Kx Ex 2162 Lecture 11

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1

Understand nominal and effective Interestrates statements

Determine equivalence calculations fordifferent payment and compoundingperiods

How commercial loans and mortgages arestructured in terms of interest andprincipal payments.

The basic of investing in financial assets

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Nominal Versus Effective Interest Rates

Nominal InterestRate:Interest rate quoted

based on an annualperiod.( Annual percentagerate-APR )

Effective InterestRate:Actual interest earned

or paid in a year orsome other timeperiod

Ex: 18% compoundedmonthly

12 12$1(1 ) $1(1 0.015)

$1.1956

$1.1956 $1.00 $0.1956

F i

I

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Effective Annual Interest Rate (Yield) Formula:

r = nominal interest rate per yeari a = effective annual interest rate

M = number of interest periods peryear

Example :18% compoundedmonthly

What It really Means

1.5% per month for 12months or19.56% compoundedonce per year

(1 ) 1M

a

r i M

120.18

1 1 19.56%12a

i

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Practice

ProblemSuppose your savingsaccount pays 9% interestcompounded quarterly .(a) Interest rate per

quarter(b) Annual effective

interest rate ( i a)(c) If you deposit

$10,000 for one year,

how much wouldyou have?

Solution:

Contemporary Engineering Economics, 5th edition, 2010

4

(a) Interest rate per quarter:

9% 2.25%

4(b) Annual effective interest rate:

(1 0.0225) 1 9.31%

(c) Balance at the end of one year (after 4 quarters)

$10,000( / ,2.

a

i

i

F F P

25%,4)

$10,000( / ,9.31%,1)

$10,931

F P

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Nominal and Effective Interest Rates withDifferent Compounding Periods

Effective RatesNominal

RateCompounding

AnnuallyCompoundingSemi-annually

CompoundingQuarterly

CompoundingMonthly

CompoundingDaily

4% 4.00% 4.04% 4.06% 4.07% 4.08%

5 5.00 5.06 5.09 5.12 5.136 6.00 6.09 6.14 6.17 6.18

7 7.00 7.12 7.19 7.23 7.25

8 8.00 8.16 8.24 8.30 8.33

9 9.00 9.20 9.31 9.38 9.42

10 10.00 10.25 10.38 10.47 10.52

11 11.00 11.30 11.46 11.57 11.62

12 12.00 12.36 12.55 12.68 12.74

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Why Do We Need an Effective Interest Rate

per Payment Period?


Payment period

Interest period

Payment period

Interest period

Whenever payment and compounding periods differ fromeach other, one or the other must be transformed so thatboth conform to the same unit of time.

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Effective InterestRate per Payment

Period ( i ) Formula:

C = number ofinterest periods perpayment period

K = number ofpayment periods peryear

CK =total number ofinterest periods peryear, or M

r / K = nominal interestrate per payment period

Functional Relationships among r , i , and i a , where interest is calculated based on 9% compoundedmonthly and payments occur quarterly


1 1C

r i

CK

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Effective Interest Rate perPayment Period with

Continuous Compounding Formula: With continuous

compoundingExample : 12% compoundedcontinuously

(a) effective interest rate per quarter

(b) effective annual interest rate


C

/

lim 1 1

1r K

C

C

r i

CK

e

0.12/4 1

3.045% per quarter

i e

0.12/1 1

12.75% per yearai e

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Case 0 : 8% compounded quarterly

Payment Period = QuarterInterest Period = Quarterly

1 interest period Given r = 8%,

K = 4 payments per yearC = 1 interest period per quarterM = 4 interest periods per year

2nd Q 3 rd Q 5th Q1st Q

1

[1 / ] 1

[1 0.08 / (1)(4)] 1

2.000% per quarter

C i r CK

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Case 1 : 8% compounded monthly

Payment Period = QuarterInterest Period = Monthly

3 interest periods Given r = 8%,K = 4 payments per yearC = 3 interest periods per quarterM = 12 interest periods per year


3

[1 / ] 1

[1 0.08 / (3)(4)] 1

2.013% per quarter

C i r CK

8/13/2019 Kx Ex 2162 Lecture 11


Case 2 : 8% compounded weekly

Payment Period = QuarterInterest Period = Weekly

13 interest periods Given r = 8%,K = 4 payments per yearC = 13 interest periods per quarterM = 52 interest periods per year


i r CK C

[ / ]

[ . / ( )( )]

.

1 1

1 0 08 13 4 1

2 0186%

13

per quarter

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Case 3 : 8% compounded continuously

Payment Period = QuarterInterest Period = Continuously

interest periods Given r = 8%,K = 4 payments per year


/

0.02

1

1

2.0201% per quarter

r K i e

e

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Summary: Effective Interest Rates per Quarter atVarying Compounding Frequencies

Case 0 Case 1 Case 2 Case 3

8%compoundedquarterly

8%compoundedmonthly

8%compoundedweekly

8%compoundedcontinuously

Payments occurquarterly




2.000% perquarter

2.013% perquarter

2.0186% perquarter

2.0201% perquarter

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Equivalence Analysis Using Effective Interest Rates.

Frequency of cash flows may or may notequal the frequency of interest compounding If the frequency of the cash flow equals thefrequency of the interest compounding NoProblem!When payment period and compoundingperiod differ, calculate an effective interestrate that covers the payment period. Thenuse the appropriate interest formulas todetermine the equivalent values

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Equivalence Calculations using Effective

Interest RatesStep 1: Identify the payment period (e.g., annual,quarter, month, week, etc)

Step 2: Identify the compounding period (e.g.,annually, quarterly, monthly, etc)

Step 3: Find the effective interest rate that coversthe payment period.


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Dollars Down

in the DrainSuppose you drink a cupof coffee ($3.00 a cup) onthe way to work everymorning for 30 years. If youput the money in the bank

for the same period, howmuch would you have,assuming your accountsearns a 5% interestcompounded daily.

NOTE: Assume you drink acup of coffee every dayincluding weekends.

Solution:Payment period = dailyCompounding period = daily


5%0.0137% per day

36530 365 10,950 days

$3( / ,0.0137%,10950)

$76,246

i

N

F F A

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Case II: When Payment Periods Differ from

Compounding PeriodsStep 1: Identify the following parameters.

M = No. of compounding periodsK = No. of payment periods per yearC = No. of interest periods per payment period

Step 2: Compute the effective interest rate per paymentperiod.

For discrete compounding

For continuous compounding

Step 3: Find the total no. of payment periods.N = K (no. of years)

Step 4: Use i and N in the appropriate equivalence formula.


[1 / ] 1C i r CK

/ 1r K i e

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Compounding Occurs More Frequently than Payments are Made .

Given : A = $1,500 per quarter, r = 6% per year, M = 12compounding periods per year,and N = 2 years

Find: F

Step 1: M = 12 compoundingperiods/year K = 4 payment

periods/year C = 3 interest periods

per quarter

Step 2:

Step 3: N = 4(2) = 8

F = $1,500 ( F/A , 1.5075%, 8)= $14,216.24


30.06

1 112

1.5075% per quarter

i

Suppose you make equal quarterly deposits of $1500 into a fund thatpays interest at a rate of 6% compounded monthly. Find the balance

at the end of year 2.

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A Decision Flow Chart on How to Compute the

Effective Interest Rate per Payment Period


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PracticeProblem

If you invest$1,000 in a savingsaccount that pays

6% annual interestcompoundedcontinuously, whatwould be the

balance at the endof 3 years?


0.06 1

6.18%

$1,000( / , 6.18%,3)$1,197.09

ai e

F F P

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Single-Payment Transactions with Continuous

Compounding Present Worth

F

0

N

P

(1 )(1 1)

N

r N

rN

P F i

F e

Fe

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Continuous-Funds Flow


0

0

( )

( )

rt

t

Nrt

P f t t e

f t e dt

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Exercise: Daily Flows and Daily Compounding with ContinuousFlows and Continuous Compounding

Consider a situation in which money flows daily.Suppose you own a retail shop and generate $200cash each day. You establish a special businessaccount and deposit your daily cash flows in an

account for 15 months. This account earns aninterest rate at 6%. Compare the accumulated cashvalues at the end of 15 months a) daily compoundingb) Continuous compounding.


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Given : A = $200 perday, r = 6% per year, M =365 compoundingperiods per year, and N =455 days

Find: F

Solution


Note: A15-month period

Is 1.25 years .

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Example 4.12 LoanBalance, Principal, andInterest: TabularApproach

Given : P = $5,000, i = 12% APR,N = 24 months

Find: A, and loan repayment

schedule

A = $5,000(A/P, 1%, 24) = $235.37


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Calculating the Remaining Loan Balance after

Making the n th Payment


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Example 4.13 LoanBalance, Principal, and

Interest: Remaining Balance Method

Given : P = $5,000, i = 12%APR, N = 24 months

Find: Loan balance, principal,and interest payment for the 6 th payment

A = $5,000(A/P, 1%, 24) = $235.37


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Example 4.15 Financing your VehicleGiven : Three financing options, r = 4.5%, paymentperiod = monthly, and compounding period =monthly Find: Which option?

Issue : Which interest rate to use in calculating theequivalent cost of financing for each option

Option A : Conventional Debt Financing:

P debt = $4,500 + $736.53( P/A , 4.5%/12, 42)- $17,817( P/F , 4.5%/12, 42)

= $17,847

Option B : Cash Financing:

P cash = $31,020 - $17,817( P / F ,4.5%/12,42)

= $15,845

Option C : Lease Financing:

P lease = $1,507.76 + $513.76( P/A , 4.5%/12, 42)+ $395( P/F , 4.5%/12, 42)

= $21,336


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Kx Ex 2162 Lecture 11

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