Students of soft engineering work with you to understand the compound interest of the time value of capital (returning interest) + basic knowledge of loans + cash flow diagram-"Engineering Economics"

The reason for studying economics is [in order to avoid being deceived by economists]-Joan Robinson

  Some time ago, the school was rectifying campus loans. This reminded me of usury. It happened that I mentioned the knowledge of compound interest in "Engineering Economics" recently. I took the time to explain it in plain and easy-to-understand terms.

Let’s take a look at the key overview
first. Cash flow chart:

1. Basic meaning of cash flow diagram
2. Drawing a cash flow diagram: three elements
3. The meaning of present value P, final value F, and annual value A
4. Drawing example

2. Time value of funds:

1. The meaning of time value of funds
2. Measurement of the time value of funds: interest, interest rate
3. Simple interest, compound interest
4. The relationship between nominal interest rate, periodic interest rate, and real interest rate

1. Cash flow chart

1.1 Basic meaning

Cash flow :

Each fund that flows into or out of the system within a certain period of time is called cash flow, or cash flow for short.
Specific to a certain project, it refers to the capital invested, cost spent, and benefits obtained during the entire life cycle of the project.

Cash flow chart :

A graph showing the size and direction of the cash flow of a project in a period of time with a vertical line segment with an arrow on the time axis is called a cash flow graph.

1.2 Basic symbols: the meaning of present value P, final value F, annual value A

Present value P: The amount of fund revenue and expenditure that occurs at the present is called "present value";
Final value F: The amount of fund revenue and expenditure that occurs in the "future" and "end point" is called "Final value";
Annual value A: The current time When the intervals are equal, the amount of capital income and expenditure occurring at the intermediate point is called the "annual value" or "annuity". If the value of each year is equal, it is also called the "equal annual value".

1.3 The three elements of a cash flow diagram

Size, direction, time point.

1.4 Drawing example one

Question: A company purchased a machine at a cost of 200,000 yuan. It is estimated that it can be used for 10 years. The residual value at the end of 10 years is 20,000 yuan. The operating cost is 10,000 yuan per year. It is overhauled every 5 years at a cost of 40,000 yuan. Try to draw a cash flow diagram for the entire life cycle of the machine.

At that time, I had a question here: Why did I not spend 40,000 yuan for the five-year overhaul in the 10th year? When the teacher woke me up, it was the 10th year that the things had been scrapped. Why fix it?

1.5 Drawing example 2

Question: It is known that a project has a construction period of 1 year and an operation period of 8 years. It is assumed that 10 million yuan will be invested at the beginning of the construction period, and 2 million yuan will be earned during the operation period. The residual value of fixed assets will be recovered in the last year of the operation period. Try to draw a cash flow diagram during the life cycle of the project. (Regulations: Except for construction investment that occurs at the beginning of the period, other cash flows occur at the end of the period)

It may be necessary to draw such a vivid cash flow diagram for the leaders one day at work in the future. It is clear hahaha.

What needs to be expanded here is: a complete project can be roughly divided into the following four phases:
initial construction (various Investigate various analyses), construction period, operation period, and scrap period

Second, the time value of funds

2.1 The meaning of time value of funds

Time value of money refers to funds in the course of the campaign, over time, generated added value .

Time value of money has two meanings: one is the currency used for investment, movement of funds through the currency appreciation ; the other is the money in the bank, individuals lose the right to use the money as compensation obtained of interest .

2.2 Interest and interest rates

Interest
Definition: The portion that the debtor pays to the creditor in excess of the principal is interest.
Meaning: The opportunity cost lost after the transfer of the right to use funds is essentially the redistribution of profits and a measure of the time value of funds.
Formula: interest = total amount currently payable (receivable)-principal
interest rate
definition: the ratio of interest paid per unit time (such as a year, half a year, quarter, month, week, day, etc.) to the amount of loan (deposit) , Which is the ratio of interest to principal.
Meaning: It is an intuitive reflection of interest.
Formula:

Examples to understand

Question: Someone borrows 1,000 yuan in principal and pays 120 yuan in interest one year later. Try to calculate the annual interest rate.
Solution: The annual interest rate is:
(120/1000)×100%=12%

Factors that affect the size of interest rates
① Social average profit rate: The level of interest rates first depends on the average social profit rate and changes accordingly. Under normal circumstances, the average profit rate is the highest limit of interest rates.
②Supply and demand situation: With the average profit rate unchanged, the interest rate depends on the supply and demand situation of borrowed capital in the financial market.
③Bearing risks: lending capital requires taking certain risks, and the size of the risk also affects the fluctuation of interest rates. The greater the risk, the higher the interest rate. (For example, stocks: the stock market is risky, and you need to be cautious when entering the market!)
④ Inflation: Inflation has a direct impact on the fluctuation of interest, and the depreciation of funds will make interest invisibly negative .
⑤The length of the term: the length of the loaned capital

2.3 Simple interest and compound interest

According to different calculation methods, it can be divided into simple interest calculation and compound interest calculation. The interest calculation period can be determined according to any time period.

Simple interest (interest-free interest)
definition: Only the principal is calculated as interest, and the interest generated by the principal is no longer calculated as interest .
Features: Interest has a linear relationship with time; the principal is the same for each interest calculation period.
Formula: I=P ╳ i ╳ n
Where:
   I–represents the amount of interest;
   P–represents the amount of principal;
   i–represents the interest rate of simple interest;
   n–represents the number of interest accrual cycles.
Compound interest

Definition: No interest is paid at the end of the previous interest accrual period, and the interest generated in the previous period will be included in the principal in the next interest accrual period to continue accruing interest.
Features: The relationship between interest and time is non-linear; the principal is different for each interest period.

We can take a look at the calculation process.

From the above picture, we can understand the so-called usury (whether the standard interest of usury exceeds 36% is the standard, that is, if the interest exceeds 36%, it is usury.) It is not a multiplication at all. Please take a look at the above formula oooh, this is an exponential increase. Combining the exponential function and the linear function of high school mathematics, we can see that the exponential function obviously grows much faster than the linear function. With a certain interest rate, the longer the date, the more interest.

2.4 loan shark

Private borrowing with interest rates higher than **24%** can be called usury. According to the latest regulations, national laws with an
annual interest of less than 24% are protected; those with an annual interest greater than 24% but less than or equal to 36% are natural claims. , That is, it is not protected by national law, but it is recognized that it can have this claim;
the portion of the annual interest exceeding 36% is not recognized by the state and not protected. It is an illegal part.

2.5 Nominal interest rate, periodic interest rate, real interest rate

(1) Nominal interest rate
Definition: When the interest calculation period is consistent with the time unit of interest rate, the specified interest rate is called the nominal interest rate under the time unit.
(2) Periodic interest rate
Definition: When the interest calculation period is inconsistent with the interest rate time unit, the interest rate obtained by dividing the nominal interest rate by the number of interest calculation periods.
(3) Real interest rate
Definition: The real interest rate obtained through compound interest equivalent conversion is also called the effective interest rate.

First, we can look at a practical example to understand:

Question: Assuming that the annual interest rate is 12%, the deposit amount is now 1,000 yuan, and the term is one year. Try to calculate the annual interest and the corresponding interest rate under the following three different interest calculation conditions: (1) The interest is calculated once a year ; (2) Calculate interest 4 times a year on a quarterly basis; (3) Calculate interest 12 times a year on a monthly basis.

Solutions:
(1) 1 year interest:
annual interest: 1000╳ (+ 12 is 1%) ¹ -1000 membered = 120 is calculated according to the above compounding publicity, since only the related interest, not the principal operator, so to reduce Go to the principal
      1000╳[(1+12%) ¹ -1]=120 yuan
Actual annual interest rate: 120÷1000╳100%=12%
(2) The interest is calculated four times a year per quarter:
annual interest: 1000╳ (1+3%) ⁴ -1000=125.51RMB
      1000╳[(1+3%) ⁴ -1]=125.51RMB
Actual annual interest rate: 125.51÷1000╳100%=12.55%
(3) Interest is calculated monthly and one year 12:
annual interest: 1000╳ (+. 1. 1%) ¹² -1000 = 126.8 membered
      1000╳ [(+. 1. 1%) ¹² -1] = 126.8 membered
APRs: 126.8 ÷ 1000╳100% = 12.68%
for When the interest calculation cycle is one year, 12% is both the actual interest rate, the nominal interest rate and the periodic interest rate. When the interest calculation cycle is quarterly and monthly, 3% and 1% can only be called periodic interest rates.

To sum up:
Nominal interest rate = periodic interest rate × number of interest accrual periods
For 4 and 12 interest accruals a year, 12% is the nominal interest rate. Cycle rate, the real interest were:
year interest 4:
1) nominal rate: 12%
2) Cycle Rate: 12%. 3% ÷ 4 =
. 3) real interest rate: (12% + 1/4) ⁴ - 1=12.55% is
calculated 12 times a year:
1) Nominal interest rate: 12%
2) Periodic interest rate: 12%÷12=1%
3) Actual interest rate: (1+12%/12) ¹² -1=12.68%

2.6 The relationship between nominal interest rates and real interest rates

1. When the interest calculation period is one year, the nominal interest rate is equal to the actual interest rate, and when the interest calculation period is shorter than one year, the actual interest rate is greater than the nominal interest rate.
2. The nominal interest rate cannot fully reflect the time value of funds, and the actual interest rate truly reflects the time value of funds.
3. Let i be the actual interest rate, r be the nominal interest rate, and m be the number of cycles of compound interest, then there is the following relationship between the actual interest rate and the nominal interest rate:
   
   There is a problem here, when m is larger, it is the number of cycles of compound interest (that is Shorten the interest cycle within a certain period of time. For example, within a year, the cycle can be monthly, daily, or milliseconds.) When the loan interest rate is infinite, will the area of ​​loan interest be infinite? ? ?
   The answer is no!

We can use this formula, combined with the extreme knowledge of advanced mathematics learned in the university, when the m area is infinite, it can be derived:

when the nominal interest rate is constant, the loan interest will also be within a certain range.

4. The larger the nominal interest rate and the shorter the period, the greater the difference between the actual interest rate and the nominal interest rate.

2.7 Combining examples to understand compound interest

Question: Set the annual interest rate to 10%, deposit 10,000 yuan at the beginning of the year, and compound interest once a day. How much 10,000 yuan can be withdrawn at the end of the year (365 days per year)?
Solution:
(1) Calculate the actual interest rate:
(1+10%/365) ³⁶⁵ -1 =10.52%
(2) Sum of principal and interest at the end of the year:
10000 ╳ (1) +10.52%) = 110.52 million yuan

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