TIME VALUE OF MONEY

BASIC TIME VALUE CONCEPTS

TIME VALUE OF MONEY

- a relationship between time and money—A DOLLAR TODAY IS ALWAYS WORTH MORE THAN A DOLLAR PROMISED AT SOME
TIME IN THE FUTURE.
- WHY? Because of the opportunity to invest today’s dollar and receive interest on the investment.
- Yet, when deciding among investment or borrowing alternatives, it is essential to be able to compare today’s dollar and
tomorrow’s dollar on the same footing—to “compare apples to apples.”

Application of Time Value Concepts

1. Notes 5. Share-based compensation

2. Leases 6. Business combinations
3. Pensions and other postretirement benefits 7. Disclosures
4. Long-term assets 8. Environmental liabilities

THE NATURE OF INTEREST

- Payment for the use of money

- Excess cash received or repaid over the amount lent or borrowed (principal).

Variables in interest computation:

1. Principal. The amount borrowed or invested.

2. Interest rate. A percentage of the outstanding principal.
3. Time. The number of years or fractional portions of a year that the principal is outstanding.

PRESENT VALUE AND FUTURE VALUE

This section discusses the essentials of (1) compound interest, (2) annuities, and (3) present value. These techniques are being used
in many areas of financial reporting where the relative values of cash inflows and outflows are measured and analyzed. The material
presented in here will provide a sufficient background for application of these techniques to topics presented in subsequent
chapters.

COMPOUND INTEREST, ANNUITY, AND PRESENT VALUE

- techniques can be applied to many of the items found in financial statements

- techniques can be used to measure the relative values of cash inflows and outflows, evaluate alternative investment
opportunities, and determine periodic payments necessary to meet future obligations.

Some of the accounting items to which these techniques may be applied are:

(a) notes receivable and payable (e) share-based compensation

(b) leases (f) business combinations
(c) pensions and other postretirement benefits (g) disclosures
(d) long-term assets (h) environmental liabilities
SIMPLE INTEREST

- Interest is computed on the amount of the principal only.

- Simple interest is applied mainly to short-term investments and debts due in one year or less.

The formula for simple interest, where:

PxIxN
P = principal | I = rate of interest for a single period | N = number of periods

Illustration: Annual Interest Interest (I) = PxIxN

Barstow Electric Inc. borrows $10,000 for 3 years at a simple interest rate of 8% per year. = $10,000 x .08 x 1
Compute the total interest to be paid for 1 year. = $800

Illustration: Total Interest Interest (I) = PxIxN

Barstow Electric Inc. borrows $10,000 for 3 years at a simple interest rate of 8% per year. = $10,000 x .08 x 3
Compute the total interest to be paid for 3 years. = $2400

Illustration: Total Interest Interest (I) = PxIxN

Barstow Electric Inc. borrows $10,000 for 3 months at a simple interest rate of 8% per = $10,000 x .08 x 3/12
year, the interest is computed as follows. = $200

COMPOUND INTEREST

- Computes interest on
o Principal and
o Interest earned that has not been paid or withdrawn
- Typical interest computation applied in business situations
- Compound interest is most common in business situations where large amounts of capital are financed over long periods
of time.
- The frequency of compounding interest can make a substantial difference in the level of return achieved.

Illustration: Tom Company deposits $10,000 in the Last National Bank, where it will earn simple interest 9% per year. It deposits
another $10,000 in the First State Bank, where it will earn compound interest of 9% per year compounded annually. In both cases,
Tom will not withdraw any interest until 3 years from the date of deposit.

Last National Bank First State Bank

Accumulated
Simple interest Simple Accumulated Compound interest Compound
year-end
computation interest year-end balance computation interest
balance
Y1 $10,000 x 9% 900.00 $10,900.00 Y1 $10,000 x 9% 900.00 $10,900.00
Y2 $10,000 x 9% 900.00 $11,800.00 Y2 $10,900 x 9% 981.00 $11,881.00
Y3 $10,000 x 9% 900.00 $12,700.00 Y3 $11,881 x 9% 1,069.29 $12,950.29
2,700.00 2,950.29
$250.29 Difference
In understanding compound interest, the term PERIOD is used in place of years because interest may be compounded daily, weekly,
monthly, and so on.

Annual interest rate → Compounding interest rate

= Annual interest rate / no. of compounding periods in a year

No. of periods over which interest will be compounded

= No. of years involved x No. of compounding periods in a year

Compound interest tables have been developed to aid in the computation of present values and annuities. Careful analysis of the
problem as to which compound interest tables will be applied is necessary to determine the appropriate procedures to follow.

The following is a summary of the contents of the FIVE TYPES OF COMPOUND INTEREST TABLES:

Description Formula
FV of 1 FV = PV (FV𝐹𝑛,𝑖)
𝑛
Contains the amounts to which 1 will FV𝐹𝑛,𝑖 = (1+i)
accumulate if deposited now at a
specified rate and left for a specified
FV𝐹𝑛,𝑖 = FV Factor of n periods at i interest
number of periods.
n = no. of periods
i = rate of interest for a single period
PV of 1 PV = FV (PV𝐹𝑛,𝑖)
Contains the amounts that must be
−𝑛 1
deposited now at a specified rate of (PV𝐹𝑛,𝑖) = (1+i) or 𝑛
(1+𝑖)
interest to equal 1 at the end of a
specified number of periods. PV𝐹𝑛,𝑖 = PV Factor of n periods at i interest

FV of ordinary annuity of 1 Contains the amounts to which periodic

rents of 1 will accumulate if the R (FVF – 𝑂𝐴𝑛,𝑖)
payments (rents) are invested at the end (1+𝑖) −1
𝑛

of each period at a specified rate of FVF – 𝑂𝐴𝑛,𝑖 = 𝑖

interest for a specified number of

periods. R = periodic rent
This table may also be used as a basis for FVF – 𝑂𝐴𝑛,𝑖 = FV Factor of an ordinary annuity
converting to the amount of an annuity factor for n periods at i interest
due of 1.
PV ordinary annuity of 1 R (PVF – 𝑂𝐴𝑛,𝑖)
Contains the amounts that must be 1−
1
𝑛
(1+𝐼)
deposited now at a specified rate of PVF – 𝑂𝐴𝑛,𝑖 = 𝑖
interest to permit withdrawals of 1 at the
end of regular periodic intervals for the R = periodic rent
specified number of periods. PVF – 𝑂𝐴𝑛,𝑖 = PV of ordinary annuity of 1 for n
periods at i interest
FV of annuity due of 1 *Simply multiply Ordinary annuity 𝑛
(1+𝑖) −1
FVF – 𝐴𝐷𝑛,𝑖 = 𝑖
(1 + i)
formula by 1+i
PV of annuity due of 1 Contains the amounts that must be
deposited now at a specified rate of 1
1− 𝑛
interest to permit withdrawals of 1 at the PVF – 𝐴𝐷𝑛,𝑖 = (1+𝐼)
(1 + i)
𝑖
beginning of regular periodic intervals for
the specified number of periods.
Number of Periods = number of years x the number of compounding periods per year

Compounding Period Interest Rate = annual rate / number of compounding periods per year

A 9% annual interest compounded daily provides a 9.42%

yield. Effective yield for a $10,000 investment.

Higher frequency of compounding, the higher the effective

interest.

Fundamental Variables

a. Rate of Interest.
o The annual rate that must be adjusted to reflect the length of the compounding period if less than a year.

b. Number of Time Periods.

o The number of compounding periods (a period may be equal to or less than a year).

c. Future Value.
o The value at a future date of a given sum or sums invested assuming compound interest.

d. Present Value.
o The value now (present time) of a future sum or sums discounted assuming compound interest.

Relationship of the 4 fundamental variables

*Future value – Present Value = Interest | Per box = 1 period

SINGLE SUM PROBLEMS

Many business and investment decisions involve a single amount of money that either exists now or will in the future. Single-sum
problems are generally classified into one of the following two categories.

● Computing the unknown FUTURE VALUE of a known single sum of money that is invested now for a certain number of
periods at a certain interest rate.
● Computing the unknown PRESENT VALUE of a known single sum of money in the future that is discounted for a certain
number of periods at a certain interest rate

When analyzing the information provided, determine first whether the problem involves a future value or a present value. Then
apply the following general rules, depending on the situation:
 ● If solving for a FUTURE VALUE, accumulate all cash flows to a future point. In this instance, interest increases the amounts
or values over time so that the future value exceeds the present value.

● If solving for a PRESENT VALUE, discount all cash flows from the future to the present. In this case, discounting reduces the
amounts or values, so that the present value is less than the future amount.

FUTURE VALUE OF A SINGLE SUM

1. Bruegger Co. wants to determine the future value of $50,000 invested for 5 years compounded annually at an interest rate
of 11%.

FV = PV (FV𝐹𝑛,𝑖)

5
FV = $50,000 (1+0.11)

FV = $50,000 (1.68506) [See PFVT.pdf for value]

FV = $84,253

2. Companies can apply this time diagram and formula approach to routine business situations. To illustrate, assume that
Commonwealth Edison Company deposited $250 million in an escrow account with Northern Trust Company at the
beginning of 2012 as a commitment toward a power plant to be completed December 31, 2015. How much will the
company have on deposit at the end of 4 years if interest is 10%, compounded semiannually?

With a known present value of $250 million, a total of 8 compounding periods (4 x 2), and an interest rate of 5% per
compounding period (.10 / 2), the company can time-diagram this problem and determine the future value as shown in
Illustration 6-8.

FV = PV (FV𝐹𝑛,𝑖)

8
FV = $250,000,000 (1+0.05)

FV = $250,000,000 (1.47746) [See PFVT.pdf for value]

FV = $369,365,000

PRESENT VALUE OF A SINGLE SUM

The present value is the amount needed to invest now, to produce a known future value.

1. What is the present value of $84,253 to be received or paid in 5 years discounted at 11% compounded annually? (Based on
previous example)

PV = FV (PV𝐹𝑛,𝑖)
PV = $84,253 (PV𝐹5,11%)
1
PV = $84,253 ( 5 )
(1+0.11)
PV = $84,253 (0.59345)
PV = $50,000 (rounded by $0.06)
 2. Assume that your rich uncle decides to give you $2,000 for a trip to Europe when you graduate from college 3 years from
now. He proposes to finance the trip by investing a sum of money now at 8% compound interest that will provide you with
$2,000 upon your graduation. The only conditions are that you graduate and that you tell him how much to invest now.

PV = $2,000 (PV𝐹3,8%)
1
PV = $2,000 ( 3 )
(1+0.08)
PV = $2,000 (0.79383)
PV = $1,587.66

SOLVING FOR OTHER UNKNOWNS IN SINGLE-SUM PROBLEMS

Computation of the Number of Periods

The Village of Somonauk wants to accumulate $70,000 for the construction of a veteran’s monument in the town square. At the
beginning of the current year, the Village deposited $47,811 in a memorial fund that earns 10% interest compounded annually. How
many years will it take to accumulate $70,000 in the memorial fund?

Future value approach

FV = PV (FV𝐹𝑛,𝑖)

$70,000 = $47,811 (FV𝐹𝑛,10%)

$70,000
FV𝐹𝑛,10% = $47,811
= 1.46410 = 4 periods

Present value approach

PV = FV (PV𝐹𝑛,𝑖)

$47,811 = $70,000 (PV𝐹𝑛,10%)

$47,811
PV𝐹𝑛,10% = $70,000
= 0.68301 = 4 periods

Computation of the Interest Rate

Advanced Design, Inc. needs $1,409,870 for basic research 5 years from now. The company currently has $800,000 to invest for that
purpose. At what rate of interest must it invest the $800,000 to fund basic research projects of $1,409,870, 5 years from now?

Future value approach

FV = PV (FV𝐹𝑛,𝑖)

$1,409,870 = $800,000 (FV𝐹5,𝑖)

$1,409,870
FV𝐹5,𝑖 = $800,000
= 1.76234 = 12%

Present value approach

PV = FV (PV𝐹𝑛,𝑖)

$800,000 = $1,409,870 (PV𝐹5,𝑖)

 $800,000
PV𝐹5,𝑖 = $1,409,870
= 0.56743 = 12%

ANNUITIES

ANNUITY REQUIRES:

(1) Periodic payments or receipts (called rents) of the same amount

(2) Same-length interval between such rents, and
(3) Compounding of interest once each interval.

TWO TYPES:

(1) Ordinary annuity – rents occur at the end of each period.

(2) Annuity due – rents occur at the beginning of each period.

FV OF ORDINARY ANNUITY

● Rents occur at the end of each period.

● No interest during 1st period.

Illustration: Assume that $1 is deposited at the end of each of 5 years (an

ordinary annuity) and earns 12% interest compounded annually.

Illustration 6-17 shows the computation of the future value, using the “future
value of 1” table for each of the five $1 rents.

Illustration 6-18 provides an excerpt from the “future value of an ordinary

annuity of 1” table.

*Note that this annuity table factor is the same as the sum of the future values
of 1 factor shown in Illustration 6-17.

Illustration: What is the future value of five $5,000 deposits made at the end of each of the next 5 years, earning interest of 12%?
FV – 𝑂𝐴𝑛,𝑖 = R (FVF – 𝑂𝐴𝑛,𝑖)

FV – 𝑂𝐴5,12% = $5,000 (FVF – 𝑂𝐴5,12%)

5
(1+0.12) −1
FV – 𝑂𝐴5,12% = $5,000 ( 0.12
)

FV – 𝑂𝐴5,12% = $5,000 (6.35285)

FV – 𝑂𝐴5,12% = $31,764.25

Illustration: Gomez Inc. will deposit $30,000 in a 12% fund at the end of each year for 8 years beginning Dec. 31, 2014. What amount
will be in the fund immediately after the last deposit?

FV – 𝑂𝐴𝑛,𝑖 = R (FVF – 𝑂𝐴𝑛,𝑖)

FV – 𝑂𝐴8,12% = $30,000 (FVF – 𝑂𝐴8,12%)

FV – 𝑂𝐴8,12% = $30,000 (12.29969)

FV – 𝑂𝐴8,12% = $368,991

FV OF AN ANNUITY DUE

● Rents occur at the beginning of each period.

● Interest will accumulate during the 1st period.
● Annuity due has one more interest period than ordinary annuity.
● Factor = multiply future value of an ordinary annuity by 1 plus the interest rate.

Computation of rent

Illustration: Assume that you plan to accumulate $14,000 for a down payment on a condominium apartment 5 years from now. For
the next 5 years, you earn an annual return of 8% compounded semiannually. How much should you deposit at the end of each
6-month period?

FV – 𝑂𝐴𝑛,𝑖 = R (FVF – 𝑂𝐴𝑛,𝑖)

FV – 𝑂𝐴10,4% = R (FVF – 𝑂𝐴10,4%)

$14,000 = R (12.00611)

R = $14,000 / 12.00611

R = $1,166.07

Computation of Number of Periodic Rents

Illustration: Suppose that a company’s goal is to accumulate $117,332 by making periodic deposits of $20,000 at the end of each
year, which will earn 8% compounded annually while accumulating. How many deposits must it make?

FV – 𝑂𝐴𝑛,8% = R (FVF – 𝑂𝐴𝑛,8%)

$117,332 = $20,000 (FVF – 𝑂𝐴𝑛,8%)

FV – 𝑂𝐴𝑛,8% = $117,332 / $20,000

FV – 𝑂𝐴𝑛,8% = 5.86660 = 5 periods

Computation of Future Value

Illustration: Mr. Goodwrench deposits $2,500 today in a savings account that earns 9% interest. He plans to deposit $2,500 every
year for a total of 30 years. How much cash will Mr. Goodwrench accumulate in his retirement savings account, when he retires in 30
years?

FV – 𝐴𝐷30,9% = R (FVF – 𝐴𝐷30,9%)

FV – 𝐴𝐷30,9% = $2,500 (148.5752)

FV – 𝐴𝐷30,9% = $371,438

Illustration: Bayou Inc. will deposit $20,000 in a 12% fund at the beginning of each year for 8 years beginning January 1. Year 1. That
amount will be in the fund at the end of Year 8?

FV – 𝐴𝐷8,912% = R (FVF – 𝐴𝐷8,12%)

FV – 𝐴𝐷8,12% = $2,000 (13.7757)

FV – 𝐴𝐷8,912% = $275,514

PV OF ORDINARY ANNUITY

● Present value of a series of equal amounts to be withdrawn or received at equal intervals.

● Periodic rents occur at the end of the period.

Illustration: Assume that $1 is to be received at the end of each of 5 periods,

as separate amounts, and earns 12% interest compounded annually.
A formula provides a more efficient way for expressing the present value of an ordinary annuity of 1.

Present value of an ordinary annuity = R (PVF – 𝑂𝐴𝑛,𝑖)

1
1− 𝑛
(1+𝐼)
PVF – 𝑂𝐴𝑛,𝑖 = 𝑖

Illustration: What is the present value of rental receipts of $6,000 each, to be received at the end of each of the next 5 years when
discounted at 12%?

PV – 𝑂𝐴𝑛,𝑖 = R (PVF – 𝑂𝐴𝑛,𝑖)

PV – 𝑂𝐴5,12% = $6,000 (3.60478)

PV – 𝑂𝐴5,𝑖12% = $21,628.68

Illustration: Jamie Yuen wins $2,000,000 in the state lottery. She will be paid $100,000 at the end of each year for the next 20 years.
How much has she actually won? Assume an appropriate interest rate at 8%.

PV – 𝑂𝐴20,8% = $100,000 (PVF – 𝑂𝐴20,8%)

PV – 𝑂𝐴20,8% = $100,000 (9.81815)

PV – 𝑂𝐴𝑛,𝑖 = $981,815

PV OF ANNUITY DUE

● Present value of a series of equal amounts to be withdrawn or received at equal intervals.

● Periodic rents occur at the beginning of the period.

Illustration: Space Odyssey, Inc., rents a communications satellite for 4 years with annual rental payments of $4.8 million to be made
at the beginning of each year. If the relevant annual interest rate is 11%, what is the present value of the rental obligations?

PV – 𝐴𝐷4,11% = $4,800,000 (PVF – 𝐴𝐷4,11%)

PV – 𝐴𝐷4,11% = $4,800,000 (3.44371)

PV – 𝐴𝐷4,11% = $16,529,808

Illustration: Jamie Yuen wins $2,000,000 in the state lottery. She will be paid $100,000 at the beginning of each year for the next 20
years. How much has she actually won? Assume an appropriate interest rate at 8%.

PV – 𝐴𝐷20,8% = $100,000 (PVF – 𝐴𝐷20,8%)

PV – 𝐴𝐷20,8% = $100,000 (10.60360)

PV – 𝐴𝐷20,8% = $1,060,360
OR

PV – 𝐴𝐷20,8% = PV – 𝑂𝐴20,8% x (1+i)

PV – 𝐴𝐷20,8% = $981,815 x (1.08)

PV – 𝐴𝐷20,8% = $1,060,360.20

Computation of the Interest Rate

Illustration: Assume you receive a statement from MasterCard with a balance due of $528.77. You may pay it off in 12 equal monthly
payments of $50 each, with the first payment due one month from now. What rate of interest would you be paying?

PV – 𝑂𝐴12,𝑖 = R (PVF – 𝑂𝐴12,𝑖)

$528.77 = $50 (PVF – 𝑂𝐴12,𝑖)

PVF – 𝑂𝐴12,𝑖 = $528.77 / $50

PVF – 𝑂𝐴12,𝑖 = 10.5754 = 2% monthly rate

Since 2% is a monthly rate, the nominal annual rate of interest is 24% (12 x 2%).
12
The effective annual rate is (1 + 0. 02) − 1 = 26.82413%.

Computation of the Periodic Rent

Juan and Marcia Perez have saved $36,000 to finance their daughter Maria’s college education. They deposited the money in the
Santos Bank, where it earns 4% interest compounded semiannually. What equal amounts can their daughter withdraw at the end of
every 6 months during her 4 college years, without exhausting the fund?

PV – 𝑂𝐴8,2% = R (PVF – 𝑂𝐴8,2%)

$36,000 = R (7.32548)

R = $36,000 / 7.32548

R = $4,914.35

MORE COMPLEX SITUATIONS

DEFERRED ANNUITIES

● Rents after a specified number of periods

● Deferred = “put off”
● Deferred annuity → You will postpone the payment of an annuity. You will only pay it after a certain period of time.

A deferred annuity does not begin to produce rents until two or more periods have expired.

“An ordinary annuity of six annual rents deferred 4 years” – means that no rents will occur during the first 4 years, and that the first
of the six rents will occur at the end of the fifth year.
“An annuity due of six annual rents deferred 4 years” – means that no rents will occur during the first 4 years, and that the first of
six rents will occur at the beginning of the fifth year.

Future Value of a Deferred Annuity – Calculation same as the future value of an annuity not deferred.

Present Value of a Deferred Annuity – Must recognize the interest that accrues during the deferral period.

FV OF DEFERRED ANNUITY

Illustration: Sutton Corporation plans to purchase a land site in 6 years for the construction of its new corporate headquarters.
Because of cash flow problems, Sutton budgets deposits of $80,000, on which it expects to earn 5% annually, only at the end of the
fourth, fifth, and sixth periods. What future value will Sutton have accumulated at the end of the sixth year?

FV – 𝑂𝐴𝑛,𝑖 = R (FVF – 𝑂𝐴𝑛,𝑖)

FV – 𝑂𝐴3,5% = $80,000 (FVF – 𝑂𝐴3,5%)

FV – 𝑂𝐴3,5% = $80,000 (3.1525)

FV – 𝑂𝐴3,5% = $252,000

PV OF DEFERRED ANNUITY

Illustration: Bob Bender has developed and copyrighted tutorial software for students in advanced accounting. He agrees to sell the
copyright to Campus Micro Systems for six annual payments of $5,000 each. The payments will begin 5 years from today. Given an
annual interest rate of 8%, what is the present value of the six payments?

Method 1 (Short method)

PV – OA = R [(PVF – 𝑂𝐴10,8%) – (PVF – 𝑂𝐴4,8%)]

PV – OA = $5,000 [(6.71008) – (3.31213)]

PV – OA = $5,000 (3.39795)

PV – OA = $16,989.75

Method 2 (Long method) Step 2: PV of single sum

Step 1: PV of OA (6 years) PV = FV (PV𝐹𝑛,𝑖)

PV – OA = R [(PVF – 𝑂𝐴6,8%) PV = $23,114.40 (PV𝐹4,8%)

PV – OA = $5,000 (4.62288) PV = $23,114.40 (0.73503)

PV – OA = $23,114.40 PV = $16,989.78
VALUATION OF LONG-TERM BONDS

- Produces two cash flows:

o Periodic interest payments (annuity) | $140,000
o Principal paid at maturity (single-sum) | $2,000,000

- At the date of issue, bond buyers determine the present value of these two cash flows using the market rate of interest
(effective interest)

Nominal interest vs. Effective interest

Nominal interest – to know the cash flows you will pay

Effective interest (market rate of interest) – to know the PV of one (cash flows?)

If face value > present value (+ discount amortization)

If face value < present value (premium amortization)

PRESENT VALUE

Illustration: Wong Inc. issues $2,000,000 of 7% bonds due in 10

years with interest payable at year-end. The current market rate
of interest for bonds of similar risk is 8%. What amount will
Wong receive when it issues the bonds?

Step 1: PV of Interest (effective interest rate)

I = $2,000,000 x 7% (nominal rate) = $140,000 Step 2: PV of Principal (effective interest rate)

PV – 𝑂𝐴10,8% = I (PVF – 𝑂𝐴10,8%) PV = P (𝑃𝑉𝐹10,8%)

PV – 𝑂𝐴10,8% = $140,000 (6.71008) PV = $2,000,000 (0. 46319)

PV – 𝑂𝐴10,8% = $939,411 PV = $926,380

PV of Interest $939,411 Account Title Debit Credit

PV of Principal $926,380 Cash 1,865,791
Bond current market value $1,865,791 Bonds payable 1,865,791

Schedule of Bond DISCOUNT Amortization

10 years | 7% Bonds sold to Yield 8%

Carrying value of
Date Cash interest paid Interest expense Increase in balance
bonds
1/1/12 1,865,791
12/31/12 140,000 149,263 9,263 1,875,054
 12/31/13 140,000 150,004 10,004 1,885,058
12/31/14 140,000 150,805 10,805 1,895,863
12/31/15 140,000 151,669 11,669 1,907,532
12/31/16 140,000 152,603 12,603 1,920,135
12/31/17 140,000 153,611 13,611 1,933,746
12/31/18 140,000 154,700 14,700 1,948,445
12/31/19 140,000 155,876 15,876 1,964,321
12/31/20 140,000 157,146 17,146 1,981,467
12/31/21 140,000 158,533 18,533 2,000,000

Interest expense = Carrying value (CA), beg x Effective interest rate (ER)

CA, end = CA, beg + Interest expense – Cash interest paid

Bond discount amortization = CA, end – CA, beg

Shortcut Computation:

CA, end = [CA, beg x (1 + i)] – Cash interest paid

FUTURE VALUE (from internet)

Illustration: Suppose a company issues $100,000 in 10-year, 9% coupon bonds at a premium to face value. Investors only demand an
8% return for owning the bond, and thus pay the company $106,710 for the bonds.

Step 1: PV of Interest (effective interest rate) Step 2: PV of Principal (effective interest rate)

I = $100,000 x 9% (nominal rate) = $9,000 PV = $100,000 (𝑃𝑉𝐹10,8%)

PV – 𝑂𝐴10,8% = $9,000 (PVF – 𝑂𝐴10,8%) PV = $100,000 (0. 46319)

PV – 𝑂𝐴10,8% = $9,000 (6.71008) PV = $46,319

PV – 𝑂𝐴10,8% = $60,391

PV of Interest $60,391
PV of Principal $46,319
Bond current market value $106,710

Schedule of Bond INCOME/PREMIUM Amortization

Carrying value of
Date Cash interest paid Interest expense Decrease in balance
bonds
0 $106,710
1 $9,000 $8,537 $463 $106,247
2 $9,000 $8,500 $500 $105,747
3 $9,000 $8,460 $540 $105,207
4 $9,000 $8,417 $583 $104,624
5 $9,000 $8,370 $630 $103,994
6 $9,000 $8,320 $680 $103,314
7 $9,000 $8,265 $735 $102,579
 8 $9,000 $8,206 $794 $101,785
9 $9,000 $8,143 $857 $100,928
10 $9,000 $8,074 $926 $100,002

[Same computations with DISCOUNT]

PRESENT VALUE MEASUREMENT

IFRS 13 explains the expected cash flow approach that uses a range of cash flows and incorporates the probabilities of those cash
flows.

Expected cash flow approach – Important for valuation purposes because we want to be as close as possible to the actual income or
expense of a particular investment or debt.

Choosing an Appropriate Interest Rate

Three components of interest:

● Pure rate
● Expected inflation rate
● Credit risk rate (credit worthiness)

Risk-free rate of return. Pure rate & Expected inflation rate

- IASB states a company should discount expected cash flows by the risk-free rate of return.

Illustration: Angela Contreras is trying to determine the amount to set aside so that she will have enough money on hand in 2 years
to overhaul the engine on her vintage used car. While there is some uncertainty about the cost of engine overhauls in 2 years, by
conducting some research online, Angela has developed the following estimates.

Estimated Cash Outflow Probability Assessment

$200 10%
$450 30%
$600 50%
$750 10%

Instruction: How much should Angela Contreras deposit today in an account earning 6%, compounded annually, so that she will
have enough money on hand in 2 years to pay for the overhaul?

Estimated Cash Outflow Probability Assessment Expected Cash Flow

$200 10% $20
$450 30% $135
$600 50% $300
$750 10% $75
$530
PV = P (𝑃𝑉𝐹2,6%)

PV = $530 (0.89)

PV = $471.70
Example: You owed money in a bank

At any given time, if nothing affects the money you

Pure rate 2%
owed, the pure rate is.. Risk-free rate of
Due to inflation, the bank will adjust the rate to return
Expected inflation rate +3%
maintain the purchasing power of the money.
To check your credit status: Do you pay regularly?
Credit risk rate Do you have enough cash to pay in case of an +1%
emergency?