A book I'm working through is J.C. Stillwell's history of mathematics. This was the happy result of a recommendation in Needham's Visual Complex Analysis -- also a worthwhile read.
Stillwell spends a bit of time on the history of the binomial theorem, which gives the expansion for a polynomial of arbitrary degree:
The coefficients of the expansion are equivalent to the number of combinations of k elements taken from a set of n elements:
These coefficients have a use in probability theory, which is probably the more familiar to most people -- namely, that the probability of getting k successes in n binary trials (e.g., coin flips) is the coefficient divided by the total number of possibilities (2^n for coin flips). They can also be calculated using Pascal's Triangle:
The binomial expansion is sometimes expressed in a simplified formula, in which y is set equal to 1, so that the equation reduces to:
The relationship between the binomial theorem and probability theory is of interest because the same formula turns up in a pretty important place in finance -- namely, in the formula for compound interest:
Here, the binomial expansion is multiplied by a fixed coefficient equal to the principal (or "present value") PV. The total expected value of the sequence of interest payments is simply the binomial expansion, with interest i as the variable x.
Curiously, one does not often see cash-flows expanded through the binomial theorem. Rather, the tendency is for each of the n payments to multiplied out. But that's of course equivalent to mutiplying the present value PV times the binomial expansion of interest payments.
What interpretation should be attached to the fact that compound interest is a binomial expansion? Here are a few observations:
- Ignore the Distant Future: The present value of compounded interest is a polynomial expansion in powers of interest; because interest < 1, the lower-order terms dominate the expansion.
- Don't Ignore the Near Future: Some higher-order terms in the expansion -- the ones that fall in the middle of Pascal's Trial -- have coefficients large enough to make up for the fact that interest rates < 1 are being multiplied. In other words, the base of the pyramid of compound interest is built of lots of combinations of principal and interest payments that occur in the immediate and near-future.
- You Can't Do Better than e: So long as your future value derives exclusively from organic growth in your present value -- i.e., no external infusions of cash -- there is no way to make the future value grow faster than exponential (sorry, Kurzweil).
The last is a consequence of the fact that as compounding occurs more and more frequently over the period in which payments are made, future value nonetheless approaches a finite limit:
Thus, if compound interest is like a binomial expansion in powers of interest, then continuously compounded interest is an exponential function of interest.
A final thought to ponder: Why do we calculate future value with this formula? In many cases, the answer is obvious -- because that's how we agree payments will be made by a borrower.
In other cases, however, that answer is less obviously satisfactory, and may even be incorrect. We can use the binomial expansion as an approximation for future value when there is some predictable process of growth occuring. But when growth occurs at different rates, or when compounding cannot be carried out efficiently, some other approximation for future value is necessary. The use of the binomial expansion as a crutch may be positively misleading.
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