Comparing two mortage rates.

Question

Say I buy a house and take a loan for amount, $a$. The monthly interest rate the bank charges me is $r$ (banks generally quote yearly rates; divide those by $12$ to get $r$), compounded monthly and there are $t$ months in the loan. Every month, I put in extra payments in the amount of $e$.

It is easy to show that at the end of month $\tau$, the total principal paid, $P_{\tau}$ will be given by equation (2) and the total interest paid, $Q_{\tau}$ will be given by equation (3) (I confirmed these on my mortgage). Also, the principal paid in month $\tau$ will be given by equation (1) and so on.

Now, let's say I'm comparing two loans with everything being the same, but the interest rates. The rate of the first loan is $r_1$ and that of the second loan is $r_2 (<r_1)$. I want to know how much money I'll save at the end of period $\tau$ by switching from the first loan to the second. This can be estimated one of two ways.

1. Set the extra payments, $e_1$ of the first loan to $0$ and those of the second loan, $e_2$ such that the total monthly payments for the two become the same. Now, use equation (2) to get the cumulative principal for the two loans and take the difference. Since the monthly payments are the same, the only thing that matters is how much principal we've paid off by the end of $\tau$.
2. Set $e_1=e_2=0$ and take the difference of the cumulative interests (each given by equation (3)) and the difference of the cumulative principals (each given by equation (2)). The difference in total interest paid up to that point and total principals should be the money saved by switching to loan-2.

Of these, I trust the first approach more. The second one doesn't give the same result as the first. What am I missing?

And even in the first approach, the cumulative interest rates are different at the end of $\tau$. What is the significance of this difference?

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$$p_{\tau} = \left(\frac{ar}{(1+r)^{t}-1} + e\right)(1+r)^\tau \tag{1}$$

$$P_{\tau} = \left(\frac{ar}{(1+r)^{t}-1} + e\right)\left(\frac{(1+r)^\tau-1}{r}\right) \tag{2}$$

$$Q_{\tau} = \tau\left(\frac{ar}{1-(1+r)^{-t}}+e\right)-P_{\tau}\tag{3}$$

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EDIT: Note that if you want to compare two loans after they've both ended, it's obvious that you simply subtract the total interest paid for the two of them. My question involves ending them prematurely at a time $\tau$ when neither of them has concluded (or one of them has and the other hasn't).

Further, we can ignore the time value of money.

Finally, to make things simple, let's consider only scenario-1. To compare the money saved, one argument would be to take the difference in total principals paid and also the total interests paid by time $\tau$. Another argument would be to just take the difference in cumulative principals since that's the money that actually changes hands between you and the bank and everything else is the same by design. Which of the arguments is correct? If its the second one, what is the significance of the delta in cumulative interests paid?

Answer 1

If you want to figure out how to save money when repaying a loan just apply the following simple principles

• The borrowed amount has to be repaid in full. There is no way to save money on that.

• Money can be saved only on the total interest paid by bringing down the outstanding pricipal faster, either by making an extra payment and/or by switching into a loan with less interest and not lowering the monthly fixed payment too much.

You don't even need those complicated formulas to do any calculations. A simple loop in a python program is enough:

T = 10 # number of periods
A = 100 # outstanding principal
F = 10 # fixed payment
E = 0 # extra payment
r = 0.01 # interest
Rtot = 0 # total interest paid
for t in range(T):
    R = A*r # interest in period t
    P = F+E-R # repayment on principal
    A -= min(P,A) # outstanding principal
    Rtot += R
    if A < 1e-5:
       break
print( Rtot )