The rule of 72: help for arguing with your mother-in-law

The rule of 72: help for arguing with your mother-in-law (which is never a good idea, you can’t win..).. she says things used to be so cheap in 1960, and you hold your head and cry – haven’t you heard of a thing called inflation? So what is the equivalent in today’s money, then? And so on.. The rule of 72 comes in handy (not that you will win the argument, of course..). The rule of 72 says that for something to double, the product of n, the number of periods, and r, the interest rate, should be 72. Since 1960, it has been 55 years. If the inflation rate has been around 8%, it means prices have been doubling every 9 years. So something that cost 10 rupees in 1960 should cost 10 x 2^6 – that is, 640 today... Works for all kinds of growth rates, discount rates, and the like. So, if the exchange rate in 1990 was $1 =Rs. 14, and the inflation differential between the US and India has been around 7%, the exchange rate should double every 10 years or so... today, 25 years later, it should be around 56 times (square root of 2) – that is, 56 x 1.4 = 78. So 63 doesn’t sound so bad, does it? For those who wonder where this 72 comes from, remember the natural log of 2 is 0.69.. So start with the equation (1+r)^n = 2 n ln(1+r) = ln 2 =.69 the Maclaurin series expansion (approximation) for ln (1+r) is then applied n (r-r^2/2+r^3/3-...) = 0.69 nr(1-r/2+..)=0.69, neglecting the higher order terms substituting 0.1 for r as the range in which we want to look for an approximation gives us nr (0.95) = 0.69, that is, nr = 0.72 roughly.. (actually gives us 0.726 but 0.72 is an easier number to work with).. If compounding is continuous, we get the rule of 69, of course, since the relevant equation is e^nr = 2 nr = ln 2 =0.69 but compounding isn’t usually continuous, so rule of 72 works well.