Lecture No



Lecture No. 4

Nominal and Effective Interest Rates

1. General

The difference between nominal and effective interest rates is that nominal means once per year and effective means compounding more than once per year.

Nominal Interest Rate

Nominal interest rate, r, is an interest rate that does not include any consideration of compounding.

r = interest rate per period x number of periods

A nominal rate may be stated for any period: 1 year, 6 months, weekly, daily.

r = 1.5% per month x 12 months = 18%

Considering 2% per month, all the following are same:

• 2% per month x 12 months = 24% per year

• 2% per month x 24 months = 48% per 2 years

• 2% per month x 6 months = 12% seminanually

• 2% per month x 3 months = 6% quarterly

• 2% per month x .231 months = .462 weekly

• 2% per month x 1/365 months = .005479 daily

Note that the nominal rates do NOT make mention of the compounding period, there is no compounding period by definition.

Given: 18% per year, compounded monthly

Find: Nominal interest rate per

a. 2 month

b. 6 months

c. 2 years

a. month

i/month = 18/12 = 1.5

r/2months = 1.5 x 2

r/2months = 3%

b. 6 months

r/6months = 1.5 x 6

r/6months = 9%

c. 2 years

r/2years = 1.5 x 24

r/years = 36%

Effective Interest Rate

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. It is commonly expressed on an annual basis as the effective annual rate ia, but any time basis can be used.

An effective rate has the compounding frequency attached to the nominal rate statement. If the compounding frequency is NOT stated, it assumed to be the same time period as r meaning that the nominal and effective rates are the same.

The following do NOT have the same effective rate over all time periods due to different compounding frequencies.

• 12% per year, compounded monthly

• 12% per year, compounded quarterly

• 3% per quarter, compounded quarterly

In the above, the 12% and 3% are nominal interest rates, the effective rate must be calculated by applying the compounding period.

The format is r % per time period, compounded m-ly. The m is any time unit. In the last example, compounded quarterly could be omitted because the periods are the same meaning that the nominal rate of 3% per quarter is the same as the effective rate of 3% per quarter compounded quarterly.

All calculations must use the effective interest rate.

The Annual Percentage Rate, APR, is the same as the nominal interest rate, and the Annual Percentage Yield, APY, is used in lieu of effective interest rate.

There are two time units associated with an interest rate statement:

• Time period – the basic time unit of the interest rate. This is the t in the statement of r % per time period t. The time unit of 1 year is by far the most common and 1 year is assumed unless otherwise stated.

• Compounding period (CP) – the time unit used to determine the effect of interest. It is defined by the compounding term in the interest rate statement. If not stated, it is assumed to be a year.

The compounding frequency is the number m, which is the number of times that compounding occurs within t, the time period. 8% per year compounded monthly has m=12. If 8% is compounded daily, m=365.

In the previous chapters t = m = 1 year meaning that the effective and nominal rates were equal.

It is common to express the effective rate on the same time basis as the compounding period.

Effective rate per CP (Compounding Period) = =

Given: r = 6% per year, compounded monthly

Find: CP

CP = =

CP = .50% per month

Note that a different time period does NOT alter the compounding period, which is monthly.

There are 3 ways to express interest rates p.129, T4-1.

1. 8% per year compounded quarterly, 8% is nominal and the effective must be calculated

2. Effective 8.243% per year compounded quarterly, 8.243% is the effective rate and may be used directly.

3. 8% per year, ambiguous because no compounding period is stated. The rate is effective only over the time period of one year; the effective rate for any other time period must be calculated.

2. Effective Annual Interest Rates

The most common period is a year by far which is considered in this section.

r = nominal interest rate per year

m = number of compounding periods per year

i = effective interest rate per compounding period CP = r/m

ia = effective interest rate per year

F = P + Pia = P(1+ia)

CP must be compounded through all m periods to obtain the total.

F = P(1+i)m

Consider of the F value for a present worth P of $1 and equating the two expressions for F and substituting $1 for P:

1+ia = (1+i)m

ia = (1+i)m –1

Solving for the effective interest rate:

i = (1+ia)1/m –1

If i = r/m

r % per year = (i% per CP)(number of CPs per year) = (i)(m)

Given: 13% per year compounded monthly

Find: effective interest rate per year

ia = (1+i)m –1

i=r/m=13/12=1.08333%=.01083333

ia = (1+.0108333)12 –1

ia = 13.80% per year

Example p.129, T4.2, T4-3

3. Effective Interest Rates for Any Time Period

The payment period, PP, is the frequency of payment or receipts. To evaluate cash flows that occur more frequently than annually, PPCP; find effective i per PP = quarter

i/quarter = ( 1+.03/3)3 –1

i/quarter = 3.03%

A=2(A/P, 3.03%,20) = 2(.0674)

A = .134805M = $134,805/quarter

PP ................
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