Compound Interest. Dan Saunders. 1 Simple Interest. Given a principal P,
interest rate r, and a time horizon of t periods, we can calculate simple interest as:
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Compound Interest Dan Saunders
1
Simple Interest
Given a principal P , interest rate r, and a time horizon of t periods, we can calculate simple interest as: I = P rt This is simply the amount of interest earned on the principal, multiplied by the number of times the interest is earned.
2 2.1
Compound Interest Compound Once Per Period
Suppose the current amount is the principal A0 = P . The simple interest for the first period will be I1 = P r. One period from now, the new amount will be: A1 = P + P r = P (1 + r) Two periods from now the amount will increase again, this time by I2 = A1 r, yielding: A2 = A1 (1 + r) = P (1 + r)2 By induction, t periods from now the principal and the compound interest will amount to: At = P (1 + r)t
2.2
Compound n Times Per Period
Suppose that we compound n times per period. Then the simple interest in the 1/nth period will be I1/n = P (r/n), and the amount will be: r A1/n = P 1 + n By the first period, this interest will have compounded n times yielding: r n A1 = P 1 + n By similar logic from the previous section, after t periods the principal will grow to: r nt At = P 1 + n 1
2.3
Continuously Compounding
First, define h = n/r. Then notice that we can rewrite the amount at time t as: " h #rt 1 At = P 1+ h Recall one definition of the Euler number: h 1 e = lim 1 + h→∞ h Since h = n/r, as n → ∞ we see that h → ∞ too. Therefore, if we compound continuously, the formula simplifies to: At = P ert
2.4
Example- “Rule of 72”
i. Suppose you want to calculate the years to double an investment: 2 = (1 + r)t ln 2 ln(1 + r) Note that ln 2 = .693, which is why it is sometimes called the “Rule of 69”. Often the number 72 is used as a close approximation because it is divisible by many small integers; hence the name “Rule of 72”. For small values of r, we have the approximation that ln(1 + r) ≈ r. This gives us the formula: t=
t≈
ln 2 r
ii. Now suppose that the interest compounds n times per period. We want to solve: r nt 2= 1+ n ln 2 t= n · ln (1 + r/n) For small values of r we have the approximation that ln(1 + r/n) ≈ r/n. Then the formula simplifies to: ln 2 t≈ r iii. If we have continuous compounding, the result is even easier: 2 = ert ln 2 t= r 2
