Lecture 05: Effective Interest Rates

1
LEGOs Theory Continued

• In learning about compound cash flows, we found
  ways to decompose and convert the cash flow
  pattern building blocks from one type to another,
  and how to relocate the block patterns to
  different points in time.
• The individual cash flow arrow periods are like
  the bumps on a LEGO brick.
• The effective interest rate per period is like
  the holes of the block the bumps and holes must
  align for the LEGO blocks to fit together.
• There are three major cases of effective interest
  rate to consider and four formulas to know.
  But first, there are some terms to learn

2
Interest Rate Terms

• Compounding Period (cp) the time between points
  when interest is computed and added to the
  initial amount.
• Payment Period (pp) the shortest time between
  payments. Interest is earned on payment money
  once per period (cost of money)
• Nominal Rate ( r ) is a simplified expression
  of the annual cost of money. It means nothing,
  unless the compounding period is stated along
  with it.
• Annual Percentage Rate (APR) is the nominal
  interest rate on a yearly basis (credit cards,
  bank loans, ). It, too, should have a
  compounding period stated.
• Effective Rate ( i ) is the rate that is used
  with the table factors or the closed form
  equations, and it converts the nominal rate
  taking into account both the compounding period
  and the payment period so that the blocks match.

3
Compounding Period is Equal to the Payment Period
r nominal annual interest rate for payments
that match the compounding period (cp lt year
and pp cp) Examples 12 per year compounded
monthly (1) 10 APR, compounded
quarterly (2) i interest rate per compounding
period r m nominal interest rate
( of compounding periods per year) Examples
12 / 12 months 1 compounded monthly
(1) 10 / 4 quarters 2.5 compounded quarterly
(2) Which would you rather have 12
compounded annually or 12 compounded monthly?
4
Compounding Period is More Frequent than the
Annual Payment Period
EFFECTIVE INTEREST RATE ia effective interest
rate per year compounded annually ( 1
interest rate per cp)( of cp per year) 1
1 r m 1 m Example r 12 per
year compounded monthly imonth 12 yearly
1 compounded monthly 12
months ia (1 .01)12 1 12.68
compounded annually
5
Another example
r 12 per year compounded semi-annually isemi-a
nnual 12 annually 2 times per
year 6 per 6 months ia (1 .06)2 1
.1236 12.36 per year compounded annually As
the compounding period gets smaller, does the
effective interest rate increase or decrease?
6
Lets Illustrate the Answer
r 12 per year compounded daily idaily
12 365 .000329 ia (1 .000329)365
1 .12747 12.747 per year compounded
annually What happens if we let the compounding
period get infinitely small?
7
Continuous Compounding
i e( r )( of years) 1 Examples r 12 per
year compounded continuously ia e( .12 )(1) 1
12.75 What would be an effective six month
interest rate for r 12 per year compounded
continuously? i6 month e( .12 )(.5) 1 6.184
8
Compounding Period is More Frequent than the
Payment Period
EFFECTIVE INTEREST RATE ie effective interest
rate per payment period ( 1 interest
rate per cp)( of cp per pay period) 1 1
r me 1 m Example r 12 APR,
compounded monthly, payments quarterly imonth
12 yearly 1 compounded monthly
12 months ie (1 .01)3 1
.0303 or 3.03 per payment
9
Summary of Effective Rates
An APR or per year statement is a Nominal
interest rate denoted r unless there is no
compounding period stated The Effective Interest
rate per period is used with tables
formulas Formulas for Effective Interest
Rate If continuous compounding, use y is
length of pp, expressed in decimal years If cp
lt year, and pp 1 year, use m is compounding
periods per year If cp lt year, and pp cp,
use m is compounding periods per year If cp lt
year, and pp gt cp, use me is cp per payment
period
10
CRITICAL POINT
When using the factors, n and i must always
match!
Use the effective interest rate formulas to make
sure that i matches the period of interest
(sum any payments in-between compounding periods
so that n matches i before using formulas or
tables)
11
Note
Interest doesnt start accumulating until the
money has been invested for the full period!
2 periods
1
0
12
Problem 1
The local bank branch pays interest on savings
accounts at the rate of 6 per year, compounded
monthly. What is the effective annual rate of
interest paid on accounts?
GIVEN r 6/yr m 12mo/yr FIND ia
13
Problem 2
What amount must be deposited today in an account
paying 6 per year, compounded monthly in order
to have 2,000 in the account at the end of 5
years?
GIVEN F5 2 000 r 6/yr m 12
mo/yr FIND P
14
Problem 2 Alternate Soln
What amount must be deposited today in an account
paying 6 per year, compounded monthly in order
to have 2,000 in the account at the end of 5
years?
GIVEN F5 2 000 r 6/yr m 12
mo/yr FIND P
15
Problem 3
A loan of 5,000 is to be repaid in equal monthly
payments over the next 2 years. The first
payment is to be made 1 month from now.
Determine the payment amount if interest is
charged at a nominal interest rate of 12 per
year, compounded monthly.
GIVEN P 5 000 r 12/yr m 12
mo/yr FIND A
16
Problem 4
You have decided to begin a savings plan in order
to make a down payment on a new house. You will
deposit 1000 every 3 months for 4 years into an
account that pays interest at the rate of 8 per
year, compounded monthly. The first deposit will
be made in 3 months. How much will be in the
account in 4 years?
17
Problem 5
Determine the total amount accumulated in an
account paying interest at the rate of 10 per
year, compounded continuously if deposits of
1,000 are made at the end of each of the next 5
years.
18
Problem 6
A firm pays back a 10 000 loan with quarterly
payments over the next 5 years. The 10 000
returns 4 APR compounded monthly. What is the
quarterly payment amount?
DIAGRAM
10 000
5 yrs 20 qtrs
0
A
19
Problem 7
Anita Plass-Tuwurk, who owns an engineering
consulting firm, bought an old house to use as
her business office. She found that the ceiling
was poorly insulated and that the heat loss could
be cut significantly if 6 inches of foam
insulation were installed. She estimated that
with the insulation she could cut the heating
bill by 40 per month and the air conditioning
cost by 25 per month. Assuming that the
summer season is 3 months (June, July, August) of
the year and the winter season is another 3
months (December, January, and February) of the
year, how much can she spend on insulation if she
expects to keep the property for 5 years?
Assume that neither heating nor air
conditioning would be required during the fall
and spring seasons. She is making this
decision in April, about whether to install the
insulation in May. If the insulation is
installed, it will be paid for at the end of May.
Anitas interest rate is 9, compounded monthly.
20
Problem 7
GIVEN SAVINGS 40/MO (DEC,JAN, FEB)
25/MO (JUN, JUL, AUG) r 9/YR,
CPD MONTHLY FIND P(SAVINGS OVER 5 YEARS)
PA Pa Pß(PPß) Aa(PA,i,na)
Aß(PA,i,nß)(PF,i,6) 25(PA,0.75,3)
40(PA,0.75,3)(PF,0.75,6) 25(2.9556)
40(2.9556)(0.9562) 186.94 at the start of
each year
21
Problem 7 cont.
GIVEN SAVINGS 40/MO (DEC,JAN, FEB)
25/MO (JUN, JUL, AUG) r 9/YR,
CPD MONTHLY FIND P(SAVINGS OVER 5 YEARS)