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6 log normality

6 log normality - What Practitioners Need to Know About...

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What Practitioners Need to Know . . . About Lognormality i CC g u I cn 10 Mark Kritzman Windbam Capital Manage- ment When reading the financial litera- ture we often see statements to the effect that a particular result depends on the assumption that retums are lognormally distrib- uted. What exactly is a lognormal distribution, and why is it relevant to financial analysis? In order to address this question, let us start with a review of logarithms. Logarithms A logarithm is simply the power to which a base must be raised to yield a particular value. For exam- ple, the exponent 2 is the loga- rithm of 16 to the base 4, because 4 squared equals 16. The loga- rithm of 8 to the base 4 equals 1.5, because 4 raised to the power 1.5 equals 8. The choice of a base depends on the context in which we use log- arithms. For simple mathematical procedures, it is common to use the base 10, which explains why logarithms to the base 10 are called common logs. The base 10 is popular because the logarithms of 10, 100, 1000 and so on equal 1, 2, 3 respectively. Why should we care about loga- rithms? In the days prior to pocket calculators ^ong before my time), logarithms were useful for performing complicated com- putations. Financial analysts would multiply large numbers by summing their logarithms, and they would divide them by sub- tracting their logarithms. For ex- ample, given a base of 4, we can multiply 16 times 8 by raising the number 4 to the 35 power, which is the sum of the logarithms 2 and 1.5. Of course, you might argue that it would have been easier to multiply large numbers direaly than to raise a base to a fractional power. In the olden days, how- ever, an analyst would use a slide rule, which is a ruler with a slid- ing central strip marked with log- arithmic scales. In most financial applications, in- stead of logarithms to the base 10, we use logarithms to the base 2.71828, which is denoted by the letter e in honor of the famous Swiss mathematician Euler. These logarithms, which are called nat- ural logs and are abbreviated as ln, have a special property. Sup- pose we invest \$100 at the begin- ning of the year at an annual interest rate of 10096. At the end ofthe year we will receive \$200— our original principal of 1100 and another \$100 of interest. Now suppose our interest is com- pounded semiannually. Our year- end payment will equal \$225. By the middle of the year we will have earned \$50 of interest, which is then reinvested to gen- erate the additional \$25.

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6 log normality - What Practitioners Need to Know About...

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