Give an example of a situation in which you might want to solve the savings plan formula to find the payments, PMT, required to achieve some goal.
One example of a situation where you might use the savings plan formula to find the payments required is when you are planning for a major purchase, such as buying a house or a car. Let's say you want to buy a car that costs $20,000, and you want to save up for it within a certain timeframe, let's say 3 years.
To calculate the monthly payments, you can use the formula for a savings plan:
PMT = (PV * r) / (1 - (1 + r)^(-n))
Where:
PMT is the monthly payment
PV is the present value or the goal amount ($20,000 in this case)
r is the monthly interest rate
n is the number of months (3 years = 36 months)
Let's assume the interest rate is 5% per year, which would translate to a monthly interest rate of 0.05 / 12 = 0.004167.
Using these values, we can plug them into the formula:
PMT = (20,000 * 0.004167) / (1 - (1 + 0.004167)^(-36))
Calculating this equation will give you the monthly payment you need to make in order to save up $20,000 within a 3-year timeframe.
Let's consider an example where you want to save a certain amount of money for a vacation. Suppose you want to save $5,000 in one year, starting from today. You estimate that you can earn an annual interest rate of 6% on your savings. Now, you want to determine the monthly payments required to achieve your savings goal.
To solve this using the savings plan formula, we can use the below information:
- PV (Present Value): Since you don't have any initial savings, this value will be zero.
- FV (Future Value): The desired savings goal, which is $5,000.
- r (Interest Rate): The annual interest rate of 6%. To calculate the monthly interest rate, we divide it by 12 (months) and convert it to a decimal.
- n (Number of Periods): Since you want to achieve your goal in one year, the number of periods will be 12 (months).
- PMT (Payment): The monthly payments required to achieve your savings goal.
The savings plan formula is: PMT = (FV - PV) / ((1 + r)^n - 1)
Plugging in the values from our example:
PMT = (5000 - 0) / ((1 + 0.06/12)^12 - 1)
Simplifying the equation will give you the monthly payment required to reach your savings goal of $5,000 in one year.