Download Chapter 22: Borrowings Models

Chapter 22: Borrowings Models
October 21, 2013
Chapter 22: Borrowings Models
Last Time
The Consumer Price Index
Real Growth
Chapter 22: Borrowings Models
The Consumer Price index
The official measure of inflation is the Consumer Price Index (CPI)
which is the determined by the Bureau of Labor Statistics (BLS).
cost of market basket in other year
CPI for other year
=
100
cost of market basket in base period
The base period used to calculated the CPI-U is 1982-1984
Chapter 22: Borrowings Models
Real Growth Under Inflation
Real rate of Growth
The real annual rate of growth of an investment at annual interest
rate r with annual inflation rate a is
g=
r −a
1+a
Chapter 22: Borrowings Models
Question
Question: In mid 2013 you put a $1000 into a savings account
with APY 1 %. Assuming there is a constant inflation rate of 2 %
for the next 3 years, how much money will you have in the account
in mid 2016 in constant mid-2013 dollars?
Chapter 22: Borrowings Models
Question
Question: In mid 2013 you put a $1000 into a savings account
with APY 1 %. Assuming there is a constant inflation rate of 2 %
for the next 3 years, how much money will you have in the account
in mid 2016 in constant mid-2013 dollars?
Answer:
g=
.01 − .02
.1
=−
= −0.0980392
1.02
1.02
A = 1000(1 − 0.0980392)3 = 970.876
Chapter 22: Borrowings Models
Question
Question: In mid 2013 you put a $1000 into a savings account.
Assuming there is a constant inflation rate of 2 % for the next 3
years, what would the APY of the savings account have to be in
order to have $1100 dollars in constant mid-2012 dollars in 3 years?
Chapter 22: Borrowings Models
Question
Question: In mid 2013 you put a $1000 into a savings account.
Assuming there is a constant inflation rate of 2 % for the next 3
years, what would the APY of the savings account have to be in
order to have $1100 dollars in constant mid-2012 dollars in 3 years?
Answer:
r − .02
1100 = 1000 1 +
1.02
r = 0.0529257
3
Chapter 22: Borrowings Models
This Time
Simple Interest
Compound Interest
Conventional Loans
Annuities
Chapter 22: Borrowings Models
Simple Interest
If a friends loans you $100 at a rate of 5 %, with no compounding.
How much money do you owe him after 2 years?
Simple Interest
For a principal P and an annual rate of interest r, the interest owed
in t years is
I = Prt
and the total amount A accumulated in the account is
A = P(1 + rt)
Answer:
A = 100(1 + .05(2)) = $110
Chapter 22: Borrowings Models
Compound Interest
Compound Interest Formula
For a principal P loaned
at a nominal annual rate of interest rate r
with m compounding periods per year (so the interest rate
i = r /m per compounding period), the amount owed
after t years with no payments of interest or principal is
r mt
A=P 1+
m
Chapter 22: Borrowings Models
Question
If you borrowed $15,000 to buy a new car a 4.9 % interest per
year, compounded monthly, and paid back all the principal and
interest at the end of 5 years, how much would you pay back?
Answer:
A = 15000(1 +
0.049 (5)12
)
= $19154.8
12
Chapter 22: Borrowings Models
Annual Percentage Rate (APR)
Annual Percentage Rate (APR)
The annual percentage rate (APR) is the number of
compounding periods per year times the rate of interest per
compounding period:
APR = m × i
Chapter 22: Borrowings Models
Question
Suppose a credit card has a APR of 18 % that compoundes
monthly. What is the effective annual rate?
Chapter 22: Borrowings Models
Question
Suppose a credit card has a APR of 18 % that compoundes
monthly. What is the effective annual rate? Answer:
0.18 12
1+
)
− 1 = 0.19562
12
So if you borrow $1000 with that credit card, will owe back $
1195.62 at the end of the year.
Chapter 22: Borrowings Models
Motivating Question
How will I have to pay off my mortgage?
Chapter 22: Borrowings Models
Amortize a loan
(1 +
(1 + i)n − 1
A=d
=d
i
where
A=
d=
n = mt
r=
m=
t=
i= r/m
r mt
m)
r
m
−1
amount accumulated
regular deposit of payment at the end of each period
number of periods
nominal annual interest rate
number of compounding periods per year
number of years
periodic rate, the interest rate per compounding period
Chapter 22: Borrowings Models
Question
Suppose that you buy a house with a $ 100,000 loan to be paid off
over 30 years in equal monthly installments. Suppose that the
interest rate for the loan is 6.00 %. How much is your monthly
payment? Answer:
How much money will you owe the bank if you wanted 30 years
and paid them all at once?
.06 12(30)
)
= 602, 257.52
12
Now to get that accumulated amount we set up the equation
"
#
12(30) − 1
)
(1 + 0.06
12
602, 257.52 = d
.06
100000(1 +
12
d = 599.55
So you have to pay $599.95 a month
Chapter 22: Borrowings Models
Amortization Payment Formula
Amortization Payment Formula
A conventional loan amount P
at a nominal annual rate of interest rate r
with m compounding periods per year (so interest rate
i = r /m per compounding period)
for t years can be paid off by uniform payments at the end of
each compounding period in the amount
r /m
d =P
1 − (1 + r /m)−mt
Chapter 22: Borrowings Models
Definitions
Equity
Equity is the amount of principal of a loan that has been repaid.
Annuity
An annuity is a specified number of equal periodic payments.
Chapter 22: Borrowings Models
Question
If you brought a house with a 30 year mortgage for $100,000 at an
8 % interest rate. After 20 years, how much equity would you have
in the house? How much of the principal had been repaid?
Answer:
So the monthly payment would be
d = 100000(
.08/12
= 733.77
(1 − (1 + .08/12)12(30)
They would still owe 10 years of payment.
733.77(
(1 − (1 + .08/12)12(10)
) = 60478.4
.08/12
So you would have 100, 000 − 60478.4 = 39, 521.60 in equity.
Chapter 22: Borrowings Models
Questions
Suppose that you buy a house with a $ 100,000 loan to be paid off
over 30 years in equal monthly installments. Suppose that the
interest rate for the loan is 6.00 %. How much money would pay?
If it were a 15 year mortgage?
Chapter 22: Borrowings Models
Questions
Suppose that you buy a house with a $ 100,000 loan to be paid off
over 30 years in equal monthly installments. Suppose that the
interest rate for the loan is 6.00 %. How much money would pay?
If it were a 15 year mortgage?
Answer:
599.95 ∗ 12 ∗ 30 = 215982
For a 15 year mortgage?
d = 100000(
.06/12
) = 843.86
1 − (1 + .06/12)−12(15)
843.86 ∗ 12 ∗ 15 = 151, 894.23
So you would end paying over $50,000 dollars less.
Chapter 22: Borrowings Models
Question
Suppose that you want to retire at 65 with an annuity that pays
$1000 per month for 25 years and the interest rate is 4 % per
compounded monthly. What amount should you have save up to
pay for this annuity?
Chapter 22: Borrowings Models
Next time
Review
Chapter 22: Borrowings Models