Need help finding a formula for:
Question:
Suppose a bank offers you the following deal:
You pay to the bank an annuity amount of $A per year over the next 10 years and the bank will in turn pay you $40,000 per year starting at the end of year 11 and ending the payments by the end of year 30.
Interest rate=10%/year throughout the 30 year period.
Find the annuity amount of $A you will be willing to pay over the next 10 yrs.
First, and excel spreadsheet is very helpful for solving these kinds of problems.
I believe you need to have a personal discount rate. How much would you pay to receive $40,000 thirty years from now. For this problem, I believe you are to assume the interest rate, r, is your personal discount rate. (While its not stated, I will also assume you pay $A at the end of the year)
So, you the value of the amount you pay at time zero will be:
A/(1.1) + A/(1.1)^2 + A/(1.1)^3 + ... A/(1.1)^10
= A * sum(i) of 1/(1.1)^i as i goes from 1 to 10
= A * 17.53117
The value of the amount you receive at time zero is 40000/(1.1)^11 + 40000/(1.1)^12 + ... 40000/(1.1)^30
= 40000 * sum(j) of 1/(1.1)^j as j goes from 11 to 30
= 40000 * 163.4123
Set these two equal and solve for A.
Thanks.
What formula did you use to calculate 17.53117? I found a formula that I thought would work to get that sum, but I got a different number. It is:
1 - 1.1^(-10)/ 0.1
and also: (40000* 163)/17.53 = $371933 is the answer?
My bad. I did'nt properly apply my own formula.
17.53 is the sum of (1.1)^i as i goes from 1 to 10. What I really want, as my original formula says, is the sum of (1/(1.1)^i) as i goes from 1 to 10. This turns out to be 6.1446. So the present value of 10 payments of A over 10 years is A*6.1446.
Likewise, my 163.4123 is the sum of (1.1)^i as i goes from 11 to 30. What I really want is the sum of 1/(1.1)^i as i goes from 11 to 30. This new sum is 3.2823.
So, set A*6.1446 = 40000*3.2823.
A = 21367.
This makes much more sense.
Sorry for the confusion.
To find the annuity amount $A, we can use the concept of present value and future value of the cash flows.
First, we need to calculate the present value of the future cash flows. Given that the interest rate is 10% per year and the period is 30 years, we can use the present value formula:
PV = FV / (1 + r)^n
Where:
PV = Present Value
FV = Future Value
r = Interest rate
n = Number of periods
In this case, we want to find the present value of the future cash flows from the bank's payment of $40,000 per year from year 11 to year 30. The future value is given as $40,000 per year, and the interest rate is 10%. The number of periods is 20 (from year 11 to year 30).
PV = 40,000 / (1 + 0.10)^20
PV ≈ 40,000 / (1.10)^20
Now, we need to calculate the present value of the annuity payments of $A over the next 10 years. Using the same formula, the future value is the annuity amount $A, and the number of periods is 10.
PV = A / (1 + 0.10)^10
PV ≈ A / (1.10)^10
Since we have two present value calculations, we can equate them to find the value of A:
40,000 / (1.10)^20 = A / (1.10)^10
To simplify, let's multiply both sides by (1.10)^10 to get:
A = 40,000 * (1.10)^10 / (1.10)^20
A ≈ 40,000 / (1.10)^10
Now, we can calculate the annuity amount A by evaluating the expression on the right side of the equation.
Using a calculator or software, we find:
A ≈ 11,496.30
Therefore, the annuity amount you will be willing to pay over the next 10 years is approximately $11,496.30.