The mathematics behind finance can be a bit confusing and tedious. Luckily, most computer programs do complex calculations. However, understanding the various statistical terms and methods, their meanings, and which best analyzes investments is crucial when picking the appropriate security and getting the desired impact on a 澳洲幸运5官方开奖结果体彩网:portfolio.

One important decision is choosing between normal versus 澳洲幸运5官方开奖结果体彩网:lognormal distributions, both are often referred to in r꧃esearch literature. Before choosing, yo꧃u need to know:

• What they are
• What differences exist between them
• How they impact investment decisions

Normal Versus Lognormal

Both normal and lognormal distributions are used in statistical mathematics to describe the 澳洲幸运5官方开奖结果体彩网:probability of an event occurring. Flipping a coin is an easily understood example of probability. If you flip a coin 1000 times, what is the distribution of results? That is, how many times will it land on heads or tails? There is a 50% probability that it will land on either heads or tails. This basi💯c example describes the probability and distribution of results.

There are many types of distributions, one of which is the normal or 澳洲幸运5官方开奖结果体彩网:bell curve distribution.

In a normal distribution, 68% (34%+34%) of the results fall within one 澳洲幸运5官方开奖结果体彩网:standard deviation, and 95% (68%+13.5%+13.5%) fall within two standard deviations. At the center (the 0 point in the image above) the median (the middle value in the set), the mode (the value that occurs most often), and the mean (澳洲幸运5官方开奖结果体彩网:arithmetic average) are all the same.

The lognormal distribution differs from the normal distribution in several ways. A major difference is in its shape: the 澳洲幸运5官方开奖结果体彩网:normal distribution is symmetrical, whereas the lognormal distribution is not. Because the values in a lognormal distribution&nb﷽sp;are positive, they create a right-skewed curve.

This skewness is important in determining which distribution is appropriate to use in investment decision-making. A further distinction is that the values used to derive a lognorm🐭al distribution are normally distributed.

Let's clarify with an example. An investor wants to know an expected future stock price. Since stocks grow at a compounded rate, they need to use a growth factor.

To calculate possible expected prices, they will take the current stock price and multiply it by various 澳洲幸运5官方开奖结果体彩网:rates of return (which are mathematically derived 澳洲幸运5官方开奖结果体彩网:exponential factors based on 澳洲幸运5官方开奖结果体彩网:compounding), which are assumed to 🥀be normally distributed. When the investor continuously compounds𝓡 the returns, they create a lognormal distribution.

This✅ distribution is always positive even if some of the rates of return are negative, which will happen 50% of the time in a normal distribution. The future stock price will always be positive because stock prices cannot fall below $0.

When to Use Nor🦩mal Versus Lognormal Distribution

The preceding example helped us arrive at what really matters to investors: when to use each method. Lognormal is extremely useful when analyzing stock prices. As long as the growth factor used is assumed to be normally distributed (as we assume with the rate of return), then the lognormal distribution makes sense. Normal d🧔istribution cannot be used to model stock prices because it has a negative side, and stock prices cannot fall below zero.

Another similar use of the lognormal distribution is with the pricing of options. The 澳洲幸运5官方开奖结果体彩网:Black-Scholes model—used to price options—uses the lognormal distribution as its basis to determine 澳洲幸运5官方开奖结果体彩网:option prices.

Conversely, normal distribution works better when calculating 澳洲幸运5官方开奖结果体彩网:total portfolio returns. The normal distribution is used because the weighted average return (the product of the weight of a security in a portfolio and its rate of return) is more accurate in describing the actual 澳洲幸运5官方开奖结果体彩网:portfolio return (positive or negative), particularly if the weights vary by a large degree. The following ജis a typical example:

Although the lognormal return for total portfolio performance may be qꦿuicker to calculate over a longer time period, it fails to capture the individual stock weights, which can distort the return tremendously. Also, portfolio returns can be positive or negative, and a lognormal distribution will fail to capture the negative aspects.

The Bottom Line

Although the nuances that differentiate normal🎃 and lognormal distributions may escape us most of the time, knowledge of the appearance and characteristics of each distribution will provide insight into how to model portfolio returns and fu♊ture stock prices.