Chapter 10
The Mathematics of Finance
Goldstein/Schnieder/Lay: Finite Math & Its Applications
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Outline
10.1 Interest
10.2 Annuities
10.3 Amortization of Loans
10.4 Personal Financial Decisions
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10.1 Interest
1.
2.
3.
4.
5.
6.
7.
Definitions for Savings Account
Common Compounding Periods
New from Previous Balance
Present and Future Value
Simple Interest
Effective Rate of Interest
Calculator Solutions
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Definitions for Savings Account
Interest is the fee a bank pays for the use of
money deposited into a savings account.
The amount deposited is called the principal.
The amount to which the principal grows (after
the addition of interest) is called the compound
amount or balance.
If interest is compounded m times per year and
the annual interest rate is r, then the interest rate
per period is i = r/m.
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Example Definitions for Savings Account
Date
Deposits
1/1/05
4/1/05
7/1/05
10/1/05
1/1/06
$100.00
Withdrawals
Interest
Balance
$1.00
1.01
1.02
1.03
$100.00
$101.00
$102.01
$103.03
$104.06
For the passbook above, determine the principal,
compound amount after 1 year, compound
interest rate and annual interest rate.
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Example Definitions - Savings Account (2)
The principal is $100.00.
The compound amount after 1 year is $104.06.
The compound interest rate is 1% since the
interest earned in the first period, $1.00, is 1% of
the principal.
Interest is compounded 4 times per year so the
annual interest rate is 4·1% = 4%.
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Common Compounding Periods
Number of
interest periods
per year
Length of each
interest period
Interest
compounded
1
2
4
12
52
365
1 year
6 months
3 months
1 month
1 week
1 day
Annually
Semiannually
Quarterly
Monthly
Weekly
Daily
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New from Previous Balance
For a savings account in which the interest rate
per period is i, the interest earned during a period
is i times the previous balance.
The new balance, Bnew is computed by adding the
interest earned during the period to the previous
balance, Bprevious.
Bnew = Bprevious + i · Bprevious
Bnew = (1 + i)Bprevious
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Example New from Previous Balance
Compute the interest and the balance for the first
two interest periods for a deposit of $1000 at 4%
compounded semiannually.
For semiannually m = 2 so i = (4/2)% = 2% = .02.
First period: interest = .02(1000) = $20
B1 = 1000 + 20 = $1020
Second period: interest = .02(1020) = $20.40
B2 = 1020 + 20.40 = $1040.40
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Present and Future Value
Let i be the interest per period, P the principal
and F the balance after n periods, then
F  1  i  P, and
n
P
F
1  i 
n
.
F is also referred to as the future value and P as
the present value.
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Example Present and Future Value
If an account pays 6% compound quarterly,
a) find the amount in the account after 5 years if
$1000 is initially deposited;
b) find the amount that must be initially
deposited if $3000 is needed in 5 years.
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Example Present and Future Value (2)
Period interest is i = .06/4 = .015.
Number of periods is n = 4*5 = 20.
a) P = $1000 so F = 1000(1 + .015)20 = $1346.86.
3000
 $2227.41.
b) F = $3000 so P 
20
(1  .015)
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Simple Interest
Simple interest is earned only on the principal
and is not compounded.
If r is the annual percentage rate and n is the
number of years, then
Interest = nrP, and
F = P + nrP = (1 + nr)P.
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Example Simple Interest
Calculate the amount after 4 years if $1000 is
invested at 5% simple interest.
F = (1 + 4(.05))1000 = $1200
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Effective Rate of Interest
The effective rate of interest is the simple interest
rate that yields the same amount after one year as
the annual rate of interest.
If r is the annual interest rate compounded m
times a year, then i = r/m and
reff = (1 + i)m – 1.
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Example Effective Rate of Interest
Calculate the effective rate of interest for a
savings account paying 3.65% compounded
quarterly.
reff = (1 + .0365/4)4 - 1 = .037
So the effective rate is 3.7%.
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Calculator Solutions
Use a calculator to determine when the balance
in a savings account in which $100 is deposited
at 4% compounded quarterly reaches $130.
For a TI-83 set
Y1 = (1 + .04/4)^X*100 and
Y2 = 130.
Find the intersection of the two graphs.
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Calculator Solutions (2)
The intersection is at X = 26.367391, so in 27
quarters the balance will exceed $130.
Graph of Y1 and Y2
with intersection
Goldstein/Schnieder/Lay: Finite Math & Its Applications
Table of Y1
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Summary Section 10.1 - Part 1
 Money deposited into a savings account earns
interest at regular time periods. Interest paid on
the initial deposit only is called simple interest.
Interest paid on the current balance (that is, on
the initial deposit and the accumulated interest)
is called compound interest.
 Successive balances of a savings account with
compound interest can be calculated with
Bnew = (1 + i)Bprevious.
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Summary Section 10.1 - Part 2
 P - principal, the initial amount of money
deposited into a savings account. P also
represents the present value of a sum of money
to be received in the future; that is, the amount of
money needed to generate the future money.
 r - annual rate of interest, interest rate stated
by the bank and used to calculate the interest rate
per period.
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Summary Section 10.1 - Part 3
 m - number of (compound) interest periods
per year, most commonly 1, 4, or 12.
 i - compound interest rate per period,
calculated as r/m.
 n - number of interest periods.
F - future value, compound amount, or
balance, value in a savings account.
F = (1 + i)nP with compound interest, and
F = (1 + nr)P with simple interest.
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Summary Section 10.1 - Part 4
reff - effective rate of interest, the simple
interest rate that yields the same amount after
one year as the annual rate of interest.
reff = (1 + i)m – 1
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10.2 Annuities
1.
2.
3.
4.
5.
Definitions of Annuity
Future Value
Rent for a Future Value
Present Value and Rent
Storing sn i and an i
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Definitions of Annuities
An annuity is a sequence of equal payments made at
regular intervals of time.
The payments are called rent.
The amount in an increasing annuity gets larger with
each payment and the final value is called the future
value of the annuity.
The amount in a decreasing annuity gets smaller with
each payment and the amount at the beginning is called
the present value of the annuity.
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Example Definitions of Annuities
Parents decide to deposit $100 at the end of each
month into a savings account for the college
education of their child. After 216 payments, the
account will contain $38,735.32.
You have just sold your house and deposit your
profit of $258,627.80 into an account so you can
withdraw $5000 at the end of each month for 5
years at which time the balance will be $0.
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Example Definitions of Annuities (2)
The first example is of an increasing annuity
with rent = $100 and future value = $38,735.32.
The second example is of a decreasing annuity
with rent = $5000 and present value =
$258,627.80.
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Future Value
Suppose that an increasing annuity consists of n
payments of $R each, deposited at the ends of
consecutive interest periods into an account with
interest compounded at a rate i per period. Then
the future value F of the annuity is
1 i

F
i
Goldstein/Schnieder/Lay: Finite Math & Its Applications
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1
R  sn i R.
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Example Future Value
Calculate the future value of an increasing
annuity of $100 per month for 2 years at 6%
interest compounded monthly.
R = 100, i = .06/12 = .005 and n = 2(12) = 24.
To calculate s24 .005 on a TI-83 calculator, key in
So s24 .005 = 25.43195524.
F = (25.43195524)(100) = $2,543.20.
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Rent for a Future Value
Suppose that an increasing annuity of n payments
has future value F and has interest compounded
at the rate i per period. Then the rent R is
F
R
.
sn i
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Example Rent for a Future Value
Ms. Adams would like to buy a $30,000 airplane
when she retires in 8 years. How much should
she deposit at the end of each half-year into an
account paying 4% interest compounded
semiannually so that she will have enough
money to purchase the airplane?
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Example Rent for a Future Value (2)
F = 30,000, i = .04/2 = .02 and n = 8(2) = 16.
F 30000
R

 (.05365013)(30000)
sn i
s16 .02
 $1609.50
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Present Value and Rent
The present value P and the rent R of a
decreasing annuity of n payments with interest
compounded at a rate i interest per period are
related by the formulas
1  i  1

P
P
R  an i R and R 
.
n
an i
i 1  i 
n
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Example Present Value and Rent
a) How much money must you deposit now at
6% interest compounded quarterly in order to be
able to withdraw $3000 at the end of each
quarter-year for 2 years?
b) How much money could you withdraw each
quarter-year for 2 years if you deposited $24,000
into the same account?
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Example Present Value and Rent
a) R = 3000, i = .06/4 = .015 and n = 4(2) = 8.
P  a8 .015 3000  (7.48592508)(3000)
 $22, 457.78
b) P = 24000, i = .06/4 = .015 and n = 4(2) = 8.
24000
R
  0.1335840245 (24000)
a8 .015
 $3, 206.02
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Storing sn i and an i
As a time-saving device, the formulas for
1
1
sn i ,
, an i , and
sn i
an i
can be assigned to the Y= editor on the TI-83
calculator.
1
Y4  s X I , Y5 
sX I
Y6  a X I , and Y7 
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1
aX I
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Example Calculating Number of Periods
Use a graphing calculator to determine when the
balance in an account in which $100 is deposited
monthly at 6% interest compounded monthly
will exceed $10,000.
Assuming Y4 contains the formula for s X I ,
store .005 into I on the home screen.
In the Y= menu, define Y1 = 100Y4.
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Example Calculating Number of Periods (2)
Scroll down the table for Y1 until Y1 exceeds
10000. This occurs when X = 82.
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Summary Section 10.2 - Part 1
 An increasing (decreasing) annuity is a
sequence of equal deposits (withdrawals) made
at the ends of regular time intervals.
 F - future value, compound amount, or
balance, value in an annuity at some point in the
future.
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Summary Section 10.2 - Part 2
 R - rent, periodic deposit into or withdrawal
from an annuity.
 sn i - s sub n angle i, future value of an
increasing annuity of n $1 payments at
compound interest rate i per period. For an
increasing annuity,
n
1  i  1

F
F  sn i R, R 
, and sn i 
.
sn i
i
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Summary Section 10.2 - Part 3
 an i - a sub n angle i, present value of a
decreasing annuity of n $1 payments at
compound interest rate i per period. For a
decreasing annuity,
n
1  i  1

P
P  an i R, R 
, and an i 
.
n
an i
i 1  i 
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Summary Section 10.2 - Part 4
 Successive balances of an increasing annuity
can be calculated with Bnew = (1 + i)Bprevious + R.
 Successive balances of a decreasing annuity
can be calculated with Bnew = (1 + i)Bprevious - R.
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10.3 Amortization of Loans
1.
2.
3.
4.
5.
6.
Amortization and Mortgage
Repayment Process
Unpaid Balance I
Unpaid Balance II
Balloon Payment
Calculator Application
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Amortization and Mortgage
Loans under consideration will be repaid in a
sequence of equal payments at regular time
intervals, with the payment intervals coinciding
with the interest periods. The process of paying
off such a loan is called amortization.
A mortgage is a long-term loan used to purchase
real estate. The real estate is used as collateral to
guarantee the loan.
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Repayment Process
1. Payments are made at the end of each interest
period.
2. The interest to be paid each interest period is
the period interest rate, i, times the unpaid
balance at the end of the previous interest period.
3. The unpaid balance at the end of the interest
period is the previous unpaid balance plus the
interest owed for the current interest period
minus the payment.
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Unpaid Balance I
For a loan amortized over n payments with
payments R, the unpaid balance at the end of the
kth payment is the present value of a decreasing
annuity with the same i and R but with n - k
payments.
 unpaid balance after k payments   ank i R
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Example Amortization
On Dec. 31, 1990, a house was purchased with
the buyer taking out a 30-year, $112,475
mortgage at 9% interest, compounded monthly.
The mortgage payments are made at the end of
each month.
a) Calculate the amount of the monthly payment.
b) Calculate the unpaid balance of the loan on
Dec. 31, 2016, just after the 312th payment.
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Example Amortization - continued
c) How much interest will be paid during the
month of January 2017?
d) How much of the principal will be paid off
during the year 2016?
e) How much interest will be paid during the
year 2016?
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Example Amortization (a)
A mortgage is a decreasing annuity. P = 112475,
i = .09/12 = .0075 and n = (30)(12) = 360.
R
1
a360 .0075
112475  .00804623112475
 $905.00
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Example Amortization (b)
There are 360 - 312 = 48 remaining payments.
Therefore,
 unpaid balance  a48 .0075  905.00 
  40.18478189  905.00 
 $36,367.23.
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Example Amortization (c)
The interest paid during January 2017 is i times
the unpaid balance at the end of December 2016
which was calculated in (b).
Interest = .0075(36367.23) = $272.75.
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Example Amortization (d)
The principal paid off during 2016 is equal to the
unpaid balance at the end of 2015 minus the
unpaid balance at the end of 2016.
The unpaid balance at the end of 2016 was
calculated in (b). At the end of 2015, there are
360 - 300 = 60 payments left.
 principal repaid   a60 .0075  905.00   36367.23
 43596.90  36367.23  $7229.67
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Example Amortization (e)
The interest paid during 2016 is the total amount
paid during 2016 minus that part of the payment
used to repay the principal during 2016 which
was calculated in (d).
interest during 2016  12 905.00   7229.67
 $3630.33
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Unpaid Balance II
If $P is borrowed at interest rate i per period and
$R is paid back at the end of each interest period,
then the formula
Bnew = (1 + i)Bprevious - R
can be used to calculate each new unpaid
balance, Bnew, from the previous unpaid balance
Bprevious.
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Example Unpaid Balance II
On Dec. 31, 1990, a house was purchased with
the buyer taking out a 30-year , $112,475
mortgage at 9% interest, compounded monthly.
The mortgage payments are made at the end of
each month.
Calculate the unpaid balance of the loan on Jan.
31, 2017, just after the 313th payment.
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Example Unpaid Balance II (2)
In the previous example, the unpaid balance at
the end of 2016 is $36,367.23 and the payment
was $905.
Bnew = (1 + .0075)(36367.23) - 905
= $35,734.98
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Balloon Payment
Sometimes amortized loans stipulate a balloon
payment at the end of the term.
The present value, P, of a loan with n payments
of R at an interest per period of i and a balloon
payment of D at the end of the loan is
P  an i R 
Goldstein/Schnieder/Lay: Finite Math & Its Applications
D
1  i 
n
.
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Example Balloon Payment
How much money can you borrow at 8% interest
compounded quarterly if you agree to pay $200
at the end of each quarter-year for 3 years and in
addition a balloon payment of $1000 at the end
of the third year?
R = 200, i = .08/4 = .02, n = 4(3) = 12 and D =
1000.
1000
P  a12 .02  200  
 $2903.56
12
1  .02 
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Example Calculator Application
Consider a loan of $112,475 at 9% interest
compounded monthly and repaid with 360
monthly payments of $905. When will the
debt-reduction portion of the payment surpass
the interest portion?
On a TI-83, store the unpaid balance in Y1, the
interest in Y2 and the debt-reduction portion of
the payment in Y3.
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Example Calculator Application (2)
Store .0075 into I.
In the Y= menu, set
Y1 = Y6(360-X)*905,
Y2 = I*Y1(X-1) and
Y3 = 905-Y2
where the formula for aX I was stored in Y6.
Graph Y2 and Y3 and find their intersection.
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Example Calculator Application (3)
The intersection is at X = 268.23423.
So the debt-reduction portion of all payments
starting with the 269th payment is greater than
the interest.
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Summary Section 10.3 - Part 1
 A mortgage is a type of loan that is paid off in
equal payments at the ends of regular time
periods.
 P - principal, the amount of money borrowed
in a loan.
 R - rent, the periodic payment on a loan.
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Summary Section 10.3 - Part 2
 At any time, the balance of a loan is the
amount of money needed to retire (that is, pay
off) the loan. It is calculated as the present value
of all future payments. A payment used to retire a
loan is called a balloon payment.
 Successive balances of a loan can be
calculated with Bnew = (1 + i)Bprevious - R.
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10.4 Personal Financial Decisions
1.
2.
3.
4.
5.
6.
7.
IRA
Equivalence of IRAs
Consumer Loans
Mortgages with Discount Points
Calculating APR
Calculating Effective Mortgage Rates
Significance of Discount Points
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IRA
An IRA is an individual retirement account
meant to shelter income from taxes.
Contributions to a traditional IRA are tax
deductible, but all withdrawals are taxed.
(Interest earned is not taxed until withdrawn.)
Contributions to a Roth IRA are not tax
deductible, but withdrawals are not taxed.
(Therefore, interest is never taxed.)
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Example IRA
In 2006 you deposit $4000 of earned income
into a traditional IRA on Jan. 1 which earns an
annual interest rate of 6% compounded annually
and suppose you are in the 30% marginal tax
bracket for the duration of the account.
a) How much income tax on earnings will you
save for the year 2006?
b) Making no more deposits, how much will you
have after 48 years and after taxes are paid on
the money?
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Example IRA (2)
a) [income tax saved] = [tax bracket] [amount]
= .30 (4000)
= $1200.
b) [balance after 48 years] = 4000(1 + .06)48
= $65,575.49.
[amount after taxes] = .70(65575.49)
= $45,902.84.
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Equivalence of IRAs
Equivalence of traditional and Roth IRAs
With the assumption that your tax bracket does
not change, the net earnings upon withdrawal
from contributing P dollars into a traditional IRA
account is the same amount as would result from
paying taxes on the P dollars but then
contributing the remaining money into a Roth
IRA.
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Example IRA
Earl and Larry each begin full-time jobs in Jan.
2006 and plan to retire in Jan. 2054 after
working for 48 years. Assume that any money
they deposit into IRAs earns 6% interest
compounded annually.
a) Earl opens a traditional IRA account
immediately and deposits $4000 into his account
at the end of the next twelve years. Then he
makes no more deposits. How much money will
Earl have in his account when he retires?
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Example IRA - continued
b) Larry waits 12 years before opening his IRA
and then deposits $4000 into the account at the
end of each year until he retires. How much
money will Larry have in his account when he
retires?
c) Who paid the most money into his IRA?
d) Who had the most money in his account upon
retirement?
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Example IRA (a)
a) At the end of 12 years, the account's balance
is
F  s12 .06  4000   $67, 479.76.
After the first 12 years, the money earns
compound interest for the final 36 years.
[amount in account]  1  .06 
36
 67479.76 
 $549, 774.61
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Example IRA (b, c, d)
b) Larry's account only earns interest for 36
years.
F  s36 .06  4000   $476, 483.47
c) Earl pays 12(4000) = $48,000.
Larry pays 36(4000) = $144,000.
d) Even though Larry paid more, he has less in
his account upon retirement than Earl.
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Consumer Loans
Consumer loans are paid off with a sequence of
monthly payments in much the same way as
mortgages. However, the most widely used
method for determining finance changes on
consumer loans is the add-on method. The
formulas for the payment R and interest rate r
are
P 1  rt 
12 Rt  P
R
and r 
.
12t
tP
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Example Consumer Loans
You take out a 2-year consumer loan of $1000 at
an annual interest rate of 6% using the add-on
method.
a) What is the monthly payment?
b) What would the monthly payment be if
computed as a mortgage?
c) What is the APR (annual percentage rate)?
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Example Consumer Loans (a and b)
a) P = 1000, r = .06 and t = 2.
R
1000 1  .06  2  
= $46.67
12  2 
b) P = 1000, i = .06/12 = .005 and n = 24.
1000
R
= $44.32
a24 .005
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Example Consumer Loans (c)
c) Using a TI-83, the period interest rate X is the
intersection of the lines Y1 = 1000 and
Y2 = ((1+X)^24-1)/(X(1+X)^24)*46.67.
The solution is i = X = .0092782 so
r = 12(.0092782) = .1113384 or 11.13%.
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Mortgages with Discount Points
Home mortgages have a stated interest rate,
called the constant rate. However, some loans
also carry points or discount points. Each point
requires that you pay up-front additional interest
equal to 1% of the stated loan amount. This has
the effect of reducing the actual loaned amount.
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Determining APR
Steps to calculate the APR for a mortgage loan having a
term of n months.
1. Calculate the monthly payment (call it R) on the
stated loan amount.
2. Subtract all up-front fees (including points) from the
stated loan amount. Denote the result by P.
3. Find the interest rate that produces the monthly
payment R for a loan of P dollars to be repaid in n
months.
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Example Determining APR
For a mortgage of $200,000 at 6% compounded
monthly that carries 1 point, find the APR.
Following the 3 steps:
1) P = 200000, i = .06/12 = .005 and
n = 12(30) = 360.
200000
R
 $1199.10
a360 .005
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Example Determining APR (2)
2) P = 200000 - .01(200000) = $198,000
3) Technology is the best way to calculate the
interest rate. Using the Excel RATE(n,-R,P,0)
function,
APR = 12*RATE(360,-1199.10,198000,0)
= .06094 = 6.094%.
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Effective Mortgage Rate
The APR is of limited use because it is relevant
only for loans that are kept for their full terms.
The average mortgage is refinanced or
terminated after around 5 years. The effective
mortgage rate takes into account the length of
time the loan will be held and the unpaid balance
at that time.
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Effective Mortgage Rate (2)
Steps to calculate the effective mortgage rate for
a mortgage loan expected to be held for m
months.
1. Calculate the monthly payment (call it R) on
the stated loan amount.
2. Subtract all up-front fees (including points)
from the stated loan amount. Denote the result
by P.
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Effective Mortgage Rate (3)
3. Determine the unpaid balance on the stated
loan amount after m months. Denote the result
by B.
4. Find the interest rate that causes a decreasing
annuity with beginning balance P dollars and
monthly payment R to decline to B dollars after
m months.
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Example Effective Mortgage Rate
For a mortgage of $200,000 at 6% compounded
monthly that carries 1 point, find the effective
mortgage rate assuming the mortgage will be
held for 5 years.
Use the 4-step process:
1. R = 1199.10 as calculated previously.
2. P = 198000 as calculated previously.
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Example Effective Mortgage Rate (2)
3. The mortgage has 360 - 60 = 300 months to
go. Therefore
B  a300 .005 1199.10   $186,108.55.
4. Using the Excel RATE(m, -R,P,-B) function,
the effective rate is
12*RATE(60,-1199.10,198000,-186108.55)
= .0624069 or 6.25%.
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Significance of Discount Points
Rule of thumb for the significance of discount points
Lifetime
Difference between effective mortgage rate
and stated interest rate, per discount point
1 year
1 percentage point
2 years
3 years
4 to 6 years
7 to 9 years
1/2 percentage point
1/3 percentage point
1/4 percentage point
1/6 percentage point
10 to 12 years
More than 12 years
1/7 percentage point
1/8 percentage point
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Summary Section 10.4 - Part 1
 An individual retirement account (IRA) is an
increasing annuity in which the annual interest
earned is either tax free (Roth IRA) or taxdeferred (traditional IRA). Contributions are tax
deductible only with a traditional IRA.
 When finance charges on a consumer loan are
calculated with the add-on method, the interest
paid each month is a fixed percentage of the
principal, rather than a fixed percentage of the
current balance.
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Summary Section 10.4 - Part 2
 For each discount point accompanying a
mortgage loan, you must pay additional interest
up-front equal to 1% of the amount borrowed.
 The APR and the effective mortgage rate,
which take up-front fees into account, are often
more useful than the contract interest rate in
appraising a loan. Each of them makes use of the
reduced loan amount obtained by subtracting the
up-front fees from the stated loan amount.
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Summary Section 10.4 - Part 3
 The APR is determined by calculating the
interest rate corresponding to the reduced loan
amount. The effective mortgage rate also takes
into account the expected lifetime of the loan and
is the interest rate corresponding to an annuity
that decreases the reduced loan amount to the
balance after the expected lifetime of the loan.
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