Hello there! It’s been more than a month now for this work, but it has been fruitful to say the least. Firstly in the previous post I have indicated that the simulation works to show Islamic Banking System (PLS) performs better than the Interest Based (Conventional) Banking System. But that was only a simulation using some algortihms. This whole research initially intended to prove that the PLS model is better than Conventional model. While that might be the case (and it is), I still have to pursue this as a scientist an mathematician so that the work isn’t biased. So going forward, the study will be more focused on the comparison of the two models than to prove that one is better than the other. I do have a few hypotheses regarding the outcome of this whole research, but I am also considering the alternate hypothesis and expecting a few to be true. Regardless that was jus the brief of the whole “change of approach” in a sense. The next section outlines the new developments and progress regarding the models themselves.
IMPORTANT
The Islamic scheme stated in the previous post was making the assumption of using the Mudarabah policy. But it turns out, I was actually using Musharakah policy.
New Developments and Works
One thing to note is that the simulation idea came later. At first, I came up with a few basic equations to model the business’s growth and other aspects, but soon I realized that my approach was too “basic” and it failed to model the core of the business’s operations properly. Hence as an alternative approach, I thought to myself “why not treat the business operations as algorithms… since they are essentially a step by step set of operations?”. That’s when it hit me (the EUREKA moment)! A business’s operations consist primarily of — Investment → Revenue → Expenses → Profit. At the end of the trading period, the cycle starts again with some of the profit from the previous period being used to invest in the next. That’s when I decided to take the algorithmic approach and come up wih the simulation program for the two models. After analyzing the results (I was shocked and didn’t expect it to work) I thought that: an algorithm in essence can be converted into a mathematical function. So I set out to convert the simulation code into actual mathematical equations and functions.
Since these are finance models and they are time dependent, the first thing I thought of is it must be a set of “time-series” models. But I didn’t know how to exactly go about them since time-series models are mostly dependent on discrete values. But hang on… that’s exactly it! Even the simulation is discrete value dependent! And the values used in the model are dependent on the previous values from the previous interation. So that can only mean (in my head) a recursive definition of each of the model’s variables that is based on discrete time intervals. Hence I set out to convert the simulation into something more analytical. I looked at how each of the variables were being updated in each iteration and incorporated them accordingly in the mathematical definitions. After the set of recurrence relations definitons were complete for both of the models, I needed to convert them into their explicit forms so that the models can take any real values ($\mathbb{R}$) as inputs. That’s where the second hurdle came about and where I had to spend most of my time.
I took the naive approach to convert the recurrence relation models into explicit form by simply thinking of them as “simple non-homogeneous recurrence relations”. Once I was “happy” that I came up with the models, I had to test the explicit solutions and graph them to check if indeed they align with the simulation. And 1, 2, 3… they didn’t! The graphs ended up being asymptotic in nature and the values went to the asymptotes where they shouldn’t have gone. I tried to recalculate and re-solve the recurrence relations but no luck. But then, I remebered from my good friend’s WhatsApp message when I asked him about the type of questions he did during his Acturial Science course in university (THANKS A BUNCH Daniel Bannister! That message gave me the hype I needed). This was roughly 5 years ago from now. I think it was something to do with modelling Donald Trump’s hair as it moved back and forth using ODEs???. I saw that the finance questions were modeled using Ordinary Differential Equations (ODE). That’s when I changed the approach and tried to convert them using ODEs. I found a fantastic Math Stack Exchange thread (link here) that was discussing how to go about it! From there, I set out to convert them explicitly into a system of ODEs and finally ended up with working models. I am not even going to discuss the amount of decoupling of equations I had to do :’) but that’s something else. I believe the strong cohesive nature of the equations would be evident from the simulation codes. Regradless, I finally have continuous analytical mathematical models for the systems.
The next problem came out when trying to run the equations as-is in code. Long story short, it’s not feasible using the compute resources I have at the moment to arrive at symbolic solutions using Python which can later be converted into a function. I had to change my approach from getting a symbolic solution and then using it for the simulation, to solving the set of ODEs numerically and simulate it that way. The speed up is more than “whatever-number-greater-than-1-you want-to-use” times faster, simply because the program couldn’t even finish solving the ODEs symbolically with +4 mins of runtime. At that point I just gave up and had to choose a different approach, hence arriving at solving the ODEs numerically. I went for using Euler’s method to solve the equations. It should be noted that symbolically it’s pratically not feasible under realistic assumptions where there are fluctuations in the models’ behaviours i.e. stochasticity and noise of the functions in the model. Regardless, I managed to arrive at the equations and am now able to solve them numerically using the computer for simulation purposes.
Conclusion for now
Right now, the core part of the research is done. But now I am simultaneously trying to use existing metrics to evaluate the business’s performance using these models and writing up a draft “thesis/research article” to make it a bit formal. When I am done with the initial draft write up, I really need to get this reviewed. I would probably consult some lecturers at my university and get some feedback regarding this. I still have a long way to go, but so far, the work looks promising! Let’s pray for the best!
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