Thoughtless use of the average can lead to absurdities. The first case - a man and his dog have an average of three legs. The second case - Mr. Adam does not drink alcohol, while his neighbor does it every day, which means that statistically they both drink alcohol every other day. Both statements, though true, do not make any deeper sense.
Suppose we want to calculate the average on a set of values. But what is the average? We have a lot of them: arithmetic, geometric, square, harmonic, logarithmic, exponential, internal, weighted and others. All of the previously mentioned are also called mathematical averages. In addition to them, we can distinguish positional averages, ie median and dominant (so-called modal value). When we calculate each of them, it may turn out that the values obtained will be extremely different. Which ones should we use then? It depends mainly on the type of data and the effect we want to achieve.
When we say „average”, we almost certainly think about the arithmetic average. In colloquial language, we can put an equal sign here. This type of average is easiest to understand and calculate, just add all observed values and divide by the number of observations. The problem with this average begins when at least one of the values is significantly lower or higher than the others. For example, the GDP of the Masovian Voivodship constitutes 109% of the EU average, but it is mainly the influence of Warsaw (where GDP reaches 147% of the European average), without which the indicator drops to only 60%.
Popularization of using arithmetic average almost everywhere causes that we do not see its defects. First of all, we cannot always use it. As long as in the school were only grades as qualitative (descriptive) variables - excellent, very good, good, etc. - we were not able to get an average out of them. Then, numbers (6, 5, 4, etc.) were assigned to each grade making it possible to calculate the average. However, the bigger problem is the occurrence of values that are significantly different from the others. Let's take a set of the following numbers: 3, 5, 5, 6, 7, 8, 8, 9, 42. Their arithmetic average is 10.3, but you can see immediately that „42” is a suspect value which „falsifies” our result. If we did not include a value that does not match the others, our average would be 6.4.
The rules are simple. From a specific set of numbers (for example: -12, -2, 0, 4, 5, 5, 7, 8, 12, 17), let's eliminate the values that differ the most. In order not to do it individually, let's assume that we remove 20% of the lowest values (in our example they will be: -12 and -2) and 10% of the highest values (12 and 17). From the remaining values, we calculate the arithmetic average. The resulting value of 4.83 is called the internal average or the average with cut-off and is usually a much better information medium than the simple arithmetic average.
Let's assume that a joint-stock company pays 0.12 PLN (12%) for each PLN of its share capital. The bank credit, which is also used by the company, costs much less, it is only 0.04 (4%) per each PLN. Can we say that the average capital cost of this enterprise is 8%? It depends on how much the share capital and the outside capital (credit) the company has. If the equity is 60% of the sum of total capital and the outside capital constitutes the rest (40%), then the 8% is a false result. There is more of more expensive capital (equity, whose cost is 12%) and less of cheaper capital (outside, at a cost of 4%). If we want to correctly calculate the total cost of capital, we should „weigh” the costs of both sources of capital by weights assigned to them which are determined by their share in the total sum (value) of capital. These weights are 60% (equity) and 40% (outside capital). This is done by multiplying the values by the weights and then summing up the results. Thus, the total cost of capital in our case is (12% x 60%) + (4% x 40%) = 8.8%, which underlines the fact that more expensive capital (share capital) has a greater impact on the total result.
The median divides all our observations into two equal groups - values lower than the median make up half of all observations and the number of results higher than the median is the same. To calculate the median of N values we must first arrange these values in a non-negligible series, from the smallest to the largest. After this sorting, all we need to do is point our finger at the value that stands exactly in the middle - to the left of it is exactly the same value as to the right. This indicated value is the median. Note that if our group consists of an even number of values (e.g. 20), we calculate the median as the arithmetic average of the two middle values, that is from the values of 10 and 11 in our series of 20 observations.
The median is also called as the second quartile. All quartiles are three. While the median divides the set into two equal parts (the same number of observations), the remaining quartiles (first and third) divide this set in a different ratio. Due to the first quartile, we divide the observation in the proportion of 25% to 75%, which means that 25% of the observations are lower or equal than the first quartile, and the remaining 75% of observations are equal or higher than it. The third quartile divides the population in a ratio of 75% to 25%. In more complex statistical analyzes, percentiles (also called centiles) are used, which determine the division of observations with an accuracy of one percent. For example, if the percentile is 34, it means the division of the set in a ratio of 34% to 66%.
The basic goal of the exponential average is to smooth out the pace of changes of a given phenomenon. The particular feature of this average is the assigning a higher and higher importance to newer observations (their significance grows exponentially). For example, the average price increase (inflation) calculated in this way to a lesser extent reflects the oldest observations, and therefore to a small extent is disturbed by the dynamic increase in prices in the economy observed in Poland in the 1990s. This causes better reflection of the current economic conditions, which we live in.
After a long reflection, Paul remembered that in the past it had not been better. Although he can not afford everything he wants, he noticed that the last years brought a greater improvement in his situation than the salary increases from the beginning of the 21st century. And in fact - when he calculated the exponential average that puts the greatest pressure on the last four years of much lower inflation, he noticed that his real salary increased by 3.0% a year on average. It's better now! :)
CAGR (Compound Annual Growth Rate) is a cumulative annual growth rate and it is used to calculate the average annual growth of a certain size in the analyzed period, e.g. the average increase of the company's revenues, value of assets or remuneration level in a given period of time. When calculating CAGR, it is assumed that the average annual increases in the considered period are added to the base value of the following year.
Paweł and Gaweł invest their savings on the stock exchange. Paweł brags that the arithmetic average annual rate of return from his portfolio is 6.7%. The average arithmetic rate of return from Gaweł's portfolio is „only” 4,0%, but it is Gaweł who has more reasons to be happy. How is this possible? The average arithmetic rate of return is not an exact measure of return on investment over time. It does not take into account fluctuations in the value of the portfolio during the investment period. If Paweł invested aggressively and in the first year he lost 60% of the portfolio value, to gain 40% in two consecutive years, its actual rate of return is negative (the portfolio lost 21.6% for three years). If Pawel calculated the compound annual growth rate (CAGR), he would learn that in fact he lost 7.8% on average in his investments.
IRR (Internal Rate of Return) presents the profitability of investments spread over time. In short, it is the discount rate for which the current value of discounted cash flows (NPV, Net Present Value) is equal to zero. The higher the IRR, the more the project is likely to be implemented. If it is higher than the capital cost of the company (the previously mentioned WACC), the decision on involvement in the project should be positive.
One of the IRR varieties is the annual percentage rate of charge (APRC), based on the same economic calculations as the IRR and used to assess the actual costs of the loan or credit. The APRC considers all factors affecting the actual cost of the credit: nominal interest, commissions, fees, insurance costs, all other fees, frequency of payments, etc. This makes the APRC a universal indicator, showing using only one value the real average annual (monthly or weekly) interest rate on the repaid debt. The higher this value, the more the client pays, and the more the lender earns.
Financial analysis, especially those based solely on values from the financial statements, is not an ideal tool, as it is based on a certain simplification, averaging reality. It allows you to bring the financial situation of the company closer, but many questions will remain unanswered: what is the degree of redemption of property, how much of the receivables are overdue, or if all stocks in the warehouse are useful?
Calculating financial indicators on the basis of data from one year is usually not difficult, it is enough to base on the values from the annual financial statements. In indicators associating flow values (e.g. revenues) with balance sheet values (e.g. assets), we can use average balance sheet values based on values from the beginning and from the end of the year. We have more possibilities of calculating in the case of indicators calculated for a longer period, let's assume a 3-year period. A simple annual average for flow/flow type indicators (e.g. ROS = net profit / revenues) is calculated as an arithmetic average of three annual ROS indicators. Similarly for the other most common types of indicators: flow/state (eg ROE = net profit / equity) and state/state (e.g. CR = current assets / short-term liabilities). For each indicator, we can calculate its average annual increase, or CAGR. However, the larger difference concerns multi-period values: flow/flow indicators are calculated by adding annual values in the numerator and dividing by the sum of annual values, flow/state indicators are calculated by adding the sum of annual flow values to the average balance sheet value, and state/state indicators are calculated only on the basis of the last annual period. The results obtained in each method can be significantly different.
The average Pole as an example of using the average has in mind the value of the average remuneration. No wonder, by typing in the popular search engine the slogan „average in Poland” as many as 9 out of the first 10 articles apply to this topic.
The average wage in Poland is PLN 4,305 (data from February 2017), but not entirely ... First, the research does not cover all employees (without self-employed, without contracts of mandate or task-specific contracts, without employees of companies employing up to 9 people, without a public sector, etc.). Secondly, the vast majority of Poles earn less than the statistics provided each month. To say this, it is enough to give the median and dominant of earnings, which are also given by the Central Statistical Office, but much less often - every two years. The last published full statistics from the end of 2015 indicate that while the average earnings amounted to 4,163.98 PLN gross, their median (the amount below which exactly half of the Poles earn) was 3,291,56 PLN, and the dominant - that is the most-received salary - only PLN 2,469.47, more than 40% below average salary.
In Poland, the National Bank of Poland is responsible for preventing high inflation. This objective results directly from the act on the NBP: „the basic objective of the activities of the National Bank of Poland is to maintain a stable level of prices (...)”. The NBP's objective is to stabilize inflation at 2.5% with an acceptable fluctuation band of +/- 1 percentage point. The history of inflation in post-war Poland includes an episode in which huge inflation put the economy of the country on the verge of bankruptcy. It was in the years 1989-1990, when prices rose at a rate of 251% and 586%!
Paul checked that for 16 years of his work his salary increased from 2,350 PLN at the end of 2000 to 4,636 PLN at the end of 2016. Knowing that the best measure to check the average annual growth of remuneration will be CAGR, he calculated it and received a result pointing an average annual growth of 4.3%. Paul wondered why, despite such a significant increase in wages, his financial situation did not improve so much. The culprit is inflation (the average price increase in the economy) leading to the loss of money value. After adjusting for inflation, that is calculating the real average annual salary increase, Paul already knows that his income actually increased by 2.1%.
When assessing the attractiveness of investment products, it is worth looking at the results achieved in the long run by products with a similar risk. For example, the average rate of return from the Polish market shares fund for 2016 was 8.7%. However, if we look at an investment from a further perspective (e.g. 5 years), we will see that such a high rate of return is above average. Equity funds earned on average only 5.1% (CAGR for 2011-2016). Unfortunately, as a large group of Poles has learned, investments in the stock market carry the risk of losing part of the value of capital. This shows CAGR for 10 years from the results of Polish equity funds. Its value for the years 2006-2016 was -0.8%, which means that the person who invested money in equity funds 10 years ago, during the investment period lost 0.8% on average per year. It also means that the average investment made at the end of 2016 has not yet regained its value. All through the bear market (period of significant declines) on the Warsaw Stock Exchange, as a result of which in 2008 alone, funds investing in Polish companies lost on average almost 50% in one year!
The example of equity funds of the Polish market shows that high-risk funds are instruments in which it is worth investing only in the long-term. Why? Just look at the CAGR of these funds in the last 15 years (ended December 31, 2016), which was as much as 7.1%! If we compare this result to the rates of return of Polish bond funds (5.3%) or money market funds (4.2%), we will see that the rate of return on investment funds is strictly dependent on the risk taken. Money market funds, which invest mainly in deposits and short-term debt securities, are characterized by the lowest average rate of return in the long-term investment horizon. In short periods, an investment with a higher risk (e.g. shares of Polish companies) is not a guarantee of satisfactory (and thus higher) rates of return than those that can be obtained by investing in secure instruments, such as treasury bonds or deposits.
The average values form the basis for the company's valuation in the comparative method (also called index or multiplier method). This is one of the most intuitive methods of valuation, based on a comparison to companies listed on stock exchange and therefore potentially transparent and insensitive to manipulation. And yet it is completely different. We only need to choose companies properly and calculate the average multipliers, and the spectrum of the finally obtained value will be very wide.
Let's assume that we are based on three comparison multipliers: P/E (price to profit), P/S (price to sales) and P/BV (price to book value). We have also selected comparative group consisting of companies similar to the X company being valued. Knowing the values of each of these indicators, for each company in the group the we calculate average value of each indicator and multiply by the basis for company X - by net profit, sales revenues and equity. We get three values, of which we take out the average value, which is our value of company X. However, we must do it with our head, and by the way - with an idea. To the average, we will not accept companies with a net loss and negative equity, and if we can influence the final result, we will replace the arithmetic average with weighted averages, medians or even (of course, justifying this approach) with one of the quartiles.