1. Compound Interest and Present Value
Chapter Twelve
Compound Interest and Present Value
McGraw-Hill/Irwin
Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.
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2. Learning unit objectives
LU 12-1 Compound Interest (Future Value) – The Big Picture
Compare simple interest with compound interest.
Calculate the compound amount and interest manually and by table lookup.
Explain and compute the effective rate (APY).
LU 12-2 Present Value -- The Big Picture
Compare present value (PV) with compound interest (FV).
Compute present value by table lookup.
Check the present value answer by compounding.
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3. Compounding Interest (Future Value)
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) –
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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4. Compounding Terms Compounding Periods Interest Calculated
Compounding Annually
Compounding Semiannually
Compounding Quarterly
Compounding Monthly
Compounding Daily
Once a year
Every 6 months
Every 3 months
Every month
Every day
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5. Future Value of $1 at 8% for Four Periods (Figure 12.1)
Compounding goes from present value to future value
Future Value
After 4 periods, $1 is worth $1.36
After 1 period, $1 is worth $1.08
After 2 periods, $1 is worth $1.17
After 3 periods, $1 is worth $1.26
Present
value
$1.00
$1.08
$1.1664
$1.2597
$1.3605
Number of periods
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6. Future Value of $1 at 8% for Four Periods (Figure 12.1)
Manual Calculation
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7. Tools for Calculating Compound Interest
Number of periods (N) Number of years multiplied by 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 4 years at 8% annually,
semiannually, or quarterly, what is N and R?
Periods
Rate
Annually:
Semiannually:
Quarterly:
4 x 1 = 4
Annually:
Semiannually:
Quarterly:
8% / 1 = 8%
4 x 2 = 8
8% / 2 = 4%
4 x 4 = 16
8% / 4 = 2%
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8. Simple Versus Compound Interest
Compounded
Bill Smith deposited $80 in a savings account for 4 years at an annual interest rate of 8%. What is Bill’s simple interest and maturity value?
Bill Smith deposited $80 in a savings account for 4 years at an annual interest rate of 8%. What is Bill’s interest and compounded amount?
I = P x R x T
I = $80 x .08 x 4
I = $25.60
MV = $80 + $25.60
MV = $105.60
Interest: $ $80.00 = $28.83
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9. Calculating Compound Amount by Table Lookup
Step 1. Find the periods: Years multiplied by number of times interest is compounded in 1 year.
Step 2. Find the rate: Annual rate divided by number of times interest is compounded in 1 year.
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 for the compound amount of $1.
Step 4. Multiply the table factor by the amount of the loan. This gives the compound amount.
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10. Future Value of $1 at Compound Interest (Table 12.1)
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11. Calculating Compound Amount by Table Lookup
Pam Donahue deposits $8,000 in her savings account that pays 6% interest compounded quarterly. What will be the balance of her account at the end of 5 years?
Periods (N) = 4 x 5 = 20
Rate (R) = 6%/4 = 1.5%
Table Factor =
Compounded Amount:
$8,000 x = $10,775.20
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12. Nominal and Effective Rates (APY) of Interest
Truth in
Savings
Law Annual
Percentage
Yield
Nominal Rate (stated rate) –
The rate on which the bank calculates interest
Effective rate (APY) = Interest for 1 year
Principal
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13. Calculating Effective Rate APY
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14. Nominal and Effective Rates (APY) of Interest Compared (Figure 12.3)
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15. Compounding Interest Daily (Table 12.2)
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16. Compounding Interest Daily
Use Table 12.2 to calculate what $1,500 compounded daily for
5 years will grow to at 7%.
N = 5
R = 7%
Factor,
$1,500 x = $2,128.50
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17. Present Value of $1 at 8% for Four Periods (Figure 12.4)
Present value goes from the future value to the present value
Future Value
$1.0000
Present
value
$.9259
$.8573
$.7938
$.7350
Number of periods
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18. Calculating Present Value by Table Lookup
Step 1. Find the periods: Years multiplied by number of times interest is compounded in 1 year.
Step 2. Find the rate: Annual rate divided by number of times interest is compounded in 1 year.
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 of the two columns is the table factor for the compound value of $1.
Step 4. Multiply the table factor by the future value. This is the present value.
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19. Present Value of $1 at End Period (Table 12.3)
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20. Comparing Compound Interest (FV) (Table 12
Comparing Compound Interest (FV) (Table 12.1) with Present Value (PV) (Table 12.3)
Compound value Table 12.1
Table Present Future
Value Value
x $80 = $108.84
(N = 4, R = 8%)
Present value Table 12.3
Table Future Present
12.3 Value Value
x $ = $80.00
(N = 4, R = 8%)
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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21. Calculating Present Value Amount by Table Lookup
Rene Weaver needs $20,000 for college in 4 years. She can earn 8% compounded quarterly at her bank. How much must Rene deposit at the beginning of the year to have $20,000 in 4 years?
Invest
Today
Periods (N) = 4 x 4 = 16
Rate (R) = 8%/4 = 2%
Table Factor = .7284
Compounded Amount:
$20,000 x = $14,568
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