Compound Interest - Rule of 70Date: Tue, 29 Nov 1994 14:12:30 -0800 (PST) From: Mizuki Nishisaka Subject: Rule of 70 There is a famous "Rule of 70" for compounded interest. What is it and why does it work? Mizuki Nishisaka Date: 7 Dec 1994 17:44:55 GMT From: Dr. Math Subject: Re: Rule of 70 Hey Mizuki, The question is: Given a compounding rate 1.r ( r is some two-digit number like 08 15 12 04 etc.), how many years will it take to double if it is compounded yearly? For example, if your savings account has an 8% interest rate, then your compounding rate is 1.08. The way to do this is the following: Let p0 be your initial investment and p1,p2,p3,... be how much you have after each year. So p1 = (1.08)p0 p2 = (1.08)(1.08)p0 p3 = (1.08)(1.08)(1.08)p0 p4 = (1.08)(1.08)(1.08)(1.08)p0 p5 = (1.08)(1.08)(1.08)(1.08)(1.08)p0 So pN equals p0 times (1.08)^n - so the question we are asking is when does (1.08)^n = 2 Using natural logs we can find that n = ln(2)/ln(1.08) and it turns out that this is very close to 70/r. So 70/r can be used to approximate the number of years needed to double your money if r is your interest rate. Here are a few comments on why this is true. Precisely what we are saying is that ln(2)/ln(1 + r/100) is approx. 70/r. Or, the value of the first derivative of the lefthand side is approximately seventy. This is true; however, for calculation purposes it is probably easier to show that the derivative of the reciprocal { ln(1 + r/100)/ln(2) } is 1/70. Once you know this it is enough to show that these two equations are close at one point, and you can say that they will be reasonably close at many points, near r = 10 for instance. Hope that this helps you. Thanks and bye. Ethan, Doctor ON CAll Date: 28 Sep 2000 17:19:55 GMT From: Dr. TWE Subject: Re: Rule of 70 Actually, in financial circles, it is usually referred to as the "rule of 72." Dividing the value 72 by the interest rate gives closer approximations when dealing with rates around 6% to 8%, which are typical in the financial industry. You can find out more about the rule of 72 by typing "rule of 72" without the quotation marks in our Ask Dr. Math searcher at: http://mathforum.org/mathgrepform.html When dealing with exponential growth in biology, the rule of 70 is more often used. When the growth rate is 3% or less, then dividing the value 70 by the growth rate gives closer approximations than 72. I hope this helps! -Doctor TWE http://mathforum.org/dr.math |
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