Transcript Document

Chapter 5
The Time Value of Money
Chapter Objectives
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Understand and calculate compound interest
Understand the relationship between compounding
and bringing money back to the present
Annuity and future value
Annuity Due
Future value and present value of a sum with nonannual compounding
Determine the present value of an uneven stream
of payments
Perpetuity
Understand how the international setting
complicates time value of money
Compound Interest
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When interest paid on an investment is
added to the principal, then during the next
period, interest is earned on the new sum
Simple Interest
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Interest is earned on principal
 $100 invested at 6% per year
 1st year interest is $6.00
 2nd year interest is $6.00
 3rd year interest is $6.00
 Total interest earned:$18
Compound Interest
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Interest is earned on previously earned interest
$100 invested at 6% with annual compounding
1st year interest is $6.00 Principal is $106
2nd year interest is $6.36 Principal is $112.36
3rd year interest is $6.74 Principal is $119.11
Total interest earned: $19.10
Future Value
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How much a sum will grow in a certain
number of years when compounded at a
specific rate.
Future Value
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What will an investment be worth in a year?
 $100 invested at 7%
 FV = PV(1+i)
 $100 (1+.07)
 $100 (1.07) = $107
Future Value
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Future Value can be increased by:
– Increasing number of years of compounding
– Increasing the interest or discount rate
Future Value
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What is the future value of $1,000 invested
at 12% for 3 years? (Assume annual
compounding)
 Using the tables, look at 12% column, 3
time periods. What is the factor?
 $1,000 X 1.4049 = 1,404.90
Present Value
What is the value in today’s dollars of a sum
of money to be received in the future ?
or
The current value of a future payment
Present Value
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What is the present value of $1,000 to be received
in 5 years if the discount rate is 10%?
 Using the present value of $1 table, 10% column,
5 time periods
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$1,000 X .621 = $621
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$621 today equals $1,000 in 5 years
Annuity
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Series of equal dollar payments for a
specified number of years.
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Ordinary annuity payments occur at the end
of each period
Compound Annuity
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Depositing or investing an equal sum of
money at the end of each year for a certain
number of years and allowing it to grow.
Compounding Annuity
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What will $1,000 deposited every year for
eight years at 10% be worth?
 Use the future value of an annuity table,
10% column, eight time periods
 $1,000 X 11.436 = $11,436
Future Value of an Annuity
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If we need $8,000 in 6 years (and the
discount rate is 10%), how much should be
deposited each year?
 Use the Future Value of an Annuity table,
10% column, six time periods.
 $8,000 / 7.716 = $1036.81 per year
Present Value of an Annuity
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Pensions, insurance obligations, and interest
received from bonds are all annuities.
These items all have a present value.
 Calculate the present value of an annuity
using the present value of annuity table.
Present Value of an Annuity
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Calculate the present value of a $100
annuity received annually for 10 years when
the discount rate is 6%.
 $100 X 7.360 = $736
Present Value of an Annuity
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Would you rather receive $450 dollars today
or $100 a year for the next five years?
 Discount rate is 6%.
 To compare these options, use present value.
 The present value of $450 today is $450.
 The present value of a $100 annuity for 5
years at 6% is XXX?
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Present Value table, five time periods, 6%
column factor is 4.2124
 $100 X 4.2124 = 421.24
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Which option will you choose?
 $450 today or $100 a year for five years
Annuities Due
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Ordinary annuities in which all payments
have been shifted forward by one time
period.
Amortized Loans
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Loans paid off in equal installments over
time
– Typically Home Mortgages
Payments and Annuities
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If you want to finance a new motorcycle with a
purchase price of $25,000 at an interest rate of 8%
over 5 years, what will your payments be?
 Use the present value of an annuity table, five time
periods, 8% column – factor is 3.993
 $25,000 / 3.993 = 6,260.96
 Five annual payments of $6,260.96
Amortization of a Loan
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Reducing the balance of a loan via annuity
payments is called amortizing.
 A typical amortization schedule looks at
payment, interest, principal payment and
balance.
Amortization Schedule
Amortize the payments on a 5-year loan for $10,000 at 6% interest.
N
1
2
3
4
5
Payment
$2,373.96
$2,373.96
$2,373.96
$2,373.96
$2,373.99
Interest
Prin. Pay
New Balance
(PxRxT)
(Payment Interest)
(Principal – Prin Pay)
$600
$493.56
$380.74
$261.15
$134.38
$1,773.96
$1,880.40
$1,993.22
$2,112.81
$2239.61
$8,226.04
$6,345.64
$4352.42
$2,239.61
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Mortgage Payments
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How much principal is paid on the first
payment of a $70,000 mortgage with 10%
interest, on a 30 year loan (with monthly
payments)
 Payment is $614
 How much of this payment goes to principal
and how much goes to interest?
 $70,000 x .10 x 1/12 = $583
 Payment of $614, $583 is interest, $31 is
applied toward principal
Compounding Interest with
Non-annual periods
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If using the tables, divide the percentage by the
number of compounding periods in a year, and
multiply the time periods by the number of
compounding periods in a year.
Example:
10% a year, with semi annual compounding for 5
years.
10% / 2 = 5% column on the tables
N = 5 years, with semi annual compounding or 10
Use 10 for Number of periods, 5% each
Non-annual Compounding
What factors should be used to calculate 5
years at 12% compounded quarterly
 N = 5 x 4 = 20
 % = 12% / 4 = 3%
 Use 3% column, 20 time periods
Perpetuity
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An annuity that continues forever is called
perpetuity
 The present value of a perpetuity is
 PV = PP/i
 PV = present value
 PP = Constant dollar amount of perpetuity
 i = Annuity discount rate
Future Value of $1 Table
N
1
2
3
4
5
6
7
8
9
10
6%
1.06
1.1236
1.1910
1.2625
1.3382
1.4185
1.5036
1.5938
1.6895
1.7908
8%
1.0800
1.1664
1.2597
1.3605
1.4693
1.5869
1.7138
1.8509
1.9990
2.1589
10%
1.1000
1.2100
1.3310
1.4641
1.6105
1.7716
1.9487
2.1436
2.3579
2.5937
12%
1.1200
1.2544
1.4049
1.5735
1.7623
1.9738
2.2107
2.4760
2.7731
3.1058
Present Value of $1
N
1
2
3
4
5
6
7
8
9
10
6%
.9434
.8900
.8396
.7921
.7473
.7050
.6651
.6274
.5919
.5584
8%
.9259
.8573
.7938
.7350
.6806
.6302
.5835
.5403
.5002
.4632
10%
.9091
.8264
.7513
.6830
.6209
.5645
.5132
.4665
.4241
.3855
12%
.8929
.7972
.7118
.6355
.5674
.5066
.4523
.4039
.3606
.3220
Future Value of Annuity
N
1
2
3
4
5
6
7
8
9
10
6%
1.000
2.060
3.1836
4.3746
5.6371
6.9753
8.3938
9.8975
11.4913
13.1808
8%
1.0000
2.0800
3.2464
4.5061
5.8666
7.3359
8.9228
10.6366
12.4876
14.4866
10%
1.000
2.100
3.310
4.6410
6.1051
7.7156
9.4872
11.4359
13.5795
15.9374
12%
1.000
2.1200
3.3744
4.7793
6.3528
8.1152
10.8090
12.2997
14.7757
17.5487
Present Value of an Annuity
N
1
2
3
4
5
6
7
8
9
10
6%
.9434
1.8334
2.6730
3.4651
4.2124
4.9173
5.5824
6.2098
6.8017
7.3601
8%
.9259
1.7833
2.5771
3.3121
3.9927
4.6229
5.2064
5.7466
6.2469
6.7101
10%
.9091
1.7355
2.4869
3.1699
3.7908
4.3553
4.8684
5.3349
5.7590
6.1446
12%
.8929
1.6901
2.4018
3.0373
3.6048
4.1114
4.5638
4.9676
5.3282
5.6502