Our formula for calculating doubling times is fairly simple, but there is an
even simpler approximate formula that does reasonably well for the small
interest rates common to everyday life. To calculate the approximate time to
doubling, simply divide 0.69 by the interest rate.
The is called the rule of 69, and it is commonly used for quick calculations.
(You may have heard of it as the rule of 70, since it is usually easier to
divide something into 70 than to divide it into 69.)
Define a function called approx that takes an interest rate as its
argument and returns the approximate time to doubling using the rule of 69.
Experiment with doubling and approx to get a feel for how good the
approximation is. You'll find that it's within 10% up to an interest rate
of around .20. This is larger than the interest rates common to such
investments as bank accounts, mutual funds, mortgages, and credit cards.
As a good Mathematica user, you might what to know at exactly what interest rate the
error in the approximate doubling formula becomes 10% of the true answer.
It's not hard to ask the question using Solve, so let's do it:
Solve[doubling[R]-approx[R] / doubling[R] == .10, R]
Unfortunately, Mathematica is unable to solve the equation. (Sometimes Mathematica will
spend a long time working on a problem that it can't solve. You can tell that
it is working on such a problem because the cursor turns into a clock face. If
you get tired of waiting for an answer, you can interrupt the computation by
typing Command. during the calculation.)
If we ask the same question using NSolve, though, we do get an
answer:
NSolve[doubling[R]-approx[R] / doubling[R] == .10, R]
What's going on? The answer is that Solve tries to solve the problem
symbolically, which essentially means that Mathematica is pushing symbols
around to try to find an exact answer. And it gets stuck. By contrast,
NSolve tries to solve the problem numerically, which means that it is
internally doing number crunching to find an approximate (but very close)
solution.
The approximate doubling formula is handy, but you may be wondering where
it came from. If you have studied Taylor series expansions in your calculus
class, you can easily derive it. Here's how.
Recall that the body of our approximation formula divides ln(2) by ln(1+R),
where R is the interest rate. As you may know, an expression such as
ln(1+R) can be approximated by taking the first few terms of its Taylor
series expansion.
What? You don't happen to know the Taylor series expansion of ln(1+R), and
you don't remember how to derive it? That's OK---Mathematica can do it for you. All
you need is enough mathematical knowledge to ask the question and understand
the answer.
Series[Log[1+R], {R, 0, 3}]
This gives us the first three terms in the Taylor series expansion. (The last
term that Mathematica gives you is simply an indication that there are more terms,
all powers of R or higher, that are not displayed. You can ask Mathematica to give
you more terms if you wish---use the help facility if you're interested how.)
Let's use the first term of the expansion, which is R, as an approximation
to ln(1+R). The body of our function now reduces approximately to the
quotient of ln(2) and R. And since ln(2) is approximately 0.69, we end
up with the rule of 69.