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Chapter 4

Nominal and Effective Interest Rates

CE 314 Engineering Economy

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.

A Nominal Interest Rate, r, is an interest Rate that does not include any consideration of the compounding of interest.

Nominal Interest Rates

r = (interest rate per period)(No. of Periods)

1.5% per month for 12 monthsSame as (1.5%)(12 months) = 18%/year

1.5% per 6 monthsSame 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 be applied…

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).

Effective Interest Rates

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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.

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 monthlyr = 18 % per year (same as nominal interest rate)m = 12 interest periods per year

Effective Interest Rates

What is the effective annual interest rate (EAIR)? It must be larger than 18% per year!

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Effective Interest RatesEffective 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 inthe 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.

Effective Interest Rates

a) r/m = 9%/4 = 2.25% per quarter

b) r/m = 9%/12 = 0.75% per month

c) r/m = 4.5%/26 = 0.173% per week

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Effective Interest RatesAll 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.

Effective Annual Interest Rates

ia = (1 + i)m – 1where:

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

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)1) ia = 12.683% per year compounded yearly

F = $10,000 (1.12683)5 = $18,167

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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!

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

Effective Interest Rates for r = 18%Effective Interest Rates for r = 18%

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Effective Annual Interest Rates for Effective Annual Interest Rates for various Nominal Interest Ratesvarious Nominal Interest Rates

Nominal Annual Rater% 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 alending institution.

CP refers to the compounding of interest by the lending institution.

Effective Interest Rates for any Time Period

Effective i = (1 + r/m)m – 1where:

r = nominal interest rate per payment period (PP)

m = number of compounding periods per payment period (CP per PP)

PP PP

CP CP

Payments every 6 months, with interest compounded every quarter

CP CP

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Equivalence Procedures

Section 4.7Section 4.7PP < CP

Section 4.6Section 4.5PP > CP

Section 4.6Section 4.5PP = CPSeries FactorsSingle FactorsTime

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Equivalence Procedures

Single Payments (P,F) when PP > or = to CPMethod 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)

Equivalence ProceduresP = 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

$2,000

F = ?

$1,000

Payments are on a yearly basis. Interest compounded twice a year. Therefore, PP > 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

Equivalence ProceduresMethod 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 semiannuallyEffective 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 ProceduresSeries (A,G and g) when PP = CPDetermine the effective interest rate over the compounding period CP or PP, and set n equal to the number of compounding periods or payment periodsbetween 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.

Equivalence ProceduresSeries (A,G and g) when PP > CPFind the effective i per payment period and determine n as the total 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 the last deposit is made?

A = $1,000 per 6-months

F = ?

X X X X

X denotes where compounding of interest is taking place.

1 2 yearsX XX X

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Equivalence ProceduresPayments are biannually. Interest is compounded quarterly. Therefore PP > 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 > CP and you are dealing with series factors, this is the only approach, which will result in the correct amount!

Equivalence ProceduresSingle Payments (P,F) and Series Amounts (A, G, g)

when PP < CP:

Bank Policy:1) Interest is not paid between compounding periods. Many

banks operate in this fashion.

2) Interest is paid or charged between compounding periods.

For a no-interperiod-interest policy, all deposits are treated as deposited at the end of the compounding period, and all withdrawals are treated as withdrawn at the beginning.