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Nominal and Effective Interest Rates and Continuous Compounding

Nominal and Effective Interest Rates and Continuous Compounding

Chapter 4Chapter 4

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Items Covered in this Items Covered in this ChapterChapterNominal and Effective Interest

RatesContinuous CompoundingEquivalence calculations for

payment periods equal to or longer than the compounding period.

Equivalence calculations for payment periods shorter than the compounding period.

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Nominal Versus Effective Interest Rates

Nominal Interest Rate: Interest rate quoted based on an annual period

Effective Interest Rate:Actual interest earned or paid in a year or some other time period

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Why Do We Need an Effective Interest Rate per Payment Period?

Payment period

Interest period

Payment period

Interest period

Payment period

Interest period

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Nominal and Effective Nominal and Effective Interest RateInterest RateCompounding at other intervals than yearly;

e.g., daily, monthly, quarterly, etc. The two terms are used when the compounding period is less than 1 year

Nominal also called Annual Percentage Rate (APR) means not actual or genuine, it must be adjusted or converted into effective rate in order to reflect time value considerations.

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

Nominal interest rate is equal to the interest rate per period multiplied by the number of periods:

Nominal rate (r) per period= i per period * number of periods.

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Nominal and Effective Interest Nominal and Effective Interest Rate (cont.)Rate (cont.)A nominal rate can be found for any time period

longer than the originally stated period. A nominal rate of 1.5%per month is expressed as a

nominal 4.5% per quarter, or 9% per semiannual period, or 18% per year, or 36% per two years, etc.

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.

An effective rate has the compounding frequency attached to the nominal rate statement.

Only effective interest rates can be used in the time value equations or formulas.

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Nominal and Effective Interest Rate Nominal and Effective Interest Rate (cont.)(cont.)

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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Nominal and Effective Interest Rate-Nominal and Effective Interest Rate-ExampleExampleConsider the rate 8% per year,

compounded monthly. It has a time period t of 1 year, a compounding period CP of 1 month, and a compounding frequency m of 12 times per year.

A rate of 6% per year, compounded weekly, has t = 1 year, CP = 1 week, and m = 52, based on the standard of 52 weeks per year.

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Nominal and Effective Interest Rate-ExampleNominal and Effective Interest Rate-Example The different bank loan rates for three separate electric

generation equipment projects are listed below. Determine tbe effective rate on the basis of the compounding period for each quote.

(a) 9% per year, compounded quarterly. (b) 9% per year, compounded monthly. (c) 4.5% per 6-montbs, compounded weekly.

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Effective interest Rates for Any Effective interest Rates for Any PeriodPeriod

Effective i =[1+ r/m]m – 1◦ i: effective interest rate per year (or certain period)◦ m: number of compounding periods per payment period◦ r: nominal interest rate per payment periods

it is possible to take a nominal rate (r% per year or any other time period) and convert it to an effective rate i for any time basis, the most common of which will be the PP time period.

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Nominal and effective Nominal and effective Interest rate-Interest rate-Consider 18 % per compounded at several

periods.

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ExamplesExamples1) Nominal rate of 18% compounded

yearly with time interval of one year (m=1) i=[1+0.18/1]1 – 1=18% per year

2) Nominal rate of 18% compounded semi-annual with a time interval of one year i=[1+0.18/2]2 – 1= 18.81% per year

3) Nominal rate of 18% compounded quarterly with a time interval of 1 year i=[1+0.18/4]4 -1= 19.252% per 1 year

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Effective interest Rate Effective interest Rate Problem 1Problem 1A company wants to buy new machine. The company

received three bids with interest rates. The company will make payments on semi-annual basis only. The engineer is confused about the effective interest rates –what they are annually and over the payment period of 6 months.

Bid #1: 9% per year, compounded quarterlyBid #2: 3% per quarter, compounded quarterlyBid #3: 8.8% per year, compounded monthly

◦ (a) Determine the effective rate for each bid on the basis of semiannual payments, and construct cash flow diagrams similar to Figure 4-3 for each bid rate.

◦ (b) What are the effective annual rates? These are to be a part of the final bid selection.

◦ (c) Which bid has the lowest effective annual rate?

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Effective Interest Rate Effective Interest Rate Problem 2Problem 2The interest rate on a credit card is 1% per month.

Calculate the effective annual interest rate and use the interest factor tables to find the corresponding P/F factor for n=8years?

1) 1% is an effective interest rate (Not nominal!!!!)Nominal rate = 0.01per month*12months/year = 0.12 i=[1+0.12/12]12 -1= 0.1268 = 12.68%2) P/F = 1/ [1+0.1268]8= 0.38483) by interpolation:

◦ 12% 0.4039◦ 12.68% P/F◦ 14% 0.3506(P/F, 12.68%, 8) = 0.4039-0.0181= 0.3858

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Effective Interest Rate for Effective Interest Rate for Continuous CompoundingContinuous Compounding

useful for modeling simplificationsIf an interest rate r is compounded m times per

year, after m periods, the result isi= lim m—∞ (1 +r/m)m -1

Since lim m-> (1 +r/m)m = er , where e 2.7818Further, ia= effective continuous interest rate = er-1Example: if the nominal annual r = 15% per

year, the effective continuous rate per year isi% = e0.15-1=16.183%

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Calculations of Effective Calculations of Effective continuous compounding of IR continuous compounding of IR

For a IR of 18% per year compounded continuously, calculate the effective monthly and annual interest rates?

Solution:◦ r= 0.18/12=0.015 per month, the effective

monthly rate = i per month= er – 1= e0.015 -1= 1.511%

◦ The effective annual rate for a nominal rate

r= 18% per year i per year = e0.18 – 1= 19.72%

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Calculations of Effective Calculations of Effective continuous compounding of IRcontinuous compounding of IR

If an investor requires an effective return of at least 15% on his money, what is the minimum annual nominal rate that is acceptable if continuous compounding takes place?

Solution◦ r =?=er -1= 0.15 er= 1.15 lner = ln 1.15 r = 0.1376 = 13.976%A rate of 13.976 per year compounded

continuously will generate an effective 15% per year return.

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Calculations for payment periods Calculations for payment periods equal to or longerequal to or longer than the than the compounding periodscompounding periods

For uniform series and gradients:For uniform series and gradient factors,

there are three cases:◦ Case 1 PP=CP◦ Case 2 PP>CP◦ Case 3 PP<CP

For cases 1 and 2 follow the following steps:◦ Step 1: count the number of payments and use

that number as n, i.e., payments made quarterly for 5 years…then n is 20 quarters

◦ Step 2: find the effective interest rate over the same time period as n in step 1. i.e., n is expressed in quarters…then the effective rate per quarter should be found and used.

◦ Step 3: use these values of n and i in the tables

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Calculations for payment periods Calculations for payment periods equal to or longerequal to or longer than the than the compounding periods (Sec 4.6)compounding periods (Sec 4.6)For single payment factors:if the compounding period (CP) and

payment period (PP) do not agree (coincide) then interest tables cannot be used until appropriate corrections are made.

For Single payment factors:◦An effective rate must be used for i◦The units on n must be the same as those on i◦If the IR is per X, then n should be in terms of X

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PP≥ CP examplea quality manager will pay $500 every 6

months for the software maintenance contract. What is the equivalent amount after the last payment, if these funds are taken from a pool that has been returning 20% per year, compounded quarterly?

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PP≥ CP example –cont.PP= 6 months, CP is quarterly = 3 months, so PP >

CP. based on PP (every 6 months), r=20% per year is

converted to semi-annual, r = 0.20/2=0.10, m based on r = 6/3=2 Use Equation (4.8) with r = 0.10 per 6-month

period and 2 CP periods per semiannual period.Effective i semi-annual =[1+ r/m]m – 1= [1+0.10/2]2 -

1=10.25%Total number of semi-annual payments = 7 yrs*2 = 14F=A(F/A,10.25%,14)= 500(28,4891)=14,244.50

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PP=CP ExampleSuppose you plan to purchase a car and carry a

loan of $12,500 at 9% per year, compounded monthly. Payments will be made monthly for 4 years. Determine the monthly payment. Compare the computer and hand solutions.

Soln:CP =monthly, PP= monthly, so PP=CP.Effective i per month=9%/12= 0.75, n= 4 yr x 12 = 48Manual: A = $12,500(A/ P,0.75%,48) =

12,500(0.02489) = $31 1.13

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PP=CP Example – cont.

Spreadsheet:Enter PMT(9%/ 12,48, - 12500) into

any cell to display $3 11.06.

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Calculations for payment periods Calculations for payment periods ShorterShorter than the compounding than the compounding periodsperiodsPayments are made on shorter periods

than Compounding Interest.Three possible scenarios:

◦ There is no interest paid on the money deposited or withdrawn between compounding periods

◦ The money deposited or withdrawn between compounding periods earns simple interest.

◦ All interperiod transactions earn compound interest

Scenario number 1 is only considered.

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0 1 2 1211109876543

Year

Month

$150$200

$75 $100

$90$120

$50

$45

Compounding period is quarterly at 3% interest rate

PP < CP examplePP < CP example

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•P= -150- 200(P/F, 3%, 1)- 175(P/F, 3%,2)+ 210(P/F, 3%,3) - 5(P/F,3%,4)

0 1 2 1211109876543

Year

Month

$150$200

$175

$210

$50

1 2 3 40

$45

PP < CP examplePP < CP example

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Non-standard Annuities and Non-standard Annuities and GradientsGradients Treat each cash flow individually Convert the non-standard annuity or gradient to

standard form by changing the compounding period Convert the non-standard annuity to standard by

finding an equal standard annuity for the compounding period

How much is accumulated over 20 years in a fund that pays 4% interest, compounded yearly, if $1,000 is deposited at the end of every fourth year?

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$1000

F? =

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Non-standard Annuities and Non-standard Annuities and Gradients-Gradients- Examples Examples

Method 1: consider each cash flows separately

F = 1000 (F/P,4%,16) + 1000 (F/P,4%,12) + 1000 (F/P,4%,8) + 1000 (F/P,4%,4) + 1000 = $7013

Method 2: convert the compounding period from annual to every four years

ie = (1+0.04)4-1 = 16.99% F = 1000 (F/A, 16.99%, 5) = $7013Method 3: convert the annuity to an

equivalent yearly annuity A = 1000(A/F,4%,4) = $235.49 F = 235.49 (F/A,4%,20) = $7012

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