In this article we show how the changing interest rate impacts the housing market, as rising interest rate shift larger portion of the monthly payment towards interest and less toward principal, i.e. price of the house. And this applies to any large loan that has substantial duration, i.e. over a year, such as automobile loans.
There is two way to buy things, cash or installments. And if you have to "finance" (the installment option) a purchase then the creditor (the person who is going to give you the loan) is concerned with two things, return on the money (interest rate of the loan) and the return of the money, i.e. the actual principal he or she let you borrow.
These two risks are usually mitigated by two things, collateral and ability to pay back the loan, (credit worthiness). The former means usually some back up plan that allows the creditor to recover the money (say foreclosure the house). The latter means some kind of income or cash flow on your part.
Since most people do not have enough cash to buy their homes out right, the vast majority of them resort to borrowing. And how much money one can borrow is usually related to the ratio of the monthly payment they have to come with, and their income.
Let us consider the following example. Let us say you come to me and ask to borrow $1000, that is the principle, or so for a year. I ask you how you are going to pay me back, you say \$100 a month and that is the maximum you can pay each month. Let us say I want to charge you 10% on that loan and let us see how the math works.
\[ principle * (1 + interest\_rate) = $100 * 12 \]And if I want to get 10% return on my money, and you are capped at \$100/month, the maximum amount you can borrow is \$1090.90
\[ $1090.90 = \frac{\$1200}{1.1} \]However if for some reason I wanted to make 20% on my money, then that means the largest amount I can let you borrow (from me at least) is $1000 even. So you see how rising rates reduced the amount you can borrow.
\[ \$1000 = \frac{\$1200}{1.2} \]Now think about it from the buyer perspective. As interest rate rise, they can borrow less. And if they can borrow less, they can afford less. So, the same person who might normally afford to bid on a \$250k house based on his/her income, now can bid on \$230k for example. That depends on what the interest rate was and what it changed to.
I have previously shown that the monthly payment on a loan is calculated using the formula:
\[ V*q^n = P* \frac{1-q^{n-1}}{1-q} \]Where:
V: Is the value of the loan (total amount borrowed).
q = 1+x with x being the monthly interest rate, annual divided by 12.
P is the monthly payment.
n is the duration of the loan in months. For 30 years it is 360.
So what happens to the "loan amount" when the interest rate changes. i.e. how much money can one borrow for the same monthly payment
but for two different interest rates?
Assume we have interest rates of \(x_1\) and \(x_2\), remember these are monthly rates.
This leads to \(q_1\) and \(q_2\). So now we have:
Now if we consider the ratio of:
\[ R = \frac{V_2}{V_1} = \frac{q_1^n*(1-q_1)*(1-q_2^{n-1})}{q_2^n*(1-q_2)*(1-q_1^{n-1})} \]The following figures look at what happened to the "loan amount" if interest rate increased from a starting \(x_1\) to \(x_2\). Each curve represent starts at 1 on the y axis, that is a normalized value of a loan. The starting point on the x axis represents the initial rate, and the final point represents the final rate. The y axis represent the percentage of the initial "loan value" that can be borrowed based on the new rate.
Based on the figure above, if interest rate was at 4% and increase to 7% (that is the green, third, line), the amount of the loan is reduced by a bit under 30%. I.e. one can borrow only 70% of what he or she was able to borrow for the same monthly payment. Of course that number depends not only on interest rate but on duration of the loan as well. Thirty years was chosen because it is a typical period of house loans nowadays.
What happens to shorter duration loans, like a car loan for example. Let us look at the figure below where the loan duration was chosen to be 6 years, which is becoming more common as automobile prices keep on rising.
In this case if interest rate went from 4% to 9%, the amount of loan becomes about 84% of the original loan. I.e. customers can afford less and must buy less expensive cars.
The typical solution for the "vendors", if they want to sell the same expensive car or house to the same person, is to extend the life of the loan to reduce the monthly payments. That is why some car loans now run over 84 months. And some people are talking about 40 year home mortgage.