Sums and approximations 3.2 Rule of 7 2 Suppose you want to double your money in n years. What annual rate of return would you need? Here are the answers for various numbers of years n=# of Years Rate Needed to double money 4 18 % 6.0% 24 2.9% 19% There's a curious pattern here: the product of the numbers on each line is always around This observation is the basis for an old investing rule that says you can double your money by investing it for n years at 72/n percent interest. For example, you could double your money by investing it for 24 years at 72/24=3 percent interest. This rule isn't exactly right, but it's almost mysteriously close. For example, a dollar invested for 24 ears at 3% becomes $2.03 In general, a dollar invested for n years at 72/n percent interest becomes dollars. This complicated expression doesnt look much like the number 2, but let's try an approximation. One standard trick when the relevant variable(n in this case)appears in the exponent is to analyze the logarithm of the expression Inm=In(1+ 100 72 nIn[1 100n Here were taking a logarithm of 1 plus something small(72/100n), so we can use the pproximation In(1+)N2-3/2 722 100m2.1002n2 162 Now let's exponentiate to undo the earlier logarith m≈e72/100-162/(625n)� � � � � � � � 12 Sums and Approximations 3.2 Rule of 72 Suppose you want to double your money in n years. What annual rate of return would you need? Here are the answers for various numbers of years: n = # of Years Rate Needed to Double Money 4 18.9% 6 12.2% 8 9.0% 12 6.0% 24 2.9% 36 1.9% There’s a curious pattern here: the product of the numbers on each line is always around 70 or 72 or so. This observation is the basis for an old investing rule that says you can double your money by investing it for n years at 72/n percent interest. For example, you could double your money by investing it for 24 years at 72/24 = 3 percent interest. This rule isn’t exactly right, but it’s almost mysteriously close. For example, a dollar invested for 24 years at 3% becomes $2.03. In general, a dollar invested for n years at 72/n percent interest becomes 72/n n m = 1 + 100 dollars. This complicated expression doesn’t look much like the number 2, but let’s try an approximation. One standard trick when the relevant variable (n in this case) appears in the exponent is to analyze the logarithm of the expression. 72/n n ln m = ln 1 + 100 72 = n ln 1 + 100n Here we’re taking a logarithm of 1 plus something small (72/100n), so we can use the approximation ln(1 + x) ≈ x − x2/2. 72 722 ln m ≈ n 2 100n − 2 100 · 2n 72 162 = 100 − 625n Now let’s exponentiate to undo the earlier logarithm: m ≈ e 72/100−162/(625n)