 Most people aren't able to invest a large amount at a single point in time. Instead, they make periodic contributions. You contribute to social security by paying in a small amount each paycheck, or you put $100 a month into a savings account, or you reinvest the dividends a stock pays out quarterly. So, we introduced two new ideas. The contributions, CT1, CT2, and so on, made at times T1, T2, and so on. The balance, B, at the end of the investment period, and the time, Tau, at which the total value of the contributions is evaluated. Note that we'll treat deposits as positive and withdrawals as negative. So let's figure this out. Suppose we contribute amounts, CT1, CT2, and so on, at times T1, T2, and so on. The value of all contributions at time Tau will be the sum of the present value of the contributions times the accumulation over the time period Tau. Now let B be the balance at time T, the present value of B will be, and the value of the balance at time Tau will be the present value times the accumulation over that period. Which we can simplify. Since the value of all contributions at time Tau should be the actual balance at time Tau, then we want, and this is called the equation of value. For example, suppose you borrow $1000 at 10% annual interest and we'll go back to assuming that our interest is compounded. Next theory you pay back $500, and two years later you pay off the rest with a payment of P. Let's write the equation of value and find the payment amount P, and also interpret the formula. Since the loan is paid off at Tau equal to 5, the balance is B equal to 0. 10% annual interest gives accumulation function. At T equal 0 you receive $1000, and because there's a disbursement of the loan amount as well as payments, we have to agree on a sign convention. So we can view this amount of the loan as a withdrawal from the loan or the bank, so our contribution at time 0 is negative 1000. At T equals 1, there's a payment of $500, so we can view this as a deposit to the loan or the bank, so C1 is 500, and at T equals 5 you pay, you deposit an additional amount P. So we can substitute these values into our equation of value, and since we know our accumulation function, we can simplify these expressions and then solve for P. Now we have this equation that tells us how to find P, and so the important question is how can we interpret this value? So here's one possibility. Note that 1000 times 1.1 to the 5th is the amount you'd have to pay on 1000 if you didn't pay anything for 5 years. Now this is reduced by 500 times 1.1 to the 4th, and we can interpret this as the amount of interest you didn't have to pay since you paid off 500 of that amount at time 1.