Transcript Document
Chapter 5
The Time Value of Money
Chapter Objectives
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
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
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
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
How much a sum will grow in a certain
number of years when compounded at a
specific rate.
Future Value
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
Future Value can be increased by:
– Increasing number of years of compounding
– Increasing the interest or discount rate
Future Value
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
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
$1,000 X .621 = $621
$621 today equals $1,000 in 5 years
Annuity
Series of equal dollar payments for a
specified number of years.
Ordinary annuity payments occur at the end
of each period
Compound Annuity
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
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
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
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
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
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?
Present Value table, five time periods, 6%
column factor is 4.2124
$100 X 4.2124 = 421.24
Which option will you choose?
$450 today or $100 a year for five years
Annuities Due
Ordinary annuities in which all payments
have been shifted forward by one time
period.
Amortized Loans
Loans paid off in equal installments over
time
– Typically Home Mortgages
Payments and Annuities
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
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
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
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
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