Is Your Student Loan Worth Paying Off Early?

If you're like me (an English or Welsh student who started an undergraduate degree anywhere in the UK on or after 1 September 2012 and you took out a student loan to help cover your fees), you probably find yourself with a significant student loan to your name. Most people I know have come to terms with this fact. They treat their loan as an additional tax that they will have to pay for 30 years after they start working.

By letting your student loan repayments take care of themselves, you subject yourself to fairly hefty interest charges on your loan. There are calculators online, such as this, which calculate exactly how much debt you will end up paying as a percentage of your initial student debt, factoring in these interest charges. Clearly, the longer it takes you to repay your loan, the longer you expose yourself to interest charges, all else being equal. This effect is counteracted by the fact that if you do not repay your loan at the end of 30 years, your debt is written off. These two competing factors mean that there is a salary threshold at which you maximise the total amount you will end up repaying. It turns out that with an initial student debt of £40,000 and an annual pay rise of 3.6%, you maximise the amount you end up repaying if you have an initial gross annual salary of £47,250. If this is your salary, you will end up paying close to an additional 100% of your initial loan in interest charges.

While this effect is interesting and perhaps shocking, in my eyes, it is not practical. Yes, if you earn £47,250 per year, you will pay the maximum in interest charges, but your alternative is to make voluntary repayments towards your student loan. In doing so, you are paying the opportunity cost of investing in the market, where your cash could potentially earn a return in excess of the interest you pay on your loan.

In the following, we will investigate whether making early voluntary repayments towards your loan is an effective strategy from an investment point of view, or whether a student loan holder is best off biting the bullet, and paying the extra interest charges in order to take advantage of the extra cash they have to invest with.

Simplifying the Terms of the Loan

Looking at the student finance website, you're presented by a confusing and messy page explaining how repayment works. In reality, it's not that complicated. Here's how it works.

How repayment works

If, like me, your course started after 1 September 2012, you have a Plan 2 loan. This means that you will start repaying your loan the first April after you finish studying, and you will only pay anything if your current gross income is over £26,575 a year (or £2,215) a month. You will then pay 9% of your gross income in excess of this threshold each payment period. If you are paid a monthly salary and you earn £2,315 per month, you would then repay 9% of £2,315 - £2,215 = £100, which is £9 per month.

If you're looking at that £9 per month and thinking 'that doesn't seem like much, I spend that much on Netflix', I'd firstly recommend jumping in on your parent's Netlfix account, but I would also point out that if you still have an outstanding student loan after 30 years, your loan is written off. Your 50 year old self can then relax in the knowledge that the last trace of your naive student years is behind you.

At any time, you also have the right to make extra, voluntary contributions to your student loan repayment on top of the automatic contribution at any time, provided that this voluntary payment is larger than £5.

How the interest works

It is important to realise that even while you are studying, your loan is accruing interest. While you study, your rate of interest is the UK RPI plus 3%. At the time of writing, UK RPI is at 2.4%. So if you are still studying right now, you are paying 5.4% in interest.

Once you are eligible for repayment, you are charged an interest of RPI, plus an extra sliding charge of between 0% for those earning £26,575 per year or less up to 3% for those earning over £47,835 per year. The exact way that the additional rate scales from 0% to 3% turns out to be incredibly difficult to find in the student finance documents. From a modeling perspective, we will assume a linear scale between 0% and 3%.

An important detail that doesn't seem to get mentioned anywhere online or on the student finance documents is the interest rate schedule of these interest payments. According to this article in the FT, the repayments are monthly, and compound. To me, this is an egregious lack of transparency from the government. I spent the good part of an hour searching for this information, and the only source I could find was in a subscription newspaper that many students have no access to. The article shares my view and states that the details of the interest payments is 'hidden in the small print'. But I digress.

Building the Model

In order to analyse the merits of different strategies of repayment, it is important to lay down both some notation, and also some criteria for comparing approaches.

Notation and framework

In the following, we will denote the total outstanding debt at the point of starting repayment by £$D$. For most students who studied a 3 year undergraduate degree after 1 September 2012 and took out a full tuition and maintenance loan, this quantity will be in the region of £40,000. We denote the total amount paid towards the loan across its lifetime by £$P$.

Since interest payments and most professional's salaries are paid monthly, we consider a debtor's gross monthly salary. In the $n$th month of repayment we will denote this by £$s(n)$. We will denote the total outstanding debt in the same month by £$d(n)$. Clearly $d(0) = D$. We then denote the automatic contributions made towards the loan in the $n$th month by $p_a(n)$, and the voluntary contributions by $p_v(n)$, giving a total contribution in month $n$ of $p(n) := p_a(n) + p_v(n)$. Finally, we denote the interest accrued on the loan in the $n$th month by $i(n)$.

Next, we have the nominal annual rate on the loan in month $n$, which we denote by $r(n)$. This is the headline figure of RPI plus an extra rate that we saw above. We break this rate down into the sum of an inflation linked component $r_i(n)$, and an additional, salary dependent component $r_a(n)$. Assuming a linear scale between the lower and upper yearly salary thresholds of £26,575 and £47,835, which we denote by their equivalent monthly salary figures $l_i$ and $u_i$ respectively, we have

Then $r(n) = r_i(n) + r_a(n)$.

Since the interest is charged monthly and is compound, we find that the interest owed in month $n+1$ is given by

Furthermore, letting the lower repayment threshold be denoted by $l_r = £2215 = £26,575/12$, the automatic repayment contribution in month $n+1$ is given by

With the above notation in place, the outstanding debt at time $n+1$ can be expressed in terms of the various quantities in month $n$. The debt in month $n+1$ is just the sum of the debt in month $n$ and the interest accrued in month $n$, minus the payment made at the start of month $n+1$. Thus, we have

As is always the case when compound interest is involved, this equation represents a discrete-time version of the exponential growth model. If your monthly repayments $p(n+1)$ are not sufficient to outpace the interest rate $r(n)$, then you will find yourself on the wrong side of an exponential growth curve, and you will very quickly find that your student debt blows so far out of proportion that early repayment becomes practically impossible.

The next step in our model is to introduce a traded asset or portfolio with a (hopefully) positive return that we can invest in. We will denote the value of this asset in month $n$ by $X(n)$ and will assume throughout that this has a constant monthly compounding return of $\mu$, i.e. we have

Clearly, this is a big assumption. Even a well-diversified portfolio experiences systematic risk, meaning there are always random factors influencing returns. However, the point of our model is to provide a tractable way to generate ballpark figures with which to compare repayment strategies. For now, we avoid complications arising from more realistic assumptions in exchange for a more tractable model that produces less realistic results, but can still hopefully compare the relative merits of different approaches.

Without loss of generality, we will assume that $X(0) = 1$ so that

In our framework, we will assume that after all outgoings (including living costs), our remaining income that we have available to invest, $I(n)$, will be a proportion, $f(n) \in [0,1]$, of that months salary $s(n)$, i.e.

Before we can use this cash, the amount $p_a(n)$ is automatically removed, leaving us with $I(n) - p_a(n)$ left to invest.

Next, we need a measure of our wealth at time $n$, which we will denote by $W(n)$. This value consists of the value of our holdings in the risky asset, which is given by

where $Q(n)$ is the quantity of the asset that we hold at time $n$.

At each timestep $n$ we have the choice of investing our remaining income $I(n) - p_a(n)$ in the asset, or using our cash to make voluntary loan repayments. We use a proportion $x(n) \ge 0$ of this cash to make voluntary prepayments, i.e.

and we invest the remainder in the risky asset, i.e. we invest

in the asset. Therefore, we have

so that

Our goal is thus to find the optimal sequence of proportions $x(n) \ge 0$ to make early repayments with, in order to maximise our terminal wealth $W(N)$. Here, $N$ is the final month of repayment, i.e. the 360th month. At this time, our debt may or may not have already been written off via earlier repayments.

The optimal strategy

Now that we have the notation in place, it is easy to define a repayment strategy. By repayment strategy, I refer to a sequence of additional repayment contribution fractions $x(1),\ldots,x(N)$.

Our goal is then to find the sequence $x := (x(1),\ldots, x(N))$ that maximises our terminal wealth, i.e. to find

If we were to consider the problem in full generality, a naive optimisation would lead to a 360 dimensional problem. A reasonable simplification that will make our problem far more tractable is that of a constant strategy, i.e. strategies where $x(n) = x$ for all $n$. From a financial perspective, this represents a set and forget approach, where a fixed proportion of ones remaining monthly income is used to make extra voluntary loan repayments.

There is an important extra consideration with this approach. If we have fully repaid our loan before the terminal date $N$, i.e. if $d(n) = 0$ for some $n$, then we stop all repayments, so if $k$ is the first index such that $d(k) = 0$ then we have

It is clear that there are two extreme approaches. The first is to pile all of our remaining income into voluntary prepayments. In this case we build up no investments until our loan is fully repaid, but as soon as the loan is written off we are able to maximally invest into the risky asset. The second is to make no voluntary prepayments. In this case we focus on building up our investment from the start, but have less cash to use to invest since we incur an automatic repayment for the longest possible period of time.

The problem is thus to deduce where the optimal strategy lies between these two extremes.

Assumptions

While the framework we have built above is fairly general and allows for a lot of flexibility in the various different parameters, in order to generate useful results that we can easily compare, we need to add some assumptions and constraints.

The aim in this analysis is to give ballpark figures for the terminal wealth of different strategies, not exact values. The goal is that the effectiveness of various strategies relative to one another is accurately modeled, not necessarily that the absolute values themselves are.

A few assumptions have already been baked in. One of these is that the maximum additional interest rate one pays will always be 3% and that the upper and lower thresholds will remain where they are. The government is capable of changing these figures and has done so periodically. Nonetheless, building a model to predict where these values are headed in the coming weeks and months is beyond the scope of this post, and would be of questionable utility anyway.

Another assumption I believe is reasonable is that RPI will remain close to its historic average. The above plot shows RPI figures for the years 1990-2019. Observe that the values tend to oscillate around a mean value of 3%. Hence, we will use an RPI of 3% in our model. I would also argue that a precise estimate of the RPI figure is unnecessary. Any increase in RPI that leads to an increase in the rate charged on the loan should be offset by a corresponding increase in one's salary.

The next assumption is perhaps the most restrictive but will be very useful in terms of simplifying our model. We will assume that salary raises occur monthly and at a constant nominal annual rate of $a_m$. Of course normally people get raises occur once a year. There will be a slight difference in the results using monthly increases compared to yearly, but again, our purpose here is to come up with a tractable model. If a reasonable yearly increase is $a$, then we can convert this to a monthly rate $a_m$ via

For example, if we assume a yearly raise of 5%, this would translate to a nominal annual raise of around 4.89%.

With this assumption, we have

Therefore, specifying $a$ and $s(0)$ fully determines the sequence of salaries $s(1),\ldots,s(N)$.

Another assumption we will use is that our disposable income, before accounting for automatic loan repayments, is a fixed value, so that $f(n) \equiv f >= 0.09$. This final constraint is just to ensure positivity of our disposable income. This assumption seems reasonable. Tax and NI contributions scale relatively linearly with income, and peoples living costs tend to increase in line with their incomes.

Lastly, as we touched on earlier, we have assumed that our investable asset returns a steady constant return month-on-month.

Implementation and results

With our framework in place, it was easy to implement the model in Python. If you're interested, you can find the code on GitHub.

In the following, the method was applied using sensible values for the various parameters.

Looking at UK average wage increases for the year of 2019, the national average rise was 3.6%. I think this is a conservative figure for an aspiring young professional, so we will use a value of $a = 0.05$.

In 2019, the average student entered repayment with a total student debt of £36,000, so we will use $D = 36000$.

We will use a disposable income of 40% of ones gross salary as the parameter $f$. I think this reflects a reasonable proportion once things like tax, National Insurance and living costs like rent are taken into account.

Finally, a return of $\mu = 0.05$ for the asset will be used. By concentrating your investments on index funds, this is historically a pretty good return. Clearly if you believe you can do better by generating a hand-picked portfolio, or if you are willing to take on more risk, you could use a larger figure than this.

Above, terminal wealth at the end of the 30 year period versus the proportion of remaining disposable income used to make voluntary prepayments for various different starting salary levels is plotted. The plots show that for starting salaries below a threshold, one is better off not making voluntary prepayments. In other words, it is a better idea to take all of your disposable income and use it to invest prudentially across a well-diversified portfolio with a steady annual return. Beyond this threshold, it may be a better choice to make steady voluntary repayments. The middle plot shows a blown up version of the case of a starting salary of £40,000. Here we see that it is slightly more effective to make voluntary repayments, and that the benefit of doing so tapers off at around 50% of available income. Don't be misled by the vertical scaling of this plot, it is still only a minor wealth increase.

In the bottom plot, the length of time to fully repay ones debt is shown. Clearly, there is an exponential drop off in the time taken to repay your full student loan once you do breach the threshold at which your loan is not simply written-off after 30 years.

Upon reflection on these plots, it seems to me that with our assumptions, for an individual earning a starting salary of £40,000 or over, it is a good idea to invest a small percentage of your available income into voluntary repayments. An investment of just 20% makes the most out of the steepest part of the terminal wealth curve and significantly deceases the time it takes for full repayment of your student loan. This gives a good balance between paying off your loan fast, allowing you to benefit from the halting of the automatic contributions at an earlier point, while giving you cash to use from day 1 to potentially benefit from riskier investments.

It is also important to consider tax implications of repaying ones loan. In our analysis, we have neglected the important point that while the money we use to make early repayments is tax-free, the capital gains we make on our investments are generally not. This might prompt us to place more weight than our model would suggest on making early repayments.

Conclusions

Clearly, the model we have used is a simplification of reality. In the real world, portfolio values rise and fall - they don't grow at a constant rate. Furthermore, people's disposable incomes ebb and flow through different points of the year.

It's also clear that restricting our proportion of remaining wealth, technically our control variable, to constant values is another big simplification (but one that probably reflects many people's set and forget mindset towards setting up monthly loan repayments). A more advanced study might leverage the power of Richard E. Bellman's dynamic programming principle, and the area of optimal control. I think this could be an interesting application of the optimal stochastic control framework, similar in vein to Merton's portfolio problem.

For now, these extensions are beyond the scope of this analysis. Interesting though they are from a technical perspective, my intuition tells me that the conclusion will not change - you are probably better off treating your student loan as another tax.