Nominal and Effective Interest Rates

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Chapter 4
CE 314 Engineering Economy

• Nominal and Effective Interest Rates

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• Interest is quoted on the basis of
• 1. Quotation using a Nominal Interest Rate
• 2. Quoting an Effective Periodic Interest Rate
• Nominal and Effective Interest rates are commonly
  quoted in business, finance, and engineering
  economic decision-making.
• Each type must be understood in order to solve
  various problems where interest is stated in
  various ways.

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Interest rates can be quoted in many ways
Interest equals 6 per 6-months Interest is
12 (12 per what?) Interest is 1 per
month Interest is 12.5 per year, compounded
monthly Interest is 12 APR You must
decipher the various ways to state interest and
to do calculations.
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Nominal Interest Rates
A Nominal Interest Rate, r, is an interest Rate
that does not include any consideration of the
compounding of interest.
r (interest rate per period)(No. of Periods)

• 1.5 per month for 12 months
• Same as (1.5)(12 months) 18/year
• 1.5 per 6 months
• Same as (1.5)(6 months) 9 per 6 months or
  semiannual period

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• A nominal rate (as quoted) does not reference the
  frequency of compounding per se.
• Nominal rates can be misleading.
• Which led to The untruth in lending law
• An alternative way to quote interest rates?
• A true Effective Interest Rate must then applied

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Effective Interest Rates

• When quoted, an Effective interest rate is a
  true, periodic interest rate.
• It is a rate that applies for a stated period of
  time.
• It is conventional to use the year as the time
  standard.
• The EIR is often referred to as the Effective
  Annual Interest Rate (EAIR).

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Effective Interest Rates

• Quote 12 percent compounded monthly is
  translated as
• 12 is the nominal rate
• compounded monthly conveys the frequency of the
  compounding throughout the year
• For this quote there are 12 compounding periods
  within a year.

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Effective Interest Rates

• r per time period, compounded m times a year.
• m denotes the number of times per year that
  interest is compounded.
• 18 per year, compounded monthly
• r 18 per year (same as nominal interest rate)
• m 12 interest periods per year

What is the effective annual interest rate
(EAIR)? It must be larger than 18 per year!
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Effective Interest Rates
Effective rate per CP r per time period
t r
m compounding periods per t
m Where Compounding Period (CP) is the time
unit used to determine the effect of interest.
It is determined by the compounding term in the
interest rate statement. If not stated, assume
one year. Time Period (t) is the basic time unit
of the interest rate. The time unit is typically
one year but can be other time periods, such as
months, quarters, semiannual periods, etc. If
not stated, assume one year.


6 per year compounded monthly is equivalent to
6/12 0.50 per month. r 6. m 12.
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Effective Interest Rates



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• r/m 9/4 2.25 per quarter
• r/m 9/12 0.75 per month
• r/m 4.5/26 0.173 per week

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Effective Interest Rates

All the interest formulas, factors, tabulated
values, and spreadsheet relations must have the
effective interest rate to properly account for
the time value of money.
The Effective interest rate is the actual rate
that applies for a stated period of time. The
compounding of interest during the time period of
the corresponding nominal rate is accounted for
by the effective interest rate ia, but any time
basis can be used.


The terms APR and APY are used in many individual
financial situations. The annual percentage rate
(APR) refers to the nominal rate and the annual
percentage yield (APY) is used in lieu of
effective interest rate.
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Effective Annual Interest Rates

ia (1 i)m 1 where m number of
compounding periods per year i effective
interest rate per compounding
period (CP) r/m r nominal interest rate
per year ia effective interest rate per year


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Effective Annual Interest Rates

Example 12 per year compounded monthly r 12
per year m 12 months per year i r/m 12/12
1 ia (1 i)m 1 ia (1 .01)12 1
12.683 per year


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Equivalence

• Example
• You borrow 10,000 at an interest rate of 12
  per year compounded monthly. How much do you owe
  after 5 years?
• F P (F/P, i, 5)
• ia 12.683 per year compounded yearly
• F 10,000 (1.12683)5 18,167

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Equivalence

Or 1 per month for 5(12) 60 months 2) ia r/m
12/12 1 per month compounded monthly F
10,000 (1.01)60 18,167
Therefore we can conclude that 1 per month
compounded monthly for 60 months is equivalent to
12 per year compounded monthly for 5 years.
Both statements imply effective interest rates!

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Effective Interest Rates for r 18


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Effective Annual Interest Rates for various
Nominal Interest Rates


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Nominal Annual Rate
r per year (i per CP)(number of CPs per year)
(i)(m)
Example i 1.5 per month compounded monthly m
12 months r 1.5(12) 18 per year (but not
compounded monthly!) ia (1 0.18/12)12 1
19.56 per year compounded yearly ia 1.5 per
month compounded monthly
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Effective Interest Rates for any Time Period
In many loan transactions or personal financial
decisions the compounding period (CP) may not be
the same as the payment period (PP). When this
occurs the effective interest rate is typically
expressed over the same time period as the
payments. Example Bank pays 4 per year
compounded quarterly and deposits are made every
month. CP 4 times per year PP 12 times per
year PP refers to the deposits and withdrawals by
an individual not a lending institution. CP
refers to the compounding of interest by the
lending institution.
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Effective Interest Rates for any Time Period
Effective i (1 r/m)m 1 where r nominal
interest rate per payment period (PP) m number
of compounding periods per payment period
(CP per PP)
Payments every 6 months, with interest compounded
every quarter
CP
CP
CP
CP
PP
PP
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Equivalence Procedures
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(No Transcript)
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Equivalence Procedures
Single Payments (P,F) when PP gt or to
CP Method 1 Determine the effective interest
rate over the compounding period CP, and set n
equal to the number of compounding periods
between P and F. P F (P/F, effective i per
CP, total number of periods n) F P (F/P,
effective i per CP, total number of periods n)
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Equivalence Procedures
P F (P/F, effective i per CP, total number of
periods n) F P (F/P, effective i per CP, total
number of periods n) Example i 6 per year
compounded semiannually
F ?
1
2
3
1,000
2,000
Payments are on a yearly basis. Interest
compounded twice a year. Therefore, PP gt
CP. Effective i per CP r/m 6/2 3 per 6
months Total number of periods m(n) 2(4) 8
semiannual periods F 2,000(F/P, 3, 8)
1,000 (F/P, 3, 4)
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Equivalence Procedures
F 2,000(F/P, 3, 8) 1,000 (F/P, 3,
4) Please note that the interest rate is quoted
over a 6-month period which corresponds with the
total number of 6-month periods. F
2,000(1.2668) 1,000(1.1255) F 3,659

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Equivalence Procedures
Method 2 Determine the effective interest rate
for the time period t of the nominal rate, and
set n equal to the total number of periods using
this same time period.
Example i 6 per year compounded
semiannually Effective i per year ( 1
0.06/2)2 1 6.09 per year F 2,000(F/P,
6.09, 4) 1,000 (F/P, 6.09, 2) F
2,000(1.0609)4 1,000(1.0609)2 F 3,659
(3,659 from Method 1)

Method 1 is preferred over Method 2 since tables
are easier to use.
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Equivalence Procedures
Series (A,G and g) when PP CP Determine the
effective interest rate over the compounding
period CP or PP, and set n equal to the number of
compounding periods or payment periods between P
and F. P A(P/A, effective i per CP or PP,
total number of periods n) F A(F/A, effective
i per CP or PP, total number of periods n) P
G(P/G, effective i per CP or PP, total number of
periods n) F G(F/G, effective i per CP or PP,
total number of periods n) P g(P/g, effective
i per CP or PP, total number of periods n) F
g(F/g, effective i per CP or PP, total number of
periods n)

See example worked in last class meeting.
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Equivalence Procedures
Series (A,G and g) when PP gt CP Find the
effective i per payment period and determine n as
the toal number of payment periods.
Example 1,000 is deposited every 6-months for
the next 2 years. The account pays 8 per year
compounded quarterly. How much money will be in
the account when then last deposit is made?
F ?

1
2 years
X
X
X
X
A 1,000 per 6-months
X denotes where compounding of interest is taking
place.
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Equivalence Procedures
Payments are biannually. Interest is compounded
quarterly. Therefore PP gt CP and the effective
interest rate must be expressed over the same
time period as the payments! Effective i (1
r/m)m 1 r nominal interest rate per payment
period (PP) 8/2 4 per 6-months m number
of compounding periods per payment period (CP per
PP) m 2 Effective i (1 0.04/2)2 1
4.04 per 6-months m(number of years) 2(2) 4
6-month periods F A (F/A, 4.04,4) F 1,000
((1.0404)4 1)(0.0404) 4,249


When PP gt CP and you are dealing with series
factors, this is the only approach, which will
result in the correct amount!
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Equivalence Procedures

• Single Payments (P,F) and Series Amounts (A, G,
  g) when PP lt CP
• Bank Policy
• Interest is not paid between compounding periods.
  Many banks operate in this fashion.
• Interest is paid or charged between compounding
  periods.
• For a no-interperiod-interest policy, deposits
  are all regarded as deposited at the end of the
  compounding period, and withdrawals are all
  regarded as withdrawn at the beginning.