# compound interest

Compound Interest ( English compound interest is) in Finance an overdue interest , of the capital added ( capitalized ) and the future at the prevailing interest rate is subject to interest together with the principal.

## General

The interest on capital (in the form of a loan / credit or as an investment ) is the price for the temporary transfer of the scarce production factor capital . If the overdue loan interest from borrowers paid or overdue credit interest from investors consumed (reversed in negative interest ), the question of compound interest does not arise because then the future only the pure capital is subject to interest. Only when the due credit or credit interest becomes part of the capital through capitalization does the compound interest take effect. Because through capitalization, the capital increases by the unpaid or unused interest, so that interest continues to be paid on this as well. The best-known example is the capitalization of the credited and unused savings interest on savings accounts .

## history

In the past, religious or secular regulations often dealt with interest prohibitions or the prohibition of compound interest. The reason for the prohibition of compound interest is that the debtor should not be crushed by the interest burden. Compound interest is as old as the interest on which it depends. Around 2400 BC , the Sumerians probably used the oldest term for interest ( maš ; German for  “calf, young goat” ). This concept of interest thus indicates wages in kind . The compound interest ( mašmaš ) also has its origin here. As a relief for the compound interest-related increase in debt , the Sumerians under their king En-metena made possible around 2402 BC. A debt relief . In the Hammurapi Codex from 1755/1754 BC compound interest could be calculated if the due and unpaid interest remained separate from the capital and was paid interest transparently for the debtor by the creditor in order to be protected from the debtor's late payment behavior .

According to ancient Roman law , compound interest ( Latin usurae usurarum ) was still allowed with Cicero until Justinian I forbade it in the 6th century AD , as soon as the backward interest reached the level of the capital and led to its doubling ( Latin ultra alterum tantum ). This regulation is still contained in Austria in § 1335 ABGB. According to Ulpian's Digest , any kind of compound interest was inadmissible, regardless of whether the backward interest was capitalized or new capital was formed from it. Justinian I demanded that compound interest should not be demanded from the debtor in any way ( Latin nullo modo usurae usurarum a debitoribus exigantur ). Emperor Diocletian demanded that in the credit repayment no disadvantages are likely to grow up and therefore not allowed to compound interest. His prohibition of compound interest was called Anatozismus ( Greek ανατοκισμός anatokismós "taking of compound interest", from aná "on" and tókos "interest"), which is still used today as a legal term in German-speaking countries . The Indian mathematician Aryabhata presented the first mathematical compound interest calculations in the 5th century.

Where there was a ban on interest, the subject of compound interest was superfluous. The Jewish federal book forbade interest on loans to the poor between 1000 and 800 BC ( Ex 22.24  EU ). The Deuteronomy demands: "You should not take any interest from your fellow citizens, neither interest on money, nor interest on food, nor interest on anything that can be borrowed" ( Dtn 23.20  EU ). The Tanach understood “national comrades” only to mean Jews . As a rule, Roman law recognized an interest-free loan with the Mutuum, mostly as a courtesy to relatives or friends, where interest could only be charged through a special stipulation . With the advent of Christianity , the interest payment met with severe criticism from the church , because needy people in need should receive interest-free loans ( Lev 25,36-37  EU ). The actual starting point for the prohibition of interest is the commandment of Deuteronomy “You should not take interest from your brother, neither for money nor for food nor for anything for which you can take interest” ( Dtn 23 : 20-21  EU ). The canon law stated interest rate for oral robbery. The Islam .. Took over the Christian ban on interest and called after 622 AD this, click, not interest ( Arabic Riba ; "growth, proliferation") to take by re-taking the creditors in multiple amounts, which have borrowed it ( Koran , Sura 3: 130). Several suras deal with the prohibition of interest. In Sura 2: 275 Allah declares the contract of sale ( bay ' ) permissible ( halāl ) and the interest forbidden ( harām ).

In the Middle Ages , compound interest was also known as “damage”. The Italian arithmetic master Leonardo Fibonacci submitted further compound interest calculations based on the Julian calendar . In Austria, in 1244, the Fridericianum allowed Jews to pay compound interest in Article 23. In Frankfurt am Main in 1368 a debtor undertook to his Jewish creditor to allow compound interest to be calculated on unpaid interest. The Archbishop of Mainz, Dietrich Schenk von Erbach , forbade the Jews of his diocese to pay compound interest in 1457, but had to revise this again in the same year. Emperor Friedrich III. declared in 1470 that trade and industry could not exist without compound interest; it is the lesser evil to allow the Jews to take compound interest than to allow the Christians to do so. The ecclesiastical ban on interest and the secular maximum interest were limited to compound interest from the 16th century onwards.

In Schleswig-Holstein , Duke Friedrich III. on March 23, 1654 a “Constitution on the compound interest of the capital of minors”, which made the calculation of compound interest a fine. In 1689, Jakob I Bernoulli requested a daily calculation of compound interest. A Trier ordinance of October 31, 1768 stipulated: "Anyone who takes interest from interest will be punished in the same way as anyone who accepts more than 6%. The moral philosopher Richard Price developed the parable of Josephspfennig in 1772 as an advice to his government on how to reorganize the English state budget, which was running into a budget deficit due to compound interest . Price calculated that if Joseph of Nazareth had invested a penny at 5% interest on the birth of his son Jesus Christ , this would have grown to the weight of 150 million earths if capitalized. He described that “money that bears compound interest grows slowly at first; but since the rate of growth is continually accelerating, after a time it becomes so rapid that it scoffs at all imagination ”.

The General Prussian Land Law (APL) of June 1794 stated: "Interest from interest may not be demanded" (I 11, § 818 APL), unless there is a judicial approval (I 11, § 820 APL). In France , the Civil Code , published in March 1804, moved away from the absolute ban on compound interest. If the interest arrears are higher than an annual amount, it could become interest-bearing by court judgment (Art. 1154 Code civil). The Austrian General Civil Code, which came into force in January 1812, followed on from this and stipulated that “Interest may never be taken from interest; but two-year or even older interest arrears can be prescribed as new capital by means of an agreement ”(§ 998 ABGB). The Higher Commercial Court (OHG) Lübeck ruled in November 1855 that compound interest is permissible on current accounts. The Saxon Civil Code of March 1865 forbade interest on arrears, even if the latter is legally recognized (Section 679 of the Saxony Civil Code). At that time, the lawyers made a distinction between interest, which earned interest as such ( Latin anatocismus separatus ) and the interest capitalized after the due date ( Latin anatocismus conjunctus ). There was therefore no anatocism if the interest was paid or used up.

In his major work Das Kapital , published in 1867, Karl Marx understood the accumulation process of capital in the economy as the accumulation of compound interest and saw compound interest as part of the surplus value that is converted back into capital. Albert Einstein is said to have remarked in 1921 that the "greatest invention of human thought was compound interest".

## Legal issues

The BGB knows the principle of freedom of contract , which is implemented as freedom from interest with regard to interest. Interest agreements are therefore generally allowed, only certain agreements that are disadvantageous for the interest debtor are prohibited. The prior agreement of compound interest ( anatocism ) is prohibited ( Section 248 (1) BGB). Agreements made nevertheless are void ( Section 134 BGB). Exceptions are made only for credit institutions (§ 248 para. 2 BGB) and the current account under merchants ( § 355 para. 1 HGB ). The ban on compound interest for all other legal entities serves to protect debtors. According to § 289 BGB no interest may be charged on default interest , which extends the prohibition of compound interest from § 248 BGB. In § 497 para. 2 BGB is the right of the lender , compound interest on consumer loan agreements to require not excluded, however, the amount of the statutory interest rate ( § 246 limited BGB).

## Economical meaning

The interest rate is a measure of risk and a risk premium when the lender or investor classifies the credit risk . On the other hand, due to his obligation to pay interest and the risk of compound interest , the debtor takes on a greater or lesser financial risk , which under certain conditions can drive him into bankruptcy . Compound interest therefore places an additional burden on debtors , favors the creditors and contributes to an exponential growth in their debts and assets . The higher the interest-bearing capital and / or the interest rate and the longer the term , the higher this growth is . As long as a debtor has debt sustainability and debt servicing ability, he can raise the debt servicing ( interest and repayment ) so that the problem of compound interest does not arise for him. If these prerequisites are no longer met and interest arrears are also paid interest, he falls into a debt trap . Above all, it consists in the fact that the exponentially growing debts are less and less covered by assets and the income tends to be insufficient to cover the interest burden ( interest coverage ratio ).

The problem of compound interest is often misrepresented in government debt . Compound interest can only occur in the case of countries that no longer pay their interest on government debt (such as government bonds ) or whose payment requires new debt . The former category includes Argentina , which already suspended the payment of debt servicing on its first government bond, issued in 1825, in 1829 for the next 28 years until 1857. This moratorium was followed by another in April 1987. If the creditors do not waive interest, the unpaid interest will lead to an increase in national debt. Since then, countries with debt crises have mostly acted according to the second variant and paid their interest by refinancing it with new debt in the state budget . These included in particular the USA , the PIIGS states, highly indebted developing countries and also Germany (until 2013). Germany has been generating budget surpluses since 2014 , so the compound interest problem no longer arises.

In the event of a payment ban or moratorium , the creditor's claim to interest is not lost, but increases the creditor's total claim and triggers compound interest when capitalized. The compound interest effect arises in countries when at least interest rates contribute to new borrowing or its increase. If backward interest is taken into account, for example in the case of debt rescheduling or consolidation , compound interest also arises. These conditions also apply to compound interest from other economic entities such as companies and private households if they have to finance their debt servicing with additional loans.

If states, companies or private households have compound interest problems, this financial situation is a clear indication of an economic problem for a debtor. Economic indicators such as economic growth (measured in terms of gross domestic product ), corporate profits and income must increase sustainably and progressively in order to guarantee payment of the interest burden.

## International

Internationally legal maximum interest rates , usury , compound interest prohibitions and absolute interest prohibitions serve to protect debtors . In Switzerland , anatocism is anchored in Art. 314 Para. 3 OR , here too there are exceptions for current accounts and credit institutions. In Austria , § 1335 ABGB allows compound interest until the interest debt has increased to the amount of the main debt. Compound interest can only be charged from the date on which the dispute is pending . In France , Art. 1343-2 CC now regulates that compound interest may be calculated from accrued interest for one year. Luxembourg, on the other hand, prohibits compound interest within one year in Art. 1154 Civil Code.

## Compound interest calculation

Compound interest calculation, a sub-area of financial mathematics , deals with the calculation of compound interest depending on the interest rate as well as the amount and duration of an investment . The compound interest calculation answers the question of which final capital an initial capital has grown after a total of time periods if  interest is paid at the fixed interest rate of % in each of these time periods . ${\ displaystyle K_ {n}}$${\ displaystyle K_ {0}}$${\ displaystyle n}$${\ displaystyle p}$

The compound interest formula with the rate of interest p is:

${\ displaystyle K_ {n} = K_ {0} \ left (1 + {\ frac {p} {100}} \ right) ^ {n}}$

or alternatively with the interest factor q :

${\ displaystyle K_ {n} = K_ {0} \ cdot q ^ {n}}$

with = final capital; = Initial capital; = Interest rate or = interest factor and = number of applicable periods / years. ${\ displaystyle K_ {n}}$${\ displaystyle K_ {0}}$${\ displaystyle p}$${\ displaystyle q}$${\ displaystyle n}$

The formula is derived from the following relationship: A saver makes a one-off investment in an account at a credit institution in the amount of an initial capital. This capital earns compound interest for a certain investment period. The investment period consists of several equally long periods of time that are continuously counted with the help of natural numbers (as an index ). This means that the investment period can be formulated as the sum of all periods: ${\ displaystyle i}$${\ displaystyle n}$

${\ displaystyle {{\ text {investment period}} = {\ text {period}} _ {1} + {\ text {period}} _ {2} + \ dots + {\ text {period}} _ {i} + \ dots + {\ text {period}} _ {(n-1)} + {\ text {period}} _ {n}}}$

At the beginning of the first period ( ), the saver's account has the initial capital : ${\ displaystyle i = 1}$${\ displaystyle K_ {0}}$

${\ displaystyle {\ text {Initial capital at the beginning of period}} _ {1} \ colon K_ {0}}$

The two index values ​​used are important. The first period is given the index value , while the initial capital is numbered with. The different numbering is due to the fact that the original initial capital does not change during the first period. The interest is only credited after the first period, i.e. at the beginning of the second period. ${\ displaystyle i = 1}$${\ displaystyle i = 0}$${\ displaystyle K_ {0}}$

The saver has decided not to access his capital for the duration of the investment. For this, the bank or, ultimately, the borrower “rewards” him with a credit of interest. It is now common practice that interest is credited repeatedly at the end of each of the periods within the investment period. ${\ displaystyle n}$

So it is z. B. the interest value paid for the first period : ${\ displaystyle Z_ {1}}$

${\ displaystyle {\ text {Interest value for the period}} _ {1} \ colon Z_ {1}}$

The specific amount of the interest value in the first period is determined as follows: The bank expresses the “reward” of the saver for the transfer of the capital in percentage form as an interest rate , e.g. B. "six percent" . The number before the percentage sign is called the rate of interest . The interest value credited at the end of the first period is related to the initial net present value in exactly the same way as the interest rate is related to the value 100. This relationship represents a relationship equation (proportion). ${\ displaystyle Z_ {1}}$${\ displaystyle \ left (6 \, \% = {\ tfrac {6} {100}} \ right)}$ ${\ displaystyle p}$${\ displaystyle Z_ {1}}$${\ displaystyle K_ {0}}$${\ displaystyle p}$

${\ displaystyle {\ frac {{\ text {Interest value for period}} _ {1}} {{\ text {Capital value at the beginning of period}} _ {1}}} = {\ frac {{\ text {Interest rate for Period}} _ {1}} {100}} \ qquad \ Leftrightarrow \ qquad {\ frac {Z_ {1}} {K_ {0}}} = {\ frac {p} {100}}}$.

This relationship equation can be transformed into:

${\ displaystyle Z_ {1} = K_ {0} \ cdot {\ frac {p} {100}}}$.

This relationship between interest value and capital value in the first period can be generalized in such a way that it applies to each and capital value in every -th period: ${\ displaystyle Z_ {i}}$${\ displaystyle K _ {(i-1)}}$${\ displaystyle i}$

${\ displaystyle Z_ {i} = K_ {i-1} \ cdot {\ frac {p} {100}}}$.

Up to this point the “interest for a period” has been considered.

To consider compound interest, it must be taken into account again that the saver is “rewarded” for “making available” the initial capital in accordance with the above interest value formula. The following interest value will be credited to his account at the end of the first period : ${\ displaystyle K_ {0}}$${\ displaystyle Z_ {1}}$

${\ displaystyle Z_ {1} = K_ {0} \ cdot {\ frac {p} {100}}}$.

Thus, the initial capital grows by exactly this interest value until the end of the first period . Your total gives the new account balance. This sum is also called the (provisional) final capital , which is consequently given the index value : ${\ displaystyle K_ {0}}$${\ displaystyle Z_ {1}}$${\ displaystyle K_ {1}}$${\ displaystyle i = 1}$

${\ displaystyle K_ {1} = K_ {0} + Z_ {1} = K_ {0} + K_ {0} \ cdot {\ frac {p} {100}} = K_ {0} \ left (1+ { \ frac {p} {100}} \ right)}$.

This (provisional) final capital is now also the initial capital for the second period ( ). It "earns" the interest value , which is added again: ${\ displaystyle K_ {1}}$${\ displaystyle i = 2}$${\ displaystyle Z_ {2}}$

${\ displaystyle {K_ {2} = K_ {1} + Z_ {2} = K_ {1} + K_ {1} \ cdot {\ frac {p} {100}} = K_ {1} \ left (1+ {\ frac {p} {100}} \ right) = K_ {0} \ left (1 + {\ frac {p} {100}} \ right) \ left (1 + {\ frac {p} {100} } \ right) = K_ {0} \ left (1 + {\ frac {p} {100}} \ right) ^ {2}}}$.

The following always applies to positive interest rates${\ displaystyle p> 0}$

${\ displaystyle 1 + {\ frac {p} {100}}> 1}$

This term is therefore called the compounding factor .

This means that the compound interest effect already takes effect during the second period: The initial capital in the first period increases with the compounding factor on the (preliminary) final capital . In the same way, the capital rises in the second period with the same compounding factor on the (provisional) final capital . Viewed over both periods, however, the initial capital has grown disproportionately, namely with the square of the compounding factor, to the (preliminary) final capital . ${\ displaystyle K_ {0}}$${\ displaystyle 1 + {\ frac {p} {100}}}$${\ displaystyle K_ {1}}$${\ displaystyle K_ {1}}$${\ displaystyle K_ {2}}$${\ displaystyle K_ {0}}$${\ displaystyle K_ {2}}$

In generalized terms, this means that at the end of the investment period, i.e. after a total of interest periods, the final capital is finally obtained by multiplying the initial capital by the compounding factor ${\ displaystyle n}$${\ displaystyle K_ {n}}$${\ displaystyle n}$${\ displaystyle K_ {0}}$

${\ displaystyle K_ {n} = K_ {0} \ left (1 + {\ frac {p} {100}} \ right) ^ {n}}$

results.

## Exponential growth

If interest is capitalized, this also results in future interest on the capitalized interest. This results in an exponential increase in total capital.

The compound interest formula

${\ displaystyle K_ {n} = K_ {0} \ left (1 + {\ frac {p} {100}} \ right) ^ {n}}$

can be determined by setting the growth constant λ ( see section Essential Terms and Notation in the article Exponential Growth )

${\ displaystyle \ lambda = \ ln \ left (1 + {\ frac {p} {100}} \ right)}$

Convert into the formula for exponential growth:

${\ displaystyle K_ {n} = K_ {0} \ cdot e ^ {\ lambda n}}$

The compound interest formula is therefore a special form of the exponential growth formula

${\ displaystyle N_ {t} = N_ {0} \ cdot (1 + p) ^ {t}}$.

An example of the extreme amounts that are arithmetically obtained by assuming growth rates that remain constant over a long period of time due to compound interest effects is the Joseph pfennig invested in the year zero .

From the compound interest formulas one can derive the 72 rule as an approximation formula, when an investment (investment of an amount at an interest rate) has doubled.

## Asset concentration

In the event of random fluctuations in individual returns , compound interest causes a concentration of assets . Joseph E. Fargione , Clarence Lehman and Stephen Polasky showed in 2011 that coincidence alone in combination with the compound interest effect can lead to an unlimited concentration of wealth.

For a population of independent capital assets, each with the same start-up capital, the -th asset currently results from the compound interest formula ${\ displaystyle n}$${\ displaystyle i}$${\ displaystyle t}$

${\ displaystyle K_ {i} (t) = K_ {0} \ left (1 + {\ frac {p_ {i, 1}} {100}} \ right) \ left (1 + {\ frac {p_ {i , 2}} {100}} \ right) \ dots \ left (1 + {\ frac {p_ {i, t}} {100}} \ right) = K_ {0} q_ {i, 1} q_ {i , 2} \ dots q_ {i, t} = K_ {0} e ^ {r_ {i, 1}} e ^ {r_ {i, 2}} \ dots e ^ {r_ {i, t}}}$.

So for ${\ displaystyle i = 1,2, \ dots, n}$

${\ displaystyle K_ {i} (t) = K_ {0} e ^ {r_ {i, 1} + r_ {i, 2} + \ dots + r_ {i, t}} = K_ {0} e ^ { x_ {i} (t)} \ quad {\ text {with}} \ quad x_ {i} (t) = \ sum _ {k = 1} ^ {t} r_ {i, k}}$.

If one assumes that the rates are drawn from a normal distribution with expected value and variance , then the exponents are normally distributed with expected value and variance ; this follows from the invariance of the normal distribution compared to the convolution . First consider the special case of the time : At this time the exponents are normally distributed with expected value and variance , therefore the powers are logarithmically normally distributed with the parameters and . The total assets can be calculated simply by adding up the individual capital assets: ${\ displaystyle r_ {i, k}}$${\ displaystyle \ mu}$${\ displaystyle \ sigma ^ {2}}$${\ displaystyle x_ {i} (t)}$${\ displaystyle \ mu t}$${\ displaystyle \ sigma ^ {2} t}$${\ displaystyle t = 1}$${\ displaystyle x_ {i}}$${\ displaystyle \ mu}$${\ displaystyle \ sigma ^ {2}}$${\ displaystyle e ^ {x_ {i}}}$ ${\ displaystyle \ mu}$${\ displaystyle \ sigma}$

${\ displaystyle W_ {Total} = \ sum _ {i = 1} ^ {n} K_ {i} = K_ {0} \ sum _ {i = 1} ^ {n} e ^ {x_ {i}} = K_ {0} n \ left ({\ frac {1} {n}} \ sum _ {i = 1} ^ {n} e ^ {x_ {i}} \ right)}$

Due to the laws of large numbers , the arithmetic mean of the powers stabilizes around the expected value of a logarithmic normal distribution with the parameters and . If the number of individual capital assets is large enough, the mean value can be replaced by the expected value and the total assets can be represented by an integral: ${\ displaystyle n}$${\ displaystyle e ^ {x_ {i}}}$${\ displaystyle \ mu}$${\ displaystyle \ sigma}$

${\ displaystyle W_ {Total} = K_ {0} n \ int _ {0} ^ {\ infty} {\ frac {1} {\ sigma y {\ sqrt {2 \ pi}}}} e ^ {- { \ frac {1} {2}} \ left ({\ frac {\ ln (y) - \ mu} {\ sigma}} \ right) ^ {2}} y \, \ mathrm {d} y = K_ { 0} n \ int _ {- \ infty} ^ {\ infty} {\ frac {1} {\ sigma {\ sqrt {2 \ pi}}}} e ^ {- {\ frac {1} {2}} \ left ({\ frac {x- \ mu} {\ sigma}} \ right) ^ {2}} e ^ {x} \ mathrm {d} x}$

In order to determine the sub-fund at the top of the population, the lower integration limit must be raised from to the value with a constant to be determined in more detail . The fraction of the population selected in this way is then added to ${\ displaystyle - \ infty}$${\ displaystyle \ mu + h \ sigma}$${\ displaystyle h}$

${\ displaystyle \ int _ {\ mu + h \ sigma} ^ {\ infty} {\ frac {1} {\ sigma {\ sqrt {2 \ pi}}}} e ^ {- {\ frac {1} { 2}} \ left ({\ frac {x- \ mu} {\ sigma}} \ right) ^ {2}} \ mathrm {d} x = 1- \ int _ {- \ infty} ^ {\ mu + h \ sigma} {\ frac {1} {\ sigma {\ sqrt {2 \ pi}}}} e ^ {- {\ frac {1} {2}} \ left ({\ frac {x- \ mu} {\ sigma}} \ right) ^ {2}} \ mathrm {d} x = 1- \ int _ {- \ infty} ^ {h} {\ frac {1} {\ sqrt {2 \ pi}}} e ^ {- {\ frac {1} {2}} z ^ {2}} \ mathrm {d} z = 1- \ Phi (h)}$.

Here, the designated error integral the distribution function of the standard normal distribution and the corresponding inverse distribution function . If one now determines the constant by , the thus selected fraction of the population is equal . The wealth share of the richest percent of the population is therefore calculated using the quotient of two integrals as follows: ${\ displaystyle \ Phi}$${\ displaystyle \ Phi ^ {- 1}}$${\ displaystyle h}$${\ displaystyle h = \ Phi ^ {- 1} (1 - {\ tfrac {a} {100}})}$${\ displaystyle {\ tfrac {a} {100}}}$${\ displaystyle a}$

${\ displaystyle P_ {Top ~ a \%} = {\ frac {\ int _ {\ mu + h \ sigma} ^ {\ infty} {\ frac {1} {\ sigma {\ sqrt {2 \ pi}} }} e ^ {- {\ frac {1} {2}} \ left ({\ frac {x- \ mu} {\ sigma}} \ right) ^ {2}} e ^ {x} \ mathrm {d } x} {\ int _ {- \ infty} ^ {\ infty} {\ frac {1} {\ sigma {\ sqrt {2 \ pi}}}} e ^ {- {\ frac {1} {2} } \ left ({\ frac {x- \ mu} {\ sigma}} \ right) ^ {2}} e ^ {x} \ mathrm {d} x}} = 1 - {\ frac {\ int _ { - \ infty} ^ {\ mu + h \ sigma} {\ frac {1} {\ sigma {\ sqrt {2 \ pi}}}} e ^ {- {\ frac {1} {2 \ sigma ^ {2 }}} \ left (x- \ mu \ right) ^ {2} + x} \ mathrm {d} x} {\ int _ {- \ infty} ^ {\ infty} {\ frac {1} {\ sigma {\ sqrt {2 \ pi}}}} e ^ {- {\ frac {1} {2 \ sigma ^ {2}}} \ left (x- \ mu \ right) ^ {2} + x} \ mathrm {d} x}} \ quad {\ text {with}} \ quad h = \ Phi ^ {- 1} \ left (1 - {\ frac {a} {100}} \ right)}$

In general, the distribution function of any normal distribution can be represented with the help of a suitable substitution by the error integral, and it is always

${\ displaystyle \ int _ {- \ infty} ^ {\ xi} {\ frac {1} {\ sigma {\ sqrt {2 \ pi}}}} e ^ {- {\ frac {1} {2 \ sigma ^ {2}}} \ left (x- \ mu \ right) ^ {2}} \ mathrm {d} x = \ int _ {- \ infty} ^ {\ frac {\ xi - \ mu} {\ sigma }} {\ frac {1} {\ sqrt {2 \ pi}}} e ^ {- {\ frac {1} {2}} z ^ {2}} \ mathrm {d} z = \ Phi \ left ( {\ frac {\ xi - \ mu} {\ sigma}} \ right)}$.

Because of

${\ displaystyle - {\ frac {1} {2 \ sigma ^ {2}}} \ left (x- \ mu \ right) ^ {2} + x = - {\ frac {1} {2 \ sigma ^ { 2}}} \ left (x ^ {2} -2 \ mu x + \ mu ^ {2} -2 \ sigma ^ {2} x \ right) = - {\ frac {1} {2 \ sigma ^ {2 }}} \ left (\ left (x - (\ mu + \ sigma ^ {2}) \ right) ^ {2} -2 \ mu \ sigma ^ {2} - \ sigma ^ {4} \ right)}$

is

${\ displaystyle \ int _ {- \ infty} ^ {\ xi} {\ frac {1} {\ sigma {\ sqrt {2 \ pi}}}} e ^ {- {\ frac {1} {2 \ sigma ^ {2}}} \ left (x- \ mu \ right) ^ {2} + x} \ mathrm {d} x = \ int _ {- \ infty} ^ {\ xi} {\ frac {1} { \ sigma {\ sqrt {2 \ pi}}}} e ^ {- {\ frac {1} {2 \ sigma ^ {2}}} \ left (x - (\ mu + \ sigma ^ {2}) \ right) ^ {2}} e ^ {\ mu + {\ frac {1} {2}} \ sigma ^ {2}} \ mathrm {d} x = \ Phi \ left ({\ frac {\ xi - ( \ mu + \ sigma ^ {2})} {\ sigma}} \ right) e ^ {\ mu + {\ frac {1} {2}} \ sigma ^ {2}}}$.

Thus applies

${\ displaystyle P_ {Top ~ a \%} = 1 - {\ frac {\ Phi \ left ({\ frac {\ mu + h \ sigma - (\ mu + \ sigma ^ {2})} {\ sigma} } \ right) e ^ {\ mu + {\ frac {1} {2}} \ sigma ^ {2}}} {e ^ {\ mu + {\ frac {1} {2}} \ sigma ^ {2 }}}} = 1- \ Phi \ left (h- \ sigma \ right) = \ Phi \ left (\ sigma -h \ right)}$.

The general case for any time results from the special case, in which through and through are replaced throughout the calculation . Thus, the wealth share of the richest percent of the population is currently calculated as follows: ${\ displaystyle t}$${\ displaystyle \ mu}$${\ displaystyle \ mu t}$${\ displaystyle \ sigma}$${\ displaystyle \ sigma {\ sqrt {t}}}$${\ displaystyle a}$${\ displaystyle t}$

${\ displaystyle P_ {Top ~ a \%} (t) = {\ frac {\ int _ {\ mu t + h \ sigma {\ sqrt {t}}} ^ {\ infty} {\ frac {1} { \ sigma {\ sqrt {2 \ pi t}}}} e ^ {- {\ frac {1} {2t}} \ left ({\ frac {x- \ mu t} {\ sigma}} \ right) ^ {2}} e ^ {x} \ mathrm {d} x} {\ int _ {- \ infty} ^ {\ infty} {\ frac {1} {\ sigma {\ sqrt {2 \ pi t}}} } e ^ {- {\ frac {1} {2t}} \ left ({\ frac {x- \ mu t} {\ sigma}} \ right) ^ {2}} e ^ {x} \ mathrm {d } x}} = \ Phi \ left (\ sigma {\ sqrt {t}} - h \ right) \ quad {\ text {with}} \ quad h = \ Phi ^ {- 1} \ left (1- { \ frac {a} {100}} \ right)}$

For example, the wealth share of the richest percentage of the population is currently through ${\ displaystyle t}$

${\ displaystyle P_ {Top ~ 1 \%} (t) = \ Phi \ left (\ sigma {\ sqrt {t}} - \ Phi ^ {- 1} (0 {,} 99) \ right) \ approx \ Phi \ left (\ sigma {\ sqrt {t}} - 2 {,} 326 \ right)}$

specified. Because of the relationship , the error integral can be replaced by the error function . This allows the wealth share of the richest percent of the population to be represented at the moment as in the source: ${\ displaystyle \ Phi}$${\ displaystyle \ Phi (\ xi) = {\ tfrac {1} {2}} \ left (1+ \ operatorname {erf} \ left ({\ tfrac {\ xi} {\ sqrt {2}}} \ right ) \ right)}$ ${\ displaystyle \ operatorname {erf}}$${\ displaystyle a}$${\ displaystyle t}$

${\ displaystyle P_ {Top ~ a \%} (t) = {\ frac {\ int _ {\ mu t + h \ sigma {\ sqrt {t}}} ^ {\ infty} {\ frac {1} { \ sigma {\ sqrt {2 \ pi t}}}} e ^ {- {\ frac {1} {2t}} \ left ({\ frac {x- \ mu t} {\ sigma}} \ right) ^ {2}} e ^ {x} \ mathrm {d} x} {\ int _ {- \ infty} ^ {\ infty} {\ frac {1} {\ sigma {\ sqrt {2 \ pi t}}} } e ^ {- {\ frac {1} {2t}} \ left ({\ frac {x- \ mu t} {\ sigma}} \ right) ^ {2}} e ^ {x} \ mathrm {d } x}} = {\ frac {1} {2}} \ left (1+ \ operatorname {erf} \ left ({\ frac {\ sigma {\ sqrt {t}} - h} {\ sqrt {2} }} \ right) \ right) \ quad {\ text {with}} \ quad h = \ Phi ^ {- 1} \ left (1 - {\ frac {a} {100}} \ right)}$

For every value, no matter how small , the wealth share of the richest percent of the population goes against the number 1 over time. This means that an arbitrarily small fraction of the population owns almost 100% of the total wealth after some time. Because of the central limit theorem , this result applies even if the rates themselves are not normally distributed. Since the concentration of assets triggered by compound interest is based on a stochastic process , the concentration effect can be illustrated using a Monte Carlo simulation . ${\ displaystyle a}$${\ displaystyle a}$${\ displaystyle r_ {i, k}}$

## example

The initial capital is 1000 €, the interest rate is 5%, 50 years are considered.

### Without compound interest

The annual 5% interest is not added to the initial capital and thus reinvested, but taken and collected separately. After 50 years, the sum of the initial capital and separately collected individual annual interest increases to € 3,500:

${\ displaystyle K_ {50} = 1000 {,} 00 \, \ mathrm {\ euro} + \ left (1000 {,} 00 \, \ mathrm {\ euro} \ cdot {\ frac {5} {100}} \ right) \ cdot 50 = 3500 {,} 00 \, \ mathrm {\ euro}}$.

### With compound interest

If the annual interest is always added to the new amount to be invested (capitalized), the initial € 1000, with otherwise unchanged parameters, becomes a total of € 11,467 in the same time:

${\ displaystyle K_ {50} = 1000 {,} 00 \, \ mathrm {\ euro} \ cdot \ left (1 + {\ frac {5} {100}} \ right) ^ {50} = 11467 {,} 40 \, \ mathrm {\ euro}}$.

### Effects

If, however, inflation of, for example, 3% is factored in over the same period, the compound interest effect is significantly reduced due to the devaluation, since after 50 years the money only has a value relative to the original value of 0.228: this value results from

${\ displaystyle {\ frac {1} {(100 \, \% + 3 \, \%) ^ {50}}} = {\ frac {1} {1 {,} 03 ^ {50}}}}$.

The 11,467 € then only have a purchasing power of 2,616 € based on the time of the initial capital. If, on the other hand, you calculate the inflation on the sum of the initial capital and the separately collected individual annual interest without compound interest totaling € 3,500, then after 50 years you only have a purchasing power of € 798 and thus significantly less than the capital employed. In order to preserve the value of a credit balance in the event of inflation, the following must be observed: since inflation causes exponential monetary devaluation, interest must also be exponential using compound interest, otherwise - without interest being added to interest - even at an interest rate that is clearly is above the inflation rate, the real value of a credit will decline in the long run.

The compound interest effect that occurs with national debt can be compensated for if there is sufficient economic growth. If, for example, a state has to pay interest on its debts at 5% and an inflation rate of 3%, real economic growth would have to be around 2% annually so that the real debt ratio does not increase if the interest is paid through new borrowing (with old debts remaining the same). In this case, inflation and real economic growth would permanently compensate for the compound interest effect, since inflation and economic growth are subject to the same exponential growth as the compound interest effect. The nominal growth rate of government revenue then corresponds to the interest rate on government debt. If economic growth is not sufficient to fully compensate for the compound interest effect, then in the long term either the interest rate must fall, inflation rise or that part of the interest burden must be paid annually that is not offset by inflation and economic growth. With real economic growth of 0%, at least the difference between the interest rate and inflation - in this example 2% - would have to be applied annually so that over-indebtedness does not arise in the long term.

## Individual evidence

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22. Hubert Beyerle: The interest cannot be grasped . In: The time. October 27, 2005.
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24. ^ Otto Palandt, Christian Grüneberg: BGB commentary . 24th edition, 2014, § 289 Rn. 1.
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28. Joseph E. Fargione et al: Entrepreneurs, Chance, and the Deterministic Concentration of Wealth. # (Results)
29. Simulation of asset concentration according to Fargione, Lehman and Polasky ( Memento from April 11, 2020 in the Internet Archive )