Contents
Topic 5
Introduction and Purpose
Time-Value-of-Money (TVM) Concepts
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—
—
—
—
—
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Time-Value of Money and Opportunity Cost
TVM Concepts: Future Value and Present Value
Compounding of Returns: Nominal versus Effective Rates of Return
Investments and Financings involving a Series of Regular Cash Flows
Terminology
Opportunity Cost: Undiscounted versus Discounted Cash Flows
Investments and Financings Involving Irregular Cash Flows
□ Effect of cash flow structure on present value of cash flows
□ Effect of cash flow frequency on present value of cash flows
□ Effect of changes in discount rates on present value of cash flows
Estimating the Fair Value of Corporate Bonds and Notes
— Fair Value of a Business' Capital Financing and Unrecognized or Under-valued Assets
Net Present Value
— Relevant Cash Flows and Sunk Costs
— Capital Rationing and Ranking Multiple Investment Opportunities
1
Topic 5
Introduction and Purpose
As a student in this course, you previously completed a course in the principles of business finance. Among
other topics, that course examined time-value-of-money (TVM), including the concepts of present value and
future value. That course introduced you to the use of TVM concepts to:
Determine the fair value of investment securities, such as bonds and stocks, and
Evaluate investment opportunities, such as the replacement of manufacturing equipment
This background paper examines TVM concepts, focusing on managers’ use of present value models to
analyze and select investment opportunities. These concepts and models are useful for analyzing a variety of
investment or financing decisions, such as whether to:
Initiate a product research-and-development project,
Launch a new product, or discontinue an aging product,
Acquire additional plant and equipment in order to expand productive capacity,
Replace production equipment with more efficient or more highly automated (cost-saving) equipment,
Acquire a competitor firm, or sell an existing division of a business, and
Refinance a business’ existing debt
Once you firmly understand the concept of opportunity cost that underlies present value, you will quickly
comprehend the models of present value analysis examined in this and the Topic 6 background papers.
However, in applying these models to investment opportunities, managers must not lose sight of the following
points:
Uncertainty of future cash flows and investment risk. The principal subject of present value analysis is
future cash flows. Of course, the future is uncertain. The uncertainty of future cash flows is the source
of investment risk. Consequently, the quality of managers’ ultimate investment decisions depends
significantly on the accuracy and completeness of their cash flow projections. These cash flows may
include, for example, sales and costs of new products, cost savings from investment in more efficient
manufacturing equipment, proceeds from the eventual disposition of investment property, and income
taxes related to these other cash flows.
Relevant cash flows. Related to the preceding point, present value analyses must include only relevant
cash flows. Sunk costs – the cash flows representing costs a business incurs before managers make
the decision to accept or reject an investment proposal – are not relevant cash flows.
Capital rationing. Frequently, managers must choose from among multiple investment proposals.
Present value analysis provides the proper model for evaluating and ranking alternative investment
opportunities. However, many businesses face a practical constraint on their investing because they
have a limited amount financial capital available during a given period. Consequently, managers must
ration capital so that they maximize the value created by all accepted investment opportunities.
The Topic 6 background paper examines the use of present value analysis in connection with the capital
budgeting process.
2
Topic 5
Learning
Objective 1
Time-Value-of-Money (TVM) Concepts
Time-Value of Money and Opportunity Cost
The time-value-of-money (TVM) concept states that the value of a stated amount of money (such as US$1,000
or €1,500, or ¥10,000) depends on the timing of its receipt or payment. This is because a business may invest
money with the expectation of earning a return on it.1 The sooner a business receives a particular amount of
money, the sooner it can invest it and begin to earn that return. During the period that a business awaits the
receipt of money, it incurs an opportunity cost because it cannot invest it.2 3 Return on investment takes the
form of future cash inflows that exceed the initial cash outflows representing the business’ initial investment.
TVM Concepts: Future Value and Present Value
Future value refers to the value of a specified amount of money on a specified future date that a business
invests between the current date (called “time zero”) and the specified future date. For example, the future
value of $1,000 in one year, if invested today at a 10 percent annual rate of return, is $1,100 (assuming no
interim compounding of returns during the year). Financial managers compute this future value as:
Future value, or FV = $1,000 x (1 + 0.10)
_____
1
A common misconception by students of finance is that the TVM concept derives from price inflation (deterioration of a currency’s
purchasing power). Of course, expectations of inflation (by businesses that invest financial capital and providers of debt and equity
capital) affect the nominal rates of return they require from those investments. However, businesses and providers of financial capital
reasonably expect a return on their investments even in the absence of price inflation, for the reason described above. Investors
determine their required nominal rate of return, which includes the anticipated effect of inflation as:
Required nominal rate of return = (1 + required real rate of return) x (1 + expected rate of inflation) – 1
where, the required real rate of return is an investor’s required rate of return excluding the effects of inflation. To illustrate, if an investor
requires a real rate of return of 5.0 percent and she expects the long-run future rate of inflation to be 3.0 percent, her required nominal
rate of return is: (1 + 0.05) x (1 + 0.03) – 1 = 0.0815, or 8.15 percent.
2
It may be useful to consider opportunity cost in another (possibly more familiar) context. Suppose you quit your job, for which you
receive a $70,000 annual salary, in order to start a business. As is often the case for new businesses, assume that yours is “strapped
for cash” in its first few years of operations while you develop a market for its products or services sufficient to generate profitable levels
of operating cash flow. To help ensure your fledgling business gets off the ground, you instruct your part-time accountant to pay you an
annual salary of only $24,000 for your services as president, CEO, CFO, VP-sales, and what not – an amount well below the market
value of these services. When you evaluate your return on investment in the business during these early years, you should include as
a cost the market value of your services as an employee of another business in excess of the salary you actually receive from your new
business. This excess ($46,000) represents the value of your services that you forego when you pass up the opportunity to earn a
market salary (“return”) in exchange for your services (“investment”). Mind you, this excess is an opportunity cost that you include in
your “spreadsheet analysis” of your investment, but is not an expenditure arising from an arms-length exchange that you actually record
in the business’ accounting records, as described in the Topic 3-4 background paper.
3
According to economists, investment opportunities exist because some of the businesses and individuals who have money prefer not
to spend it immediately on items that do not increase an economy’s stock of income-producing assets. (These assets include, for
example, manufacturing equipment, roads, aircraft, office buildings, and telecommunications infrastructure.) Instead, these “savers”
deposit their money into interest-earning bank accounts and purchase corporate bonds, stocks, and other securities that yield returns
(or into professionally managed retirement accounts and mutual funds that acquire these securities on behalf of these savers).
Provided the rate of return on these investments is sufficiently high, savers continue to save, rather than spend their money currently
(on, say, entertainment and travel). Economists refer to this discussion as “inter-temporal consumption preferences.”
3
Present value is the “inverse” of future value. That is, present value refers to value today (i.e., at “time zero”)
of a specified amount of money an investor expects to receive on a specified future date. For example, the
present value of $1,000 a business expects to receive in one year at a 10 percent annual rate of return is
$909.09 (assuming no interim compounding of returns):
Present value, or PV = $1,000 / (1 + 0.10)
Present value answers the question, “What amount of money must my business invest today, assuming an
annual rate of return of 10 percent, in order for the value of that money to grow to a future value of $1,000 in
one year?” In general,
FV = PV x (1 + r)n
PV = FV / (1 + r)n
where,
FV is the future value of a specified amount of money invested for n periods, beginning “today,”
PV is the present value of a specified amount of money to be received n periods from “today,”
r is the periodic rate of return that an investor requires or expects to earn on a given investment, and
n is the number of periods for which the money is invested (such as, 5 years, 12 quarters, 48 months, or 180 days)
The Topic 6 background paper examines the determination of businesses’ required rate of return.
Compounding of Returns: Nominal versus Effective Rates of Return
Managers, capital markets participants, and financial institutions generally describe investments by referring to
their nominal (or stated) annual rates of return (such as “the 8 percent debenture bonds that mature in 20X9,”
or “the 6.5 percent note maturing next June”). However, most investments generate their returns (in the form
of cash distributed to investors) more frequently than annually. For example,
Most bonds issued by U.S. corporations and government entities pay semi-annual interest to their holders;
and
Most business loans made by banks require borrowers to pay interest monthly or quarterly.
Investors (businesses and their capital providers) may reinvest those interim returns in the same or similar
investments. This reinvestment opportunity gives rise to compound returns – that is, returns on previous
returns4. In the case of investments that generate returns more frequently than annually (or borrowings that
require interest payments more frequently than annually), the opportunity to compound returns requires
investors to recast the nominal or stated rates in the form of an effective rate of return. The effective rate of
return permits managers to compare alternative investments (or financing arrangements) that have different
return frequencies. Managers must also know the effective rate of return in order to compute properly FV and
PV amounts.
Effective annual rate of return, re = (1 + r / n)n – 1
where,
r is the nominal (stated) rate of return, and
n is the number of return periods during a year
_____
4
On the subject of TVM, Nobel prize-winning physicist, Albert Einstein, apparently once said, "The most powerful force in the universe
2
is compound interest." (More powerful than E = MC ?! We can forgive Benjamin Franklin for making a similar observation two
centuries earlier, well before Einstein shared his insights.)
4
Consider these two illustrations:
Corporate bond. The effective rate of return for a corporate bond having a nominal (stated or “coupon”)
interest rate of 10.0 percent and paying interest semi-annually is:
re = (1 + 0.10 / 2)2 – 1 = 10.25 percent
The bondholders’ effective annual rate of return, 10.25 percent, exceeds the nominal (or stated) rate of
10.0 percent because the bondholder may reinvest the first semiannual interest payment before the end of
a year during which the bond is outstanding (say, in additional corporate bonds). Likewise, the bondissuing corporation’s effective borrowing rate exceeds the contractually stated (annual) rate of 10 percent
because the requirement to pay money to bondholders (in the form of interest) during a year that the bonds
are outstanding imposes an opportunity cost on the issuer. This is because the money representing the
first semi-annual interest payment during a year the bond is outstanding is not available to the bond issuer
for other investment opportunities.
Bank note. The effective rate a business pays on a bank loan whose nominal (or stated) interest rate is 10
percent and which requires monthly interest payments is:
re = (1 + 0.10 / 12)12 – 1 = 10.471 percent
The business’ effective borrowing rate exceeds the contractually stated annual rate of 10 percent because
the requirement to pay money to the bank (in the form of interest) during the year the loan is outstanding
imposes an opportunity cost on the borrower. That is, the money (interest payments) that the business
pays the bank during the year is not available to the borrower for other investments. Likewise, the bank’s
effective rate of return on the loan exceeds the contractually stated annual rate of 10 percent because it
may reinvest the borrower’s monthly interest payments in (say) additional interest-paying loans to other
businesses.
In the illustrations above, note that the effective rate on the 10 percent loan requiring monthly interest
payments (10.471 percent) exceeds the effective rate on the 10 percent bond requiring (only) semi-annual
interest payments. This illustrates the “power” of compounding. The more frequent the opportunity (for a
bondholder or lending bank) to reinvest periodic returns, the greater is effective rate of return. For a borrower
(such as a business that issues bonds or bank notes), as the frequency of required interest payments
increases, so does the effective borrowing rate of the bond or bank note.
In light of the preceding discussion, the future value and present value equations, defined more precisely, are:
FV of cash flow occurring in period n
=
Cash flow at “time zero” x (1 + r)n
PV of cash flow occurring in period n
=
Cash flow in period n
(1 + r)n
where,
r is the periodic rate of return, stated on a basis consistent with n, and
n is the number periods – years, semi-annual periods, quarters, months, days, etc. – until the occurrence of
the future cash flow
5
Learning
Objective 2
Investments and Financings Involving a Series of Regular Cash Flows
The future value and present value equations above are suitable for analyzing investments or financing
arrangements involving a single cash inflow and a single cash outflow. Of course, most investments or
financing arrangements undertaken by businesses involve more than a single cash inflow and cash outflow.
Financial managers use the following equations to determine the future value and present value of annuities –
investments or financing arrangements involving a series of equal, periodic cash flows:
FVannuity
= Periodic cash flow
(1 + r)n – 1
r
x
1–
PVannuity 5
= Periodic cash flow
x
1
(1 + r)n
r
where,
r is the investor’s required periodic rate of return from an investment, stated on a basis consistent with the
frequency of the periodic cash flow – annual, semi-annual, quarterly, monthly, etc.
n is number of periodic – annual, semi-annual, quarterly, monthly, etc. – cash flows comprising the annuity
_____
5
While the PVannuity equation may seem confounding, the underlying reasoning is compelling and not too difficult to comprehend. First,
consider an “annuity-in-perpetuity,” in which an investor and her heirs are to receive a regular, periodic cash flow, C . . . well, in
perpetuity (i.e., forever). The PV of this annuity-in-perpetuity is: PV = C / r. Notice that this expression is algebraically identical to C =
PV x r, where an investment in the amount of PV yielding a perpetual annual return of r would produce annual cash flows of C until “the
end of time.” Now, to determine the PV of an annuity that is not perpetual, the PVannuity equation subtracts from the PV of an annuity-inperpetuity, C / r, the PV of a second annuity-in-perpetuity whose periodic cash flows, also C, begin later. Exploiting a bit of algebra:
PVannuity
=
C
1
n
(1 + r)
1–
x
r
C
=
r
–
C
r
1
x
(1 + r)
n
To illustrate, assume that C = $100, r = 10 percent, and the annuity period, n, is 5 years
$100
$100
1
=
–
x
6
0.10
0.10
(1 + 0.10)
=
$1,000
–
$564.47
=
PV of annuity-inperpetuity that begins
at end-of-year 1
–
PV of an identical annuity-in-perpetuity,
except that it begins at end-of-year 6 (5
years after the first annuity-in-perpetuity)
=
$435.53
=
PV of a 5-year
annuity beginning at
the end of year 1
6
To illustrate a manager’s use of the PVannuity equation, consider a proposal by ABC Company’s managers to license
to another firm certain intellectual property – a manufacturing process on which ABC Company holds a legally
registered patent. Under a proposed 10-year licensing agreement, the licensee may use ABC Company’s process
in exchange for an annual, end-of-year payment of $100,000. Management of ABC Company determined that the
company’s required rate of return on investments, r, is 8.0 percent. Accordingly, the present value of the
contractually specified license payments is 6 7:
1–
PVannuity
1
(1 + 0.08)10
0.08
=
$100,000
x
=
$100,000
x 6.7100814 8
=
$671,008 (rounded)
However, most managers find computerized spreadsheets (or a handheld financial calculator) easier to use
for this purpose.9 Using an MS Excel spreadsheet, a manager may determine easily the present value of an
annuity using the following function (formula):
=PV(rate,nper,-pmt,-fv,0) , where
Rate
Nper
Pmt
FV
0
=
=
=
=
=
r, Company ABC’s required rate of return: 0.08 (or, 8.0%)
Number of periodic cash inflows: the 10 license payments
Amount of periodic cash flow: the $100,000 annual license payment
0, in the case of “straight” annuity, such as Company ABC’s proposed license agreement
Indicates that payments are due at end of each period throughout the proposed license agreement
In the ABC Company illustration:
MS Excel present value function (formula)
Formula result
=PV(0.08,10,-100000,0,0)
$671,008
Terminology
Financial managers refer to the process of determining the present value, or PV, of future cash flows as
discounting, refer to the required rate of return, r, used in PV analysis as the discount rate10, and refer to the
computed results as discounted cash flows, or simply DCF. Comparison of the equations above for the
future value and present value of cash flows reveals that discounting is the mathematical inverse of
compounding.
_____
6
Throughout this background paper (and the Topic 6 background paper), assume that cash flows occur at the end of the periods
indicated (so-called cash flows “in arrears”), rather than the beginning of the periods. Application of the TVM equations examined in
this background paper requires some minor adjustment in the case where the investment analyzed involves beginning-of-period cash
flows.
7
The examination of investments and financing arrangements in this background paper ignores the effects of income taxes. The Topic
6 background paper examines income tax effects on investment decisions in connection with capital budgeting analysis.
8
Financial managers refer to this term as the annuity factor.
9
Before computer spreadsheets and handheld financial calculators came into widespread use in the 1980s, financial managers used
tables that summarized discount factors and annuity factors for the possible combinations of discount rates and number of periods,
which they applied to investment cash flows.
10
Some managers use the terms hurdle rate, target rate, or cut-off rate to refer to a business’ required rate of return, r.
7
Opportunity Cost: Undiscounted versus Discounted Cash Flows
In the ABC Company illustration, above, the proposed license agreement requires the licensee to pay the
company a total of $1,000,000 in licensing fees over the term of the 10-year agreement. This amount is the
undiscounted (total) cash flows under the agreement. Notice that the discounted cash flows (DCF), or
present value of these cash flows, $671,008, is about one-third less than the undiscounted total cash flows.
The difference, or $328,992, represents the opportunity cost to ABC Company of not having the money
representing the licensing payments “now” so that it can invest it another project that would yield a return. As a
result, ABC Company managers would be indifferent between:
An agreement that requires the licensee to pay 10 annual end-of-year payments of $100,000, and
An agreement that requires the licensee to pay the company a single upfront payment of $671,008 (and
permits the licensee to use the company’s patented process for the ensuing 10 years)11
Investments and Financings Involving Irregular Cash Flows
Many investments by businesses do not involve a series of regular cash flows.12 In these situations, managers
cannot use the PVannuity equation and the related MS Excel function (formula), =PV(rate,nper,-pmt,-fv,0) above to
evaluate investments. When investment opportunities involve irregular cash flows, the most effective and
convenient means for evaluating their present value is a computer spreadsheet that:
Discounts each of the cash flows using the “single cash flow” equation, PV = FV / (1 + r)n, defined earlier,
and
Sums up the results of these individual PV computations
To illustrate this approach to evaluating the present value of investments involving irregular cash flows, reconsider
ABC Company’s proposal to license its patented manufacturing process. Instead of a fixed annual end-of-year
license payment of $100,000, managers are considering an annual royalty payment based on the licensee’s annual
sales:
Annual royalty payment
=
Licensee’s annual sales of products manufactured using
the licensed manufacturing process
x
5 percent
Management prepared the following present value analysis based on projected annual sales for each of the next 10
years, as provided by the prospective licensee:
_____
11
Now, whether the licensee would be willing and able to make a large upfront payment to ABC Company is an open question!
12
The Topic 6 background paper examines the capital budgeting process, which considers investments typically involving multiple,
irregular cash inflows and cash outflows over several periods.
8
Year
(n )
Projected
Royalty
sales (1)
rate
Projected
Discount
Discounted
MS Excel
royalty
factor (2)
PV of royalty
formula for
payment
discount factors
payment
1 / (1 + r )
n
73,611
=1/(1+0.08)^1
0.85734
71,588
=1/(1+0.08)^2
87,500
0.79383
69,460
=1/(1+0.08)^3
0.05
92,000
0.73503
67,623
=1/(1+0.08)^4
1,930,000
0.05
96,500
0.68058
65,676
=1/(1+0.08)^5
6
2,030,000
0.05
101,500
0.63017
63,962
=1/(1+0.08)^6
7
2,130,000
0.05
106,500
0.58349
62,142
=1/(1+0.08)^7
8
2,240,000
0.05
112,000
0.54027
60,510
=1/(1+0.08)^8
9
2,350,000
0.05
117,500
0.50025
58,779
=1/(1+0.08)^9
10
2,470,000
0.05
123,500
0.46319
57,204
=1/(1+0.08)^10
1
1,590,000
x
0.05
2
1,670,000
0.05
83,500
3
1,750,000
0.05
4
1,840,000
5
Total
=
79,500
x
0.92593
1,000,000
=
650,556
This column displays the
formula input in the
"Discount factor" column of
this worksheet. Use the "^"
(carat) character - find it
above the "6" key on your
PC keypad - to indicate that
the term, (1 + r), is to be
taken to the n power. Of
course, your Excel formula
may refer to the contents of
the "n" column (the first
column in this worksheet)
for the corresponding row ,
instead of using a fixed
integer (1, 2, 3, etc.) in each
row, as shown here to
ensure your understanding.
(1) Projection of annual sales of products manufactured using licensed manufacturing process
(2) The discount rate is Company ABC's required rate of return, r : 0.08 (or, 8.0 percent)
Effect of cash flow structure on present value of cash flows. Note that the total of projected royalty
payments (undiscounted cash flows) in the above analysis, $1,000,000, is the same as the total of the
undiscounted cash flows in the proposed licensing agreement. However, the total discounted present value
of the royalty payments in this analysis, $650,556, is $20,452 less than the total discounted present value of
the original $100,000-per-year license agreement, $671,008. To understand this difference, study carefully the
“structure” or profile of the projected royalty payments and their PVs in the table above; compare these
amounts to the payments and PVs in the table, below, which analyzes the original proposed licensing
arrangement using the same approach and compares the PVs of each cash flow under both proposals:
Year
(n )
Licensing
fee (original
proposal)
Discount
factor
Discounted
PV of license
payment (1)
1
100,000
2
100,000
0.85734
85,734
71,588
14,146
3
100,000
0.79383
79,383
69,460
9,923
4
100,000
0.73503
73,503
67,623
5,880
5
100,000
0.68058
68,058
65,676
2,382
6
100,000
0.63017
63,017
63,962
(945)
7
100,000
0.58349
58,349
62,142
(3,793)
8
100,000
0.54027
54,027
60,510
(6,483)
9
100,000
0.50025
50,025
58,779
(8,754)
10
100,000
0.46319
46,319
57,204
(10,885)
Total
1,000,000
671,008
650,556
20,452
1 / (1 + r )n
x
0.92593
=
92,593
Discounted
PV of royalty
payment (2)
–
73,611
Diff. in PV
of alternative
proposals
=
18,981
(1) PV of licensing payments included in original proposal
(2) PV of royalty payments included in alternate proposal
9
Notice that, because of “inverse compounding” effects, the discount factors (in the third column of the table
immediately above), are smaller in each succeeding period. As a result, holding constant the expected or
contractual amount of a future cash flow and the discount rate, the farther into the future that the cash flow
occurs, the lower is its present value. That is, managers discount distant cash flows more “severely” than
near-term cash flows. Again, the greater discounting of cash flows that are more distant reflects the greater
opportunity cost to a business of waiting longer to receive them so it can invest the money at a return, r.
Next, notice that the annual licensing payment in the original proposal, $100,000, exceeds the
projected royalty payment in each of the first five years of the 10-year arrangement. The total
difference in payments for years 1 – 5 is $61,000. As a result, the present value of licensing
payments (original proposal) in each of those five years exceeds the present value of projected
royalty payments (alternative proposal). The total present value of the licensing payments
(original proposal) in years 1 – 5 exceeds the total present value of the royalty payments
(alternate proposal) in those years by about . . .
In years 6 – 10, however, the $100,000 annual licensing payment (original proposal) is less than
the projected annual royalty payments in the alternate proposal. The total difference in
payments for years 6 – 10 is ($61,000). As stated above, managers discount these more distant
cash flows more severely. As a result, the total present value of the royalty payments (alternate
proposal) in years 6 – 10 exceeds the total present value of licensing payments (original
proposal) in those years by about . . .
For years 1 – 10 in total, the PV of the payments under the original proposal exceeds the PV of
the payments under the alternate proposal by about . . .
$51,300
($30,800)
$20,500
As a result, managers of Company ABC will prefer the original proposal that requires a $100,000 annual
licensing payment over the alternate sales royalty arrangement.13
Effect of cash flow frequency on present value of cash flows. To illustrate further the effect of cash flow
structure on present value, reconsider once more Company ABC’s original proposed licensing agreement,
requiring a $100,000-per-year license payment.
Instead of requiring an annual $100,000 payment, managers have proposed a requirement that the licensee pay
Company ABC $25,000 at the end of each quarter (totaling $100,000 annually) throughout the term of the 10-year
agreement. Using the PVannuity equation examined above, managers determined the present value of this
alternative cash flow structure as follows:
1–
PVannuity
1
(1 + 0. 019426547)40
0.019426547
=
$25,000
x
=
$25,000
x 27.6326263
=
$690,816 (rounded)
Effective rate of return for cash flow series more frequent than annually. Recall that Company ABC’s required
annual effective rate of return is 8.0 percent (or, 0.08). In order to evaluate a proposed investment having a
series of (regular or irregular) cash flows occurring more frequently than annually, managers must recast the
company’s annual effective required rate of return, as follows:
_____
13
In the case that the PV of the alternate (sales royalty) proposal exceeded the PV of the original (licensing fee) arrangement,
managers of Company ABC must also consider seriously the inherent uncertainty of any ten-year sales forecast and the obvious
incentive for the licensee to purposefully bias upward the amount of projected sales in each year. If those projections ultimately prove
to be excessively optimistic, the licensee will benefit – at Company ABC’s expense – from a lower-than-projected royalty payment
obligation.
10
rn, required rate of return for non-annual period, n = (1 + required annual effective rate of return, re) 1/n – 1
In the case of Company ABC’s proposed series of quarterly cash flows, managers may compute the discount
rate using the above equation, as follows:
rQuarterly = (1 + 0.08)1/4 – 1, or 4√(1 + 0.08) – 1 = 0.019426547
They may also compute the periodic rate of return, rn, in an MS Excel spreadsheet using the following
formula: =(1+.08)^(1/4)-1. (Recall that “^” – the carat symbol on your computer’s “6” key – indicates that the
term, (1 + r), is to be taken to the 1/n power.)
Using an MS Excel spreadsheet, managers may determine the present value of this revised annuity using
the function (formula) described earlier:
=PV(rate,nper,-pmt,-fv,0) , where
Rate
Nper
Pmt
FV
0
=
=
=
=
=
rn, Company ABC’s required quarterly rate of return, 0.019426547 (or, 1.9426547%)
Number of periodic cash inflows: the 40 quarterly license payments
Amount of periodic cash flow: the $25,000 quarterly license payment
0, in the case of “straight” annuity, such as Company ABC’s license agreement
Indicates that payment due at end of annual periods through the end of the license agreement
In the Company ABC illustration:
MS Excel present value function (formula)
Formula result
=PV(0.019426547,40,-25000,0,0)
$690,816
The table below uses the spreadsheet approach to compute the PV of the quarterly cash flow series included
in the revised proposal of the original licensing agreement:
Licensing
Quarter fee (revised
(n )
Discount
Discounted
MS Excel
factor
PV of license
formula for
1 / (1 + r n)n
proposal)
payment
discount factors
24,524
=1/(1+0.01942655)^1
0.96225
24,056
=1/(1+0.01942655)^2
25,000
0.94391
23,598
=1/(1+0.01942655)^3
25,000
0.92593
23,148
5
25,000
0.90828
22,707
=1/(1+0.01942655)^4
=1/(1+0.01942655)^5
6
25,000
0.89097
22,274
=1/(1+0.01942655)^6
7
25,000
0.87399
21,850
8
25,000
0.85734
21,433
=1/(1+0.01942655)^7
=1/(1+0.01942655)^8
9
25,000
0.84100
21,025
10
25,000
0.82497
=1/(1+0.01942655)^9
20,624 =1/(1+0.01942655)^10
11
25,000
0.80925
20,231 =1/(1+0.01942655)^11
12
25,000
0.79383
19,846 =1/(1+0.01942655)^12
1
25,000
2
25,000
3
4
:
:
x
0.98094
:
=
:
:
36
25,000
0.50025
12,506 =1/(1+0.01942655)^36
37
25,000
0.49072
12,268 =1/(1+0.01942655)^37
38
25,000
0.48136
12,034 =1/(1+0.01942655)^38
39
25,000
0.47219
11,805 =1/(1+0.01942655)^39
25,000
0.46319
40
Total
1,000,000
11,580 =1/(1+0.01942655)^40
690,816
11
The total undiscounted cash flows are $1,000,000 over a 10-year period in both the original and
revised licensing agreement proposals. However, the present value of quarterly license
payments under the revised proposal, $690,816, is $19,808 greater than the present value of
annual license payments under the original proposal, $671,008. This increase in present value
represents the reduction in the opportunity cost to Company ABC because it does not wait as
long to receive money (so that it can invest the money at a rate of return, r).
The graphic below illustrates this difference in opportunity cost by examining the timing and present value of
the cash flows in the first year of Company ABC’s original proposed licensing agreement (requiring annual
payments of $100,000) and the revised proposal (requiring quarterly payments of $25,000):
First Year’s Cash Flows under Original and Revised Licensing Agreement Proposals
Original proposed licensing agreement – Annual end-of-year license payments of $100,000
Present value at time zero
Annual payment 1
$92,593
$100,000
Revised proposed licensing agreement – End-of-quarter license payments of $25,000
Quarterly payment 1
24,524
Quarterly payment 2
Quarterly payment 3
Quarterly payment 4
$25,000
24,056
$25,000
23,598
$25,000
23,148
$25,000
95,326
$ 2,733
Difference
When negotiating the terms of the ultimate licensing agreement with the prospective licensee, Company ABC’
managers will be aware that, while the revised (quarterly payment) proposal is a “better deal” for Company
ABC, the prospective licensee will prefer the original (annual payment) proposal. This is because the
licensee’s opportunity costs under the original proposal are less than under the revised proposal. That is,
the present value of the original proposal is $19,808 less than the present value of the revised proposal.
Effect of changes in discount rates on present value of cash flows. Careful examination of the PV
equation examined earlier in this background paper reveals that the present value of cash flows changes
inversely with changes in the discount rate (the required rate of return on investment, r). Holding constant the
amounts and timing of cash flows from an investment, if a manager increases the discount rate, the present
value of cash flows will decrease, and vice versa.
PV of cash flow occurring in period n
=
Cash flow in period n
(1 + discount rate)n
(Managers’ primary interest in TVM concepts is the application of present value, rather than future value,
analysis. This is because they confront the need to plan for a business’ future operations and make current
decisions in anticipation of expected future conditions or events.)
12
To illustrate the effect of changes in discount rates on the present value of cash flows, consider once more the
original proposed licensing agreement by ABC Company’s managers. That proposal requires the licensee to pay
the company $100,000 at the end of each year for 10 years. The PV analysis of the cash flows assumed that
Company ABC’s required rate of return, r, is 8.0 percent (or, 0.08). If, instead of 8.0 percent, Company ABC’s
required rate of return is 10.0 percent, using the PVannuity equation, the present value of the cash flows under the
originally proposed agreement decreases (by nearly 8.5 percent) from $671,008 to $614,457:
1–
PVannuity
1
(1 + 0.10)10
0.10
=
$100,000
x
=
$100,000
x 6.144567106
=
$614,457 (rounded)
Or, using an MS Excel spreadsheet and its present value function (formula), as described earlier:
MS Excel present value function (formula)
Formula result
=PV(0.10,10,-100000,0,0)
$614,457
The chart below summarizes the effect on the present value of a 10-year, $100,000 annuity of changes in
discount rates, ranging from 6.0 percent to 12.0 percent.
Present Value of Cash Flows under Proposed Licensing Agreement
(10-year, $100,000 Annuity)
$750,000
$700,000
$650,000
$600,000
$550,000
$500,000
6%
7%
8%
9%
10%
11%
12%
Company ABC's Required Rate of Return, r (Discount Rate)
The chart above suggests two important observations:
It is critical that managers select the proper discount rate to compute the PV of cash flows.
For example, in the chart, the PV of the 10-year $100,000 annuity ranges from about
$736,000 (using a 6.0 percent discount rate) to $565,000 (using a 12.0 percent discount
rate) – a difference of $171,000 and a 30 percent change from the PV6% value!
As examined further in the Topic 6 background paper, as a business’ required rate of return
increases, fewer investment opportunities will appear sufficiently attractive to justify
accepting them.
13
Estimating the Fair Value of Corporate Bonds and Notes
Businesses finance their investments – in property, plant, and equipment (PP&E), product research,
manufacturing technology and processes, and other firms – with the cash flows they generate from their daily
operations and a variety of equity and debt financial instruments. Equity instruments include principally
common stock and preferred stock. Examples of debt instruments that businesses use to finance their
investments include:
Corporate bonds (both collateralized and unsecured debentures)
Notes issued to financial institutions in connection with loans (both collateralized and unsecured)
Mortgages (collateralized by PP&E)
Commercial paper (short-term, unsecured borrowings)
Leases of real property, equipment, and vehicles
Businesses are also investors in the bonds and commercial paper issued by other firms.
At the time a business issues bonds or notes, it receives cash from bond investors or the financial institutions
accepting the notes (a cash inflow). Once issued, the governing agreement (bond indentures and note
agreements) require their issuers to make two kinds of cash payments (cash outflows):
Periodic payment of interest on the bond or note while it remains outstanding, and
Repayment of the principal (par or face14) amount borrowed under the instrument upon its maturity
Most corporate bonds and many bank notes require the issuer to repay the entire principal (par or face)
amount borrowed in full upon its maturity. However, other debt instruments – including mortgages and many
equipment leases – require the borrower to repay the principal amount borrowed throughout the term of
borrowing.15
Managers can readily estimate the fair value of bonds or notes by computing the present value of their cash
flows, using the PV equations defined earlier in this background paper, as the sum of the:
Present value of the bond’s or note’s interest payments (an annuity), plus
Present value of the principal (par or face) amount that the issuer will repay at maturity
The discount rate used to compute these present values is based on:
For publicly traded16 bonds, the current market interest rate for the particular bond issue
For non-publicly traded bonds and notes, the current market interest rate for publicly traded bonds
substantially similar to the non-publicly traded instrument under consideration. “Substantially similar”
means that the market interest rate used to determine the discount rate corresponds to publicly traded
bonds having a debt rating and remaining maturity similar to the non-publicly traded bond or note under
consideration.
_____
14
Corporations issue bonds in individual denominations of $1,000 par or face amount. For instance, if the par or face amount of an
entire bond issue is $250,000,000, the corporation issued 250,000 bonds each having a $1,000 par value.
15
Homeowners who acquired their homes subject to a traditional (say, 30-year or 15-year) mortgage are familiar with this debt
repayment structure, called a “fully amortizing” loan.
16
In the U.S., publicly traded securities are those listed and traded on a national securities exchange (such as the New York Stock
Exchange (NYSE) or the National Association of Securities Dealers Automated Quotation (NASDAQ) system, and registered, as
required by federal law, with the U.S. Securities Exchange Commission (SEC). Bond dealers at several investment firms maintain a
less formal market in many non-publicly traded bonds.
14
For example, in order to estimate the current fair value of non-publicly traded notes previously issued by a
corporation in connection with a bank loan, a manager would first look up the current debt rating of the
corporate issuer (using a financial Web search engine, such as Yahoo! Finance). Assume the manager
learned that the corporation’s current debt rating is “AA” and the notes have a maturity date that is slightly
less than six years away. She would next look up (using the financial Web search engine) the current
market interest rate for AA-rated, five-year bonds (current annual market yields are generally available only
for maturities of 2, 5, 10, and 20 years).17
Present value of bond or note interest payments. As indicated earlier in this background paper, the typical
fixed-rate U.S. corporate bond requires the issuer to pay interest to bondholders semi-annually. Notes issued
to financial institutions in connection with business loans typically require the borrower to make quarterly or
monthly interest payments. Accordingly, financial managers may compute the present value of the interest
cash flows using the PVannuity equation:
1–
PV of interest payments (annuity)
=
Periodic interest payment
x
1
(1 + y)n
y
Present value of bond or note principal (or face) amount. On the maturity date of a bond or note, the issuer
repays the bondholders the principal (or, face) amount of the bonds. The maturity date of a bond or note
coincides with the date of the final interest payment.
PV of principal (face) amount
=
Principal (face) amount
(1 + y)n
where,
y =
Current annual market rate (yield) on bond, or on “substantially similar” bonds or note
Number of periodic interest payments in each annual period (for most corporate bonds, this is 2)
n = Number of interest payment periods until the bond’s or note’s maturity date (e.g., for a 5-year bond,
this is 10)
and
Periodic
interest =
payment
Principal (face)
amount of bond
or note
x
Bond’s “coupon” rate or note’s stated (contractual) annual rate
Number of interest payments in each year
Total present value of bond or note interest payments and principal (or face) amount. The total present value
of a bond’s or note’s interest and principal cash flows represents the fair value of a publicly traded bond or the
estimated fair value of a non-publicly traded bond or note.
_____
17
Other finance courses examine the operations of financial markets, including the determination of market prices (fair market values)
and market rates of return for publicly traded securities, including corporate and government bonds, corporate stocks, and derivative
instruments (such as options and futures). The Financial Accounting Standards Board in the U.S. (discussed in the Topic 3-4
background paper) defined fair value as “the price that would be received to sell an asset or paid to transfer a liability in an orderly
transaction between market participants at the measurement date.” Note that, a given business’ required rate of return on investments,
r, and the market rate of return on a particular debt security are different concepts. As examined in the Topic 6 background paper,
managers seek out investments whose rate of return is equal to or greater than their business’ required rate of return, r. However, as
examined in the Topic 7 background paper, businesses often invest in securities whose returns are less than their required rate of
return, r, in order to achieve their day-to-day working capital management (liquidity) objectives.
15
To illustrate, Company MNO previously issued 20-year bonds having an 8 percent “coupon” rate, principal (face)
amount of $100,000,000, and maturity date of June 30, 20X5 – which is 5 years, or 10 interest payment periods
from today, July 1, 20X0 The bonds are non-publicly traded. Management believes that current market interest
rates (yields) are at a historical low-point and is currently considering refinancing the bonds with new, lower-rate
bonds. Management must first estimate the fair value of the existing bonds. The current fair value of the bonds
indicates the likely minimum amount that the company will have to pay the current bondholders to induce them to
redeem the existing, relatively high-rate bonds. The company’s current debt rating is “single-A” (accordingly to
credit-rating agencies Standard & Poor’s and Moody’s Investor Services) and the current annual market rate (yield)
for 5-year, A-rated bonds is 6.0 percent (or, 0.06).
1
(1 + .06 / 2)10
.06 / 2
1–
PV of interest payments (annuity)
PV of principal (face) amount
Total PV of bond’s cash flow (bond’s
estimated fair value)18
= $100,000,000 x 0.08 / 2
x
=
$4,000,000
x
=
$34,120,811
=
$100,000,000
(1 + .06 / 2)10
=
$ 74,409,392
= PV of interest payments +
(annuity)
=
$ 34,120,811
=
$108,530,203
8.53020283
PV of principal (face)
amount
+
$74,409,392
Company MNO managers may also use an MS Excel spreadsheet to determine the present value of
the bond’s cash flows using the function (formula) described earlier: =PV(rate,nper,-pmt,-fv,0)
where:
Rate
=
Nper
Pmt
=
=
FV
0
=
=
Current annual market rate (yield) on A-rated 5-year debt divided by the number of periodic interest
payments in each annual period (for most corporate bonds, this is 2): 6.0% / 2 = 3.0%
Number of remaining semi-annual interest payment periods until maturity: 5 years x 2 = 10
Periodic interest payment i.e., the coupon rate times the outstanding principal balance of the bonds
divided by 2: 0.08 x $100,000,000 / 2 = $4,000,000
Future value – the amount of the debt principal outstanding, due at maturity: $100,000,000
Payment due at end of semi-annual periods through maturity
The present value of the bond’s cash flows, and estimated fair value of the bonds, is:
MS Excel present value function (formula)
Formula result
=PV(6%/2,5*2,-100000000*0.08/2,-100000000,0)
$108,530,203
_____
18
Other finance courses examine the valuation of bonds on dates between interest payment dates.
16
Fair Value of a Business' Capital Financing and Unrecognized or Under-valued Assets
The Topic 3-4 Background Paper examines the limitations on the usefulness of financial statements.
Recall that two kinds of such limitations are:
— Trade-offs between the relevance and representational faithfulness (completeness or freedom
from error) or verifiability of financial statements caused by the combination of conservatism and
articulation. Under U.S. GAAP, accounting for such items as research and development (“R&D”) costs,
internally generated intangible assets, and inventory reflect this trade-off and may impair the relevance
of certain asset values reported in businesses' balance sheets.
— Definitions of financial statement elements and measurement uncertainty. Measurement uncertainty
and exclusion of most executory contracts from the definitions of “assets” and “liabilities” may impair the
completeness of businesses' balance sheets. Following the revenue recognition principle,
businesses generally must complete arms-length transactions in order to record increases in net
assets. In accordance with the historical cost principle, businesses report most assets at amounts
that reflect their original exchange value, rather than their current fair values.
Managers may use present value concepts, together with an examination of other information about a
business to identify:
The extent to which its balance sheet may fail to recognize “probable future economic benefits” (the
Financial Accounting Standards Board's definition of “assets”) or under-value recognized assets, and
The possible kinds or categories of assets that could be unrecognized or under-valued
Other information that may be useful in making this analysis is included in annual reports on Form 10K filed by public companies with the U.S. Securities and Exchange Commission. (The Topic 3-4
Background Paper examines the regulatory reporting framework of public companies in the U.S.,
including the periodic reports these companies must file with the SEC.) To perform this analysis,
recall the basic accounting equation, examined in the Topic 3-4 Background Paper:
Left side of balance sheet
ASSETS
Right side of balance sheet
=
LIABILITIES + STOCKHOLDERS’ EQUITY
or
INVESTMENTS
=
SOURCES OF FINANCING FOR THOSE INVESTMENTS
Accordingly, managers can estimate the implied fair value of a business’ assets – recognized and
unrecognized – by determining or estimating the fair value of its sources of financing, comprised of debt and
equity. For publicly traded shares of common or preferred stock and certain bonds, managers may locate
quoted market prices, as published by the securities exchanges. For non-publicly traded stocks and bonds,
managers may prepare valuations of the business or its individual issues of securities using present value
concepts. The table below illustrates the general framework for this analysis:
Estimated fair value of or quoted market price for ABC Company’s:
Common stock (10.0 million shares at $67.67 per share)
Preferred stock, if any
Debt, including bonds and notes
$US in millions
$ 676.7
–
583.3
Accounts payable, accrued and other liabilities, as reported in company’s balance sheet
240.0
Total fair value of company’s financing sources (liabilities and shareholders’ equity) and,
therefore, the implied fair value of company’s investments (total assets)
1,500.0
Less total assets reported in company’s balance sheet as of the valuation date
1,400.0
Excess difference
$ 100.0
17
The difference between the implied fair value of a business’ assets and the reported balance of total
assets in the business’ balance sheet as of the valuation date reflects the limitations of the financial
reporting model underlying U.S. GAAP (or IFRS). An excess difference represents the estimated fair
value of assets unrecognized or under-valued in the balance sheet. (Other courses examine the
accounting and financial implications of a deficiency, including the possibility of unrecognized asset
impairment and, in the case that the amount of a business’ liabilities exceed the fair value of its
assets, insolvency.)
Managers may estimate the fair value of a business’ non-publicly traded bonds and notes using the
present value methods described above and information contained in the business’ financial
statements, as illustrated below.
The financial statements of ABC Company for the fiscal year ended December 31, 20X2 include the following balance sheet
and footnote information related to the company’s debt and shareholders’ equity.
ABC Company
Consolidated Balance Sheet
December 31, 20X2
$US in millions
Assets:
Cash and cash equivalents
Liabilities and shareholders' equity:
$
Investments securities, at cost (to be held to maturity)
18 Accounts payable and accrued expenses
$
108 Income taxes payable
Accounts receivable, net
213
27
37 Notes payable - current portion (Note 6)
20
Inventory, at lower of LIFO cost or market value
182 Bonds payable - current portion (Note 6)
70
Total current assets
345 Total current liabilities
Property, plant, and equipment, at cost
Less accumulated depreciation
Property, plant, and equipment, net
1,420 Notes payable - noncurrent portion (Note 6)
140
(390) Bonds payable - noncurrent portion (Note 6)
350
1,030 Total liabilities
Investments in affiliated businesses
Total assets
330
820
25 Total common shareholders' equity (Note 9)
$
1,400 Total liabilities and shareholders' equity
580
$
1,400
18
Excerpts from FY 20X2 financial statement footnotes . . .
Note 6 - Debt
The company's debt at December 31, 20X2 is comprised of notes and
bonds payable, as follows:
($US in millions)
The amounts of debt outstanding as of December 31, 20X2 that
is payable in each of following five years and for all remaining
years thereafter in the aggregate are:
FYE Dec. 31,
Weighted average Principal balance
interest rate
Working capital loans
4.625%
Term loans
7.0 percent serial bonds
outstanding
$
90
40
20X4
90
5.375%
120
20X5
85
7.000%
420
20X6
75
Total
580
20X7
60
Less current portion
(90)
Thereafter
490
Total
Noncurrent portion
$
20X3
Principal due
$
180
$
580
In January 20X1, the company issued a $25 million note to a syndicate of banks in connection with a seven-year term loan that bears a
fixed 5.0 percent interest rate. The loan is secured by the company's assets and subject to financial and other covenants, including a
requirement that the company maintain a total debt ratio not exceeding 1.5:1 (or 1.50).
Note 9 - Shareholders' equity
The company's articles of incorporation authorize it to issue up to 15 million shares of the company's common stock, par value $20 per
share. At December 31, 20X2, 11.5 million shares of the company's common stock were issued and 1.5 million of those shares were
held by the company as treasury stock. The company's articles authorize it to issue up to 1.0 million shares of 7.5 percent cumulative
preferred stock, $100 par value. The company had issued no shares of preferred stock as of December 31, 20X2.
Market Information
The company’s bonds are currently rated "single A" (Standard & Poor’s) and "A2" (Moody's Investor Services).A Managers
separately located current market yields on "single A" corporate debt securities, by term to maturity, as follows:
1 year
4.750%
6 years
6.625%
2 years
5.125%
7 years
7.000%
3 years
5.500%
8 years
7.375%
4 years
5.875%
9 years
7.750%
5 years
6.250%
10 years
8.125%
The company's common stock currently trades at $67.67 per share on the NASDAQ.
Managers estimated the fair value of the company’s outstanding debt “portfolio” using the information obtained above, as
set forth below:
19
Remaining semi-
Estimated fair
Current yield on comparably rated debt value (PV of debt
(3)
principal
annual interest
payment
Outstanding
periods (Years debt principal at
Year of
to maturity
Dec. 31, 20X2
maturity
TIMES 2)
($US millions)
20X3
2
20X4
$
Semi-annual Interest (1)
Rate
Payment (2)
Term
Annual yield
Semi-
outstanding and
annual
interest payments)
yield (4)
(5)
90.0
3.250% $
2.93
1 year
4.750%
2.3750% $
91.5
4
90.0
3.250%
2.93
2 years
5.125%
2.5625%
92.3
20X5
6
85.0
3.250%
2.76
3 years
5.500%
2.7500%
87.3
20X6
8
75.0
3.250%
2.44
4 years
5.875%
2.9375%
76.6
20X7
10
60.0
A
3.250%
1.95
5 years
6.250%
3.1250%
60.6
180.0
B
Thereafter
3
Years (B / A) (6)
20X8
12
60.0
1 Yr
3.250%
1.95
6 years
6.625%
3.3125%
59.6
20X9
14
60.0
2
3.250%
1.95
7 years
7.000%
3.5000%
58.4
20Y0
16
60.0
3
3.250%
1.95
8 years
7.375%
3.6875%
56.9
20Y1
18
-
3.250%
-
9 years
7.750%
3.8750%
-
20Y2
20
-
3.250% $
-
10 years
8.125%
4.0625%
-
Total
$
580.0
$
583.3
(1) Assume all debt outstanding requires semi-annual interest payments and matures December 31 of years disclosed.
The weighted average interest rate on company's "portfolio" of debt is estimated using footnote information, as follows:
Debt footnote information
Weighted
Principal
Wghtd avg int
average
balance
Percent of rate X Percent
interest rate outstanding
Working capital loans
4.625% $
Term loans
7.0 percent serial bonds
total
of total
40.0
6.9%
0.319%
5.375%
120.0
20.7%
1.112%
7.000%
420.0
72.4%
5.069%
580.0
100.0%
$
6.500% Semi-annually, 3.25%
(2) Semi-annual interest payments computed as: debt principal outstanding TIMES estimated semi-annual interest rate, in (1)
(3) Company's debt rating and current annual yields on comparably-rated debt obtained via financial Websearch.
(4) Semi-annual yield (discount rate) used to compute present value of debt principal outstanding and interest payments
through maturity is the annual yield DIVIDE 2. [The effective annual yield = 1 + annual yield / 2 ) 2 - 1]
(5) Present value of debt principal outstanding and interest payments for each maturity year computed using the MS Excel formula:
=PV(rate,nper,-pmt,-fv,0)
where:
Rate = Semi-annual yield on comparably-rated debt for the maturity period indicated in (4)
Nper = Remaining semi-annual interest payment periods until maturity (years to maturity TIMES 2)
Pmt = Semi-annual interest payment, computed in (2) above
FV
= Future value - i.e., the debt principal outstanding for the maturity year indicated
0
= Payment due at end of semi-annual periods through maturity
(6) Financial statement footnotes of companies disclose annual maturities of debt only for the first 5 years following the
balance sheet date and disclose the remaining maturities of debt only as a "thereafter" total. For the purposes of
estimating the fair value of a company's total debt, analysts estimate the amount of annual maturities after Year 5 by
dividing the "thereafter" total by the amount of the debt maturing in Year 5 to estimate the average number of
years over which the "thereafter" total matures. Analysts then assume that an amount similar to that maturing in
Year 5 will also mature in each subsequent year (here, 20X8 - 20Y0).
20
A
The table below summarizes the long-term debt ratings assigned to corporate debt securities by three
principal debt-rating agencies. The rating agencies call securities rated below BBB- (S&P and Fitch) or Baa3
(Moody’s) “non-investment grade” or “speculative” because of the issuer presents pronounced credit (default)
risk.
S&P
Moody’s
Fitch
Credit-worthiness category
AAA
Aaa
AAA
Prime
AA+
Aa1
AA+
High grade
AA
Aa2
AA
AA-
Aa3
AA-
A+
A1
A+
A
A2
A
A-
A3
A-
BBB+
Baa1
BBB+
BBB
Baa2
BBB
BBB-
Baa3
BBB-
Upper medium grade
Lower medium grade
21
Net Present Value
So far, this background paper has examined arrangements or financial instruments for which the cash flows
are all inflows or all outflows. In the Company ABC licensing agreement proposal, all the cash flows examined
were inflows to the company; in the Company MNO illustration, all the bond-related cash flows examined were
outflows to the company. Most investments examined by a business require cash outflows (the initial
investment), followed by cash inflows (returns on investment and, eventually, return of the initial investment).
For example, a business may:
Acquire additional plant and equipment in order to expand productive capacity,
Replace production equipment with more efficient or more highly automated (cost-saving) equipment,
Launch a product research-and-development project,
Introduce a new product, or
Acquire a competitor or supplier business
Managers evaluate the acceptability of these and similar proposed investments based on their net present
value (NPV). NPV analysis is central to capital budgeting, examined in the Topic 6 background paper.
However, to clarify the NPV concept at this point, consider the following brief illustration:
Management of Company XYZ has proposed that the company acquire
an additional drill press. Demand for the company’s products has risen
to a level that exceeds its present productive capacity. An additional
drill press will make it possible for the company to increase its
production and sales by 20 percent, resulting in projected additional
operating cash flows (OCF) in each of the next five years, as indicated
at right:
The equipment will cost $640,000 to purchase and install.
Management estimates that its economic life will be five years, after
which it will have no residual value.
Year
Projected increase in OCF
20X1
$150,000
20X2
180,000
20X3
200,000
20X4
170,000
20X5
140,000
Total
$840,000
The company’s vice president of operations is confident that this proposal will “pencil out” favorably, because the
projected additional OCF, $840,000, exceeds the cost of the equipment by $200,000. She prepared an NPV
analysis of the drill press proposal using a computer spreadsheet, which is set forth below. In doing so, she
discounted the relevant cash flows at the company’s required rate of return on investments, 9.5 percent (or, 0.095).
(To simplify her analysis, she assumed that the projected increases in OCF occur at the end of years indicated.)
22
Relevant
cash
flows (1)
Year (n)
Discount
factor (2)
1 / (1 + r )
n
20X0
0
20X1
1
150,000
0.91324
136,986
20X2
2
180,000
0.83401
150,122
20X3
3
200,000
0.76165
152,331
20X4
4
170,000
0.69557
118,248
20X5
5
140,000
0.63523
88,932
Net cash flows
(640,000) x
Discounted
cash flows
1.00000
200,000
=
MS Excel
formulas for
discount factors
(640,000)
NPV
=1/(1+0.095)^0
=1/(1+0.095)^1
=1/(1+0.095)^2
=1/(1+0.095)^3
=1/(1+0.095)^4
=1/(1+0.095)^5
6,619
(1) Relevant cash flows are those which are incremental to the proposed investment
(2) The discount rate is Company XYZ's required rate of return, r : 0.095 (or, 9.5%)
Note that this analysis includes the initial investment (the cost of the drill press) as a negative cash flow at
“time zero.” Managers do not discount cash flows occurring at the outset of an investment. To ensure this
result, they use a discount factor is 1.0 in their spreadsheet analyses.
An expression of the NPV analysis of the drill press proposal in the form of an equation is:
NPV of drill press
=
proposal
– Initial investment cash
Year 1 OCF incr.
+
+
outflow (cost of drill
(1 + r)1
press)
=
($640,000)
=
$6,619
+
$136,986
+
Year 2 OCF incr.
(1 + r)2
$150,122
+
+
Year 3 OCF incr.
(1 + r)3
$152,331
+
+
Year 4 OCF incr.
(1 + r)4
$118,248
+
+
Year 5 OCF incr.
(1 + r)5
$88,932
The analysis shows that the NPV of the drill press proposal is positive $6,619. It is immediately apparent that
the NPV of the proposed investment is very small in comparison to the net undiscounted cash flows, $200,000.
However, the important conclusion that managers should draw from this analysis is that, because the NPV is
zero or positive, the expected rate of return on the proposed investment exceeds the company’s required rate
of return on new investments. Therefore, the company should approve the investment proposal, provided that:
Management is confident that the projected increases in OCFs are substantially accurate19, and
The company has a sufficient amount of financial capital available to finance the acquisition of the drill
press20
The general decision rules in NPV analysis are:
If the NPV of a proposed investment is zero or positive, accept the proposal
If the NPV of a proposed investment is negative, reject the proposal
Positive-NPV investments increase the value of a business to its owners, while negative-NPV
investments destroy its value.
______
19
The Topic 6 background paper examines sensitivity analysis, used in connection with capital budgeting. In the case of the proposed
drill press investment, if actual results later prove that the projected increases in OCF were overstated by more than 1 percent in each
year, 20X1 – 20X5, the NPV of the investment is negative, indicating (in hindsight) that managers should have rejected the proposal!
20
Management may need to ration a limited amount of available financial capital among multiple investment opportunities, as
discussed below. The Topic 6 background paper examines businesses’ required rate of return (cost of capital). Company XYZ’s cost
of capital establishes its required rate of return on new investments.
23
Relevant Cash Flows and Sunk Costs
In the previous illustrations that examined Company ABC’s proposed licensing agreement, the PV analyses
properly ignored the company’s initial costs of developing the manufacturing process that was the subject of
the proposed agreement. This is because the cash flows representing those development costs are sunk
costs. That is, the company developed and patented its manufacturing process for its own use some time
before the prospect of licensing it arose. Therefore, those initial costs are not relevant to the analyses in these
illustrations.21 (Hopefully, managers included those initial cash outflows in an earlier NPV analysis of the
proposed investment in the manufacturing process.) For purposes of analyzing proposed investments, only
incremental cash flows are relevant cash flows. Incremental cash flows are those that are:
Directly related to the proposed investment and
Occur only if the business accepts the proposal
As stated at the outset of this background paper, cash flows representing any costs a business incurs before
managers make their decision to accept or reject an investment proposal are not relevant cash flows.
The Topic 7 background paper examines the application of NPV analysis to setting customer credit policy.
Learning
Objective 2
Capital Rationing and Ranking Multiple Investment Opportunities
In theory, a business should be able to obtain financing for all investments for which the capital budgeting
analysis indicates the NPV is positive. This is because positive-NPV investments promise rates of return that
exceed the business’ cost of capital and, therefore, they increase the value of the business to its owners.
However, businesses generally face practical constraints on the amount of financial capital that is available to
them in the short-term (say, one year). For example, the process of issuing common or preferred stock or
bonds, or negotiating loans from a syndicate of commercial banks is time-consuming and costly. As a result,
many businesses must ration available financial capital, accepting some opportunities while rejecting others.
The proper method for ranking possible investments and rationing capital to the highest-ranked opportunities
uses the profitability index (PI) (or, excess present value index):
Profitability index (PI)
=
PV of cash flows subsequent to initial investment
Initial investment cash flow
=
NPV of investment + Initial investment cash flow
Initial investment cash flow
Close examination of the PI equation above reveals that the PI for a proposed investment will be greater than
1.0 if its projected NPV is positive. Accordingly, following the general NPV decision rules listed above,
businesses should accept investment proposals for which the PI is greater than 1.0. However, when a
business has only limited capital available, it may ration that capital to investment opportunities by ranking
them according their PI.22
_____
21
The illustrations involving the proposed licensing agreement implicitly assumed that the prospective licensee would not use
Company ABC’s manufacturing process to make products that would compete with those of Company ABC. In the case where such an
arrangement would result in erosion of sales by Company ABC, managers’ NPV analysis should include the contribution margin
related to projected lost sales. (The Topic 6 background paper examines sales erosion in connection with capital budgeting analysis of
proposed new products.) Of course, if Company ABC’s managers expected the licensing agreement to lead to increased competition
for the company, they would be extremely skeptical about “making the deal.”
22
However, the PI is not effective for rationing capital over more than one period (say, beyond one year).
24
To illustrate, assume that Company RST has an approved total capital budget of $12,000,000 for the coming fiscal year and
managers have identified and estimated the NPV of the following eight investment opportunities, A through H:
(1)
(2)
(2) - (1)
(2) / (1)
Project
Initial
investment
cash flow
PV of
cash flows
subsequent
to initial
investment
NPV
Profitability
index (PI)
A
$7,200,000
$ 9,350,000
$2,150,000
1.30
B
1,260,000
1,440,000
180,000
1.14
C
1,440,000
2,250,000
810,000
1.56
D
960,000
1,260,000
300,000
1.31
E
1,140,000
1,300,000
160,000
1.14
F
2,100,000
2,415,000
315,000
1.15
G
2,400,000
2,150,000
(250,000)
0.90
H
1,080,000
1,090,000
10,000
1.01
Total
17,580,000
Absent any capital constraints, the company should pursue each of these projects, except Project G, for which the
estimated NPV is negative. The remaining opportunities all have a positive NPV and therefore serve to increase the value
of the company. In light of the company’s capital constraint, managers ranked the proposed investments (projects) based
on their PIs, as follows:
(1)
(2)
(2) - (1)
(2) / (1)
Project
Initial
investment
cash flow
Cumulative
total initial
investment
cash flow
PV of
cash flows
subsequent
to initial
investment
NPV
C
$1,440,000
$ 1,440,000
$ 2,250,000
$ 810,000
$ 810,000
1.56
D
960,000
2,400,000
1,260,000
300,000
1,110,000
1.31
A
7,200,000
9,600,000
9,350,000
2,150,000
3,260,000
1.30
F
2,100,000
11,700,000
2,415,000
315,000
3,575,000
1.15
E
1,140,000
12,840,000
1,300,000
160,000
3,735,000
1.14
B
1,260,000
14,100,000
1,440,000
180,000
3,915,000
1.14
H
1,080,000
15,180,000
1,090,000
10,000
3,925,000
1.01
G
2,400,000
17,580,000
2,150,000
(250,000)
3,675,000
0.90
Cumulative
total
Profitability
NPV
index (PI)
Of course, management immediately rejected project G because its estimated NPV is negative (and consequently, the rate
of return on this investment is less than the company’s cost of capital).
Initially, it appears that managers should approve projects C, D, A, and F (and reject the remaining proposals) because
These are the highest-ranking opportunities based on their respective PIs, and
The cumulative total initial investment required by these four projects, $11,700,000, is within the approved budget of
$12,000,000
The estimated cumulative total NPV from these four investments is $3,575,000.
25
However, on closer inspection, managers noticed that, if the company approved Projects B and E, and rejected Project F,
the company would be able to:
Invest the entire $12,000,000 of capital available, and
Maximize the cumulative total NPV from all five accepted projects at $3,600,000 (note that the total NPV of Projects B
and E is $340,000, $25,000 more than the NPV of Project F, $315,000):
(1)
(2)
(2) - (1)
Project
Initial
investment
cash flow
Cumulative
total initial
investment
cash flow
PV of
cash flows
subsequent
to initial
investment
NPV
C
$1,440,000
$ 1,440,000
$ 2,250,000
D
960,000
2,400,000
1,260,000
A
7,200,000
9,600,000
E
1,140,000
B
F
(2) / (1)
Cumulative
total
Profitability
NPV
index (PI)
Decision
$ 810,000
$ 810,000
1.56
Approve
300,000
1,110,000
1.31
Approve
9,350,000
2,150,000
3,260,000
1.30
Approve
10,740,000
1,300,000
160,000
3,420,000
1.14
Approve
1,260,000
12,000,000
1,440,000
180,000
3,600,000
1.14
Approve
2,100,000
14,100,000
2,415,000
315,000
3,915,000
1.15
Reject
H
1,080,000
15,180,000
1,090,000
10,000
3,925,000
1.01
Reject
G
2,400,000
17,580,000
2,150,000
(250,000)
3,675,000
0.90
Reject
Of course, strategic, operating, or other non-financial factors may affect managers’ decision to select Project F
over Projects B and E. For example, Projects B and E may present political or reputational issues that it is not
practical for managers to factor into their NPV analyses.
*****
Course developer’s note on content of and sources used in preparing course background papers:
In selecting the content and determining the organization of material for this course, the developer considered a
number of factors, including the university’s MBA program outcomes, material examined in the program’s
accounting and finance foundation courses, and material examined in subsequent MBA accounting and finance
courses, in particular MBA A602 (Interpreting Accounting Information) and MBA F602 (Financial Decision-making),
which, in turn, are prerequisites for other MBA accounting and finance courses. The course developer reviewed
several accounting and finance texts, listed below, to ensure that the examination of models, concepts, methods,
and terminology in the background papers is generally consistent with a variety of such texts over time. The course
developer noticed that, on the one hand, there is substantial similarity among texts in the material (models,
concepts, methods, and terminology) examined. The developer also noticed that, on the other hand, in spite of this
similarity, none of these texts appears to include references to other texts (just original research articles,
authoritative accounting literature, and the occasional federal statute, internal revenue code section, or IRS
regulation). In addition, much of the material examined in the texts and the background papers is the subject of
articles on several unrestricted Websites, such as Wikipedia.com. As such, the material contained in the
background papers represents both essential and common knowledge for business managers.
Atkinson, A. A., Banker, R. D., Kaplan, R. S. & Young, S. M. (1995). Management accounting. Englewood Cliffs, New Jersey:
Simon & Schuster Co./Prentice-Hall.
Horngren, Charles T. (1977). Cost accounting: A managerial emphasis. (4th ed.). Englewood Cliffs, NJ: Prentice-Hall.
Kieso, D. E., Weygandt, J. J., & Warfield, T. D. (2004). Intermediate accounting. (11th ed.). New York: Wiley & Sons.
Ross, S. A., Westerfield, R. W. & Jaffe, J. E. (1993). Corporate finance. (3rd ed.). Burr Ridge, IL: Richard D. Irwin.
Wild, J. J., Subramanyam, K. R. & Halsey, R. F. (2004). Financial statement analysis. (10th ed.). New York: McGraw-Hill/Irwin.
Wolk, H. I., Dodd, J. L. & Tearney, M. G. (2004). Accounting theory: Conceptual issues in a political and economic environment.
(6th ed.). Mason, OH: Thomson Learning/South-Western.
26
Instructions:
Using the company and market information provided below, complete the following two tabs in this MS Excel Workbook:
– Computation of the estimated current fair value of the note issued by the company in 20X3
– Computation of the estimated current fair value of the company's "portfolio" of debt (notes and bonds) outstanding, as
reported in its fiscal year end (FYE) 20X4 balance sheet
The background papers, Present Value Concepts and Bond Valuation, and Financial Statement Concepts and Financial
Reporting provide useful guidance for completing this assignment.
Based on the results of your computed estimate of the fair value of the company's debt "portfolio" (the second tab following this one)
and the additional information provided below, indicate (i) the apparent total fair value ($US) of assets unrecognized or under-valued in
the FYE 20X4 balance sheet and (ii) the possible kinds or categories of assets that could be unrecognized or under-valued and the
apparent reasons for this. (In formulating your response, it may be helpful to review pages 17 - 18 of the background paper, Present
Value Concepts and Bond Valuation , including the illustration on those pages and pages 38 and 41 - 47 of the background paper,
Financial Statement Concepts and Financial Reporting .) Limit the length of your response to 150 words.
Replace the text in this cell with your response.
Advanced Technology and Services Company
Consolidated Balance Sheet
December 31, 20X4
$US in millions
Assets:
Liabilities and shareholders' equity:
Cash and cash equivalents
15
Accounts payable and accrued expenses
Investments securities, at cost (to be held to maturity)
75
Income taxes payable
20
Accounts receivable, net
18
Notes payable - current portion (Note 5)
10
Inventory, at lower of LIFO cost or market value
192
Bonds payable - current portion (Note 5)
50
Total current assets
300
Total current liabilities
250
Notes payable - noncurrent portion (Note 5)
100
(380) Bonds payable - noncurrent portion (Note 5)
250
$
Property, plant, and equipment, at cost
1,045
Less accumulated depreciation
Property, plant, and equipment, net
665
Investments in "strategic partner" suppliers
35
Total assets
$
1,000
$
170
Total liabilities
600
Total common shareholders' equity (Note 7)
400
Total liabilities and shareholders' equity
$
1,000
Excerpts from FY 20X4 financial statement footnotes . . .
Note 5 - Debt
The company's debt at December 31, 20X4 is comprised of notes and
bonds payable, as follows:
($US in millions)
Weighted
average interest Principal balance
rate
outstanding
Notes payable to banks
4.00%
Notes payable to other lenders
6.0 percent serial bonds
$
The amounts of debt outstanding as of December 31, 20X4
that is payable in each of following five years and for all
remaining years thereafter in the aggregate are:
FYE Dec. 31,
20X5
Principal due
$
60
54
20X6
65
5.00%
56
20X7
60
6.00%
300
20X8
60
Total
410
20X9
55
Less current portion
(60)
Thereafter
110
Noncurrent portion
$
350
Total
$
410
In January 20X3, the company issued a $30 million note to a syndicate of banks in connection with a seven-year term loan that bears a
fixed 4.25 percent interest rate. The loan is secured by the company's assets and subject to financial and other covenants, including a
requirement that the company maintain a total debt ratio not exceeding 1.5:1 (or 1.50).
Note 7 - Shareholders' equity
The company's articles of incorporation authorize it to issue up to 25 million shares of the company's common stock, par value $5 per
share. At December 31, 20X4, 22.0 million shares of the company's common stock were issued and 2.154 million of those shares
were held by the company as treasury stock. The company's articles authorize it to issue up to 2.5 million shares of 8 percent
preferred stock, $100 par value, for which dividends shall be cumulative. At December 31, 20X4, the company had issued no shares of
preferred stock.
Market information
The company's 6.0 percent serial bonds are currently rated "single A" (Standard & Poors) and "A2" (Moody's Investor Services)
Current market yields on "single A" corporate securities, by term to maturity are:
1 year
4.625%
6 years
6.500%
2 years
5.000%
7 years
6.875%
3 years
5.375%
8 years
7.250%
4 years
5.750%
9 years
7.625%
5 years
6.125%
10 years
8.000%
The company's common stock currently trades at $25 per share on the NASDAQ.
The facilitator will grade this assignment, assigning up to 100 points for it as follows:
Maximum
Earned
Complete, accurate, and clear presentation of calculations of the estimated fair value of:
– The specified individual company bond or note issue
10
– The company’s aggregate debt “portfolio”
75
(1) Clear and accurate indication of the apparent total fair value ($US) of assets
unrecognized or under-valued in the balance sheet and (2) clear, concise, and
complete description of the possible kinds or categories of assets that could be
unrecognized or under-valued and the apparent reasons for this
15
Total points
100
points
-
S17S8W2
A1
Instructions: Use the information in the first worksheet tab (Instructions and company information) to complete the
analysis in this tab. Show all computations in good form and label properly all amounts presented .
Compute the current fair value of the bank note syndicated in 20X3, showing separately to the nearest whole $US
dollar (1) the present value of the principal (face) amount of the note and (2) the present value of related interest
payments due under the note. You may assume that the note interest payments are either annual or semiannual. Your choice!!!
Present Value (PV) of a Bond Example:
Given:
Face Value of Bond
$30,000,000
Discount Rate (YTM)
Interest %
Number of Periods
PThe PV of a Bond is calculated below, using the Example information listed above
PV Calculations Assuming Annual Payments
PV or Price of bond
=
PV Calculations Assuming Semi-Annual Payments
PV or Price of bond
=
17S8W2
B1
Remaining
semi-annual
interest
Outstanding
payment
debt principal at
Year of
periods (Years Dec. 31, 20X4
maturity (1)
to maturity
($US millions) 17F8W2
Semi-annual Interest
Rate
Payment
Estimated fair
value (PV of debt
principal
Current yield on comparably rated debt outstanding and
Semiinterest
Annual yield
annual
payments) ($US
Term
17S8W2
Use the in
workshee
company
complete
tab.
A
Thereafter
B
Years (B / A)
Enter num
labels, as
shaded
leaving th
workshee
Total
(1) Assume all debt outstanding requires semi-annual interest payments and matures December 31 of years disclosed.
(2) Show your computation of the estimated weighted average interest rate on the debt "portfolio" using footnote information, below:
Debt footnote information
Weighted
average
Principal
interest
balance
rate
outstanding
Percent of
total
%
Wghtd avg int
rate X Percent
of total
To ensure
your work
whenever
help enab
understan
reasoning
"partial cr
Year of
maturity (1)
Use the information in the first
worksheet tab (Instructions and
company information) to
complete the analysis in this
Remaining
semi-annual
Outstanding
interest
debt principal
payment
at Dec. 31,
periods (Years 20X4 ($US
17F8W2
to maturity
millions)
To ensure maximum credit for
your work, use formulas
whenever possible. This will
help enable the faciliator's
understanding of your
reasoning, which may justify
"partial credit" for inaccurate
Semi-annual Interest
Rate
Payment
Term
Annual
yield
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
A
Error
Error
Error
Error
Error
B
Error
Years (B / A)
Thereafter
Enter numbers, formulas, or
labels, as appropriate, in
shaded worksheet cells only,
leaving the remainder of this
worksheet tab unchanged.
Current yield on comparably rated
debt
17S8W2
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Total
Error
(1) Assume all debt outstanding requires semi-annual interest payments and matures December 31 of years d
(2) Show your computation of the estimated weighted average interest rate on the debt "portfolio" using footno
Debt footnote information
Weighted Principal
average
balance
Percent of
interest rate outstanding
total
Wghtd avg
int rate X
Percent of
total
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Estimated
fair value
n comparably rated (PV of debt
debt
principal
Semi- outstanding
annual and interest
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
Error
ber 31 of years disclosed.
olio" using footnote information, below:
17S8W2
D1
D5
Purchase answer to see full
attachment