Time Value of Money Module An electronic presentation by Norman Sunderman Angelo State University An electronic presentation by Norman Sunderman Angelo.

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)


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


9


f = p(1 + i)n

fn=4, i=6 = (1.06)4

f = $1,000(1.2624796) = $1,262.48


10

Table ApproachTable ApproachTable ApproachTable Approach


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


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


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


13


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.


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


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


The quarterly rate is 4%, which makes the annual rate 16%.

17

1(1 + i) np = f


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


18

p = $1,262.48 (0.792094) = $1,000.00

p n=4, i=6 =1

(1 .06)4 = 0.792094



19


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.


20


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


21


$1,262.48 times 0.792094 equals

$1,000.

$1,262.48 times 0.792094 equals

$1,000.


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


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


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%.


26


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


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


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%.


32


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


33


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.


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


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


$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.


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.


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%



40


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


41

One thousand dollars times 3.673012 equals $3,673.01.

One thousand dollars times 3.673012 equals $3,673.01.



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


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


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


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


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


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

49

TVM

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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.

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