Polynomial functions involve powers like xk, where the exponent k is an integer number. We recall some fundamental rules that are quite handy when dealing with powers and should be familiar from high school mathematics:
In a monomial function f(x) = αxk, the basis x is the independent variable and the exponent k is a fixed parameter. In exponential functions we reverse their roles and deal with expressions such as
One problem is that while it is quite clear what this expression means when x is an integer number, the same cannot be said when x is a real number. Actually, we may easily find a meaning when x is a rational number, i.e., a ratio m/n of integer numbers. We recall that
It is easy to see that defining the square root this way agrees with the general rules above. We may also define a cubic root
i.e., a number such that x = a3. In general
which is again consistent with the rule for powers. However, if the exponent x is a generic real number, it is not clear how ax can be computed, and it is not clear why it should be useful at all. The best way to answer the second question is by a practical example.
Example 2.8 (Continuous compounding in finance) If we invest an amount B at an annual interest rate r, we will end up with a capital B(l + r) after 1 year. If interest is paid after the first year and we reinvest capital plus accrued interest, thus earning interest on interest, we will own B(1 + r)2 after 2 years. Now imagine that interests are paid semiannually, i.e., every 6 months (let us ignore complications due to the fact that months do not consist of the same number of days). In this case, the annual rate r is used just for quotation purposes, but in practice we earn a rate r/2 every 6 months. Hence, after 1 year, we own B(1 + r/2)2. For instance, if we invest $100 at r = 10% with annual compounding, after 1 year our wealth will be
whereas semiannual compounding yields a slightly greater wealth:
As you may imagine, the smaller the time interval at which interest is compounded, the faster our capital grows. You may see this by evaluating
Table 2.1 Discovering Euler’s number e.
What happens if k → +∞, i.e., if compounding occurs so often that we have a continuously compounded interest? To provide a clue, Table 2.1 shows values of function
for increasing values of k. It seems that this sequence does converge to a number. This number is so important that it has been named Euler’s number and is denoted as
Whatever base we use, we know that a power like en has a clear meaning for an integer exponent n. Rather surprisingly, it turns out that
where x is a generic real number. A proof of this equality requires some advanced concepts that we introduce later, so we have to defer this to Example 2.31. The equality (2.6) provides us with one clear and well-defined procedure for computing exponentials with base e. This need not be the best one, but we need not worry about that, since the exponential function is implemented in many software tools, including spreadsheets. We show a plot of the exponential function ex and the negative exponential function e−x = 1/ex in Fig. 2.10. The exponential ex grows quite rapidly even for moderately large values of x; this is where the term exponential growth comes from.
Let us go back to our financial example and consider the growth of one dollar invested over one year at a nominal annual rate r compounded k times per year, when k goes to infinity. We rewrite the expression a bit, introducing a new variable y = k/r, which goes to +∞ when k does so:
Hence, continuous compounding results in an exponential growth of capital. An initial capital B0, earning a continuously compounded interest rate r, grows in time according to the following exponential function:
Fig. 2.10 Graphs of exponential and negative exponential functions.
The negative exponential function comes into play when going the other way around, i.e., when discounting cash flows. If the discount factor r is compounded semiannually, a cash flow C occurring in one year should be discounted as
If there are k periods per year, we use
With continuous compounding, the present value of a cash flow C occurring at time t is
Note that with this concept we may easily discount cash flows occurring at arbitrary time instants. Indeed, continuous compounding does a great job at simplifying calculations in financial mathematics.
Now we have made a little step forward, since we know how to compute an exponential function with base e; still, we do not know how to compute something like ax for an arbitrary value a. To do this, we need to introduce the logarithm as an inverse of the exponential function (see Section 2.6).
Fig. 2.11 A discontinuous function.
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