1. Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

2. Prepared by Dr. Elena Skliarenko
Chapter 14
Compound Interest:
Present Value; Future Value; Equivalent Payments; Effective, Nominal, and Equivalent Interest Rates
Prepared by Dr. Elena Skliarenko
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3. Present the timing and amount of payments using a time line diagram
#14
Compound Interest
LU14.1
Learning Unit Objectives
Basic Concepts, Terminology, and Time Line Diagrams
Present the timing and amount of payments using a time line diagram
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4. #14 Learning Unit Objectives Compound Interest
LU14.2
Future Value – The Big Picture
Compare simple interest with compound interest
Calculate compound amount and interest manually
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5. #14 Learning Unit Objectives Compound Interest
LU14.3
Present Value – The Big Picture
Compare present value (PV) with compound interest (FV)
Compute present value
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6. Learning Unit Objectives
#14
Compound Interest
Learning Unit Objectives
LU14.4
Equivalent Payments
Compute the equivalent value of a single payment or serial payments on any date
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7. #14 Learning Unit Objectives Compound Interest Effective Interest Rate
LU14.5
Effective Interest Rate
Explain and compute the effective rate
Define an effective interest rate when a nominal interest rate is given
Copyright © 2005 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved.

8. #14 Learning Unit Objectives Compound Interest
LU14.6
Equivalent Interest Rates
Compute the equivalent interest rate when nominal interest rate is given
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9. Compounding Interest (Future Value)
Compounding - involves the calculation of interest periodically over the life of the loan or investment
Compound interest - the interest on the principal plus the interest of prior periods
Future value (compound amount) - is the final amount of the loan or investment at the end of the last period
Present value - the value of a loan or investment today
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10. Future Value of $1 at 8% for Four Periods
Compounding goes from present value to future value
Future Value
After 4 periods $1 is worth $1.36
After 3 periods $1 is worth $1.26
After 2 periods $1 is worth $1.17
After 1 period $1 is worth $1.08
Present
value
$1.1664
$1.2597
$1.3605
$1.00
$1.08
Number of periods
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11. Compounding Terms Compounding Periods Interested Calculated
Compounding Annually Once a year
Compounding Semiannually Every 6 months
Compounding Quarterly Every 3 months
Compounding Monthly Every month
Compounding Daily Every day
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12. Simple Versus Compound Interest
Compounded
Al Jones deposited $1,000 in a savings account for 5 years at an annual simple interest rate of 10%. What is Al’s simple interest and maturity value?
Al Jones deposited $1,000 in a savings account for 5 years at an annual compounded rate of 10%. What is Al’s interest and compounded amount?
I = P x R x T
I = $1,000 x .10 x 5
I = $ MV = $1,000 + $500
MV = $1,500
Compound amount $1, $1,000 = $610.51
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13. Tools for Calculating Compound Interest
Number of periods (N) Number of years times the number of times the interest is compounded per year
Rate for each period (R) Annual interest rate divided by the number of times the interest is compounded per year
If you compounded $100 for 3 years at 6% annually, semiannually, or quarterly What is N and R?
Periods
Rate
Annually: 3 x 1 = 3
Semiannually: 3 x 2 = 6
Quarterly: 3 x 4 = 12
Annually: 6% / 1 = 6%
Semiannually: 6% / 2 = 3%
Quarterly: % / 4 = 1.5%
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14. Calculating Compound Amount by Table Lookup
Step 4. Multiply the table factor by the amount of the loan.
Step 3. Go down the period column of the table to the number desired; look across the row to find the rate. At the intersection is the table factor
Step 2. Find the rate: Annual rate divided by number of times interest is compounded in 1 year
Step 1. Find the periods: Years multiplied by number of times interest is compounded in 1 year
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15. Future Value of $1 at Compound Interest
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16. Calculating Compound Amount by Table Lookup
Steve Smith deposited $1,000 in a savings account for 4 years at an semiannual compounded rate of 8%. What is Steve’s interest and compounded amount?
N = 4 x 2 = 8
R = 8% = 4% 2
Table Factor =
Compounded Amount:
$1,000 x = =$1,368.60
I = $1, $1,000 = =$368.60
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17. Nominal and Effective Rates of Interest
Nominal Rate (Stated Rate) - The rate on which the bank calculates interest.
Effective Rate = Interest for 1 year
Principal
Truth in
Savings
Law Annual
Percentage
Yield
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18. Nominal and Effective Rates of Interest Compared
Beginning Nominal rate Compounding End Effective rate
balance of interest period balance of interest
Annual
Semiannual
Quarterly
Daily
$1,060.00
$1,060.90
$1,061.40
$1,061.80
6.00
6.09%
6.14%
6.18%
$1,000
+ 6%
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19. Compounding Interest Daily
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20. Compounding Interest Daily
Calculate what $2,000 compounded daily for 7 years will grow to at 6%
N = 7
R = 6%
Factor
$2,000 x = $3,043.80
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21. Present Value of $1 at 8% for Four Periods
Future Value
Present value goes from the future value to the present value
$1.0000
$.9259
Present
value
$.8573
$.7938
$.7350
Number of periods
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22. Calculating Present Value by Table Lookup
Step 4. Multiply the table factor by the future value. This is the present value.
Step 3. Go down the period column of the table to the number desired; look across the row to find the rate. At the intersection is the table factor.
Step 2. Find the rate: Annual rate divided by number of times interest is compounded in 1 year
Step 1. Find the periods: Years multiplied by number of times interest is compounded in 1 year
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23. Present Value of $1 at End Period
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24. Calculating Present Value Amount by Table Lookup
Steve Smith needs $1, in 4 years. His bank offers 8% interest compounded semiannually. How much money must Steve put in the bank today (present) to reach his goal in 4 years?
N = 4 x 2 = 8
R = 8% = 4% 2
Table Factor = .7307
Compounded Amount:
$1, x = $1,000.18
Invest
Today
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25. Comparing Compound Interest (FV) with Present Value (PV)
Compound value (FV) Present value (PV)
Table Present Future Table Future Present
Value Value Value Value
x $1,000 = $1, x $1, =$1,000
(N = 8, R = 4) (N = 8, R = 4)
We know the present dollar amount and find what the dollar amount is worth in the future
We know the future dollar amount and find what the dollar amount is worth in the present
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26. Short Cut Formulas
Periodic interest rate is an interest rate per one compounding period and is calculated using formula
To calculate nominal interest rate when periodic interest rate and compounding frequency are given, multiply these values:
j = i · m
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27. Short Cut Formulas
Where FV is the maturity value at the end of term of an investment or a loan,
PV is the present (initial) value or principle of an investment or a loan,
i – periodic interest
n – number of compounding periods during the term of the investment or loan
14.5
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28. Short Cut Formulas
You can find the formula for PV through formula rearrangement
14.7
A more common form of this formula may be obtained, using property of powers:
14.8
PV – present value FV – future value
n- number of compounding periods in the term i-periodic interest rate
Then
Where
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29. Equivalent Payments
We already know how to work with equivalent payments at simple interest. In this unit we will learn how to calculate equivalent payments at compound interest.
The approach is a very similar: it is a two-step procedure.
Step 1: Bring all the values of payments to the same point in time and calculate their equivalent values, using formulas 14.5 and 14.8
Step 2. Add them up and calculate the unknown replacement value
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30. Short Cut Formulas
There is a short-cut formula for calculation of an effective interest rate:
14.9
where f is the effective interest rate, m is the compounding frequency and i is the periodic interest rate.
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31. Short Cut Formulas
There is a short-cut formula for calculating equivalent interest rates.
To use this formula you have to remember the following two things:
To calculate nominal interest rate you should first calculate periodic interest rate. This is a two-step approach.
Using shortcut formula assign variables with code “1” to the given values and variables with code “2” to the values to be defined.
i2 = (1+i1)m1/m2 –
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