One of those times you would 'never' need math: part 1

Imagine back to the days you were in high school math lamenting the absent utility of the algebra you were being forced to learn. Hell, maybe you even took calculus and wondered when you would actually use something other than the basic orders of operation—PEMDAS. Well, this post will be about one of those circumstances. Let's figure out an optimal strategy for purchasing a new vehicle.

Our goal in this scenario is to maximize our future account balance given some annual net income and some arbitrary car value. We should first consider the down payment. The down payment on a vehicle is used to reduce the principle amount, \(P\), being financed. Financing rates are generally given as an annual percent interest (APR). Current rates (September, 2017) are \(3-4\%\) according to Wells Fargo. So, let us assume the rate is \(3.5\%\). If we remember back to high school math, the interest gained on a principle amount compounded annually is given by

\[  A(t) = P(1+r/n)^{n*t}, \:\:\:\:  (1)  \]

where\(A(t)\) is the account balance after some time \(t\), \(n\) is the number of compounds per year, and \(r\) is the APR. However, you must remember that interest is actually compounded every month at a rate of \(i=r/12\%\). Therefore, we are more accurately compounding at a rate of \(i=3.5/12\%\) per month. That's why financing options are provided in terms of months. You can use this same equation to determine the interest on your credit card(s), too. 

Now that we know how to calculate the interest on a principle amount, we need to determine our monthly note. And, we can derive our monthly note with \(Eq. 1\) in mind. Recall that we submit a monthly payment, which reduces the principle amount for a given month, and the interest is then calculated on the remaining account balance. Additionally, we assume that the monthly payments are equivalent. But, if we are paying down the principle amount every month and the interest is calculated based on the principle amount, how is it possible that our monthly payments are equivalent? 

Start with the most basic truth: the sum of all monthly payments must equate to the MSRP of the vehicle plus the total interest paid over the term,

\[ \Sigma_{j=1}^{n}xj = P + I = P + \Sigma_{j+1}^{n}I_j,  \:\:\:\:  (2) \]

where \(P\) is the principle amount of the vehicle (MSRP), \(I\) is the total interest paid, and \(n\) is the number of months financed. We also know that each monthly payment is equivalent, therefore we deduce \(x_j = x\) and

\[ \Sigma_{j=1}^{n}xj =nx. \:\:\:\: (3) \]

Now we assume that the monthly interest is calculated on the current principle and not the principle plus the previous month's interest. In other words, every payment reduces the remaining account balance and interest is calculated on that balance. This can be more easily understood by working out the first few months

\[ \begin{align} I_1 &= iP \\ I_2 &= i(P-P_1). \:\:\:\: (4) \\ I_3 &= i(P- P_2-P_1) \end{align} \]

We must substitute an expression for \(P_j\) if we are to eventually solve for \(x\), which we can do by recognizing that any given monthly payment is equivalent but the principle amount and interest paid for that month is unique:

\[ x_j = x = P_j+I_j. \:\:\:\: (5) \]

Now, rearrange \(Eq. 5\) and substitute \(Eq. 4\), to yield a recurrence relation of 

\[ P_j=(x-iP)(1+i)^{j-1}. \:\:\:\: (6) \]

Using \(Eq. 2-6\), we find a relation for \(x\) in terms of the principle value, the monthly interest rate, and the term length, 

\[ \begin{align} nx &= P +\Sigma_{j+1}^{n}I_j \\  &= P +\Sigma_{j+1}^{n}[x-(x-iP)(1+i)^{j-1}]\\  P &= (x-iP)\Sigma_{j=1}^n(1+j)^{j-1},\:\:\:\: (7) \end{align}\]

and we are finally ready to solve for the monthly payments. Use \(Eq. 7\) to solve for \(x\), but first allow \(j\rightarrow j+1 \):

\[ \begin{align} P &= (x-iP) \Sigma_{j=0}^{n-1}(1+j)^{j} \\ x-iP &= \frac{P}{\Sigma_{j=0}^{n-1}(1+j)^{j}} \\  x &= P \big(  \frac{1}{\Sigma_{j=0}^{n-1}(1+j)^{j}}+i \big) \\ x&= P \big( \frac{1-(1+i)}{1-(1+i)n}+i \big),\:\:\:\: (8)\end{align} \]

and we have found our monthly payments as a function of interest rate and term length. A better illustration is the 2-D contour plot shown below

Figure 1. 2-D contour plot of the monthly payments for an auto loan. The principle value is \(P=\$30,000.00\) and the variable interest rate and term length are provided on the vertical and horizontal axis, respectively. The bar legend provides the monthly payments given a rate and term

The plot above illustrates the monthly payments for a principle value of \( \$30,000.00\), which is approximately the MSRP of a 2018 Subaru Crosstreck, and a range of term lengths and interest rates. Therefore, without a down payment and an annual interest rate of \( 3.5\% \), the monthly payments would be \( \$546\) for \(60\) months and \( \$ 463\) for \(72\) months. The total amounts paid for these payments after the terms are \( \$32,745.10\) (60 months) and \( \$33,303.74 \) (72 months). So, you can "save" \($80\) per month if you choose to continue payments for an additional year, but this will cost you an additional \($600\) in the long run.

Now that we have an idea of our payments and a few options, we should ask if there is any incentive to including additional payments per month. This idea is predicated on having a less required per month and paying more as if it were required. For example, what if we paid \( \$546\) per month even if our monthly payments were \( \$ 463\)—is there an incentive? ... See part II for the analysis above.