Nominal and Effective

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• Nominal and Effective
• Interest Rates

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Nominal Interest Rate

• A nominal interest rate, r, is an interest rate
  that does not include any consideration of
  compounding
• Mathematically we have the following definition
• r (interest rate per period)(No. of Periods)

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APR and APY

• The terms APR and APY are used in many financial
  situations instead of nominal and effective
  interest rates.
• The Annual Percentage Rate (APR) is the same as
  the nominal interest rate, and Annual Percentage
  Yield (APY) is used instead of effective interest
  rate.

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Example Wachovia
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Examples Nominal Interest Rates

• 1.5 per month for 24 months
• Same as (1.5)(24) 36 per 24 months
• 1.5 per month for 12 months
• Same as (1.5)(12 months) 18 / year
• 1 per week for 1 year
• Same as (1)(52 weeks) 52 per year

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There are always 3 time based units associated
with an interest rate statement.

• Time period - the period over which the interest
  is expressed. This is the t in the statement of r
  per time period t, for example, 1 per month.
  The time unit of 1 year is by far the most
  common. It is assumed when not stated otherwise.
• Compounding period (CP) - the shortest time unit
  over which interest is charged or earned. This is
  defined by the compounding term in the interest
  rate statement, for example, 8 per year
  compounded monthly. If not stated, it is assumed
  to be 1 year.
• Compounding frequency - the number of times that
  m compounding occurs within the time period t. If
  the compounding period CP and the time period t
  are the same, the compounding frequency is 1, for
  example, 1 per month compounded monthly.

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4.5 per 6 months compounded weekly

• Nominal Rate 4.5.
• Time Period 6 months.
• Compounded weekly
• Assume 52 weeks per year
• 6-months then equal 52/2 26 weeks per 6 months
• The effective weekly rate is
• (0.045/26) 0.00173 0.173 per week

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• The derivation of an effective interest rate
  formula directly parallels the logic used to
  develop the future worth relation
• The future worth F at the end of 1 year is the
  principal P plus the interest P(i) through the
  year.
• Since interest may be compounded several times
  during the year, replace i with the effective
  annual rate ia and write the relation for F at
  the end of 1 year.

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The rate i per CP must be compounded through all
m periods to obtain the total effect of
compounding by the end of the year.
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Symbols used for nominal and effective interest
rates
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Compound Interest 1 Period

• Consider a one-year time period.

FP(1i)1
P 1.00
Invest 1 of principal at time t 0 at interest
rate i per year.
One year later, F P(1i)1
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Example 12 per year compounded semi-annually

• i/6 month 6
• F1100(1 0.06)106
• F2106(1 0.06)112.36 or
• 100(1 0.06)2 112.36
• To calculate ia, we can write
• 100(1ia) 100(10.12/2)2
• ia (10.12/2)2 - 10.1236 or 12.36

112.36
0
1
2
100
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Example Interest is 8 per year compounded
quarterly.

• What is the annual interest rate?
• Calculate ia (1 0.08/4)4 1
• ia (1.02)4 1 0.0824 8.24 / year
• Example 18 / year, comp. monthly
• The effective annual rate is
• (1 0.18/12)12 1 0.1956 19.56 / year

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Example 12 Nominal
12 nominal for various compounding periods
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Equivalence Comparing PP to CP

• CP is the compounding period
• PP is the payment period
• PP and CPs do not always match
• Cash flows may be monthly, but compounding period
  is not monthly.
• Savings Accounts for example
• Monthly deposits with,
• Quarterly interest earned or paid

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One-year cash flow diagram for a monthly payment
period (PP) and semiannual compounding period
(CP).
It is important to distinguish between the
compounding period and the payment period because
in many instances the two do not coincide. For
example, if a company deposits money each month
into an account that pays a nominal interest rate
of 14 per year, compounded semiannually, the
payment period is 1 month while the compounding
period is 6 months (Figure 43).
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Single Amounts PP CP

• Example
• r 15, (compounded monthly)
• Let P 1500.00
• Find F at t 2 years.
• 15 0.15/12 0.0125 1.25/month.
• n 2 years or 24 months
• Work in months or in years

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Single Amounts PP CP

• Approach 1. (work with months)
• F24 1,500 (F/P, 0.15/12, 24)
• F24 1,500 (F/P, 1.25, 24)
• F24 1,500 (1.0125)24 2,021.03
• Approach 2. (work with years)
• Assume n 2 years and i/month 0.0125
• Eff i (1.0125)12 1 0.16076 (16.076)
• F2 1,500 (F/P, 16.076, 2)
• F2 1,500 (1.16076)2 2,021.05

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Series Analysis
A 500 every 6 months
r 10 / 6 mos (qtr compound) 2-qtrs in a
6-month period. i for 6-months (1.05)2 1
10.25/6-months. Now, the interest matches the
payments. F year 7 F period 14
500(F/A,10.25,14) F 500(28.4891) 14,244.50
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Continuous Compounding

• Example
• What is the effective annual interest rate if the
  nominal rate is given as
• r 18, compounded continuously

Solve e0.18 1 1.1972 1 19.72 / year
The 19.72 represents the MAXIMUM effective
annual interest rate for 18 compounded.
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Example

• An investor requires an effective return of at
  least 15 per year. What is the minimum annual
  nominal rate that is acceptable if interest on
  his investment is compounded continuously?
• Solution
• er 1 0.15
• er 1.15
• ln(er) ln(1.15)
• r ln(1.15) 0.1398 13.98

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• Since many real-world situations involve cash
  flow frequencies and compounding periods other
  than 1 year, it is necessary to use nominal and
  effective interest rates.
• When a nominal rate r is stated, the effective
  interest rate per payment period is determined by
  using the effective interest rate equation.
  However, when series cash flows (A, G, and g) are
  present, only one combination of the effective
  rate i and number of periods n is correct for the
  factors.
• This requires that the relative lengths of PP and
  CP be considered as i and n are determined. The
  interest rate and payment periods must have the
  same time unit for the factors to correctly
  account for the time value of money.

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