Nominal%20-vs-%20Effective%20Interest%20Rates

1
Nominal -vs- Effective Interest Rates

• Nominal interest rate, r, is an interest rate
  that does not include any consideration of
  compounding. This rate is often referred to as
  the Annual Percentage Rate (APR).
• r interest rate per period x number of periods
• Effective interest rate is the actual rate that
  applies for a stated period of time. The
  effective interest rate is commonly expressed on
  an annual basis as the effective annual interest,
  ia. This rate is often referred to as the Annual
  Percentage Yield (APY).

2
Nominal -vs- Effective Interest Rates

• The following are nominal rate statements
• Nominal Rate (r) Time Period (t)
  Compounding Period (CP)
• 1) 12 interest per year, compounded monthly.
• 2) 12 interest per year, compounded quarterly.
• 3) 3 interest per quarter, compounded monthly.
• What are the corresponding effective annual
  interest rates?

3
Nominal -vs- Effective Interest Rates

• Corresponding effective annual interest rates
• Let compounding frequency, m, be the number of
  time the compounding occurs within the time
  period, t.
• 12 interest per year, compounded monthly.
• m 12
• Effective rate per CP, iCP r/m 1 (per
  month).
• Effective annual rate (1 iCP)12 1 12.68

4
Nominal -vs- Effective Interest Rates

• Corresponding effective annual interest rates
• 12 interest per year, compounded quarterly.
• m 4
• Effective rate per CP, iCP r/m 3 (per
  quarter).
• Effective annual rate (1 .03)4 1 12.55
• 3 interest per quarter, compounded monthly.
• m 3
• Effective rate per CP, iCP r/m 1 (per
  month).
• Effective annual rate (1 .01)12 1 12.68

5
Nominal -vs- Effective Interest Rates

• Example You are purchasing a new home and have
  been quoted a 15 year 6.25 APR loan. If you
  take out a 100,000 mortgage using the above
  rates, what is your monthly payment?
• Compound period monthly
• iCP 6.25 / 12 .521 (per month)
• n 15(years) x 12(months/year) 180 months
• A 100,000(A/P, .521, 180).

6
Nominal -vs- Effective Interest Rates

• Determining n
• Given a stated APR and APY can you determine the
  compounding frequency?
• Example A Certificate of Deposit has a stated
  APR of 8 with an Annual Yield of 8.3. What is
  the compounding period?
• Compound Period Effective Annual Interest
• 1 day 8 / 365 .022 / day
• ia (1.00022)365 1 8.36
• 1 week 8 / 52 .15 / week
• ia (1.0015)52 1 8.322
• 1 month 8 / 12 .67 / month
• ia (1.0067)12 1 8.30
• 6 months 8 / 2 4 / semi-annual
• ia (1.04)2 1 8.16

7
Nominal -vs- Effective Interest Rates

• Effective interest rates for any time period
• Let PP represent the payment period (period of
  time between cash flows).
• And m is the number of compounding periods per
  payment period.
• Effective i (1r/m)m 1
• r nominal interest rate per payment period,
  PP.
• m number of compounding periods per payment
  period.

8
Nominal -vs- Effective Interest Rates

• Effective interest rates for any time period
• Example If cash flows are received on a
  semi-annual basis, what is the effective
  semi-annual interest rate under the following
  conditions
• 9 per year, compounded quarterly
• Effective isa (14.5/2)2 1 4.55
• 3 per quarter, compounded quarterly
• Nominal is 6 per semi-annual.
• Effective isa (16/2)2 1 6.09
• 8.8 per year, compounded monthly.
• Effective isa (14.4/6)6 1 4.48

9
Nominal -vs- Effective Interest Rates

• Equivalence Relations
• Example Consider the following cash flow. Find
  the present worth if the cash flows earn a) 10
  per year compounded quarterly, or b) 9 per year
  compounded monthly.
• (Time in Years)

1 2 3 4 5 6 7
300
500
700
10
Nominal -vs- Effective Interest Rates

• Equivalence Relations
• Example a) 10 per year compounded quarterly
• (Time in Years)
• ia (110/4)4 1 10.38
• P 500(P/F,10.38,3) 700 (P/F,10.38,6)
  300 (P/F,10.38,7)

1 2 3 4 5 6 7
300
500
700
11
Nominal -vs- Effective Interest Rates

• Equivalence Relations
• Example a) 9 per year compounded monthly.
• (Time in Years)
• ia (19/12)12 1 9.38
• P 500(P/F,9.38,3) 700 (P/F,9.38,6)
• 300 (P/F,9.38,7)

1 2 3 4 5 6 7
300
500
700
12
Nominal -vs- Effective Interest Rates

• Equivalence Relations (PP gt CP)
• Find P for the following in standard factor
  expressions
• Cash Flow Interest Rate Standard Notation
• 500 semiannually 16 per year,
  P
• for 5 years compounded
• 
  semi-annually
• 75 monthly for 24 per year,
  P
• 3 years compound
  monthly
• 180 quarterly for 5 per quarter
  P
• 15 years
• 25 per month 1 per month
  P
• increase for 4 years
• 5000 per quarter 1 per month
  P
• for 6 years

13
Nominal -vs- Effective Interest Rates

• Equivalence Relations (PP gt CP)
• Find P for the following in standard factor
  expressions
• Cash Flow Interest Rate Standard Notation
• 500 semiannually 16 per year,
  P 500(P/A,8,10)
• for 5 years compounded
• 
  semi-annually
• 75 monthly for 24 per year,
  P 75(P/A,2,36)
• 3 years compound
  monthly
• 180 quarterly for 5 per quarter
  P 180(P/A,5,60)
• 15 years
• 25 per month 1 per month
  P 25(P/G,1,48)
• increase for 4 years 25(P/A,1,48)
• 5000 per quarter 1 per month
  P 5000(P/A,3.03,24)
• for 6 years

14
Nominal -vs- Effective Interest Rates

• Equivalence Relations (PP lt CP)
• If payments occur more frequently than the
  compounding period, do these payments compound
  within the compounding period?
• Answer Depends

15
Nominal -vs- Effective Interest Rates

• Equivalence Relations (PP lt CP)
• No compounding within compound period
• All deposits are regarded as occurring at the end
  of the compounding period.
• All withdrawals are regarded as occurring at the
  beginning of the period.

16
Nominal -vs- Effective Interest Rates

• Equivalence Relations (PP lt CP)
• No compounding within compound period
• Example A company has the following monthly
  cash flows. If the company expects an ROR of 12
  per year, compounded quarterly, what is the
  present value of the cash flows?

800
500
P
600
500
400
1 2 3 4 5 6
7 8 9 10 11
12
300
500
500
500
700
700
17
Nominal -vs- Effective Interest Rates

• Equivalence Relations (PP lt CP)
• No compounding within compound period
• gt

800
500
P
600
500
400
1 2 3 4 5 6
7 8 9 10 11
12
300
500
500
500
1400
700
700
P
500
500
400
1 2 3 4 5 6
7 8 9 10 11
12
500
700
1000
1000
18
Nominal -vs- Effective Interest Rates

• Equivalence Relations (PP lt CP)
• No compounding within compound period
• P 1400 - 100(P/F,3,1) - 200(P/F,3,2) -
  500(P/F,3,3)- - 1000(P/F,3,4)

1400
P
500
500
400
1 2 3 4 5 6
7 8 9 10 11
12
500
700
1000
1000
19
Nominal -vs- Effective Interest Rates

• Equivalence Relations (PP lt CP)
• With interperiod compounding
• If interest is compounded within the period,
  treat interest on cash flows the same as the
  treatment of nominal interest rates.

20
Nominal -vs- Effective Interest Rates

• Equivalence Relations (PP lt CP)
• With interperiod compounding
• Example A company has the following monthly
  cash flows. If the company expects an ROR of 12
  per year, compounded quarterly, what is the
  present value of the cash flows?

800
500
P
600
500
400
1 2 3 4 5 6
7 8 9 10 11
12
300
500
500
500
700
700
21
Nominal -vs- Effective Interest Rates

• Equivalence Relations (PP lt CP)
• With interperiod compounding
• Example Interest is compounded monthly at the
  rate of 1.
• P 800(P/F,1,1) 600(P/F,1,2) -
  500(P/F,1,3) etc.

800
500
P
600
500
400
1 2 3 4 5 6
7 8 9 10 11
12
300
500
500
500
700
700
22
Nominal -vs- Effective Interest Rates

• Continuous compounding
• Recall,
• Effective i (1r/m)m 1
• Where m number of compounding periods per
  payment period.
• As m approaches infinity,
• i er 1
• Example A 15 APR compounded continuously is
  effectively i e0.15 1 16.183