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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.

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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.