An insurance company is offering a new policy to its customers. Typically, the policy is bought by a parent or grandparent for a child at the child's birth. The details of the policy are as follows: The purchaser (say, the parent) makes the following six payments to the insurance company:
First birthday: $920
Second birthday:920
Third birthday: 1020
Fourth birthday: 1020
Fifth birthday: 1,120
Sixth birthday: 1,120
After the child's sixth birthday, no more payments are made. When the child reaches age 65, he or she receives $430,000. The relevant interest rate is 12 percent for the first 6 years and 7 percent for all subsequent years. What is the value of the policy at the child's 65th birthday?
Value at the end of 6 years
= 1120 + 1120(1.12) + 1020(1.12)^2 + 1020(1.12)^3 + 920(1.12)^4 + 920(1.12)^5
= 8155.91
now the rate goes down to 7% for the next
59 years.
Final value at age 65
= 8155.91(1.07)^59
= 441,687.52
but they only paid out $430,000
while using that money to invest in other business ventures. Mmmmh, can you see why "insurance" is a profitable business to be in?
Hi Reiny,
I came up with the same answer using the same logic you did. Unfortunately, I am doing this question through an online system that checks my answer automatically, and the system says that the answer is wrong. I have tried every variation that I can think of, and still can't come up with the right answer. :( Any other ideas?
Never mind!!! It helps to put the answer to the correct question! :)
Why did the clown go to the insurance company?
Because he wanted to juggle with numbers and policies!
Now, let's calculate the value of the policy at the child's 65th birthday. To do this, we'll need to calculate the present value of each payment made by the parent and then add up all the present values.
The first thing we need to do is determine the present value of the payments made in the first six years. We'll use the formula for the present value of an ordinary annuity:
PV = P * (1 - (1 + r)^(-n)) / r
Where:
PV = Present Value
P = Payment amount
r = Interest rate
n = Number of periods
Using this formula for each payment, we get:
First birthday payment: PV = 920 * (1 - (1.12)^(-1)) / 0.12 = 827.80
Second birthday payment: PV = 920 * (1 - (1.12)^(-2)) / 0.12 = 1474.13
Third birthday payment: PV = 1020 * (1 - (1.12)^(-3)) / 0.12 = 2771.01
Fourth birthday payment: PV = 1020 * (1 - (1.12)^(-4)) / 0.12 = 2819.59
Fifth birthday payment: PV = 1120 * (1 - (1.12)^(-5)) / 0.12 = 3067.93
Sixth birthday payment: PV = 1120 * (1 - (1.12)^(-6)) / 0.12 = 3111.87
Next, we need to calculate the present value of the final payment at the child's 65th birthday. Since this payment is made after the first six years, we'll use the interest rate of 7 percent:
PV = 430,000 / (1 + 0.07)^59 = 29,182.11
Finally, we add up all the present values:
827.80 + 1474.13 + 2771.01 + 2819.59 + 3067.93 + 3111.87 + 29,182.11 = 40,274.44
Therefore, the value of the policy at the child's 65th birthday is approximately $40,274.44.
Remember, though, that this calculation assumes the insurance company will keep their promise and not pull a disappearing act!
To find the value of the policy at the child's 65th birthday, we need to calculate the present value of all the payments made and the future value of the policy at age 65.
First, let's calculate the present value of the six payments made by the parent. We will use the formula for the present value of an annuity:
PV = PMT * ((1 - (1 + r)^-n) / r)
Where:
PV = Present value
PMT = Payment amount
r = Interest rate per period
n = Number of periods
For the first six years, the interest rate is 12 percent, so r = 12% = 0.12.
Let's calculate the present value of the payments up to the sixth birthday:
PV_payments = (920 * ((1 - (1 + 0.12)^-6) / 0.12)) +
(920 * ((1 - (1 + 0.12)^-5) / 0.12)) +
(1020 * ((1 - (1 + 0.12)^-4) / 0.12)) +
(1020 * ((1 - (1 + 0.12)^-3) / 0.12)) +
(1120 * ((1 - (1 + 0.12)^-2) / 0.12)) +
(1120 * ((1 - (1 + 0.12)^-1) / 0.12))
After the sixth birthday, the interest rate changes to 7 percent, so r = 7% = 0.07.
Now, let's calculate the value of the policy at the child's 65th birthday, using the future value formula:
FV = PV * (1 + r)^n
Where:
FV = Future value
PV = Present value
r = Interest rate per period
n = Number of periods
We will calculate the future value for a period of 59 years, from the child's 6th birthday to their 65th birthday:
FV_policy = PV_payments * (1 + 0.07)^59
Finally, we will add the future value of the policy to the value of the payments made:
Value_at_65 = FV_policy + 430,000
Now, we can calculate the value of the policy at the child's 65th birthday by plugging in the values and calculating the equation.