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Chapter 6 Accounting and the Time Value of Money

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Chapter 6 Accounting and the Time Value of Money ...

    Chapter 6 page 1 of 18

    Chapter 6: Accounting and the Time Value of Money

    1) Basic Time Value Concepts: Time Value (TV) of Money: A dollar received today is worth more than a dollar promised at some time in the future. This relationship exists because of the opportunity to invest today’s

    dollar and receive interest on the investment.

a Applications of Time Value Concepts:

    i) Used for making decisions about Notes, Leases, Pensions and Other Postretirement

    Benefits, Long-Term Assets, Sinking Funds, Business Combinations, Disclosures,

    and Installment Contracts.

    ii) Also, TV concepts are very important in personal finance and investment decisions.

    For example, TV of Money is used when purchasing home or car, planning for

    retirement, and deciding on investments.

    b The Nature of Interest:

    i) Interest:

    ii) Amount of Interest in transaction is function of three variables:

    (1) Principal:

    (2) Interest Rate:

    (3) Time:

    The larger the principal the larger the dollar amount of interest.

    The higher the interest rate the larger the dollar amount of interest.

    The longer the time period the larger the dollar amount of interest.

c Simple Interest:

d Compound Interest:

    i)

    Chapter 6 page 2 of 18

    ii) Example: Simple vs. Compound Interest (Illustration 6-1, page 255)

    Deposit $10,000 at bank.

    Let simple interest = 9%.

    Let compound interest = 9% compounded annually.

    Assume no withdrawal until 3 years.

    Illustration 6-1 on page 255:

    Year Simple Interest Calculation Compound Interest Calculation

     Simple Simple Accumulated Compound Compound Accumulated

    Interest Interest Year-end Interest Interest Year-end

    Calculation Balance Calculation Balance

    Yr 1 $10,000 x $900 $10,900 $10,000 x 9% $900 $10,900

    9%

    Yr 2 $10,000 x $900 $11,800 $10,900 x 9% $981.00 $11,881.00

    9%

    Yr 3 $10,000 x $900 $12,700 $11,880.10 x $1069.29 $12,950.29

    9% 9%

    Total $2700 $2950.29

Note that the Compounded Interest is $250.29 higher than the Simple Interest ($2950.29 -

    $2,700 = $250.29)

Simple Interest Calculation:

    ? Uses the initial principal of $10,000 to compute interest in all 3 years.

Compound Interest Calculation:

    ? Uses the accumulated balance (principal plus interest to date) at end of each year

    to compute interest for the next year. (This explains why compounded interest is

    larger.)

Compounding assumes that unpaid interest becomes a part of the principal. The

    accumulated balance at the end of each year becomes the new principal, which is used to

    calculate interest for the next year.

Simple interest

iii) Compound Interest Tables: Five different types of compound interest tables are

    presented at the end of the chapter.

    (1) Future Value of $1 Table (Single Sum Table): Amount $1 will equal if

    deposited now at a specified rate and left for a specified number of periods.

    Example: Can be used to answer the question:

    (Table 6-1; page 302 and 303.)

    Chapter 6 page 3 of 18

    (2) Present Value of $1 Table: Amount that must be deposited now at a specified

    rate of interest to equal $1 at the end of a specified number of periods. Example:

    Can be used to answer the question:

     (Table 6-2; page 304 and 305.)

    (3) Future Value of an Ordinary Annuity of $1 Table: Amount to which payments

    of $1 will accumulate if payments are invested at END of each period at specified

    rate of interest for specified number of periods. Example: Can be used to

    answer the question:

     (Table 6-3; page 306 and 307.)

    (4) Present Value of an Ordinary Annuity of $1 Table: Amount that must be

    deposited now at a specified rate of interest to permit withdrawals of $1 at the

    END of regular periodic intervals for specified number of periods. Example:

    Can be used to answer the question:

     (Table 6-4; page 308 and 309.)

    (5) Present Value of an Annuity due of $1 Table: Amounts that must be deposited

    now at a specified rate of interest to permit withdrawals of $1 at the BEGINNING

    of regular periodic intervals for the specified number of periods. Example: Can

    be used to answer the question:

    (Table 6-5; page 310 and 311.)

    Chapter 6 page 4 of 18

    (6) General:

    (a) Compound tables are computed using basic formulas.

    (b)

    (i) TO CONVERT ANNUAL INTEREST RATE TO COMPOUNDING

    PERIODIC INTEREST RATE:

    (ii) TO DETERMINE THE NUMBER OF PERIODS:

(c) Frequency of Compounding: (Illustration 6-4 page 257)

    This illustration shows how to determine:

    (1) Interest rate per compounding period.

    (2) Number of compounding periods in four different scenarios.

    12% Annual Interest Rate per Number of Compounding Periods

    Interest Rate over Compounding

    5 years Period

    Compounded

    Annually (1) 0.12/1 = 0.12 5 yrs x 1 period per yr = 5 periods

    Semiannually (2) 0.12/2 = 0.06 5 yrs x 2 period per yr = 10 periods

    Quarterly (4) 0.12/4 = 0.03 5 yrs x 4 period per yr = 20 periods

    Monthly (12) 0.12/12 = 0.01 5 yrs x 12 period per yr = 60 periods

(d) Definitions:

    Example: Assume 9% annual interest compounded DAILY provides a 9.42%

    yield, or a difference of 0.42%.

    Effective rate: The 9.42 % is referred to as the effective yield.

    Stated rate (or nominal rate or face rate): The 9% is referred to as the

    stated rate.

    Relationship between effective and stated rate: When compounding

    frequency is greater than once a year, the effective interest rate will always be

    greater than the stated rate.

    Chapter 6 page 5 of 18

    e Fundamental Variables: The following four variables are fundamental to all compound

    interest problems:

    i) Interest Rate: Unless otherwise stated, the rate given is the annual rate that must be

    adjusted to reflect length of compounding period if less than a year.

    ii) Number of Time Periods: Number of compounding periods (An individual period

    may be equal to or less than 1 year.)

    iii) Future Value: Value at a future date given sum(s) invested assuming compound

    interest.

    iv) Present Value: Value now (present time) of future sum(s) discounted assuming

    compound interest.

    In some cases, all four variables are known. However, many times at least one

    variable is unknown.

    2) Single-Sum Problems:

    Two categories of single-sum problems:

    a Future Value of a Single Sum:

i) Compute unknown future value of known single sum of money invested now for

    certain number of periods (n) at a certain interest rate (i).

    ii)

iii) Determine future value of single sum: Multiply the future value factor (FVF) by its

    present value (principal).

    FV?PV(FVF) n,i

    where FV = future value; PV = present value; FVF= future value factor for n periods

    at i interest.

    iv) Example 1: (p 260)

    What is future value of $50,000 invested for 5 years compounded annually at 11%?

    FV = PV(FVF)

    FV =

    FV =

    (To get the ___________FVF, look at Table 6-1 on page 303. The ___% column and

    ____-period row gives the future value factor of ___________.)

    Chapter 6 page 6 of 18

    v) Example 2: (p260)

    What is the future value of $250 million if deposited in 2002 for 4 years if interest is

    10%, compounded semi-annually?

    FV=PV(FVF)

    FV=

    FV=

    (To get the ________, look at Table 6-1 which is the Future Value of $1 table. This

    is the table used to figure out the future value of $1 invested today. To get the

    number of

    periods,______________________________________________________. Thus,

    there are ______ periods. To get the correct semi-annual interest rate,

    ____________________________________. This gives us a

    ______________________________________We use n= ___ (number of periods)

    and i=____% (interest rate) to find the correct FVF.)

    b Present Value of a Single Sum:

i)

    ii) Compute unknown present value of known single sum of money in the future that is

    discounted for n periods at i interest rate.

    iii)

iv) Determine present value of single sum:

    PV?FV(PVF) n,i

    where PV = present value; FV = future value; PVF = present value factor for n

    periods at i interest.

    Chapter 6 page 7 of 18

    v) Example 1: (page 261-262)

    What is the present value of $84,253 to be received or paid in 5 years discounted at

    11% compounded annually?

    PV=FV(PVF)

    PV=

    PV=

    (To get the __________ PVF, look at Table 6-2 on page 305. The ___% column and

    ____-period row gives the present value factor of _________.)

    vi) Example 2: (page 262)

    If we want $2,000 three years from now and the compounded interest rate is 8%, how

    much should we invest today?

    PV=FV(PVF)

    PV=

    PV=

    (To get the __________ PVF, look at Table 6-2, page 305. The ____% column and

    the ___-period row give the PVF of __________.)

    c Solving for Other Unknowns in Single-Sum Problems: Unlike the examples given

    above, many times both the future value and present value are known, but the number of

    periods or the interest rate is unknown. If any three of the four values (FV, PV, n, i) are

    known, the remaining unknown variable can be derived.

i) Illustration Computation of the Number of Periods:

    How many years will it take for a deposit of $47,811 at 10% compounded annually to

    accumulate to $70,000?

    Solution 1:

    FV?PV(FVF) n,10%

    FVF=

    FVF=

    Look at Table 6-1, Future Value of $1. Look at ____% column and find the

    calculated FVF of _________. We find this factor in the row n= ____. Thus, it will

    take ___________.

    Chapter 6 page 8 of 18

    Solution 2:

    PV?FV(PVF) n,10%

PVF =

PVF =

Look at Table 6-2, Present Value of $1. Look at ____% column and find the

    calculated PVF of _________. We find this factor in the row n=___. Thus, it will

    take ____________.

    ii) Illustration Computation of the Interest Rate:

    What is the interest rate needed if we invest $800,000 now and want to have

    $1,409,870 five years from now?

Solution 1:

    FV?PV(FVF) 5,i

FVF =

FVF =

Look at Table 6-1 p 303, Future Value of $1. Look at row n=____ and find the

    calculated FVF of _______. We find this factor in the column i = ___%. Thus, we

    would need an interest rate of ___%.

Solution 2:

    PV?FV(PVF) 5,i

PVF =

PVF =

Look at Table 6-2 p 305, Present Value of $1. Look at row n=____ and find the

    calculated PVF of ________. We find this factor in the column for ____%. Thus,

    we would need an interest rate of _____%.

    Chapter 6 page 9 of 18

    3) Annuities:

    a General:

    i) Up to this point, we have only worked with discounting a single sum. However,

    many times a series of dollar amounts are to be paid (received) periodically (ex: loans,

    sales on installments, invested funds recovered in intervals)

    ii) Annuity:

    Requires

    (1)

    (2)

    (3)

iii) The future value of an annuity is the sum of all the rents plus the accumulated

    compound interest on them.

    iv) NOTE:

    (1) Ordinary Annuity:

    (2) Annuity Due:

    (3) Deferred Annuity:

b Future Value of an Ordinary Annuity:

    i) Can compute future value of an annuity by computing value to which each rent in

    series will accumulate. Total their individual values. (See Illustration 6-11.)

    ii)

iii) The future value of an ordinary annuity is computed as follows.

    Future_value_of_an_ordinary_annuity?R(FVF?OA) n,i

    where: R = periodic rent; FVF-OA = future value of an ordinary annuity factor for n

    periods at i interest.

    Chapter 6 page 10 of 18

    iv) Example 1: What is the future value of five $5,000 deposits made at the end of each

    of the next 5 years, earnings 12%?

    FV-OA = R(FVF-OA)

    FV-OA =

    FV-OA =

    (To get the ___________ FVF-OA, look at Table 6-3, page 307. The ___% column

    and the ___-period row give the FVF-OA of _________.)

    v) Example 2: If we deposit $75,000 at the end of 6 months for 3 years earnings 10%

    interest, what will the future value be?

    FV-OA = R (FVF-OA)

    FV-OA =

    FV-OA =

    Note: Because we are making semi-annual deposits, n =

    ____________________________and i =

    ___________________________________

    (To get the ____________ FVF OA, look at Table 6-3, page 306. Use column

    ____% and row ____.)

    c Future Value of an Annuity Due:

    i)

ii)

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