# OPC Calculations 2

Oh, it’s $\LaTeX$ equations day again. Haha…

In my last post, I mentioned that high COE is more ‘beneficial’ for OPC owners from the savings point of view. But of course, it is often more practical to discuss actual spendings instead as savings can be very misleading. So I’ll try to formulate some equations now to calculate what is the actual amount spent for low and high COE. Let’s use this very simplified equation to calculate amount spent $S$.

$S=\mbox{downpayment}+\mbox{principal loan}+\mbox{interest}-\mbox{scrap value}$

Let $C$ be the COE of the car, $O$ be the OMV of the car, and $M$ be the price of the car without COE. Hence, we assume the actual selling price of the car to be $C+M$. Also assume $C+O>17000$.

Let us further assume that we will scrap the car at the end of the 10 year COE period. Let $D$ be the downpayment paid and we assume that $D$ is a fixed value becos the buyer only has $D$ amount of available cash. Let $I$ be the interest rate, and $T$ be the loan period.

First, we consider the case that person $A17$ bought an OPC at $C=17000$.

$\begin{array}{rcl}S_{A17}&=&D+(C+M-17000-D)\\&&+(C+M-17000-D)\times I\times T-0.5\times O\\&=&M+(M-D)IT-0.5O\end{array}$

Next, we consider the case that person $B0$ bought an OPC at $C=0$.

$\begin{array}{rcl}S_{B0}&=&D+(C+M-17000-D)\\&&+(C+M-17000-D)\times I\times T\\&&-0.5\times(O-17000)\\&=&(M-17000)+(M-17000-D)IT-0.5O+8500\\&=&M+(M-D)IT-0.5O-17000IT-8500\\S_{B0}&=&S_{A17}-(17000IT+8500)\end{array}$

Since both $I$ and $T$ are positive values, therefore $17000IT+8500>0$. Hence, we arrive at the following result.

$S_{B0}

In other words, all else equal, the actual amount spent by person $B0$ is lesser than that by person $A17$. In fact, the difference in amount spent takes the form of $17000IT+8500$. That is, the interest incurred by person $A17$ in borrowing the extra 17000 from the bank, and an additional 8500. This additional 8500 can be interpreted as person $A17$ spending 17000 more than person $B0$ on COE and recovering back 8500 more than person $B0$ (due to person $A17$ having higher OMV) at the end of 10 years.

It is interesting to note that this result is independent of the amount of downpayment, price of the car (without COE), and the OMV of the car. Also, the amount of extra money spent by person $A17$ is determined by the interest rate and the period of loan. At best, even if we assume both persons may have sufficient cash for full downpayment such that $D=C+M-17000$, then $I=T=0$, and person $A17$ is still worse off by 8500.

There you have it. It is more practical to buy an OPC when COE is low becos the actual amount spent is lesser. :)

-Dear1