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Unformatted text preview: a is the investment plan called an annuity.
Annuities diﬀer from the kind of investments we studied in Section 6.5 in that payments are
deposited into the account on an on-going basis, and this complicates the mathematics a little.6
Suppose you have an account with annual interest rate r which is compounded n times per year.
We let i = n denote the interest rate per period. Suppose we wish to make ongoing deposits of P
dollars at the end of each compounding period. Let Ak denote the amount in the account after k
compounding periods. Then A1 = P , because we have made our ﬁrst deposit at the end of the ﬁrst
compounding period and no interest has been earned. During the second compounding period, we
earn interest on A1 so that our initial investment has grown to A1 (1 + i) = P (1 + i) in accordance
with Equation 6.1. When we add our second payment at the end of the second period, we get
A2 = A1 (1 + i) + P = P (1 + i) + P = P (1 + i) 1 + 1
1+i The reason for factoring out the P (1 + i) will become apparent in short order. During the third
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