time value of money module an electronic presentation by norman sunderman angelo state university an...
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Time Value of Money Module
An electronic presentation by Norman Sunderman Angelo State University
An electronic presentation by Norman Sunderman Angelo State University
COPYRIGHT © 2007 Thomson South-Western, a part of The Thomson Corporation. Thomson, the Star logo, and South-Western are trademarks used herein under license.
TVM
Intermediate AccountingIntermediate Accounting 10th edition 10th edition
Nikolai Bazley JonesNikolai Bazley Jones
2
Some of the accounting items to which these techniques maybe applied are:
1. Receivables and payables
2. Bonds
3. Leases
4. Pensions
5. Sinking funds
6. Asset valuations
7. Installment contracts
Uses of Time Value of Money
3
Simple interest is interest on the original principal regardless of
the number of time periods that have passed.
Simple interest is interest on the original principal regardless of
the number of time periods that have passed.
Interest = Principal x Rate x TimeInterest = Principal x Rate x Time
Simple Interest
4
Compound interest is the interest that accrues
on both the principal and the past unpaid
accrued interest.
Compound interest is the interest that accrues
on both the principal and the past unpaid
accrued interest.
Compound Interest
5
Value at Beginning of Quarter
Compound Interestx Time
1st qtr. $10,000.00 x 0.12 x 1/4 $ 300.00 $10,300.002nd qtr. 10,300.00 x 0.12 x 1/4 309.00 10,609.003rd qtr. 10,609.00 x 0.12 x 1/4 318.27 10,927.274th qtr. 10,927.27 x 0.12 x 1/4 327.82 11,255.095th qtr. 11,255.09 x 0.12 x 1/4 337.65 11,592.74Compound interest on $10,000 at 12% compounded quarterly for 5 quarters………………………...$1,592.74
1st qtr. $10,000.00 x 0.12 x 1/4 $ 300.00 $10,300.002nd qtr. 10,300.00 x 0.12 x 1/4 309.00 10,609.003rd qtr. 10,609.00 x 0.12 x 1/4 318.27 10,927.274th qtr. 10,927.27 x 0.12 x 1/4 327.82 11,255.095th qtr. 11,255.09 x 0.12 x 1/4 337.65 11,592.74Compound interest on $10,000 at 12% compounded quarterly for 5 quarters………………………...$1,592.74
Period x Rate =
Value at End of Quarter
Quarterly Compounded Interest
6
One thousand dollars is invested in a savings account on December 31, 2007. What will be the amount in the savings account on December 31, 2011 if interest
at 6% is compounded annually each year?
One thousand dollars is invested in a savings account on December 31, 2007. What will be the amount in the savings account on December 31, 2011 if interest
at 6% is compounded annually each year?
Dec. 31, 2007
Dec. 31, 2008
Dec. 31, 2009
Dec. 31, 2010
Dec. 31, 2011
$1,000 is invested on this date
How much will be in the savings account (the future
value) on this date?
Future Value of a Single Sum at Compound Interest
7
2008 $1,000.00 $ 60.00 $1,060.002009 1,060.00 63.60 1,123.602010 1,123.60 67.42 1,191.022011 1,191.02 71.46 1,262.48
2008 $1,000.00 $ 60.00 $1,060.002009 1,060.00 63.60 1,123.602010 1,123.60 67.42 1,191.022011 1,191.02 71.46 1,262.48
Annual Future Value Value at Compound at End Beginning of Interest of YearYear Year (Col. 2 x 0.14) (Col. 2 + Col. 3)
(1) (2) (3) (4)
Future Value of a Single Sum at Compound Interest
The future value of $1,000 compounded at 6% for four years is shown below.
8
Formula ApproachFormula Approach
ƒ = p(1 + i)n
where ƒ = future value of a single sum at compound interest i and n periods
p = principal sum (present value)
i = interest rate for each of the stated time periods
n = number of time periods
Future Value of a Single Sum at Compound Interest
9
Formula ApproachFormula Approach
f = p(1 + i)n
fn=4, i=6 = (1.06)4
f = $1,000(1.2624796) = $1,262.48
Future Value of a Single Sum at Compound Interest
10
Table ApproachTable ApproachTable ApproachTable Approach
Future Value of a Single Sum at Compound Interest
This time we will use a table to determine how much $1,000 will
accumulate to in four years at 6% compounded annually.
This time we will use a table to determine how much $1,000 will
accumulate to in four years at 6% compounded annually.
11
Table ApproachTable Approach
Future Value of a Single Sum at Compound Interest
Using Table 1 (the future value of 1) at the end of the
TVM Module, determine the future value interest factor
for an annual interest rate of 6 percent and four periods.
Using Table 1 (the future value of 1) at the end of the
TVM Module, determine the future value interest factor
for an annual interest rate of 6 percent and four periods.
12
Table ApproachTable Approach
n 6.0% 8.0% 9.0% 10.0% 12.0% 14.0% 1 1.060000 1.080000 1.090000 1.100000 1.120000 1.140000
2 1.123600 1.166400 1.188100 1.210000 1.254400 1.299600
3 1.191016 1.259712 1.295029 1.331000 1.404928 1.481544
4 1.262477 1.360489 1.411582 1.464100 1.573519 1.688960
5 1.338226 1.469328 1.538624 1.610510 1.762342 1.925415
6 1.418519 1.586874 1.677100 1.771561 1.973823 2.194973
1.262477
Future Value of a Single Sum at Compound Interest
13
Table ApproachTable ApproachTable ApproachTable Approach
One thousand dollars times 1.262477 equals the future
value, or $1,262.48.
One thousand dollars times 1.262477 equals the future
value, or $1,262.48.
Future Value of a Single Sum at Compound Interest
14
If $1,000 is worth $1,262.48 when it earns 6% compounded annually for 4 years, then it follows that $1,262.48 to be received in 4 years from now
is worth $1,000 now at time period zero.
If $1,000 is worth $1,262.48 when it earns 6% compounded annually for 4 years, then it follows that $1,262.48 to be received in 4 years from now
is worth $1,000 now at time period zero.
Dec. 31, 2007
Dec. 31, 2008
Dec. 31, 2009
Dec. 31, 2010
Dec. 31, 2011
$1,000 (the present value)
must be invested on this date
$1,262.48 will be received on this date
Present Value of a Single Sum
15
Interest Rate Unknown
If $1,000 is invested on December 31, 2007, to If $1,000 is invested on December 31, 2007, to earn compound interest and if the future earn compound interest and if the future value on December 31, 2014 is $2,998.70, value on December 31, 2014 is $2,998.70,
what is the what is the quarterlyquarterly interest rate? interest rate?
If $1,000 is invested on December 31, 2007, to If $1,000 is invested on December 31, 2007, to earn compound interest and if the future earn compound interest and if the future value on December 31, 2014 is $2,998.70, value on December 31, 2014 is $2,998.70,
what is the what is the quarterlyquarterly interest rate? interest rate?
Future ValuePresent Value
= Future value factor 28 periods
$2,998.70$1,000
= 2.99870
16
Table ApproachTable Approach
n 1.5% 4.0% 4.5% 5.0% 5.5% 6.0%
1 1.015000 1.040000 1.045000 1.050000 1.055000 1.060000
2 1.030228 1.081600 1.092025 1.102500 1.113025 1.123600
3 1.045678 1.124864 1.141166 1.157625 1.174241 1.191016
28 1.517222 2.998703 3.429700 3.920129 4.477843 5.111687
29 1.539981 3.118651 3.584036 4.116`36 4.724124 5.418388
30 1.563080 3.243398 3.745318 4.321942 4.983951 5.743491
2.998703
Future Value of a Single Sum at Compound Interest
The quarterly rate is 4%, which makes the annual rate 16%.
17
1(1 + i) np = f
Formula ApproachFormula Approach
Where p = present value of any given future value due in the future ƒ = future value i = interest rate for each of the stated time periodsn = number of time periods
Present Value of a Single Sum
18
p = $1,262.48 (0.792094) = $1,000.00
p n=4, i=6 =1
(1 .06)4 = 0.792094
Formula ApproachFormula Approach
Present Value of a Single Sum
19
Table ApproachTable Approach
Find Table 3, the present value of 1, at the end of the Time Value of Money Module.
Find Table 3, the present value of 1, at the end of the Time Value of Money Module.
Use 6% and four periods to obtain the future value
interest factor.
Use 6% and four periods to obtain the future value
interest factor.
Present Value of a Single Sum
20
Table ApproachTable Approach
n 6.0% 8.0% 9.0% 10.0% 12.0% 14.0%
1 0.943396 0.925926 0.917431 0.909091 0.892857 0.877193
2 0.889996 0.857339 0.841680 0.826446 0.797194 0.769468
3 0.839619 0.793832 0.772183 0.751315 0.711780 0.674972
4 0.792094 0.735030 0.708425 0.683013 0.635518 0.592080
5 0.747258 0.680583 0.649931 0.620921 0.567427 0.519369
6 0.704961 0.630170 0.596267 0.564474 0.506631 0.455587
0.792094
Present Value of a Single Sum
21
Table ApproachTable Approach
$1,262.48 times 0.792094 equals
$1,000.
$1,262.48 times 0.792094 equals
$1,000.
Present Value of a Single Sum
22
Debbi Whitten wants to calculate the future value of four cash flows of $1,000, each with interest
compounded annually at 6%, where the first cash flow is made on December 31, 2007.
Debbi Whitten wants to calculate the future value of four cash flows of $1,000, each with interest
compounded annually at 6%, where the first cash flow is made on December 31, 2007.
$1,000 $1,000 $1,000 $1,000
Dec. 31, 2007
Dec. 31, 2008
Dec. 31, 2009
Dec. 31, 2010
The future value of an ordinary annuity is determined immediately after the last
cash flow
Future Value of an Ordinary Annuity
23
Formula ApproachFormula ApproachFormula ApproachFormula Approach
(1 + i) - 1 n
Fo= Ci
Where F = future value of an ordinary annuity of a series of cash flows of any amountC = amount of each cash flown = number of cash flows i = interest rate for each of the stated time periods
o
Future Value of an Ordinary Annuity
24
Formula ApproachFormula ApproachFormula ApproachFormula Approach
Fo= n=4, i=6 =(1 .06) – 14
= 4.374620.06
Fo = $1,000(4.37462) = $4,374.62
Future Value of an Ordinary Annuity
25
Table ApproachTable ApproachUsing the same data—four equal annual cash flows of
$1,000 beginning on December 31, 2007, and an interest rate of 6 percent.
Using the same data—four equal annual cash flows of
$1,000 beginning on December 31, 2007, and an interest rate of 6 percent.
Go to Table 2, the future value of an ordinary annuity of 1.
Read the table value for n equals 4 and i equals 6%.
Go to Table 2, the future value of an ordinary annuity of 1.
Read the table value for n equals 4 and i equals 6%.
Future Value of an Ordinary Annuity
26
Table ApproachTable ApproachTable ApproachTable Approach
n 6.0% 8.0% 9.0% 10.0% 12.0% 14.0%
1 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000
2 2.060000 2.080000 2.090000 2.100000 2.120000 2.140000
3 3.183600 3.246400 3.278100 3.310000 3.374400 3.439600
4 4.374616 4.506112 4.573129 4.641000 4.779328 4.921144
5 5.637093 5.866601 5.984711 6.105100 6.352847 6.610104
6 6.975319 7.335929 7.523335 7.715610 8.115189 8.535519
4.374616
Future Value of an Ordinary Annuity
27
So, cash flows of $1,000 each at 6% at the end of 2007, 2008,
2009, and 2010 will accumulate to a future value of $4,374.62.
So, cash flows of $1,000 each at 6% at the end of 2007, 2008,
2009, and 2010 will accumulate to a future value of $4,374.62.
$1,000 x 4.374616 = $4,374.62$1,000 x 4.374616 = $4,374.62
Future Value of an Ordinary Annuity
28
Cash Flows Unknown
At the beginning of 2007, the Rexson Company issued 10-year bonds with a face value of $1,000,000 due on December 31,
2016. The company will accumulate a fund to retire these bonds at maturity. It will
make annual deposits to the fund beginning on December 31, 2007. How much must the company deposit each year, assuming that
the fund will earn 12% interest?
29
Cash Flows Unknown
Maturity value $1,000,000
Periods 10 years
Interest rate 12%
Future ValueFV Annuity factor
= Annual Cash flows for 10 periods
$1,000,00017.548735
= $56,984.16
30
Kyle Vasby wants to calculate the present value on January 1, 2007, (one period before the first cash flow) of four future withdrawals (cash flows) of $1,000 each, with the first withdrawal being made on December 31,
2010. Assume again an interest rate of 6%.
Kyle Vasby wants to calculate the present value on January 1, 2007, (one period before the first cash flow) of four future withdrawals (cash flows) of $1,000 each, with the first withdrawal being made on December 31,
2010. Assume again an interest rate of 6%.
$1,000 $1,000 $1,000 $1,000
Present Value of an Ordinary Annuity
Dec. 31, 2007
Dec. 31, 2008
Dec. 31, 2009
Dec. 31, 2010
Jan. 1, 2007
31
Go to Table 4, the present value of an ordinary annuity of 1. Read
the table value for n equals 4 and i equals 6%.
Go to Table 4, the present value of an ordinary annuity of 1. Read
the table value for n equals 4 and i equals 6%.
Present Value of an Ordinary Annuity
32
Table ApproachTable Approach
n 4.0% 5.0% 6.0% 7.0% 8.0% 9.0%
1 0.961538 0.952381 0.943396 0.934579 0.925926 0.917431
2 1.886095 1.859410 1.833393 1.808018 1.783265 1.759111
3 2.775091 2.723248 2.673012 2.624316 2.577097 2.531296
4 3.62895 3.545951 3.465106 3.387211 3.312127 3.239720
5 4.451822 4.329477 4.212364 4.100197 3.992710 3.889651
6 5.242137 5.075692 4.917324 4.766540 4.622880 4.485919
3.465106
Present Value of an Ordinary Annuity
33
Table ApproachTable ApproachTable ApproachTable Approach
One thousand dollars times 3.46511 equals $3,465.11 So,
the present value of this ordinary annuity is $3,465.11.
One thousand dollars times 3.46511 equals $3,465.11 So,
the present value of this ordinary annuity is $3,465.11.
Present Value of an Ordinary Annuity
34
Cash Flows UnknownOn January 1, 2007, Rex Company borrows
$100,000 at 12% interest to finance a plant expansion project. Ten equal payment are to be made starting on December 31, 2007.
What are the annual payments?
Jan. 1, 2007Dec. 31,
2007Dec. 31,
2008Dec. 31,
2009Dec. 31,
2016
? ? ??
The present value of 10 payments
with first payment made
one period later.
35
Table ApproachTable Approach
n 4.0% 5.0% 6.0% 7.0% 8.0% 12.0%
1 0.961538 0.952381 0.943396 0.934579 0.925926 0.892857
2 1.886095 1.859410 1.833393 1.808018 1.783265 1.690051
3 2.775091 2.723248 2.673012 2.624316 2.577097 2.401831
4 3.62895 3.545951 3.465106 3.387211 3.312127 3.037349
5 4.451822 4.329477 4.212364 4.100197 3.992710 3.604776
10 8.110896 7.721735 7.360087 7.023582 6.710081 5.6502235.650223
Present Value of an Ordinary Annuity
$100,000 = Cash flows X 5.65023
Present value = Cash flows X PVA factor
36
The present value of $100,000 divided by the present value of an annuity factor of 5.605223 equals an annual payment of
$17,698.42, which includes both principle and interest.
The present value of $100,000 divided by the present value of an annuity factor of 5.605223 equals an annual payment of
$17,698.42, which includes both principle and interest.
Present Value of an Ordinary Annuity
37
The present value of an annuity due is determined on the date of the first cash flow in the series.
The present value of an annuity due is determined on the date of the first cash flow in the series.
Present Value of an Ordinary Annuity
38
Barbara Livingston wants to calculate the present value of an annuity on December 31, 2007, which will permit four annual future receipts of $1,000
each, the first to be received immediately on December 31, 2007.
Barbara Livingston wants to calculate the present value of an annuity on December 31, 2007, which will permit four annual future receipts of $1,000
each, the first to be received immediately on December 31, 2007.
$1,000 $1,000 $1,000 $1,000
Dec. 31, 2007
Dec. 31, 2008
Dec. 31, 2009
Dec. 31, 2010
Present Value of an Annuity Due
39
Look up the factor in the present value of an annuity due table (Table 5), for
four periods at 6%
Table ApproachTable ApproachTable ApproachTable Approach
Present Value of an Annuity Due
40
Table ApproachTable ApproachTable ApproachTable Approach
n 5.0% 6.0 7.0% 8.0% 9.0% 10.0%
1 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000
2 1.952381 1.943396 1.934579 1.925926 1.917431 1.909091
3 2.859410 2.833393 2.808018 2.783265 2.759111 2.735537
4 3.723248 3.673012 3.624316 3.577097 3.531295 3.486852
5 4.545951 4.465106 4.387211 4.312127 4.239720 4.169865
6 5.329477 5.212364 5.100197 4.992710 4.889651 4.790787
3.673012
Present Value of an Annuity Due
41
One thousand dollars times 3.673012 equals $3,673.01.
One thousand dollars times 3.673012 equals $3,673.01.
Table ApproachTable ApproachTable ApproachTable Approach
Present Value of an Annuity Due
42
Cash Flow Unknown
Suppose that on Jan. 1, 2007, Katherine Spruill purchases an item that costs $10,000 and agrees to make 10 annual installments with interest of 8% starting immediately.
What are her payments?
Present value = Annual cash flow X PVAD factorOR
Present value / PVAD factor = annual cash flow
43
Table ApproachTable ApproachTable ApproachTable Approach
n 5.0% 6.0 7.0% 8.0% 9.0% 10.0%
1 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000
2 1.952381 1.943396 1.934579 1.925926 1.917431 1.909091
3 2.859410 2.833393 2.808018 2.783265 2.759111 2.735537
4 3.723248 3.673012 3.624316 3.577097 3.531295 3.486852
5 4.545951 4.465106 4.387211 4.312127 4.239720 4.169865
6 8.107822 7.801692 7.515232 7.246888 6.995247 6.7590247.246888
Present Value of an Annuity Due
44
The cash flow can be calculated by dividing $10,000
by the PVAD factor of 7.246888. Therefore, the
annual payments, starting immediately, are $1,379.90
The cash flow can be calculated by dividing $10,000
by the PVAD factor of 7.246888. Therefore, the
annual payments, starting immediately, are $1,379.90
Present Value of an Annuity Due
45
Helen Swain buys an annuity on January 1, 2007, that yields her four annual receipts of $1,000 each,
with the first receipt on January 1, 2011. The interest rate is 6% compounded annually. What is
the cost of the annuity?
Helen Swain buys an annuity on January 1, 2007, that yields her four annual receipts of $1,000 each,
with the first receipt on January 1, 2011. The interest rate is 6% compounded annually. What is
the cost of the annuity?
Present Value of a Deferred Ordinary Annuity
46
$1,000 $1,000 $1,000 $1,000
Jan. 1, 2011
Jan. 1, 2012
Jan. 1, 2013
Jan. 1, 2014
Jan. 1, 2010
Jan. 1, 2009
Jan. 1, 2008
Jan.1, 2007
The present value of the
deferred annuity is
determined on this date
$1,000 x 3.465106 (n=4, i=6) = $3,465.11
Present Value of a Deferred Ordinary Annuity
47
Jan. 1, 2010
Jan. 1, 2009
Jan. 1, 2008
Jan.1, 2007
The present The present value of the value of the
deferred deferred annuity is annuity is
determined determined on this dateon this date
$3,465.11
$3,465.11 x 0.839619 = $2,909.37
Present Value of a Deferred Ordinary Annuity
48
If Helen buys an annuity for $2,909.37 on January 1, 2007,
she can make four equal annual $1,000 withdrawals (cash flows) beginning on
January 1, 2011.
If Helen buys an annuity for $2,909.37 on January 1, 2007,
she can make four equal annual $1,000 withdrawals (cash flows) beginning on
January 1, 2011.
Present Value of a Deferred Annuity