The mathematics behind Singapore's student loans (19/20)

Being a student about to enter university, student loans would undoubtedly play a substantial part of one’s finances during higher level education. Its sole purpose is to allow students to concentrate on their education without taking on part-time jobs simultaneously while financing living expenses, tuition, books, or other necessities required to maintain a modest lifestyle.

Tuition (annual fees), particularly for universities, has been increasing worldwide, especially in developed countries, with a statistical estimate of 1.6 times increase in real monetary value (Digest of education, 2006) because salaries and benefits of university personnel tend to rise faster than inflationary growth (‘Baumol’s cost disease’). Attributed with education being a fundamental role in functioning society, it gives room to those equipped with the power of nurturing new members of the labour force to accept higher costs, with hopes of funding a sound education investment. With tertiary education having significant importance in the lives of many, it is no wonder many students end up taking loans without much considerations of the future. In the US alone, this affects over 44 million borrowers, with an approximate USD $1.4 trillion of outstanding student loan debt.

Therefore, this necessitates the importance of students and even their family members alike should be aware of the amount of money owed to specific organizations (be it government or private), such that they are better equipped to make appropriate financial decisions when taking up loans, which would also include the repayment portion afterwards when they graduate. Even though most students’ loans include subsidies, it might seem daunting to owe such amounts of money without a significant net worth as a student, especially to poorer households.

This sparked my interest in exploring the mathematical nature of student loans and is applicable given that banks are economically profit driven where they usually strike a balance between income and attractiveness of the loan itself. By exploring loans in a mathematical perspective, it would foster a better understanding and finally justifying which type of loans are preferable and furthermore, even possibly apply the mathematical nature of loans to future expenditures. I would also apply mathematics used here into analysing current student loan packages in Singapore and evaluating their suitability. First, we look into how interest is calculated over a fixed sum of money borrowed.

When money is borrowed, we must make a percentage payment for the loan in addition to repaying the principal, which is the initial sum borrowed, which is the interest, and is usually expressed as a percentage of the principal to be paid with subjective frequency (either monthly or yearly). Simply put, if $1000 is borrowed for a year, and the annual interest rate is 10%, in a year, the amount owed will be $1000 (1+ interest rate)= $1000(1+0.10)=$1100. In typical bank loans, interest is usually compounded monthly (called compound interest). This interest rate is predominantly dependent on a variety of factors, most importantly the risk level of the loan, and the maturity of the loan (the longer the time period, the higher the interest rate because it is a higher risk loan for banks).

Compound interest, meanwhile, means that the interest rate (usually per annum) is multiplied as a percentage of the principal amount taken as a loan from the first compounded of interest, then subsequently, to the compounded amount in the previous compound of interest plus its principal. Mathematically, it can be written as:

\(S_n=P(1+i)^n\)

Where

𝑃 is the principal (initial) amount of money loaned;

𝑖 is the interest rate set by the institution (e.g. 50% = 0.5); and

𝑛 being the number of times the principal has been compounded by an interest rate of 𝑖 (usually varied depending on the length of repayment time and can be expressed in a percentage)

The validity of the interest rate formula can be proven by mathematical induction:

\(S_1=P+iP=P(1+i)\)

Where 𝑆₁, means the first time the principal sum 𝑃 is compounded by interest rate 𝑖, written as 𝑖𝑃 (total interest paid) since 𝑖 is actually a percentage of i𝑃 which is added to the principal amount.

Secondly, to prove \(S_n=P(1+i)^n\) is true, we have to prove that for \(S_t=P(1+i)^t,\) \(S_{t+1}=P(1+i)^{t+1}\) is true, where 𝑡 is an arbitrary value replacing 𝑛.​​​​​​​

The amount of money owed, after S_t has been compounded once more by 𝑖, gives:

\(S_{t+1}=S_t+iS_t=S_t(1+i)=P(1+)^t(1+i)=P(1+i)^{t+1}\ QED.\)

In reality, yearly tuition fees might be vary, due to changes in curriculum, etc. However, for simplicity, we can assume this to be constant. Even though financing an entire education upfront might be convenient, it is usually not done because loan eligibility would be undetermined at a fixed point of time. Furthermore, even if tuition fees are fixed yearly throughout the student’s candidature, it is commonly financially unwise to defer payments after borrowing more funds than would actually be utilized at that point in time, because interest will still be added (compounded) to the original amount borrowed even though it is unutilized.

Taking an general example of a loan without further expenses, of an eligible student taking a 3-year undergraduate program with a fixed tuition fee per annum of $10,500 (for years 2016/2017). We assume the loan quantum is\(\left(\frac{90}{100}\right)\) = $9450, which is usually 90% of total course fees, and the fixed annual percentage rate (APR) of 3.5%, with the sum compounded monthly at a rate of\(\frac{(3.5%)}{12}=\left(\frac{7}{24}\right)\%,\)the student will owe the financial institution (monetary values are rounded up to 2 d.p.) :

𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑤𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒

[Maybank education loan is used here]

During loan repayment, most students will opt for monthly instalments, given a monthly income, as compared to paying a lump sum. Usually, the principal is compounded monthly, with the rate of (APR/12), as a year consists of 12 months, however, in some countries, banks might compound it daily. Consistently repaying a loan allows the student to repay the loan over comfortable periods, such as ensuring a smaller amount is compounded afterwards, leading to a lower cumulative interest accrued. However, some loans, when paid earlier before maturity (avoidance of additional interest) might incur a charge called the early repayment fee (Maybank’s education loan has it at 1%). Separately, there is also a processing fee (usually 1-3%), which will be net off from the disbursed loan amount. Regardless 100% of the principal will still be compounded with the APR. (even though 97% to 99% is disbursed.)

The monthly instalment, known as the EMI (equated monthly instalment), is the amount that will be deducted from the student or the guarantor’s account in a monthly basis, usually after a grace period and this number is usually provided by the bank.

This value is important such that it allows the one to place aside a portion of their monthly salary to pay of his/her debt, and is also called the amortization schedule.

To further understand how much money we would owe and pay back per month, based on interest, we can derive a formula based on the steps on how the EMI will be determined. The rationale behind this formula is that it offers the buyer an alternative to pay small amounts over time, instead of paying off the complete sum upfront, assuming the interest remains unchanged. Suppose a student borrows 𝐿 dollars as a student loan, with an APR of 𝑖%. This means that, in a month, the financial institution charges an interest of \((\frac{i}{12})%\)%. At the end of the 1st month, the student owes the initial sum of 𝐿 dollars with the added interest of\( (\frac{i}{12})% \)%, which is:

\( L\biggl(1+\frac{i}{12}\biggl) \)

For simplicity, let\( \bigg(1+\frac{i}{12}\bigg) \) 𝑅, the EMI as 𝐸 and let \(L_n\) be the amount of money owed after the 𝑛th month. (after the 𝑛th month payment of 𝐸 is completed)

The student will pay 𝐸 dollars at the end of the first month, assuming the EMI is exactly followed, and will therefore still owe a total amount of \( L_1=L_0R-E. \). At the end of the second month (𝑛=2), the student will owe:

\(L_1=L_1\bigg(\frac{i}{12}\bigg)=L_1\bigg(1+\frac{i}{12}\bigg)=L_1R\)

After paying 𝐸 dollars again in the \(2^{nd} \)month, the student owes:

\( L_2=L_1R-E=(L_0R-E)R-E = L_0R^2-E(1+R) \)

In the 3^rd month, the student owes:

\(L_3=L_2R-E=[L_0R^2-E(1+R)]R-E=L_0R^3-E(1+R+R^2)\)

By induction, we could notice that at the end of 𝑛th month, in other words, the tenure of the loan, the student will owe an amount corresponding to 𝐿" :

\(L_n=L_0R^n-E\begin{matrix}n-1\\{\LARGE\sum}\\t=0\\\end{matrix}{R^t=L_0R^n-E(1+R+R^2+...+R^{n-1})}\)

From the expression above, we can see that \(1+R+R^2+...+R^{n-1}\)  is a geometric series, with 1 as the first term, and 𝑅 as the common ratio. Therefore,

\( 1+R+R^2+...+R^{n-1}=\frac{R^n-1}{R-1},where\ R\neq1 \)

\(L_n=L_0R^n-E\left(\frac{R^n-1}{R-1}\right)\)

Pragmatically, 𝑅 will never by one or lesser, because of the interest rate,\(\frac{i}{12}\) . Also, if\(R-1\) is negative, this means that it is possible that \(E\left(\frac{R^n-1}{R-1}\right)\) will be lesser than zero, which will mean the loan is increasing after a payment, which simply does not make sense as it is impractical.

If the loan is expected to be fully paid at the 𝑛th month, this simply means \(L_n=0.\)From equation (3), it will therefore give: 

\(E\left(\frac{R^n-1}{R-1}\right)=L_0R^n\Rightarrow E=\frac{L_0R^n(R-1)}{R^n-1}\)

Which can be manipulated into by re-substituting\(\bigg(1+\frac{i}{12}\bigg)=R:\)

\(E=\frac{L_0\left(1+\frac{i}{12}\right)^n\left(\frac{i}{12}\right)}{\left(1+\frac{i}{12}\right)^n-1}\)

Finally, replacing the equation with conventional terms:

\(E=\frac{pr(1+r)^n}{\left(1+r\right)^n-1}\)

Where

 r is the monthly [fixed] interest rate, APR divided by 12 (12 months in a year)

𝑛 is the number of payments [in months] (the longer the tenure, the lower the EMI,

however total amount of money paid to the bank subsequently is higher due to many more times of interest compounded)

𝑃 is the initial sum of money borrowed from the institution (principal) 𝐸 is the EMI

This formula gives the EMI for the student loan or any type of loan in particular with a similar repayment mechanism. (for fixed interest rate and equal repayments)

After further investigation into the amortization formula, I found that an interesting aspect of the EMI is that a portion of the EMI paid monthly is deduced from the principal, as well as the interest. Students who might have taken a student loan might have realized this, where the amount repaid covers a larger portion of the interest accrued initially, as compared to the principal. This information might be useful in the future when taking on mortgage loans because the interest portion paid might be tax deductible which can be claimed directly on tax returns to reduce taxable income. This can be pictured using mathematics, to find out the proportion of EMI towards interest or principal, we first recall the interest (per annum), 𝑖, at the end of the first month, where total initial principal,\( 𝐿_0\) and \(\bigg(1+\frac{i}{12}\bigg),\) denoted by R, which is:

\(L_0\bigg(\frac{i}{12}\bigg)=L_0(R-1)\)

The value from the above expression is deducted from the EMI, 𝐸, and the remaining balance is deducted from the principal. Mathematically, this will mean the amount that goes towards paying off the principal loan is\(E-L(R-1)\) after the \(1^{st}\) month. In the second month, the interest is now:

\((L_0R-E)\bigg(\frac{i}{12}\bigg)\)

The amount that goes towards paying the new principal, 𝐿𝑅−𝐸, for the \(2^{nd}\)​​​​​​​ month is now:

\( E-[{(L}_0R-E)(R-1)]=R[E-L0(R-1)] \)​​​​​​​

In the third month, the interest is:

\([L_0R^2-E(1+R)]\bigg(\frac{i}{12}\bigg)=[L0R2-E(1+R)](R-1)\)

And the amount of money that goes towards paying the principal in the 3^rd month’s EMI is, therefore:

\(E-[L_0R^2-E(1+R)](R-1)=R2[E-L0(R-1)]\)

To generalize it, one might notice that by continuing the months ahead, out of 𝑘th month’s APR, this amount is then directed towards paying the principal, where it increases exponentially (since k is the only variable):

\({Principal\ paid\ off\ in\ month\ k\ =R}^{k-1}[E-L_0(R-1)]\)

Whilst the rest is towards the interest incurred in the kth month. Afterwards, total principal amount after kth month’s EMI, as illustrated above, is now:

\( R^kL_0-E\left(\frac{R^k-1}{R-1}\right) \)

Equation (4) can also tell us that, since R must be more than 1 (R > 1) as the interest rate is a percentage over the principal, the amount of money contributed towards the principal increases each month, until all of it has been repaid; therefore, the amount due to interest added on each time after every payment will naturally decrease as well.

Furthermore, a graph can be drawn which distinguishes the payment (EMI) paid, whether it goes towards the principal or the interest section, by substituting numerical values into equation (4). We assume the amount loaned to be $29370.28 (after 3 years of tuition with interest covered in the first two years; calculation in page 3) with an annual interest rate of 3.5% or a monthly rate of \( \frac{0.035}{12}=0.292%\ (to\ 3\ s.f.) \) % (to 3.f.).As a control, the tenure decision will be 20 years, the maximum allowed tenure. The EMI will then be:

\(E\frac{pr\ (1+r)^n}{(1+r)^n-1}=\frac{29370.28(0.00292)\ (1+0.00292)^{20(12)}}{(1+0.00292)^{20(12)}-1}= $170.40 (2 𝑑. 𝑝. )\)

The calculated values for the principal and the interest portion will be in the appendix. Regardless on the tenure, the trend of the interest against the principal paid off would be similar since the same equation (4) is used, with only different values.

From figure 3, one can see that as the number of years the loan is taken increases, the difference between the normal instalment scheme and the DBS education loan becomes smaller, until a tenure of 13 years where the DBS education loan overtakes the normal instalment and continues to increase until it intercepts the interest scheme total payment at around year 19. So, therefore, even though the normal instalment scheme requires a highest amount of money to be paid off initially, even during education, if the tenure undertaken is high, it might be more preferable to take up the normal instalment loan, if there is a lack of funds because of other expenses that might be incurred. To be able to pay off the initial EMI during education, I might have to undertake part-time jobs to pay off the excess EMI ($305 monthly for 10 year period of normal instalment scheme as compared to $204.51 for the first two years of education with the interest servicing option) However, if I were eligible for the MOE’s education grant to take the education loan, it would be the better option if I were to have a tenure for 10 years. The interest scheme meanwhile, might only be preferable over if the DBS education loan’s total payment increases above the normal instalment loan as the amount of money paid throughout the years are more spread out, as compared to the DBS loan where the first 3 years will be added to the subsequent years. However, one should also note, in the event of sudden economic fluctuations, it is possible that the interbank lending rate (variable rate) loans which include the NUS-DBS tuition fee loan, which is currently the most suitable and cost-effective loan available, might be less cost-friendly over the long run. Hence in the event that interest rates do happen to rise in the future, the costs must be recalculated, to ensure personal indebtedness is kept to a minimum.

One limitation of this exploration, however, in a practical sense, is that for certain aspects of a bank loan, to factor in scenarios of early repayment, the entire sum has to be recalculated, and the values for EMI has to be evaluated again since early repayment covers a proportion of the principal as well as the interest. (as illustrated in Figure 2) Conversely, during instances where the monthly instalment is unable to be paid on time, the principal portion will be carried forward and added into the subsequent months and divided again, such that the EMI will be increased and the total amount of money paid over the tenure will increase as well.

Overall, the multiple variables that can be adjusted accordingly to the type of loan, as well as the flexibility of the formula such that it can be easily adjusted in many applications, as be it, student or even mortgage loans in the future does effectively utilise this amortisation formula together with formula (2), and also the foresight it gives into projecting a certain amount of money makes it a powerful tool into personal finance tool for many.