start early. If you remember the graph of compound interest in section 2.2, we can see that as time goes on, the balance increases exponentially. Deriving the Savings Plan Formula (Optional) If you are interested in where the savings plan formula came from, we will explain it here. A savings plan with regular payments can be described recursively. Recall that basic compound interest follows from the relationship for each compound period. For a savings plan, we need to add a deposit, d, to the account with each compounding period: Taking this equation from recursive form to explicit form is a bit trickier than with compound interest. It will be easiest to see by working with an example rather than working in general. Suppose we will deposit $100 each into an account paying 6% interest. We assume that the account is compounded with the same frequency as we make deposits unless stated otherwise. In this example: r = 0.06 6% interest n = 12 12 compounds/deposits per year d = $100 our deposit per month Writing out the recursive equation gives where A is exchanged with where m is the number of compounding periods. Assuming we start with an empty account, we can begin using this relationship: Continuing this pattern, after m deposits, we’d have saved: In other words, after m months, the first deposit will have earned compound interest for m-1 months. The second deposit will have earned interest for months. Last month’s deposit would have earned only one month worth of interest. The most recent deposit will have earned no interest yet. This equation leaves a lot to be desired, though it make calculating the ending – balance any easier! To simplify things, multiply both sides of the equation by 1.005: Distributing on the right side of the equation gives Now we’ll line this up with like terms from our original equation, and subtract each side
start early. If you remember the graph of compound interest in section 2.2, we can see that as time goes on, the balance increases exponentially. Deriving the Savings Plan Formula (Optional) If you are interested in where the savings plan formula came from, we will explain it here. A savings plan with regular payments can be described recursively. Recall that basic compound interest follows from the relationship for each compound period. For a savings plan, we need to add a deposit, d, to the account with each compounding period: Taking this equation from recursive form to explicit form is a bit trickier than with compound interest. It will be easiest to see by working with an example rather than working in general. Suppose we will deposit $100 each into an account paying 6% interest. We assume that the account is compounded with the same frequency as we make deposits unless stated otherwise. In this example: r = 0.06 6% interest n = 12 12 compounds/deposits per year d = $100 our deposit per month Writing out the recursive equation gives where A is exchanged with where m is the number of compounding periods. Assuming we start with an empty account, we can begin using this relationship: Continuing this pattern, after m deposits, we’d have saved: In other words, after m months, the first deposit will have earned compound interest for m-1 months. The second deposit will have earned interest for months. Last month’s deposit would have earned only one month worth of interest. The most recent deposit will have earned no interest yet. This equation leaves a lot to be desired, though – it make calculating the ending balance any easier! To simplify things, multiply both sides of the equation by 1.005: Distributing on the right side of the equation gives Now we’ll line this up with like terms from our original equation, and subtract each side
Chapter2-Financial-Math9-4-18.docx