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How can compound interest and loan amortization models evaluate the repayment burden of student loans in Japan?
This IB Internal Assessment in Mathematics investigates how compound interest, geometric sequences, and loan amortization formulas can be applied to evaluate the repayment burden of Japanese student loans under different repayment schemes. Using JASSO data, it models both interest-free and interest-bearing loans to compare long-term financial outcomes.
August 20, 2025
* The sample essays are for browsing purposes only and are not to be submitted as original work to avoid issues with plagiarism.
Mathematics Internal Assessment
International Baccalaureate (IB)
Research Question:
To what extent can mathematical models such as compound interest and loan
amortization be used to evaluate the total repayment burden of student loans in Japan under
different repayment schemes?
!"
Table of Contents
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1. Introduction
In today's world economy, the cost of pursuing higher education continues to rise, and Japan is
no exception. University students face higher charges for tuition fees, along with accommodation
and living costs that many families can hardly afford. To resolve this, the Japan Student Services
Organization (JASSO) provides two main types of student loans, which include Type 1 (interest-
free) and Type 2 (interest-bearing)(Çetinkaya et al.202). These loans have a vital function in
enabling students to access higher education opportunities. Nevertheless, the loans trigger long-
term financial obligations that might that students may not fully understand when they take the
loans. The majority of the students are not very familiar with how interest rates and repayment
terms affect the total repayment amount. This raises a very important question of how
mathematics can help one to understand the implications of different loan repayment structures.
Thus, the research question is as follows;
To what extent can mathematical models such as compound interest and loan
amortization be used to evaluate the total repayment burden of student loans in Japan under
different repayment schemes?
My reason for studying this topic originates from personal relevance. As I am preparing
for university, I will most probably need to utilize student loans. The majority of students,
including myself, do not think about the effect of interest on total repayment. This investigation
will allow me to apply mathematical principles like compound interest, sequences, and
amortization formulas to a real-life financial issue, making mathematics relevant and real in
terms of personal finance planning.
!-
2.0 Aim
This Internal Assessment mathematically aims to analyze the Japanese repayment burden of
student loans. I will utilize financial formulas such as compound interest and the Equated
Monthly Installment formula to model the different repayment scenarios. The analysis will be
based on real JASSO data and will analyze how changes in the interest rates, loan types, and
repayment periods affect the total payments. In this exploration, I predict that the Type 2 loan at
3% interest would involve a much higher overall repayment than the Type 1 interest-free loan. I
also expected that, for the first few months of repayment, a larger portion of EMI would go
towards interest than principal. These assumptions will guide my investigation, and I will test
them with modelling below.
3.0 Mathematical Background
This investigation applies three mathematical concepts from financial mathematics, which
include compound interest, geometric series, and the loan amortization formula, often referred to
as the EMI (Equated Monthly Installment). These concepts assist in the accurate modeling of
student loan repayment simulations and in comparing various financial scenarios.
Compound Interest
Compound interest is when the interest is added to the original borrowed amount (principal) and
the future interest that is calculated on both the principal and the accumulated interest. This is
especially applicable in the interest-bearing student loans, where payment is deferred, and this
allows the debt to accrue (grow) with time. The compound interest formula is:
A=P(1 + r)n(Tingtingetal . 170)
!3
Where:
The above formula is adopted to determine the future value of each disbursement in case
the interest is applied monthly or annually after a grace period.
Geometric Sequences
Student loans are often disbursed in annual installments over a number of years. Each installment
accumulates interest over a different number of years depending on when it was disbursed. This
can be modeled using geometric sequences. The future value of the loan of each year is treated as
the term in a geometric sequence(Putri et al.270):
Where:
A = Thetotalamountafterncompoundingperiods,
P = Theoriginalprincipal,
r−Theinterestrateperperiod,
n = Thenumberofcompoundingperiods .
Tk=P(1 + r)n−k
Tk= Thefuturevalueoftheloandisbursedinyeark,
P = Theannualdisbursement,
r = Theannualinterestrate,
n−k= Thenumberofyearstheloanaccruesinterest .
!L
It is important to note that the amount owed at the end of the degree is the sum of the
above geometric sequence.
Loan Amortization and EMI Formula
Students pay their loans on a monthly basis after graduating, and these payments include both
interest and principal, and they are normally the same every month. The EMI (Equated Monthly
Installment) formula is used to calculate the monthly payment:
Where:
The above formula is important in the modelling of repayment over time and also in
comparing the burdens of loans under different terms.
4.0 Mathematical Modelling
To effectively analyze the use of mathematics in real financial decisions, I collected data from
the Japan Student Services Organization (JASSO), Japan's primary student lender. JASSO offers
two simple types of loans:
EMI =P⋅r(1 + r)n
n(1 + r)n−1(Reddyetal . 820)
EMI = Thefixedmonthlyrepayment,
P = Thetotalloanamount,
r = Themonthlyinterestrate(annualrate÷12),
n = Thetotalnumberofmonthlypayments .
!:
1. Type 1 Loan: Interest-free loans to students who are eligible as per academic and income
standards(Japan Student Services Organization, Second Party Opinion).
2. Type 2 Loan: Interest-bearing loans to a broader cohort of students, with interest typically
ranging between 0.1% and 3% per year. The interest is applied after graduation and is
compounded monthly(Kobayashi).
Loan disbursements typically occur monthly during the study period, but for simplicity, I
will assume that the loan is issued annually at the start of each academic year. Repayment begins
six months after graduation, with a loan term extending up to 15 years in equal monthly
installments(Japan Student Services Organization).
4.1 Assumptions for Modeling
To build realistic yet manageable models, I made the following assumptions:
1. The student is enrolled in a 4-year undergraduate program.
2. The student borrows ¥700,000 annually, totaling ¥2,800,000 over four years.
3. Loans are disbursed at the start of each academic year.
4. No repayments are made during the study period.
5. Repayment begins 6 months after graduation, over 15 years (180 months).
6. For the Type 1 loan, 0% interest is applied (interest-free).
7. For the Type 2 loan: 3% annual interest, compounded monthly after graduation.
4.2 Scenario 1 – Type 1 Loan (Interest-Free):
In this scenario, the student receives an interest-free loan from JASSO. The student borrows
¥700,000 per year over the course of a 4-year undergraduate program, resulting in a total loan of:
TotalL oa n = 4 ×¥700,000 = ¥2,800, 000
!>
Since this is a Type 1 loan, no interest is charged, and the student begins repayment six
months after graduation. The repayment is structured as equal monthly payments over 15 years
(180 months). Therefore, the monthly installment is calculated as:
Each monthly payment goes entirely toward reducing the principal, since no interest is
added. For instance, I did the calculation for the first 3 months as follows;
The formula used to calculate the remaining balance for any month mmm is:
Month 1:
Month 2:
Month 3:
The same formula was applied repeatedly for months 4 through 180. The result is a
linearly decreasing schedule of remaining balances, where each month the loan is reduced by
exactly ¥15,555.56. The values below represent the repayment schedule for the first 10 months.
For the rest of the table, see Appendix A, which is a summary for every 12 months.
EMI =2,800, 000
180 =¥15,555 . 56
RemainingBalancem= 2,800, 000 −(m−1) ×15,555 . 56
RemainingBalance=¥2,800, 000 − 0 × ¥15,555 . 56
=¥2, 800,000 . 00
RemainingBalance=¥2,800, 000 − 1 × ¥15,555 . 56
=¥2,784, 444.44
RemainingBalance=¥2,800, 000 − 2 × ¥15,555 . 56
=¥2,768, 888.89
!Q
Table 1: Type 1 Loan Repayment Schedule (First 10 Months)
Afterwards, I generated the following graph with the help of Excel software;
Figure 1:Type 1 Loan Repayment Schedule (Interest-Free)”
The graph for the Type 1 loan shows a straight, linearly decreasing trend, reflecting equal
monthly repayments of ¥15,555.56 over 180 months. Since the loan is interest-free, each
Month
Monthly payment (¥)
Principal Paid (¥)
Remaining Balance (¥)
1
15,555.56
15,555.56
¥2,800,000.00
2
15,555.56
15,555.56
¥2,784,444.44
3
15,555.56
15,555.56
¥2,768,888.89
4
15,555.56
15,555.56
¥2,753,333.33
5
15,555.56
15,555.56
¥2,737,777.78
6
15,555.56
15,555.56
¥2,722,222.22
7
15,555.56
15,555.56
¥2,706,666.67
8
15,555.56
15,555.56
¥2,691,111.11
9
15,555.56
15,555.56
¥2,675,555.56
10
15,555.56
15,555.56
¥2,660,000.00
Type 1 Loan Repayment Schedule (Interest-Free)
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payment goes entirely toward reducing the principal. The balance starts at ¥2,800,000 and
decreases by a fixed amount each month, forming a uniform slope. This repayment behaves like
an arithmetic sequence, where:
The graph reaches zero at month 180, indicating full repayment. There are no fluctuations
or curvature, as the model excludes interest or variable rates. This makes the Type 1 loan simple,
predictable, and budget-friendly, offering a clear advantage over interest-bearing loans with
complex, nonlinear repayment paths.
3. Scenario 2 – Type 2 Loan (3% Interest, Compounded Monthly):
The JASSO Type 2 loan charges interest, typically ranging from 0.1% to 3%. In this
model, I assume a 3% annual interest rate, which is common and realistic. The interest begins
accumulating after graduation, and the loan is repaid in equal monthly installments (EMIs) over
a 15-year (180-month) term. Below is the loan information;
Total loan amount: ¥2,800,000 (¥700,000/year × 4 years)
Interest rate: 3% annually → 0.0025 monthly
Loan term: 180 months (15 years)
Now, I calculated the standard EMI (Equated Monthly Installment) using the following formula:
Where:
180 ×¥15,555 . 56 = ¥2,800, 000
EMI =P.r(1 + r)n
(I+r)n−1
P=2,800, 000(loanpr incipal )
!"/
So, the monthly payment is ¥19,336 for 180 months. For the first month, as an example, I
calculated it as follows;
Initial loan balance: ¥2,800,000
Interest (Month 1):
Principal Paid:
Remaining Balance:
This method is repeated monthly, where interest is calculated from the current balance,
subtracted from the fixed EMI, and the remaining portion goes toward reducing the principal.
r=3
100 ×12 = 0.0025(monthlyinterestrate)
n=180months
EMI =2,800, 000 ×0.0025(1 + 0.0025)180
(I+ 0.0025)180 −1
(I+ 0.0025)180 = 1.5687
EMI =2,800, 000 ×0.0025(1 + 0.0025)180
1.5687 −1
EMI = ¥19, 336
Interest = 2,800, 000 ×0.0025 = ¥7,000
Pr incipal =EMI −Interest = 19,336 −7,000 = ¥12,336
NewBalance = 2,800, 000 −12,336 = ¥2,787, 664
!""
The result obtained is presented in the following table. For the result of the values, see Appendix
B, which shows a reduced version of amortization for student loans (Japan) with a summary of
data shown every 12 months and also the final month(month 180).
Table 2: Type 2 Loan Repayment Breakdown (First 10 Months)
Afterwards, I generated the following graph for effective analysis with the help of Excel software
;
Month
EMI (¥)
Interest Paid (¥)
Principal Paid (¥)
Remaining Balance (¥)
1
19,336
7,000.00
12,336.29
¥2,787,663.71
2
19,336
6,969.16
12,367.13
¥2,775,296.59
3
19,336
6,938.24
12,398.04
¥2,762,898.54
4
19,336
6,907.25
12,429.04
¥2,750,469.50
5
19,336
6,876.17
12,460.11
¥2,738,009.39
6
19,336
6,845.02
12,491.26
¥2,725,518.13
7
19,336
6,813.80
12,522.49
¥2,712,995.64
8
19,336
6,782.49
12,553.80
¥2,700,441.84
9
19,336
6,751.10
12,585.18
¥2,687,856.66
10
19,336
6,719.64
12,616.64
¥2,675,240.02
!".
Figure 2:Type 2 Loan EMI Breakdown (3% Interest)
Graph 2 illustrates how the fixed monthly payments (EMIs) for the interest-bearing Type
2 loan are divided between interest and principal over 180 months. In the early months, a larger
portion of the EMI goes toward paying interest because the outstanding balance is high. As the
loan balance decreases, the interest portion gradually declines, and the principal portion
increases. This creates two intersecting curves: the interest curve slopes downward, while the
principal curve rises. This pattern reflects the nature of loan amortization, where early payments
contribute more to interest and later payments primarily reduce the principal. The graph
highlights how interest accumulation significantly affects repayment structure, resulting in
slower early progress in reducing the debt compared to an interest-free loan.
!"-
5.0 Conclusion
From the repayment models and graphs presented, one can see that interest plays a significant
role in the total repayment burden of student loans. The Type 1 loan, which is interest-free,
provides plain repayment terms on which the repayment equals the amount taken as a loan. The
Type 2 loan, even at a modest interest rate of 3% per annum, provides the equivalent of
¥681,684.20 as additional payments within a period of 15 years. This increased cost is because of
compound interest, which is observed in the EMI breakdown graph, where payments made
earlier mostly cover interest rather than principal.
Type 2 loans end up being more expensive, even though the monthly EMI of 19336 yuan
is affordable. In case a student fails to meet the criteria for approval of a Type 1 loan, the loan
term length should be considered deeply. Short-term loans involve lower interest and higher
monthly payments. Some students might be forced to work part-time or cut down expenses to
pay their loans. One of the major limitations of this study is the assumption of fixed interest and
timely payments. An improvement to resolve this limitation will be to include models with
shorter repayment periods to observe how they change the total repayment. Another approach in
the exploration could have involved modelling variable interest over time instead of a fixed 3%
annual rate. This will provide a more realistic and dsynamic analysis. Real-life situations include
late payments, early payments, inflation, and variable interest, all of which might affect the
outcomes. Nevertheless, this exploration demonstrates how mathematical models can be applied
in order to make educated financial choices and educate students on their responsibility toward
borrowing. If given another opportunity to conduct this exploration, I will investigate different
repayment periods, early repayment strategies, and refinancing periods.
!"3
6.0 Work Cited
Çetinkaya, Reşit, and Ertuğrul Çavdar. "A comparative perspective on fee policies used in OECD
countries for financing higher education." Journal of Social Sciences and Education 7.1
(2024): 201-233.https://s-space.snu.ac.kr/bitstream/10371/220291/1/000000188120.pdf
Japan Student Services Organization (JASSO). Outline of Scholarship Programs. JASSO,
www.jasso.go.jp/shogakukin/about/taiyo/index.html. Accessed 16 July 2025.
Japan Student Services Organization (JASSO). Second Party Opinion: JASSO Social Bonds.
J A S S O , 2 0 2 2 , www.jasso.go.jp/en/about/ir/__icsFiles/afieldfile/
2022/11/30/70second_party_opinion_en_1.pdf. Accessed 16 July 2025.
Kobayashi, Toshiyuki. Japan Student Services Organiza=on: Social Finance and Bond Issuance.
International Capital Market Association (ICMA), Oct. 2017, www.icmagroup.org/assets/
documents/Events/1110_JASSO_Toshiyuki%20Kobayashi.pdf. Accessed 16 July 2025.
Putri, Reni Albertin, Susiswo Susiswo, and Makbul Muksar. "Students' Learning Obstacles on
Sequences and Series Viewed by Pirie Kieren's Theory." Jurnal Didaktik
Matematika 11.2 (2024): 269-286.https://jurnal.usk.ac.id/DM/article/download/
37855/22069
Reddy, Challa Ashok Kumar, et al. "A Client-Side Web Application for Loan Eligibility and EMI
Calculation." International Journal of Computational Learning & Intelligence 4.4
!"L
(2025): 818-822.https://milestoneresearch.in/JOURNALS/index.php/IJCLI/article/
download/235/234
Tingting, Wang. "Simple Interest and Compound Interest." Dictionary of Contemporary Chinese
Economics. Singapore: Springer Nature Singapore, 2025. 163-684.https://
link.springer.com/rwe/10.1007/978-981-97-4036-9_839
!":
7.0 Appendices
Appendix A
Table 1:Type 1 Loan Repayment Schedule( Continoued)
Month
Monthly Payment (¥)
Principal Paid (¥)
Remaining Balance (¥)
12
15,555.56
15,555.56
2,628,888.84
24
15,555.56
15,555.56
2,442,222.12
36
15,555.56
15,555.56
2,255,555.40
48
15,555.56
15,555.56
2,068,888.68
60
15,555.56
15,555.56
1,882,221.96
72
15,555.56
15,555.56
1,695,555.24
84
15,555.56
15,555.56
1,508,888.52
96
15,555.56
15,555.56
1,322,221.80
108
15,555.56
15,555.56
1,135,555.08
120
15,555.56
15,555.56
948,888.36
132
15,555.56
15,555.56
762,221.64
144
15,555.56
15,555.56
575,554.92
156
15,555.56
15,555.56
388,888.20
168
15,555.56
15,555.56
202,221.48
180
15,555.56
15,555.56
15,554.76
!">
Appendix B
Table 2:Type 2 Loan Repayment Schedule(Continued)
Month
EMI (¥)
Interest Paid (¥)
Principal Paid (¥)
Remaining Balance (¥)
12
19336
6656.49
12679.51
2,649,915.51
24
19336
6270.83
13065.17
2,495,266.05
36
19336
5873.44
13462.56
2,335,912.78
48
19336
5463.96
13872.04
2,171,712.62
60
19336
5042.03
14293.97
2,002,518.15
72
19336
4607.27
14728.73
1,828,177.49
84
19336
4159.28
15176.72
1,648,534.07
96
19336
3697.66
15638.34
1,463,426.64
108
19336
3222.01
16113.99
1,272,688.99
120
19336
2731.88
16604.12
1,076,149.86
132
19336
2226.85
17109.15
873,632.78
144
19336
1706.46
17629.54
664,955.98
156
19336
1170.24
18165.76
449,932.07
168
19336
617.72
18718.28
228,367.99
180
19336
48.38
19287.62
64.85
!"Q
!"T
August 20, 2025
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Academic level:
High school
Type of paper:
IB Internal Assessment
Discipline:
Mathematics
Citation:
MLA
Pages:
7 (1800 words)
Spacing:
Double
* The sample essays are for browsing purposes only and are not to be submitted as original work to avoid issues with plagiarism.
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