- 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 = e^{X}. Therefore, LnY = Ln(e^{X}) = 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.

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**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 |

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