Market efficiency and portfolio theory

MGB Portfolio Management I
MARKET EFFICIENCY

RANDOM WALK

Random Walk
In 1973 when author Burton Malkiel wrote "A Random Walk Down Wall Street", which
remains on the top-seller list for finance books.
 Strict Definition
─ Successive stock returns are independent and identically distributed. This implies that past
movement or trend of a stock price or market cannot be used to predict its future movement.
 Common Definition
─ Price changes are essentially unpredictable
This is the idea that stocks take a random and unpredictable path. A follower of the
random walk theory believes it's impossible to outperform the market without
assuming additional risk.
Critics of the theory, however, contend that stocks do maintain price trends over time
- in other words, that it is possible to outperform the market by carefully selecting
entry and exit points for equity investments.

Random Walk
Financial Economists were disturbed as this seemed to
imply that stock markets were dominated by some
erratic market psychology or some “animal spirit” that
followed no logical rules.
It soon became apparent however, that random price
movements indicated a well-functioning or efficient market,
not an irrational one.

Random Walk
And why is that?
Because any new information that could be used to predict stock performance must
already reflect in the stock price. As soon as any new information is available that can
impact stock prices, investors will buy/sell the security immediately to its fair level
where only ordinary return can be expected (rate of return commensurate with the
risk).
However, if prices are bid immediately to fair levels. On getting new information, it
must be that the increase/decrease is due to only that new information. But New
information, by definition, must be unpredictable. If not, then the information would
already be priced into the price of the security!
So, stock prices should follow a random walk, that is, price changes should be random
and unpredictable. Randomly evolving prices are a result of intelligent investors
discovering relevant information and by their action moving the prices.

THE EFFICIENT MARKET HYPOTHESIS

The Efficient Market Hypothesis
 Expectations are very important in our financial system.
─ Expectations of returns, risk, and liquidity impact asset demand
─ Inflationary expectations impact bond prices
─ Expectations not only affect our understanding of markets, but also
how financial institutions operate.
 To better understand expectations, we examine the efficient
markets hypothesis.
─ Framework for understanding what information is useful and what is
not
─ However, we need to validate the hypothesis with real market data.
The results are mixed, but generally supportive of the idea.

The Efficient Market Hypothesis
 In sum, we will look at the basic reasoning behind the efficient
market hypothesis. We also examine empirical evidence
examining this idea:
─ The Efficient Market Hypothesis
─ Evidence on the Efficient Market Hypothesis
─ Behavioral Finance

Efficient Market Hypothesis
• The rate of return for any position is the sum of the capital gains (Pt+1 – Pt)
plus any cash payments (C):
• At the start of a period, the unknown element is the future price: Pt+1. But,
investors do have some expectation of that price, thus giving us an
expected rate of return.

The Efficient Market Hypothesis views the expectations as equal
to optimal forecasts using all available information. This implies:
Assuming the market is in equilibrium:
Re = R* [market’s equilibrium return]
Put these ideas together: efficient market hypothesis
Rof = R*

Rof = R*
•This equation tells us that current prices in a financial market
will be set so that the optimal forecast of a security’s return
using all available information equals the security’s
equilibrium return.
•As a result, a security’s price fully reflects all available
information in an efficient market.
•Note, R* depends on risk, liquidity, other asset returns …

Rationale Behind the Hypothesis
 When an unexploited profit opportunity arises on a
security (so-called because, on average, people would
be earning more than they should, given the
characteristics of that security), investors will rush to
buy until the price rises to the point that the returns
are normal again.
 Investors do not leave $ bills lying on the sidewalk.

• Why efficient market hypothesis makes sense
If Rof > R* → Pt ↑ → Rof ↓
If Rof < R* → Pt ↓ → Rof ↑
Until Rof = R*
• All unexploited profit opportunities eliminated
• Efficient market condition holds even if there are uninformed,
irrational participants in market

 In an efficient market, all unexploited profit
opportunities will be eliminated.
 Not every investor need be aware of every security
and situation.
 Only a few investors (even 1 big one) are needed to
eliminate unexploited profit opportunities and push
the market price to its equilibrium level.

Efficient Capital Markets
• In an efficient capital market, security prices adjust rapidly to
the arrival of new information, therefore the current prices of
securities reflect all information about the security
• Whether markets are efficient has been extensively
researched and remains controversial

Why Should Capital Markets Be Efficient?
The premises of an efficient market
– A large number of competing profit-maximizing participants analyze and
value securities, each independently of the others
– New information regarding securities comes to the market in a random
fashion
– Profit-maximizing investors adjust security prices rapidly to reflect the
effect of new information
Conclusion: the expected returns implicit in the current price of a
security should reflect its risk

Alternative Efficient Market Hypotheses (EMH)
• Random Walk Hypothesis – changes in security prices occur
randomly
• Fair Game Model – current market price reflect all available
information about a security and the expected return based
upon this price is consistent with its risk
• Efficient Market Hypothesis (EMH) - divided into three sub-hypotheses
depending on the information set involved

Efficient Market Hypotheses (EMH)
• Weak-Form EMH - prices reflect all security-market
information
• Semistrong-form EMH - prices reflect all public
information
• Strong-form EMH - prices reflect all public and
private information

Weak-Form EMH
• Current prices reflect all security-market information,
including the historical sequence of prices, rates of return,
trading volume data, and other market-generated information
• This implies that past rates of return and other market data
should have no relationship with future rates of return

Semistrong-Form EMH
• Current security prices reflect all public information,
including market and non-market information
• This implies that decisions made on new information
after it is public should not lead to above-average
risk-adjusted profits from those transactions

Strong-Form EMH
• Stock prices fully reflect all information from public
and private sources
• This implies that no group of investors should be able
to consistently derive above-average risk-adjusted
rates of return
• This assumes perfect markets in which all
information is cost-free and available to everyone at
the same time

Tests and Results of
Weak-Form EMH
• Statistical tests of independence between rates of
return
– Autocorrelation tests have mixed results
– Runs tests indicate randomness in prices

Weak-Form EMH
• Comparison of trading rules to a buy-and-hold policy is
difficult because trading rules can be complex and
there are too many to test them all
– Filter rules yield above-average profits with small filters, but
only before taking into account transactions costs
– Trading rule results have been mixed, and most have not
been able to beat a buy-and-hold policy

Weak-Form EMH
• Testing constraints
– Use only publicly available data
– Include all transactions costs
– Adjust the results for risk

Weak-Form EMH
• Results generally support the weak-form EMH, but results are
not unanimous

Tests of the Semistrong Form of Market Efficiency
Two sets of studies
• Time series analysis of returns or the cross section
distribution of returns for individual stocks
• Event studies that examine how fast stock prices
adjust to specific significant economic events

Semistrong-Form EMH
• Test results should adjusted a security’s rate of return for the
rates of return of the overall market during the period
considered
Arit = Rit - Rmt
where:
Arit = abnormal rate of return on security i during period t
Rit = rate of return on security i during period t
Rmt =rate of return on a market index during period t

Semistrong-Form EMH
• Time series tests for abnormal rates of return
– short-horizon returns have limited results
– long-horizon returns analysis has been quite successful
based on
• dividend yield (D/P)
• default spread
• term structure spread
– Quarterly earnings reports may yield abnormal returns due
to
• unanticipated earnings change

Semistrong-Form EMH
• Quarterly Earnings Reports
– Large Standardized Unexpected Earnings (SUEs) result in
abnormal stock price changes, with over 50% of the
change happening after the announcement
– Unexpected earnings can explain up to 80% of stock drift
over a time period
• These results suggest that the earnings surprise is
not instantaneously reflected in security prices

Semistrong-Form EMH
• The January Anomaly
– Stocks with negative returns during the prior year had
higher returns right after the first of the year
– Tax selling toward the end of the year has been mentioned
as the reason for this phenomenon
– Such a seasonal pattern is inconsistent with the EMH

Semistrong-Form EMH
• Other calendar effects
– All the market’s cumulative advance occurs during the
first half of trading months
– Monday/weekend returns were significantly negative
– For large firms, the negative Monday effect occurred
before the market opened (it was a weekend effect),
whereas for smaller firms, most of the negative
Monday effect occurred during the day on Monday (it
was a Monday trading effect)

Semistrong-Form EMH
• Predicting cross-sectional returns
– All securities should have equal risk-adjusted returns
• Studies examine alternative measures of size or
quality as a tool to rank stocks in terms of risk-adjusted
returns
– These tests involve a joint hypothesis and are dependent
both on market efficiency and the asset pricing model
used

Semistrong-Form EMH
• Price-earnings ratios and returns
– Low P/E stocks experienced superior risk-adjusted results
relative to the market, whereas high P/E stocks had
significantly inferior risk-adjusted results
– Publicly available P/E ratios possess valuable information
regarding future returns
– This is inconsistent with semistrong efficiency

Semistrong-Form EMH
• Price-Earnings/Growth Rate (PEG) ratios
– Studies have hypothesized an inverse relationship between
the PEG ratio and subsequent rates of return. This is
inconsistent with the EMH
– However, the results related to using the PEG ratio to
select stocks are mixed

Semistrong-Form EMH
• The size effect (total market value)
– Several studies have examined the impact of size on the
risk-adjusted rates of return
– The studies indicate that risk-adjusted returns for
extended periods indicate that the small firms consistently
experienced significantly larger risk-adjusted returns than
large firms
– Firm size is a major efficient market anomaly
– Could this have caused the P/E results previously studied?

Semistrong-Form EMH
• The P/E studies and size studies are dual tests of the
EMH and the CAPM
• Abnormal returns could occur because either
– markets are inefficient or
– market model is not properly specified and provides
incorrect estimates of risk and expected returns

Semistrong-Form EMH
• Adjustments for riskiness of small firms did not
explain the large differences in rate of return
• The impact of transactions costs of investing in small
firms depends on frequency of trading
– Daily trading reverses small firm gains
• The small-firm effect is not stable from year to year

Semistrong-Form EMH
• Neglected Firms
– Firms divided by number of analysts following a stock
– Small-firm effect was confirmed
– Neglected firm effect caused by lack of information
and limited institutional interest
– Neglected firm concept applied across size classes
– Another study contradicted the above results

Tests and Results of Semistrong-form EMH
• Trading volume
– Studied relationship between returns, market value, and
trading activity.
– Size effect was confirmed. But no significant difference
was found between the mean returns of the highest and
lowest trading activity portfolios

Semistrong-Form EMH
• Ratio of Book Value of a firm’s Equity to Market Value of
its equity
– Significant positive relationship found between current
values for this ratio and future stock returns
– Results inconsistent with the EMH
• Size and BV/MV dominate other ratios such as E/P ratio or
leverage
• This combination only works during expansive monetary
policy

Semistrong-Form EMH
• Firm size has emerged as a major predictor of future returns
• This is an anomaly in the efficient markets literature
• Attempts to explain the size anomaly in terms of superior risk
measurements, transactions costs, analysts attention, trading
activity, and differential information have not succeeded

Semistrong-Form EMH
• Event studies
– Stock split studies show that splits do not result in
abnormal gains after the split announcement, but
before
– Initial public offerings seems to be underpriced by
almost 18%, but that varies over time, and the price is
adjusted within one day after the offering
– Listing of a stock on an national exchange such as the
NYSE may offer some short term profit opportunities
for investors

Semistrong-Form EMH
• Event studies (continued)
– Stock prices quickly adjust to unexpected world events
and economic news and hence do not provide
opportunities for abnormal profits
– Announcements of accounting changes are quickly
adjusted for and do not seem to provide opportunities
– Stock prices rapidly adjust to corporate events such as
mergers and offerings
– The above studies provide support for the semistrong-form
EMH

Summary on the
Semistrong-Form EMH
• Evidence is mixed
• Strong support from numerous event studies with
the exception of exchange listing studies

Summary on the
Semistrong-Form EMH
• Studies on predicting rates of return for a cross-section
of stocks indicates markets are not
semistrong efficient

Summary on the
Semistrong-Form EMH
• Studies on predicting rates of return for a cross-section
of stocks indicates markets are not
semistrong efficient
– Dividend yields, risk premiums, calendar patterns, and
earnings surprises
• This also included cross-sectional predictors such as
size, the BV/MV ratio (when there is expansive
monetary policy), E/P ratios, and neglected firms.

Strong-Form EMH
• Strong-form EMH contends that stock prices fully
reflect all information, both public and private
• This implies that no group of investors has access to
private information that will allow them to
consistently earn above-average profits

Testing Groups of Investors
• Corporate insiders
• Stock exchange specialists
• Security analysts
• Professional money managers

Corporate Insider Trading
• Corporate insiders include major corporate officers,
directors, and owners of 10% or more of any equity
class of securities
• Insiders must report to the SEC each month on their
transactions in the stock of the firm for which they
are insiders
• These insider trades are made public about six weeks
later and allowed to be studied

• Corporate insiders generally experience above-average
profits especially on purchase transaction
• This implies that many insiders had private
information from which they derived above-average
returns on their company stock

• Studies showed that public investors who traded
with the insiders based on announced transactions
would have enjoyed excess risk-adjusted returns
(after commissions), but the markets now seem to
have eliminated this inefficiency (soon after it was
discovered)

• Other studies indicate that you can increase returns
from using insider trading information by combining
it with key financial ratios and considering what
group of insiders is doing the buying and selling

Stock Exchange Specialists
• Specialists have monopolistic access to information
about unfilled limit orders
• You would expect specialists to derive above-average
returns from this information
• The data generally supports this expectation

Security Analysts
• Tests have considered whether it is possible to
identify a set of analysts who have the ability to
select undervalued stocks
• This looks at whether, after a stock selection by an
analyst is made known, a significant abnormal return
is available to those who follow their
recommendations

The Value Line Enigma
• Value Line (VL) publishes financial information on
about 1,700 stocks
• The report includes a timing rank from 1 down to 5
• Firms ranked 1 substantially outperform the market
• Firms ranked 5 substantially underperform the
market

The Value Line Enigma
• Changes in rankings result in a fast price adjustment
• Some contend that the Value Line effect is merely
the unexpected earnings anomaly due to changes in
rankings from unexpected earnings

Security Analysts
• There is evidence in favor of existence of superior
analysts who apparently possess private information

Professional Money Managers
• Trained professionals, working full time at
investment management
• If any investor can achieve above-average returns, it
should be this group
• If any non-insider can obtain inside information, it
would be this group due to the extensive
management interviews that they conduct

Performance of
Professional Money Managers
• Most tests examine mutual funds
• New tests also examine trust departments, insurance
companies, and investment advisors
• Risk-adjusted, after expenses, returns of mutual
funds generally show that most funds did not match
aggregate market performance

Conclusions Regarding the
Strong-Form EMH
• Mixed results, but much support
• Tests for corporate insiders and stock exchange
specialists do not support the hypothesis (Both
groups seem to have monopolistic access to
important information and use it to derive above-average
returns)

Conclusions Regarding the
Strong-Form EMH
• Tests results for analysts are concentrated on Value Line
rankings
– Results have changed over time
– Currently tend to support EMH
• Individual analyst recommendations seem to contain
significant information
• Performance of professional money managers seem to
provide support for strong-form EMH

Behavioral Finance
It is concerned with the analysis of various
psychological traits of individuals and how these
traits affect the manner in which they act as
investors, analysts, and portfolio managers

Implications of
Efficient Capital Markets
• Overall results indicate the capital markets are
efficient as related to numerous sets of information
• There are substantial instances where the market
fails to rapidly adjust to public information

Efficient Markets
and Technical Analysis
• Assumptions of technical analysis directly oppose the
notion of efficient markets
• Technicians believe that new information is not
immediately available to everyone, but disseminated
from the informed professional first to the aggressive
investing public and then to the masses

Efficient Markets
• Technicians also believe that investors do not analyze
information and act immediately - it takes time
• Therefore, stock prices move to a new equilibrium after the
release of new information in a gradual manner, causing
trends in stock price movements that persist for periods

Efficient Markets
• Technical analysts develop systems to detect
movement to a new equilibrium (breakout) and
trade based on that
• Contradicts rapid price adjustments indicated by the
EMH
• If the capital market is weak-form efficient, a trading
system that depends on past trading data can have
no value

Efficient Markets
and Fundamental Analysis
• Fundamental analysts believe that there is a basic
intrinsic value for the aggregate stock market,
various industries, or individual securities and these
values depend on underlying economic factors
• Investors should determine the intrinsic value of an
investment at a point in time and compare it to the
market price

Efficient Markets
and Fundamental Analysis
• If you can do a superior job of estimating intrinsic value you
can make superior market timing decisions and generate
above-average returns
• This involves aggregate market analysis, industry analysis,
company analysis, and portfolio management
• Intrinsic value analysis should start with aggregate market
analysis

Aggregate Market Analysis with Efficient Capital
Markets
• EMH implies that examining only past economic events is not
likely to lead to outperforming a buy-and-hold policy because
the market adjusts rapidly to known economic events
• Merely using historical data to estimate future values is not
sufficient
• You must estimate the relevant variables that cause long-run
movements

Industry and Company Analysis with Efficient Capital
Markets
• Wide distribution of returns from different industries and
companies justifies industry and company analysis
• Must understand the variables that effect rates of return and
• Do a superior job of estimating future values of these relevant
valuation variables, not just look at past data

Industry and Company Analysis with Efficient
Capital Markets
• Important relationship between expected
earnings and actual earnings
• Accurately predicting earnings surprises
• Strong-form EMH indicates likely existence of
superior analysts
• Studies indicate that fundamental analysis based
on E/P ratios, size, and the BV/MV ratios can lead
to differentiating future return patterns

How to Evaluate Analysts or Investors
• Examine the performance of numerous securities
that this analyst recommends over time in relation to
a set of randomly selected stocks in the same risk
class
• Selected stocks should consistently outperform the
randomly selected stocks

Efficient Markets
and Portfolio Management
• Portfolio Managers with Superior Analysts
– concentrate efforts in mid-cap stocks that do not receive
the attention given by institutional portfolio managers to
the top-tier stocks
– the market for these neglected stocks may be less efficient
than the market for large well-known stocks

Efficient Markets
and Portfolio Management
• Portfolio Managers without Superior Analysts
– Determine and quantify your client's risk preferences
– Construct the appropriate portfolio
– Diversify completely on a global basis to eliminate all
unsystematic risk
– Maintain the desired risk level by rebalancing the portfolio
whenever necessary
– Minimize total transaction costs

The Rationale and
Use of Index Funds
• Efficient capital markets and a lack of superior analysts imply
that many portfolios should be managed passively (so their
performance matches the aggregate market, minimizes the
costs of research and trading)
• Institutions created market (index) funds which duplicate the
composition and performance of a selected index series

Insights from Behavioral Finance
• Growth companies will usually not be growth stocks
due to the overconfidence of analysts regarding
future growth rates and valuations
• Notion of “herd mentality” of analysts in stock
recommendations or quarterly earnings estimates is
confirmed

Evidence on Efficient Market Hypothesis
 Favorable Evidence
1. Investment analysts and mutual funds don't beat
the market
2. Stock prices reflect publicly available info:
anticipated announcements don't affect stock price
3. Stock prices and exchange rates close to random walk; if
predictions of DP big, Rof > R*  predictions
of DP small
4. Technical analysis does not outperform market

Evidence in Favor of Market Efficiency
• Performance of Investment Analysts and Mutual Funds
should not be able to consistently beat the market
– The “Investment Dartboard” often beats investment managers.
– Mutual funds not only do not outperform the market on average, but
when they are separated into groups according to whether they had
the highest or lowest profits in a chosen period, the mutual funds that
did well in the first period do not beat the market in the second
period.
– Investment strategies using inside information is the only “proven
method” to beat the market. In the U.S., it is illegal to trade on such
information, but that is not true in all countries.

 Do Stock Prices Reflect Publicly Available
Information as the EMH predicts they will?
─ Thus if information is already publicly available, a positive
announcement about a company will not, on average,
raise the price of its stock because this information is
already reflected in the stock price.
─ Early empirical evidence confirms: favorable earnings
announcements or announcements of stock splits (a
division of a share of stock into multiple shares, which is
usually followed by higher earnings) do not, on average,
cause stock prices to rise.

 Random-Walk Behavior of Stock Prices that is, future
changes in stock prices should, for all practical purposes, be
unpredictable
─ If stock is predicted to rise, people will buy to equilibrium
level; if stock is predicted to fall, people will sell to
equilibrium level (both in concert with EMH)
─ Thus, if stock prices were predictable, thereby causing the
above behavior, price changes would be near zero, which
has not been the case historically

 Technical Analysis means to study past stock price data and
search for patterns such as trends and regular cycles,
suggesting rules for when to buy and sell stocks
─ The EMH suggests that technical analysis is a waste of time
─ The simplest way to understand why is to use the random-walk result
that holds that past stock price data cannot help predict changes
─ Therefore, technical analysis, which relies on such data to produce its
forecasts, cannot successfully predict changes in stock prices

• 2: Empirical Evidence for TA is Negligible
• Much of the faith in TA hinges on anecdotal experience, not any kind of long-term statistical
evidence, unlike value investing or other quantitative/fundamental methodologies we discuss on
this site. Most of the statistical work done by academics to determine whether the chart patterns
are actually predictive has been inconclusive at best. Indeed, a recent study by finance professors at
Massey University in New Zealand examined 49 developed and emerging markets to see if TA
added value. They looked at more than 5,000 technical trading rules across four rule families :
• Filter Rules - These rules involve opening long (short) positions after price increases (decreases)
by x% and closing these positions when price decreases (increases) by x% from a subsequent high
(low).
• Moving Average Rules - These rules generate buy (sell) signals when the price or a short moving
average moves above (below) a long moving average.
• Channel Break-outs - These rules involve opening long (short) positions when the closing price
moves above (below) a channel. A channel (sometimes referred to as a trading range) can be said to
occur when the high over the previous n days is within x percent of the low over the
previous n days, not including the current price.
• Support and Resistance Rules - These “Trading Range Break” rules involve opening a long (short)
position when the closing price breaches the maximum (minimum) price over the
previous n periods.
• The result? Using statistical methods to adjust for data snooping bias, the authors concluded that
there wasno evidence that the profits [attributed] to the technical trading rules considered were
greater than those that might be expected due to random data variation.

But wait – it’s not all bad….
• As you can tell, trading purely on the basis of TA is a mug’s game. However, despite inconsistencies
in predictive value,
• 1. TA may be a useful tool as part of a broader strategy for managing holdings (e.g. to help you
time any investments that are decided on other, hopefully fundamentally-focused, criteria).The fact
is that many (misguided) market participants use TA to drive their investment decisions. These
collective actions result in tangible changes in asset values, so they need to be understood even by
less mis-guided investors. A fundamental investor need not agree that a stock should be moving but
it’s worth understand why a stock is nevertheless moving. As Birinyi, a research and money-management
firm, noted in a research note:
• 2. “technical approaches can and should be a useful adjunct to every investor’s — amateur and
professional — arsenal, if and only if used properly and with understanding… Technicals detail
and hopefully illuminate, but do not predict.”
• 3. TA may be particularly useful on the sell-side where it is deemed (according to William
O’Neill) prudent to sell based on “unusual market action such as price and volume movement”…
• 4. Good investing is about managing your losses too, and here TA can be a useful tool to
determine where best to place a stop-loss (given the number of TA practitioners out there that are
likely to be anchoring around certain price points).

Case: Foreign Exchange Rates
• Could you make a bundle if you could predict FX
rates? Of course.
• EMH predicts, then, that FX rates should be
unpredictable.
• That is exactly what empirical tests show—FX rates
are not very predictable.

Evidence on Efficient Market Hypothesis
 Unfavorable Evidence
1. Small-firm effect: small firms have abnormally high returns
2. January effect: high returns in January
3. Market overreaction
4. Excessive volatility
5. Mean reversion
6. New information is not always immediately incorporated into stock
prices
 Overview
─ Reasonable starting point but not whole story

Evidence Against Market Efficiency
• The Small-Firm Effect is an anomaly. Many empirical studies
have shown that small firms have earned abnormally high
returns over long periods of time, even when the greater risk
for these firms has been considered.
– The small-firm effect seems to have diminished in recent years but is
still a challenge to the theory of efficient markets
– Various theories have been developed to explain the small-firm effect,
suggesting that it may be due to rebalancing of portfolios by
institutional investors, tax issues, low liquidity of small-firm stocks,
large information costs in evaluating small firms, or an inappropriate
measurement of risk for small-firm stocks

 The January Effect is the tendency of stock prices to
experience an abnormal positive return in the month of
January that is predictable and, hence, inconsistent with
random-walk behavior
– Investors have an incentive to sell stocks before the end of the
year in December because they can then take capital losses on
their tax return and reduce their tax liability. Then when the
new year starts in January, they can repurchase the stocks,
driving up their prices and producing abnormally high returns.
– Although this explanation seems sensible, it does not explain
why institutional investors such as private pension funds, which
are not subject to income taxes, do not take advantage of the
abnormal returns in January and buy stocks in December, thus
bidding up their price and eliminating the abnormal returns.

 Market Overreaction: recent research suggests that stock
prices may overreact to news announcements and that the
pricing errors are corrected only slowly
─ When corporations announce a major change in earnings, say, a large
decline, the stock price may overshoot, and after an initial large
decline, it may rise back to more normal levels over a period of several
weeks.
─ This violates the EMH because an investor could earn abnormally high
returns, on average, by buying a stock immediately after a poor
earnings announcement and then selling it after a couple of weeks
when it has risen back to normal levels.

 Excessive Volatility: the stock market appears to display
excessive volatility; that is, fluctuations in stock prices may be
much greater than is warranted by fluctuations in their
fundamental value.
─ Researchers have found that fluctuations in the S&P 500 stock index
could not be justified by the subsequent fluctuations in the dividends
of the stocks making up this index.
─ Other research finds that there are smaller fluctuations in stock prices
when stock markets are closed, which has produced a consensus that
stock market prices appear to be driven by factors other than
fundamentals.

 Mean Reversion: Some researchers have found that stocks
with low returns today tend to have high returns in the future,
and vice versa.
─ Hence stocks that have done poorly in the past are more likely to do
well in the future because mean reversion indicates that there will be
a predictable positive change in the future price, suggesting that stock
prices are not a random walk.
─ Newer data is less conclusive; nevertheless, mean reversion remains
controversial.

 New Information Is Not Always Immediately Incorporated
into Stock Prices
─ Although generally true, recent evidence suggests that, inconsistent
with the efficient market hypothesis, stock prices do not
instantaneously adjust to profit announcements.
─ Instead, on average stock prices continue to rise for some time after
the announcement of unexpectedly high profits, and they continue to
fall after surprisingly low profit announcements.

Implications for Investing
1. How valuable are published reports by investment advisors?
2. Should you be skeptical of hot tips?
3. Do stock prices always rise when there is good news?
4. Efficient Markets prescription for investor

 How valuable are published reports by investment advisors?

1. Should you be skeptical of hot tips?
─ YES. The EMH indicates that you should be skeptical of hot tips since, if
the stock market is efficient, it has already priced the hot tip stock so
that its expected return will equal the equilibrium return.
─ Thus, the hot tip is not particularly valuable and will not enable you to
earn an abnormally high return.
– As soon as the information hits the street, the unexploited profit
opportunity it creates will be quickly eliminated.
– The stock’s price will already reflect the information, and you should
expect to realize only the equilibrium return.

3. Do stock prices always rise when there is
good news?
– NO. In an efficient market, stock prices will respond to announcements
only when the information being announced is new and unexpected.
– So, if good news was expected (or as good as expected), there will be no
stock price response.
– And, if good news was unexpected (or not as good as expected), there
will be a stock price response.

 Efficient Markets prescription for investor
─ Investors should not try to outguess the market by constantly buying
and selling securities. This process does nothing but incur
commissions costs on each trade.
─ Instead, the investor should pursue a “buy and hold” strategy—
purchase stocks and hold them for long periods of time. This will lead
to the same returns, on average, but the investor’s net profits will be
higher because fewer brokerage commissions will have to be paid.
─ It is frequently a sensible strategy for a small investor, whose costs of
managing a portfolio may be high relative to its size, to buy into a
mutual fund rather than individual stocks. Because the EMH indicates
that no mutual fund can consistently outperform the market, an
investor should not buy into one that has high management fees or
that pays sales commissions to brokers but rather should purchase a
no-load (commission-free) mutual fund that has low management
fees.

All mutual funds sold to the public – performance of all
general equity mutual funds compared to the Wilshire
5000 Index. In most years more than ½ of the funds
were outperformed by the index. Over the 26.5 year
period about 2/3 of the funds proved inferior to the
market as a whole. Same result holds for professional
pension managers.
Cost Compare
Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6-97

Case: Any Efficient Markets Lessons from Black Monday of 1987
and the Tech Crash of 2000?
 Does any version of Efficient Markets Hypothesis (EMH) hold
in light of sudden or dramatic market declines?
 Strong version EMH?
 Weaker version EMH?
 A bubble is a situation in which the price of an asset differs
from its fundamental market value?
 Can bubbles be rational?
 Role of behavioral finance

Behavioral Finance
BF argues that a few psychological phenomena pervade
financial markets:
1. Practitioners rely on rules of thumb called heuristics to process
information.
– Heuristic—a process by which people find things out for themselves,
usually by trial and error. Leads to the development of rules of thumb
which are imperfect and result in errors which lead to heuristic-driven
bias.
2. In addition to objective considerations, practitioners perception of risk
& return are highly influenced by how decision problems are framed
 frame dependence.
3. Heuristic-driven bias and framing effects cause market prices to
deviate from fundamental values, i.e. markets are inefficient.

Heuristic Driven Bias
• Representativeness—reliance on stereotypes
– Example of High School GPA as predictor of College GPA
and reversion to the mean.
• Overconfidence
– People set overly narrow confidence bands, high guess is
too low and low guess is too high.
– Results in being surprised too often.
• Anchoring to old information
– Security analysts do not revise their earnings estimates
enough to reflect new info.

Frame Dependence
• EMH assumes framing is transparent—If you move a $ from your right pocket to
your left pocket, you are no wealthier! (Merton Miller)
… In other words, practitioners can see through all the different ways that cash
flow might be described.
• But if frame is opaque, a difference in form (which pocket) is also a difference in
substance and affects behavior.
• Loss Aversion
– Choose between
• Sure loss of $7,500 or
• 75% chance of loosing $10K or 25% chance of loosing $0.
• Hedonic editing
– Organizing Gains and Losses in separate mental accounts.
• One loss and one gain are netted against each other.
• Two gains are savored separately
• But multiple losses are difficult to net out against moderate gains.

Frame Dependence
• Hedonic editing
1. Imagine that you face the following choice. You can accept
a guaranteed $1500 or play a lottery. The outcome of the
lottery is determined by the toss of a fair coin.
Heads—> you win $1950
Tails—> you win $1050
Which would you chose?
Are you risk averse?

Frame Dependence
• Hedonic editing
2. Imagine that you face the following choice. You can
accept a guaranteed loss of $750 or play a lottery. The
outcome of the lottery is determined by the toss of a fair
coin.
Heads—> you lose $750
Tails—> you lose $525
Which would you chose?

Frame Dependence
• Hedonic editing
3. Now imagine that you have just won $1500 in one lottery,
and you can choose to participate in another. The outcome
of this second lottery is determined by the toss of a fair
coin.
Heads—> you win $450
Tails—> you win $450
Would you choose to participate in the second lottery?

Frame Dependence
• Hedonic editing has both cognitive and emotional
causes
– Main cognitive issue in choice 3 above—Do you ignore the
preliminary $1500 winnings or not?
– Those that begin by seeing themselves $1500 ahead then
experience the emotion of loosing $450 as the equivalent
of winning $1050 (i.e. a smaller gain, not a loss).
– Those that ignore the $1500 are less prone to accept the
gamble because they will feel a $450 loss as a loss.

Assignment
Q1: If the weak form of the efficient market is valid must the strong
form also hold? Conversely, does strong-form efficiency imply weak-form
efficiency?
Q2: What would happen to market efficiency if every investor followed
a passive strategy?
Q3. A portfolio manager outperforms the market in 11 of 14 years.
Does this violate the concept of market efficiency?
Q4. A segment of the market believes that continued economic worries
brought about the stock market crash of 1987. Is this explanation for
the crash consistent with the Efficient Market Hypothesis?

PORTFOLIO THEORY

cov(X,Y)=E(XY)−E(X)E(Y).
Proof:
Let μ=E(X) and ν=E(Y). Then
cov(X,Y)=E[(X−μ)(Y−ν)]=E(XY−μY−νX+μν)=E(XY)−μE(Y)−νE(X)+μν=E(XY)−μν

The "Population Standard
Deviation":
The "Sample Standard
Deviation":

In probability theory and statistics, the mathematical concepts of covariance and
correlation are very similar. Both describe the degree to which two random
variables or sets of random variables tend to deviate from their expected values in
similar ways.
correlation
covariance
where E is the expected value operator and σx and σy are the standard
deviations of X and Y, respectively. Notably, correlation is dimensionless while
covariance is in units obtained by multiplying the units of the two variables. The
covariance of a variable with itself (i.e. σxx ) is called the variance and is more
commonly denoted as σ2
x the square of the standard deviation. The correlation
of a variable with itself is always 1

Last year, five randomly selected students took a math aptitude test before they began their statistics
course. The Statistics Department has three questions.
What linear regression equation best predicts statistics performance, based on math aptitude scores?
If a student made an 80 on the aptitude test, what grade would we expect her to make in statistics?
How well does the regression equation fit the data?
How to Find the Regression Equation
In the table below, the xi column shows scores on the aptitude test. Similarly, the yi column shows
statistics grades. The last two rows show sums and mean scores that we will use to conduct the
regression analysis.
Student xi yi (xi - x) (yi - y) (xi - x)2 (yi - y)2 (xi - x)(yi - y)
1 95 85 17 8 289 64 136
2 85 95 7 18 49 324 126
3 80 70 2 -7 4 49 -14
4 70 65 -8 -12 64 144 96
5 60 70 -18 -7 324 49 126
Sum 390 385 730 630 470
Mean 78 77
The regression equation is a linear equation of the form: ŷ = b0 + b1x . To conduct a regression analysis,
we need to solve for b0 and b1. Computations are shown below.
b1 = Σ [ (xi - x)(yi - y) ] / Σ [ (xi - x)2]
b1 = 470/730 = 0.644
Therefore, the regression equation is: ŷ = 26.768 + 0.644x .
b0 = y - b1 * x
b0 = 77 - (0.644)(78) = 26.768

The Coefficient of Determination
Whenever you use a regression equation, you should ask how well the equation fits
the data. One way to assess fit is to check the coefficient of determination, which can
be computed from the following formula.
R2 = { ( 1 / N ) * Σ [ (xi - x) * (yi - y) ] / (σx * σy ) }2
where N is the number of observations used to fit the model, Σ is the summation
symbol, xi is the x value for observation i, x is the mean x value, yi is the y value for
observation i, y is the mean y value, σx is the standard deviation of x, and σy is the
standard deviation of y. Computations for the sample problem of this lesson are shown
below.
σx = sqrt [ Σ ( xi - x )2 / N ]
σx = sqrt( 730/5 ) = sqrt(146) = 12.083
σy = sqrt [ Σ ( yi - y )2 / N ]
σy = sqrt( 630/5 ) = sqrt(126) = 11.225
R2 = { ( 1 / N ) * Σ [ (xi - x) * (yi - y) ] / (σx * σy ) }2
R2 = [ ( 1/5 ) * 470 / ( 12.083 * 11.225 ) ]2 = ( 94 / 135.632 )2 = ( 0.693 )2 = 0.48
A coefficient of determination equal to 0.48 indicates that about 48% of the variation
in statistics grades (the dependent variable) can be explained by the relationship to
math aptitude scores (the independent variable). This would be considered a good fit
to the data, in the sense that it would substantially improve an educator's ability to
predict student performance in statistics class.

Portfolio Mathematics
114
• Of course, in practice, assets are not correlated in this
simplistic way. Let us look at how portfolio risk is
affected when we put two arbitrarily correlated assets in
a portfolio. Let us call the two assets, a bond, D, and a
stock (equity), E.
• Then, we can write out the following relationship:
r
r
w
r
w
r
p D D E E
Portfolio Return
Bond Weight
Bond Return
Equity Weight
Equity Return
P
D
D
E
E
 w r w r





( ) ( ) ( ) p D D E E E r  w E r  w E r

115
The expected return
on a portfolio
consisting of several
assets is simply a
weighted average of
the expected returns
on the assets
comprising the
portfolio.

116
• If we denote variance by s2, then we have the
relationship:
D D E E D E  D E  w w 2w w Cov r ,r 2 2 2 2 2
p s  s  s 
 where Cov(rD, rE) represents the covariance between the
returns on assets D and E.
 If we use DE to represent the correlation coefficient between
the returns on the two assets, then
 Cov(rD,rE) = DEsDsE
 The formula for portfolio variance can be written either
with covariance or with correlation.

• The correlation coefficient can take values between
+1 and -1.
• If DE = +1, there is no diversification and the
portfolio standard deviation equals wDsD + wEsE,
i.e. a linear combination of the standard deviations
of the two assets.
• If DE= -1, the portfolio variance equals (wDsD –
wEsE)2. In this case, we can construct a risk-free
combination of D and E.
• Setting this equal to zero and solving for wD and wE,
we find
117
s
E w   w
D
D
 1
s 
s
D E

118
For intermediate values of r,
the portfolio standard
deviations fall in the
middle, as shown on the
graph to the right.
In this example, the stock
asset has a standard
deviation of returns of 20%
and the bond asset, of 12%.

Problem
Seventy-five percent of a portfolio is invested in Honeybell stock and the
remaining 25% is invested in MBIB stock. Honeybell stock has an expected
return of 6% and an expected standard deviation of returns of 9%. MBIB stock
has an expected return of 20% and an expected standard deviation of 30%.
The coefficient of correlation between returns of the two securities is
expected to be 0.4. Determine the following:
(a) the expected return of the portfolio;
(b) the expected variance of the portfolio;
(c) the expected standard deviation for the portfolio.

Measuring Mean: Scenario
or Subjective Returns
Subjective returns
s


E(r)  p 
r
i i i 1
‘s’ = number of scenarios considered
pi = probability that scenario ‘i’ will occur
ri = return if scenario ‘i’ occurs

Numerical example:
Scenario Distributions
Scenario Probability Return
1 0.1 -5%
2 0.2 5%
3 0.4 15%
4 0.2 25%
5 0.1 35%
E(r) = (.1)(-.05)+(.2)(.05)...+(.1)(.35)
E(r) = .15 = 15%

Measuring Variance or
Dispersion of Returns
Subjective or Scenario Distributions
s 2
Variance  s 2  p(i)  [r(i) 
E(r)]
i 1

Standard deviation = [variance]1/2 = s
Using Our Example:
s
2=[(.1)(-.05-.15)2+(.2)(.05- .15)2+…]
=.01199
s = [ .01199]1/2 = .1095 = 10.95%

W = 100
W1 = 150; Profit = 50
W2 1-p = .4 = 80; Profit = -20
E(W) = pW1 + (1-p)W2 = 122
2 = p[W1 - E(W)]2 + (1-p) [W2 - E(W)]2
s
2 = 1,176 and s = 34.29%
s
Risk - Uncertain Outcomes

Risky Investments
with Risk-Free Investment
W1 = 150 Profit = 50
1-p = .4 W2 = 80 Profit = -20
100
Risky
Investment
Risk Free T-bills Profit = 5
Risk Premium = 22-5 = 17

Risk Aversion & Utility
• Investor’s view of risk
– Risk Averse
– Risk Neutral
– Risk Seeking
• Utility
• Utility Function
U = E ( r ) – .005 A s 2
• A measures the degree of risk aversion

Risk Aversion and Value:
The Sample Investment
U = E ( r ) - .005 A s 2
= 22% - .005 A (34%) 2
Risk Aversion A Utility
High 5 -6.90
3 4.66
Low 1 16.22
T-bill = 5%

Dominance Principle
2 3
1
4
Expected Return
Variance or Standard Deviation
• 2 dominates 1; has a higher
return
• 2 dominates 3; has a lower risk
• 4 dominates 3; has a higher

Utility and Indifference Curves
• Represent an investor’s willingness to trade-off return and
risk
Example (for an investor with A=4):
Exp Return
(%)
St Deviation
(%)
10 20.0
15 25.5
20 30.0
25 33.9
U=E(r)-.005As2
2
2
2
2

Indifference Curves
Expected Return
Increasing Utility
Standard Deviation

Portfolio Mathematics:
Assets’ Expected Return
Rule 1 : The return for an asset is the
probability weighted average return
in all scenarios.
s


E(r)  p 
r
i i i 1

Assets’ Variance of Return
Rule 2: The variance of an asset’s return is the
expected value of the squared
deviations from the expected return.
s 2
2 p [r E(r)]
s   
i 1
i i


Portfolio Mathematics: Return
on a Portfolio
Rule 3: The rate of return on a portfolio is a
weighted average of the rates of
return of each asset comprising the
portfolio, with the portfolio
proportions as weights.
rp = w1r1 + w2r2

Risk with Risk-Free Asset
Rule 4: When a risky asset is combined with a
risk-free asset, the portfolio standard
deviation equals the risky asset’s
standard deviation multiplied by the
portfolio proportion invested in the
risky asset.
s  s
p risky asset riskyasset w

Risk with two Risky Assets
Rule 5: When two risky assets with variances
2 and s2
s1
2 respectively, are combined
into a portfolio with portfolio weights
w1 and w2, respectively, the portfolio
variance is given by:
sp 2
 2
s 2
 2
s 2
1
1
2
2

w w 2w1w2Cov(r1,r2)

Asset Mix Decision
Asset mix decisions consider both investment opportunities and investor
preferences. These are best described within a risk-reward framework.
Investment Opportunities
The goal of assessing investment opportunities can be expressed in terms
of:
• Expected investment returns and
• Potential deviations from these expectations
Asset returns are typically viewed in a probabilistic sense as:
푛
E(R) =
푖=0
푃푖*ri n= number of possible outcomes
Pi is the probability that outcome I will occur
ri= Realized returns if outcome I occurs

Asset Mix Decision
The expected return on portfolio is written as
E(Rm) =
푘
푖=0
푥푖*E(Ri) k= number of assets in the portfolio
xi is the proportion of the portfolio invested in asset i
E(Ri)= Realized returns if outcome i occurs
The variability of the returns about the expectations is measured by the
standard deviation of the returns:
The right hand side of the equation is collectively known as the capital
market conditions. The resulting risk return characteristics of each mix can
be plotted on a return-standard deviation graph to get a chart of all the
portfolios that are constructed.

Asset Mix Decision
The Efficient Frontier
It's clear that for any given value of standard deviation, you would like to choose a portfolio that gives you
the greatest possible rate of return; so you always want a portfolio that lies up along the efficient frontier,
rather than lower down, in the interior of the region. This is the first important property of the efficient
frontier: it's where the best portfolios are.
The second important property of the efficient frontier is that it's curved, not straight. This is actually
significant -- in fact, it's the key to how diversification lets you improve your reward-to-risk ratio. To see why,
imagine a 50/50 allocation between just two securities. Assuming that the year-to-year performance of these
two securities is not perfectly in sync -- that is, assuming that the great years and the lousy years for Security
1 don't correspond perfectly to the great years and lousy years for Security 2, but that their cycles are at least
a little off -- then the standard deviation of the 50/50 allocation will be less than the average of the standard
deviations of the two securities separately. Graphically, this stretches the possible allocations to the left of
the straight line joining the two securities.
In statistical terms, this effect is due to lack of covariance. The smaller the covariance between the two
securities -- the more out of sync they are -- the smaller the standard deviation of a portfolio that combines
them. The ultimate would be to find two securities with negative covariance.

Asset Mix Decision
Investor Preferences
Investor preference is quantified in terms of utility derived from owning a
security. They
• Like Return
• Dislike Risk
2 = Return – “Risk Penalty”
tk
• Umk = E(Rm) – σm
Umk = Expected utility of asset mix m derived by investor k
tk = Investor k’s risk tolerance

Asset Mix Decision
Utility Curves
• An investor is indifferent between any two portfolios that lie on the same
indifference curve.
• Investors want to be on the highest indifference curve that is available given
current capital market conditions.
• Indifference curves do not intersect.
• Flatter indifference curves indicate that the investor has higher tolerance for risk
• Certainty equivalent rate of return is given by the y intercept and is greater than
the risk free rate of return.

Utility Functions
• Utility is a measure of well-being.
• A utility function shows the relationship between utility
and return (or wealth) when the returns are risk-free.
• Risk-Neutral Utility Functions: Investors are
indifferent to risk. They only analyze return when making
investment decisions.
• Risk-Loving Utility Functions: For any given rate of
return, investors prefer more risk.
• Risk-Averse Utility Functions: For any given rate of
return, investors prefer less risk.

Utility Functions (Continued)
• To illustrate the different types of utility functions, we will
analyze the following risky investment for three different
investors:
Possible Return (%)
(ri)
_________
10%
50%
Probability
(pi)
_________
.5
.5
  
E(r ) .5(10%) .5(50%) 30%
2 2
i
σ(r ) .5(10% 30%) .5(50% 30%) 20%
i
    

Risk-Neutral Investor
• Assume the following linear utility function:
ui = 10ri
Return (%)
(ri)
__________
0
10
20
30
40
50
Total Utility
(ui)
__________
0
100
200
300
400
500
Constant
Marginal Utility
__________
100
100
100
100
100

Risk-Neutral Investor (Continued)
• Expected Utility of the Risky Investment:
 
E(u) .5*u(10%) .5*u(50%)
  
E(u) .5(100) .5(500) 300
• Note: The expected utility of the risky investment
with an expected return of 30% (300) is equal to
the utility associated with receiving 30% risk-free
(300).

Risk-Neutral Utility Function
ui = 10ri
Total Utility
600
500
400
300
200
100
0
0 10 20 30 40 50 60
Percent Return

Risk-Loving Investor
• Assume the following quadratic utility function:
ui = 0 + 5ri + .1ri
2
Return (%)
(ri)
__________
0
10
20
30
40
50
Total Utility
(ui)
__________
0
60
140
240
360
500
Increasing
Marginal Utility
__________
60
80
100
120
140

Risk-Loving Investor (Continued)
 
E(u) .5*u(10%) .5*u(50%)
  
E(u) .5(60) .5(500) 280
• Note: The expected utility of the risky investment with an
expected return of 30% (280) is greater than the utility
associated with receiving 30% risk-free (240).
33.5%
- 5 + 25- 4(.1)(-280)
Certainty Equivalent : 
2(.1)
• That is, the investor would be indifferent between
receiving 33.5% risk-free and investing in a risky asset that
has E(r) = 30% and s(r) = 20%

Risk-Loving Utility Function
2
ui = 0 + 5ri + .1ri
Total Utility
600
0
0 60
Percent Return
500
280
240
60
10 30 33.5 50

Risk-Averse Investor
• Assume the following quadratic utility function:
ui = 0 + 20ri - .2ri
2
Return (%)
(ri)
__________
0
10
20
30
40
50
Total Utility
(ui)
__________
0
180
320
420
480
500
Diminishing
Marginal Utility
__________
180
140
100
60
20

Risk-Averse Investor (Continued)
 
E(u) .5*u(10%) .5*u(50%)
  
E(u) .5(180) .5(500) 340
• Note: The expected utility of the risky investment with an
expected return of 30% (340) is less than the utility
associated with receiving 30% risk-free (420).
- 20+ 400- 4(-.2)(-340)
Certainty Equivalent : 

2( .2)
• That is, the investor would be indifferent between
21.7%
receiving 21.7% risk-free and investing in a risky asset that
has E(r) = 30% and s(r) = 20%.

Risk-Averse Utility Function
2
ui = 0 + 20ri - .2ri
Total Utility
600
0
0 60
Percent Return
500
420
340
180
10 21.7 30 50

Indifference Curve
• Given the total utility function, an indifference curve can
be generated for any given level of utility. First, for
quadratic utility functions, the following equation for
expected utility is derived in the text:
   
E(u) a a E(r) a E(r) a σ (r)
2
a E(r)
2
1
0
2
Solving for σ(r) :
E(u)
2
2
2
2
0 1 2
E(r)
a
a
a
a
σ(r) =
  

Indifference Curve (Continued)
• Using the previous utility function for the risk-averse
investor, (ui = 0 + 20ri - .2ri
2), and a given level of utility
of 180:
2 E(r)
20E(r)
σ(r) 
.2
180
.2




• Therefore, the indifference curve would be:
E(r)
10
20
30
40
50
s(r)
0
26.5
34.6
38.7
40.0

Risk-Averse Indifference Curve
2
When E(u) = 180, and ui = 0 + 20ri - .2ri
Expected Return
60
50
40
30
20
10
0
0 10 20 30 40 50
Standard Deviation of Returns

Maximizing Utility
• Given the efficient set of investment possibilities and a
“mass” of indifference curves, an investor would maximize
his/her utility by finding the point of tangency between an
indifference curve and the efficient set.
Expected Return
60
50
40
30
20
10
0
E(u) = 380 E(u) = 280
Portfolio That
Maximizes
Utility
E(u) = 180
0 10 20 30 40 50
Standard Deviation of Returns

Problems With Quadratic Utility Functions
Quadratic utility functions turn down after they reach a
certain level of return (or wealth). This aspect is obviously
unrealistic:
Total Utility
600
500
400
300
200
100
0
Unrealistic
0 20 40 60 80
Percent Return

Problems With Quadratic Utility
Functions (Continued)
• With a quadratic utility function, as your wealth level
increases, your willingness to take on risk decreases (i.e.,
both absolute risk aversion [dollars you are willing to
commit to risky investments] and relative risk aversion [%
of wealth you are willing to commit to risky investments]
increase with wealth levels). In general, however, rich
people are more willing to take on risk than poor people.
Therefore, other mathematical functions (e.g.,
logarithmic) may be more appropriate.

What do you think about the move to a more active stock-picking strategy?
stock standard deviation return
Index fund 4.61% 1.10%
California R.E.I.T. 9.23% -2.27%
Brown Group 8.17% -0.67%
Portfolio of 99% index
4.57% 1.07%
fund and 1 % California
R.E.I.T.
Portfolio of 99% index
fund and 1 % Brown
Group
4.61% 1.08%
Thus we see that the index fund has the highest return of 1.10%
with the standard deviation of 4.61%
By including California REIT the standard deviation (risk) is reduced to 4.57%
but the return also reduces to 1.07%
Thus there can be a tradeoff between these two strategies
However including Brown Group is not a good idea as return drops but the risk (standard deviation remains the same)

Asset Mix Decision
Optimal Portfolio - Where the Efficient frontier and Utility curve meet

Estimating Risk Aversion
• Use questionnaires
• Observe individuals’ decisions when
confronted with risk
• Observe how much people are willing to
pay to avoid risk

Risk Aversion and Capital Allocation to
Risky Assets

The Investment Decision
• Top-down process with 3 steps:
1. Capital allocation between the risky portfolio and
risk-free asset
2. Asset allocation across broad asset classes
3. Security selection of individual assets within each
asset class

Allocation to Risky Assets
• Investors will avoid risk unless there is a
reward.
– i.e. Risk Premium should be positive
• Agents preference (taste) gives the
optimal allocation between a risky
portfolio and a risk-free asset.

Speculation vs. Gamble
• Speculation
– Taking considerable risk for a commensurate gain
– Parties have heterogeneous expectations
• Gamble
– Bet or wager on an uncertain outcome for
enjoyment
– Parties assign the same probabilities to the possible
outcomes

Available Risky Portfolios (Risk-free Rate = 5%)
Each portfolio receives a utility score to
assess the investor’s risk/return trade off

Utility Function
U = utility of portfolio
with return r
E ( r ) = expected
return portfolio
A = coefficient of risk
aversion
s2 = variance of
returns of portfolio
½ = a scaling factor
2 1
U  E ( r )
 As
2

Utility Scores of Alternative Portfolios for Investors with
Varying Degree of Risk Aversion
IN CLASS EXERCISE. Answer: How high
does the risk aversion coefficient (A) has to be
so that L is preferred over M and H?

Mean-Variance (M-V) Criterion
• Portfolio A dominates portfolio B if:
• And
 A   B  E r  E r
A B s s
• As noted before: this does not determine the choice of one
portfolio, but a whole set of efficient portfolios.

Capital Allocation Across Risky and Risk-Free Portfolios
Asset Allocation:
• Is a very important part
of portfolio
construction.
• Refers to the choice
among broad asset
classes.
– % of total Investment in risky
vs. risk-free assets
Controlling Risk:
• Simplest way:
Manipulate the fraction
of the portfolio invested
in risk-free assets versus
the portion invested in
the risky assets

Basic Asset Allocation Example
Total Amount Invested $300,000
Risk-free money market
$90,000
fund
Total risk assets $210,000
Equities $113,400
Bonds (long-term) $96,600
$113,400
W   0.54
0.46
E $210,000
$96,600
  B W
$210,00
Proportion of Risk assets
on Equities
Proportion of Risk assets
on Bonds

Basic Asset Allocation
• P is the complete portfolio where we have y as
the weight on the risky portfolio and (1-y) =
weight of risk-free assets:
$90,000
1 y  
y   0.7
0.3
$210,000
$300,000
$113,400
$96,600
B : 
E :  .322
• Complete Portfolio is:
(0.3, 0.378, 0.322)
$300,000
.378
$300,000
$300,000

The Risk-Free Asset
• Only the government can issue default-free
bonds.
– Risk-free in real terms only if price indexed
and maturity equal to investor’s holding
period.
• T-bills viewed as “the” risk-free asset
• Money market funds also considered risk-free
in practice

Figure 6.3 Spread Between 3-Month
CD and T-bill Rates

Portfolios of One Risky Asset and a Risk-Free Asset
• It’s possible to create a complete portfolio by
splitting investment funds between safe and
risky assets.
– Let y=portion allocated to the risky portfolio, P
– (1-y)=portion to be invested in risk-free asset, F.

Example
rf = 7% srf = 0%
E(rp) = 15% sp = 22%
y = % in p (1-y) = % in rf

Example (Ctd.)
The expected
return on the
complete portfolio
is the risk-free
rate plus the
weight of P times
the risk premium
of P
E(rc )  rf  y E(rP )  rf 
Er   7  y15  7 c

Example (Ctd.)
• The risk of the complete portfolio is
the weight of P times the risk of P:
y y C P s  s  22
– This follows straight from the formulas we saw
before and the fact that any constant random
variable has zero variance.

Feasible (var, mean)
• Taken together this determines the set
of feasible (mean,variance) portfolio
return:
Er   7  y15  7 c
y y C P s  s  22
– This determines a straight line, which we call
Capital Allocation Line. Next we derive it’s
equation completely

Example (Ctd.)
• Rearrange and substitute y=sC/sP:
s
8
  C
    P f C
E r  r  E r  r  7 
s
C f s
P
22
– The sub-index C is to stand for complete portfolio
 
8
22

E r 
r
P f 
s
P
Slope
– The slope has a special name: Sharpe ratio.

The Investment Opportunity Set

Capital Allocation Line - Changing Allocation
Increasing he fraction of the overall portfolio
invested in the risky asset increases the
expected return by the risk premium of the
equation (which is 8%) but also increases
portfolio standard deviation at the rate of 22%.
The extra return per extra risk is 8/22 = 0.36

Capital Allocation Line Changing Allocation
I have invested 300,000 risky assets and if I borrow
120,000 and invest it into the risky asset as well
y = 420,000/300,000 = 1.4
1-y = 1-1.4 = -0.4
E(rc) = 7% + (1.4 x 8%) = 18.2%
σc = 1.4 X 22% = 30.8%
S= E(rc) – rf = 18.2 – 7 = 0.36
σc 30.8

Capital Allocation Line with Leverage
• Lend at rf=7% and borrow at rf=9%
– Lending range slope = 8/22 = 0.36
– Borrowing range slope = 6/22 = 0.27
• CAL kinks at P

The Opportunity Set with Differential Borrowing and Lending
Rates

Utility Levels for Various Positions in Risky Assets (y) for an
Investor with Risk Aversion A = 4

Utility as a Function of Allocation to the Risky Asset, y

Table 6.5 Spreadsheet Calculations of
Indifference Curves

Portfolio problem
• Agent’s problem with one risky and one risk-free
asset is thus:
• Pick portfolio (y, 1-y) to maximize utility U
– U(y,1-y) = E(rC) -0.005*A*Var(rC)
• Where rC is the complete portfolio
– This is the same as
– rf + y[E(r) – rf] -0.5*A*y2*Var(r)
– Solution: y* = E(r) – rf )/0.01A*Var(rC)

Indifference Curves for
U = .05 and U = .09 with A = 2 and A = 4

Finding the Optimal Complete Portfolio Using
Indifference Curves

Expected Returns on Four Indifference Curves and the CAL

Risk Tolerance and Asset Allocation
• The investor must choose one optimal
portfolio, C, from the set of feasible choices
– Expected return of the complete portfolio:
E(rc )  rf  y E(rP )  rf 
– Variance:
2 2 2
C P s  y s

Summary
The Asset Allocation process has 2 steps:
1. Determine the CAL
2. Find the point of highest utility along that
line

One word on Indifference Curves
• If you see the IC curves over (mean,st. dev) you will
note that these are all nice smooth concave curves.
– This is an assumption.
– Note that investors have preference over random variables
(representing payoff/return). A random variable, in general, is
not completely described by (mean, variance).
• That is, in general, we can have X and Y with mean(X) < mean (Y)
and var(X)=var(Y) BUT X is ranked better than Y nonetheless.

Passive Strategies:
The Capital Market Line
• A natural candidate for a passively held risky
asset would be a well-diversified portfolio of
common stocks such as the S&P 500.
• The capital market line (CML) is the capital
allocation line formed from 1-month T-bills
and a broad index of common stocks (e.g. the
S&P 500).

Passive Strategies:
• The CML is given by a strategy that involves
investment in two passive portfolios:
1. virtually risk-free short-term T-bills (or a
money market fund)
2. a fund of common stocks that mimics a
broad market index.

Passive Strategies:
• From 1926 to 2009, the passive risky
portfolio offered an average risk premium of
7.9% with a standard deviation of 20.8%,
resulting in a reward-to-volatility ratio of
.38.

Diversification and Portfolio Risk
• Suppose there is a single common source of risk in the economy.
• All assets are exposed both to this single common source of risk and a separate
197
idiosyncratic source of risk that is uncorrelated across assets.
• Then the insurance principle says that if we construct a portfolio of a very large
number of these assets, the combined portfolio will only reflect the common
risk. The idiosyncratic risk will average out and tend to zero as the number of
securities grows very large.
• Thus, if there are many home fire insurance policyholders and the risk of fire is
uncorrelated across similarly sized homes, then if the number of policy holders
is very large, the actual losses in the portfolio tends to the expected loss per
home times the number of homes.
• This means that homeowners, by pooling their risk, can remove their exposure
to risk completely.
• In practice, the risks are not completed uncorrelated across homes but a fair
amount of risk reduction is possible.
• The next slide shows graphically how portfolio risk would be affected in these
conditions.

198

199

Investment Opportunity Sets: Risky Assets
This graph shows the portfolio opportunity
set for different values of .
That is, the combination of portfolio E(r)
and s than can be obtained by combining
the two asset.
In our example, the equity asset has an
expected return of 13%, while the bond
asset has an expected return of 8%.
The curved line joining the two assets D
and E is, in effect, part of the opportunity
set of (E(R), s) combinations available to
the investor. To get the entire opportunity
set, we simply extend this curve both
beyond E and beyond D.

Optimal Portfolio: Two Risky Assets

7-202
Covariance and Correlation
• Portfolio risk depends on the correlation
between the returns of the assets in the
portfolio
• Covariance and the correlation coefficient
provide a measure of the way returns of
two assets vary

7-203
Two-Security Portfolio: Return
r
 p w r w r
D D E E
r

Portfolio Return
P
w

Bond Weight
D
r

Bond Return
D
w

Equity Weight
E
r

Equity Return
E
E ( r )  w E ( r )  w E ( r
) p D D E E

7-204
Two-Security Portfolio: Risk
p s  s  s 
= Variance of Security D
= Variance of Security E
= Covariance of returns for
Security D and Security E
2
D s
2
E s
  D E Cov r , r

Two-Security Portfolio: Risk
• Another way to express variance of the
portfolio:
2 ( , ) ( , ) 2 ( , ) P D D D D E E E E D E D E w s  w Cov r r  w w Cov r r  w w Cov r r

Covariance
Cov(rD,rE) = DEsDsE
D,E = Correlation coefficient of
returns
sD = Standard deviation of
returns for Security D
sE = Standard deviation of
returns for Security E

Correlation Coefficients: Possible Values
Range of values for 1,2
+ 1.0 >  > -1.0
If  = 1.0, the securities are perfectly
positively correlated
If  = - 1.0, the securities are perfectly
negatively correlated

Correlation Coefficients
• When ρDE = 1, there is no diversification
P E E D D s  w s  w s
• When ρDE = -1, a perfect hedge is possible
s
E w   w
D
D
 1
s 
s
D E

Computation of Portfolio Variance From the Covariance Matrix

Three-Asset Portfolio
1 1 2 2 3 3 E(rp )  w E(r )  w E(r )  w E(r )
2 s ws ws ws p   
2
3
2
3
2
2
2
2
2
1
2
1
1 2 1,2 1 3 1,3 2 3 2,3  2w ws  2w ws  2w ws

Portfolio Expected Return as a Function of Investment
Proportions

Portfolio Standard Deviation as a Function of Investment
Proportions

7-213
The Minimum Variance Portfolio
• The minimum variance
portfolio is the portfolio
composed of the risky
assets that has the
smallest standard
deviation, the portfolio
with least risk.
• When correlation is less
than +1, the portfolio
standard deviation may
be smaller than that of
either of the individual
component assets.
• When correlation is -1,
the standard deviation
of the minimum
variance portfolio is
zero.

Portfolio Expected Return as a Function of Standard Deviation

Correlation Effects
• The amount of possible risk reduction through
diversification depends on the correlation.
• The risk reduction potential increases as the
correlation approaches -1.
– If  = +1.0, no risk reduction is possible.
– If  = 0, σP may be less than the standard deviation
of either component asset.
– If  = -1.0, a riskless hedge is possible.

Optimal Portfolio Selection
• We can solve the optimization problem to
compute the following useful formulas:
• The minimum variance portfolio of risky assets
D, E is given by the following formula:
• The optimal portfolio for an investor with a risk
aversion parameter, A, is given by this formula:
푤퐷 =
퐸 푟퐷 − 퐸 푟퐸 + 0.01퐴[휎2
퐸 − 퐶표푣 푟퐷, 푟퐸 ]
0.01퐴[휎2
퐸 + 휎2
퐷 − 퐶표푣 푟퐷, 푟퐸 ]

Numerical Example
Debt Equity
2 mutual funds
Expected Return, E(r) 8% 13%
Standard deviation, σ 12% 20%
Covariance, Cov (rD, rE) 72
Correlation Coefficient, ρDE 0.30
Wmin(D) = σ2
E – Cov (rD,rE) = 202 -72 = 0.82
σ2
D + σ2
E – 2Cov (rD,rE) 122 + 202 – 2x72
Wmin(E) = 1-0.82 = 0.18
The minimum variance portfolio
σ = [0.822 x 122 + 0.182 x 202 + 2x0.82x0.18x72]1/2 = 11.45%
Sharpe Ratio SA= E(rA) – rf = 8.9 – 5 = 0.34
σA 11.45

We now introduce a risk free asset.
The expected return on a portfolio
consisting of a risk free asset and a risky
portfolio is, of course, a weighted
average of the expected returns on the
component assets. But the standard
deviation of the portfolio is also linear in
the standard deviation of the risky asset.
Hence the CAL if there is one risk free
asset and a risky portfolio is simply a
straight line passing through the two
assets, as shown in the figure on the
right.

Numerical Example
Debt Equity
2 mutual funds
Expected Return, E(r) 8% 13%
Standard deviation, σ 12% 20%
Covariance, Cov (rD, rE) 72
Correlation Coefficient, ρDE 0.30
B has an E(r) = 9.5% and a σ of 11.7% giving it a risk premium of 4.5%
Its Sharpe Ratio is SB = 9.5 – 5.0 = 0.38
11.7
SB – SA = .38 - .34 = 0.04. We get 4 basis points per percentage point increase in risk.

The slope of each of the CALs drawn
in the previous figure is a reward-to-volatility
(Sharpe) ratio. Since we
want this ratio to be maximized, the
single CAL for the set of risky and risk
free assets is the CAL with the
steepest slope, i.e. the highest
Sharpe ratio.

If we now superimpose the
indifference curve map on the
CAL, we can compute the
complete optimal portfolio.

• The formula for the tangency portfolio (shown as
portfolio C on the picture in the previous slide) is:
Max Sp =
퐸 푟푃 −푟푓
휎푃
• Note that the investor risk aversion coefficient does not
show up in this formula.
• Once the tangency portfolio is available, all investors
choose a combination of this portfolio (denoted p in the
formula below) and the risk-free asset. The formula for
this, which we know already, is:
푦 ∗ =
퐸 푟푃 − 푟푓
0.01퐴휎2
푃

• WD = (8-5)400 – (13-5)72__________ = 0.40
(8-5)400 + (13-5)144 – (8-5+13-5)72
• WE = 1-0.4 = 0.60
• σp = (0.42 x 144 + 0.62x400 + 2 x 0.4 x 0.6 x 72)1/2 = 14.2
• Sp = 11-5/14.2 = 0.42
• Note that the investor risk aversion coefficient does not show up in
this formula.
• Once the tangency portfolio is available, all investors choose a
combination of this portfolio (denoted p in the formula below) and
the risk-free asset. The formula for this, which we know already, is:
푦 ∗ =
퐸 푟푃 −푟푓
0.01퐴휎푃
2 = 11 – 5 /(0.01 x 4 x 14.22) = -.7439

224
The investor will invest 74.39% of the wealth in Portfolio P and 25.62 in T-bills.
Portfolio P consists of 40% bonds and 60% stocks so 0.4x74.39 = 29.76% of the
wealth will be in bonds and 0.6 x 74.39 = 44.63% of the wealth will be in stocks.

Numerical Example
You have available to you, two mutual funds, whose returns have a correlation of
0.23. Both funds belong to the fund category “Balanced – Domestic.” Here is
some information on the fund returns for the last six years (obtained from
http://www.financialweb.com/funds/):
In addition, you can also invest in a risk free 1-year T-bill yielding 6.286%. The
expected return on the market portfolio is 20%.
a.If you have a risk aversion coefficient of 4, and you have a total of $20,000 to
invest, how much should you invest in each of the three investment vehicles?
b.What is the standard deviation of your optimal portfolio?
226
Year
Capital Value
Fund
Green Century
Balanced Year
Capital Value
Fund
Green Century
Balanced
1999 21.32% -10.12% 1996 21.48% 18.26%
1998 21.44% 18.91% 1995 0.91% -4.28%
1997 9.86% 24.91% 1994 10.79% -0.47%
average 14.30% 7.87%
stdev 8.52% 14.57%

Solution
• a. Using the formula, we can find the portfolio weights for the tangent
portfolio of risky assets as follows:
which works out to 1641.13/1529.31 = 1.073. Hence wGCB = 1-(1.073) = -
0.073.
In order to find the optimal combination of the tangent portfolio and the
risk free asset for our investor, we need to compute the expected return
on the tangent portfolio and the variance of portfolio returns.
E(Rtgtport) = 1.073(14.3) + (-0.073)(7.87) = 14.77%
Var(Rtgtport) = (1.073)2(8.52)2 + (-0.073)2(14.57)2 +
2(-0.073)(1.073)(8.52)(14.57)(0.23) = 82.47. Hence, stgtport = 9.08%

Solution (Contd.)
Using the formula y* = [E(Rport) – Rf]/0.01AVar(Rtgtport), we get y* = = 2.57;
hence the proportion in the riskfree asset is -1.57. In other words, the
investor borrows to invest in the tangent portfolio.
If the investor’s total outlay is $20,000, the amount borrowed equals
(20000)(1.57) = $31,400. This provides a total of $51,400 for investment
in the tangent portfolio. However, the tangent portfolio itself consists of
shortselling Green Century Balanced to the extent of (0.073)(51,400) =
3752.20, providing a total of 51,400 + 3752.2 = $55,152.20 for investment
in Capital Value Fund.
b. The standard deviation of the optimal portfolio is 2.57(9.08) =
23.34%. The expected return on the optimal portfolio is 2.57(14.77) + (-
1.57)(6.286) = 28.09%

• Until now, we have dealt with the case of two risky assets. We now
increase the number of risky assets to more than two.
• In this case, graphically, the situation remains the same, as we will
see, except that the opportunity set instead of being a simple
parabolic curve becomes an area, bounded by a parabolic curve.
• However, since all investors are interested in higher expected return
and lower variance of returns, only the northwestern frontier of this
set is relevant, and so the graphic illustration remains comparable.
• Mathematically, the computation of the tangency portfolio is a bit
more complicated, and will require the solution of a system of n
equations. We will not go further into it, here.
• We now look at the graphical illustration of the problem

Markowitz Portfolio Selection
• The first step is to
determine the risk-return
opportunities
available.
• All portfolios that lie
on the minimum-variance
frontier
from the global
minimum-variance
portfolio and
upward provide the
best risk-return
combinations

• We now search
for the CAL with
the highest
reward-to-variability
ratio

• The separation property tells us that the portfolio choice
problem may be separated into two independent tasks
– Determination of the optimal risky portfolio is purely
technical.
– Allocation of the complete portfolio to T-bills versus the risky
portfolio depends on personal preference.
• Thus, everyone invests in P, regardless of their degree of risk
aversion.
– More risk averse investors put more in the risk-free asset.
– Less risk averse investors put more in P.

More on Diversification
• We have seen that
p s  s  s 
• If we have three assets, portfolio variance is given by:
2 s ws ws ws p   
2
3
2
3
2
2
2
2
2
1
2
1
1 2 1,2 1 3 1,3 2 3 2,3  2w ws  2w ws  2w ws
• If we generalize it to n assets, we can write the formula as:
• Defining the average variance and the average covariance, we then get
• That is, the portfolio variance is a weighted average of the average variance and
the average covariance.
• However, as the number of assets increases, the relative weight on the variance
goes to zero, while that on the covariance goes to 1.
• Hence we see that it is the covariance between the returns on the component
assets that is important for the determination of the portfolio variance.

Market efficiency and portfolio theory

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