Suppose that you want to buy a house 5 years from now, and you estimate that an initial down payment of $30,000 will be required at that time. To accumulate the $30,000, you wish to make equal annual end of year deposits into an account paying annual interest of 6%. What should your payments be?
Well, if you want to buy a house in 5 years and need $30,000 for the down payment, you'll need to start saving some money.
Let's break it down: You're looking to save up over 5 years and earn 6% annual interest on your account.
To calculate the equal annual end-of-year deposits, we can use the formula for the future value of an ordinary annuity:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future Value (the amount you want to accumulate - $30,000 in this case)
P = Annual Payment (what we're trying to find)
r = Interest Rate per period (6% annual interest = 0.06)
n = Number of periods (5 years)
Plugging in the given values, we have:
$30,000 = P * ((1 + 0.06)^5 - 1) / 0.06
Now, let's solve for P (the payment):
$30,000 * 0.06 = P * ((1.06)^5 - 1)
$1,800 = P * (1.338225 - 1)
$1,800 = P * 0.338225
P ≈ $5,324.92
So, your annual payments should be approximately $5,324.92. Just remember, no clowns in the house!
To calculate the equal annual end-of-year payments needed to accumulate $30,000 in 5 years, we can use the future value of an ordinary annuity formula.
The formula to calculate the future value of an ordinary annuity is:
FV = P * [(1 + r) ^ n - 1] / r
Where:
FV = Future Value
P = Payment or annual deposit
r = Interest rate per period
n = Number of periods
In this case, the future value (FV) is $30,000, the interest rate (r) is 6% or 0.06, and the number of periods (n) is 5 years.
So the formula becomes:
$30,000 = P * [(1 + 0.06) ^ 5 - 1] / 0.06
Now, we need to solve this equation for P to find the annual payment needed.
$30,000 * 0.06 = P * [(1 + 0.06) ^ 5 - 1]
1,800 = P * [1.338225 - 1]
1,800 = P * 0.338225
P = 1,800 / 0.338225
P ≈ $5,328.20
Therefore, you would need to make annual payments of approximately $5,328.20 to accumulate $30,000 in 5 years with an annual interest rate of 6%.
To determine the equal annual end of year deposits needed to accumulate $30,000 in 5 years, you can use a financial concept called the future value of an ordinary annuity.
The formula for calculating the future value of an ordinary annuity is:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = Future value of the annuity
P = Annual payment or deposit
r = Interest rate per period (in this case, 6% or 0.06)
n = Number of periods (in this case, 5 years)
In this case, you know the future value (FV) is $30,000, and you need to find the annual payment (P).
Plugging in the values into the formula, we get:
$30,000 = P * [(1 + 0.06)^5 - 1] / 0.06
Now we can solve for P:
$30,000 * 0.06 = P * [(1 + 0.06)^5 - 1]
$1,800 = P * [(1.06)^5 - 1]
To simplify the equation, we calculate (1.06)^5 = 1.338225, and substitute it back into the equation:
$1,800 = P * (1.338225 - 1)
$1,800 = P * 0.338225
Now we can solve for P by dividing both sides of the equation by 0.338225:
P = $1,800 / 0.338225
P ≈ $5,324.50
So, your annual payments should be approximately $5,324.50 to accumulate $30,000 in 5 years with an interest rate of 6%.
n = 5
payment --- p
i = .06
p(1.06^5 - 1)/.06 = 30000
I get p = $5321.89