What Does Platykurtic Mean?

The term “platykurtic” refers to a statistical distribution in which the excess kurtosis value is negative. For this reason, a platykurtic distribution will have thinner tails than a normal distribution will, resulting in fewer extreme positive or negative events. The opposite of a platykurtic distribution is a leptokurtic distribution, in which excess kurtosis is positive.

Investors will consider which statistical distributions are associated with different types of investments when deciding where to invest. More risk-averse investors might prefer assets and markets with platykurtic distributions because those assets are less likely to produce extreme results.

Understanding Platykurtic Distributions

There are three basic kinds of statistical distributions: leptokurtic, mesokurtic, and platykurtic. These distributions differ depending on their amount of excess kurtosis, which relates to the probability of extreme positive or negative events. The normal distribution, which is a type of mesokurtic distribution, has a kurtosis of three. Therefore, distributions with kurtosis greater than three are said to have “positive excess kurtosis,” while those with kurtosis of less than three are said to have “negative excess kurtosis.”

Though mesokurtic distributions have a kurtosis of three, leptokurtic and platykurtic distributions have positive and negative excess kurtosis, respectively. Therefore, leptokurtic distributions have a relatively high probability of extreme events, whereas the opposite is true for platykurtic distributions.

The following figures show charts of these three types of distributions, all with the same standard deviation. Although the figure on the left does not reveal much of the differences between these distributions’ tails, the figure on the right gives a clearer view by plotting the quantiles of the distributions against each other. This technique is known as a quantile-quantile plot, or Q-Q for short.

Special Considerations

Most investors believe that equity market returns more closely resemble a leptokurtic distribution than a platykurtic one. That is, while most returns are likely to be similar to the average return for the market as a whole, returns will occasionally deviate widely from the mean. These dramatic and unpredictable events, sometimes referred to as black swans, are less likely to occur in markets that are platykurtic.

For this reason, more cautious investors might avoid investing in leptokurtic markets and focus on investments offering platykurtic returns. On the other hand, some investors deliberately pursue investments with leptokurtic returns, believing that their extreme positive returns will more than compensate for their extreme negative returns.

Real-World Example of a Platykurtic Distribution

Morningstar published a research paper that featured information on the excess kurtosis levels of different types of assets, as observed between February 1994 and June 2011. The list included a wide range of investments, from U.S. and international equities to real estate, commodities, cash, and bonds.

The levels of excess kurtosis were similarly varied. On the low end of the spectrum were cash and international bonds, which had excess kurtosis of -1.43 and 0.58, respectively. On the other end of the spectrum were U.S. high-yield bonds and hedge-fund arbitrage strategies, offering excess kurtosis of 9.33 and 22.59.

Asset classes with intermediate levels of excess kurtosis included international real estate (2.61), equities from international emerging economies (1.98), and commodities (2.29).

An investor looking at this data could quickly discern what kinds of assets they wish to invest in, given their tolerance for potential black swan events. Risk-averse investors who want to minimize the likelihood of extreme events could focus on low-kurtosis investments, while investors more comfortable with extreme events could focus on high-kurtosis ones.

What Is Average Return?

The average return is the simple mathematical average of a series of returns generated over a specified period of time. An average return is calculated the same way that a simple average is calculated for any set of numbers. The numbers are added together into a single sum, then the sum is divided by the count of the numbers in the set.

Understanding Average Return

There are several return measures and ways to calculate them. For the arithmetic average return, one takes the sum of the returns and divides it by the number of return figures.

$\text{Average Return} = \dfrac{\text{Sum of Returns}}{\text{Number of Returns}}$Average Return=Number of ReturnsSum of Returns​

The average return tells an investor or analyst what the returns for a stock or security have been in the past, or what the returns of a portfolio of companies are. The average return is not the same as an annualized return, as it ignores compounding.

Average Return Example

One example of average return is the simple arithmetic mean. For instance, suppose an investment returns the following annually over a period of five full years: 10%, 15%, 10%, 0%, and 5%. To calculate the average return for the investment over this five-year period, the five annual returns are added together and then divided by 5. This produces an annual average return of 8%.

Now, let’s look at a real-life example. Shares of Walmart returned 9.1% in 2014, lost 28.6% in 2015, gained 12.8% in 2016, gained 42.9% in 2017, and lost 5.7% in 2018. The average return of Walmart over those five years is 6.1%, or 30.5% divided by 5 years.

Calculating Returns From Growth

The simple growth rate is a function of the beginning and ending values or balances. It is calculated by subtracting the ending value from the beginning value and then dividing by the beginning value. The formula is as follows:

$\begin{aligned} &\text{Growth Rate} = \dfrac{\text{BV} -\text{EV}}{\text{BV}}\\ &\textbf{where:}\\ &\text{BV} = \text{Beginning Value}\\ &\text{EV} = \text{Ending Value}\\ \end{aligned}$​Growth Rate=BVBV−EV​where:BV=Beginning ValueEV=Ending Value​

For example, if you invest $10,000 in a company and the stock price increases from $50 to $100, then the return can be calculated by taking the difference between $100 and $50 and dividing by $50. The answer is 100%, which means you now have $20,000.

Average Return Alternatives

Geometric Average

When looking at average historical returns, the geometric average is a more precise calculation. The geometric mean is always lower than the average return. One benefit of using the geometric mean is that the actual amounts invested need not be known. The calculation focuses entirely on the return figures themselves and presents an apples-to-apples comparison when looking at two or more investments’ performances over more various time periods.

The geometric average return is sometimes called the time-weighted rate of return (TWR) because it eliminates the distorting effects on growth rates created by various inflows and outflows of money into an account over time.

Money-Weighted Rate of Return (MWRR)

Alternatively, the money-weighted rate of return (MWRR) incorporates the size and timing of cash flows, making it an effective measure for returns on a portfolio that has received deposits, dividend reinvestments, and/or interest payments, or has had withdrawals.

The MWRR is equivalent to the internal rate of return (IRR), where the net present value equals zero.