Series are another important topic in classical calculus. They have limited use in the remainder, so we will offer a very limited treatment, covering what is strictly necessary. To motivate the study of series, let us consider once again the price of a fixed-coupon bond, with coupon C and face value F, maturing at time T. If we discount cash flows with rate r, we know that its price is
If we have to compute the price of a long-term bond, the formula above results in a quite tedious calculation. Can we find a more compact expression for the bond price? Furthermore, suppose that maturity goes to infinity. This may sound weird and unrealistic, but years ago bonds were issued in the UK with infinite maturity; a bond like this was called a console. How can we calculate the price of a console, since this requires us to discount and add an infinite number of cash flows?
Before proceeding further, let us define a series. Consider an infinite sequence of numbers:
For a finite n, the partial sum of the first n terms in the sequence is
Apparently, summing an infinite number of terms in the sequence makes no sense. It is a reasonable bet that the result will be either +∞ or −∞. However, if an goes to zero fast enough, when n → +∞, the partial sum sn can converge to a finite limit. In such a case, the value of the infinite series is defined as the limit of the sum.
DEFINITION 2.21 Given an infinite sequence an, n = 1, 2, 3, …, the corresponding series is defined as the limit of the partial sums (2.21):
provided that the limit exists.
A very useful case is the so-called geometric series.
Example 2.37 (Geometric series) Let α be a real number and consider the series
This kind of series is called geometric. If we choose α = 2, there is no doubt that the series goes to infinity, as each element of the sequence αi is increasing and we sum an infinite number of larger and larger terms. If we choose α = 0.5, the elements of the sequence tend to zero; if they do so fast enough, maybe, the series will converge to a finite limit. If α is negative, there is the additional complication of oscillations of terms αi, which are positive for even powers and negative for odd powers.
To keep it simple, let us just consider 0 < α < 1 and rewrite the series as follows:
Solving for S yields
which makes sense if α < 1; if α > 1 the formula above yields a negative number, which is not reasonable as the series is a sum of positive terms. Sometimes, the starting element of the series we are interested in is α, rather than α0 = 1. Then, it is easy to adapt formula (2.22):
Alternatively, we may regard the series starting from i = 1 as αS to obtain the same result.
Example 2.38 (Pricing a console bond) A console bond is a bond with infinite maturity. Hence, its price is obtained by discounting an infinite sequence of cash flows:
All we have to do is using the result for the geometric series by plugging α = 1/(1 + r) into Eq. (2.23):
In order to check whether this result makes financial sense, consider the value of a console bond paying $1 per year, when r = 10%. The formula yields P = 1/0.1 = $10. Indeed, if we invest $10 at an interest rate r = 10%, each year we will get the dollar we need to pay the coupon, while maintaining the capital.
The basic result from the geometric series can be applied to find a compact expression for a finite sum. A finite sum can be expressed as the difference of two infinite series:
Example 2.39 (Pricing a fixed-coupon bond) To find the price of a bond paying a coupon C per year, we plug α = 1/(1 + r) into Eq. (2.24):
In order to get the bond price, we should also include the discounted value of the face value refunded at maturity:
If we let T → +∞, we recover the pricing formula for the console.
In this book, we will just use variations of the geometric series. Hence, we will not proceed further with series theory. We just mention a useful result stating that, under suitable conditions, a series can be differentiated term by term. We clarify what we mean by a useful example.
Example 2.40 (Derivative of the geometric series) We know that the geometric series converges to S(α) = 1/(1 − α). Let us take its derivative with respect to α:
Now, let us consider a slightly different series
With a little intuition, we see that this series is related to the derivative of the geometric series. In fact, using term-by-term differentiation, we get
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