INDR 202
ENGINEERING ECONOMICS
CHAPTER 3
MONEY MANAGEMENT
SPRING 2015
INSTRUCTOR: BORA ÇEKYAY
MONEY MANAGEMENT
2
Interest Rates: Nominal vs. Effective Calculating Effective Interest Rate
Understanding Money and Its
Management – Main Focus
1. If payments occur more frequently than annual, how do you calculate economic equivalence?
2. If interest period is other than annual, how do you calculate economic
equivalence?
Nominal Versus Effective Interest Rates
q
Nominal Interest
Rate:
Interest rate quoted
based on an annual
period
q
Effective Interest
Rate:
Actual interest earned
or paid in a year or
NOMINAL VS. EFFECTIVE INTEREST
5
Financial institutions often quote interest rate as
annual percentage rate (APR). [NOMINAL]
Financial analysis of situations where interest is not compounded annually requires converting
APR into annual percentage yield (APY) based on
payment period. [EFFECTIVE]
Calculating effective interest rate is necessary
NOMINAL VS. EFFECTIVE INTEREST
6
Nominal Interest Rate (APR) interest rate quoted on annual basis
Effective Annual Interest Rate (APY) actual interest earned (paid) in a year
NOMINAL VS. EFFECTIVE INTEREST
Financial Jargon
7
15%
compounded monthly
Nominal
Interest Rate
APR
NOMINAL VS. EFFECTIVE INTEREST
8
15% compounded monthly
𝑀 = 12 interest periods per year
Monthly interest rate:
𝑖 = 15%
12 = 1.25%
1.25% interest charged each month on unpaid balance of borrowed money
NOMINAL VS. EFFECTIVE INTEREST
9
15% compounded monthly
1.25% 1.25% 1.25% 1.25% 1.25% 1.25% 1.25% 1.25% 1.25% 1.25% 1.25% 1.25%
𝐹 = $1 1 + 𝑖 ,- = $1 1 + 0.0125 ,- = $1.160755
𝑖1 = 16.0755% (APY)
Question: Suppose that you invest $1 for 1 year at 15%
NOMINAL VS. EFFECTIVE INTEREST
10
15% compounded monthly
1.25% 1.25% 1.25% 1.25% 1.25% 1.25% 1.25% 1.25% 1.25% 1.25% 1.25% 1.25%
Effective interest rate per year = 16.0755% 16.0755% compounded annually
EFFECTIVE ANNUAL INTEREST RATE
11
𝑟 nominal annual interest rate
𝑀 number of interest periods per year
𝑖1 effective annual interest rate (APY)
𝑖1 = 1 + 𝑟
𝑀
9
− 1
Practice Problem
Suppose your savings
account pays 9% interest compounded quarterly.
(a) Interest rate per quarter
(b) Annual effective interest rate (ia)
(c) If you deposit
$10,000 for one year, how much would you have?
Solution:
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
= =
EXAMPLE 1: CREDIT CARD APR
13
18% APR compounded monthly
Monthly effective interest rate
18%
12 = 1.5%
Interest charged for skipping payments for 3 months for a balance of $10,000
$10,000 1 + 0.015 = − 1 = $456.7837
Effective annual interest rate
𝑖1 = 1 + 18% 12
ANNUAL PERCENTAGE YIELD COMPOUNDING FREQUENCY APR % Annual % Semi-Annual % Quarterly % Monthly % Daily %
4 4 4.04 4.06 4.07 4.08
5 5 5.06 5.09 5.12 5.13
6 6 6.09 6.14 6.17 6.18
7 7 7.12 7.19 7.23 7.25
8 8 8.16 8.24 8.30 8.33
9 9 9.20 9.31 9.38 9.42
10 10 10.25 10.38 10.47 10.52
11 11 11.30 11.46 11.57 11.62
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 from each other, one or the other must be transformed so that
EFFECTIVE INTEREST RATE
16
𝑟 nominal annual interest rate
𝐶 number of interest periods per payment period
𝐾 number of payment periods per year
𝑀 = 𝐶𝐾 number of interest periods per year
𝑖 effective interest rate per payment period
𝑖 = 1 + 𝑟
𝐶𝐾
C
EFFECTIVE INTEREST RATE
17
12% compounded monthly
Payment Period = QUARTERLY
Interest Period = MONTHLY
1% 1% 1%
Quarter 1 Quarter 2 Quarter 3 Quarter 4
1% 1% 1% 1% 1% 1% 1% 1% 1%
EFFECTIVE INTEREST RATE
18
Effective interest rate per quarter
𝑖 = 1 + 12% 3 4
=
− 1 = 3.03%
Effective annual interest rate
𝑖1 = 1 + 12% 12
,-− 1 = 12.68%
EXAMPLE 3: CASE 0
19
8% compounded quarterly
Payment Period = QUARTERLY
Interest Period = QUARTERLY
Quarter 1 Quarter 2 Quarter 3 Quarter 4
𝑟 = 8%
𝐾 = 4 payments per year
𝐶 = 1 interest period per quarter 𝑀 = 4 interest periods per year
𝑖 = 1 + E%
, D
,
EXAMPLE 3: CASE 1
20
8% compounded monthly
Payment Period = QUARTERLY
Interest Period = MONTHLY
Quarter 1 Quarter 2 Quarter 3 Quarter 4
𝑟 = 8%
𝐾 = 4 payments per year
𝐶 = 3 interest period per quarter 𝑀 = 12 interest periods per year
𝑖 = 1 + E%
= D
=
EXAMPLE 3: CASE 2
21
8% compounded weekly
Payment Period = QUARTERLY
Interest Period = WEEKLY
Quarter 1 Quarter 2 Quarter 3 Quarter 4
𝑟 = 8%
𝐾 = 4 payments per year
𝐶 = 13 interest period per quarter 𝑀 = 52 interest periods per year
𝑖 = 1 + E%
,= D
,=
CONTINUOUS COMPOUNDING
22
Effective interest rate per payment period:
K: Number of payment period per year.
𝑖 = lim
C→J 1 +
𝑟
𝐶𝐾
C
Effective Interest Rate per Payment Period with Continuous Compounding
q Formula: With
continuous compounding Example: 12% compounded
continuously
(a) effective interest rate per quarter
(b) effective annual interest rate
C → ∞
/
lim 1 1
1 r K C C r i CK e →∞ ⎡ ⎤ = ⎢ + ⎥ − ⎣ = − ⎦ 0.12/4 1
3.045% per quarter
i e= −
=
0.12/1
1
12.75% per year
a
i
=
e
−
EXAMPLE 3: CASE 3
24
8% compounded continuously
Payment Period = QUARTERLY
Interest Period = CONTINUOUSLY
Quarter 1 Quarter 2 Quarter 3 Quarter 4
𝑟 = 8%
𝐾 = 4 payments per year
𝐶 = ∞ interest period per quarter 𝑀 = ∞ interest periods per year
EXAMPLE 3: SUMMARY
25
EFFECTIVE QUARTERLY INTEREST RATE
CASE 0 CASE 1 CASE 2 CASE 3
8% compounded quarterly 8% compounded monthly 8% compounded weekly 8% compounded continuously quarterly
payments paymentsquarterly paymentsquarterly paymentsquarterly
EXAMPLE 4: CASE 1
26
Savings account with 8% APR compounded weekly
Effective quarterly interest rate
𝑖 = 1 + 0.08 52
,=
− 1 = 2.0186%
Interest earned in 3 years for $10,000
EXAMPLE 4: CASE 2
27
8% APR compounded daily
Effective quarterly interest rate
𝑖 = 1 + 0.08 365
=QR/D
− 1 = 2.0199%
Interest earned in 3 years for $10,000
EXAMPLE 4: CASE 3
28
8% APR compounded continuously
Effective quarterly interest rate
𝑖 = 𝑒 P.PE/D − 1 = 2.0201%
Interest earned in 3 years for $10,000
EXAMPLE 5: LOAN
29
6% compounded monthly Monthly interest rate
6%
12 = 0.5%
Effective annual interest rate
𝑖1 = 1 + 0.06
12
,-− 1 = 6.168%
Effective quarterly interest rate
𝑖 = 1 + 0.06
12
=
EXAMPLE 5: LOAN
30
Interest charged in 3 years for 40,000 TL
40,000 1 + 0.005 =Q − 1 = 7,867.221
40,000 1 + 0.06168 = − 1 = 7,867.517
EQUIVALENCE CALCULATIONS
31
Identify the interest period.
Identify the payment period.
Compute effective interest rate that covers the
EFFECTIVE INTEREST RATE
32
CASE 1: interest period = payment period
CASE 2: interest period < payment period
INTEREST PERIOD = PAYMENT PERIOD
33
STEP 1: Identify the number of interest periods per year. STEP 2: Compute the effective interest rate per payment period.
STEP 3: Find the total number of payment periods.
EXAMPLE 6: AUTO LOAN
34
STEP 1: 𝑀 = 12
STEP 2:
𝑖 =
E.R%,-
= 0.7083%
per month
STEP 3: 𝑁 = 48 months
STEP 4: 𝐴 = $20,000 𝐴|𝑃, 0.7083%, 48 = $492.97
48 0
1 2 3 4
$20,000
𝐴
𝑟 = 8.5% compounded monthly
EXAMPLE 7: SAVINGS FROM NOT SMOKING
35
Assume one pack of cigarettes costs 10 TL & the interest rate is 4% compounded weekly. How much would each smoker have at the end of 10 years if she/he invested the money instead of buying cigarettes?
Level of smoker Would have had 1 pack a day
2 packs a day
3 packs a day
44,732.32
89,464.64
EXAMPLE 7: SAVINGS FROM NOT SMOKING
36
STEP 1: 𝑀 = 52
STEP 2:
𝑖 =
D%R-
= 0.0769%
per week
STEP 3: 𝑁 = 520 weeks
STEP 4: 𝐹 = 70 𝐹|𝐴, 0.0769%, 520 = 44,732.32
INTEREST PERIOD < PAYMENT PERIOD
37
STEP 1: Identify the numbers of interest periods per
year, of payment periods per year, of interest periods per payment period.
STEP 2: Compute the effective interest rate per payment period.
STEP 3: Find the total number of payment periods.
EXAMPLE 8: CASE 1
38
Balance at the end of 3 years if quarterly deposits of
$1,000 are made in a fund with interest rate 12%
compounded monthly
STEP 1: 𝑀 = 12, 𝐾 = 4, 𝐶 = 3
STEP 2: 𝑖 = 1 + ,-%
,-=
− 1 = 3.03% per quarter
STEP 3: 𝑁 = 12 quarters
EXAMPLE 8: CASE 2
39
Balance at the end of 3 years if quarterly deposits of
$1,000 are made in a fund with interest rate 12%
compounded continuously
STEP 1: 𝑀 = ∞, 𝐾 = 4, 𝐶 = ∞
STEP 2: 𝑖 = 𝑒 P.,-/D − 1 = 3.045% per quarter
STEP 3: 𝑁 = 12 quarters
INTEREST PERIOD > PAYMENT PERIOD
40
If compounding does not start until the next interest period after the payment:
STEP 1: Identify the numbers of interest periods per year, of payment periods per year, of payment periods per interest period.
STEP 2: Compute the effective interest rate per interest
period.
STEP 3: Find the total number of interest periods.
EXAMPLE 9: CASE 1
41
Balance at the end of 10 years if monthly deposits of
$500 are made in an account with interest rate 10%
compounded quarterly
STEP 1: 𝑀 = 4, 𝐾 = 12, 𝐵 = 3 STEP 2: 𝑖 = ,P%
D = 2. 5% per quarter
STEP 3: 𝑁 = 40 quarters
INTEREST PERIOD > PAYMENT PERIOD
42
If compounding starts immediately after the payment:
STEP 1: Identify the numbers of interest periods per year, of payment periods per year, of interest periods per payment period.
STEP 2: Compute the effective interest rate per payment
period.
STEP 3: Find the total number of payment periods.
EXAMPLE 9: CASE 2
43
Balance at the end of 10 years if monthly deposits of
$500 are made in an account with interest rate 10%
compounded quarterly
STEP 1: 𝑀 = 4, 𝐾 = 12, 𝐶 = ,
=
STEP 2: 𝑖 = 1 + ,P%
D
,/=
− 1 = 0. 826% per month
STEP 3: 𝑁 = 120 months
EXAMPLE 10: VARYING INTEREST RATES
44
Payment period: annual
$150
$450 $450
8% compounded annually
0 1 2 3 4
𝐹
6% compounded monthly
EXAMPLE 10: VARYING INTEREST RATES
45
Years 0-2: 𝑟 = 8%
𝑖1 = 8%
𝐵- = $450 𝐹|𝑃, 8%, 2 + $150 = $674.88
Years 2-4: 𝑟 = 6%
𝑖1 = 1 + 6% 12
,-− 1 = 6. 168%
DEBT MANAGEMENT
46
Credit Cards
The total cost depends on annual fees, APR, grace period, credit & payment amounts, finance charges.
Loans
EXAMPLE 6: AUTO LOAN
47
STEP 1: 𝑀 = 12 STEP 2: 𝑖 = E.R%
,- = 0.7083% per month
STEP 3: 𝑁 = 48 months
STEP 4: 𝐴 = $20,000 𝐴|𝑃, 0.7083%, 48 = $492.97
48 0
1 2 3 4
$20,000
𝐴
𝑟 = 8.5% compounded monthly
EXAMPLE 6: AUTO LOAN
48
Remaining balance after 32nd payment
𝑃 = 𝐵=- = $492.97 𝑃|𝐴, 0.7083%, 16 = $7,431.12
Interest component of 33rd payment
𝐼== = $7,431.12 0.007083 = $52.76
Principal component of 33rd payment
P== = $492.97 − $52.76 = $440.21
Example 4.13 Loan Balance, Principal, and Interest: Remaining –Balance Method
qGiven: P = $5,000, i = 12%
APR compounded monthly,
N = 24 months
q Find: Loan balance,
principal, and interest payment for the 6th
payment
A = $5,000(A/P, 1%, 24) = $235.37
Example 4.13 Continued
To compute I6, we first need to find B5,
B5 = $235.37 x (P/A, 1%, 19) = $ 4,054.44
Then, I6 = $ 4,054.44 x 0,01 = $ 40,54. Note that the principal payment is the remaining part of the total monthly payment amount $235.37. Thus,
P6 = $235.37 – 40.54 = $194.83
The remaining balance after the 6th payment, B
6, is equal to $3,859.62 as
EXAMPLE 11: BUYING VS. LEASING A CAR
51
DEBT FINANCING LEASE FINANCING
Price $14,695 $14,695
Down payment $2,000
APR 3.6%
Monthly payment $372.55 (end of month)
(12695(A/P,0.003,36))
$236.45
(beginning of month)
Length 36 months 36 months
Fees $495
Cash due at lease end $300
Purchase option at lease end $8,673.10
EXAMPLE 11: BUYING VS. LEASING A CAR
52
Cost of Debt Financing
𝑃YZ = $2,000 + $372.55 𝑃|𝐴, 0.5%, 36 − $8,673.10 𝑃|𝐹, 0.5%, 36
𝑃YZ = $6,998.47
Cost of Lease Financing
𝑃[Z = $731.45 + $236.45 𝑃|𝐴, 0.5%, 35 + $300 𝑃|𝐹, 0.5%, 36
𝑃[Z = $8,556.90
EXAMPLE 11: BUYING VS. LEASING A CAR
53
Let 𝑆 be the resale value at the end of 36 months.
Cost of Debt Financing
𝑃YZ = $2,000 + $372.55 𝑃|𝐴, 0.5%, 36 − 𝑆 𝑃|𝐹, 0.5%, 36
Cost of Lease Financing: 𝑃[Z = $8,556.90
Debt financing if 𝑆 > $6,808.14
SUMMARY
54
Interest is typically compounded more frequently than annually even though quoted on annual basis.
Nominal interest rate is the stated rate for a given time period (often a year).
Effective interest rate is the actual interest charged. Economic equivalence calculations require using effective interest rates rather than nominal ones.
