### The Rational, Risk Averse Investor

```CHAPTER 3:
THE DECISION USEFULNESS
APPROACH TO FINANCIAL
REPORTING
Francis Moniz, Catherine Koene, Josh Proksch, James
Wells, Pamela Feldkamp, Lorcan Duffy
The Decision Usefulness Approach


Contrasted by stewardship
Two questions:
 Identifying
constituencies
 Decision problems


Single-person theory of decision
Theory of investment
Single-Person Decision Theory





Theory viewpoint
Payoff
Ethical issues
Expected Utility
Bayes’ Theorem
The Information System




Predict future investment returns
Conditional Probabilities
Transparent, Precise, High Quality
Trade off between relevance and Reliability
Example


A student has \$1000 to invest. With two possible
investment possibilities: government bonds yielding
10% or share of Company A.
Company A has two states of nature:
 State
1: future performance is high
 Probability
 State
60% P(H) = 0.60
2: future performance is low
 Probability
40% P(L) = 0.40
Example Cont’d




State 1: P(H) = 0.60 State 2: P(L)=0.40
Act
High
Low
225
0
100
100
Utility Function: EU(X)=√(X)
EU(A)= (0.6)√(225)+(0.4)√(0)=0.6(15)+0.4(0)=9
EU(B)=1.00√(100)=1.00(10)=10
Example Cont’d


Student’s research find that if Company A is a HighState firm
 There
is a 65% chance of good news (GN) and 35%
of bad news (BN)
 If Company A is a low state firm then there is a 5%
chance of GN and 95% of BN
Example Cont’d



P(GN|H)=0.65
P(BN|H)=0.35
P(GN|L)=0.05
P(BN|L)=0.95
Posterior Probabilities of high performance state:
P(H|GN)= P(H)*P(GN|H)
= 0.60(0.65)
P(H)*P(GN|H) + P(L)*P(GN|L) 0.60(0.65)+0.40(0.05)
0.39 = 0.95122
0.41
P(H|GN)=0.951
.
P(L|GN)=1-0.951=0.049
EU(A)=15(0.951)+0(0.049)=14.265
EU(B)=10(1.00)=10
EU(A)>EU(B), annual report has changed
the decision and the student will buy shares.
The Rational, Risk Averse Investor


According to decision theory Rational Investors make
their decisions based on the act that yields the highest
expected utility.
In reality not all investors may make their decisions
according to this “rational” basis but the theory
suggests that this is the general behaviour of investors
who want to make good investments.
The Rational, Risk Averse Investor


With rational investors, another assumption is that they are
risk averse
What is meant by risk averse?

One definition is:

Being risk averse means that an individual will want to
minimize risks even when the potential benefit of an action is
large.
As risk decreases, a risk averse person is willing to accept a
situation or make a decision with a higher expected return.
There is a trade off between expected return and risk.


Risk aversion is the reluctance of a person to accept a bargain with
an uncertain payoff rather than another bargain with a more certain,
but possibly lower, expected payoff
The Rational, Risk Averse Investor
-modeling risk aversion

To model risk aversion one must use a utility function which
shows an individual’s payoff amounts as it relates to the
individuals utility for those amounts.
Consider the example were an investor has the option to either
invest their money in shares of a company or buy bonds.

The following table shows the payoff table of the above options and
the probabilities of these outcomes.
Act
State
Probability of Payoffs
High
Low
High
Low
\$225
\$0
60%
40%
\$100
\$100
100%
The Rational, Risk Averse Investor
-modeling risk aversionU(x)
B
15
10
9
A
For this example
the rational
investor’s utility
function is:
U(x) = x , x≥0
D
C
100
135
A: (0.6 X \$225)+ (0.4 X \$0) = \$135
U(x) = ( 225 X 0.6) + ( 0 X 0.4) =
(15*0.6) = 9
225 X (payoff)
B: (1.00 X \$100) = \$100
U(x) = 100 =10
The Principle of Portfolio Diversification



Typical investors: risk-averse
Risk adverted by investment strategy
Mean-variance utility function (  )

=   , 2
a
= the investment act
 i = the investor
  = the expected rate of return

= the variance of risk
Portfolio Diversification: Example



A risk-averse investor has \$400 to invest and is
considering investing all of it in the share of firm A,
currently trading for \$25.
Assume that the investor assesses a 0.65 probability
that these shares will increase in market value to
\$29 over the coming period and a 0.35 probability
that they will decrease to \$21.
Assume also that A will pay a dividend of \$2 per
share at the end of the period.
Example cont’d


\$400 divided by \$25 = 16 shares
If shares increase:
 \$29

x 16 shares + \$32 dividend = \$496
If shares decrease:
 \$21
x 16 shares +\$32 dividend = \$368
Payoff Rate of Return
Probability
Expected Rate
of Return
Variance
\$496
(496-400) / 400 =
0.24
0.65
0.156
(0.24 – 0.128) 2x 0.65 =
0.0082
\$368
(368-400) / 400 =
-0.08
0.35
-0.028
(-0.08 – 0.128) 2x 0.35 =
0.0151
= 0.128
2 = 0.0971
Example Cont’d

Assume the investor’s utility function (  ) can be
represented by:


=  − 22
Therefor, their utility for this investment is:
 0.128

– (2 x 0.0971) = -0.0662
The investor now has to decide whether to take this
investment or not.
Optimal Investment Strategy


Assumes no transaction fees or brokerage fees
Invest in every single security on market then
 Cancels

market security risk
Risk not eliminated still Systemic Risk
 Economy
wide factors that cause unavoidable risk
Risk Free Asset


Ensures diversification yet lowers risk (treasury
bonds, or T-bills)
Sell a little of each security in portfolio invest in risk
free asset
Risk free investment with treasury bonds
Probability of 0.8 for 10% increase and 0.2probaility of 2.5% increase
xm =(0.10*0.8)+(0.0250*0.2)=0.0850
σm2=[(0.10-0.0850)2*0.8]+[(0.0250-0.0850)2*0.2
=0.0002 + 0.0007
=0.0009
Utility of
2xM – σM2=0.1700 – 0.0009
= 0.1691
Risk Free with Market Risk




Toni borrows \$100 at 0.04 and buys additional \$100 in market share
\$300 in market portfolio
Return of 0.0850 and owes \$100 at 4% interest
Xa = (300/200 * 0.0850) –(100/200 *0.0400)
= (0.1275 – 0.0200)
= 0.1075
Variance
σa2 =(300/200)2 *0.0009
= 0.0020
Utility
= (2*0.1075) -0.0020
= 0.2130
Optimal Investment Graph
Beta

Measures changes in the price of a security and
changes in the market value of market portfolio
Β= Cov (A,M)
Var (M)



Cov A,M is covariance of returns on A to returns on
market portfolio M
Dividing by Var (M) is done to express Cov (A,M) in
units of market variance
High beta security undergoes wide swings when
market conditions change.
Beta Results




Transaction costs not ignored when using Beta
Buy relatively few securities instead of market
securities
Important to know expected returns and betas
Assess expected return and risk of portfolios
Reaction of Professional Accounting Bodies to
the Decision Usefulness Approach

The objective of the financial statements:
 To
provide financial information that is “useful to
present and potential equity investors, lenders, and
other creditors in making decisions in their capacity as
capital providers.”

Primary user group
```