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LEGOs Theory Continued… ● In learn ing about compound cash f lows , we found
ways to decompose and convert the cash f low pattern bu i ld ing b locks from one type to another, and
how to relocate the block patterns to dif ferent points
in t ime.
●The ind ividual cash f low arrow per iods are l ike the bumps on a LEGO br ick.
● The effectiv e interest rate per per iod is l ike the holes
of the block – the bumps and holes mus t al ign for the
LEGO block s to f i t to gether.
● There are three major cases of effectiv e interest rate
to consider – and four formulas to know. But f i rs t ,
there are some terms to learn…
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Interest Rate Terms… ● Compounding Per iod (cp) – the t ime between p oints
wh en interest is computed and added to th e ini t ia l amoun t.
● Payment Per iod (pp) – the sho rtest t ime between
payments. Interest is earned on payment money once per
per iod (cost of money)
● Nom inal Rate ( r ) – is a simpl i f ied expression o f the
annual cost of money. It means no thing, unless the compounding period is stated along w i th i t .
● Annual Percentage Rate (APR) – is the nom inal interest
rate on a yearly basis (credit cards, bank loans, …). It, too,
shou ld have a compounding per iod stated.
● Effective Rate ( i ) – is the rate that is used w ith the table
factors or the closed form equations, and i t con verts the
nom inal rate tak ing in to account bo th the compound ing
per iod and the payment per iod so that the blocks m atch.
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Compounding Period is Equal to thePayment Period
r = nom inal annu al interest rate for paym ents that match th e
compound ing per iod : (cp < year and pp = cp )
Examp les: 12% per year compounded month ly (1)
10% APR, compou nded quarter ly (2)
i = interest rate per compou ndin g per iod = r = m
nom inal interest rate
( # of com poun ding periods per year)
Examples: 12% / 12 month s = 1% com pou nded mon thly (1)
10% / 4 quarters = 2.5% compounded quarter ly (2)
Which w ould you rather have: 12% com pounded annually
or 12% com pounded month ly?
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EFFECTIVE INTEREST RATE
i a = effect ive interest rate per year com pou nded annual ly
= ( 1 + interest rate per cp ) (# of cp per year) – 1
= 1 + r m – 1 m
Example:
r = 12% per year com poun ded month ly
i month = 12% year ly = 1 % compounded month ly
12 months
i a = (1 + .01) 12 – 1 = 12.68% compounded annually
Compounding Period is More Frequent thanthe Annual Payment Period
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Another example…
r = 12% per year compounded semi-annual ly
i semi-annual = 12% annually
2 times per year
= 6% per 6 months
i a = (1 + .06) 2 – 1 = .1236
= 12.36% per year compounded annual ly
As th e compound ing per iod gets smal ler, does the effect ive interest rate inc rease or decrease?
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Let’s Illustrate the Answer…
r = 12% per year compounded daily
i daily = 12%
365
= .000329
i a = (1 + .000329) 365 – 1 = .12747
= 12.747% per year compounded annual ly
What happens i f we let the com pound ing per iod
get inf ini tely small?
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Continuous Compounding
i = e ( r )(# of years ) – 1
Examples:
r = 12% per year compounded cont inuous ly i a = e ( .12 )(1) – 1 = 12.75%
What wou ld be an effect ive six m on th interest rate for r = 12% per year com pounded cont inuous ly?
i 6 month = e ( .12 )(.5) – 1 = 6.184%
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EFFECTIVE INTEREST RATE
i e = effective interest rate per payment period
= ( 1 + interest rate per cp ) (# of cp per pay per iod) – 1
= 1 + r m e – 1 m
Example:
r = 12% APR, com pound ed monthly, payments quarter ly
i month = 12% year ly = 1 % compounded month ly
12 months
i e = (1 + .01) 3 – 1 = .0303 – or – 3.03% per payment
Compounding Period is More Frequent thanthe Payment Period
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Summary of Effective Rates An “APR” or “% per year” statement is a Nominal interest rate
– denoted r – unless there is no compounding per iod stated
The Effective Interest rate per per iod is u sed with tables & formulas
Formulas fo r Effect ive Interest Rate:
If cont inuous com pounding, use
y is length o f pp, expressed in decimal years
If cp < year, and pp = 1 year, use m is # compound ing per iods per year
If cp < year, and pp = cp , us e m is # compounding p eriods per year
If cp < year, and pp > cp , us e
m e is # cp per payment per iod
11
m
am
r i
m
r
i
1ei )y ( r
11
em
em
r i
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CRITICAL POINTWhen us ing the factors ,
n and i mus t always match! Use the effect ive in terest rate
formulas to make sure that i
matches the per iod o f interest (sum any payments in-between com pounding per iods so
that n matches i before usin g formulas or tables)
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Note:
Interest doesn’t start accumulating
un t i l the money has been
invested fo r the ful l per iod !
0 1 2 periods
Shows up hereon CFD…
(End of Period
Convention)
X
Depositmadehere …
i
Returnsinteresthere!
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Problem 1The local bank branch pays interest on savings accounts at
the rate of 6% per year, com pou nded mon thly. What is the effect ive annu al rate of interest paid on accounts?
GIVEN:r = 6%/yr
m = 12mo/yr FIND ia:
DIAGRAM:
NONE NEEDED!
%17.6112
06.01
11
12
m
am
r i
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Problem 2What amount m ust be deposi ted today in an account paying
6% per year, compounded month ly in order to have $2,000 in the account at the end of 5 years?
GIVEN:F5 = $2 000r = 6%/yr
m = 12 mo/yr
FIND P:
0 1 2
5 yrs
P?
$2 000
DIAGRAM: 12
5
5
0 061 1 1 1 6 17
12
2000 6 17 5
2000 1 0 0617 2000 74129
1482 59
.. % /
$ ( | , . %, )
$ ( . ) $ (. )
$ .
m
a
n yrs
r i yr
m
P P F
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Problem 2 – Alternate SolnWhat amount m ust be deposi ted today in an account paying
6% per year, compounded month ly in order to have $2,000 in the account at the end of 5 years?
GIVEN:F5 = $2 000r = 6%/yr
m = 12 mo/yr
FIND P:
0 1 2
60mos
P?
$2 000
DIAGRAM:
80 .1482 $
)7414.0 ( 2000 $ )60 %,5 .0 ,F |P ( 2000 $ P
mo / %5 .0 mo12
yr 1
yr
06 .0
m
r i
mos60 )yrs5 ( yr
mos12 )yrs )(# m( n
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Problem 3A loan of $5,000 is to be repaid in equal month ly payments
over the next 2 years. The f irst payment is to be made 1 month from now. Determine the payment amount i f interest
is charged at a nom inal interest rate of 12% per year,
compounded month ly.
0
1 2 yrs
$5 000
A ?
DIAGRAM:
GIVEN:P = $5 000r = 12%/yr
m = 12 mo/yr FIND A:
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Problem 4You h ave decided to begin a savings plan in order to make a
dow n payment on a new hou se. You w il l depos it $1000 every 3 months for 4 years into an accoun t that pays interest
at the rate of 8% per year, compou nded month ly. The f irst
deposi t wi l l be made in 3 months. How much w i l l be in the
account in 4 years?
0 1 2 3 4 yrs
$1 000
DIAGRAM:F ?
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Problem 5Determine the total amount accum ulated in an account
paying interest at the rate of 10% per year, compou nded con t inuo usly i f depos its of $1,000 are made at the end of
each of th e next 5 years.
0 1 2 3 5 yrs
$1 000
DIAGRAM: F ?
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Problem 6A firm pays b ack a $10 000 loan w ith quarter ly payments
over the next 5 years. The $10 000 return s 4% APR compoun ded month ly. What is the quar ter ly payment
amount?
0
1 2 3 5 yrs = 20 qtrs
$A
DIAGRAM:
$10 000
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Problem 7Anita Plass-Tuwurk, who owns an engineer ing con sul t ing f i rm,
bought an old house to use as her business off ice. She found that
the ceil ing was poor ly in sulated and th at the heat loss cou ld be cut
signi f icant ly i f 6 inches of foam insu lat ion were instal led. She
est imated th at with the insulat ion she cou ld cut th e heat ing b i l l by
$40 per month and the air condi t ioning co st by $25 per month.
Ass um ing that the summer season is 3 months (June, July,
Aug ust) of the year and the w inter season is another 3 mon ths
(December, January, and February) of the year, how much can she
spend on insu lat ion i f sh e expects to keep the pro perty for 5 years?
Assume that nei ther heat ing no r air condi t ioning would be required
dur ing the fal l and spr ing season s.
She is making this d ecis ion in Apri l , abou t wh ether to ins tal l the
insu lation in May. If the insulation is in stal led, it wil l be paid for at
the end of May. Anita’s interest rate is 9%, compounded monthly.
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Problem 7GIVEN:
SAVINGS = $40/MO (DEC,JAN, FEB); $25/MO (JUN, JUL, AUG)r = 9%/YR, CPD MONTHLY
FIND P(SAVINGS OVER 5 YEARS):
$40PA ?
1ST YR DIAGRAM:
1 2 3 4 5 6 7 8 9 10 11 12 MO0
$25
P ?1 2 3 4 5 YRS
0
5 YR DIAGRAM:
PA
PA = Pα + Pβ(PPβ) = Aα(P|A,i,nα) + Aβ(P|A,i,nβ)(P|F,i,6)
= $25(P|A,0.75%,3) + $40(P|A,0.75%,3)(P|F,0.75%,6)
= $25(2.9556) + $40(2.9556)(0.9562) = $186.94 at the start of each year…
i = r = 0.09 = 0.75% / MO
m 12
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