- CFA Exams
- 2023 Level I
- Topic 1. Quantitative Methods
- Learning Module 4. Common Probability Distributions
- Subject 9. The Lognormal Distribution
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Subject 9. The Lognormal Distribution PDF Download
The key properties of the normal distribution have been presented in the LOS above. As a summary, some of the key properties of this distribution are:
- It is symmetrical about the mean.
- It has zero skewness.
- It has a kurtosis of 3.
A random variable, Y, follows a lognormal distribution if its natural logarithm, lnY, is normally distributed. You can think of the term lognormal as "the log is normal." For example, suppose X is a normal random variable, and Y = eX. Therefore, LnY = Ln(eX) = X. Because X is normally distributed, Y follows a lognormal distribution.
- Like the normal distribution, the lognormal distribution is completely described by two parameters: mean and variance.
- Unlike the normal distribution, the lognormal distribution is defined in terms of the parameters of the associated normal distribution. Note that the mean of Y is not equal to the mean of X, and the variance of Y is not equal to the variance of X. In contrast, the normal distribution is defined by its own mean and variance.
- The lognormal distribution is bounded below by 0. In contrast, the normal distribution extends to negative infinity without limit.
- The lognormal distribution is skewed to the right (i.e., it has a long right tail). In contrast, the normal distribution is bell-shaped (i.e., it is symmetrical).

The reverse is also true; if a random variable Y follows a lognormal distribution, then its natural logarithm, lnY, is normally distributed.
User Contributed Comments 15
User | Comment |
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surob | Perfect explanation why continuosly compounded return is better than HPY. Good job!!! |
achu | That is a good explanation of why Cnts compouding is better. I wonder if it's used in the real world, though. |
rufi | this is a good tool for BSOPM |
bahodir | what is BSOPM? |
bobert | Black-Scholes Option Pricing Model |
aakash1108 | nice. |
Seemorr | What kind of variable would be lognormally distributed, but not normally? |
riouxcf | Some variables which have frequent outliers can be made more normal by taking the log. The normal distribution tends to underestimate extremes. |
jpducros | Is Continuously coumpounding yield always < HPY ? More generally, can we always write : Cont Compound Y < HPY < MMY < BEY < EAY And is BEY = BDiscountYield ? So many Yields it becomes complicated to remember everything, and the logic behind. |
pbielstein | You should keep in mind though that in this example the arithmetic average for the discretely compounded returns is taken. If you take the geometric average then you obtain sqrt{(1-0.2)*(1+0.25)} - 1 = 0 |
czar | Seemor: stock prices (log) and stock returns (normal) as stock prices lowest value can only be 0 while returns can be negative |
johntan1979 | Shouldn't the discretely compounded quarterly rate of return = (1 + HPR/4)^4 ? 1, not (1 + HPR)^1/4 - 1? |
johntan1979 | How can the quarterly compounding be less than the annual HPR? |
fanDango | The quarterly compounding rate is not 4*X = annual HPR because you are compounding the principle and interest each quarter. |
Streberli | @riouxcf lognormal distributions can't go negative they are used in stockprices not returns. what you mean is student-t distribution with this one you can include the fat tails in of return distributions |

You have a wonderful website and definitely should take some credit for your members' outstanding grades.

Colin Sampaleanu
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