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-marketinformation

• Semistrong-form EMH - prices reflect all publicinformation

• 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



• 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



• Testing constraints

– Use only publicly available data

– Include all transactions costs

– Adjust the results for risk



• 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


Tests and Results of 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



• 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



• 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



• 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



• Other calendar effects

– All the market’s cumulative advance occurs during thefirst 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)



• 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



• 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



• 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



• 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?



• 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



• 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



• 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



• 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



• 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



• 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



• 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



• Studies on predicting rates of return for a cross-section of stocks indicates markets are not semistrong efficient



• 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.


Tests and Results of 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



• 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



• 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 Evidence1. 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.


Copyright © 2006 Pearson Addison-Wesley. All rights reserved. 6-97


Cost Compare

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.

https://personal.vanguard.com/VGApp/hnw/FundsCostCompare?FW_Event=Results&CurrentState=FundsCostCompareInvestmentInfoState&invest=10,000&years=20&aveReturn=12


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 $1950Tails—> you win $1050

Which would you chose?

Are you risk averse?

http://en.wikipedia.org/wiki/Risk_aversion


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 $750Tails—> 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 $450Tails—> 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.

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

correlation

covariance


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 EquationIn 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.

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.

Therefore, the regression equation is: ŷ = 26.768 + 0.644x .

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

b1 = Σ [ (xi - x)(yi - y) ] / Σ [ (xi - x)2]

b1 = 470/730 = 0.644

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.

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.

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


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:

Portfolio Return

Bond Weight

Bond Return

Equity Weight

Equity Return

p D ED E

P

D

D

E

E

r

r

w

r

w

r

w wr r

( ) ( ) ( )p D D E EE 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:

EDEDEEDD rrCovwwww ,222222

p sss

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.



117

• 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

D

ED

DE ww

1

ss

s



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.


Subjective returns

‘s’ = number of scenarios consideredpi = probability that scenario ‘i’ will occur ri = return if scenario ‘i’ occurs

Measuring Mean: Scenarioor Subjective Returns

s

1i

ii rp)r(E


E(r) = (.1)(-.05)+(.2)(.05)...+(.1)(.35)E(r) = .15 = 15%

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%


Using Our Example:s2=[(.1)(-.05-.15)2+(.2)(.05- .15)2+…]=.01199

s = [ .01199]1/2 = .1095 = 10.95%

Subjective or Scenario Distributions

Measuring Variance orDispersion of Returns

2s

1i

2 )]r(E)i(r[)i(pVariance

s

Standard deviation = [variance]1/2 = s


W = 100W1 = 150; Profit = 50

W2 = 80; Profit = -201-p = .4

E(W) = pW1 + (1-p)W2 = 122

s2 = p[W1 - E(W)]2 + (1-p) [W2 - E(W)]2

s2 = 1,176 and s = 34.29%

Risk - Uncertain Outcomes


W1 = 150 Profit = 50

W2 = 80 Profit = -201-p = .4100

Risky Investment

Risk Free T-bills Profit = 5

Risk Premium = 22-5 = 17

Risky Investments with Risk-Free Investment


• 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 & Utility


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

1

2 3

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 return


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

Standard Deviation

Increasing Utility


Portfolio Mathematics:Assets’ Expected Return

Rule 1 : The return for an asset is the probability weighted average return in all scenarios.

s

1i

ii rp)r(E


Portfolio Mathematics: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.

2s

1i

ii

2 )]r(Er[p

s


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


Portfolio Mathematics: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.

ss assetriskyassetriskyp w


Rule 5: When two risky assets with variances s1

2 and s22 respectively, are combined

into a portfolio with portfolio weights w1 and w2, respectively, the portfolio variance is given by:

Portfolio Mathematics:Risk with two Risky Assets

)r,r(Covww2ww 21212

22

22

12

12

p sss


Asset Mix Decision

Asset mix decisions consider both investment opportunities and investor preferences. These are best described within a risk-reward framework.

Investment OpportunitiesThe 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:

E(R) = 𝑖=0

𝑛

𝑃𝑖*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) = 𝑖=0

𝑘

𝑥𝑖*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.

http://www.moneychimp.com/glossary/covariance.htm


Asset Mix Decision

Investor Preferences

Investor preference is quantified in terms of utility derived from owning a security. They• Like Return• Dislike Risk

• Umk = E(Rm) – σm2 = Return – “Risk Penalty”

tk

Umk = Expected utility of asset mix m derived by investor ktk = 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

20%30%).5(50%30%).5(10%)σ(r

30%.5(50%).5(10%))E(r

22i

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:

• 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).

300.5(500).5(100)E(u)

u(50%)*.5u(10%)*.5E(u)


Risk-Neutral Utility Function

ui = 10ri

0

100

200

300

400

500

600

0 10 20 30 40 50 60

Total Utility

Percent Return


Risk-Loving Investor

• Assume the following quadratic utility function:

ui = 0 + 5ri + .1ri2

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)


• 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).

• 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%

280.5(500).5(60)E(u)

u(50%)*.5u(10%)*.5E(u)

33.5%2(.1)

)4(.1)(-280-25+5- :Equivalent Certainty


Risk-Loving Utility Function

ui = 0 + 5ri + .1ri2

0

600

0 60

Total Utility

Percent Return

500

280

240

60

10 30 33.5 50


Risk-Averse Investor

• Assume the following quadratic utility function:

ui = 0 + 20ri - .2ri2

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)


• 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).

• That is, the investor would be indifferent between receiving 21.7% risk-free and investing in a risky asset that has E(r) = 30% and s(r) = 20%.

340.5(500).5(180)E(u)

u(50%)*.5u(10%)*.5E(u)

21.7%.2)2(

0)4(-.2)(-34-400+20- :Equivalent Certainty


Risk-Averse Utility Function

ui = 0 + 20ri - .2ri2

0

600

0 60

Total Utility

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:

2

2

1

2

0

2

22

2210

E(r)a

E(r)a

a

a

a

E(u)=σ(r)

:σ(r)for Solving

(r)σaE(r)aE(r)aaE(u)


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:

• Therefore, the indifference curve would be:

2E(r)

.2

E(r)20

.2

180σ(r)

E(r)

10

20

30

40

50

s(r)

0

26.5

34.6

38.7

40.0


Risk-Averse Indifference Curve

When E(u) = 180, and ui = 0 + 20ri - .2ri2

0

10

20

30

40

50

60

0 10 20 30 40 50

Expected Return

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.

0

10

20

30

40

50

60

0 10 20 30 40 50

Expected Return

Standard Deviation of Returns

Portfolio That

Maximizes

Utility

E(u) = 380 E(u) = 280

E(u) = 180


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:

0

100

200

300

400

500

600

0 20 40 60 80

Total Utility

Percent Return

Unrealistic


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

fund and 1 % California

R.E.I.T.

4.57% 1.07%

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

21( )

2U E r As


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

• As noted before: this does not determine the choice of oneportfolio, but a whole set of efficient portfolios.

BA rErE

BA ss


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

fund

$90,000

Total risk assets $210,000

Equities $113,400

Bonds (long-term) $96,600

54.0000,210$

400,113$EW 46.0

00,210$

600,96$BW

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:

• Complete Portfolio is: (0.3, 0.378, 0.322)

7.0000,300$

000,210$y 3.0

000,300$

000,90$1 y

378.000,300$

400,113$: E 322.

000,300$

600,96$: B


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


• 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.

Portfolios of One Risky Asset and a Risk-Free Asset


rf = 7% srf = 0%

E(rp) = 15% sp = 22%

y = % in p (1-y) = % in rf

Example


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

( ) ( )c f P f

E r r y E r r

7157 yrE c


Example (Ctd.)

• The risk of the complete portfolio is the weight of P times the risk of P:

– This follows straight from the formulas we saw before and the fact that any constant random variable has zero variance.

yy PC 22 ss


Feasible (var, mean)

• Taken together this determines the set of feasible (mean,variance) portfolio return:

– This determines a straight line, which we call Capital Allocation Line. Next we derive it’s equation completely

yy PC 22 ss

7157 yrE c


Example (Ctd.)

• Rearrange and substitute y=sC/sP:

– The sub-index C is to stand for complete portfolio

– The slope has a special name: Sharpe ratio.

CfP

P

CfC rrErrE s

s

s

22

87

22

8

P

fP rrESlope

s


The Investment Opportunity Set


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 Changing Allocation


• 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

Capital Allocation Line with Leverage


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:

– Variance:

( ) ( )c f P f

E r r y E r r

2 2 2

C Pys 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).



• 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.



• 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

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• 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

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.



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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


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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

Portfolio Return

Bond Weight

Bond Return

Equity Weight

Equity Return

p D ED E

P

D

D

E

E

r

r

w

r

w

r

w wr r

( ) ( ) ( )p D D E E

E r w E r w E r


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= Variance of Security D

= Variance of Security E

= Covariance of returns forSecurity D and Security E

Two-Security Portfolio: Risk


p sss

2

Es

2

Ds

ED rrCov ,


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 Ew w Cov r r w w Cov r r w w Cov r rs


D,E = Correlation coefficient of returns

Cov(rD,rE) = DEsDsE

sD = Standard deviation ofreturns for Security D

sE = Standard deviation ofreturns for Security E

Covariance


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: Possible Values


Correlation Coefficients

• When ρDE = 1, there is no diversification

• When ρDE = -1, a perfect hedge is possible

DDEEP ww sss

D

ED

DE ww

1

ss

s


Computation of Portfolio Variance From the Covariance Matrix


Three-Asset Portfolio

1 1 2 2 3 3( ) ( ) ( ) ( )

pE r w E r w E r w E r

2

3

2

3

2

2

2

2

2

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Portfolio Expected Return as a Function of Investment Proportions


Portfolio Standard Deviation as a Function of Investment Proportions


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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


• 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.

Correlation Effects


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:

𝑤𝐷 =𝐸 𝑟𝐷 − 𝐸 𝑟𝐸 + 0.01𝐴[𝜎2𝐸 − 𝐶𝑜𝑣 𝑟𝐷, 𝑟𝐸 ]

0.01𝐴[𝜎2𝐸 + 𝜎2𝐷 − 𝐶𝑜𝑣 𝑟𝐷, 𝑟𝐸 ]


Numerical Example

Debt EquityExpected Return, E(r) 8% 13%Standard deviation, σ 12% 20%Covariance, Cov (rD, rE) 72Correlation Coefficient, ρDE 0.30

2 mutual funds

Wmin(D) = σ2E – Cov (rD,rE) = 202 -72 = 0.82

σ2D + σ2

E – 2Cov (rD,rE) 122 + 202 – 2x72

Wmin(E) = 1-0.82 = 0.18

The minimum variance portfolioσ = [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 EquityExpected Return, E(r) 8% 13%Standard deviation, σ 12% 20%Covariance, Cov (rD, rE) 72Correlation Coefficient, ρDE 0.30

2 mutual funds

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 = 0.3811.7

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 = 𝐸 𝑟𝑃 −𝑟𝑓

𝜎𝑃

• 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:

𝑦 ∗ =𝐸 𝑟𝑃 − 𝑟𝑓

0.01𝐴𝜎𝑃2



• 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:

𝑦 ∗ =𝐸 𝑟𝑃 −𝑟𝑓

0.01𝐴𝜎𝑃2= 11 – 5 /(0.01 x 4 x 14.22) = -.7439



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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

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YearCapital 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%

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?


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

• If we have three assets, portfolio variance is given by:

• 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.


p sss

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