## Interest

Interest is the monetary charge for the privilege of borrowing money, typically expressed as an annual percentage rate (APR). Interest is the amount of money a lender or financial institution receives for lending out money. Interest can also refer to the amount of ownership a stockholder has in a company, usually expressed as a percentage.

### Simple interest

Simple interest is a quick and easy method of calculating the interest charge on a loan. Simple interest is determined by multiplying the annual interest rate by the principal by time that elapse between payments expressed as a part of the year.

This type of interest usually applies to automobile loans or short-term loans, although some mortgages use this calculation method.

When we talk about interest, we are usually more interested in the total of our principal and interest more than interest itself. We can calculate this total using simple formula $$FV = PV \cdot (1 + i \cdot t),$$ where $FV$ is the future value of a principal, $PV$ determines present value of a principal i.e. principal itself, $i$ indicates the annual interest rate expressed as a decimal number and $t$ is maturity expressed in years (or a part of the year).

Sometimes we can come across the modified formula which is used when we work with maturity counted in days: $$FV = PV \cdot \left(1 + i \cdot \frac{d}{360}\right),$$ where values for $FV$, $PV$ and $i$ are the same as in the previous formula and $d$ is maturity expressed in days.

For better visualisation of simple interest we can use figure below, where we use simplification in form of $PV$ equals 1.

### Compound interest

Compound interest (or compounding interest) is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. Thought to have originated in 17th-century Italy, compound interest can be thought of as "interest on interest," and will make a sum grow at a faster rate than simple interest, which is calculated only on the principal amount.

Similarly to simple interest in case of compound interest we are more focused on future value of a principal. Future value of principal using compound interest can be calculated using formula $$FV = PV \cdot (1 + i)^t,$$ where $FV$ is a future value of the principal, $PV$ determines present value of a principal, $i$ indicates annual interest rate expressed as a decimal number and $t$ is maturity expressed in years.

In a real-world scenarios, we are more frequently confronted with situations, where interest is compounded on different frequency schedules. There are no limits to this schedules and interest can be compounded from daily to annually.

To take these options into account the modified formula can be used: $$FV = PV \cdot \left(1 + \frac{i}{m}\right)^{t \cdot m},$$ where values for $FV$, $PV$, $i$ and $t$ are the same as in the previous formula and $m$ is number compounding periods per year.

For better visualisation of compound interest we can use figure below, where we use simplification in form of $PV$ equals 1.

If the value of $m$ is changed then the future value of a principal $FV$ is changed as well. The higher the value of $m$, the higher the future value of a principal. What if the value of $m$ was enormous or even infinite?

#### Continuous compound

Continuous compounding is the mathematical limit that compound interest can reach if it's calculated and reinvested into an account's balance over a theoretically infinite number of periods. While this is not possible in practice, the concept of continuously compounded interest is important in finance. It is an extreme case of compounding, as most interest is compounded on a monthly, quarterly, or semiannual basis.

Formula for continuous compound come from slight modification of the formula for future value of principal using compound interest and to get the formula in usable form, basic knowledge of differential calculus is needed. First the information about infinite periods needs to be added to original formula: $$FV = PV \cdot \lim_{m \to \infty} \left(1 + \frac{i}{m}\right)^{t \cdot m}.$$ Now the known limit for Euler constant $$e = \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x$$ can be utilised. When two above mentioned formulas are compared, it is easy to see, that they are not too different. Main difference lays in the numerators, where in the formula for continuous compound the numerator is $i$ and in the second formula the numerator is 1. All what needs to be done is to replace the interest rate $i$ with 1 using basic mathematical tools $$FV = PV \cdot \lim_{m \to \infty} \left(1 + \frac{\frac{i}{i}}{\frac{m}{i}}\right)^{t \cdot m} = PV \cdot \lim_{m \to \infty} \left(1 + \frac{1}{\frac{m}{i}}\right)^{\frac{m}{i} \cdot i \cdot t} .$$ Now the last step is to substitute $\frac{m}{i}$ with $x$ and use the limit for Euler number $$FV = PV \cdot \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x \cdot i \cdot t} = PV \cdot \lim_{x \to \infty}\left[\left(1+\frac{1}{x}\right)^x \right]^{i \cdot t} = PV \cdot e^{i \cdot t}$$

To see effect of the infinite periods and differences between compound interest and continuous compound we can analyse figure below. To simplify this process, we will count with present value $PV$ equals to 1.

As we can see the effect of the continuous compound is the most noticeable when high interest rates are introduced and can be mitigated by introducing more frequent compounding.

We can also compare simple interest and compound interest. These two approaches are presented in the below, where in case of compound interest we assume annual compounding.

As we can see compound interest gets us more interest over time, which is even more apparent for higher interest rates. It is given by its exponential growth which is steeper than linear growth of the simple interest. However, this difference in interest does not apply to the entire maturity period. If we take a closer look at the first year we can find that compound interest is not always preferable if a main goal is profit. This fact is displayed in figure below.

As we can see for a maturity smaller than 1 is simple interest more profitable than compound interest. On the other hand, for maturity equal to or larger than 1 the compound interest is preferable by investor. The best-case scenario would be to combine these two properties to maximalise the profit regardless the maturity.

### Mixed interest

Mixed interest represents the most profitable form of interest which is used. Mixed interest is a generalization of both above mentioned types of interest (simple and compound). Practically, the length of the loan can be, for example, 1.5 years. If we wanted a simple (linear) interest rate, by the end of the first year the interest would be higher than for compound (exponential) interest, but in subsequent years it would be more advantageous to use compound interest. The mixed method takes these factors into account - full interest rates for compound years and simple interest for less than a part of the year. It is also clear from the form of the equations that by substituting the appropriate values, the equation automatically changes into a simple, compound or mixed type of interest. $$FV = PV \cdot \left( 1 + \frac{i}{m}\right)^{n_m} \cdot (1+l \cdot i),$$ where $FV$ is the future value of a principal, $PV$ determines present value of a principal, $i$ indicates the annual interest rate expressed as a decimal number, $m$ is number of compounding periods per year, $n_m$ is natural number representing number of completed $m$ parts of the year for which a principal is deposited and $l$ is number smaller than $m$ part of the year expressed as a part of the year. Total maturity $n$ is equal to sum $n_m + l$.

Effect of the mixed interest and its comparison with compound interest is displayed in the figure below.

Compound interest itself is profitable so at the firt glance it can seem as mixed interest and compound interest are the same. But if take a closer look, we can see that mixed interest is slightly more profitable. Thus this form of interest, if we will not take it into account continuous compound which is not practicaly used, is the best for every investor.

### Effect of inflation

Inflation is the decline of purchasing power of a given currency over time. A quantitative estimate of the rate at which the decline in purchasing power occurs can be reflected in the increase of an average price level of a basket of selected goods and services in an economy over some period of time. The rise in the general level of prices, often expressed as a percentage, means that a unit of currency effectively buys less than it did in prior periods. Inflation affects deposits too.

If we denote $FV_r$ real future value of a principal, $PV$ present value of a principal, $i$ interest rate, $i_r$ real interest rate and $i_i$ inflation rate (all rates are expressed as a decimal number) than we can calculate real future value with formula for future value using simple interest. Future value than has to be discounted with inflation rate. $$FV_r = PV \cdot (1+i) \cdot \frac{1}{1+i_i}.$$ We can also calculate real future value using real interest rate $i_r$ $$FV_r = PV \cdot (1+i_r).$$ Now we have two options how to calculate real future value and we can compare them $$PV \cdot (1+i) \cdot \frac{1}{1+i_i} = PV \cdot (1+i_r).$$ After some adjustmens we get formula known as Fisher equation $$i=i_r + i_i + i_r \cdot i_i.$$ Because the product $i_r \cdot i_i$ is relatively small for small values of inflation rate and real interest rate it is often excluded from the formula and relation between real interest rate and inflation rate is simplified to formula $$i_r = i - i_i.$$

Effect of inflation is demonstrated in figure bellow on simple interest.

We can see that if interest rate is higher than inflation rate then value of a principal grows in time. On the other hand, if inflation rate is higher then value of principal declines. This means that in future value of a principal is smaller and thus it cannot be used to buy the same amount of goods and/or services. The value of principal has grown by some amount due to positive iterest rate but at the same time prices of goods and services have also grown due to inflation rate but more than the value of the principal.