# How are the interest rates determined

## Calculate interest: simple interest, compound interest & types of interest

### The interest calculation at a glance

Definition: Interest calculation describes how to calculate interest as consideration for borrowed capital.

Calculation method: A distinction is made between simple interest calculation and compound interest calculation (exponential interest calculation).

Base rate: represents the current market interest rate; recalculated and published by the Bundesbank every six months

problem: Interest calculations are complicated; many variables have to be taken into account for the qualitative assessment of various loan offers.

### Interest calculation: simple and exponential interest

Interest plays one part in our society leading role. Whether in the media or in financial investments: interest is mentioned again and again. There is no getting around this concept for investors. But very few people know exactly how the mathematical procedure for calculating interest actually works.

With the simple interest calculation, the interest amount and the interest-bearing capital (credit or savings) are not added together. In the Compound interest calculation (exponential interest rate), however, the interest is added to the basic amount. They are then taken into account in the further interest calculation. The difference seems negligible at first, but it is serious.

#### Explained using an example

If an investor puts in an amount of 10 euros at an annual interest rate of 5 percent, he will receive one after one year Interest gain in the amount of 0.50 euros. Since the investor wants to invest his money over a longer period of time, he does not have the return paid out, but adds it to the balance. This then increased to 10.50 euros for the second year.

At the end of the second period, our investor will again receive the 5 percent interest on the total balance. The interest gain now corresponds to EUR 0.52. It can now be dated compound interest be spoken. Because the profit from the first period is also compounded in the second year. Even if the actual additional profit in this example is only 0.02 euros, the effects are extensive.

The thought experiment from Josephspfennig shows what potential there is in exponential interest rates - especially with longer investment horizons.

### The Josephspfennig

The mental game of Josephspfennig was set up in 1772 by the English economist Richard Price and illustrates the growth of one long-term assets through compound interest. He assumed that the Father of Christ invested one cent each time his son was born at an annual interest rate of 5 percent. This money is still in the bank to this day. The compound interest accelerates the rate of growth more and more.

One cent, the interest profits of which would be added to the credit balance and further interest paid, would add up to the unbelievable sum of 23,911,022,046,136,196,279,738,609,537,102,053,900,288 euros after a term of 2,000 years. That’s close 24 sextillion. If Joseph had paid out his interest income every year, his balance today would be just 1.01 euros. The example of Josephspfennig makes it clear: With exponential interest, the long-term is decisive.

### Formula for simple compounding

The formula for calculating simple interest without compound interest (linear interest) is:

K n = K 0 + K 0 ⋅ n ⋅ i = K 0 ⋅ (1 + n ⋅ i) {\ displaystyle K_ {n} = K_ {0} + K_ {0} \ cdot n \ cdot i = K_ {0 } \ cdot (1 + n \ cdot i)}

Starting capital x (term years x interest + 1) = final capital

If the investor has invested start-up capital of 10,000 euros with an annual interest rate of 2 percent over two years, the following interest calculation would result:

10,000 x (2 x 0.02 + 1) =

= 10,000 x 1.04 = 10,400

The investor receives a total of the conditions mentioned 400 euros interest after two years.

### Formula for calculating compound interest (exponential interest)

The formula for calculating exponential interest rates is a little more complicated:

K n = K 0 ⋅ (1 + i) n = K 0 ⋅ qn {\ displaystyle K_ {n} = K_ {0} \ cdot (1 + i) ^ {n} = K_ {0} \ cdot q ^ { n}}

Value at the end of the term = investment value * (1 + interest) term

For the example (10,000 euros start-up capital, 2 percent interest, two years term) the following calculation would result:

10,000 x [(1 + 0.02) x (1+ 0.02)]

= 10,000 x [1.02 x 1.02]

= 10,000 x 1.0404 = 10,404

The investor receives a total of 404 euros in interest under the conditions mentioned, which is a Compound interest effect of "only" 4 euros. Using the Josephspfennig example, however, it becomes clear that this effect would become more significant after a few years.

The interest calculation is not as easy as in the example above? If this is the case, it is worth resorting to one of the many Interest calculator, that can be found on the Internet.

### What is interest?

The term interest in general denotes that angel, which is required for the temporary transfer of capital. Since there are different types of interest, there are consequently different bases for calculating them. In order for interest to come about, you always need one Lender as well as one borrower. That is, on the one hand there is a demand for credit, on the other hand there is a credit supply.

Here is also between Savings Interest and Lending Interest differentiated. The saver makes his savings available to the bank freely. In return, he receives savings interest from the bank. If, on the other hand, a private person wants a loan from a bank - for financing, for example - the bank has to pay loan interest, which the bank receives for making the capital available.

### The different types of interest

The borrower pays the creditor for that Surrender of the principal to interest. Most of the time, the interest rate is calculated for one year and shown as a percentage. From the borrower's point of view, the money paid to the bank is debit interest. In order to modify this borrowing rate, the legislator has introduced other types of interest with a view to consumer protection:

• tied borrowing rate
• variable borrowing rate
• effective interest
• Real interest rate

### Bound borrowing rate and nominal rate: what's the difference?

The fixed borrowing rate is a fixed interest rate for a certain period of the loan term. You can also use it for the entire term of the credit. It is “tied” because one or more borrowing rates are fixed for the entire term or for certain intervals. Debit interest and nominal interest are the same. In the past, the term “nominal interest” was common.

The pegged rate offers one better predictability over the entire term of the loan. With a classic installment loan, the debit interest is tied for the entire term of the loan.

### Variable borrowing interest and differentiation from the nominal interest

In contrast, a variable borrowing rate for the borrower is always with the Risk of a short-term rate hike connected. Nevertheless, some borrowers decide to do this because variable debit interest rates are initially lower than the fixed interest rate at the time the contract is signed. Loan agreements with variable borrowing interest also have the advantage that they can be terminated at any time with three months' notice.

Typical loan agreements with variable borrowing interest are Overdraft facilitiesand credit lines. Here the interest rates can change daily. In practice, however, this theoretical risk is somewhat reduced by making certain agreements on interest intervals. For example, you can agree that the variable borrowing rate is adjusted to the market interest rate every three months.

### What is the APR?

The effective annual interest rate (effective interest rate) is the interest rate that makes different loan offers best comparable. In the calculation, all credit costs are taken into account, which are annually on the nominal loan amount attack. The Price Indication Ordinance determines which costs must be taken into account in the effective interest rate. These precise specifications enable private borrowers to better compare different offers.

The following costs flow into the Calculation of the effective interest rate with a:

• Borrowing rate
• Repayment
• Discount
• Solvency
• Processing costs
• Agency commissions
• with a real estate loan: costs for an appraiser
• further costs according to §6 PankV

The regulation also includes costs that not included in the effective interest rate may be. These include, for example, certain account management fees or insurance costs. The costs for residual debt insurance may only be taken into account if they are absolutely necessary for the conclusion of the contract. However, this is rarely the case.

### What is a real interest rate?

The real interest gives the Change in value of an interest-bearing investment on, after subtracting inflation (or deflation). It indicates the difference between the credit interest on an investment and the inflation rate in percent.

When consumers invest in interest-bearing investments, they need to keep an eye on real interest rates to avoid losses. Because an investment with high credit interest rates can still be a net asset after deducting the inflation rate loss generate.

### What is the base rate?

Finally, the key interest rate should be mentioned. This is the interest rate set unilaterally by central banks at which the banks pay their Interbank transactions transact. It is the main instrument of central bank monetary policy. The key rate has no direct impact on credit market rates. However, it can be observed that if the key interest rate falls, at some point lending rates will follow suit and will also fall.

The key interest rate is usually adjusted to stimulate the economy or reduce inflation. For the euro area controls the European Central Bank (ECB) this award. Adjusting the key interest rate has an immediate effect on the value of the currency.

### What is the base rate?

The base interest rate is a variable interest rate that is published every six months by the Deutsche Bundesbank. This serves the Assessment of capital services. The Deutsche Bundesbank recalculates it every six months according to the specifications of the European Central Bank and then makes it official.

With the official announcement, the base rate applies as the official market rate. In particular, it forms the valuation basis for the default interest rate and thus defines that of a bank maximum interest on arrears to be charged. The default interest rate is made up of the base rate of the Deutsche Bundesbank and a statutory or a contractually agreed interest rate. The base interest rate represents the current market level and thus the market interest rate. The base interest rate is currently -0.88 percent (valid since January 1, 2019).

### Interest calculation: application of the calculation bases

There are several uses for interest calculation. Common products are loans, mortgages, call money accounts and the good old savings book.

### Calculating a loan

A special interest rate is paid for credits and loans. This one stands for Financing costs. A distinction is made in the credit system between the fixed borrowing interest and the effective interest. The tied borrowing rate results from the current market interest rate on the credit market.

Debit interest

The creditworthiness of the applicant, the loan amount and the term of the loan all affect the borrowing rate. The effective interest rate is influenced by other criteria. The processing fees for the loan are also included in the comparison there. There are many online Interest calculator, whether for a personal loan, car loan or as a mortgage calculator. These interest calculators compare different offers in terms of borrowing rate and effective interest rate. The borrowing interest is usually included in the loan installment. These are usually paid monthly by borrowers.

To calculate the borrowing rate, you need the amount of the capital, the period and the interest:

1. Calculate interest: p = (z * 100) / K
2. Interest by months: p = (z * 100 * 12) / (K * m)
3. Interest by days: p = (z * 100 * 360) / (K * t)

z = interest / K = capital /p = interest rate /t = period

Calculation of the repayment

The basis of the interest calculation is remaining debt. Since the borrower reduces his debt on a monthly basis, the base for the interest calculation is reduced on a monthly basis. This means that the repayment of the loan increases with constant loan installments. A monthly repayment is preferable to an annual repayment, as it allows the borrower the financial advantage described.

Effective interest rate

In addition to the borrowing rate, the effective interest rate is calculated other factors, such as the processing fees for the loan or a possible discount (deduction from the loan amount).

The following costs may not be included in the effective interest rate:

• Bank account management fees
• Costs for appraisals
• Agency commissions
• Payment protection insurance

The effective interest rate is well suited to various loan offers to compare with each other, since credit costs are taken into account here.

### Interest calculation for overnight money

A bank's overnight money account is all the more attractive, the higher the interest rate. The annual interest on the overnight money is determined as follows:

Interest per year = (investment capital * interest rate) / 100

For example, if an overnight deposit of 20,000 euros earns interest at 2.3 percent for a year, the following applies:

Interest per year = (20,000 euros * 2.3) / 100 = 460 euros

If the balance on the call money account does not remain constant throughout the year, you can calculate the daily interest. The formula for calculating the daily interest rate is:

Interest for t days = (investment capital * interest rate * t) / (100 * days per year)

For example, if you had invested 20,000 euros from January 1, 2019 to April 1, 2019 with 2.3 percent, then the following would apply to the calculation of the daily interest:

Interest for 90 days = (20,000 euros * 2.3 * 90) / (100 * 365) = 113.42 €

For the sake of simplicity, some banks work with reduced formulas for the daily interest rate. For example, you assume 360 ​​days for the calendar year and 30 days for each month.

### The mortgage and its interest rates

There are many free mortgage calculators on the Internet to make a flexible calculation of real estate financing. Compared to the simple loan calculator, which maps simple installment loans, the mortgage calculator can kmore complex annuity loans map with constant repayment rate and fixed interest rate. Interested parties can use the online mortgage calculator to plan their real estate or construction financing. It would be too complex to use formulas to calculate all variables and special cases yourself.

The following Components of the mortgage loan the interest calculator can determine if necessary:

• Initial repayment
• Installment amount
• Loan duration
• maximum loan amount
• Debit interest
• Special repayments
• Remaining debt after the fixed interest rate has expired

The result enables you good comparison between different loan offers. A detailed repayment schedule with interest and fees and the annual percentage rate is calculated. Different time periods, follow-up financing or individual special payments can also be taken into account.

### The savings book

With the online savings account calculators you have flexible interest calculator for irregular deposits into the savings book and for variable interest rates. This allows the saver to calculate interest income and credit for a longer period of time. It doesn't matter whether it is a one-time deposit of money or monthly savings payments. Any withdrawals can also be taken into account. The interest periods are flexible. In addition to the usual annual interest credit, you can also switch to semi-annual, quarterly or monthly interest credits.

You can also choose between simple interest and exponential interest. However, the interest credit usually flows into the credit at the end of the year and is also subject to interest. There is thus a calculation of compound interest. As a result, the saver receives his Credit development optionally in annual or monthly slices submitted.