounded annually
Interest, i=9%=0.09 p.a.
Future value, S
Ordinary annuity for 6 years,
n=6
yearly payment, R = $20,000
Find future value when child will be 24 years old:
S = R((1+i)^n-1)/i
= $20,000*(1.09^6-1)/0.09
= $20,000*(7.523335)
= $150,466.69
Present value (when child is 17)
P= S/(1+i)^7
= $82310.43
To determine the amount of money that must be removed from the estate to purchase the annuity, we can use the present value of an ordinary annuity formula.
The formula to calculate the present value of an ordinary annuity is:
PV = C * [(1 - (1 + r)^(-n)) / r]
Where:
PV = Present Value
C = Cash flow received each period ($20,000 per year)
r = Interest rate per period (9% or 0.09)
n = Number of periods (7 years, from age 18 to age 24)
Plugging in the values into the formula:
PV = $20,000 * [(1 - (1 + 0.09)^(-7)) / 0.09]
Calculating the present value:
PV = $20,000 * [(1 - 1.790847050) / 0.09]
PV = $20,000 * [(-0.790847050) / 0.09]
PV = $176,738.33
So, approximately $176,738.33 must be removed from the estate to purchase the annuity.
To calculate the amount of money that needs to be removed from the estate to purchase the annuity, we need to use the formula for the present value of an ordinary annuity. Here's how you can calculate it step by step:
Step 1: Determine the number of years the child will receive the annuity. In this case, it is from age 18 to age 24, which means a total of 7 years.
Step 2: Determine the interest rate. In this case, the interest rate is 9% compounded annually.
Step 3: Use the present value of an ordinary annuity formula:
PV = A * [(1 - (1 + r)^(-n)) / r]
Where:
PV = Present value (amount to be removed from the estate)
A = Annuity payment per year ($20,000)
r = Interest rate per compounding period (9% or 0.09)
n = Number of compounding periods (7 years)
Step 4: Plug in the values into the formula and calculate:
PV = $20,000 * [(1 - (1 + 0.09)^(-7)) / 0.09]
Calculating the expression inside the square brackets:
(1 + 0.09)^(-7) = 0.548
1 - 0.548 = 0.452
PV = $20,000 * (0.452 / 0.09)
PV = $20,000 * 5.022
PV = $100,440
Therefore, approximately $100,440 needs to be removed from the estate to purchase the annuity.