Mathematics and Statistics for Financial Risk Management

CHAPTER 1

Some Basic Math

In this chapter we review three math topics—logarithms, combinatorics, and geometric series—and one financial topic, discount factors. Emphasis is given to the specific aspects of these topics that are most relevant to risk management.

LOGARITHMS

In mathematics, logarithms, or logs, are related to exponents, as follows:

(1.1)

We say, “The log of a, base b, equals x, which implies that a equals b to the x and vice versa.” If we take the log of the right-hand side of Equation 1.1 and use the identity from the left-hand side of the equation, we can show that:

(1.2) logb(bx) = logb a = x

logb(bx) = x

Taking the log of bx effectively cancels out the exponentiation, leaving us with x.

An important property of logarithms is that the logarithm of the product of two variables is equal to the sum of the logarithms of those two variables. For two variables, X and Y:

(1.3)

Similarly, the logarithm of the ratio of two variables is equal to the difference of their logarithms:

(1.4)

If we replace Y with X in Equation 1.3, we get:

(1.5)

We can generalize this result to get the following power rule:

(1.6)

In general, the base of the logarithm, b, can have any value. Base 10 and base 2 are popular bases in certain fields, but in many fields, and especially in finance, e, Euler's number, is by far the most popular. Base e is so popular that mathematicians have given it its own name and notation. When the base of a logarithm is e, we refer to it as a natural logarithm. In formulas, we write:

(1.7)

From this point on, unless noted otherwise, assume that any mention of logarithms refers to natural logarithms.

Logarithms are defined for all real numbers greater than or equal to zero. Exhibit 1.1 shows a plot of the logarithm function. The logarithm of zero is negative infinity, and the logarithm of one is zero. The function grows without bound; that is, as X approaches infinity, the ln(X) approaches infinity as well.

Exhibit 1.1 Natural Logarithm

LOG RETURNS

One of the most common applications of logarithms in finance is computing log returns. Log returns are defined as follows:

(1.8) rt ≡ ln(1 + Rt) where

Here rt is the log return at time t, Rt is the standard or simple return, and Pt is the price of the security at time t. We use this convention of capital R for simple returns and lowercase r for log returns throughout the rest of the book. This convention is popular, but by no means universal. Also, be careful: Despite the name, the log return is not the log of Rt, but the log of (1 + Rt).

For small values, log returns and simple returns will be very close in size. A simple return of 0% translates exactly to a log return of 0%. A simple return of 10% translates to a log return of 9.53%. That the values are so close is convenient for checking data and preventing operational errors. Exhibit 1.2 shows some additional simple returns along with their corresponding log returns.

Exhibit 1.2 Log Returns and Simple Returns

To get a more precise estimate of the relationship between standard returns and log returns, we can use the following approximation:1

(1.9)

As long as R is small, the second term on the right-hand side of Equation 1.9 will be negligible, and the log return and the simple return will have very similar values.

COMPOUNDING

Log returns might seem more complex than simple returns, but they have a number of advantages over simple returns in financial applications. One of the most useful features of log returns has to do with compounding returns. To get the return of a security for two periods using simple returns, we have to do something that is not very intuitive, namely adding one to each of the returns, multiplying, and then subtracting one:

(1.10)

Here the first subscript on R denotes the length of the return, and the second subscript is the traditional time subscript. With log returns, calculating multiperiod returns is much simpler; we simply add:

(1.11)

By substituting Equation 1.8 into Equation 1.10 and Equation 1.11, you can see that these definitions are equivalent. It is also fairly straightforward to generalize this notation to any return length.

LIMITED LIABILITY

Another useful feature of log returns relates to limited liability. For many financial assets, including equities and bonds, the most that you can lose is the amount that you've put into them. For example, if you purchase a share of XYZ Corporation for $100, the most you can lose is that $100. This is known as limited liability. Today, limited liability is such a common feature of financial instruments that it is easy to take it for granted, but this was not always the case. Indeed, the widespread adoption of limited liability in the nineteenth century made possible the large publicly traded companies that are so important to our modern economy, and the vast financial markets that accompany them.

That you can lose only your initial investment is equivalent to saying that the minimum possible return on your investment is −100%. At the other end of the spectrum, there is no upper limit to the amount you can make in an investment. The maximum possible return is, in theory, infinite. This range for simple returns, −100% to infinity, translates to a range of negative infinity to positive infinity for log returns.

(1.12)

As we will see in the following chapters, when it comes to mathematical and computer models in finance it is often much easier to work with variables that are unbounded—that is, variables that can range from negative infinity to positive ­infinity. This makes log returns a natural choice for many financial models.

GRAPHING LOG RETURNS

Another useful feature of log returns is how they relate to log prices. By rearranging Equation 1.10 and taking logs, it is easy to see that:

(1.13)

where pt is the log of Pt, the price at time t. To calculate log returns, rather than taking the log of one plus the simple return, we can simply calculate the logs of the prices and subtract.

Logarithms are also useful for charting time series that grow exponentially. Many computer applications allow you to chart data on a logarithmic scale. For an asset whose price grows exponentially, a logarithmic scale prevents the compression of data at low levels. Also, by rearranging Equation 1.13, we can easily see that the change in the log price over time is equal to the log return:

(1.14)

It follows that, for an asset whose return is constant, the change in the log price will also be constant over time. On a chart, this constant rate of change over time will translate into a constant slope. Exhibits 1.3 and 1.4 both show an asset whose price is increasing by 20% each year. The y-axis for the first chart shows the price; the y-axis for the second chart displays the log price.

For the chart in Exhibit 1.3, it is hard to tell if the rate of return is increasing or decreasing over time. For the chart in Exhibit 1.4, the fact that the line is straight is equivalent to saying that the line has a constant slope. From Equation 1.14 we know that this constant slope is equivalent to a constant rate of return.

Exhibit 1.3 Normal Prices

Exhibit 1.4 Log Prices

In Exhibit 1.4, we could have shown actual prices on the y-axis, but having the log prices allows us to do something else. Using Equation 1.14, we can easily estimate the average return for the asset. In the graph, the log price increases from approximately 4.6 to 6.4 over 10 periods. Subtracting and dividing gives us (6.4...