Nominal and Effective Interest Rates
The term “nominal” Nominal means, “in name only”, not the real rate in this case.
Over View 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, r, is an interest rate that does not include any consideration of compounding.
A
given principal set deposited at the same nominal rate of interest will earn different effective rates of interest depending on the type of compounding.
Nominal
r = interest rate per period x number of periods
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 1.5% per 6 months Same as: (1.5%)(6 months) = 9%/6 months or semiannual period 1% per week for 1 year Same as: (1%)(52 weeks) = 52% per year
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 2 month 6 months 2 years month i/month = 18/12 = 1.5 r/2months = 1.5 x 2 r/2months = 3% 6 months r/6months = 1.5 x 6 r/6months = 9% 2 years r/2years = 1.5 x 24 r/years = 36%
Normal Format
Focus on the Differences Nominal Rates: Format: “r% per time period, t” Ex: 5% per 6-months” Effective Interest Rates: Format: “r% per time period, compounded ‘m’ times a year. ‘m’ denotes or infers the number of times per year that interest is compounded. Ex: 18% per year, compounded monthly
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.
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.
Example Given, “9% per year, compounded quarterly” Qtr. 1 Qtr. 2 Qtr. 3 Qtr. 4 One Year: Equals 4 Quarters CP equals a quarter (3 months)
Given, “9% per year, compounded quarterly” Qtr. 1 Qtr. 2 Qtr. 3 Qtr. 4 What is the Effective Rate per Quarter? iQtr. = 0.09/4 = 0.0225 = 2.25%/quarter 9% rate is a nominal rate; The 2.25% rate is a true effective monthly rate
Statement: 9% compounded monthly r = 9% (the nominal rate). “compounded monthly means “m” =12. The true (effective) monthly rate is: 0.09/12 = 0.0075 = 0.75% per month
Statement: 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 true, effective weekly rate is: (0.045/26) = 0.00173 = 0.173% per week
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
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.
Time Periods Associated with Interest
Payment Period, Tp - Length of time during which cash flows are not recognized except as end of period cash flows. Compounding Period, Tc - Length of time between compounding operations. Interest Rate Period, T - Interest rates are stated as % per time period.
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 =r/m = 6/12 CP = .50% per month
3 ways to express interest rates
8% per year compounded quarterly, 8% is nominal and the effective must be calculated Effective 8.243% per year compounded quarterly, 8.243% is the effective rate and may be used directly. 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.
It can be unclear as to whether a stated rate is a nominal rate or an effective rate. Three different statements of interest follow………
8% per year, compounded quarterly” Nominal rate is stated: 8% Compounding Frequency is given Compounded quarterly True quarterly rate is 0.8/4 = 0.02 = 2% per quarter Here, one must calculate the effective quarterly rate!
Effective Rate Stated
“Effective rate = 8.243% per year, compounded quarterly: No nominal rate given (must be calculated) Compounding periods – m = 4 No need to calculated the true effective rate! It is already given: 8.243% per year!
Only the interest rate is stated “8% per year”.
Note: No information on the frequency of compounding. Must assume it is for one year! “m” is assumed to equal “1”. Assume that “8% per year” is a true, effective annual rate! No other choice!
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
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, PP<1 year, the effective interest rate over the PP must be used in the engineering economy relations. Substituting r/m for the period interest rate in eq. 4.5 yields
Given: interest is 8% per year compounded quarterly”. What is the true annual interest rate? Calculate: EAIR = (1 + 0.08/4)4 – 1 EAIR = (1.02)4 – 1 = 0.0824 = 8.24%/year
Example: “18%/year, comp. monthly” What is the true, effective annual interest rate? r = 0.18/12 = 0.015 = 1.5% per month. 1.5% per month is an effective monthly rate. The effective annual rate is: (1 + 0.18/12)12 – 1 = 0.1956 = 19.56%/year
Example: EAIR given a nominal rate. Given: interest is 8% per year compounded quarterly”. What is the true annual interest rate? Calculate: EAIR = (1 + 0.08/4)4 – 1 EAIR = (1.02)4 – 1 = 0.0824 = 8.24%/year
Example: “18%/year, comp. monthly” What is the true, effective annual interest rate? r = 0.18/12 = 0.015 = 1.5% per month. 1.5% per month is an effective monthly rate. The effective annual rate is: (1 + 0.18/12)12 – 1 = 0.1956 = 19.56%/year
APY & APR
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.
Examples An interest rate of effective 12% per year compounded monthly is = ? If you deposit $1000 per month into an account which pays interest at a rate of 12% per year compounded monthly, the amount of money you would have at the end of five years is =?
The owner of a small business borrowed $70,000 with an agreement to repay the loan with quarterly payments over a five year time period. If the interest rate is 12% per year compounded quarterly, his loan payment each quarter is=?
A metal plating company wants to set aside money now to prepare for a lawsuit it expects to face in four years. If the company wants to have $1,000,000 available at that time, how much must it set aside now in one lump sum if the account will earn 1% per month?
For an interest rate of 2% per month, the effective semiannual rate is=?