Scott, Financial Accounting Theory, 7th Edition Instructor’s Solutions Manual Chapter 3
CHAPTER 3
THE DECISION USEFULNESS APPROACH TO FINANCIAL REPORTING
3.1 Overview
3.2 The Decision Usefulness Approach
3.2.1 Summary
3.3 Single–Person Decision Theory
3.3.1 Decision Theory Applied
3.3.2 The Information System
3.3.3 Information Defined
3.3.4 Summary
3.4 The Rational, Risk–Averse Investor
3.5 The Principle of Portfolio Diversification
3.5.1 Summary
3.6 Increasing the Usefulness of Financial Reporting
3.6.1 Introduction
3.6.2 Objectives of Management Discussion and Analysis
3.6.3 An Example of MD&A Disclosure
3.6.4 Is MD&A Decision useful?
3.6.5 Conclusion
3.7 The Reaction of Professional Accounting Bodies to the Decision Usefulness
Approach
3.7.1 The Conceptual framework
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3.7.2 Summary
3.8 Conclusions on Decision Usefulness
LEARNING OBJECTIVES AND SUGGESTED TEACHING APPROACHES
1. Decision Usefulness
The main purpose of this Chapter is to provide a framework for understanding the
concept of decision usefulness of financial reporting. Consistent with the conceptual
framework, I assume that the major decision problem to which financial reporting is
oriented is the investment decision. I then argue that if accountants are to produce
financial statements that are useful for investment decisions, they need to understand
how rational investors make such decisions.
2. Single–person Decision Theory
I use this theory, including the revision of beliefs by means of Bayes’ theorem, as a
model of rational investment decision making. Prior to getting into the theory itself, I
usually discuss with the class how they would proceed to make investment decisions if
they had a sum of money to invest, and steer the discussion to make the point that
single–person decision theory provides a systematic and formal way to do what many of
them would do anyway. Some instructors and students may disagree with this
argument, in view of increasing acceptance by academics that securities markets are
not fully efficient, and that investors may not be rational in an economic sense. These
issues are discussed in Section 6.2. I argue there that securities markets are sufficiently
close to full efficiency that the efficient markets model is still the most useful one to use
for studying the information needs of investors, and that if some of the underlying
assumptions of many economic models are relaxed, the rational investment decision
model can explain security price behaviour that is often attributed to non–rational
investor behaviour. I also argue that to the extent securities markets are less than fully
efficient, the scope for decision useful information is increased.
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I stick quite close to the text when illustrating the decision theory model, since this
model and the concepts that go into it are new to most students. I go over in detail
either the text example or some other similar example such as one of the end–of–
chapter problems. I always end up by asking the class how realistic they think the model
is (see point 4 below for additional discussion). If a student is particularly critical, I fall
back on asking again how he or she would make an investment decision under similar
circumstances. I do not particularly try to defend the extent to which the model can be
operationalized. However, as stated above, I do make the argument that whether it is
operational or not, it is a very useful conceptual device to help us understand what
information is and how investors may find financial statement information to be useful.
It is important to emphasize that the decision theory model is a model of an average
investor. There is no implication that all investors act this way. The real question is
whether investors on average behave as the model predicts or whether on average they
are biased away from the model’s predictions.
3. The Concept of an Information System
This is one of the most important concepts in the text. While the idea of the financial
statements being represented as a table of objective, conditional probabilities may take
some getting used to, the information system provides the crucial link between current
reported performance and the future performance of the firm. It conceptualizes the
quality of the financial statements with respect to their usefulness for investment
decisions.
Many reporting issues can be conceptualized by their effect on the main diagonal
probabilities of the information system. For example, a switch from historical cost to
current value accounting, or earlier recognition of revenue, increases these probabilities
by increasing relevance. This tightens up the relationship between current and future
performance and, other things equal, increases decision usefulness. However, to the
extent that fair value accounting and early revenue recognition are less reliable than
historical cost, this would have the opposite effect on the main diagonal probabilities.
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The net effect on decision usefulness is thus not clear, but the information system is
helpful in conceptualizing the nature of the trade–offs in accounting policy choice and
the concept of earnings quality—higher main diagonal probabilities, higher quality.
There are numerous ways of measuring earnings quality empirically, such as analyst
forecast revisions, market response to net income, accruals. All of them can be tied
back conceptually to the main diagonal probabilities.
I find that the students’ understanding of the information system is helped if the
instructor spends some time on the two extremes — a perfect system and a useless
system. See Problem 1 of this chapter. This problem pushes understanding by showing
what happens when state probabilities are revised by Bayes’ theorem using the
conditional probabilities from the perfect and from the useless information systems.
I have structured the states of nature in Example 3.1 in terms of future firm
performance, rather than some more primitive states such as good economy or bad
economy. Future firm performance can be conceptualized in terms of future cash flows,
earnings, or dividends
Many discussions and models of firm performance and value are based on expected
future dividends or cash flows, particularly in the finance literature. I include future
earnings as an alternate measure of performance and value to be consistent with
Ohlson’s Clean Surplus Theory, which is discussed in Section 6.10. The Ohlson theory
shows that the market value of the firm can equally be expressed in terms of expected
future dividends, cash flows or financial statement variables. Since this is an accounting
text, it seems natural to take financial statement variables such as earnings as a
fundamental determinant of firm performance and value, on an equal footing with cash
flows and dividends. Also, the Ohlson theory and modifications and extensions of it,
have become quite common in empirical accounting research and in accounting
practice. Empirical implications of clean surplus theory are discussed and illustrated in
Section 6.10.4.
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4. Does it Work?
While, as stated above, I do not particularly try to defend the decision theory model as
an operational way to make decisions, I do spend some time discussing with the class
whether they would be willing to make an investment decision this way. The following
notes, which could be distributed to the students, discuss some of the issues in applying
the procedure.
Issues in Applying the Decision Theory Model
Specifying the states of nature. States of nature are specific to the decision problem
at hand. For example, if my decision is whether or not to take my raincoat, relevant
states would be rain or no rain. That is, the relevant states of nature are simply those
random events whose outcome matters to the decision at hand. In an investment
context, these can be taken as different levels of future firm performance, since it is
future performance that determines investment payoff.
Specifying prior probabilities of the states of nature. These capture everything the
decision maker knows up to the beginning of the decision analysis. There are
techniques to help specify these probabilities. One technique is to conceptualize an urn
containing 100 coloured balls, of which a certain number are red and the remainder
black. To illustrate, suppose an investor wants to assess his/her prior probability of high
future firm performance next year. Envisage a bet of, say, $50 on this state—if future
performance is high you win $50, otherwise you lose $50. Now consider another bet.
You will draw 1 ball from the (opaque) urn. If you draw a red ball you win $50, otherwise
you lose $50. How many red balls should there be in the urn so that you are indifferent
between the 2 bets? Suppose you decide you would be indifferent if the urn contains 6
red balls. Then, your subjective probability of high future firm performance is 0.06.
Since prior probabilities are subjective, we cannot say that this probability is “correct.”
The point is, however, that in deciding on the number of red balls you are forced to
consider everything you know about the firm’s future prospects.
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Specifying payoffs. For each state of nature, specification of your payoff if a particular
state happens should be relatively straightforward. For example, suppose you invest
$10,000 in shares of X Ltd. and the high performance state happens. Analysis of past
share price behaviour of X Ltd. when the firm is performing well may reveal an average
share return of 16%, that is, a net payoff of $1,600.
Of course, if you decide to invest your $10,000 in a riskless asset instead, the states of
nature for X Ltd. do not affect your payoff—if you buy a government bond yielding
21/4%, your payoff will be $225 regardless of X’s performance. That is, states of nature
only apply to decisions with uncertain payoffs. Nevertheless, in deciding between a risky
and a riskless investment, you need to evaluate the payoff from the risky asset even if
your decision turns out to be to buy the riskless one. In other cases, your decision may
be between 2 or more risky investments.
Specifying your utility function. Since most decision makers are risk averse, the
expected utility of a risky payoff depends on how risky it is. The text uses the device of a
utility function to calculate expected utility. There are techniques available to interrogate
yourself to estimate your utility function. A related approach is to estimate your expected
utility for a given risky investment directly. For example, suppose you intend to invest
$10,000 and are considering a risky gamble of a 0.30 probability of a payoff of $1,600
and a 0.70 probability of a payoff of zero. Ask yourself, what certain payoff would you
need to be indifferent between this payoff and the risky gamble just described?
Suppose you feel the certain payoff is $200. Then, you could use $200 (called a
certainty equivalent) as your expected utility for the risky gamble. If an alternative
investment yields a certainty equivalent of, say, $225, you would take the alternative.
Note that a riskless investment is an alternative, such as a government bond (of a
financially secure country), yielding a return of $225, you could take the $225 payoff as
your certainty equivalent for this investment.
Versions of this approach are used by investment advisors, who ask clients whether
their tolerance for risk is low, medium, or high. This helps them evaluate the client’s
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certainty equivalents for investments of differing risks. Thus, if a client has high risk
tolerance, his/her certainty equivalent for the above gamble might be $250. Then, the
advisor would advise a risky gamble, whereas a lower–risk gamble would be advised for
a low risk tolerance investor.
The information system. If you decide to gather more information before acting,
specification of the information system is a difficult aspect of your decision problem.
Unlike prior probabilities, information system probabilities are objective. If your
additional evidence is to be obtained from financial statements, the information system
probabilities are determined by the quality of GAAP. Thus, if X Ltd. is in the high
performance state, the probability that the financial statements show GN will be higher
the higher is the quality of GAAP. Nevertheless, it is still possible that the financial
statements show BN since GAAP cannot completely rule out errors and biases in
accounting estimates.
However, the information system probabilities are also affected by the integrity of the
manager, who may, within GAAP, or even in violation of GAAP, manage the financial
statements opportunistically. Thus the probabilities also depend on the quality of the
firm’s corporate governance. In this regard, see Note 8 of this Chapter.
One approach to estimating information system probabilities is to use a sampling
approach to analyze the past relationship between financial statements and subsequent
firm performance. When past financial statements have shown GN, how many times
has next year’s firm performance been high, etc.? Another approach is to estimate
information system probabilities based on analyst reaction to the financial statements,
as outlined in Section 3.3.2 of the text. The stronger is analyst reaction per dollar of GN
or BN, the higher the information system main diagonal probabilities.
Conclusion. You may feel that there are so many issues surrounding the inputs into the
decision theory model that the procedure is not viable. If so, ask yourself how else you
would make a decision under uncertainty. By forcing careful consideration of the
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variables that really matter to a decision, the decision theory approach may well lead to
better decision making on average.
Of course, you may instead turn your decision making over to an expert, such as a
financial institution or advisor. However, if you do, you still face a decision problem—
which financial advisor, how much to invest, do you accept the advisor’s advice, etc.
The issues described above still apply.
Finally, whether or not you accept the model, a major argument of Chapter 3 is that the
model reasonably captures the behaviour of the average investor, even though
individual investors may not follow the procedures exactly. As such, the model provides
guidance to accountants about the information needs of investors and the crucial role of
information in facilitating these decisions.
5. Portfolio Theory and the Optimal Individual Investment Decision
The text then goes on to the principle of portfolio diversification, since an understanding
of diversification is crucial to understanding many of the empirical and theoretical
discussions later in the text.
Note: The following is a more complete illustration and explanation of the theory of the
optimal investment decision than that given in Sections 3.5 and 4.5 of the text. It was
included in earlier editions of the text, and is reproduced here for instructors who may
wish to consider this theory more thoroughly.
The Principle of Portfolio Diversification
In Section 3.4, we stated that individual investors are typically assumed to be risk–
averse. Consequently, for a given expected payoff from investments the rational
investor wants the lowest possible risk or, equivalently, for a given risk, will want the
highest possible expected payoff. In effect, the investor adopts a trade–off between risk
and return; greater risk will be borne only if expected return is higher and vice versa.
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One way investors can lower risk for a given expected return is to adopt a strategy of
diversification, that is, to invest in a portfolio of securities. The principle of portfolio
diversification shows us that some, but not all, risk can be eliminated by appropriate
investment strategy. This principle has important implications for the nature of the risk
information that investors need. The risk reported on by many common accounting–
based risk measures, such as debt to equity, times interest earned (ratio of net income
before interest and taxes to interest expense), or the current ratio, can be reduced or
eliminated a priori by appropriate diversification.
Before illustrating the diversification principle, we return briefly to our risk–averse
investor. Note that before we can calculate an individual’s expected utility for different
investment acts, we need to know what that individual’s utility function looks like. For
example, Bill Cautious’ utility function in Example 3.1 was U(x) = √𝑥 , x ≥ 0. With this
utility function and payoff probabilities, Bill’s expected utilities for different acts were
calculated and compared.
One might reasonably ask, “How do we know what an individual’s utility function is?” To
avoid this question, we shall now assume mean–variance utility, which provides a model
utility function of a typical risk–averse investor:
𝑈𝑖(𝑓
𝑖)=𝑓
𝑖(𝑥̅𝑎 , 𝜎𝑎
2 )
where symbol a represents an investment act. For example, investment act a could be
an investment in a riskless government bond, or in a firm’s shares, as in Example 3.1.
Alternatively, it could be an investment in a portfolio of securities.
The equation states that the utility of an investment act a to investor i is a function fi of
the expected rate of return from that act 𝑥̅a and the risk as measured by its variance σa2.
We assume that fi is increasing in 𝑥̅a and, to capture risk aversion, decreasing in σa2. A
specific example of a mean variance utility function is:
𝑈𝑖(a)= 2𝑥̅ɑ – 𝜎𝑎
2
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which can be seen to increase in 𝑥̅ɑ and decrease in σɑ2. Individuals will have different
trade–offs between expected rate of return and risk—for example, a more risk–averse
investor might have –2σɑ2 rather than –σɑ2 as shown above. It is not true in general that
the utility of an act depends only on its mean and variance. However, investigation of
this is beyond our scope.
The significance of a mean–variance utility assumption to accountants is that it makes
investors’ decision needs more explicit—all risk averse investors need information about
the expected values and riskiness of returns from investments, regardless of the
specific forms of their utility functions. Without such an assumption, specific knowledge
of investors’ utility functions would be needed to fully deduce their information
requirements.
With this background in mind, we now illustrate the principle of portfolio diversification
by means of two examples.
Suppose that Toni Difelice, a risk–averse investor has $200 to invest and is considering
investing all of it in the shares of firm A, currently trading for $20. Assume that Toni
assesses a 0.74 probability1 that the shares will increase in market value to $22 over
the coming period and a 0.26 probability that they will decrease to $17. Assume also
that A will pay a dividend of $1 per share at the end of the period (we could also make
the dividend uncertain, but this would just add complexity without affecting the point to
be made).
Example 3.2
The Principle of Portfolio Diversification (Part 1)
As in our decision theory Example 3.1, Toni’s subjective probabilities could be posterior
to her analysis of firm A’s financial statements and the resulting application of Bayes’
theorem. Alternatively, they could be her prior probabilities based on whatever other
information is at her disposal. For present purposes, the extent to which Toni may have
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become informed does not matter. The important point is that she has assessed
probabilities.
The gross payoffs from Toni’s proposed investment are as follows:
If shares increase: $22 × 10 shares + $10 dividend = $230
If shares decrease: $17 × 10 shares + $10 dividend = $180
Table 3.3 shows the calculation of the expected rate of return and variance of this
investment. Henceforth, we will work with the rate of return. As can be seen from Table
3.3, this just involves dividing net returns by the amount of original investment ($200).
The division by original investment is a standardization device—rates of return can be
directly compared across securities while amounts of returns cannot. Also, rate of return
fits in nicely with the assumption of mean–variance utility, which is in terms of the
expected value and variance of rate of return.
Table 3.3 Calculating Expected Rate of Return and Variance
Expected
Payoff Rate of Return Probability Rate of Return Variance
$230 230−200
200 = 0.15 0.74 0.1110 (0.15 – .0850)2 × 0.74 = 0.0031
$180 180−200
200 =−0.10 0.26 – 0.0260 (-0.10 – 0.0850)2 × 0.26 = 0.0089
𝑥̅a = 0.0850 𝜎𝑎
2 = 0.0120
The variance of return is 0.0120. The variance of an investment return serves as a
measure of its riskiness. Since Toni is risk–averse, increasing riskiness will lower her
utility, other things equal.
Assume that Toni’s utility function is:
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𝑈𝑖(a)= 2𝑥̅ɑ – 𝜎𝑎
2
as given above. Then, her utility for this investment is:
(2 × 0.0850) – 0.0120 = 0.1580
Toni now has to decide whether to take this investment act. If she feels that this utility is
not sufficiently high, further research would be necessary to find a more attractive
investment, or some other use for the $200 of capital.
Example 3.3
The Principle of Portfolio Diversification (Part 2)
It turns out that Toni would not be rational to accept the above investment—a more
attractive investment can be found. It is possible to find another investment decision that
has the same expected return but lower risk. This is because of the principle of
portfolio diversification.
To illustrate, assume that shares of firm B are also traded on the market, with a current
market value of $10. These shares also pay a dividend of $1. Assume there is a 0.6750
probability that firm B’s shares will increase in market value to $10.50 at the end of the
period, and a 0.3250 probability that they will decrease to $8.50.
Now suppose that Toni decides to invest $200 in six shares of firm A at $20 and eight
shares of firm B at $10. We must calculate Toni’s expected utility for the portfolio
consisting of six shares of firm A and eight shares of firm B. Notice that the same
amount ($200) is invested, but that it is now spread over two different securities.
Four possible payoffs now exist from the portfolio: both shares increase in market value,
one share increases and the other decreases, or both shares decrease. The amounts of
the payoffs and their assumed probabilities are as follows in Table 3.4:
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Table 3.4 Payoffs and Their Probabilities
A B Dividends Total Payoff Probability
132 + 84 + 14 = $230 0.5742
132 + 68 + 14 = $214 0.1658
102 + 84 + 14 = $200 0.1008
102 + 68 + 14 = $184 0.1592
1.0000
Recall that six shares of firm A and eight shares of firm B are held, and that the high
gross payoff is $22 per share for firm A and $10.50 for firm B, plus a $1 dividend from
each share. This gives the $230 payoff on the first line of the table. The other payoffs
are similarly calculated.
Now let us consider more closely the probabilities we have assumed for the four
possible payoffs. The returns from shares of firm A and firm B are correlated in our
example. To see this, consider the first row in Table 3.4 with a total payoff of $230. This
payoff will be realized if both shares A and B realize their high–payoff values. On the
basis of our assumption about the probabilities of the individual payoffs of shares A and
B, the probabilities of these two payoffs, when each share is considered separately, are
0.74 for A and 0.6750 for B. If the payoffs of shares A and B were independent, the
probability of both shares realizing their high payoffs would be 0.74 × 0.6750 = 0.4995.
However, in any economy, there are states of nature, also called factors, which affect
the returns of all shares, such as levels of interest rates, foreign exchange rates, the
level of economic activity, and so on. These are called market–wide or economy–wide
factors. Their presence means that if the return on one share is high, it is more likely
that the returns on most other shares in the economy will also be high—more likely, that
is, than would be the case if the returns on shares were independent. Thus, we have
assumed that the probability that both shares A and B realize their high payoffs is
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0.5742, which is greater than the 0.4995 that we would obtain under independence, to
reflect these underlying common factors.
Similar reasoning applies to the last row of Table 3.4 with a payoff of $184. Here we
have assumed that the joint probability of both firm A and firm B realizing their low
payoffs is 0.1592, greater than the (0.26 × 0.3250 = 0.0845) probability under
independence. If market–wide state realizations are such that they work against high
returns (i.e., if the economy is performing poorly), then the probability that both shares
realize low payoffs is greater than what would be expected under independence.
Table 3.5 Calculating Expected Rate of Return and Variance
Expected
Payoff Rate of Return Probability Rate of Return Variance
$230 230−200
200 = 0.15 0.5742 0.0861 (0.15 – 0.0850)2 × 0.5742 = 0.0024
$214 214−200
200 = 0.07 0.1658 0.0116 (0.07 – 0.0850)2 × 0.1658 = 0.0000
$200 200−200
200 = 0.00 0.1008 0.0000 (0.00 – 0.0850)2 × 0.1008 = 0.0007
$184 184−200
200 =−0.08 0.1592 –0.0127 (-0.08 – 0.0850)2 × 0.1592= 0.0043
[ 𝑥̅a = 0.0850 σ2a = 0.0074
Of course, while share returns may be correlated due to common factors, they will not
be perfectly correlated. It is still possible that one firm realizes a high return and another
a low return—witness the two middle rows of Table 3.4. This is because, in addition to
economy–wide factors, there are also firm–specific factors, also called idiosyncratic
factors, that affect the return of one firm only. Examples include the quality of a firm’s
management, new patents, strikes, machine breakdowns, and so on. Thus, the second
row of the table represents a situation where firm A realizes a high return (say, because
of a new invention it has just patented) and firm B realizes a low return (say, because of
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a critical machine failure in its assembly line). However, due to the presence of
economy–wide factors, the probabilities for these high/low payoff realizations will also be
different than under independence. This is true of Example 3.3.
Thus, if all factors were economy–wide, returns on firms’ shares would be perfectly
correlated. If all factors were firm–specific, returns would be independent. As is usually
the case, the truth lies somewhere in between. Consequently, the probabilities given in
Table 3.4 assume that both types of factors are present.2
The expected rate of return and variance of Toni’s portfolio of A and B shares are
calculated in Table 3.5 using the correlated probabilities. Thus, the expected rate of
return of the portfolio is 0.0850, as before (we have forced this result by appropriate
choice of the probabilities, to facilitate comparison), but the variance has decreased to
0.0074, from 0.0120. Since Toni is risk–averse, she would be better off buying the
portfolio of A and B shares rather than just A, because the expected return is the same,
but the risk is lower.
In fact, her utility now is:
Ui(ɑ) = (2 × 0.0850) – 0.0074
= 0.1626
up from 0.1580 for the single–share investment.
3.5.1 Summary
Risk–averse investors can take advantage of the principle of portfolio diversification to
reduce their risk, by investing in a portfolio of securities. This is because realizations of
firm–specific states of nature tend to cancel out across securities, leaving economy–wide
factors as the main contributors to portfolio risk.
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While individual attitudes to risk may differ, we can see investors’ decision needs with
particular clarity if we assume mean–variance utility. Then, regardless of the degree of
risk aversion, we know that utility increases in expected rate of return and decreases in
variance of the portfolio.
3.6 The Optimal Investment Decision
If a portfolio of two shares is better than one, then a three–share portfolio should be
better than two, and so on. Indeed, this is the case and, assuming there are no
transaction costs such as brokerage fees, Toni should continue buying until the portfolio
includes some of every security traded on the market. This is called “holding the market
portfolio.” Note again that the total amount invested remains at $200, but is spread over
a greater number of securities.
Be sure you understand why the same amount invested in a portfolio can yield lower
risk than if it were invested in a single firm for the same expected rate of return. To
repeat, when more than one risky investment is held, the firm-specific risks tend to
cancel out. If one share realizes a low return, there is always the chance that another
share will realize a high return. The larger the number of different firms’ shares in the
portfolio, the more this effect can operate. As a result, the riskiness of returns is
reduced, which we have illustrated above by means of our variance calculations. Of
course, in the presence of economy–wide risk, there is not a complete cancelling out. At
a minimum, that is, when the market portfolio is held, the economy–wide factors will
remain to contribute to portfolio risk. Such non–diversifiable risk is called systematic
risk.3
Conceptually, the market portfolio includes all assets available for investment in the
economy. As a practical matter, the market portfolio is usually taken as all the securities
traded on a major stock exchange. The return on the market portfolio can then be
proxied by the return on a market index for that exchange, such as the Dow Jones
Industrial Average index of the New York Stock Exchange, the S&P/TSX Composite
Index, etc.
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Now return to our investor, Toni Difelice. Toni decides to buy the market portfolio after
hearing about the benefits of diversification. Her first task is to assess the expected
return and variance of the market portfolio. She subjectively assesses a 0.8 probability
that the S&P/TSX Composite Index will increase by 10% for the coming period and a
0.2 probability that it will increase by 2 1/2%. Then, denoting the expected return and
variance of the market portfolio by
M
x
and σM2 respectively:
𝑥̅M = (0.10 × 0.8) + (0.0250 × 0.2) = 0.0850
σ2M = [(0.10 – 0.0850)2 × 0.8] + [(0.0250 – 0.0850)2 × 0.2]
= 0.0002 + 0.0007
= 0.0009
This gives Toni a utility of:
2𝑥̅M – σ2M = 0.1700 – 0.0009
= 0.1691
which is greater than the 0.1626 utility of the two–share portfolio in Example 3.3.
The question now is: Is this Toni’s optimal investment decision? The answer is probably
not. If Toni were quite risk–averse, she might prefer a portfolio with lower risk than
0.0009, and would be willing to have a lower expected return as a result.
One strategy she might follow would be to sell some of the high–risk stocks in her
portfolio. But, if she does this, she is no longer holding the market portfolio, so some of
the benefits of diversification are lost. How can Toni adjust portfolio risk to her desired
level without losing the benefits of diversification?
The answer lies in the risk–free asset. If a risk–free asset, such as treasury bills
yielding, say, 4%, is available, an investor could sell some of the market portfolio (that
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is, sell some of each security, so that the market portfolio is still held but total
investment in it is lower) and use the proceeds to buy the risk–free asset.
Conversely, if Toni were less risk–averse, she may prefer to borrow at the risk–free rate
and buy more of the market portfolio, thereby moving to higher expected return and risk.
In this way, each investor can secure a desired risk–return trade–off while continuing to
enjoy the maximum risk–reduction effects of diversification.
To illustrate, suppose that Toni borrows $100 at a rate of 0.04 and buys an additional
$100 of the market portfolio. Toni now has $300 of market portfolio, on which she
expects to earn 0.0850, and owes $100 at 4% interest. But her own investment is still
$200. Consequently, her expected return is now:
𝑥̅ɑ = (300/200 × 0.0850) – (100/200 × 0.0400)
= 0.1275 – 0.0200
= 0.1075
The variance of her return also increases, since she now has $300 at risk on an
investment of $200. There is no variance attached to the $100 borrowed, of course,
since interest and principal payments are fixed. The variance of her return is now:
σ2ɑ = (300/200)2 × 0.0009)
= 0.0020
yielding utility of (2 × 0.1075) – 0.0020 = 0.2130. This yields Toni a higher utility than
simply holding the market portfolio (0.1691). Toni will continue to borrow until the
amount borrowed and reinvested yields an
a
x
and σɑ2 that maximizes her utility. In fact,
if she can borrow all she wants at 4%, she would borrow $9,800, which would yield her
utility of 2.33.
3.6.1 Summary
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86
When transaction costs are ignored, a risk–averse investor’s optimal investment
decision is to buy that combination of market portfolio and risk–free asset that yields the
best trade–off between expected return and risk. This trade–off is individual–specific—it
depends on the investor’s utility function. Some investors may wish to reduce their
investment in the market portfolio and buy the risk–free asset with the proceeds. Others
may wish to borrow at the risk–free rate and increase their investment. Either way, all
investors can enjoy the full benefits of diversification while at the same time attaining
their optimal risk–return trade–off.
3.7 Portfolio Risk
3.7.1 Calculating and Interpreting Beta
The principle of diversification leads to an important risk measure of a security in the
theory of investment. This is beta, which measures the co–movement between changes
in the price of a security and changes in the market value of the market portfolio. To
illustrate, we will calculate the betas of shares of firms A and B in Example 3.3, in
relation to the market portfolio M given in Section 3.6.
Beta is an important and useful concept in financial accounting. As we shall see in
Chapter 5, a stock’s beta is a crucial component of empirical studies of the usefulness
to investors of financial accounting information. Also, it is a “launching pad” for reporting
on firm risk. Consequently, an understanding of what a stock’s beta is and what it tells
us about firm risk is an important part of an accountant’s knowledge base.
Example 3.4
Calculating Beta
The beta of A shares, denoted by βA , is given by:
βA =
𝐶𝑜𝑣(𝐴,𝑀)
𝑉𝑎𝑟(𝑀)
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where Cov(A,M) is the covariance of the returns on security A with the returns on the
market portfolio M. In effect, βA measures how strongly the return on A varies as the
market varies. For example, a high–beta security would undergo wide swings in its rate
of return as market conditions change. Shares of airlines and aircraft manufacturers are
examples, since these industries are sensitive to economic conditions. Shares of
electric utilities and fast food firms would be low–beta, since the returns of such firms are
less subject to the state of the economy.
Division by Var(M) is simply a standardization device, to express Cov(A,M) in units of
market variance. For example, if the returns on the Toronto and New York Stock
Exchanges have different variances, standardization by the variance of returns on the
respective exchanges makes betas of Canadian and U.S. firms more comparable.
To calculate the beta of security A, assume that the conditional payoff probabilities of
security A are as follows:
■ When return on M is high:
Probability that return on A is high = 0.90
Probability that return on A is low = 0.10
■ When return on M is low:
Probability that return on A is high = 0.10
Probability that return on A is low = 0.90
These probabilities could be estimated by examining past data on the returns on A
shares in relation to the returns on M. Cov(A,M) is calculated in Table 3.6.
Table 3.6 Calculation of Covariance
Returns Joint