FINANCIAL LITERACY OF GRADUATED STUDENTS

Acta Didactica Universitatis Comenianae Mathematics, Issue 10, 2010, pp. 121-147

FINANCIAL LITERACY OF GRADUATED STUDENTS MICHAELA REGECOVÁ, MÁRIA SLAVÍČKOVÁ

Abstract. The paper deal with problem of financial literacy of students graduated on secondary schools. We tested sample of 25 students at university level in the 1st year of study to be a teacher of mathematics. Because of the new school reform in year 2008 the basic financial mathematics is obligatory part of the mathematics education at secondary school. In our research we tested students – future mathematics teachers – without any additional financial education (to the previous mathematics curriculum on secondary school), if they have sufficient knowledge from financial mathematics because of need in their future praxes. The results imply that their knowledge is not sufficient and therefore there is need to prepare new curricula for preparation of future mathematics teachers. Résumé. Cet article sóccupe du problème des capacités financières des anciens élevés après finir des écoles secondaires. Des mathématiques financières font une partie obligatoire des mathématiques secondaires selon la réforme scolaire dánnée 2008. Nous avons observé un groupe de 25 élevés de premier année d´études à lúniversité dans la spécialisation "enseignant des mathématiques" qui ont fini l´école secondaires sauf une étude spéciale des mathématiques financières. L'objectif de notre recherche était dóbserver des capacités d´étudiants pour lútilisation des connaissances financières dans leur practice future. Les résultats indiquent que les élèves ne disposent pas des connaissances suffisantes alors il est nécessaire de commencer à un changement des programmes scolaires universitaires dans la préparation des futures enseignants. Zusammenfassung. Im Artikel widmen wir uns dem Finanzbildungsniveau von Schülern, die eine mittlere Ausbildung beendet haben. Wir haben eine Gruppe von 25 Studenten des ersten Semesters eines Bakkalaureatsstudiums für Mathematiklehrer getestet. Nach einer Reform der Schule aus dem Jahr 2008 ist Finanzlehre ein fester Pflichtteil des Mathematikunterrichts an Mittelschulen in unserem Land. Wir testeten jedoch diejenigen, die Finanzlehre nich erfahren haben, es jedoch unterrichten werden müssen. Daher stellte sich die Frage ob sie auf ihre Praxis mit genügend Kenntnis vorbereitet sind. An Hand der Resultate geht hervor, dass ihre Kenntnis nich auf ausreichendem Niveau ist und deshalb eine Änderung des Kurikulums in der Vorbereitung werdender Lehrer von Nöten ist. Riassunto. Il paper affronta il problema della literacy finanziaria degli studenti diplomati alla scuola Secondaria. Sono stati esaminati 25 studenti universitari iscritti al 1 anno del corso di preparazione per insegnati di Matematica. A causa della riforma del 2008 della scuola, la Matematica finanziaria di base è parte obbligatoria dell'istruzione matematica nella scuola Secondaria. Nella nostra ricerca abbiamo esaminato studenti – future insegnanti di Matematica – senza alcuna istruzione finanziaria (seguendo il precedente

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M. REGECOVÁ, M. SLAVÍČKOVÁ

curriculum di Matematica della scuola Secondaria), verificando se avessero sufficienti conoscenze di Matematica finanziaria per le loro necessità future. I risultati implicano che la loro conoscenza non è risultata sufficiente. Ne consegue quindi una necessaria strutturazione di nuovi curricula per la preparazione dei futuri insegnati di Matematica. Abstrakt. V článku sa venujeme problému finančnej gramotnosti absolventov stredných škôl. Testovali sme vzorku 25 študentov prvého ročníka vysokoškolského štúdia odbor učiteľstvo matematiky. Podľa školskej reformy z roku 2008 je finančná matematika povinná časť matematického vzdelania na strednej škole. V našom výskume sme testovali študentov – budúcich učiteľov matematiky – bez absolvovania kurzu finančnej matematiky (absolvovali matematiku podľa pôvodných platných učebných osnov), či majú dostatočné vedomosti z finančnej matematiky pre potreby svojej budúcej praxe. Z výsledkov vyplýva, že tieto ich vedomosti nie sú na postačujúcej úrovni, a preto je potrebná kurikulárna zmena v príprave budúcich učiteľov matematiky. Key words: financial mathematics, financial literacy, school reform, preparation of future mathematics’ teachers

1

INTRODUCTION

The importance of financial education has increased in recent years, in particular as a result of developments in the financial market and demographic, economic and strategic changes. The financial markets have become more sophisticated and offer more and more complex services, many of them designated only for a narrow target group. Currently, consumers have better access to a whole range of credit and saving structures that differ with regard to fees, interest rates, maturity or other parameters. And it is certainly not easy for the consumer to evaluate the quality of these structures or to compare them. This is especially difficult in the case of long-term financial services – e.g. the quality of life insurance will be revealed only after attaining a certain age or in the event of an insurance claim. (Ministry of Education & Ministry of Finance of the Slovak Republic, 2008) Today more than ever, citizens need to be financially aware, to know the characteristics and risks of financial products, to choose them correctly. (Tarantola, 2010) The crisis has demonstrated the need for a new equilibrium that favour strengthened consumer financial empowerment and protection. There was survey of public opinion organized by Slovak bank association in 2007. Survey was focus on evaluation of financial literacy of Slovak consumers. The result shows the necessity to give stress on education on the field of finance. (Slovenská banková asociácia 1, 2007) Respondents in this survey reach value of I-FiG (index of financial literacy, the ability make effective decisions on field of management and control personal information base on analysis of accessible information) equal to 0.56, it means that 56 % questions was answered correct. (Slovenská banková asociácia 2, 2007)

FINANCIAL LITERACY OF GRADUATED STUDENTS

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Tangible results from the survey shows that more than half of the respondents do not choose more profitable term deposit from two offers, 84 % do not choose advantageous credit from two offers, 65 % respondents who already have mortgage get wrong answer to the question concerning annual proportional rate of costs. There was also information about disability of analysis information by customers mostly because of misunderstanding basic financial terms in this survey. In 2003 OECD launched project to support increasing financial literacy of inhabitants. Project is focused on the data collection, research and international analytical framework, internationally recognized standards and international dissemination and co-operation. (OECD, 2003) In 2008 within this project was launched portal International Gateway for Financial Education (IGFE) (www.financial-education.org) which provide access to a comprehensive range of information, data, resources, research and news on financial education issues and programs around the globe. The main objectives of the gateway are to (OECD, 2003):  Serve as the first global clearinghouse on financial education gathering an extensive and comprehensive range of information, data, resources, research and news on financial education issues and programs around the globe.  Raise awareness on, and visibility of, financial education issues worldwide and ensure the wide dissemination of research, best practices and guidelines on financial education.  Promote the access to and exchange of information, knowledge, experience and expertise on financial education.  Support the creation of the International Network on Financial Education through the establishment of a secure area for members only.  Encourage the elaboration of further best practices and principles with the guidance of the network and under the aegis of the OECD. “Financial education is the process by which financial consumers/ investors improve their understanding of financial products and concepts and, through information, instruction and/or objective advice, develop the skills and concepts and, through information, instruction and/or objective advice, develop the skills and confidence to become aware of (financial) risks and opportunities, to make informed choices, to know where to go for help, and to take other effective actions to improve their financial well-being and protection.” OECD (2003) Main OECD Recommendations how to Improve Financial Literacy (OECD, 2003):  Governments and all concerned stakeholders should promote unbiased, fair financial education.  Programs should be coordinated and developed with efficiency.

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

The development of methodologies to assess existing financial education programs should be promoted.  Financial education should start at school, for people to be educated as early as possible.  Financial education should be part of the good governance of financial institutions, whose accountability and responsibility should be encouraged.  Financial education should be clearly distinguished from commercial advice; codes of conduct for the staff of financial institutions should be developed.  Financial institutions should be encouraged to check that clients read and understand information, especially when related to long-term commitments or financial services with potentially significant financial consequences: small print and abstruse documentation should be discouraged.  Financial education programs should focus particularly on important lifeplanning aspects, such as basic savings, debt, insurance or pensions.  Programs and delivery strategies should be oriented towards financial capacity building, where appropriate targeted on specific groups and made as personalized as possible.  Future retirees should be made aware of the need to assess the finance adequacy of their current public and private pensions schemes.  National campaigns, specific Web sites, free information services and warning systems on high-risk issues for financial consumers (such as fraud) should be promoted. Ministry of education of Slovak Republic in compliance with recommendation OECD prepare National standard of financial literacy (Ministerstvo školstva SR & Ministerstvo financií SR, 2008) in October 2008. The main goal was start with financial education at schools according with new State educational programs (SEP) within new school reform. New school reform started in September 2008 at primary and secondary schools. This reform innovated content of many subjects, financial education was integrated mostly into mathematics education. Implementation of this reform unfortunately did not count with necessary changes on the universities that prepare future mathematics teachers. The effect of this situation is that universities continue with preparation of future teachers that are not appropriately prepared for teaching according the new State educational programs.

2

RESEARCH

The aim of our research was to find out the level of financial literacy of the students in the 1st year of study on university to be teachers of mathematics. These students were just graduated from secondary schools; they followed the


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old system of education – it means without the financial mathematics course and other changes that are part of the new school reform from 2008. In our research we proceeded from the proposal of content financials’ mathematics subject (Regecová, 2009) that we planed integrated into the subject Didactical seminar from school mathematics. Content of this subject at Faculty of mathematics, physics and informatics Comenius University in Bratislava is solving more difficult tasks on secondary school mathematics level. The effort is to cover wider area of secondary school mathematics and to keep our students in touch with secondary school mathematics level. Therefore we decided to integrated financial mathematics into this subject in accord with State educational program. There are courses of Financial mathematics taught at our faculty (Fakulta matematiky, fyziky a informatiky Univerzity Komenského v Bratislave 1, 2008; Fakulta matematiky, fyziky a informatiky Univerzity Komenského v Bratislave 2, 2008) on master level of study for future mathematics teachers, but they are focused on higher level. These courses are not obligatory so it is possible that absolvent of our teaching program start to learn more about financial mathematics when he/she start to teach it. The main idea of State educational program is to know use knowledge from financial mathematics in real life and on secondary school mathematics level. Therefore we tried to suggest tasks, problems and examples in accord with State educational program. 2.1

THEORETICAL FRAMEWORK

Our research was realized following the analysis of students´ solutions in accordance with Theory of didactic situations. Brousseau (1998) defines the didactic situation as a situation for which is possible to describe the social intention of acquirement of student’s knowledge. This situation is realized in system called the didactic system (didactic triangle) that is composed from three subsystems: learner (student), learning (teacher), information and from relations between them. The relations – the didactic contract is possible defined like a result of negotiation (intervention) explicitly or implicitly defined relations between students or student’s group. It is the environment and the educational system that prepare the students to accept completed or nascent knowledge. They are exactly the rules of game to activate the student. The basic notion of theory of didactic situation is the didactic milieu (Brousseau, 1990). Following the Piaget’s theory the milieu is source of contradictions and non-steady states of learner (subject) by process of adaptation (by Brousseau (1986) it is assimilation and accommodation). The environment is specific for each of knowledge. The different levels of milieu are embedded one inside the other, a situation at one level becoming a milieu for a situation at the next higher level – action at an upper level supposes reflection on the previous level:

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Table 1. Vertical structure of the milieu in a didactic situation (Perrin-Glorian, 2007) M1 E1 P1 S1 didactic milieu universal subject teacher preparing Metadidactic the class Situation M0 E0 P0 S0 learning milieu Student Teacher Didactic Situation M-1 reference milieu M-2 objective milieu M-3 material milieu

E-1 epistemic subject E-2 acting subject E-3 objective actors

S-1 Learning Situation S-2 Reference Situation S-3 Objective Situation

At the level M-3, there is no didactic intention; objective actors act in a material milieu, this action will be the object of the reference situation S-2; E-2 is the student acting with his prior knowledge, he has to understand the rules of the game (possible states and final state to reach) and to play; E-1 is the student reflecting on his action and learning: he has to elaborate a strategy to win. (Perrin-Glorian, 2007) The first important issue is to identify the target knowledge (it is not always explicit and not always the one expressed by the teacher) and how it appears in the problem to solve. The second one is to identify what could be the milieu: data and all actual givens usable by students without any intervention of the teacher. The third one is to identify prior knowledge of students, to foresee actions students may undertake on this milieu, and how they could interpret feedback coming from it. Doing this, we can elaborate an a priori analysis of the class situation (even yet carried out). (Perrin-Glorian, 2007) Material milieu should activate the student to activity which leads to obtain the target knowledge. This milieu consists of 3 parts:  Material: self-existence object, material and non material information, tools included (usually the same of all students)  Cognitive (potential and dynamic part): consist of students’ knowledge which they have and which are necessary for solving the problem, this knowledge is institutionalized it means that it’s in compliance with educational program and the school books.  Social: consist of the other participants (teacher, other students,…) who can intervene to the solving the problem. (Berekova et al., 2001) We have assumed that the didactic situation can be described as a game between a (person in the role or position of the) teacher and the student-milieu system. Every game has its rules and strategies. The rules and strategies of the game between the teacher and the student-milieu system, which are specific of the knowledge taught, are called the „didactic contract“. The rules of the didactic contract are not explicit and they can be slightly different from classroom to


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classroom, culture to culture, and they can even change in the history of a single classroom with the same teacher and the same students. The fact remains, nevertheless, that in every didactic situation there is a didactic contract, and that across different cultures, classrooms and time some rules remain constant, such as, for example, that the teacher is expected to perform teaching actions such as giving the students tasks that are specific to the knowledge he or she aims at, and the student is expected to attend to the tasks given by the teacher. (Sierpinska, 2003) 2.2

PARAMETERS OF OUR RESEARCH

The research was realized in March/April 2009 with students in the 1 st year of study on Bachelor degree at Faculty of Mathematics, Physics and Informatics (FMPI) and Faculty of Natural Sciences (FNS). There were involved only students to be teachers. Students wrote test on the subject Didactical seminar from school mathematics (DS). There were 16 students from FMPI and 9 students from FNS. Duration of the test was 45 minutes. The aim of our research was to find out financial literacy of the graduated secondary school students (in our case students in the 1st year of study on University). Our sample of the students did not follow course of financial mathematics at secondary schools. Basis for preparing the test was the proposal of the study material for DS, which obtain 42 tasks. We choose 4 of them to cover content of the study material and the second important criteria was to choose tasks from real life, which every absolvent of secondary school should solve without problems. Chosen tasks are in Appendix 1. We made detailed analysis a priori of possible students’ strategies of solving, understanding the terms from financial mathematics, possible obstacles with solving etc. We prepared students for a-didactical situation; they can use only paper, pen/pencil and calculator. We compare our projection from a priori analysis of students’ solutions and made evaluation in a posteriori analysis. 2.3

A PRIORI ANALYSIS

Increasing analysis a priori according the Theory of didactical situations (TDS) started in Objective situation S-3 and can stop even in Metadidactic situation S1 (see Table 1). Since in our case the interruption by the teacher in the 1st phase of the research was not suitable to fulfill the aims of the research, in solving the tasks the Didactical situation S0 is not reach. Students can reach at the most the level S-1 Learning situation. We assume these possible strategies and models of solving given tasks:

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Task 1: We would like to set up new bank account and deposit 100 EUR every month starting at the beginning of the year. The deposit will always be made at the beginning of the month. We know that “our bank“ offers a 2.3 % fixed interest rate (the interest rate will not change through the year). The interest rate is calculated and deposited only once to our account, and always at the end of the calendar year. How many EUR would we have in the account by December 31st with a 2.3 % interest rate? How many EUR would we save by the end of the year if we would put 100 EUR every month to our piggy bank? The aim: find out which model for solving chooses students to solve standard tasks from financial mathematics, it means, if they know terminology and models of financial mathematics. Terminology: bank account, annual interest, interest rate, interest tax. Models: according to the aim for the task 1 we will discriminate two basic models of solution. We assume 2 models of solving: M1 (incorrect solution) and M2 (correct solution). In model M1 students do not think about periodicity of deposits and in model M2 they count with periodicity of deposits. There can be more strategies in every model. To each strategy we write also process of solving in steps. M1: Methods and procedures that use different knowledge concerning percentages and algorithms for working with them

B1. Sum of deposits: 12100 = 1 200 EUR

B2. Value of interest from the deposit 100 EUR: x  100  0.023  2.30 EUR

C1. Value of interest for whole year: x  1200  0.023  27.60 EUR

C2. Profit for whole period: 12  2.3  27.60 EUR

D. Counting with tax 19 %? yes E1. Tax y  27.60  0.19  5.24 EUR F1. Value of the interest add to the account z  27.60  5.24  22.36 EUR G1. Amount at the end of the year s  1200  22.36  1222.36 EUR

no

H. Amount at the end of the year y  1200  27.60   1227.60 EUR

E2. Tax y  2.3  0.19  0.44 EUR F2. Interest add to the account in 1 month y  2.3  0.44  1.86 EUR G2. Amount at the end of the year s  12  100  1.86   1222.36 EUR

I. Piggy bank money: 12  100   1200 EUR


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M2: Using financial mathematics mechanism

K1. Interest in each month separately: 12 2.3 January: 100    2.30 EUR 12 100 February: 100 

11 2.3   2.11 EUR 12 100

K2. Interest in each month in EUR 1 2.3 January: 100    0.19 EUR 12 100 February: 200 



2.3

 0.38 EUR

12 100

... December: 100 

1

... 1



2.3

 0.19 EUR

1

December: 1200 

12 100



2.3

 2.30 EUR

12 100

L. Counting with tax 19 %? yes N. Value of the interest for whole period in EUR 2.3  0.81  2.11  0.81    0.19   1.86  1.71    0.16  12.11 EUR

no Q. Amount at the end of year in EUR 12  100  2.3  2.11    0.19

P. Amount at the end of year 12  100  12.11  1212.11 EUR

H. Piggy bank amount: 12  100  1200 EUR

Task 2: Mr. Smoliar needs an emergency loan of 3000 EUR only for one year. The bank offers a loan with a 15.6 % interest rate. His neighbour Mr.Podnikavy is willing to lend him the same amount under different terms. Mr. Podnikavy would add 1.5 % from the loan every month for the term of the loan. a) Guess, which option would be more beneficial for Mr. Smoliar. b) Calculate the total interest that needs to be paid back to the bank and to Mr. Podnikavy. How much is the difference? c) How much higher can Mr. Podnikavy raise his interest rate and still be better than the bank? a) The aim: find out if students are able to guess the better alternative. Terminology: credit, loan, interest rate, interest. Models: according to the aim we assume these two models:

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M1: Solving/Consideration

A1: conversion month neighbors’ surcharge to years’

A2: conversion years’ interest rate of bank to months’

M2: Guessing

C1: Bank

C2: Neighbor

M3: Answer after the solving part b) B: comparing

b) The aim: find out if students know basic financial terminology and mechanism of loan interest or credit and if they know count which of to offers is more profitable Terminology: credit, loan, interest rate, interest. Model: we assume one model with three different strategies to find out the solution. All models lead to correct solution. M1. Calculating with interest/ surcharge only

B1. Interest we pay to the bank: 3000  0.156  468 EUR C1. Interest paid to the neighbor in one month 3000  0.015  45 EUR D1. Interest paid to neighbor for whole year 12  45  540 EUR E1. Difference in interests: 540  468  72 EUR

M2. Calculating with whole owed amount

B2. Amount we owe to the bank: 3000  1.156  3468 EUR

C2. Conversion neighbors months’ surcharge to years’ 1.5%  12  18 %

D2. Amount we should pay to neighbor back: 3000  1.18  3540 EUR

E2. Difference: 3540  3468  72 EUR

M3. Converting the interest rate

B3. Months’ banks’ surcharge: 15.6 : 12  1.3%

C3. Month banks’ interest: 3000  0.013  39 EUR

D3. Difference in months’ interest rates: 1.5%  1.3%  0.2 %

E3. Years’ interest difference: 3000  0.002  12  72 EUR


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c) The aim: find out if students know basic financial terminology and mechanism of loan interest or credit which have influence on profitability of the loan Terminology: credit, loan, interest rate, interest. Models: we assume two basic models to find the solution in accord of the frame of students’ solution (both are correct). M1: Using strategy S2 or S3 from previous part of the task

M2: In-equation

A. 468  12 

p

 3000

100

A1. Bank: 15,6 % Neighbor: 1.5  12  18%

A2. Bank: 15.6 : 12  1.3% Neighbor: 1,5 %

B. p  1.3%

B: comparing

Task 3: A brokerage firm received a loan for 99 000 EUR with a 13 % interest rate over 3 years. According to the contract with the bank, they will start the payments one year from the origination of the loan. Payment is yearly and the bank collects interest once a year. How much would the yearly payment be? Round your answer to the cents. Terminology: credit, interest rate, annual payment, annuity, patching up credit The aim: find out if students know basic terminology and mechanism of interest in credits and if they know use it correctly for solving the task. Models: in accord with the aim we assume two basic models for the solving of the task. Model M1 represent correct solutions and model M2 incorrect solutions. M2: fairness in payments

B2. Interest paid: 99000  0.13  12870 EUR

B3. Years‘debt: 99000 : 3  33000 EUR C3. Interest paid: 33000  0.13  4290 EUR

C2. Different incorrect solutions of the task, for example:  9900012870 : 3  37290 EUR 33000  4290  37290 EUR

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M1. Financial mathematics procedures

 

  100  13

B1. Debt at the end of the 1st year: 99000  1 

 

C1. Subtracting the payment: 99000  1 

 

 

D1. Debt at the end of the 2nd year:  99000  1 

13

 a

100 

   13    a  1   100    100  13

E1. Debt at the end of the 3rd year: 99000  1.13  a 1.13  1.13  1 3

2

F1. Debt at the end of the 3rd year should be equal to zero: 3 2 99000  1.13  a 1.13  1.13  1  0

Task 4: Mr. Mrkvicka received a four year loan at the beginning of the year. He will pay it off in four annual payments. The bank calculates and collects the interest rate once a year. Mr. Mrkvicka’s payment plan chart is shown below. Payment amount Initial amount The end 1st year The end 2nd year The end 3rd year The end 4th year

loan of the of the of the

– 23 500EUR

Interest paid –

Principal

Status Debt –

of

the

66 600 EUR

9 990 EUR

19 500EUR 16 500EUR

of the

a) Find using the chart: - The size of the loan, - The payment amount at the end of the first, second, and the third years, - The interest rate of the loan. b) Fill in the missing information on the chart. c) How many EUR will the last payment amount be? d) What will be the total amount paid by Mr.Mrkvicka at the end of the loan? e) How many EUR total is in interest?


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a) The aim: find out if the students understand basic terminology from financial mathematics Terminology: credit/mortgage, repayment/principal, repayment plan, interest, interest rate Models: we assume correct solution by this model -

Find in the table “Status of the debt”, row “Initial loan amount”: 66 600 EUR Find in the table payments from first column, rows “end of the xth year”: 23 500 EUR, 19500 EUR and 16500 EUR Count value of interest rate using the value of the interest at the end of 9990 the first year and the Initial loan amount: x  .100%  15% 66600

b) The aim: find out if the students know mechanism of interest if there is credit/mortgage for more than 1 year Terminology: credit/mortgage, repayment/principal, repayment plan, interest, interest rate Models: in accord of the aim we assume the one model of correct solution: A. Student start to solve the task

B. Principal at the end of the 1st year 23500  9990  13510 EUR

D. Interest at the end of the 2nd year 0.15  53090  7963.50 EUR

E. Principal at the end of the 2nd year 19500  7963.50   11536.50 EUR

G. Interest at the end of the 3rd year 0.15  41553.50  5233 EUR

H. Principal at the end of the 3rd year 16500  6233   10267 EUR

J. Interest at the end of the 4th year 0.15  31286.50  4693 EUR

K. Principal at the end of the 4th year is equal to the debt at the end of the 3rd year – 31286.50 EUR

C. Debt at the end of the 1st year 66600  13510  53090 EUR

F. Debt at the end of the 2nd year: 53090  11536.50  41553.50 EUR

I. Debt at the end of the 3rd year: 41553.50  10267  31286.50 EUR

L. Payment amount: 4693  31286.50   35979.50 EUR

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c) The aim: find out if the students understand the basic terminology from financial mathematics. Terminology: credit/mortgage, repayment/principal, repayment plan, interest, interest rate Models: In accord to the aim we assume the one model of correct solution of the task. M. From the last step of the solution we can see, that the value of the last payment amount is 35979.50 EUR. d) The aim: find out if the students understand basic mechanism and algorithm of financial mathematics Terminology: credit/mortgage, repayment/principal, repayment plan, interest, interest rate Models: In accord with the aim we assume two strategies of finding correct solution. M1. Total payment is sum of the four annual payments amount, it means: 23500  19500  16500  35979.50  95479.50 EUR. M2. These amounts we can determine by add the sum of mortgage and interests for whole period in the row Interest paid. e) The aim: Find out if students understand basic terms from financial mathematics. Terminology: credit/mortgage, repayment/principal, repayment plan, interest, interest rate Models: in accord with the aim we assume this strategy of correct solution: M. Interest amount for whole period we find as sum of the values in the column Interest paid: 9990  7963.50  6233  4693  28879.50 EUR. 2.4

ANALYSIS A POSTERIORI

In accord to the Theory of didactical situation Objective situation S-3 is the same in every strategy. Student acquaint with the tasks and Material milieu M-3. Material part of M-3 include basic writing tools (paper, pen…), mathematical and real context of the task. Cognitive part of M-3 contains knowledge and understanding of on different levels. The level depends on secondary school where the students graduated, age of the students and used strategy. This is mostly knowledge concerning interest, interest rate, payment amount, mortgage, knowledge of necessary mathematical tool for solving given tasks. Social part


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of the M-3 was minimized because of is not desirable inroad other persons during the solving the tasks. In the Reference situation S-2 is the student the active who try to solve the tasks in already known milieu M-3. He/she use only known schemas, relations, procedures, algorithms. In the situation S-1 – Learning situation, student is in the position of solver of the problem. The character of the activity is losing, student start to thing more and start to formulate answers to the questions which he/she asks himself (herself). Realization of the basic research do not assume the interruption by teacher to students solution, students do not reach Didactical situation S0. We provide analysis of the students’ solution of given task and identify the steps of solution (see Appendix 2). Obtained strategies we analyze in accord with the reference situation reached by the students. We state the analysis of every task separately. Task 1 In accord to our presumption in analysis a priori we notice that most students use strategies in model M1, it means model which do not lead to the correct answer to the question. Most frequent was Strategy 1 (ABEFLO), it was used by 4 students. Students in this strategy are in the reference situation and used well known mathematics algorithm to calculate with percentages. Students also think over tax 19 % of interest. The wrong part of the solution was that students did not think over periodicity of the deposit, it means not whole amount is interest by the same time. The 2nd very frequent strategy of solving this task was Strategy 2 (ABKO), again used by 4 students. Difference between these two strategies is that students who used Strategy 2 did not think over the tax. The procedure and methods used by students in reference situation are only on the level of calculating with percentages. Strategy 3 (ACO) used by 3 students stay at reference situation calculating with percentages. Difference between this strategy and Strategy 2 is that students who used Strategy 3 calculate the final amount of the money directly. These two strategies are from this point of view different. But the strategy – find out the amount of the money by use of procedure to calculate with percentages, do not think over tax is basis of both of them. Difference is only in way of finding the solution. Other strategy used by students was Strategy 4 (ABO). The reference situation is the same like in Strategy 2 and 3. The difference is that students calculate only the value of the interest, not the amount of the money at the end of year. We notice 10 different strategies of solving the task in 10 other students’ solutions. We notice here Strategy 5 (DMHLO) and Strategy 6 (DGO). Students in reference situation did not work only with well known procedures for calculating

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with percentages but they also try to consider single interest rate for every month (step D). From the viewpoint of the work with percentages is very interesting Strategy 7 (IJLO). Students in this strategy calculate new interest rate: 2.3%0.81%. There was the same problem as in other Strategies 1 – 4, students do not figure out the periodicity of the deposit, student stay in model M1 (see a priori analysis). There was only 1 student calculating in model M2 describing in a priori analysis, the model of financial mathematics mechanism. His strategy labeled Strategy 7 (P). In the reference situation student figured out the periodicity of deposit, but did not calculate with tax 19 %. Only 1 student did not start to calculate this task. From the written analysis of the task 1 we can see, that students are not well informed about the mechanism of term deposits, most of them calculate with whole amount of deposit, they did not think over the periodicity of the deposit, approximately half of the students (13) did not calculate with tax. The correct answer to the question: „How many EUR would we save by the end of the year if we would put 100 EUR every month to our piggy bank?“ had 22 students. Other problems in solving this task (expect of the mentioned Model M1) were numerical errors and errors in order of magnitude. In this case was the very surprising disability of the students to know if their result is appropriate/real. The one of the reasons of this disability could be luck of the personal experiences with financial products and transactions. Task 2 a) The aim of this part of the task was to find out, if students are able to guess the better alternative. From analysis of the solution we can see that only 5 of them were able to guess without additional calculations. Three students guess bank (step A) and 2 guess a neighbor (step B). 15 students get and answer to the question after solving the part b) (step C). Other students did not start with solving (5 of them). b) The aim of this part was similar as in the part a) – find out, if students are able to choose better alternative but with calculations. The most frequented strategy to solve this part was Strategy 1 (ACFI). In this strategy students calculate surcharge for whole year they should pay to neighbor and to the bank, then they find difference between them. Reference situation in this strategy is calculating proportion (interest) from the whole amount. They use knowledge from lover secondary school. Strategy 1 is one of our assume strategy describe in a priori analysis. This strategy used 8 students; other 7 students used strategy which is equivalent to the S1 from a priori analysis. These 7 students use other steps (AFI, AF, ACF, JKBHM, JCFBI, JCFPM).


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Strategy 2 (JDKI) is the 2nd frequent strategy is equal to our M2 from a priori analysis. Reference situation of this strategy is similar as in Strategy 1, but there is expect calculating total surcharge to neighbor and to the bank also calculate total “interest rate” to neighbor. Application of this strategy is like in Strategy 1 on the level of using the knowledge from lower secondary school concerning the percentages and calculating with them. Strategy 2 used 3 students, but only one of them solves this task to the end. Strategy 3 (ADGI) is almost equal to M2 from a priori analysis. This strategy used only 1 student and from the viewpoint of reference situation it is equal to the Strategy 2. Other possibility of solving the task was comparing neighbors and banks interest rates we labeled Strategy 4 (DNO). We can recognize application of percentages not as a part of whole, but percentage points in reference situation. This strategy belongs to M3 from a priori analysis. The student compare years’ interest rates, not months (like is written in M3). The 4 of 25 students use step L. The present of this step determine that students did not understand to the task because they did not add 1,5 % from the loan, but 1,5 % from outstanding amount. In this step we can see indication of mechanism how interest runs in financial institutions. The correct solution of this part had 16 students from 25; they correctly calculate difference in interest in bank and neighbor. This part did not start to solve 1 student. Therefore successfulness of solving is 64%. Problems connected to the finding the solution expect of the numerical errors was apply tax to interest (2 students), power of the outstanding amount (1 students). c) The aim of this part was to find out if are able to manipulate with parameters that have influence on profitability of financial offer. In concrete we would like to know, if students can determine more profitable (or equal) surcharge in percentages comparing to interest rate offer by other subject. According to the part b) we presume in a priori analysis one of the strategies describe in models M1 of a priori analysis. Analysis of students solution shows that only 3 students use strategy in model M1 – Strategy 1 (AB), other 3 students write down answer without any calculations, therefore we can not assume using the M1 model strategy. From viewpoint of reference situation in Strategy 1 the knowledge of students are on the level of using mathematical operation with percentage, they work with the concept of percentage point. Strategy from the a priori analysis model M2 used 4 students but only 2 of them write down correct answer. Strategy 2 (CDB) reference situation require knowledge how to make in-equation and find its solution. Therefore we think about this model as about more difficult one for students. In solution of this part of the task we determine also other strategies. Strategy 3 (EFB) where students set the maximum months’ payment to neighbor and from this information they calculate interest rate. Reference situation in this strategy is

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on the level of ability to calculate percentage parts. From the 6 students who used this strategy half had correct answer. Strategy 4 (GH) is analogy to Strategy 3 used by 1 student. Base of this strategy is in month difference in interest paid in bank and neighbor. Then it is easy to find out difference in percentages (percentage points) – interest rates. Reference situation is the same as in Strategy 3. This part of the task did not start calculate 8 students. Problems and errors in solving this part: power of debt (2 students), month difference in interest paid was set as a maximum percentage surcharge for neighbor. The first problem pointing to the mechanism of financial mathematics and shift these students to the Learning situation S-1, where the students try to set themselves as a solver of the problem. The second problem points to didactical contract in students’ behavior, because they often have no idea what they are calculating but they know kind of the expected result. Task 3 The task was focused on basic terms and financial mechanism of interest rate of loan, and if the students are able to apply this knowledge to calculating the tasks. Because of higher level of the students knowledge is required we presume mostly solution in model M2 described in a priori analysis. Strategies in this model also pointed to the natural effort of the human for fairness, uniformity, proportional dividing etc. The 11 students of 25 calculate interest from owed amount (step P), 3 of them try to add this amount to the loan (Strategy 1, steps PR), 2 students divide interest to 12 equal parts (Strategy 2, steps PU). The number representing the value of the interest tried 2 students used by any way (steps Cʹ, Dʹ) but from their solution is obvious that they had no idea about paying off the loan. There was no student who was able to continue in calculating, their reference situation was using basic knowledge about interests. The knowledge concerning the annual payment was not applied here. The rule of “fairness” (model M2 a priori analysis) used in solution 5 students but in different way (steps PS, AV, PQ,…) One of these students use Strategy 3 (HIJKLMNO) on reference situation used sequential interest of owed amount during the 3 years, but divided into 3 equivalent parts. Between the same students we also determine troubles with financial mathematics terminology; they mistook annual payment with sum of interests in 3 years (step O). There was only one student in model M2 at whom we can recognize in his steps indication of the mechanism of pay of the loan (Strategy 4, steps PTVWXYZAʹBʹ). Students in reference situation used knowledge of interest in loan, pay off the loan (sequential decreasing the owed amount). The problem was that these students apply the fairness dividing the amount of loan and interest into 3 equal parts (step V).


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There was only one student who used analysis a priori model M1 to solve the task – Strategy 5 (ABCDEFG). The reference situation for him was knowledge from financial mathematics concerning pay off the loan by annual payments. Analysis of the solution of the task 3 imply that more than half of the students (13) have no knowledge from financial mathematics concerning pay off the loan and are not familiar with term annual payment. There were only 2 students where we identify the mechanism of pay off the loan in their solutions. We find in their solutions also very illogical using of mathematical operation with given, or calculated data as well. We think that students did not have appropriate model in their knowledge net to solve this task together with effect of didactical contract where students tried to use input information from the task attempt to find the solution. 10 students did not start to solve this task. Task 4 a) 8 students did not solve this part of the task, even that in this part was only necessary to look to the table and write down the numbers from the correct column. The value of the loan successful writes down 13 students (step A). Surprising fact was that students had problems with finding the payments at the end of the 1st, the 2nd and the 3rd year (step B) from the table with column “Payment”. Correct answer had only 6 students, other 5 students thought that payment is a sum of “Payment” and “Interest” (step C). We can assume that this fact pointing on students’ idea that interest we have to pay back to the bank is part of the payment. Interest rate (step D) has been correctly calculated by 9 students from 17 whom this part start to solve. b) The interest procedure was well known only to 1 student from 25, other 6 who tried to solve this task assign debt status from columns “Debt status” of previous year and “Payment” of actual year (step B). Reference situation represented known procedure of decreasing of the debt by payment, not by principal. We can conclude that most of the students do not know the relation between payment, interest and principal. 18 students (72%) did not solve this part at all. The student who solves this task correctly graduated on secondary school focus on economics. Solution in 4c, 4d and 4e depend on successful solution of 4b, therefore no one answer the question correctly.

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c) The most frequent answer was 7 100 EUR (5 students, step A), the reason was the step B in part 4b. Two students get answer 14 500 EUR (step B) because of the pattern decreasing by 2 000 EUR every year. 18 students did not solve this part. d) The answer to the question get 4 students but without any explanation (step B – 2 students, step A – 1 student, step C – 1 student). This part did not solve 21 students. e) The sum of the total amount paid to the bank solves only 3 students. Two of them only guess the result (step A), 1 solve it wrong by mathematical and logical point of view (step B) In general, task 4 solved only 72 % of students. The reason could be also that it was the last one. During the analysis of this task we find in some solution effort to use mechanism to set interest, payment, principal and debt status in single year. But students had a problem to identify the relations between these terms. We also determine the effect of didactical contract since we find illogical calculations in many solutions that pose using of different mathematical operations with the given numbers (without any reasonable procedure, sequence or indication of using financial mathematics mechanism). In general from the solution on whole test we observed that graduated students on secondary schools (followed the old system of education – it means without the financial mathematics course) are not familiarized with basic terms of financial mathematics, their understanding (from the advertisements concerning mortgages, credits, loans, leasing and other financial products of the banks and saving banks). This limitation has influence on choosing from the concrete offer of financial institution. This chooses is often without any logical correct counting or guesses. Therefore students failed on this field of mathematics and real life situations.

3

CONCLUSION

Financial literacy entails the ability to make professionally qualified judgments and effective decisions on the utilization and administration of financial resources. It is becoming a prerequisite for existence in a modern society and it is more and more important for the sustainable welfare of individuals and of the whole of society. (Ministry of Education, Ministry of Finance of the Slovak Republic, 2008)


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Financial education must have a long term vision through ongoing and repetitive actions: measurable impacts won’t necessarily occur today or tomorrow. A series of conditions can ensure that financial education programs do make a difference:  Programs must start with an assessment of areas where financial literacy is most urgently needed as well as an identification of those sectors of the population most at risk. Only this will permit policy makers to single out priority areas for action and set objectives.  Programs must include benchmarking and evaluation components to ensure appropriate and informed decision-making process. Evaluation should be planned at the outset of the program, during the design phase and it should be tailored to the program and its objective.  Tying in learning from behavioral economics and psychology can strengthen and ensure the relevancy of financial education programs.  Financial education programs that improve consumers’ knowledge and attitudes towards retirement planning are critical. Much research remains to be done in spite of the work already carried out. The OECD together with partners like the International Organization of Pension Supervisors and its network are committed to this key priority area and welcome your inputs and insight to identify tangible and appropriate policy options. (Boucher 2010) The one of the factors of this fail could be that students that had no financial mathematics in secondary level education know about it very few. The information from the media and everyday life are not sufficient. Therefore they are not prepare for it and need to learn more on secondary school from this very important area of the life. As we can see, the one of the aim of the teaching mathematics develop the analytical thinking and ability to solve the task in real life was not reached – students are able to memorize, but they do not understand, they cannot make their own construction; they need to be lead by teacher/tutor. The other ability – make a guess is still not on adequate level. As was written (Brisudová & Slavíčková, 2006) students are not able to make a guess and their calculation always depend on calculator. The ability of guessing is missing on mathematics lessons. The solving of the problems from financial mathematics those are problems from the real life could improve students’ attitudes towards mathematics education (Vankúš & Kubicová, 2010). Therefore we realize to prepare new curricula for preparation of future mathematics teachers oriented on development of pedagogical content knowledge for them (Kohanová & Slavíčková, 2009). The financial mathematics should be taught from the very basic level.

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REFERENCES Bereková, H. et al. (2001). Slovník teórie didaktických situácií, 1. časť. Zborník Bratislavského seminára z teórie vyučovania matematiky, No.4, Bratislava 2001, ISBN 80-223-1704-7, p.95-103 Boucher, R. (2010). Concluding Remarks. OECD-Bank of Italy Symposium on Financial Literacy: Improving Financial Education Efficiency, Rome, Italy. Retrieved July 24, 2010, from http://www.oecd.org/dataoecd/36/12/45486183.pdf Brisudová, Z., Slavíčková, M. (2006). Problems with using calculators on mathematical lessons, Acta Didactica Universitatis Comenianae - Mathematics, Issue 6, Comenius University, Bratislava 2006, ISBN 80-223-2138-9, p.19-31 Brousseau, G. (1986). Fondaments et méthodes de la didactique des mathématiques. Recherches en Didactique des Mathématiques. Grenoble, La Pensée sauvage Brousseau, G. (1990). Le contrat didactique: le milieu. Recherches en Didactique des Mathématiques, Vol 9/3, Grenoble, La Pensée sauvage Brousseau, G. (1998). Théorie des situations didactiques. Grenoble, La Pensée sauvage Fakulta matematiky, fyziky a informatiky Univerzity Komenského v Bratislave 1 (2008). Úvod do financií a základy účtovníctva. Retrieved August 01, 2010, from http://www.fmph.uniba.sk/ fileadmin/user_upload/editors/studium/bc_mgr/IL/1-UMA-305.html Fakulta matematiky, fyziky a informatiky Univerzity Komenského v Bratislave 2 (2008). Základy finančnej a poistnej matematiky 1. Retrieved August 01, 2010, from http://www.fmph.uniba.sk/ fileadmin/user_upload/editors/studium/bc_mgr/IL/2-UMA-255.html Kohanová, I., Slavíčková, M. (2009). Development of Pedagogical Content Knowledge in preparation of future mathematics' teachers, IMEM Congress 2009, Ružomberok: Catholic University, p. 572-581 Ministerstvo školstva SR, Ministerstvo financií SR (2008). Národný štandard finančnej gramotnosti verzia 1.0. Retrieved July 24, 2010, from http://www.mpc-edu.sk/library/ files/narodny_standard.pdf Ministry of Education, Ministry of Finance of the Slovak Republic (2008). Consumer protection in the area of financial services, provision of financial education and regulation of mediatory and advisory activities in the financial market. Retrieved July 24, 2010, from http://www.financial-education.org/dataoecd/4/42/40512698.pdf OECD (2003). OECD Project on Financial Education and its International Network on Financial Education. Retrieved July 24, 2010, from http://www.financial-ducation.org/dataoecd/8/28/ 44409678.pdf Perrin-Glorian, M.J. (2007). From producing optimal teaching to analysing usual classroom situations. Development of a fundamental concept in the theory of didactic situations: the notion of milieu. Retrieved July 26, 2010, from http://www.unige.ch/math/EnsMath/ Rome2008/ WG5/Papers/PERRIN.pdf Regecová. M. (2009). Návrh kurzu finančnej matematiky pre študentov odboru učiteľstvo akademických predmetov: prieskum finančnej gramotnosti študentov. Vzdelávanie - veda spoločnosť: sekcia matematické vedy (CD ROM), Komárno : Univerzita J. Selyeho, ISBN 978-80-89234-84-4, S. 62-81 Sierpinska, A. (2003). The notion of ‘didactic contract’, Lecture 3. Retrieved August 16, 2010, from http://annasierpinska.wkrib.com/pdf/TDSLecture%203.pdf Slovenská banková asociácia 1 (2007, November 8). Kvíz: Otestujte svoju finančnú gramotnosť. Retrieved July 25, 2010, from http://www.sbaonline.sk/sk/presscentrum/aktuality/kvizotestujte-svoju-financnu-gramotnost.html


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Slovenská banková asociácia 2 (2007, November 5). Finančná gramotnosť nie je na dostatočnej úrovni. Retrieved July 25, 2010, from http://www.sbaonline.sk/sk/presscentrum/ aktuality/financna -gramotnost-nie-je-na-dostatocnej-urovni.html Tarantola, A. M. (2010). Concluding Remarks. OECD-Bank of Italy Symposium on Financial Literacy: Improving Financial Education Efficiency, Rome, Italy. Retrieved July 24, 2010, from http://www.oecd.org/dataoecd/36/57/45486368.pdf Vankúš, P., Kubicová, E. (2010). Postoje žiakov 5. a 9. Ročníka ZŠ k matematike, Acta Mathematica, Vol. 13, UKF Nitra, Nitra

MICHAELA REGECOVÁ, Department of Algebra, Geometry and Didactics of Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, 842 48 Bratislava, Slovakia E-mail: [email protected] MÁRIA SLAVÍČKOVÁ, Department of Algebra, Geometry and Didactics of Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, 842 48 Bratislava, Slovakia E-mail: [email protected]

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Appendix 1 1.

We would like to set up new bank account and deposit 100 EUR (3 012,60 SKK ) every month starting at the beginning of the year. The deposit will always be made at the beginning of the month. We know that “our bank“ offers a 2.3 % fixed interest rate (the interest rate will not change through out the year). The interest rate is calculated and deposited only once to our account, and always at the end of the calendar year. How many EUR would have in the account by December 31th with a 2.3% interest rate? How many EUR would we save by the end of the year if we would put 100 EUR every month to our piggy bank?

2.

Mr. Smoliar needs an emergency loan of 3000 EUR only for one year. The bank offers a loan with a 15.6 % interest rate. His neighbour Mr.Podnikavy is willing to lend him the same amount under different terms. Mr. Podnikavy would add 1.5 % from the loan every month for the term of the loan. a) Guess, which option would be more beneficial for Mr. Smoliar. b) Calculate the total interest that needs to be paid back to the bank and to Mr. Podnikavy. How much is the difference? c) How much higher can Mr. Podnikavy raise his interest rate and still be better than the bank?

3.

A brokerage firm received a loan for 99 000 EUR with a 13 % interest rate over 3 years. According to the contract with the bank, they will start the payments one year from the origination of the loan. Payment is yearly and the bank collects interest once a year. How much would the yearly payment be? Round your answer to the nearest cent.

4.

Mr. Mrkvicka received a four year loan at the beginning of the year. He will pay it off in four annual payments. The bank calculates and collects the interest rate once a year. Mr. Mrkvicka’s payment plan chart is shown below. Payment amount Initial amount The end 1st year The end 2nd year The end 3rd year The end 4th year a)

b) c) d) e)

loan of the of the of the

Interest paid –

23 500EUR

–

Principal –

Status of the Debt 66 600 EUR

9 990 EUR

19 500EUR 16 500EUR

of the

Find using the chart:  The size of the loan,  The payment amount at the end of the first, second, and the third years,  The interest rate of the loan. Fill in the missing information on the chart. How many EUR will the last payment amount be? What will be the total amount paid by Mr.Mrkvicka at the end of the loan? How many EUR total is in interest?


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Appendix 2 TASK 1 Model 1 A. Whole year credit: 12.100  1200 EUR B. Interest paid for whole year: 1200  0.023  27.60 EUR C. Direct counting of the profit: 1200  1.023  1227.60 EUR D. Interest paid from single credit: 100  0.023  2.30 EUR E. Paid tax for whole year: 27.60  0.19  5.24 EUR F. Final Interest paid (subtract tax): 27.60  5.24  22.36 EUR G. Interest paid for whole year – use one month interest paid: 12  2.30  27.60 EUR H. Interest paid for whole year – subtract 1 month tax:  2.30  0.44   12  22.32 EUR I. 19% tax from Interest rate: 0.19  2.3  0.437 EUR. New interest rate: 2.3  0.437  1,863% . J. Interest paid for whole year – use new interest rate (1,863 %): 1200  0.01863  22.36 EUR K. Amount of money at the end of the year (count without tax): 1200  27.60  1227.60 EUR L. Amount of money at the end of the year (count with tax): 1200  22.36  1222.36 EUR M. Total interest paid (count with tax 19 %): 2.30  0.81  1.86 EUR N. Interest paid (count with tax 19 %): 27.60  0.81  22.36 EUR O. Piggy bank money: 12  100  1200 EUR Model 2

 1  2  ...  12   12  

P. Direct counting of the profit without tax: 1200  100  0.023   TASK 2 a) A. Guess – bank is better B. Guess – neighbor is better C. Answer after solving part b)

b) A. Interest paid to the bank: 3000  0.156  468 EUR B. Interest paid to the bank: 3468  3000  468 EUR C. Interest paid to the neighbor per month: 3000  0.015  45 EUR D. Conversion neighbors months’ surcharge to years’: 1.5  12  18% E. Conversion neighbors months’ surcharge to year’:  3540 : 3000   100%  118% it means, years’ surcharge is 18 %. F. Neighbors’ years’ surcharge: 45  12  540 EUR G. Neighbors’ years’ surcharge (using his years’ interest rate): 3000  0.18  540 EUR H. Surcharge paid to neighbor: 3450  3000  540 EUR I. Difference between those offers: 540  468  72 EUR J. Amount we should pay back to the bank: 3000  1.156  3468 EUR K. Amount we should pay back to the neighbor: 3000  1.18  3540 EUR L. Amount we pay back to the neighbor:

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1. month 2. month 3. month 4. month 5. month 6. month 7. month 8. month 9. month 10. month 11.month 12. month

30001.015 = 3045 30451.015 = 3090.68 3090.681.015 = 3137.04 3137.041.015 = 3184.10 3184.101.015 = 3231.86 3231.861.015 = 3280.34 3280.341.015 = 3329.55 3329.551.015 = 3379.49 3379.491.015 = 3430.18 3430.181.015 = 3481.63 3481.631.015 = 3533.85 3533.851.015 = 3586.86

Together: 3586.86 EUR M. Difference between bank and neighbor: 3540  3468  72 EUR N. Conversion banks’ years’ interest rate to months’: 15.6 : 12  1.3% O. Difference in interest rates of bank and neighbor: 18  15.6  2.4% P. Amount we should pay back to the neighbor: 3000  540  3540 EUR c) A. Conversion banks’ years’ interest rate to months’: 15.6 : 12  1.3% B. Neighbors surcharge per month max. 1.3 % p C. Try to find p % surcharge: 468  12   3000 100 D. Solution of in-equation: p  1.3% E. Conversion banks’ years’ interest paid to months’: 468 : 12  39 EUR F. Month interest rate equal to 39EUR:

39

.100%  1.3% 3000 G. Difference in interest paid: 540  468  72 EUR H. Difference in interest paid per month: 72 : 12  6 EUR

I. Guessed answer. TASK 3 A. Debt status at the end of the 1st year: 99000  1.13  111870 EUR B. Debt status at the end of the 1st year after subtracting of the payment a: 99000  1.13  a C. Debt status at the end of the 2nd:  99000  1.13  a   1.13 D. Debt status at the end of the 2nd year after subtracting of the payment a:

 99000  1.13  a   1.13  a  99000  1.13  a 1  1.13 E. Debt status at the end of the 3rd:  99000  1.13  a 1  1.13  1.13 rd 2

2

F. Debt status at the end of the 3 year after subtracting of the payment a: 3 2 99000  1.13  a 1  1.13  1.13  rd G. Debt at the end of the 3 year should be equal to the zero – we got and equation. H. Fairness of the debt division: 1. year… ….3000 EUR 2. year… …33000 EUR 3. year …... 33000 EUR I. Interest paid in one year: 33000  0.13  4290 EUR J. Debt at the end of the 1st year: 33000  4290  37290 EUR


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K. Interest paid in the 2nd year: 37290  0.13  4847.70 EUR L. Debt at the end of the 2nd year: 37290  4847.70  37847.70 EUR M. Interest paid in the 3rd year: 37847.70  0.13  4920.20 EUR N. Debt at the end of the 3rd year: 33000  4920.20  37920.90 EUR O. Annuity: 4290  4847.70  4920.20  14057.90 EUR P. Interest paid in the 1st year: 99000  0.13  12870 EUR Q. Every year payment: 12870  3  38610 EUR R. Other incorrect answers: 99000  12870 EUR, 99000  12870  2 EUR, 99000  12870  3 EUR S. 33000  12870  45870 EUR T. 99000  12870  111870 EUR U. 12870 : 12  1076.50 EUR V. 111870 : 3  37290 EUR W. Balance after the 1st payment: 111870  37290  74580 EUR X. Balance in the 2nd year: 74580  74580  0.13  84275.40 EUR Y. The 2nd payment: 84275.40 : 2  42137.70 EUR Z. Debt after the 2nd payment: 84275.40  42137.70  42137.70 EUR Aʹ. Balance in the 3rd year: 42137.70  42137.70  0.13  47615.60 EUR Bʹ. Last payment: 47615.60 EUR Cʹ. 99000  12870  1.13  126413.10 EUR Dʹ. 99000  0.13  2  99000 : 36  3465EUR

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TASK 4 a) A. Find in the table “Status of the debt”, row “Initial loan amount”: 66600 EUR B. Find in the table payments from first column: 23500 EUR, 19500 EUR and 16500 EUR C. Count of payments as sum of “Payment amount“ and „Interest paid“. D. Count of the interest x 

9990

.100 %  15%

66600

b) A. Debt is decreasing by principal: 66600  13510  53090 EUR B. Debt status in previous year is equal subtract of payment and previous debt, for example debt at the end of the 1st year is equal to 66600  23500  43100 EUR C. Count interest paid in the 2nd year: 0.15  53090  7963.50 EUR D. Debt status in the 2nd year: 53090  11536.50  41553.50 EUR E. Interest paid in the 3rd year 0.15  41553.50  6233 EUR and debt status in the 3rd year: 41553.50  10267  31286.50 EUR F. Interest pain in the 4th year: 0.15  31286.50  4693 EUR c) A. Last payment 7100 EUR (debt = previous debt – payment) B. Last payment 14500 EUR (payment is decreasing by 2000) d) A. 66000 EUR B. 74000 EUR C. 94911 EUR e) A. guess: 12 560 or 28 311 EUR B. Interest rate 2.3 %, so 19500  0.977  19041 EUR