# The Magic Number You Must Know to Find Your Perfect Mortgage Size

How much should you be willing to spend on your next home?

The good news is that there is a simple number that answers this question very easily. Over the rest of this article, I will explain how the number came to be, but before that for those who want the quick and dirty so they can move on with their days, I will give the number, how to use it, and a table of examples.

### Why did we calculate the “Magic number”

At Phoenix Homes, we help save people who have “gotten in over their heads”. When the mortgage is too much for them to handle, and they are risking the loss of their home to the bank, people call Phoenix Homes to bail them out. It is for this reason that we feel that it is essential to have an easy-to-use rule of thumb for estimating a “right-sized” mortgage. When you buy a home that is right for your budget, you can avoid having to use a service such as ours.

That being said, if you know anyone who has a home that is more expensive then their budget will allow, have them call us, please. We can help them get out of that situation and into an appropriately priced home for their income level so that they have years of happiness and positive growth ahead. A right-priced home is an investment, not a burden.

## The magic number and how to use it

### The Number is 4.63

Write it down: “4.63”.  You can write down the longer more accurate number, but the longer number makes it less easy… 3 digits is easy to remember and apply time and time again. The much longer and full value is “4.6331004”.

* Before we continue, I will note that this number is calculated on the assumptions of a mortgage at 30 years with 6% interest with a monthly payment equalling 1/3 of the household’s monthly gross income — 1/3 is the traditional advice for most financial advisors for the amount that should be spent on mortgage or rent, and 6% is a conservative estimate during this time (2018) when interest rates fluctuate between 4% and 5.5% — I always use conservative estimates and make my calculations based on higher interest rates, because reality is then so much nicer than the calculations.

### How to use it

Take your annual income and the income of your spouse and add it together. Because it is tax season, this number can easily be found on your W-2 (provided by your employer). The result of adding these two numbers together is your Gross Annual Income.  To calculate your ideal mortgage, multiply your Gross Annual Income by the magic number.

For example, if you and your spouse together earn \$100,000/year, then your ideal mortgage is \$463,000.

Add to this number any cash that you are willing to pay for the down payment to determine the price of the home you can shop for.  For example, if you have \$15,000 for the down payment that you could buy a home at \$478,000 or less, and your monthly payment would be appropriate for your income level.  If you have, \$25,000 available for the down payment, then you can easily shop for a home at \$488,000 or less and your monthly mortgage payment will be just 33% of your monthly income, which is a safe values according to all financial advisors.

Keep in mind that in addition to the mortgage payment that there will be taxes and interest, but this is a conversation for another day, because right now we are just looking at the mortgage principal and not the extra non-mortgage related payments that often get lumped together with the mortgage payment.

### Tables of examples

Below, I have demonstrated how this works by performing calculations on Annual Incomes between \$100k and \$600k. In these tables, I use the full number for more precise detail. The minimum downpayment calculation assumes an FHA loan (for “First-time Homebuyers”), which is 3% of the total property value, FHA will mortgage 97% of a property’s value.  This table also assumes 6% interest rate and a 30 year mortgage, which means that your payment will actually be LESS than what is displayed…. this is the benefit of using conservative values in these estimations.

 Annual Gross Income Target Mortgage Min Down Target Home Value Monthly Payment Percent of Monthly Income \$ 100,000.00 \$ 463,310.04 \$ 14,329.18 \$ 477,639.22 \$ 2,777.78 33% \$ 150,000.00 \$ 694,965.06 \$ 21,493.76 \$ 716,458.82 \$ 4,166.67 33% \$ 200,000.00 \$ 926,620.08 \$ 28,658.35 \$  955,278.43 \$  5,555.56 33% \$ 250,000.00 \$ 1,158,275.10 \$ 35,822.94 \$ 1,194,098.04 \$ 6,944.44 33% \$ 300,000.00 \$ 1,389,930.12 \$ 42,987.53 \$ 1,432,917.65 \$ 8,333.33 33% \$ 350,000.00 \$ 1,621,585.14 \$ 50,152.12 \$ 1,671,737.26 \$ 9,722.22 33% \$ 400,000.00 \$ 1,853,240.16 \$ 57,316.71 \$ 1,910,556.87 \$ 11,111.11 33% \$ 450,000.00 \$ 2,084,895.18 \$ 64,481.29 \$ 2,149,376.47 \$ 12,500.00 33% \$ 500,000.00 \$ 2,316,550.20 \$ 71,645.88 \$ 2,388,196.08 \$ 13,888.89 33% \$ 550,000.00 \$ 2,548,205.22 \$ 78,810.47 \$ 2,627,015.69 \$ 15,277.78 33% \$ 600,000.00 \$ 2,779,860.24 \$ 85,975.06 \$ 2,865,835.30 \$ 16,666.67 33%

What this table means is that if you and your spouse are earning a combined total of \$100,000, you can safely shop for a home at \$470,000, and your mortgage payment will be affordable.  Nothing prevents you from shopping for a \$280,000 home and leaving more room in your budget for other things, but we find that when people shop in a price range close to their right priced home that their neighbors tend to share similar values about what is the right amount of noise to make at various hours of the day, what is the appropriate amount of yard maintenance, and other values that just make living in your new home more enjoyable.

### Where to get the down payment money if you don’t have it in savings

We get this question all the time. Later we will make an article to address this concern. The question really should be, when does it make sense to borrow against the 401k or take from the investment account in order to fund an investment into your home. It does make sense in many situations, and if you want us to write that article sooner rather than later, email us at info@phoenixhomesltd.com, and we will be sure to send you a link as soon as that article is up and available. (NOTE: The more emails that we get, the sooner it will be that we write the article)

<LINK WILL GO HERE WHEN ARTICLE IS WRITTEN>

## The boring math stuff

If you want the details on how we calculate the “magic number”, this is the section with the details. We put it at the end, because most people see math and get scared off, but the savvy investor wants to know that the “magic number” is based on something solid and not something “magical”.  The table of values above provides a quick check to verify that the magic number works, and here we will go into the technical details.

To calculate the magic number, we need several equations:

#1) A = kx, where A is the original mortgage principal value, k is the magic number and x is the annual gross income.  That’s what this is all about, right! We want a quick way for you to figure out your mortgage size based on your annual income, this is the equation that started everything.

#2) The equation for calculating the amortized payment of a mortgage. This equation is reproduced from BrownMath, which provides easy to use reference formulas: https://brownmath.com/bsci/loan.htm

P = iA / ( 1 – ( 1 + i ) ^ -N ), where P is the period payment, i is the period rate, A is the original loan value, and N is the total number of periods in the loan’s life.  Period rate is calculated by determining the amount of interest that is accrued during the payment period. Our payment is monthly, and the rates are usually stated annually (12 months), so that leads us to equation #3.

#3) i = r / 12, where i is the period rate used in #1 and r as the annual percentage rate (APR) of the mortgage. Standard mortgage right now is around 5%. In the last 6 months, I have seen new mortgages issued between 4.5% and 5.5%, so we will assume 6% in our calculations, but it is important to note that the rate changes the ‘magic number’ which is why we list it in the assumptions and disclaimers.

#4) N = y * 12, where N is the total number of periods in the loans life, y is the number of years of the loan, and “12” is the number of months in a year. This means that in a 30 year loan that there will be 360 months, it is just the mathematical way to represent the same.

#5) P = Xs, where P is the period payment, X is the monthly gross income, and s is the standard rate recommended by financial advisors for how the rent or mortgage payment should relate to monthly income. The standard that we are familiar with and we hear most frequently is 1/3 of monthly income should be spent on housing.  This is the value we will use in calculating the “magic number”

#6) x = X * 12, where X is the monthly gross income, x is the annual gross income, and 12 is the number of months in a year.

With these equations, we are able to find the value of k.

First, we take equation #2 and relate it to values that are more frequently communicated when talking about mortgages. No one talks about a mortgage with the monthly interest rate, we use the annual rate!  So, we substitute equations #1, #3 and #4 into #2 to get equation #7:

#7) P = ( r / 12 ) * ( k * x ) / ( 1 – ( 1 + ( r / 12 ) ^ ( -1 * ( 12 * y ) ) )

Now, we have two equations to calculate P: #7 and #5.  We can make equation #8 by setting these equations equal to each other.

#8) X * s = P = ( r / 12 ) * ( k * x ) / ( 1 – ( 1 + ( r / 12 ) ^ ( -1 * ( 12 * y ) ) )

We have both X and x in this equation, we can fix this by using equation #6 to replace one of the two values. We are interested in keeping x, because more people are familiar with their annual income rather than their monthly income, so we will use equation #6 to substitute for X.  Equation #6 substituted into equation #8 makes equation #9:

#9) ( x / 12 ) * s = ( r / 12 ) * ( k * x ) / ( 1 – ( 1 + ( r / 12 ) ^ ( -1 * ( 12 * y ) ) )

We now simplify in a step by step process. For example, we can multiply both sides of equation #9 with ( 12 / x ) to make equation #10

#10)  s = r * k / ( 1 – ( 1 + (r / 12) ^ ( -1 * (12 * y ) ) )

Equation #10 has our magic number k in terms of s ( the standard recommended housing payment as a ratio of monthly income ), r ( the annual interest rate of the mortgage), and y (the number of years that the mortgage will take to be paid off). We must re-arrange Equation #10 so that k is the output in terms of the remaining variables that we will use to calculate k at any given time. To get from equation #10 to #11 there are many ways to do this, but here are two ways:

### Method 1

• Divide both sides by ‘s’
• Divide both sides by ‘k’
• Reciprocate

### Method 2

• Divide both sides by ‘r’
• Multiply both sides by ( 1 – (r / 12) ^ ( -1 * 12 * y ) )

#11) k = s * ( 1 – ( r / 12 ) ^ ( -1 * 12 * y ) ) / r

‘k’ is our magic number, and Equation #11 allows us to calculate the magic number any time we want using values that are appropriate for our environment. Right now, we choose to use the following conservative values:

• r = 6%= 0.06 (APR)
• y = 30 (Most common mortgage length)
• s = 1/3 (Most recommended relation between mortgage payment and gross monthly income)

Using these values, we produce k = 4.6331004, which simlifies to 4.63.  We will not worry about a discussion about “significant digits”, because this is a rule of thumb which is an estimate anyway and not an ‘exact’ value, so rounding to a reasonable value is assumed to be part of the process.

## Disclaimers

Phoenix Homes is a real estate investment company. We use math to inform our business decisions, but we are not certified financial advisors or certified public accountants. Before making decisions about your financial future, please consult appropriate professionals to verify the information provided above and find a mortgage that is right sized for you.