8.2. GEOMETRIC SERIES
465
(d) Based on these assumptions, how much stimulus will a 200 billion dollar tax
cut/rebate to consumers add to the economy, assuming consumer spending
remains consistent forever.
5. Like stimulus packages, home mortgages and foreclosures also impact the economy. A
problem for many borrowers is the adjustable rate mortgage, in which the interest rate
can change (and usually increases) over the duration of the loan, causing the monthly
payments to increase beyond the ability of the borrower to pay. Most financial analysts
recommend fixed rate loans, ones for which the monthly payments remain constant
throughout the term of the loan. In this exercise we will analyze fixed rate loans.
When most people buy a large ticket item like car or a house, they have to take out
a loan to make the purchase. The loan is paid back in monthly installments until
the entire amount of the loan, plus interest, is paid. With a loan, we borrow money,
say P dollars (called the principal), and pay off the loan at an interest rate of r%. To
pay back the loan we make regular monthly payments, some of which goes to pay off
the principal and some of which is charged as interest. In most cases, the interest is
computed based on the amount of principal that remains at the beginning of the month.
We assume a fixed rate loan, that is one in which we make a constant monthly payment
M on our loan, beginning in the original month of the loan.
Suppose you want to buy a house. You have a certain amount of money saved to make
a down payment, and you will borrow the rest to pay for the house. Of course, for the
privilege of loaning you the money, the bank will charge you interest on this loan, so
the amount you pay back to the bank is more than the amount you borrow. In fact, the
amount you ultimately pay depends on three things: the amount you borrow (called the
principal), the interest rate, and the length of time you have to pay off the loan plus
interest (called the duration of the loan). For this example, we assume that the interest
rate is fixed at r%.
To pay off the loan, each month you make a payment of the same amount (called
installments). Suppose we borrow P dollars (our principal) and pay off the loan at
an interest rate of r% with regular monthly installment payments of M dollars. So in
month 1 of the loan, before we make any payments, our principal is P dollars. Our
goal in this exercise is to find a formula that relates these three parameters to the time
duration of the loan.
We are charged interest every month at an annual rate of r%, so each month we pay
r
12 % interest on the principal that remains. Given that the original principal is P dollars,
we will pay
0.0r
12
P dollars in interest on our first payment. Since we paid M dollars
in total for our first payment, the remainder of the payment (M −
r
12
P) goes to pay
down the principal. So the principal remaining after the first payment (let’s call it P 1 ) is
the original principal minus what we paid on the principal, or
P 1 = P −
M −
r
12
P
=
1 +
r
12
P − M.
465
(d) Based on these assumptions, how much stimulus will a 200 billion dollar tax
cut/rebate to consumers add to the economy, assuming consumer spending
remains consistent forever.
5. Like stimulus packages, home mortgages and foreclosures also impact the economy. A
problem for many borrowers is the adjustable rate mortgage, in which the interest rate
can change (and usually increases) over the duration of the loan, causing the monthly
payments to increase beyond the ability of the borrower to pay. Most financial analysts
recommend fixed rate loans, ones for which the monthly payments remain constant
throughout the term of the loan. In this exercise we will analyze fixed rate loans.
When most people buy a large ticket item like car or a house, they have to take out
a loan to make the purchase. The loan is paid back in monthly installments until
the entire amount of the loan, plus interest, is paid. With a loan, we borrow money,
say P dollars (called the principal), and pay off the loan at an interest rate of r%. To
pay back the loan we make regular monthly payments, some of which goes to pay off
the principal and some of which is charged as interest. In most cases, the interest is
computed based on the amount of principal that remains at the beginning of the month.
We assume a fixed rate loan, that is one in which we make a constant monthly payment
M on our loan, beginning in the original month of the loan.
Suppose you want to buy a house. You have a certain amount of money saved to make
a down payment, and you will borrow the rest to pay for the house. Of course, for the
privilege of loaning you the money, the bank will charge you interest on this loan, so
the amount you pay back to the bank is more than the amount you borrow. In fact, the
amount you ultimately pay depends on three things: the amount you borrow (called the
principal), the interest rate, and the length of time you have to pay off the loan plus
interest (called the duration of the loan). For this example, we assume that the interest
rate is fixed at r%.
To pay off the loan, each month you make a payment of the same amount (called
installments). Suppose we borrow P dollars (our principal) and pay off the loan at
an interest rate of r% with regular monthly installment payments of M dollars. So in
month 1 of the loan, before we make any payments, our principal is P dollars. Our
goal in this exercise is to find a formula that relates these three parameters to the time
duration of the loan.
We are charged interest every month at an annual rate of r%, so each month we pay
r
12 % interest on the principal that remains. Given that the original principal is P dollars,
we will pay
0.0r
12
P dollars in interest on our first payment. Since we paid M dollars
in total for our first payment, the remainder of the payment (M −
r
12
P) goes to pay
down the principal. So the principal remaining after the first payment (let’s call it P 1 ) is
the original principal minus what we paid on the principal, or
P 1 = P −
M −
r
12
P
=
1 +
r
12
P − M.