Chapter 22: Borrowings Models

Chapter 22: Borrowings Models

October 21, 2013

Last Time

The Consumer Price Index

Real Growth

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).

CPI for other year

100 = cost of market basket in other year

cost of market basket in base period

The base period used to calculated the CPI-U is 1982-1984

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

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

1.02 = − .1

1.02 = −0.0980392

A = 1000(1 − 0.0980392)

= 970.876

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

1.02 = − .1

1.02 = −0.0980392

A = 1000(1 − 0.0980392)

= 970.876

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:

1100 = 1000



1 + r − .02 1.02



r = 0.0529257

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:

1100 = 1000



1 + r − .02 1.02



r = 0.0529257

This Time

Simple Interest

Compound Interest

Conventional Loans

Annuities

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

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

A = P

 1 + r

m



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

12 )

= $19154.8

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

Question

Suppose a credit card has a APR of 18 % that compoundes monthly. What is the effective annual rate?

Answer:



1 + 0.18 12 )



− 1 = 0.19562

So if you borrow $1000 with that credit card, will owe back $

1195.62 at the end of the year.

Question

Suppose a credit card has a APR of 18 % that compoundes monthly. What is the effective annual rate? Answer:



1 + 0.18 12 )



− 1 = 0.19562

So if you borrow $1000 with that credit card, will owe back $

1195.62 at the end of the year.

Motivating Question

How will I have to pay off my mortgage?

Amortize a loan

A = d  (1 + i )

− 1 i



= d  (1 +

)

− 1

r m



where

A = amount accumulated

d = regular deposit of payment at the end of each period n = mt number of periods

r= nominal annual interest rate

m = number of compounding periods per year t= number of years

i= r/m periodic rate, the interest rate per compounding period

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?

100000(1 + .06

12 )

= 602, 257.52

Now to get that accumulated amount we set up the equation

602, 257.52 = d

"

(1 +

)

− 1

.06 12

#

d = 599.55

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

d = P

 r /m

1 − (1 + r /m)



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.

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

(1 − (1 + .08/12)

= 733.77 They would still owe 10 years of payment.

733.77( (1 − (1 + .08/12)

.08/12 ) = 60478.4

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

1 − (1 + .06/12)

) = 843.86 843.86 ∗ 12 ∗ 15 = 151, 894.23

So you would end paying over $50,000 dollars less.

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

1 − (1 + .06/12)

) = 843.86

843.86 ∗ 12 ∗ 15 = 151, 894.23

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?

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