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The importance of financial mathematics

A few words about mathematics financial to start.

Undoubtedly, everyone makes use of mathematical to do basic calculations in the life of every day, either with consciousness or not. For a financial tool that is invaluable is essential for proper management of flows of money.

One of the interests of mathematics in finance is to simplify the complexity of the calculations and financial problems. Take a small example to understand the meaning:

Mr. Y is 50000 Euro bank at 6%. We want to know how much it will ave the first year, second year, third year ...

By process of spontaneous calculations:

• First year: It will be the 50,000 Euro plus interest. To calculate interest by multiplying the percentage of 6% interest (ie 0, O6) by the amount deposited in the bank 50000 Euro, which will be: 50 000x0, 06 = 3000 Euro. So in total there will be at the end of the first year: 50,000 + 6% = 53,000 Euro x50000

• Second year: He will have the 53,000 plus interest calculated on the 53000, ie: 53000 + 6% = 56,180 Euro x53000

• The third year: 56,180 + 6% = 59,550.8 Euro x56180

So you see it's very tiring make these calculations, especially if you want to know how Mr. Y will be after 10 or 20 years for example. Well that's where financial mathematics will intervene to simplify any calculations of this kind.

could well have a mathematical formula that allows us to calculate a time and at any time the principal amount plus interest earned.

Noting in general, C 0 is the amount of capital deposited at time 0 (today) so C n and the amount of capital available to the bank after the n th year and also note i = the interest rate.

• The first year we have: C = C 1 0 + C 0 x i = 0 C (1 + i)

• The second year we have: C 2 = C + C 1 1 x i = C 1 (1 + i) = C 0 (1 + i) (1 + i) = C 0 (1 + i) ²

....

• By inference, the n th years we have: C n = 0 C (1 + i) n

So it suffices to replace in this formula, n the number of years. For example, after the 20 th year, Mr. Y will ave 50000 (1 +6%) 20 = Euro 160,356.77.

This case seems more or less simple because it can easily happen to calculate, with a little patience, that Mr. Y strike at the end of the 20 th year. But if one asks a question such as: how can we calculate, for example, when Mr. Y reaches 100,000 Euro. mentally, it is impossible to calculate, but mathematically speaking, it is very simple. Just solve the equation C = C n 0 (1 + i) n No stranger to one that represents the number of years on the capital deposited in the bank. Here we asked when C n = 100000 So by replacing in formula (C n = 0 C (1 + i) n ) i by 6%, C No. 100000 and by C 0 by 50,000 we would have:

100000 = 50000 (1 +0.06) n

==> 1.06 ⁿ 100000/50000 =

==> 1.06 n = 2 To determine the value of n, we must use a function called the sign logarithm "ln" (because one of the characteristics and that this function lnx n = n . lnx)

==> LN1, 06 n = ln2

==> x n LN1, 06 = ln2

==> No = ln2/ln1, 06 = 11.895660996576

That is Mr. Y will be Euro 100,000 in the bank in 11 years , 0.895660996576x12m = 10 months and 22 days 0.747931958912x30j =

To check the answer, just replace n 11.895660996576 by the formula 50000 (1 +0.06) n and see if it gives good Euro 100,000.

This is a simple example to show you how important points is to use the formulas mathematics to solve practical problems .

In this blog, I would put more emphasis on calculations with bank loan explanations and practical examples.

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Calculate a borrowing base formulas

In this part of the blog, I will try to simplify understanding of financial mathematics applied to various calculations of a loan or a loan. I'm not going to prove the formulas that I'll explain here but I will concentrate more on their meaning and usefulness in practice every day to prevent this from being an abstract mathematical or private practice almost every direction.

In the following formulas, you have the essentials of mathematical finance , you can deduct all other forms with some simple mathematical keys.

Vo Present value of future value discounted at Vn n periods rate i

Value Vn acquired by capital Vo n placed for periods at a rate i



Update / Capitalization :



The first formula represents the present value V 0 ( updating) a future value V n over n periods discounted at a rate i. From this formula we deduce the future value or acquired V n ( capitalization) versus V 0 capital invested for n periods at a rate i.


Some practical examples of these formulas:

1. How do I place today at 8% for 10 000 Euro in 3 years?


2. I want to buy a car in 5 years at a price of 50,000 Euro, the bank offered me an interest rate of 12%. How do I place a monthly basis so I can have that exact amount in 5 years?

To better answer this question, we must learn to correctly translate the data on a timeline and ask the same question but as a diagram.

With V n = 50 000 Euro and n = 60

So the question rephrased in this scheme is how much I have to place monthly now that I have, thanks to these products investments after the fifth year, ie in 60 months, amounting to 50 000 Euro?

He must know that the rate proposed by the bank is an annual rate it is necessary to convert to monthly rate. To do this we must distinguish between a proportional rate and an equivalent rate. Monthly rate proportional

= annual rate / 12 months (banks used incorrectly because this rate is equivalent to the rate of interest but not simple compounds. Indeed, the placement of a Euro in simple interest monthly at i will give at the end of the year an amount of interest 12i. On the other hand, the product of that interest placement for a year at the rate r would give an amount of interest equal r. So, assuming the equivalence of these two amounts of interest we get the formula for the proportional rate)

rate equivalent: it is in fact the monthly rate that would produce the same result as the annual rate in the case of investing the same amount.

Just for information, the level of bank loans and mortgages especially, we use the proportional rate and not the equivalent of what benefits most banks since the proportional rate (1 / 12 = 1 % in our example) is always greater than the equivalent (0.95% in our case) reflecting the true monthly rate equal to the rate proposed by the bank.

Returning to our example, it suffices to replace in the formula Vn values: t = 0.95%, n = 60 and V n = 50 000 to find the amount has it takes place monthly for 5 years.

So if I put a monthly amount of 622.12 euro at the rate of 12% would have after the fifth year due to 50 000 Euro 37 Euro 327.2 (= 622.12 x 60) repayment installments and the rest placed 12 672.8 Euro as interest on investments.

More practical cases for calculating loans further.

Response to Comments:

_____________________
To answer your question, we apply the formula of earned value cited below as shown on the image below:
This formula gives us what I would have at the end of the Xth year of an investment of 1 € monthly at annual T%.

If I place 1 € every 3 months, I would replace T% / 12% by T / 4 and 12.X by 4.X.

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Mortgages: amortization schedule financial

An amortization schedule loans or amortization is a plan to repay a loan by means of disbursement in installments. It helps to identify, at all maturities, the share of the loan repaid, the loan outstanding, the amount of interest and monthly payment (loan + interest paid). These four elements are crucial for any comparison of supply and making a better decision.

I am at your disposal: an array of automatic calculation of borrowing . It allows you to easily compare different offers on the market and see the different elements that let you make your decision to borrow.

How can it be used?

I'll put an example below of a loan of Euro 5000 at the rate of 12% to be repaid by 14 monthly payments. Once the data is entered simply click to get the amortization schedule.


In the table you will have four main columns that I will explain below, giving as an example the calculations of the first line:

• Payment Amount: this is the monthly pay = amount + interest share of capital payable (principal). To calculate it, we need another formula for the capital share is unknown at this time, so to calculate it we have another formula: This formula is applied at the level of mortgages, although it did is not the only method of calculating the monthly
  
• Amount interest: the share of interest = Still to pay monthly rate proportional x = 5000 x 12% / 12 = 50
  
• Main : The share of capital to pay = monthly - Interest amount = 384.51-50 = 334.51
  
• Still to pay : That remains to be paid by deducting the shares of capital already paid = 5000 to 334.51 = 4665.49

Response to Comments:

marie said ...

hello on your example, can you tell me what is the number 1 in your formula for calculating the monthly payment and can we make this formula on a calculator;

thank you

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The 1 in the formula quoted here comes from the formula of the present value of a series of annual (or monthly) should be equal to the capital borrowed.

If you replace V by 0 K, and by m i by T/12 you get to the same formula mentioned in this article, which comes from the discount amount of each "a" to the period 0.

To calculate a calculator, yes it can and must learn to do it at once to get the result as fair as possible.

I hope this answers your question.

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Bank loan: problems and solutions

In this part of the blog, you will learn to use mathematical formulas to solve practical problems related to finance.

I will appreciate any problems or questions posted in the comments, I will try to give him a good answer. So do not hesitate.

3. I want to borrow 20 000 Euro to be repaid over five years to fund an investment project. For that, I received two offers from two different banks:

• Bank A: 1% (monthly rate), monthly repayments
  
• Bank B: 3% per quarter (quarterly rate) Which is the

Does he choose?

It should be noted that there is not a possible solution for solve problems using mathematical financial .

First solution:

can compare the rates of the two banks. But beware, one must compare apples to apples. In this case, rates for different periods. For this, we must calculate the equivalent of one relative to another. Ie, compare, for example, the monthly rate of 1%, the equivalent monthly rate (to find) rate of 3% (quarterly)

To ease the problem formulation, one wonders what is the interest rate Monthly i (unknown) give the same result as the quarterly rate of 3% for the placement of an identical amount M for 3 months (1 quarter).

• placement of an amount M i at the monthly rate for 3 months: M x (1 + i) ⁴

• placement of an amount M quarterly rate of 3% for 1 quarter: M x (1 +3%)

So to assume equivalence between the two results we have: M x (1 + i) M = 4 x (1 +3%) is to say that i = (1 +3%) ^{1 / 4} - 1 = 0.74% monthly rate is equivalent to a quarterly rate of 3% to be compared with the monthly rate of 1% proposed by bank A.

So you see that the proposed bank B (0.74% monthly) is less expensive so better than that of A (1% monthly).

Second solution:

could solve this problem the same way but in comparison, this time, the quarterly rate of 3% with the quarterly rate equal to the monthly rate of 1%.

placement of an amount M at 1% monthly for 3 months: M x (1 +1%) 4

Placing the same amount M t quarterly rate for a quarter (4 months): M x (1 + t)

The equivalence between the two interest rates would give the same result: M x (1 +1%) M = ⁴ x (1 + t)

That is t = (1 +1%) 4 - 1 = 4.06% So you

see that you get the same result. Whether you use the first solution or the second proposal of the bank B is the best.

Third solution:

could solve this problem by comparing all simply the amount of interest each proposal and select the solution that would pay less interest. For this, we calculate the capitalization 20 000 Euro for 5 years for both proposals.

• Offer Bank A:

Bank A offers us the monthly amount to 1% for 5 years ie for 60 months = 5x12

20,000 (1 .01) = 60 36 333.9 Euro. This gives an interest amount of 16 Euro 333.9 (36 333.9 to 20 000)

• Offer Bank B:

20,000 (1 +0.03) 3x5 = 31 159.3 Euro. Ie an interest amount of 11 Euro 159.3

The offer of bank B would pay less attention to the supply of bank A so we always prefer a quarterly interest rate of 3% that a monthly rate of 1%.

4. A couple just had a baby and they wonder how they can offer a better education in university 18 years later. Why he want to know how much to place annually, starting today, until the 17 th anniversary of their son to get back, after the age of 18 years and for four years, four annuities 20 000 Euro, which will be used to finance his university studies.

The bank offers a APR of 10%.

To resolve this problem, we can always proceed in different ways and necessarily get the same result. The key in all the solutions is to search for unknown (the amount to be placed annually in this exercise) by comparing the amounts returned to a specific date in time (finance any comparison must be done at the same time capitalizing on it and updating are dispersed in time) .

Looking scheme, the question is that different amounts to invest must be comparable, financially speaking, the same amount I receive from the bank in 18 years for 4 years. This issue gives us much to formulate the equation to find the unknown but it remains to choose a date comparison (you can choose any date, the solution will be the same).

I will choose spontaneously date 0. So we update the various investment has at time 0 to obtain a present value of individual investments that must equalize the current value at time 0 for different amounts receivable from the age 18.


Hence we 9.20 to 542.82 = 12 ==> a = 1 363.4 Euro , ie an average of approximately 114 Euro monthly.

5. Using redemption or consolidation credit in the situation we are facing a series of monthly payments of different loans to finance various projects, is the best solution offered by banks so far . For more information click the concept of redemption credit . Access links above for practical examples.

continued ...

Response to Comments: Laetitia

Thanks for your comment. I was actually thinking of the confusion in many quarters in the year instead of the number of months in the quarter.

For info useful :

A semester is a period of 6 months [ The word semester comes from the Latin sex (six) and lied (months) ] Source: Wiktionary


A semester is a 4 months, one year therefore consists of 3 semesters. A quarter is a space of three months, There are 4 on the year.