Activity Ten: Household Finance
Using the Reference Book
Mortgages
Assignment
Read pages 72-78 in The Only Math Book You'll Ever Need about monthly loan payments. If it is
confusing, switch from reading carefully to skimming.
Unfortunately, The Only Math Book You'll Ever Need really shows its age in its fourth chapter. There
are many tasks for which people now use computer software, and no longer care about the formulas
and graphs the reference book presents.
Because of this, that fourth chapter is more confusing to read than otherwise necessary. However, the
math it describes is not overly difficult, and even the obsolete applications are worth discussing.
(As an interesting historical note we could compare how loan payments are found using an
old Book's Amortization Tables, loan formulas, various computer software programs, and a
TI-83 Calculator. That calculator is owned my many LCC students and its menu path
APPS→Finance→TVM Solver does mortgages. But we won't do pursue this comparison.)
Commentary
It is recommended that you spend between 30% and 50% of your income on your mortgage.
If you are spending more than 50% of your income on your mortgage then you probably bought too
big a house. Your quality of living is being compromised by your high mortgage payments.
If you are spending less than 30% of your income on your mortgage then you are missing two
investment benefits. First, mortgage interest is tax deductible and it is prudent to take advantage of
this. Second, inflation "helps pay off" a mortgage: if each dollar is worth less, owing the same
number of them is less significant. So for most households the value of their home is a major tax-
friendly and inflation-friendly investment that compliments their savings in retirement accounts and
educational savings accounts (which are not tax-friendly or inflation-friendly).
How large a mortgage can you afford? To answer this question consult the Amortization Table.
This table tells us a loan's monthly payment. Look up the value corresponding to the annual interest
rate and years of the mortgage and then multiply this "result" by how many $1,000 are in the loan.
years 5% 6% 7% 8% 9% 10%
5 $18.90 $19.30 $19.80 $20.30 $20.80 $21.20
10 $10.60 $11.10 $11.60 $12.10 $12.70 $13.20
15 $7.91 $8.44 $8.99 $9.56 $10.10 $10.70
20 $6.60 $7.16 $7.75 $8.36 $9.00 $9.65
25 $5.85 $6.44 $7.07 $7.72 $8.39 $9.09
30 $5.36 $6.00 $6.65 $7.34 $8.05 $8.78
Activity Ten, Page 1
Where did this table come from? The answer is provided in our reading on pages 72-78 of the
reference book. The formula on the top of page 73 tells us that to find the monthly payment for a
mortgage we need to multiply the amount of the loan by a scale factor.
That scale factor is complicated but only depends upon two variables: the interest rate and the
number of payments to be made. The table in the middle of page 76 provides one way to skip the
formula: knowing the two variables just look up the scale factor in a table!
The Amortization Table is another table of these scale factors. It is adjusted so that its entries are
multiplied not by the entire loan amount but how many thousands of dollars are in the loan amount.
Example: Find the monthly payment for a 30 year loan at 7% annual interest for $150,000.
According to the Amortization Table, 7% and 30 years means $6.65 per thousand dollars.
The loan is for 150 thousand dollars.
$6.65 × 150 = $997.50
If you worked at a mortgage lending company, you would have computer software that used the
actual formula on page 73 to find the precise monthly payment for any loan. But you would also
have (produced by that computer software) a copy of the Amortization Table at your desk. Your
copy would be similar to this one, but with more appropriate interest rates. For example, if mortgage
interest rates were currently between 6% and 7% then you would print out a table with 6%, 6.1%,
6.2%, etc. all the way to 7% across the top. With it you could quickly estimate monthly payments
when talking with potential clients, without having to turn on your computer.
For people who are considering buying a home, the Amortization Table above is very useful to
answer the question "How big a mortgage can I afford?"
Example: Jordan wants to purchase a house, and after examining her budget decides she can
spend $630 per month on mortgage payments. How big a mortgage can she afford if current
mortgage interest rates are 6%?
(Step 1) Notice that the monthly payments are smallest with 30-year mortgages. So Jordan
will want a 30-year loan if she is trying to get the biggest mortgage for her money.
(Step 2) The monthly payment for a tiny $1,000 mortgage, at 6% for 30 years, is $6.00.
(Step 3) Jordan can pay that many times over. In fact, she can pay it
$630 ÷ $6 = 105 times over. So the biggest mortgage she can afford is $105,000.
A related question is how much interest you will pay during all the years of your mortgage.
Example: How much interest does Jordan pay during the 30 years of her mortgage?
If she pays $630 per month for 30 years (with 12 months in each year) the total amount paid
to the mortgage company will be $630 × 30 × 12 = $226,000.
Of this, $105,000 was repaying her loan.
Subtract to find the remaining amount, which is the total interest: $121,000.
Activity Ten, Page 2
Charge Options
Assignment
Unfortunately, The Only Math Book You'll Ever Need does not discuss charge options.
Commentary
For a large expense, there are often four different charge options.
The simplest is to pay cash up front. By saving up until you can pay cash you avoid paying any
interest. However, you have to wait to own the item you want to purchase.
The most common source of a loan is to use a credit card. However, credit cards tend to have very
high interest rates, because compared to other lenders they frequently have to deal with clients who
cannot pay back the loan. This is expensive for the credit card company, and the expense gets passed
along to the credit card user.
Many stores offer an installment plan. With an installment plan the store checks the customer's
credit rating, to eliminate much of the risk that makes credit cards expensive. Furthermore, with an
installment plan the seller retains legal ownership of the item until the entire loan has been paid off.
This reduces the risk to the store to a very little amount. Thus an installment plan usually has a much
lower interest rate than a credit card loan.
Some stores offer a special zero down plan. This is a special kind of installment plan in which the
buyer pays $0 per month for a certain number of months, and then begins making monthly payments.
Because the seller does not get money as soon it is slightly more expensive and riskier to the seller,
and so these plans tend to have higher interest rates than a normal installment plan. These rates are
usually still lower than a credit card if you have a good credit rating; beware of zero down plans with
very high interest rates designed for desperate people who have a low credit rating.
You have seen that credit card payments can be solved using a table with one row for each month.
The math for installment plans and zero down plans works very similarly. The interest rates will be
lower, but how to use them does not change. The zero down plan will have several months with no
payment (but interest builds up).
Activity Ten, Page 3
Activity Ten, Page 4
The Real-Life Tasks
10-1) Practice multiplying with the Amortization Table values to find monthly payments.
a) You borrow $60,000 for 15 years at 8% interest.
What is each payment amount?
What is the total amount you pay during all 15 years?
What is the total amount of interest you pay?
b) You borrow $60,000 for 30 years at 8% interest.
What is each payment amount?
What is the total amount you pay during all 30 years?
What is the total amount of interest you pay?
10-2) Compare the answers from the previous problem. When we lengthen the term (years) of a loan
payment each payment ___________ but the total amount paid ___________.
10-3) Two more examples, this time with a larger mortgage amount.
a) You borrow $120,000 for 15 years at 9% interest.
What is each payment amount?
What is the total amount you pay during all 15 years?
What is the total amount of interest you pay?
b) You borrow $120,000 for 30 years at 9% interest.
What is each payment amount?
What is the total amount you pay during all 30 years?
What is the total amount of interest you pay?
10-4) Now practice dividing with the Amortization Table values to find how large a mortgage fits a
budget.
a) Brenda can afford to spend $719.20 per month on mortgage payments. Currently mortgage
rates are 7% per year. How big a 15-year mortgage can she afford? Round the answer to the nearest
thousand dollars.
b) If the mortgage rate was instead up at 8%, then how big a 15-year mortgage could Brenda
afford? Round the answer to the nearest thousand dollars.
10-5) The actual situation is slightly more complicated.
a) Susan can afford to spend $900 per month on mortgage payments. Currently mortgage
rates are 7% per year. How big a 30-year mortgage can she afford? Round the answer to the nearest
thousand dollars.
b) Since the previous answer was rounded to the nearest thousand dollars, Susan's payment is
close to $900 but not exactly $900. What is the exact amount of her monthly payment?
c) What will be the total amount Susan pays during the 30 years?
d) What will be the total amount of interest Susan pays?
Activity Ten, Page 5
10-6) Consider the budget show as pie graph below.
What are the total monthly expenses for this household?
What percent of the total budget is each expense category? Round to the tenth of a percent.
Category Dollar Amount Percent
Housing $800
Food $200
Transportation $100
Luxuries $150
Other $60
What type of household might have a budget like this one?
Activity Ten, Page 6
10-7) Here is a second budget for a different household.
The total monthly expenses for this household are $2,250. What is the dollar amount for each
expense category? Round to the nearest cent.
Category Dollar Amount Percent
Housing 57%
Food 16%
Transportation 11%
Luxuries 9%
Other 7%
What type of household might have a budget like this one?
Activity Ten, Page 7
10-8) Fendrick wants to buy a new dining room set for $900. He is considering four methods of
payment. After looking at his budget as well as his actual expenses for the past few months, he
thinks he can save $80 per month towards this purchase.
He has four options for how to pay for the dining room set.
(i) He could pay cash up front. If he is willing to wait almost a year he can buy the dining room set
without any loan, and thus without paying any interest. But he would prefer not to wait that long.
(ii) He could use the furniture store's normal installment plan with a 15% annual interest rate. With
this plan he puts $100 down, and then pays $73 each month for a year (except for the last month in
which he will pay off the balance, which will be less than $73).
(iii) He could use the furniture store's special zero down plan with an 18% annual interest rate. With
this plan he pays $0 per month for six months, and then $180 each month for six months (except for
the last month in which he will pay off the balance, will which be less than $180). Note that the debt
earns interest during the first six months even though the buyer does not make payments.
(iv) He could use his credit card. For the sake of easy bookkeeping, he would open a new credit card,
buy the dining room set with it, and then not use it again. This credit card has a 22% annual rate.
With the credit card he is free to pay $80 per month for as long it necessary to pay off the debt.
a) How much interest would he pay using each of these four plans?
b) Do any of the four plans not work for him?
c) What would you do in Fendrick's situation?
Activity Ten, Page 8
10-9) Consider the Speck family budget below for the first trimester of the year. Most of their
expenses are the same from month to month. Garbage pickup is billed quarterly. Their home has
about $3,000 of property tax due each November, which they budget for by saving throughout the
year. In March they receive some anniversary money from family. In April they need to fly to a
wedding, and income tax is due.
EXPENSES January February March April
House Mortgage 1,000 1,000 1,000 1,000
Garbage pickup 53 - - 53
Electric and Water Bill 120 110 100 90
Phone Bill 40 40 40 40
Cable Modem Bill 55 55 55 55
House Insurance 40 40 40 40
Property Tax 250 250 250 250
Stuff for the House 50 50 50 50
Stuff for the Garden 30 30 30 30
Food Groceries, Drug Store 200 200 200 200
Farmers' Markets 0 0 0 50
Transportation Car/Bike Maintenance 20 20 20 20
Gasoline 40 40 40 40
Auto Insurance 80 80 80 80
Travel 0 0 0 600
Luxuries New Clothes 30 30 30 30
Eating Out 50 50 50 50
Recreation/Books 20 20 20 20
Postage 5 5 5 5
Gifts 10 10 10 50
Other Income Tax - - - 900
Haircuts 30 - 30 -
Medical/Glasses 10 10 10 10
Miscellaneous 40 40 40 40
Total Expenses = $2,173 $2,080 $2,100 $3,703
INCOME His Paycheck 1,300 1,300 1,300 1,300
Her Paycheck 1,300 1,300 1,300 1,300
Other Income - - 100 -
Total Income = $2,600 $2,600 $2,700 $2,600
Income - Expenses = $427 $520 $600 -$1,103
Standard financial advice is to save 10% if your income for retirement, and to spend between 30%
and 50% of your income on your mortgage.
a) Are the Specks saving 10% of their income?
b) Are the Specks spending between 30% and 50% of their income on their mortgage?
Activity Ten, Page 9
c) For each of their five expense categories find a monthly average for each of the five
expense categories, as a dollar amount and as the percentage of their total expenses.
Category 4-Month Total 4-Month Average Percent
Housing $6,386
Food $850
Transportation $1,160
Luxuries $500
Other $1,160
Activity Ten, Page 10
10-10) To reinforce our understanding of pie graphs, let's make one out of a bar graph.
Complete the table below of how many vowels are in the full (first and last) name of each person in
your group.
Person's Full Name Number of Vowels
1)
2)
3)
4)
Next make a bar graph below to represent the same information. For each vowel in that person's
name, color in one square. Use a different color for each person. Label each axis. Label each bar.
Your instructor will provide a second copy of the bar graph, so you can cut apart one of the copies
and still have an original.
Finally, make a pie chart by following five steps:
• cut out the bars in the copy of the bar graph
• tape the bars together into a long strip (like a "train" of bars)
• tape the two ends of the long strip together to make a circle
• center this circle on the empty pie chart below
• draw the "spokes" at the edges of the bars
• color and labeling the appropriate sections
Activity Ten, Page 11
Activity Ten, Page 12