Nominal & effective Interest Rates

Nominal & effective Interest Rates in Engineering economy and management

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Salahaddin University-Erbil
College of Engineering
Geomatics(Surveying)Department
Nominal & effective Interest Rates

Student Name: Copyright
Supervisor: Dr.Kamaran Kakil Hamad
Class: 4
nd
stage
Course Title: Engineering Economy & Management


Academic year_ 2019-2020

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Abstract
In all engineering economy relations developed thus far, the interest rate has
been a constant, annual value. For a substantial percentage of the projects
evaluated by professional engineers in practice, the interest rate is
compounded more frequently than once a year; frequencies such as
semiannually, quarterly, and monthly are common. In fact, weekly, daily,
and even continuous compounding may be experienced in some project
evaluations. Also, in our own personal lives, many of our financial
considerations—loans of all types (home mortgages, credit cards,
automobiles, boats), checking and savings accounts, investments,
stock option plans, etc.—have interest rates compounded for a time period
shorter than1 year. This requires the introduction of two new terms—
nominal and effective interest rates .This chapter explains how to understand
and use nominal and effective interest rates in engineering practice and in
daily life situations. Equivalence calculations for any compounding
frequency in combination with any cash flow frequency are presented.

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Table of Content
Subject Page
Title of Raport……………………………………………..... 1
Abstract…………………………………………………….... 2
Table of Content…………………………………………… 3
Introduction………………………………………………… 4
Background & Review…………………………………….. 7
Methods……………………………………………………. 9
Theory……….……………………………………………. 12
Conclusion………………………………………………..... 16
Reference………………………………………… ………... 17

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Introduction
Engineering economics, previously known as engineering economy, is a
subset of economics concerned with the use and "...application of economic
principles"in the analysis of engineering decisions. As a discipline, it is
focused on the branch of economics known as microeconomics in that it
studies the behavior of individuals and firms in making decisions regarding
the allocation of limited resources. Thus, it focuses on the decision making
process, its context and environment. It is pragmatic by nature, integrating
economic theory with engineering practice. But, it is also a simplified
application of microeconomic theory in that it avoids a number of
microeconomic concepts such as price determination, competition and
demand/supply. As a discipline though, it is closely related to others such
as statistics, mathematics and cost accounting. It draws upon the logical
framework of economics but adds to that the analytical power of
mathematics and statistics.
1.The concept of real interest rate is useful to account for the impact of
inflation. In the case of a loan, it is this real interest that the lender
effectively receives. For example, if the lender is receiving 8 percent from a
loan and the inflation rate is also 8 percent, then the (effective) real rate of
interest is zero: despite the increased nominal amount of currency received,
the lender would have no monetary value benefit from such a loan because
each unit of currency would get devaluated due to inflation by the same
factor as the nominal amount gets increased.
The relationship between the real interest value r, the nominal interest rate
value R, and the inflation rate value i is given by (1+r)=(1+R)/(1+i)

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 When the inflation rate i is low, the real interest rate is approximately
given by the nominal interest rate minus the inflation rate, i.e.,
r ≈ R- i
In this analysis, the nominal rate is the stated rate, and the real interest rate is
the interest after the expected losses due to inflation. Since the future
inflation rate can only be estimated, the ex ante and ex post (before and after
the fact) real interest rates may be different; the premium paid to actual
inflation (higher or lower).
2.The nominal interest rate (also known as an Annualised Percentage Rate or
APR)*{ASIDE: This doesn't look right: the APR is an annualized rate that
lumps in all charges (fees, initial costs, and so on) and is always a rate used
for comparison between lenders, rather than the nominal interest rate, which
is quoted by lenders and is the actual rate used in the calculation of, say,
monthly payments} is the periodic interest rate multiplied by the number of
periods per year. For example, a nominal annual interest rate of 12% based
on monthly compounding means a 1% interest rate per month
(compounded). A nominal interest rate for compounding periods less than a
year is always lower than the equivalent rate with annual compounding (this
immediately follows from elementary algebraic manipulations of the
formula for compound interest). Note that a nominal rate without the
compounding frequency is not fully defined: for any interest rate,
the effective interest rate cannot be specified without knowing the
compounding frequency and the rate. Although some conventions are used
where the compounding frequency is understood, consumers in particular
may fail to understand the importance of knowing the effective rate.

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Nominal interest rates are not comparable unless their compounding periods
are the same; effective interest rates correct for this by "converting" nominal
rates into annual compound interest. In many cases, depending on local
regulations, interest rates as quoted by lenders and in advertisements are
based on nominal, not effective interest rates, and hence may understate the
interest rate compared to the equivalent effective annual rate.
Confusingly, in the context of inflation, 'nominal' has a different meaning. A
nominal rate can mean a rate before adjusting for inflation, and a real rate is
a constant-prices rate. The Fisher equation is used to convert between real
and nominal rates. To avoid confusion about the term nominal which has
these different meanings, some finance textbooks use the term 'Annualised
Percentage Rate' or APR rather than 'nominal rate' when they are discussing
the difference between effective rates and APR's.
The term should not be confused with simple interest (as opposed to
compound interest) which is not compounded.
The effective interest rate is always calculated as if compounded annually.
The effective rate is calculated in the following way, where r is the effective
rate, i the nominal rate (as a decimal, e.g. 12% = 0.12), and m the number of
compounding periods per year (for example, 12 for monthly compounding):
r=(1+i/m)
m
-1

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Background & Review
In the past two centuries, interest rates have been variously set either by
national governments or central banks. For example, the Federal
Reserve federal funds rate in the United States has varied between about
0.25% and 19% from 1954 to 2008, while the Bank of England base rate
varied between 0.5% and 15% from 1989 to 2009, and Germany
experienced rates close to 90% in the 1920s down to about 2% in the
2000s. During an attempt to tackle spiraling hyperinflation in 2007, the
Central Bank of Zimbabwe increased interest rates for borrowing to 800%.
The interest rates on prime credits in the late 1970s and early 1980s were far
higher than had been recorded – higher than previous US peaks since 1800,
than British peaks since 1700, or than Dutch peaks since 1600; "since
modern capital markets came into existence, there have never been such
high long-term rates" as in this period.
Possibly before modern capital markets, there have been some accounts that
savings deposits could achieve an annual return of at least 25% and up to as
high as 50%.

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The History of Interest Rates Over 670 Years
Today, we live in a low-interest-rate environment, where the cost of
borrowing for governments and institutions is lower than the historical
average. It is easy to see that interest rates are at generational lows, but did
you know that they are also at 670-year lows?
This week’s chart outlines the interest rates attached to loans dating back to
the 1350s. Take a look at the diminishing history of the cost of debt—money
has never been cheaper for governments to borrow than it is today.











Fig.(1)history of interest rate.

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Method
Nominal interest rate is also defined as a stated interest rate. This interest
works according to the simple interest and does not take into account the
compounding periods. Effective interest rate is the one which caters the
compounding periods during a payment plan. It is used to compare the
annual interest between loans with different compounding periods like week,
month, year etc. In general stated or nominal interest rate is less than the
effective one. And the later depicts the true picture of financial payments.
The nominal interest rate is the periodic interest rate times the number of
periods per year. For example, a nominal annual interest rate of 12% based
on monthly compounding means a 1% interest rate per month
(compounded). A nominal interest rate for compounding periods less than a
year is always lower than the equivalent rate with annual compounding (this
immediately follows from elementary algebraic manipulations of the
formula for compound interest). Note that a nominal rate without the
compounding frequency is not fully defined: for any interest rate, the
effective interest rate cannot be specified without knowing the compounding
frequency and the rate. Although some conventions are used where the
compounding frequency is understood, consumers in particular may fail to
understand the importance of knowing the effective rate.
Nominal interest rates are not comparable unless their compounding periods
are the same; effective interest rates correct for this by "converting" nominal
rates into annual compound interest. In many cases, depending on local
regulations, interest rates as quoted by lenders and in advertisements are
based on nominal, not effective interest rates, and hence may understate the
interest rate compared to the equivalent effective annual rate.

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The term should not be confused with simple interest (as opposed to
compound interest) which is not compounded.
The effective interest rate is always calculated as if compounded annually.
The effective rate is calculated in the following way, where i
e is the effective
rate, r the nominal rate (as a decimal, e.g. 12% = 0.12), and “m” the number
of compounding periods per year (for example, 12 for monthly
compounding):
i
e = (1 + r/m)
m
- 1
The following two tables will illustrate the terminologies commonly used for
i
e and r.
Where :
m=number of compounding periods per year
Table(1) Below table show the relation between period of effective (i) , and
time period for (r).

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Fig.(2) Interest is the price that borrowers pay to obtain capital. This graph
compares the changing interest rates of first mortgages for house loans with
the interest the banks pay those who invest in a six-month savings bond. In
general the difference between the two rates is what the bank earns. It was
about 2.5% through this period. The rise in rates from the early 1970s to the
late 1980s represented the period of general price inflation in New Zealand.

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Theory
Here we can show some examples about nominal and effective intrest rate :
Example 1.
If a lender charges 12% interest, compounded quarterly, what effective
annual interest rate is the lender charging?
Solution :
A i
a = [ 1 + (0.12 / 12) ] 12 - 1 = (1.01)12 - 1 = 1.1268 - 1 = .1268 = 12.68%
B i
a = [ 1 + 0.12 ] 12 - 1 = (1.12)12 - 1 = 3.8960 - 1 = 2.8960 = 289.6%
C i
a = [ 1 + (0.12 / 12) ] 4 - 1 = (1.01)4 - 1 = 1.0406 - 1 = .0406 = 4.06%
D i
a = [ 1 + (0.12 / 4) ] 4 - 1 = (1.03)4 - 1 = 1.1255 - 1 = .1255 = 12.55%
Example 2.
If a lender charges 12% interest, compounded monthly, what is the effective
interest rate per quarter?
Note : m = number of compounding periods per quarter
i
e = effective interest rate per quarter.
Solution :
A i = [ 1 + (0.12 / 3) ] 3 - 1 = (1.04)3 - 1 = 0.1249 = 12.49%
B i = [ 1 + 0.03 ] 12 - 1 = (1.03)12 - 1 = 0.4258 = 42.58%
C i = [ 1 + (0.03 / 3) ] 3 - 1 = (1.01)3 - 1 = 0.0303 = 3.03%
D i = [ 1 + (0.03 / 12) ] 3 - 1 = (1.0025)3 - 1 = 0.0075 = 0.75%

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Example 3.Janice is an engineer with Southwest Airlines. She purchased
Southwest stock for $6.90 per share
and sold it exactly 1 year later for $13.14 per share. She was very pleased
with her investment earnings. Help Janice understand exactly what she
earned in terms of
( a ) effective annual rate and
( b ) effective rate for quarterly compounding, and for monthly
compounding. Neglect any commission fees for purchase and selling of
stock and any quarterly dividends paid to stockholders.
Solution :
(a) The effective annual rate of return ia has a compounding period of 1
year, since the stock purchase and sales dates are exactly 1 year apart. Based
on the purchase price of $6.90 per share and using the definition of interest
rate.

i
a


=


= 90.43% per year
(b) For the effective annual rates of 90.43% per year, compounded quarterly,
and 90.43%, compounded monthly to find corresponding effective rates on
the basies of each compounding period
quarter : m = 4 times per year i=(1.9043)
¼
- 1 = 1.17472 – 1 = 0.17472
this is 17.472% per quarter compounded quarterly .
month: m = 4 times per year i=(1.9043)
¼
- 1 = 1.17472 – 1 = 0.17472
this is 5.514% per month, compounded monthly.

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Note: that these quarterly and monthly rates are less than the effective
annual rate divided byThe number of quarters or months per year. In the
case of months,this would be 90.43%/12 =7.54% per month.This
computation is incorrect because it neglects the fact that compounding
Takes place 12 times during the year to result in the effective annual rate
of 90.43%.
Example: Tesla Motors manufactures high-performance battery electric
vehicles. An engineer is on a Tesla committee to evaluate bids for new-
generation coordinate-measuring machinery to be directly linked to the
automated manufacturing of high-precision vehicle components. Three bids
include the interest rates that vendors will charge on unpaid balances. To get
a clear understanding of fi nance costs, Tesla management asked the
engineer to determine the effective semiannual and annual interest rates for
each bid. The bids are as follows:
Bid 1: 9% per year, compounded quarterly
Bid 2: 3% per quarter, compounded quarterly
Bid 3: 8.8% per year, compounded monthly
(a) Determine the effective rate for each bid on the basis of semiannual
periods.
(b) What are the effective annual rates? These are to be a part of the final bid
selection.
(c) Which bid has the lowest effective annual rate?

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Solution:
(a) Convert the nominal rates to a semiannual basis, determine m,
(b) calculate the effective semiannual interest rate i. For bid 1,
r _ 9% per year _ 4.5% per 6 months
m _ 2 quarters per 6 months
Effective i% per 6 months = (1+ 0.045/2 )
2
– 1 = 1.0455 – 1 = 4.55%
Table(2):Effective simeanuual and anuual intreast rates for three bid rates,example

(b) For the effective annual rate, the time basis in Equation [4.7] is 1 year.
For bid 1,
r = 9% per year m = 4 quarters per year
Effective i% per year = (1+ 0.09/4)
4
– 1 = 1.0931 – 1 = 9.31%
The right section of Table 4–4 includes a summary of the effective annual
rates.
(c) Bid 3 includes the lowest effective annual rate of 9.16%, which is
equivalent to an effective semiannual rate of 4.48% when interest is
compounded monthly.

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Conclusion
In this report we explained all about economic engineering and we defined it
after that in background we discussed about those scientists that created
books about engineering economy and when and their names.
After that in method we explained the part of economy which is nominal and
effective interest rate we briefly defined them , and we wrote there
equations and how to work with the equations and how to solve there
problems in real life.
in the method i was proved the equations of nominal and effective with
some examples that goes to real life problems and explained briefly
And in the Fig.(2) it shows a simple example about interest rate in real life
about mortgages for house loans and in the graph shows everything .

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References…
 ENGINEERING ECONOMY 7
th
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American University of Sharjah, United Arab Emirates.
 Anthony Tarquin , P. E. University of Texas at El Paso.
 https://www.visualcapitalist.com/the-history-of-interest-rates-over-670-years/
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 https://www.researchgate.net/publication/315432395_Interest_Rates_1_What_are_Interest_R
ates
 https://www.soa.org/globalassets/assets/Files/Edu/2017/fm-determinants-interest-rates.pdf