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      Part VII Mathematics and Statistics for Financial Risk Management: Time Series Models

      Learning Objectives

      ■ Describe the concept of “random walks” and how it might apply to investment markets.

      ■ Explain autocorrelation as it relates to measuring risk/return and random walks.

      ■ Discuss why over-estimations and under-estimations of risk occur based on the assumptions a) that variance is linear in time and b) that no serial correlation exists.

      Part I Some Basic Math

      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:

(3.1)

      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:

(3.2)

      It is fairly straightforward to generalize this notation to any return length.

      Sample Problem

      Question:

Using Equation 3.1, generalize Equation 3.2 to returns of any length.

      Answer:

      Note that to get to the last line, we took the logs of both sides of the previous equation, using the fact that the log of the product of any two variables is equal to the sum of their logs.

      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 percent. 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 percent to infinity, translates to a range of negative infinity to positive infinity for log returns.

      (3.3)

      As we will see in the following sections, 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.

      Graphing Log Returns

      Another useful feature of log returns is how they relate to log prices. By rearranging Equation 3.1 and taking logs, it is easy to see that:

(3.4)

      where p_t is the log of P_t, 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 3.4, we can easily see that the change in the log price over time is equal to the log return:

(3.5)

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. Figures 3.1 and 3.2 both show an asset whose price is increasing by 20 percent each year. The y-axis for the first chart shows the price; the y-axis for the second chart displays the log price.

FIGURE 3.1 Normal Prices

FIGURE 3.2 Log Prices

For the chart in Figure 3.1, it is hard to tell if the rate of return is increasing or decreasing over time. For the chart in Figure 3.2, the fact that the line is straight is equivalent to saying that the line has a constant slope. From Equation 3.5 we know that this constant slope is equivalent to a constant rate of return.

      In the first chart, the y-axis could just have easily been the actual price (on a log scale), but having the log prices allows us to do something else. Using Equation 3.4, we can easily estimate the log return. Over 10 periods, the log price increases from approximately 4.6 to 6.4. Subtracting and dividing gives us (6.4 – 4.6)/10 = 18 percent. So the log return is 18 percent per period, which – because log returns and simple returns are very close for small values – is very close to the actual simple return of 20 percent.

      Continuously Compounded Returns

      Another topic related to the idea of log returns is continuously compounded returns. For many financial products, including bonds, mortgages, and credit cards, interest rates are often quoted on an annualized periodic or nominal basis. At each payment date, the amount to be paid is equal to this nominal rate, divided by the number of periods, multiplied by some notional amount. For example, a bond with monthly coupon payments, a nominal rate of 6 %, and a notional value of $1,000, would pay a coupon of $5 each month: (6 % × $1,000)/12 = $5.

      How do we compare two instruments with different payment frequencies? Are you better off paying 5 percent on an annual basis or 4.5 percent on a monthly basis? One solution is to turn the nominal rate into an annualized rate:

      (3.6)

      where n is СКАЧАТЬ