Introduction to Financial Mathematics Wu Lan, Second Edition.

Several original definitions of interest

The basic function of interest

The following story is purely virtual, just for fun.

Definition 1: the value expression of the original investment after time, which is called the total function when it changes.

Definition 2: the change of the aggregation function in time is called the interest of the initial currency in time, which is recorded as, that is, especially the interest of the first time period.

Note: interest is always realized at the end of the period. For example, if I lend you money, I will definitely return the principal and interest to you when it expires; Bank deposits do not generate interest until they are due.

Cumulative function,

Let's think about this question: For the above example, I used 500 yuan's principal to buy watermelons, and I can get back 1000 yuan. Similarly, 2000 can be recovered with 1000. Simply put, 1 can become two. The law of this process has nothing to do with the size of the principal. In order to reveal this rule, we consider the following definitions:

Definition 3: Let the principal value of 1 currency unit be, and when it changes, it is called the cumulative function.

This shows that the time value of money can be expressed by an accumulation function. Generally, the accumulation function has the following properties:

Is an incremental function. At least I have cash, and I won't be short after a while.

interest rate

In order to express the relative change range of currency value, the common method to measure interest is to calculate the so-called interest rate. The interest rate is defined as follows:

Define 1.4 The change ratio of the set function to the initial currency in a given time interval is called the interest rate in this time interval, and it is recorded as

For example, I now have 100 yuan, which will become 1 10 yuan one year later, so my interest rate this year is:

Of course, this is just an example. The actual deposit bank can't have such a high interest rate.

I am the dividing line:

Simple interest and compound interest

simple interest

Definition 5. Simple interest: It is considered that the interest generated by 1 monetary unit in any 1 unit time is constant, and the corresponding interest calculation method is simply called simple interest calculation method.

To put it simply, I have 100 yuan now, and the annual interest is 10 yuan, so I will have 1100 yuan after1year, and 120 yuan after two years.

Then, according to the calculation method of simple interest, the cumulative function is:

compound interest

Different from simple interest, the basic idea of compound interest calculation is that interest income should be automatically included in the next principal. For example, if the annual interest rate is 10%, it will become 110 after I deposit it in the bank1year; The principal at the end of 1 is 1 10, and it will become 12 1 two years later. Compound interest here every year.

Definition 6. Compound interest: it is considered that the interest rate of 1 unit currency is unchanged after any unit time. This interest calculation method is called compound interest calculation method, which is called compound interest method for short.

We can also draw the conclusion that the calculation method of compound interest is:

Note: compound interest needs a compound interest period, and it needs to be specified that compound interest is annual, semi-annual, quarterly, monthly and so on.

There is also a short story about compound interest about the natural constant E. We don't need to pay attention to whether this problem is reasonable or not, but more importantly, we need to understand the thinking of studying the problem. )

Suppose the annual interest rate in a certain place is 100%, that is, I deposit 1 in the bank, so I can get 1 in one year.

What if I take it out for half a year and put it for another half a year? It is not difficult to know that I will get a piece in a year.

Even we can compound interest four times quarterly. Even a little crazy, compound interest by the day. It is not difficult to know that if we compound interest n times, we will get it one year later, and obviously, the more times we compound interest, the greater the number on it.

So when will this number tend to infinity? The answer is no.

Finally, in the17th century, the Swiss mathematician Jacob Bernoulli proved that time tends to be a constant, which is what we often call a natural constant.

About the above content, friends who have studied calculus can easily understand, because this is one of the two important limits. In addition, we will use it later when explaining continuous compound interest.