A $1.2 million state lottery pays $5,000 at the beginning of each month for 20 years. How much money must the state actually have in hand to set up the payments for this prize if money is worth 7.7%, compounded monthly?
(a) Decide whether the problem relates to an ordinary annuity or an annuity due.
1
annuity due
ordinary annuity
.
(b) Solve the problem.
1.
Note: "...pays $5,000 at the beginning of each month..."
Should that ring a bell?
2.
To find the present value of the annuity due, we can consider it as the sum of the first payment (immediate) and an ordinary annuity with one payment less (the last one).
Thus:
n=20*12-1=239
R=5000
i=7.7%/12
P=R + R(1-(1+0.077/12)^(-239))/(0.077/12)
=5000+5000(1-0.21682)/0.006416667
=5000+5000*122.0539
=5000+610269.5
=$615,269.50
(a) To determine whether the problem relates to an ordinary annuity or an annuity due, we need to consider when the payments are made. In an ordinary annuity, payments are made at the end of each period, while in an annuity due, payments are made at the beginning of each period.
Here, the problem states that $5,000 is paid at the beginning of each month for 20 years. Since the payments are made at the beginning of each period, this problem relates to an annuity due.
(b) To solve the problem, we need to calculate the present value of the annuity due, which represents the amount of money the state must have in hand to set up the payments for this prize.
To calculate the present value of an annuity due, we can use the formula:
PV = PMT * [(1 - (1 + r)^(-n)) / r]
where PV is the present value, PMT is the payment amount, r is the interest rate per period, and n is the number of periods.
In this case, PMT = $5,000, r = 7.7% or 0.077 (since the interest rate is compounded monthly), and n = 20 years * 12 months/year = 240 months.
Plugging these values into the formula, we get:
PV = $5,000 * [(1 - (1 + 0.077)^(-240)) / 0.077]
Now, we can calculate the present value.
(a) The problem relates to an ordinary annuity because the payments are made at the beginning of each month.
(b) To solve the problem, we can use the present value of an ordinary annuity formula:
PV = PMT x [1 - (1 + r)^-n] / r
where:
PV = Present value
PMT = Payment amount per period
r = Interest rate per period
n = Number of periods
In this case, the payment amount per period is $5,000, the interest rate per period is 7.7%/12 = 0.64167%, and the number of periods is 20 x 12 = 240.
Plugging these values into the formula, we have:
PV = $5,000 x [1 - (1 + 0.0064167)^-240] / 0.0064167
Calculating this equation will give us the amount of money the state must actually have in hand to set up the payments for this prize.