Nominal%20and%20Effective%20Interest%20Rate

1
Chapter 4

• Nominal and Effective Interest Rate

2
Recall

• Nominal interest rate an interest rate that does
  not include any consideration of compounding.
• 10 per year 20 per quarter
• r per period t

3
Recall

• Effective interest rate the interest rate taking
  into account the effect of any compounding during
  the time period.
• 10 per year, compounding monthly
• r per time period t, compounding m-ly

4
How to calculate effective interest rate

• r Nominal interest rate per payment period
• (PP)
• i Effective interest rate per payment period
• m Number of compounding periods (CP) per
• payment period (PP)
• ia Effective interest rate per year

5
Equations

• Interest rate per compounding period (i)
• i r/m
• ia (1i)m -1
• And in general for any time period
• i (effective) (1r/m)m -1
• ie (1r/m)m -1

6
Example

• An interest rate of 13 per year, compounded
  monthly, is equivalent to what effective interest
  rate per year?

7
Solution

• r13
• m12
• ia?
• ia (1r/m)m -1
• (10.13/12)12 1
• 13.80

8
Example

• To help prevent the spread of the AIDS virus
  around the world, the United States budgeted 300
  million to be spent in 1 year. At an interest
  rate of 12 per year, compounded monthly, what is
  the equivalent monthly expenditure in that year?

9
Solution

• Number of Compounding Periods 12
• i12
• Payment period is a month. So you need to find
  the interest rate per month
• ir/m
• 0.12/12
• 0.01
• AP(A/P,1,12)
• 300,000,000(0.08885)
• 26,655,000

10
Example

• A manufacturer of precision laboratory equipment
  borrowed 2 million to renovate one of its
  testing labs. The loan was repaid in 5 years in
  equal quarterly payments of principal and
  interest. At an interest rate of 1 per month,
  what was the size of each quarterly payment?

11
Solution

• r1
• i (quarter) (1 imonth)3 1
• (1 0.01)3 -1
• 3.03
• A 2,000,000 (P/A, i, n)
• 2,000,000 i(1i)n / (1i)n 1
• 2,000,000 .0303(1.0303)20 /
  (1.0303)20 1
• 134,895 per quarter

12
Effective Interest Rate for Continuous Compounding

• In continuous compounding, the compounding period
  is very short, i.e. each day, where the interest
  is compounded infinite number of times per year.
• ia er - 1

13
Example

• What nominal interest rate per year, compounded
  continuously, is equivalent to an effective rate
  of 13.2 per year?

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Solution

• i13.2
• ia er 1
• 0.132 er 1
• er 1.132
• Ln (er ) Ln (1.132)
• r .1239 or 12.39