Effective Interest Rate Problems

7/28/2019 Effective Interest Rate Problems

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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…



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.



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?



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



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?



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?



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%



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



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



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)



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!



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



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



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



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:



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 ?



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 ?



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



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.



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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Effective Interest Rate Problems