investing in securities markets - hebrew university of...

Investing in SecuritiesMarkets

Professor Doron Avramov

Prof. Doron Avramovדורון אברמוב ' פרופ

Investment Management1

Investing in Securities Markets: Security Types

Basic Types Major Subtypes

Interest-bearingMoney market instruments

Fixed-income securities

Equities Stocks

DerivativesOptions, Futures, Swaptions

Interest rate derivatives



Investing in Securities Markets: A few Key Questions

• Long term perspective on the risk return tradeoff in US and International financial markets.

• Computing investment return, risk premium, volatility, and Sharpe ratio.

• Stock indexes

a. why do we need them?

b. How can we construct indexes?

c. What is the difference between DJIA and S&P500?



Calculating Investment Return: A Single Period Example

Suppose you invested $1,000 in stock at $25 per

share. After one year, the price increases to $35. For

each share, you also receive $2 in dividends during the

end of that year.

What is your investment rate of return?



Calculating Investment Return: A Single Period Example

Dividend Yield = $2/$25 = 8%

Capital Gain = ($35 - $25)/$25 = 40%

Total Percentage Return = 8% + 40% = 48%

Total Dollar Return = 48% of $1,000 = $480

At the end of the year, the value of your $1,000

investment becomes $1,480.



Multi-Period Rate of Return

• The previous example computes a single period (e.g., one day, one month, or one year) rate of return.

• Suppose your investment spans three periods, e.g., three years.

• The annual returns are R1, R2, and R3.

• The total three-year holding period return (HPR) is

HPR=(1+ R1)(1+ R2)(1+ R3)-1.



Multi-period Return: Example

• Let R1 =10%, R2 =5%, and R3 =7%.

HPR=(1+0.1)(1+0.05)(1+0.07)-1=23.59%

So if you start with $1, after three years you have $1.2359.

A common mistake is to compute HPR as

HPR=10+5+7=22%.

What if the returns are R1 =-10%, R2 =5%, and R3 =7%?



A Note on Annualizing Returns

• Often you want to compare various investments, each of which applies for a different time period.

• Then you should express returns on a per-year, or annualized, basis.

• Such a return is often called an effective annual return (EAR).

• (1 + EAR) = (1 + holding period percentage return)m

(m is the number of holding periods in a year)



Example: Annualizing Returns

• You buy AIG at $34 and sell it 3 months later for $38.

• There were no dividends paid, and suppose those

prices are net of commissions.

• What is your holding period percentage return and

your EAR?



Example: Annualizing Returns

56%.about or ,(1.560338)

0.117647) (1

Return)e Percentag Period Holding (1 EAR 1

0.11764734

4

34

34 - 38 Returne Percentag PeriodHolding

4

m

=

+=

+=+

===



Historical Average Returns

• A useful number to help us summarize historical

financial data is the simple, or arithmetic, average.

For instance, if you have annual return on some

investment the historical mean is given by

N

returnyearly

Return Average Historical

N

1i

∑==



Return Volatility –A Measure of Risk

Variance of return

( )( )

1σ 1

2

2

−

−==

∑=

N

RR

RVar

N

i

i

Standard deviation of return

( ) ( )RVarRSD == σ



Example• To illustrate, the spreadsheet below shows us how to

calculate the average, the variance, and the standard

deviation (the long way…).

1 2 3 4 5

Average Difference: Squared:

Year Return Return: (2) - (3) (4) x (4)

1926 13.75 12.12 1.63 2.66

1927 35.70 12.12 23.58 556.02

1928 45.08 12.12 32.96 1086.36

1929 -8.80 12.12 -20.92 437.65

1930 -25.13 12.12 -37.25 1387.56

Sum: 60.60 Sum: 3470.24

Average: 12.12 Variance: 867.56

29.45Standard Deviation:



Equity, Bond, or Cash? The Empirical Evidence

• We next present an extensive evidence about long run

investment payoffs corresponding to investments in

stocks, bonds, and cash.

• You will find out that stocks are much more profitable

but also more risky for the short run.

• All investment instruments have been able to hedge

against inflation risk over the long investment horizon.



A $1 Investment in Different Types

of Portfolios, 1926—2005.



New Important Concept: Risk Premium

Risk Premium

(%)

Average Return (%)Investment

8.512.3Large stocks

13.617.4Small stocks

2.86.2Long-term corporate

bonds

2.05.8Long-term government

bonds

0.03.8U.S treasury bills



An International Perspective: 1990-2009 vs. longer Periods



Averaging Methods of Returns

Returns over three periods are 20%, 0%, and -10%.

Arithmetic Average

[0.20 + 0 + (-0.1)] / 3 = 3.33%

Geometric Average

[(1+0.2) (1+0) (1-0.1)]1/3 - 1 = 2.60%



Geometric versus Arithmetic Averages

Standard

Deviation

(%)

AA (%)GA (%)Series

20.212.310.4Large-company

stocks

32.917.412.6Small-company

stocks

8.56.25.9Long-term

corporate bonds

9.25.85.5Long-term

government bonds



Geometric versus Arithmetic Averages

Standard

Deviation

(%)

AA (%)GA (%)Series

9.25.85.5Long-term

government bonds

5.75.55.3Intermediate-term

government bonds

3.13.83.7U.S treasury bills

4.33.13.0Inflation



• Track average returns.

• Compare performance of managed portfolios

(benchmarks).

• Base of derivatives.

• Base of ETF-s and other passive funds.


Investment Management

Stock Indexes –Why are they Useful?

23

Int’l Equity Indexes

• S&P 100,500

• Nasdaq

• The Shanghai Composite Index

• Nikkei 225

• FTSE (Financial Times of London)

• DAX (Germany)

• Hang seng (HKG)

• TSX (CANDA)



Construction of Indexes

• How are stocks weighted in an index?

– Price weighted (DJIA).

– Market-value weighted (S&P500, NASDAQ).

– Equally weighted (Value Line Index).



Constructing Indexes: A Three-Security Example

JPM AAPL KO

# of Outstanding Stocks 100 500 200

Price as of Oct. 12 15 5 10

Price as of Nov. 12 18 5 9

Market Value as of

Nov. 12

1500

(1/4)

2500

(5/12)

2000

(1/3)



Constructing the Dow Jones Index: An Example

Weight the prices equally:

Dow Jones level in OCT 12: (15+5+10)/3=10

The level in Nov 12: (18+5+9)/3=10.67

Monthly return on the Dow Jones index is

10.67/10-1=6.7%



Constructing the Dow Jones Index: An Example

• What if there are stock splits?

• How would you compute the current divisor of the dow jones?

• Before answering these questions let us show that the DJ is indeed a price weighted index.



Is it Really a Price Weighted Index?

Weight the prices equally:

The prices of JPM, AAPL, and KO are 15, 5, and 10,

which means that the corresponding weights in the DJ

index are 1/2, 1/6, and 1/3.



Is it Really a Price Weighted Index?

Thus, the price weighted average is

1/2×20%+1/6×0%+1/3(×-10%)=6.7%.

Next, we discuss some adjustments to the DJ divisor.



Changing the Divisor: An Example

Day 1 of Index: Company Price

GM 40.56

Nordstrom 25.91

Lowe's 53.68

Sum: 120.15

Index: 40.05 (Divisor = 3)

Before Day 2 starts, you want to replace Lowe's with Home Depot,

selling at $32.90.

To keep the value of the Index the same, i.e., 40.05:

GM 40.56

Nordstrom 25.91

Home Depot 32.90

Sum: 99.37

99.37 / Divisor = 40.05, Divisor is: 2.481



Changing the Divisor: An Example

What would have happened to the divisor had Home

Depot shares been sold at $65.72 per share?



Constructing the S&P Index: An Example

• The monthly returns on JPM, AAPL, and KO are 20%, 0%, and -10%, respectively.

• The value weighted return (S&P and NASDAQ) is computed as:

1/4×20%+5/12×0%+1/3×(-10%)= 1.67%



Constructing the S&P Index: An Example

• If the S&P index level in OCT. 12 is 1000, then the index level in NOV. 12 would then be 1000×(1+1.67%)=1016.7

• What if at least one of those three stocks pays a dividend?



Biggest Daily Price Appreciation and Depreciation since 1980 - S&P

500 (SOURCE: YAHOO FINANCE)

DATEChangeDATE

28 /10 /200810.79 %-20.47 %19 /10 /1987

13 /10 /20089.93 %-8.72 %15 /10 /2008

21 /10 /19879.10 %-8.49 %29 /09 /2008

13 /11 /20086.82 %-8.27 %26 /10 /1987

23 /03 /20096.55 %-8.15 %01 /12 /2008



Biggest Daily Price Appreciation and Depreciation since 1984-

NASDAQ (SOURCE: YAHOO FINANCE)

DATEChangeDATE

03 /01 /200116.06 %-8.96 %31 /08 /1998

17 /04 /20009.47 %-8.69 %19 /10 /1987

13 /10 /20008.58 %-7.68 %14 /04 /2000

28 /10 /19977.97 %-7.61 %29 /09 /2008

27 /04 /20007.24 %-7.37 %02 /01 /2001



Largest Corporate America Annual Earnings of All Time (SOURCE: WIKIPEDIA)

USD real earning (Bn $)

Year Corporate

48.552008ExxonMobile

41.732012Apple

30.391998Ford Motor Company

28.22005Citigroup

25.692008Chevron



Largest Corporate America Annual Losses of All Time (SOURCE: WIKIPEDIA)

USD real loss (Bn $)

Year Corporate

123.160022AOL Time Warner

106.622008AIG

77.772009Fannie mae

71.662001JDS Uniphase

54.542008Freddie Mac



Financial Market Anomalies



Case Studies: The DFA and the AQR’s Momentum Funds

Quantitative asset management

• This lecture covers topics related to the Dimensional

Fund Advisors (DFA) as well as the AQR’s momentum

funds – both are HBS’ case studies.

• Quantitative asset management does not rely on hands-

on individual company analysis (investment based on

fundamental values does). This facilitates investments

in a large spectrum of stocks, bonds, etc.



• To be a successful quant you need good forecasting

models, a powerful computer, and $$$$ to manage.

• Currently DFA manages over and above 200 Billion

Dollar – a fast growing investment corporation.

דורון אברמוב ' פרופ



Prof. Doron Avramov


• The DFA’s case study deals with the size and book to

market effects with some special treatment of market

liquidity.

• The AQR’s case deals with price momentum. The second

pdf of this case study is basically a one line file indicating the

momentum crash over the year 2009.




Prof. Doron Avramov


The Overall Agenda

• We will start with some dry, albeit essential, definitions

about the investment process.

• We will explain some important concepts relevant for

evaluating investment performance including benchmarks,

alpha, Sharpe Ratio, IR, the drawdown measure, VaR, and

shortfall probability.



The Overall Agenda

• We will present evidence from financial markets in Israel

and the US on the inability of active management to beat

the market.

• We will then present performance of passive strategies,

such as the size and value effects as well as momentum,

that do seem to work well (but not recently…).



The Overall Agenda

• My own take: be a passive investor. Don’t be too active,

and don’t over-trade based on emotions, sentiments, and so

on and so forth… More later!

• Let us begin…



What is Asset Allocation?

• How to invest in major asset classes - equities, fixed-

income, commodities, currencies, and cash and cash

equivalents?

• Asset allocation aims to balance risk and reward based

on our goals, risk tolerance, taxation, and investment

horizon.



What is Asset Allocation?

• The consensus among financial professionals is that

asset allocation is the most important decision that

investors make.

• Alas, the profession lacks good enough models that

communicate reliable hints about how to optimally

invest in the major classes.



What is Security Selection?

• Once you decide upon a policy portfolio that invests in stocks, bonds, commodities, currencies, and cash (e.g., 30%, 20%, 30%, 15%, and 5%) you have to choose the particular securities.

• Stocks: big versus small market cap, high versus low book-to-market, biotech versus real estate, high versus low past return, etc.

• Bonds: Government versus corporate, high versus low quality (credit risk), long versus short time to maturity (duration), etc.



What is Security Selection?

• Commodities: Gold, Oil, Soybeans, Citrus, etc.

• Currencies: Euro, Dollar, Yen, Pound, etc.

• Cash: money market funds versus bank CDs, etc.

• I believe security selection is the place where one could detect profitable opportunities.



What is the Ultimate Question in Finance?

• Could you beat the market?

• Beating the market means consistently accomplishing

higher risk-adjusted return, relative to a simple investment

in the market index.

A valid question: good past performance – is it really skill

or just a pure luck?



Market Efficiency

• Market efficiency summarizes the relation between

stock prices and information available to investors.

• If markets are efficient, then all information is already

in the stock price; the price is right.



Market Efficiency

• If the price is right it is impossible to “beat the

market,” except by luck.

• If the price isn’t right, you can buy underpriced stocks

and short sell overpriced stocks.



Are Markets Efficient?

• You will never know for sure.

• If you want to make your career in the investment

management industry you better believe that you can

beat the market.

• It is difficult to believe that over the short run markets

are able to digest all the huge amount of information

affecting asset prices.



Three Forms of Market Efficiency

• Weak-form efficient market

A market in which past prices and volume figures are of no use in beating the market.

• Semi-strong-form efficient market

A market in which all publicly available information is of no use in beating the market.

• Strong-form efficient market

A market in which information of any kind, public or private, is of no use in beating the market.



Information Sets for Market Efficiency

Strong-form information set:

All information of any kind, public or private.

Semistrong-form information set:

All publicly available information.

Weak-form information set:

Past price and volume.



Starting Point: Could Actively Managed Funds Beat the Market?

• Over the next slides we will show that actively managed

funds in the US and Israel DO NOT beat the market, on

average, over the long run...

• Some do over short investment horizons.



Starting Point: Could Actively Managed Funds Beat the Market?

• However, this could be due to luck, not skill, as

performance virtually vanishes over longer periods – an

inherent lack of persistence in performance.

• Let us look at Legg Mason Value to display such lack of

persistence.



:Legg Mason Value ביל מילר ב"קרן הנאמנות הטובה בארה

2006 - 1991



So would you Invest in Legg Mason in 2006?

• Probably you would!

• Based on empirical research - money is chasing

performance.

• But will the investment be successful over the following

years?

• Coming next!



:Legg Mason Value ביל מילר ב "קרן הנאמנות הטובה בארה

2006-2011



:Legg Mason Value ביל מילר ב "קרן הנאמנות הטובה בארה

1991-2011



And what about performance in the overall universe of U.S. equity

mutual funds? A One-Year Perspective.



Here is a Longer Horizon (10 years) Perspective



Summary: The Lack of Mutual Fund Performance over Various

Investment HorizonsPercent (%)Number of

Investment Periods

Three-Fourth of the Funds

Beat Vanguard

Percent (%)Number of Investment

PeriodsHalf of funds

Beat Vanguard

Number of Investment

Periods

SpanLength of each Investment Period (Years)

6.7246.714301977-20061

7.1246.413281979-20063

7.7234.69261981-20065

0.009.52211986-200610



What about Israel?

Quite similar… just take a look into the

figures



Draw

Down

סטית

תקן

שחם 1 6.4-2.84.81.530.85אלטשולר

2 AlphaBeta3.9-4.64.50.33

4-9.45.71.850.28ילין לפידות 3

3.3-6.14.11.560.25אפסילון 4

2.8-5.131.380.12מנורה מבטחים 5

2.6-6.14.31.580.06אלומות 6

2.4-8.24.51.40סיגמא 7

1.9-14.37.61.29-0.06.אי.בי.אי 8

1.7-9.25.31.54-0.07ממוצע מנהלים 9

1.8-126.41.87-0.08א.ד. רוטשילד 10

1.9-9.14.90.65-0.09דש 11

1.9-7.14.61.3-0.1אקסלנס 12

1.4-8.25.61.54-0.1מיטב 13

1.5-7.54.81.51-0.12פסגות 14

1.5-74.21.39-0.17מגדל 15

1.4-7.94.61.5-0.18הראל-פיא 16

ננסים 17 1-10.36.11.94-0.19כלל פי

0.9-10.15.71.8-0.2אנליסט 18

0.6-15.17.21.9-0.36-מאור לוסקי 19

0.8-115.31.8-0.63-איילון 20

3.2-17.68.11.55-0.66-יובנק 21

מדד

שארפשם קרן

תשואה

שנתית

(%)

ממוצע

דמי

ניהול

(%)

רמת סיכון (%)

מניות 30%-10%ח כללי "מנהלי אג) שנתיים (



מניות 30%-10%ח כללי "מנהלי אג) שנים8(

Draw

Down

סטית

תקן

1 AlphaBeta6.7-85.70.51

5.8-10.84.81.420.42הראל-פיא 2

5.8-22.86.62.290.31אפסילון 3

5.9-28.681.30.28.אי.בי.אי 4

4.4-21.46.21.680.13ממוצע מנהלים 5

ננסים 6 4.4-23.36.72.180.11כלל פי

4.1-15.65.21.920.08אקסלנס 7

4.1-13.75.21.760.08פסגות 8

4-24.36.41.50.05א.ד. רוטשילד 9

3.8-21.64.61.270.02מגדל 10

2-31.68.41.51-0.17יובנק 11

מדד

שארפשם קרן

תשואה

שנתית

(%)

ממוצע רמת סיכון (%)

דמי

ניהול

(%)



Draw

Down

סטית

תקן

1 AlphaBeta-1.3-21.717.7-0.12

3.2-27.220.82.32-0.17-מיטב 2

3.5-24.519.31.55-0.21-פסגות 3

4.6-26.620.21.1-0.25-.אי.בי.אי 4

3.9-25.218.53.02-0.26-א.ד. רוטשילד 5

ננסים 6 4.3-26.919.42.98-0.26-כלל פי

5.6-27.520.21.89-0.3-מגדל 7

5.8-27.419.62.47-0.33-ממוצע מנהלים 8

6.3-2720.72.16-0.34-הראל-פיא 9

6.5-27.920.52.37-0.34-אקסלנס 10

6-26.818.22.99-0.38-מנורה מבטחים 11

7.6-29.5193.1-0.45-דש 12

8.6-29.719.13.54-0.5-אנליסט 13

9.1-30.418.92.68-0.54-איילון 14

מדד

שארפשם קרן

תשואה

שנתית

(%)

ממוצע רמת סיכון (% )

דמי

ניהול

(%)

120% עד 100א "מנהלי מניות ת) שנתיים(מניות



Draw

Down

סטית

תקן

8.2-45.320.22.770.3א.ד. רוטשילד 1

2 AlphaBeta7.8-40.218.90.29

ננסים 3 6.8-47.420.42.330.24כלל פי

6.6-42.319.32.690.24פסגות 4

6.5-39.520.33.10.23אקסלנס 5

6.1-50.2231.90.21.אי.בי.אי 6

6.1-46.319.72.80.21מנורה מבטחים 7

5.9-47.920.62.750.2ממוצע מנהלים 8

5.2-49.921.12.580.17מיטב 9

5.2-45.920.62.930.17מגדל 10

4.3-52.721.32.640.13הראל-פיא 11

3.8-59.420.43.780.1אנליסט 12

ממוצע

דמי

ניהול

(%)

מדד

שארפשם קרן

תשואה

שנתית

(%)

רמת סיכון (%)

120% עד 100א "מנהלי מניות ת) שנים8(מניות



Active versus Passive Investment Management

• Even when one family of mutual funds in Israel did

perform well – would you be able to identify that family

upfront?

• It is fairly safe to conclude that active management does

not beat the market, on average, on a consistent basis.

• Nevertheless, in the following I will analyze several

trading strategies in the US and Israel that had

performed well.



Active versus Passive Investment Management

• All such strategies are passive – they don’t require an in-

depth analysis of the corporations, and they are easy to

implement as long as you could access asset returns as

well as financial statement data and then embark on a

relatively tractable data analysis.

• We start with the size, value, and momentum effects, then

switch to other interesting strategies.



Anomaly I: The size effect

• Size effect: higher average returns on small stocks than

large stocks. Beta cannot explain the difference.



Anomaly II: The value effect

• Value effect: higher average returns on value stocks

than growth stocks. Beta cannot explain the

difference.

• Value firms: Firms with high E/P, B/P, D/P, or CF/P.

The notion of value is that physical assets can be

purchased at low prices.

• Growth firms: Firms with low ratios. The notion is

that high price relative to fundamentals reflects

capitalized growth opportunities.



The Value Effect: The International Evidence



As of 31/12/2010 1 year 2 years

Asset NameAnn.

ret

Ann.

ret

Ann.

ret

Ann.

std

Ann.

ret

Ann.

std

Sharpe

ratio

Ann.

ret

Ann.

std

Sharpe

ratio

Ann.

ret

Ann.

std

Sharpe

ratio

DD

(12

months)

DD

(24

months)

DD

(36

months)

Info

ratio

Pr. α

(12

months)

Pr. α

(24

months)

Pr. α

(36

months)

CV3000 - TOP 27.95% 22.93% 1.96% 30.48% 4.55% 24.33% 0.10 10.77% 19.94% 0.43 11.9% 17.72% 0.49 -47.49% -54.55% -45.43% 0.60 69% 75% 75%

RUSSELL3000 14.75% 19.99% -4.08% 22.94% 0.71% 18.45% -0.08 0.32% 16.78% -0.11 5.17% 16.62% 0.12 -44.85% -48.28% -43.09%

Excess return 13.2% 2.95% 6.04% 3.83% 10.45% 6.73%

CV3000 - BTM 25.72% 41.19% -2.53% 31.2% 0.66% 26.46% -0.06 -4.71% 32.16% -0.21 -0.98% 37.39% -0.11 -76.01% -83.85% -86.99% -0.23 40% 33% 26%

RUSSELL3000 14.75% 19.99% -4.08% 22.94% 0.71% 18.45% -0.08 0.32% 16.78% -0.11 5.17% 16.62% 0.12 -44.85% -48.28% -43.09%

Excess return 10.97% 21.21% 1.54% -0.06% -5.03% -6.15%

CV1000 - TOP 21.67% 23.36% 1.81% 24.18% 4.34% 19.39% 0.11 9.38% 16.08% 0.45 11.67% 15.49% 0.55 -43.93% -48.77% -40.91% 0.62 71% 75% 82%

RUSSELL1000 13.87% 19.53% -4.49% 22.6% 0.51% 18.17% -0.09 -0.05% 16.57% -0.13 5.13% 16.53% 0.12 -44.99% -48.01% -43.56%

Excess return 7.8% 3.83% 6.3% 3.83% 9.43% 6.53%

CV1000 - BTM 31.57% 46.7% 0.02% 28.89% 5.39% 24.63% 0.13 -2.29% 31.57% -0.14 4.23% 35.61% 0.03 -79.13% -84.77% -84.9% -0.04 49% 62% 57%

RUSSELL1000 13.87% 19.53% -4.49% 22.6% 0.51% 18.17% -0.09 -0.05% 16.57% -0.13 5.13% 16.53% 0.12 -44.99% -48.01% -43.56%

Excess return 17.7% 27.16% 4.51% 4.88% -2.24% -0.91%

3 years 5 years 10 years 15 years

The value Effect: Recent Evidence in the US



The size and value effects during the post discovery period

• In 1993 Dimensional Fund Advisors (DFA) began a

mutual fund that focuses on small firms with high

B/M ratios (the DFA US 6-10 Value Portfolio).

• This portfolio would have earned significantly

positive abnormal return of about 0.5% per month

over the period 1963-1993.

• The estimate of the risk adjusted return to that

portfolio from 1994-2002 is about –0.2% per month,

with a t-statistic of -0.59%.



Anomaly III: Price Momentum

• Jegadeesh and Titman (1993) found that recent past

winners (stocks with high returns in the last 3, 6, 9,

and 12 months) outperform recent past losers.

• Keep in mind that practitioners already were using

momentum strategies prior to the academic discovery.

• Take a look at the next plot.




Asset NameAnn.

ret

Ann.

ret

Ann.

ret

Ann.

std

Ann.

ret

Ann.

std

Sharpe

ratio

Ann.

ret

Ann.

std

Sharpe

ratio

Ann.

ret

Ann.

std

Sharpe

ratio

DD

(12

months)

DD

(24

months)

DD

(36

months)

Info

ratio

Pr. α

(12

months)

Pr. α

(24

months)

Pr. α

(36

months)

Momentum 1000 -TOP 27.55% 26.36% -5.21% 27.28% 4.09% 22.98% 0.08 6.42% 19.83% 0.22 13.98% 23.04% 0.47 -51.5% -46.97% -41.07% 0.56 74% 77% 89%

RUSSELL1000 13.87% 19.53% -4.49% 22.6% 0.51% 18.17% -0.09 -0.05% 16.57% -0.13 5.13% 16.53% 0.12 -44.99% -48.01% -43.56%

Excess return 13.68% 6.83% -0.72% 3.58% 6.47% 8.84%

Momentum 1000 -BTM 25.97% 47.92% 5.68% 34.28% 5.73% 27.3% 0.13 4.22% 28.45% 0.07 4.43% 26.2% 0.05 -51.66% -57.21% -51.11% -0.05 50% 49% 50%

RUSSELL1000 13.87% 19.53% -4.49% 22.6% 0.51% 18.17% -0.09 -0.05% 16.57% -0.13 5.13% 16.53% 0.12 -44.99% -48.01% -43.56%

Excess return 12.1% 28.39% 10.16% 5.22% 4.27% -0.7%

10 years 15 years3 years 5 years

Stock Price Momentum: Recent Evidence in the US



Momentum Robustness

• Momentum is fairly robust in a cross-industry analysis,

cross-country analysis, cross-style analysis, and within

other countries virtually all over the world.

• Momentum also seems to appear in bonds, currencies,

and commodities, as well as in mutual funds and hedge

funds.



Momentum Robustness

• The momentum phenomena has invoked many academic

studies trying to understand its source.

• Momentum could be rationally based, it could be

triggered by behavioral biases, or it could be attributable

to under-reaction to financial news.

• But in 2009 – momentum delivers a -85% payoff!!!

• There are some other momentum crash periods.



Cumulative Return of 1 USD Invested 1927 - 2012

Mkt-RF SMB HML WML

$ 129.83 $ 6.97 $ 27.06 $ 322.97



Cumulative Return of 1 USD Invested Last 2 Decades (1992 – 2012)

Mkt-RF SMB HML WML

$ 2.98 $ 1.54 $ 1.69 $ 2.73



Economic Links

• Often times investors do not recognize economic links

between economically related firms.

• Something that can be exploited to establish profitable

strategies due to such limited attention.

• Here is a striking example.



Economic Links

• Coastcast Corporation was a leading manufacturer of

golf club heads.

• Since 1993 its major customer (50% of the sales) had

been Callaway Golf Corporation

• On June 2001, Callaway was downgraded – expected

EPS was down from 70 cents per share to only 35.



Economic Links

•On the very same day the Callaway’s stock price is down

about 30% which makes sense.

•However, no impact on Coastcast's stock price.

•The ultimate adjustment occurred only two months later.



Down Side Risk Measures

•Value at risk (VaR), shortfall probability, drawdown,

semi-variance, downside beta.

•All such measures focus on the left tail (the unappealing

part) of the return distribution – common practice in

investment management.




Asset NameAnn.

ret

Ann.

ret

Ann.

ret

Ann.

std

Ann.

ret

Ann.

std

Sharpe

ratio

Ann.

ret

Ann.

std

Sharpe

ratio

Ann.

ret

Ann.

std

Sharpe

ratio

DD

(12

months)

DD

(24

months)

DD

(36

months)

Info

ratio

Pr. α

(12

months)

Pr. α

(24

months)

Pr. α

(36

months)

VaR 5% 1000 -TOP 20.04% 20.37% 0.1% 18.62% 5.51% 15.39% 0.22 7.87% 13.31% 0.43 13.11% 14.91% 0.67 -34.82% -33.42% -26.35% 0.71 79% 89% 92%

RUSSELL1000 13.87% 19.53% -4.49% 22.6% 0.51% 18.17% -0.09 -0.05% 16.57% -0.13 5.13% 16.53% 0.12 -44.99% -48.01% -43.56%

Excess return 6.17% 0.84% 4.59% 5.0% 7.91% 7.98%

VaR 5% 1000 -BTM 31.67% 54.99% 5.31% 37.26% 5.65% 29.88% 0.12 1.93% 32.01% -0.01 5.26% 29.53% 0.07 -57.23% -66.27% -60.6% 0.01 50% 54% 51%

RUSSELL1000 13.87% 19.53% -4.49% 22.6% 0.51% 18.17% -0.09 -0.05% 16.57% -0.13 5.13% 16.53% 0.12 -44.99% -48.01% -43.56%

Excess return 17.8% 35.46% 9.8% 5.14% 1.98% 0.13%

ShortFall 10% 1000 -TOP 20.38% 19.53% -3.66% 21.1% 2.83% 17.17% 0.04 5.98% 14.3% 0.27 11.82% 16.67% 0.52 -43.0% -42.44% -36.15% 0.54 81% 80% 92%

RUSSELL1000 13.87% 19.53% -4.49% 22.6% 0.51% 18.17% -0.09 -0.05% 16.57% -0.13 5.13% 16.53% 0.12 -44.99% -48.01% -43.56%

Excess return 6.51% 0.0% 0.83% 2.32% 6.02% 6.69%

ShortFall 10% 1000 -BTM 27.6% 44.21% 2.71% 34.25% 3.93% 27.48% 0.06 2.39% 28.85% 0.01 4.63% 26.58% 0.06 -53.39% -58.41% -52.8% -0.03 47% 52% 59%

RUSSELL1000 13.87% 19.53% -4.49% 22.6% 0.51% 18.17% -0.09 -0.05% 16.57% -0.13 5.13% 16.53% 0.12 -44.99% -48.01% -43.56%

Excess return 13.73% 24.68% 7.19% 3.42% 2.43% -0.5%


Trading on Down Side Risk Measures: Evidence in the US




Asset NameAnn.

ret

Ann.

ret

Ann.

ret

Ann.

std

Ann.

ret

Ann.

std

Sharpe

ratio

Ann.

ret

Ann.

std

Sharpe

ratio

Ann.

ret

Ann.

std

Sharpe

ratio

DD

(12

months)

DD

(24

months)

DD

(36

months)

Info

ratio

Pr. α

(12

months)

Pr. α

(24

months)

Pr. α

(36

months)

FScore 3000 - TOP 18.53% 23.08% 1.25% 23.12% 4.74% 18.94% 0.14 9.66% 16.59% 0.45 13.2% 16.0% 0.63 -38.35% -42.22% -34.55% 1.03 82% 83% 84%

RUSSELL3000 14.75% 19.99% -4.08% 22.94% 0.71% 18.45% -0.08 0.32% 16.78% -0.11 5.17% 16.62% 0.12 -44.85% -48.28% -43.09%

Excess return 3.77% 3.1% 5.33% 4.03% 9.34% 8.03%

FScore 3000 - BTM 34.84% 47.42% 2.13% 34.28% 2.76% 28.23% 0.02 1.06% 29.11% -0.04 1.7% 30.07% -0.05 -54.13% -67.92% -69.45% -0.19 33% 29% 28%

RUSSELL3000 14.75% 19.99% -4.08% 22.94% 0.71% 18.45% -0.08 0.32% 16.78% -0.11 5.17% 16.62% 0.12 -44.85% -48.28% -43.09%

Excess return 20.09% 27.44% 6.21% 2.05% 0.74% -3.47%

FScore 1000 - TOP 17.28% 25.01% 0.83% 22.39% 5.58% 18.2% 0.19 8.58% 16.09% 0.40 11.92% 15.54% 0.56 -39.35% -39.61% -30.82% 1.02 78% 80% 84%

RUSSELL1000 13.87% 19.53% -4.49% 22.6% 0.51% 18.17% -0.09 -0.05% 16.57% -0.13 5.13% 16.53% 0.12 -44.99% -48.01% -43.56%

Excess return 3.41% 5.48% 5.32% 5.07% 8.62% 6.78%

FScore 1000 - BTM 29.76% 38.47% 0.21% 29.84% 4.13% 23.89% 0.08 2.38% 23.16% 0.01 7.24% 23.05% 0.18 -52.75% -57.68% -53.6% 0.20 59% 64% 70%

RUSSELL1000 13.87% 19.53% -4.49% 22.6% 0.51% 18.17% -0.09 -0.05% 16.57% -0.13 5.13% 16.53% 0.12 -44.99% -48.01% -43.56%

Excess return 15.89% 18.94% 4.69% 3.62% 2.43% 2.1%


Using the F-Score: Evidence in the US




Asset NameAnn.

ret

Ann.

ret

Ann.

ret

Ann.

std

Ann.

ret

Ann.

std

Sharpe

ratio

Ann.

ret

Ann.

std

Sharpe

ratio

Ann.

ret

Ann.

std

Sharpe

ratio

DD

(12

months)

DD

(24

months)

DD

(36

months)

Info

ratio

Pr. α

(12

months)

Pr. α

(24

months)

Pr. α

(36

months)

CV/FScore/Mom 3000 - TOP 18.48% 17.45% 5.98% 24.97% 6.4% 20.27% 0.21 14.54% 17.4% 0.71 15.42% 16.07% 0.76 -31.57% -38.89% -30.33% 0.91 72% 78% 82%

RUSSELL3000 14.75% 19.99% -4.08% 22.94% 0.71% 18.45% -0.08 0.32% 16.78% -0.11 5.17% 16.62% 0.12 -44.85% -48.28% -43.09%

Excess return 3.72% -2.54% 10.06% 5.68% 14.21% 10.25%

CV/FScore/Mom 3000 - BTM 25.49% 52.63% 1.74% 35.9% 2.2% 29.77% 0.00 -3.98% 38.26% -0.16 -6.46% 37.0% -0.26 -65.32% -80.74% -85.41% -0.45 28% 24% 15%

RUSSELL3000 14.75% 19.99% -4.08% 22.94% 0.71% 18.45% -0.08 0.32% 16.78% -0.11 5.17% 16.62% 0.12 -44.85% -48.28% -43.09%

Excess return 10.74% 32.64% 5.82% 1.48% -4.31% -11.63%

CV/FScore/Mom 1000 -TOP 8.89% 13.27% 2.83% 20.66% 5.9% 17.01% 0.22 9.47% 14.84% 0.49 11.91% 15.05% 0.58 -27.43% -27.99% -18.78% 0.63 67% 75% 84%

RUSSELL1000 13.87% 19.53% -4.49% 22.6% 0.51% 18.17% -0.09 -0.05% 16.57% -0.13 5.13% 16.53% 0.12 -44.99% -48.01% -43.56%

Excess return -4.99% -6.26% 7.32% 5.39% 9.52% 6.78%

CV/FScore/Mom 1000 -BTM 18.07% 37.51% -1.75% 31.38% 2.7% 25.28% 0.02 -5.55% 35.36% -0.22 -3.44% 33.14% -0.20 -75.47% -83.44% -82.93% -0.40 38% 36% 28%

RUSSELL1000 13.87% 19.53% -4.49% 22.6% 0.51% 18.17% -0.09 -0.05% 16.57% -0.13 5.13% 16.53% 0.12 -44.99% -48.01% -43.56%

Excess return 4.2% 17.98% 2.73% 2.19% -5.51% -8.58%


Combining F-Score, Momentum, and Value Effects



One more Interesting Anomaly is Earnings Momentum: Cumulative Returns in Response to

Earnings Announcements




Asset NameAnn.

ret

Ann.

ret

Ann.

ret

Ann.

std

Ann.

ret

Ann.

std

Sharpe

ratio

Ann.

ret

Ann.

std

Sharpe

ratio

Ann.

ret

Ann.

std

Sharpe

ratio

DD

(12

months)

DD

(24

months)

DD

(36

months)

Info

ratio

Pr. α

(12

months)

Pr. α

(24

months)

Pr. α

(36

months)

Value/Momentum - TOP -13.55% 1.57% 22.62% 17.15% 6.49% 20.04% 0.17 17.43% 19.4% 0.67 15.4% 22.51% 0.39 -39.86% -37.06% -41.52% 0.59 70% 78% 83%

TA-MAAGAR -16.4% -2.48% 19.12% 17.69% 0.42% 21.85% -0.13 8.78% 19.72% 0.22 10.08% 20.74% 0.17 -52.05% -40.85% -44.45%

Excess return 2.84% 4.05% 3.5% 6.06% 8.65% 5.33%

Value/Momentum - BTM -28.27% -10.04% 20.17% 29.08% -8.26% 33.75% -0.34 -0.56% 27.37% -0.18 1.98% 26.04% -0.18 -70.01% -69.08% -65.72% -0.68 20% 9% 6%

TA-MAAGAR -16.4% -2.48% 19.12% 17.69% 0.42% 21.85% -0.13 8.78% 19.72% 0.22 10.08% 20.74% 0.17 -52.05% -40.85% -44.45%

Excess return -11.88% -7.56% 1.06% -8.68% -9.34% -8.1%

Momentum TA100 - TOP -6.3% 6.12% 32.93% 19.25% 10.77% 25.8% 0.29 20.31% 22.58% 0.70 17.94% 24.1% 0.47 -50.75% -39.96% -38.68% 0.72 70% 82% 90%

TA100 -15.79% -2.08% 19.1% 17.51% 1.13% 21.67% -0.09 8.79% 19.69% 0.22 10.01% 21.02% 0.16 -51.14% -39.78% -42.7%

Excess return 9.49% 8.2% 13.82% 9.64% 11.52% 7.93%

Momentum TA100 - BTM -33.43% -15.29% 13.72% 29.76% -12.62% 35.07% -0.45 -4.6% 28.86% -0.31 -1.46% 28.24% -0.28 -72.6% -71.83% -71.19% -0.78 22% 13% 2%

TA100 -15.79% -2.08% 19.1% 17.51% 1.13% 21.67% -0.09 8.79% 19.69% 0.22 10.01% 21.02% 0.16 -51.14% -39.78% -42.7%

Excess return -17.64% -13.2% -5.39% -13.75% -13.39% -11.47%

Value-at-Risk TA100 - TOP -5.68% 7.44% 33.05% 17.62% 13.26% 24.18% 0.42 21.83% 21.01% 0.83 20.6% 22.63% 0.62 -47.05% -33.8% -26.11% 1.08 84% 100% 100%

TA100 -15.79% -2.08% 19.1% 17.51% 1.13% 21.67% -0.09 8.79% 19.69% 0.22 10.01% 21.02% 0.16 -51.14% -39.78% -42.7%

Excess return 10.11% 9.52% 13.95% 12.13% 13.04% 10.59%

Value-at-Risk TA100 - BTM -28.36% -14.18% 19.15% 30.75% -13.85% 37.02% -0.46 -5.48% 30.0% -0.33 -1.78% 29.22% -0.29 -77.71% -77.16% -77.16% -0.76 22% 17% 8%

TA100 -15.79% -2.08% 19.1% 17.51% 1.13% 21.67% -0.09 8.79% 19.69% 0.22 10.01% 21.02% 0.16 -51.14% -39.78% -42.7%

Excess return -12.57% -12.1% 0.04% -14.98% -14.28% -11.79%


What about Israel? Strategies Involving Stocks




Asset NameAnn.

ret

Ann.

ret

Ann.

ret

Ann.

std

Ann.

ret

Ann.

std

Sharpe

ratio

Ann.

ret

Ann.

std

Sharpe

ratio

Ann.

ret

Ann.

std

Sharpe

ratio

DD

(12

months)

DD

(24

months)

DD

(36

months)

Info

ratio

Pr. α

(12

months)

Pr. α

(24

months)

Pr. α

(36

months)

Accruals TA100 - TOP -18.15% 2.68% 20.51% 17.84% 10.58% 20.04% 0.37 16.89% 18.57% 0.67 16.44% 20.22% 0.49 -35.24% -28.98% -24.9% 1.01 83% 97% 98%

TA100 -15.79% -2.08% 19.1% 17.51% 1.13% 21.67% -0.09 8.79% 19.69% 0.22 10.01% 21.02% 0.16 -51.14% -39.78% -42.7%

Excess return -2.37% 4.76% 1.41% 9.45% 8.1% 6.43%

Accruals TA100 - BTM -17.58% -7.1% 21.18% 24.82% -10.11% 32.81% -0.40 0.73% 28.79% -0.13 3.79% 28.91% -0.10 -74.58% -70.15% -70.86% -0.52 34% 27% 11%

TA100 -15.79% -2.08% 19.1% 17.51% 1.13% 21.67% -0.09 8.79% 19.69% 0.22 10.01% 21.02% 0.16 -51.14% -39.78% -42.7%

Excess return -1.79% -5.02% 2.08% -11.24% -8.07% -6.22%


What about Israel? Strategies Involving Stocks




Asset NameAnn.

ret

Ann.

ret

Ann.

ret

Ann.

std

Ann.

ret

Ann.

std

Sharpe

ratio

DD

(12

months)

DD

(24

months)

DD

(36

months)

Info

ratio

Pr. α

(12

months)

Pr. α

(24

months)

Pr. α

(36

months)

Spillover Momentum - TOP 3.4% 7.6% 14.79% 5.01% 9.74% 5.73% 1.07 -2.39% 2.08% 0.15 94.97% 73.77% 83.67% 100.0%

TelBond 60 0.59% 4.78% 11.6% 5.86% 6.09% 8.24% 0.30 -12.96% -8.48% 0.03

Excess return 2.81% 2.82% 3.19% 3.65%

Spillover Momentum - BTM -8.74% 1.1% 13.89% 10.72% 2.55% 16.88% -0.06 -40.27% -36.98% -0.21 -34.69% 40.98% 20.41% 10.81%

TelBond 60 0.59% 4.78% 11.6% 5.86% 6.09% 8.24% 0.30 -12.96% -8.48% 0.03

Excess return -9.32% -3.68% 2.29% -3.54%

Spillover VaR - TOP 3.8% 7.05% 11.18% 4.45% 8.62% 5.92% 0.85 -2.41% 2.74% 0.19 65.48% 67.21% 75.51% 97.3%

TelBond 60 0.59% 4.78% 11.6% 5.86% 6.09% 8.24% 0.30 -12.96% -8.48% 0.03

Excess return 3.21% 2.27% -0.41% 2.52%

Spillover VaR - BTM -10.48% 0.85% 16.39% 12.57% 3.44% 18.28% -0.01 -41.19% -38.81% -0.22 -23.0% 44.26% 28.57% 18.92%

TelBond 60 0.59% 4.78% 11.6% 5.86% 6.09% 8.24% 0.30 -12.96% -8.48% 0.03

Excess return -11.07% -3.93% 4.79% -2.66%

Spillover ShortFall - TOP 4.12% 7.12% 13.17% 5.47% 9.03% 6.34% 0.86 -3.89% 1.14% 0.16 97.98% 68.85% 81.63% 100.0%

TelBond 60 0.59% 4.78% 11.6% 5.86% 6.09% 8.24% 0.30 -12.96% -8.48% 0.03

Excess return 3.53% 2.34% 1.58% 2.94%

Spillover ShortFall - BTM -9.43% 0.85% 11.83% 8.63% 1.83% 15.16% -0.12 -37.93% -35.12% -0.21 -48.76% 42.62% 24.49% 10.81%

TelBond 60 0.59% 4.78% 11.6% 5.86% 6.09% 8.24% 0.30 -12.96% -8.48% 0.03

Excess return -10.01% -3.93% 0.23% -4.26%

3 years 6 years

What about Israel? Strategies Involving Bonds



Portfolio Optimization and Asset Classes

96 Prof. Doron Avramovדורון אברמוב ' פרופ


W1 = 150 Profit = 50

W2 = 80 Profit = -20

p =0.6

1-p = 0.4100

Risky Inv.

Risk-Free T-bills W=105 Profit = 5

Risky and Risk-free Investments

If the two investments are mutually exclusive (you cannot

mix), which one would you select?



Comparing the Investments

Risky Investment:

E(R) = pR1 + (1-p)R2 = 0.6×0.50 + 0.4×(-0.2) = 0.22

σ2 = p[R1 - E(R)]2 + (1-p) [R2 - E(R)]2 =

0.6 (0.50-0.22) 2 + 0.4(-0.2-0.22) 2 =0.1176.

σ = 0.3429

Risk-less Investment:

E(R) =0.05 and σ = 0

Risk Premium: 22%-5%=17%.



Characterizing Worldwide Investors

• Investors worldwide are, on average, risk averse.

• In the US and other countries, average rate of return

on equities has exceeded T-Bill rate.

• Such a premium, often termed the equity premium,

reflects a compensation for risky investments.

• The equity premium is typically computed as the

return on a market-wide index, e.g., the S&P500, in

excess of the return on the three-month T-Bill.



Risk Measures

• This lecture deals with the volatility of investment return as a risk measure.

• There are other well used measures.

• Shortfall probability, conditional tail expectation, Value At Risk (VaR), and lower partial standard deviation are often used for risk management and investment.

• Our investment framework is the mean variance methodology.



Utility Function in a Mean Variance Setup

U = E(r) – 0.5Aσ2

Where:

U = utility or certainty equivalent return.

E ( r ) = expected return on the asset or portfolio.

A = coefficient of risk aversion.

σ2 = variance of returns.



Decision Rule in a Mean-Variance Setting

• The larger A the more risk averse the investor is.

• Very intuitive and heavily used among academic

scholars as well as fund managers.

• Example: assume that a stock has E(r) =0.22, σ=0.34,

and the return on T-bill is 5%.

• Suppose you can invest in the stock or T-bill. No mix!

What will you do?



Risk Aversion and Investment Strategies

• It depends on the risk aversion level.

• Different risk aversion levels lead to different investment strategies.

• Recall, U=0.22-0.5A(0.34)2

• Hence:

A Utility Investment Decision

5 -6.90% Buy T.bill (utility = 5%)

3 4.66% Buy T.bill (utility =5%)

1 16.22% Buy stock (utility>5%)



A Point of Indifference

• What is A such that investors are indifferent between

the risky and risk-less investments?

• Solve:

0.05=0.22-0.5×A×(0.34)2

which yields

A=2.9412.

If A is smaller than 2.9412 invest in the risky asset.

Otherwise, invest in the risk-free T.bill.



Risk Aversion Leads to Diversification

• A sensible strategy for a risk averse agent would be to diversify funds across several securities.

• I will give examples formally displaying how diversification reduces risk.

• To study the optimal diversification, or the optimal portfolio a risk-averse investor would choose, you should know how to compute expected return and volatility of portfolios.

• This is what we do next.



Diversification and Asset Allocation

• The role and impact of diversification were formally

introduced in the early 1950s by professor Harry

Markowitz.

• Based on his work, we will look at how diversification

works, and how we can create efficiently diversified

portfolio.

• An efficient portfolio is one having the highest expected

return for a certain level of risk, OR having the lowest

risk for a certain level of expected return.



Portfolios

In the analysis that follows we will master the following

key concepts that establish the notion of diversification:

• portfolio weights.

• portfolio expected return and volatility.

• Correlation.

Portfolios are group of assets such as stocks

and bonds held by an investor.



Computing Expected Return: An Example

• You invest $4,000 in domestic equities, $3,000 in real estate, $2,000 in commodities, and $1,000 in foreign bonds.

• Portfolio weights are 0.4, 0.3, 0.2, and 0.1.

• Expected returns are 12%, 10%, 9%, and 8%.

• What is the portfolio expected rate of return?

• Solve:

0.4×12%+0.3×10%+0.2×9%+0.1×8%=10.4%.



What about Volatility?

• More challenging!

• We will start with a relatively simple case of a two-

security portfolio.

• We will then generalize the concept to portfolios

consisting of more than two securities.



When two risky assets are combined to form a portfolio with

weights wA and wB=1- wA, the portfolio variance is computed as:

Volatility of a Two-Security Portfolio


( ) ( ) ABBAAABAAAp wwww ρσσσσσ −+−+= 121 22222

Prof. Doron Avramov


Example

• The volatilities of JP Morgan and Goldman Sachs are 16% and 20%, respectively. The covariance is 0.01.

• What is the variance of a portfolio that invests 75% in JP and 25% in Goldman:

σp2 = 0.752×0.162 + 0.252×0.22

+ 2×0.75 ×0.25×0.01=0.0207.

• The portfolio volatility is 14.37% -- the square root of the variance -- which is lower than 16% (JP) or 20% (Goldman).



Example

•This is a diversification benefit.

•Interestingly, I did not even come up with the lowest volatility portfolio (GMVP).



The GMVP

• What is the investment in JP Morgan that minimizes

the portfolio volatility?

Answer: 65.79%.

• What is the portfolio volatility?

Answer: 14.23%, even better.



Why Diversification Works? Correlation

• Positively correlated assets tend to move up and down

together, while negatively correlated assets tend to

move in opposite directions.

• Imperfect correlation is why diversification reduces

portfolio risk.

Correlation: the tendency of the returns on

two assets to move together.



Why does Diversification Work?

Portfolio AB (%)

Stock B (%)

Stock A (%)

Year

12.515102001

10-10302002

7.525-102003

12.52052004

12.515102005

11139Average return

2.213.514.3Standard

deviation



Why Diversification Works?



More about Correlation



Combining Stock and Bond



Efficient Frontier for Stock and Bond



Extending to a Three-Security Portfolio


2

3

2

3

2

2

2

2

2

1

2

1

2

332211

σσσσ

µµµµ

www

www

p

p

++=

++=

( )( )( )3232

3131

2121

,cov2

,cov2

,cov2

rrww

rrww

rrww

+

+

+

Prof. Doron Avramov


The Case of Multiple Asset Classes

• Computing volatility in the presence of multiple asset

classes is challenging.

• Both HMC case studies are about multiple asset

classes.

• Our focus here is a four-security portfolio.

• Generalizations follow the same vein.



Computing Volatility

• The volatilities of domestic equities, real estate,

commodities, and foreign bonds are 18%, 17%, 16%,

and 17%, respectively.

• Investment weights are 0.4, 0.3, 0.2, and 0.1

• Correlations are

1 0.8 0.6 0.5

0.8 1 0.7 0.8

0.6 0.7 1 0.4

0.5 0.8 0.4 1



Computing Volatility – Array # 1

You need to create three arrays.

• Array # 1 is the product of portfolio weights:

0.4×0.4 0.4 ×0.3 0.4 ×0.2 0.4 ×0.1

0.3×0.4 0.3 ×0.3 0.3 ×0.2 0.3 ×0.1

0.2×0.4 0.2 ×0.3 0.2 ×0.2 0.2 ×0.1

0.1×0.4 0.1 ×0.3 0.1 ×0.2 0.1 ×0.1




• Array # 2 is the covariance matrix:

0.18×0.18 0.8×0.18×0.17 0.6×0.18×0.16 0.5×0.18×0.17

0.8×0.18×0.17 0.17×0.17 0.7×0.17×0.16 0.8×0.17×0.17

0.6×0.18×0.16 0.7×0.17×0.16 0.16×0.16 0.4×0.16×0.17

0.5×0.18×0.17 0.8×0.17×0.17 0.4×0.16×0.17 0.17×0.17




• Array # 3 is obtained by multiplying array 1 by array

2, element by element:

0.0052 0.0029 0.0014 0.0006

0.0029 0.0026 0.0011 0.0007

0.0014 0.0011 0.0010 0.0002

0.0006 0.0007 0.0002 0.0003



Benefits from Diversification

• The portfolio variance follows by summing over all the elements in array # 3, which gives 0.0229.

• The portfolio volatility is 15.13% -- the square root of 0.0229.

• Diversification gain: the volatility of single securities ranges between 16% and18%, whereas the portfolio volatility is smaller.

• Once again, I did not even come up with the lowest volatility portfolio (GMVP).

• What is the GMVP? Coming soon!



The Contribution of a Single Security to the Overall Risk

• Column number 1 denotes the contribution of

domestic equities to the total risk.

• Example: summing up the components of column one

yields 0.0101

• The relative contribution of domestic equities to the

total risk is 0.0101/0.0229=44.11%.



The Principle of Diversification



GMVP

• In risk-management applications, investors typically care only about reducing risk, even at the cost of having low payoffs.

• Such investors seek the minimum-variance combination.

• How can we find the minimum-variance combination when you can invest only in two-risky securities? This is easy!



1 2

σ 22 - Cov(r1r2)

w1 =

+ - 2Cov(r1r2)

w2 = (1 - W1)

σ2

E(r2) = 0.14Sec 2

E(r1) = 0.10Sec 1 = 0.151σσσσ2 = 0.20σσσσ

12 = 0.2ρρρρ

σ 2

Finding Minimum-Variance Combinations



w1 =(0.2)2 - (0.2)(0.15)(0.2)

(0.15)2 + (0.2)2 - 2(0.2)(0.15)(0.2)

w1 = 0.6733

w2 = (1 - 0.6733) = 0.3267

Minimum-Variance Combination: ρρρρ = 0.2



GMVP in more complex cases

• What if you impose portfolio constraints?

• What if you have more than two security classes?

• The first case is not analytically tractable, and the second

case obeys a complex reduced form expression.

• What can you do?

• Use Solver installed in Excel



• The optimal combinations result in the lowest level of risk for a given expected return.

• The optimal trade-off is described as the efficient frontier of risky securities.

• An optimizing mean-variance investor would pick only portfolios that lie on the efficient frontier.

• Exhibit 12 in the HMC case displays 22 different portfolios along the efficient frontier. Every portfolio is a combination of the 12 asset classes.

The Efficient Frontier



Figure 7.10 The Minimum-Variance Frontier of Risky Assets



A New Asset Class

• In January 1997, the US Treasury started auctioning a

new type of bond, TIPS – Treasury Inflation Protected

Securities.

• Should TIPS be considered an additional asset class in

Harvard’s Policy Portfolio?

• The next slide gives some hints.



Criteria for Determining Asset Class Status

• The asset class should be relatively independent of other asset classes already in the portfolio.

• An asset class should be comprised of homogeneous investments.

• An asset class should have the capitalization capacity to absorb a meaningful fraction of the investor’s portfolio.

• An asset class should be expected to raise the utility of the investor’s portfolio without selection skill on the part of the investor.



Derivatives Securities



Overview

• Option terminology.

• Call and put options.

• Forward and futures contracts.

• Speculation with derivatives.

• Hedging with derivatives.

• The Put-Call Parity.

• The Binomial Tree.

• The B&S formula for pricing Call, Put, and Exotic

Options.



What are Derivatives?

Primary assets: Securities sold by firms or

government to raise money (stocks and bonds) as well

as stock indexes (e.g., S&P, Nikkei), exchange rates ($

versus ₤), and commodities (e.g., gold, coffee).

Derivative assets: Options, forward and futures

contracts, as well as swaptions. These financial assets

are derived from existing primary assets.



• Buy = Long = Hold

• Sell = Short = Write

• Call - option to buy underlying asset

• Put - option to sell underlying asset

• So we have: buy call, buy put, sell call, sell put

• Key Elements:

– Exercise or Strike Price

– Maturity or Expiration

– Premium or Price

– Zero Sum Game

Option Terminology



Definition and Terminology

• A call option gives the owner the right but not the obligation to buy the underlying asset at a predetermined price during a predetermined time period. Note the right belongs to the buyer not the seller.

• Strike (or exercise) price: the amount paid by the option buyer for the asset if he/she decides to exercise.

• Exercise: the act of paying the strike price to buy the asset.



Definition and Terminology

• Expiration: the date by which the option must be exercised or become worthless.

• Exercise style: specifies when the option can be exercised.

– European-style: can be exercised only at expiration date.

– American-style: can be exercised at any time before expiration.

– Bermudan-style: Can be exercised during specified periods (e.g., on the first day of each month. Bermuda is located between the US and Europe.



Examples

• Example: S&R (special & rich) index

– Today: call buyer acquires the right to pay $1,020 in six months for the index, but is not obligated to do so

– In six months at contract expiration: if spot price is

• $1,100, call buyer’s payoff = $1,100 – $1,020 = $80

• $900, call buyer walks away, buyer’s payoff = $0



Examples

• Example: S&R index

– Today: call seller is obligated to sell the index for $1,020 in six months, if asked to do so

– In six months at contract expiration: if spot price is

• $1,100, call seller’s payoff = $1,020 – $1,100 = ($80)

• $900, call buyer walks away, seller’s payoff = $0



Payoff/Profit of a Purchased Call

• Payoff = Max [0, spot price at expiration–strike price]

• Profit = Payoff – future value of option premium



Payoff/Profit of a Purchased Call

• Examples:

– S&R Index 6-month Call Option

• Strike price = $1,000, Premium = $93.81, 6-month risk-free rate = 2%

– If index value in six months = $1100

• Payoff = max [0, $1,100 – $1,000] = $100

• Profit = $100 – ($93.81 x 1.02) = $4.32


• Payoff = max [0, $900 – $1,000] = $0

• Profit = $0 – ($93.81 x 1.02) = – $95.68



Diagrams for Purchased Call

• Payoff at expiration • Profit at expiration



Payoff/Profit of a Written Call

• Payoff = – max [0, spot price at expiration–strike price]

• Profit = Payoff + future value of option premium



Payoff/Profit of a Written Call

• Example:

– S&R Index 6-month Call Option



• Payoff = – max [0, $1,100 – $1,000] = – $100

• Profit = – $100 + ($93.81 x 1.02) = – $4.32


• Payoff = – max [0, $900 – $1,000] = $0

• Profit = $0 + ($93.81 x 1.02) = $95.68



Put Options

• A put option gives the owner the right but not the obligation to sell the underlying asset at a predetermined price during a predetermined time period.

• The seller of a put option is obligated to buy if asked.

• Payoff/profit of a purchased (i.e., long) put

– Payoff = max [0, strike price – spot price at expiration]

– Profit = Payoff – future value of option premium



Put Options

• Payoff/profit of a written (i.e., short) put

– Payoff = – max [0, strike price – spot price at expiration]

– Profit = Payoff + future value of option premium



Put Option Examples

• Examples:

– S&R Index 6-month Put Option



• Payoff = max [0, $1,000 – $1,100] = $0

• Profit = $0 – ($74.20 x 1.02) = – $75.68



Put Option Examples

• Examples:


• Payoff = max [0, $1,000 – $900] = $100

• Profit = $100 – ($74.20 x 1.02) = $24.32



A Few Items to Note

• A call option becomes more profitable when the underlying asset appreciates in value.

• A put option becomes more profitable when the underlying asset depreciates in value.

• Moneyness is an important concept in option trading.



In the Money - exercise of the option would be profitable.

Call: market price>exercise price (denoted by K or X).

Put: exercise price>market price.

Out of the Money - exercise of the option would not be

profitable.

Call: market price<exercise price.

Put: exercise price<market price.

At the Money - exercise price and market price are equal.

“Moneyness”



Options on IBM



Homeowner’s Insurance is a Put Option

• You own a house that costs $200,000.

• You buy a $15,000 insurance policy.

• The deductible amount is $25,000.

• Let us graph the profit from this contact.



Options and Insurance

• Homeowner’s insurance as a put option.



Call Options are also Insurance

• Banks and insurance companies offer investment

products that allow investors to benefit from a rise in a

stock index and provide a guaranteed return if the

market falls.

• The equity linked CD provides a zero return if the

index falls (refund of initial investment) and a return

linked to the index if the index rises.



−×+× 17.01000,10$

1300

S 0, max

final

Structure: Equity Linked CDs

• The 5.5-year CD promises to repay initial invested

amount plus 70% of the gain in S&P500 index.

– Assume $10,000 is invested when S&P 500 = 1300

– Final payoff=

– Where S final= the value of the

S&P500 that will be in 5.5 years



The Economic Value of the Equity Linked CD

• We paid $10,000 and we get $10,000 in 5.5 years plus some extra amount if the S&P500 index level exceeds 1300.

• That payoff structure is equivalent to buying a zero coupon bond and x call options.

• Why? Assuming that the annual effective rate is 6%, the present value of $10,000 to be received in 5.5 years is $7,258.



The Economic Value of the Equity Linked CD

• Thus, we practically paid $7,258 for a zero coupon bond and $2,742 for x call options.

• What is x? And what is the implied value of one call option?

• What is the fraction of the gain in the S&P500 index we should get if the current value of one call option is $450?



Much More Challenging

• In the 1980s, Bankers Trust developed index currency option notes (ICONs).

• ICONs are bonds in which the amount received by the holder at maturity varies with a foreign exchange rate. One example was its trade with the Long Term Credit Bank of Japan.

• The ICON specified that:

– If the yen-US dollar exchange rate is greater than 169 yen per dollar at maturity (in 1995), the holder of the bond will receive 1,000$.

– If it is less than 169 yen per dollar, the amount received by the holder of the bond is:



Index Currency Option Notes

o If the exchange rate is below 84.5, nothing is received

by the holder at maturity.

Objective: show that this ICON is a combination of a regular zero coupon bond and two options.

−×− 1

169000,1,0max000,1

TS



payoff

The Payoff Function

ST

84.5 169

1,000



Strategy:

•Buy zero coupon bond which pays 1,000$ at time T.

•Sell 11.83432 put options, maturing at time T, and with

strike price equal to 169 (11.83432=1000/84.5).

•Buy 11.83432 put options, maturing at time T, and with

strike price equal to 84.5.


Resolving the Complex Strategy

Prof. Doron Avramov


Resolving the Complex Strategy

Strategy:

S=84.5 Payoff=0

S=169 Payoff=1,000

ST>16984.5<ST<169ST<84.5

1,0001,0001,000Bond

001,000-11.83432 STPUT (84.5)

011.83432 ST-2,00011.83432 ST-2,000PUT (169)

1,00011.83432 ST-1,0000TOTAL



Forward Contract

• A forward contract is an agreement made todaybetween a buyer and a seller who are obligated to

complete a transaction at a pre-specified date in the

future.

• The buyer and the seller know each other. The

negotiation process leads to customized agreements:

What to trade; Where to trade; When to trade; How

much to trade?



Futures Contract

• A Futures contract is an agreement made todaybetween a buyer and a seller who are obligated to

complete a transaction at a pre-specified date in the

future.

• The buyer and the seller do not know each other. The

"negotiation" occurs in the fast-paced frenzy of a

futures pit.



Futures Contract

• The terms of a futures contract are standardized. The

contract specifies what to trade; where to trade; When

to trade; How much to trade; what quality of good to

trade.



Understanding Price Quotes



Payoff on a Futures Contract

• Payoff for a contract is its value at expiration.

• Payoff for

– Long forward = Spot price at expiration – Forward

price

– Short forward = Forward price – Spot price at

expiration



Payoff on a Futures Contract

• Example: S&R index:

– Today: Spot price = $1,000, 6-month forward

price = $1,020

– In six months at contract expiration: Spot price =

$1,050

• Long position payoff = $1,050 – $1,020 = $30

• Short position payoff = $1,020 – $1,050 = ($30)



Payoff Diagram for Futures

• Long and short forward positions on the S&R 500

index.



Example: Speculating in Gold Futures, Long

• You believe the price of gold will go up. So,

– You go long 100 futures contract that expires in 3 months.

– The futures price today is $400 per ounce.

– Assume interest rate is zero.

– There are 100 ounces of gold in each futures contract.

• Your "position value" is: $400×100×100 = $4,000,000



Example: Speculating in Gold Futures, Long

• Suppose your belief is correct, and the price of gold is $420 when the futures contract expires.

• Your "position value" is now: $420×100×100 = $4,200,000

Your "long" speculation has resulted in a gain of $200,000



Example: Speculating in Gold Futures, Short

• You believe the price of gold will go down. So,

– You go short 100 futures contract that expires in 3 months.

– The futures price today is $400 per ounce.

– Assume interest rate is zero.

– There are 100 ounces of gold in each futures contract.

• Your "position value" is: $400 × 100 × 100 = $4,000,000



Example: Speculating in Gold Futures, Short

• Suppose your belief is correct, and the price of gold is $370 when the futures contract expires.

• Your “position value” is now: $370 × 100 × 100 = $3,700,000

Your "short" speculation has resulted in a gain of $300,000



Risk management: The Producer’s Perspective

• We can also use futures contracts for hedging.

• A producer selling a risky commodity has an inherent long position in this commodity.

• When the price of the commodity increases, the profit typically increases.

• Common strategies to hedge profit:

– Selling forward.

– Buying puts.

– Selling Calls.



Producer: Hedging With a Forward Contract

• A short forward contract allows a producer to lock in a price for his output.

– Example: a gold-mining firm enters into a short forward contract, agreeing to sell gold at a price of $420/oz. in 1 year. The

cost of production is $380



Producer: Hedging With a Put Option

• Buying a put option allows a producer to have higher

profits at high output prices, while providing a floor

on the price.

Example: a gold-mining

firm purchases a 420-strike

put at the premium of $8.77

interest rate is 5%. FV=9.21



Producer: Insuring by Selling a Call

• A written call reduces losses through a premium, but

limits possible profits by providing a cap on the price.

– Example: a gold-

mining firm sells

a 420-strike call and

receives an $8.77

premium (FV=$9.21)



The Buyer’s Perspective

• A buyer that faces price risk on an input has an

inherent short position in this commodity.

• When the price of, say raw material, is up the firm’s

profitability falls.

• Some strategies to hedge profit:

– Buying forward.

– Buying calls.



Buyer: Hedging With a Forward Contract

• A long forward contract allows a buyer to lock in a

price for his input.

– Example: a firm, using gold

as an input, purchases a

forward contract, agreeing

to buy gold at a price of

$420/oz. in 1 year. The

product is selling for $460



Buyer: Hedging With a Call Option

• Buying a call option allows a buyer to have higher

profits at low input prices, while being protected

against high prices.

– Example: a firm,

which uses gold as

an input, purchases

a 420-strike call at

the premium (future value)

of $9.21/oz



Yet Another Example: Short Hedging with Futures Contracts

• Suppose Starbucks has an inventory of about 950,000pounds of coffee, valued at $0.57 per pound.

• Starbucks fears that the price of coffee will fall in the short run, and wants to protect the value of its inventory.



Yet Another Example: Short Hedging with Futures Contracts

• How best to do this? You know the following:

– There is a coffee futures contract at the New York Board of Trade.

– Each contract is for 37,500 pounds of coffee.

– Coffee futures price with three month expiration is $0.58 per pound.

– Selling futures contracts provides current inventory price protection.

• 25 futures contracts covers 937,500 pounds.

• 26 futures contracts covers 975,000 pounds.



Example: Short Hedging with Futures Contracts, Cont.

• Starbucks decides to sell 25 near-term futures contracts.

• Over the next month, the price of coffee falls. Starbucks sells its inventory for $0.51 per pound.

• The futures price also falls, to $0.52. (There are two months left in the futures contract)

• How did this short hedge perform?

• That is, how much protection did selling futures contracts provide to Starbucks?



The Short Hedge Performance

Date

Starbucks

Coffee Inventory

Price

Starbucks Inventory

Value

Near-Term Coffee

Futures Price

Value of 25 Coffee

Futures Contracts

Now $0.57 $541,500 $0.58 $543,750

1-Month

From now

$0.51 $484,500 $0.52 $487,500

Gain (Loss) ($0.06) ($57,000) $0.06 $56,250



The Short Hedge Performance

• The hedge was not perfect.

• But, the short hedge “threw-off” cash ($56,250) when Starbucks needed some cash to offset the decline in the value of their inventory ($57,000).

What would have happened if prices had increased by $0.06 instead?



Empirical Evidence on Hedging

• Half of nonfinancial firms report using derivatives.

• Among firms that do use derivatives, less than 25% of

perceived risk is hedged, with firms likelier to hedge

short-term risk.

• Firms with more investment opportunities are more

likely to hedge.

• Firms that use derivatives have a higher market value

and more leverage.



Option versus Futures Contracts

• Options differ from futures in major ways:

– Holders of call options have no obligation to buy the

underlying asset.

Holders of put options have no obligation to sell the

underlying asset.

– To avoid this obligation, buyers of calls and puts must

pay a price today.

– Holders of futures contracts do not pay for the

contract today, but they are obligated to buy or sell the

underlying asset, depending upon the position.



Option Relations and Option Pricing

• We will show the important put call parity then study

two ways for pricing options:

– The binomial Tree.

– The B&S Formula.



Put-Call Parity

• If the underlying asset is a stock and Div is the

dividend stream, then

C(K ,T ) = P(K ,T ) + [S0

− PV0,T

(Div)] − e−rT

(K )



Stock Price = 110 Call Price = 17Put Price = 5 Effective interest rate 5%.Maturity = .5 yr K = 105

C - P > S0 - K / (1 + rf)T

17- 5 > 110 - (105/1.05)12 > 10

Since the leveraged equity is less expensive, acquire the low cost alternative and sell the high cost alternative.

Put Call Parity: An arbitrage Opportunity



Put-Call Parity Arbitrage(assuming no dividends)

Immediate Cashflow in Six MonthsPosition Cashflow ST<105 ST> 105

Buy Stock -110 ST ST

BorrowK/(1+r)T = 100 +100 -105 -105

Sell Call +17 0 -(ST-105)

Buy Put -5 105-ST 0

Total 2 0 0Prof. Doron Avramovדורון אברמוב ' פרופ


Parity for Options on Stocks (cont’d)

• Examples:

– Price of a non-dividend paying stock: $40, r=8%,

option strike price: $40, time to expiration: 3

months, European call: $2.78, European put: $1.99.

$2.78=$1.99+$40 – $40e -0.08x0.25

– Additionally, if the stock pays $5 just before

expiration, call: $0.74, and put: $4.85.

$0.74=$4.85+($40 – $5e-0.08x0.25) – $40e-0.08x0.25



Parity for Options on Stocks (cont’d)

• Synthetic security creation using parity

– Synthetic stock: buy call, sell put, lend PV of strike and dividends

– Synthetic T-bill: buy stock, sell call, buy put (interesting tax issues)

– Synthetic call: buy stock, buy put, borrow PV of strike and dividends

– Synthetic put: sell stock, buy call, lend PV of strike and dividends



The Relation between Future, Strike, Call, and Put prices

• Suppose we have call and put options on the same underlying asset with the same time expiration and with the same strike price. The put call parity implies that

• Suppose now we have a future contract on the same underlying asset and with the same tome to expiration. The futures price is F. what is the relation between F and K?

)( KPVSPC −=−




• To answer this question let us consider the following strategy. You buy call, sell put, and sell the futures contract.

0Buy Call

0Sell Put

Sell Futures

Total

KS T < KS T ≥KS T −

TSF −

KS T −

TSF −

KF − KF −




• The bottom line is that you pay C-P today and you get F-K in the future. Thus

• This is actually a generalization of the put call parity.

)( KFPVPC −=−



The Generalized Put-Call Parity

• For European options with the same strike price and time to expiration the parity relationship is

Call – Put = PV (forward price – strike price)

or

C(K ,T ) − P(K ,T ) = PV0,T

(F0,T

− K ) = e−rT

(F0,T

− K )



Example

• Consider buying the 6-month 1000 strike call for a premium of $93.808 and selling a similar put for $74.201. What must the future price (set up today) be if the 6-month interest rate is 2%?

93.808-74.201=PV(F-1000). Thus F=1020.

• Are our previous hedging examples consistent with the relation we derived?



Binomial Option Pricing

• So far we have discussed how the price of one option

is related to the price of another, but we have not

addressed the question: how is the price of an option

related to the price of the underlying asset.

• The binomial tree, a very general approach to valuing

options, explicitly addresses this question.



Binomial Option Pricing

•We assume that in a given period of time, the price of the

underling asset (e.g., a stock) will either increase to a

particular value or decrease to another value.

•We will illustrate this approach with an example, similar

yet different from the text example.



Binomial Option Pricing Example

•Suppose that we have a stock with a price of $60 today

and that in one year, the price will be either $90 or $30.

The risk-free rate is 10% (simple annualized rate).

Consider the payoffs to a call option with a strike price of

$60.



60

90

30

Stock Price

C

30

0

Call Option with

K = 60

Binomial Option Pricing Example



Alternative Portfolio

Buy .5 shares of stock for $30

Borrow $13.64 (@10% Rate)

Net outlay $16.36

Payoff to this strategy:

S=30 S=90

Value of .5 shares 15 45

Repay loan -15 -15

Net Payoff 0 30

16.36

30

0Payoffs are exactly the

same as the Call.

Therefore the call

price should be

$16.36. If not, then

arbitrage is possible!

A Replicating Portfolio



Another View of Replication of Payoffs and Option Values

•This example also shows us how options can be used to

hedge the underlying asset. What if we held one share

of stock and wrote 2 calls (K = 60)?

•Portfolio is perfectly hedged

Stock Value 30 90

Two written Calls 0 -60

Net payoff 30 30



Another View of Replication of Payoffs and Option Values

•The combined portfolio has a riskless return (note that

this portfolio would cost (60-2×16.36=27.28), so the

return is 10%, i.e. the riskless rate of return.



S

uS

dS

Stock Price

C

Cu

Cd

Call Option with

strike price = K

Binomial Option Pricing:General One-period Case



Alternative Portfolio

Buy ∆ shares of stock for $ ∆S Put $B

into bonds (@ r% Rate.

(if B is negative, then borrow)

Net outlay $ ∆S + B

Payoff to this strategy:

S=uS S=dS

Value of ∆ shares ∆uS ∆dS

Bond (R=1+r) RB RB__

Net Payoff ∆uS+RB ∆dS+RB

∆S+B

∆uS+RB

∆dS+RB

A General Replicating Portfolio



Solving for ∆ and B

• We want the replicating portfolio to give us the same

payoff as the option in both states.

C

C

d

u

RBdS

RBuS

=+∆

=+∆

Rdu

duB

dSuS

CC

CC

ud

du

)( −

−=

−

−=∆



The One-Period Binomial Option Value Formula

• The value of the call option today must be equivalent

to the value of the replicating portfolio to prevent

arbitrage.

du

dRpwhere

R

pp

Rdu

du

du

BSC

CC

CCCC

du

uddu

−−

=−+

=

−

−+

−

−=

+∆=

,)1(

)(



Risk-adjusted probabilities

• Note that the valuation formula for the call is just a

probability weighted average of the payoffs

discounted by the risk-free rate. The adjustment for

risk is taken into account in the probabilities that are

used: p and (1-p).

• These probabilities can be backed out by considering

the underlying asset.

du

dRp

R

dSppuSS

−−

=⇒−+

=)1(



Generalizing the Binomial Approach

• While it may seem unrealistic that the price of the

stock can only go up or down in a given period, we

can break the life of the option into as many (very

short) periods as we like!

• This point can be illustrated by using a two-period

example (but can be generalized to as many periods as

deemed necessary).



Generalizing the Binomial Approach

• Consider a stock that is selling today for 80, and

which will go up or down by 50% in each of the next

two periods (each of which is one-year long). The

risk-free rate is 10% p.a. (i.e. R=1.10).



A Two-Period Example

80

120

180

40

20

60



The binomial tree for the call option

• Consider a call option with a strike price of $80.

Cud = 0C

Cu

Cd

Cuu= 180-80 = 100

Cdd = 0



Valuing Cu and Cd

• If we are in the up-state at the end of the first period, we are looking at a one-period tree.

• First, we need to calculate p = (R-d)/(u-d) = (1.10-.5)/(1.5-.5) = 0.6

• Using our one-period formula:

– Cu = [(.6) 100 + (.4) 0] / 1.10 = 54.54

– Cd = [(.6) 0 + (.4) 0] / 1.10 = 0



Valuing C

• Now, consider the first period on its own. We now

know the values of the options at the end of this first

period (Cu and Cd).

• Using the same one-period valuation technique:

C = [(.6)(54.54) + (.4)(0)] / 1.10 = 29.75



The replicating portfolios

• Recall that we have formulas for ∆ and B. We can use

these at each point in the tree, based on the call and

stock values in the up and down state at the end of

each period.

• For example, at the beginning of the first period, we

can calculate:

793.241.1)5.5.1(

)54.54)(5(.0

)(

682.40120

054.54

−=−

−=

−

−=

=−

−=

−

−=∆

Rdu

duB

dSuS

CC

CC

ud

du



Option Prices and Deltas

• The tree below shows the prices and deltas (in

parentheses) at each node.

29.75

(.682)

54.54

(.833)

0

(0)

100

0

0



Self Financing Strategy

• If the stock price is up you buy more stocks.

• You pay (0.833-0.682)*120=18.2

• You also pay interest on your loan 0.1×24.793=2.5

• Who brings the money?

• Note that the new B=-45.45

• So the amount of debt increases by 20.7



Self Financing Strategy

• What if the stock price drops to 40.

• Then you sell the stock for 0.682×40=27.3

• You pay interest 2.5

• You net inflow is 24.8

• You also repay the loan 24.8

• Thus your net inflow is zero.



Put Options

• The analysis we have just gone through works exactly the same way for put options, or for any derivative security that depends on the stock price.

• Interpret Cu and Cd as the payoffs to a derivative security such as a call or put.

• In replicating a put, delta will be negative and B will be positive - i.e. we would short stock and lend out $B.

• Note that P= ∆S+B, or B= P- ∆S, so the amount lent out is equal to the money from the short position plus the cost of the put option.



American Options• The binomial valuation approach can also be used for

valuing American Options (in fact, it must be used since

formulas such as the Black-Scholes don’t allow for early

exercise).

• The only difference in the procedure is that at every point

in the tree we need to check to see whether it is

preferable to exercise the option early.

• In the following example, we value both a European and

an American put option on the stock we just looked at,

where the strike price equals $80.



European Put Option

• Find the payoffs at maturity, and work backwards to

find option and delta values.

0

20

60

7.27

(-.167)

32.73

(-1.0)

15.87

(-.318)



American Put Option

• In this case, check at each node to see whether it would

be preferable to exercise (it is when the stock price goes

down to $40 in the first year). Note that the American

option is worth 2.64 more than the European option.

0

20

60

7.27

(-.167)

40

(-1.0)

18.51

(-.409)



Exercising Options “Early”

• Why may options be exercised early?

• As we just saw, puts may be exercised early if they are deep in-the-money. Since it is likely that the option will remain in-the-money, the holder would prefer to get the fixed strike price today rather than later in time.

• For call options, since the exercise price is paid out, there is an advantage, rather than disadvantage, to waiting in this regard (one would prefer to delay payment of a fixed amount).



Exercising Options “Early”

• However, if (and only if) the stock pays out a

dividend, the option holder may want to exercise early

in anticipation of the stock price dropping at the ex-

dividend date. However, the holder will only do so if

the stock is deep enough in-the-money (or the option

is near maturity) since he is giving up the time value

of the option.



S

S +

S + +

S -S - -

S + -

S + + +

S + + -

S + - -

S - - -

Expanding the number of periods



The Binomial Model - summary

• The binomial model can break down the time to

expiration into potentially a very large number of time

intervals, or steps.

• A tree of stock prices is initially produced working

forward from the present to expiration.

• At each step it is assumed that the stock price will

move up or down by an amount calculated using

volatility and time to expiration. This produces a

binomial distribution of underlying stock prices.




• The tree represents all the possible paths that the stock

price could take during the life of the option.

• At the end of the tree , at expiration of the option, all

the terminal option prices for each of the final possible

stock prices are known as they simply equal their

intrinsic values.




• Next the option prices at each step of the tree are

calculated working back from expiration to the

present.

• The option prices at each step are used to derive the

option prices at the next step of the tree using risk

neutral valuation based on the probabilities of the

stock prices moving up or down, the risk free rate and

the time interval of each step.

• At the top of the tree you are left with one option

price.



To get a feel …

• To get a feel of how the binomial tree work you can use the

on-line binomial tree calculators:

• http://www.hoadley.net/options/binomialtree.aspx?tree=B



Nice features

• The big advantage the binomial model is that it can be

used to accurately price American options.

• This is because it's possible to check at every point in

an option's life (at every step of the binomial tree) for

the possibility of early exercise

• Where an early exercise point is found it is assumed

that the option holder would elect to exercise, and the

option price can be adjusted to equal the intrinsic

value at that point. This then flows into the

calculations higher up the tree and so on.



Nice features

•The on-line binomial tree graphical option calculator

highlights those points in the tree structure where early

exercise would have have caused an American price to

differ from a European price.

•The binomial model basically solves the same equation,

using a computational procedure that the Black-Scholes

model solves using an analytic approach and in doing so

provides opportunities along the way to check for early

exercise for American options.



The Black-Scholes Model

• What happens if the number of periods in the binomial

tree are greatly increased, and the periods become

extremely short?

• If the volatility and interest rate in each period is the

same, then we arrive at the model developed by

Fischer Black and Myron Scholes.



dln S / K r T

T1

21

2=+ − +( ) ( )δ σ

σd d T2 1= − σ

C S,K, ,r,T, Se N d Ke N d- T -rT( ) = ( ) ( )σ δ δ1 2−

P S,K, ,r,T, Ke N d Se N d-rT - T ( ) = ( ) ( )σ δ δ− − −2 1

Black-Scholes Formula (cont’d)

• Call Option price

• Put Option price

Where and



The Put Call Parity Revisits

• Note that the put price easily follows by the

implementing the put call parity and using the

relations

N(-d1)=1-N(d1)

N(-d2)=1-N(d2)



Option Pricing Parameters

• The formula requires six parameters:

– Stock price, dividend, and volatility (stock level).

– Strike price and time to expiration (option level).

– Risky free rate (economy level).



Black-Scholes (BS) Assumptions

• Assumptions about stock return distribution:

– Continuously compounded returns on the stock are normally distributed and independent over time (no “jumps”).

– The volatility of continuously compounded returns is known and constant.

– Future dividends are known, either as dollar amount or as a fixed dividend yield.



Black-Scholes (BS) Assumptions (cont’d)

• Assumptions about the economic environment:

– The risk-free rate is known and constant.

– There are no transaction costs or taxes.

– It is possible to short-sell costlessly and to borrow

at the risk-free rate .



Implied Volatility

• Volatility is unobservable.

• Choosing a volatility to use in pricing an option is difficult but important.

• One approach to obtaining a volatility is to use history of returns.

• However history is not a reliable guide to the future.

• Alternatively, we can invert the Black-Scholes formula to obtain option implied volatility.



Implied Volatility

• IV is the volatility implied by the option price

observed in the market.

• Implied volatilities are not constant across strike

prices and over time, in contrast to the B&S

assumptions.



Volatility Index –VIX

• It provides investors with market estimates of

expected volatility.

• It is computed by using near-term S&P 100 index

options.



VIX Options

• A type of non-equity option that uses the VIX as the underlying asset.

• This is the first exchange-traded option that gives individual investors the ability to trade market volatility.

• A trader who believes that market volatility will increase can purchase VIX call options.

• Sharp increases in volatility generally coincide with a falling market, so this type of option can be used as a natural hedge.



Implied Volatility (cont’d)

• In practice implied volatilities of in, at, and out-of-the

money options are generally different resulting in the

volatility skew.

• Implied volatilities of puts and calls with same strike

and time to expiration must be the same if options are

European because of put-call parity.



Implied Volatility (cont’d)



Pricing Exotic Options

• An exotic option is a derivate which has features making

it more complex than the commonly traded call and put

options (vanilla options); These products are usually

traded over-the-counter (OTC), or are embedded in

structured notes.

• One can use the B&S structure to price a large set of such

options.



Pricing Exotic Options

• Below we display the price of four exotic options while

the starting point is the B&S formula C=BS(S,K,σ,r,T,δ).

• That is, we will use the BS formula but will change one

or more of the six underlying parameters.. More details

are coming up!



Option Payoff:

• That is, at the time of expiration it is not the price of the

underlying stock that matters – rather it is the squared

stock price.

• The option price is then given by

( ))(,,,2,,Pr 22 σσ +−= rTrKSBSice

[ ]0,max 2 KS −



Option Payoff:


underlying stock that matters – rather it is the product of

two stock prices.

• The option price is then given by:

( ))(,,,,,Pr 21

*

21 σρσσ +−= rTrKSSBSice

[ ]0,max 21 KSS −

21

2

2

2

1

* 2 σρσσσσ ++=



Option Payoff:


underlying stock that matters – rather it is the ratio of two

stock prices.


( ))(,,,,,/Pr 122

*

21 ρσσσσ −−= rTrKSSBSice

[ ]0,/max 21 KSS −

21

2

2

2

1

* 2 σρσσσσ −+=



Payoff:


underlying stock that matters – rather it is the geometric

average of two stock prices.


−+−= )2(8

1,,,,,Pr 21

2

2

2

1

*

21 σρσσσσ TrKSSBSice

[ ]0,)(max 5.0

21 KSS −

21

2

2

2

1

* 22/1 σρσσσσ ++=



Shortfall Probability and Option Pricing

• We now use concepts coming from option pricing to

understand the risk of return of long horizon

investment decisions.

• Previously, we were dealing with the mean variance

setup to make invest decisions.

• Here, we introduce a new concept: investing based on

shortfall probability.

• Let us denote by R the return on the investment over

several years (say T years).



Shortfall Probability in Long Horizon Asset Management

• Let us denote by R the cumulative return on the

investment over several years (say T years).

• Rather than finding the distribution of R we analyze

the distribution of

which is the continuously compounded return over the

investment horizon.

• The investment value after T years is

( )Rr += 1ln

( )( ) ( )TT RRRVV +++= 111 210 K



• Dividing both sides of the equation by we

get

• Thus

0V

( )( ) ( )TT RRR

V

V+++= 111 21

0

K

( )( ) ( )TRRRR +++=+ 1111 21 K




• Taking natural log from both sides we get

• Using properties of the normal distribution, we

get

Trrrr +++= K21

( )2,~ σµ TTNr




Shortfall Probability and Long Horizon

• Let us now understand the concept of shortfall

probability.

• We ask: what is the probability that the investment

yields a return smaller than the riskfree rate, or any

other threshold level?

• To answer this question we need to compute the value

of a riskfree investment over the T year period.




• The value of such a riskfree investment is

where is the continuously compounded risk free rate.

( )T

frf RVV += 10

( )fTrV exp0=

fr




• Essentially we ask: what is the probability that

• This is equivalent to asking what is the probability

that

• This, in turn, is equivalent to asking what is the

probability that

rfT VV <

00 V

V

V

V rfT <

<

00

lnlnV

V

V

V rfT




• So we need to work out

• Subtracting and dividing by both sides of

the inequality we get

• We can denote this probability by

( )fTrrp <

µT σT

−<

σ

µfrTzP

−=

σ

µfrTNShortfall probabilit y



Shortfall Probability and long horizon

• Typically which means the probability

diminishes with increasing T.

µ<fr



Example

• Take r=0.04, µ=0.08, and σ=0.2 per year. What is the

Shortfall Probability for investment horizons of 1, 2, 5, 10,

and 20 years?

• Use the excel normdist function.

• If T=1 SP=0.42

• If T=2 SP=0.39

• If T=5 SP=0.33

• If T=10 SP=0.26

• If T=20 SP=0.19



One more Example

• Example: A stock portfolio’s monthly continuously

compounded return has a mean of 0.01 and a volatility

of 0.05. if the current portfolio value is $50 million,

what is the probability that the portfolio’s value will

be less than $65 millions in five years?



One more Example

3.150

65<=

< tt VprobVprob

( )( )

192.0

872.0

05.060

01.060262.0

262.0

262.0ln

3.1

2

=

−<=

−

<=

−

<=

<=

<=

zprob

zprob

T

Tzprob

Vprob

Vprob

t

t

σ

µ



Cost of Insuring against Shortfall

• Let us now understand the mathematics of insuring

against shortfall.

• Without loss of generality let us assume that

• The investment value at time T is a given by the

random variable

10 =V

TV

Prof. Doron Avramov

Investment Management269דורון אברמוב ' פרופ


• Once we insure against shortfall the investment value

after T years becomes

– If you get

– If you get

TV( )fT TRV exp>

( )fT TRV exp< ( )fTrexp

Prof. Doron Avramov



• So you essentially buy an insurance policy that pays 0

if

Pays if

• You ultimately need to price a contract with terminal

payoff given by

( )fT TrV exp>

( )fT TrV exp<( ) Tf VTr −exp

( ) Tf VTr −exp,0max

Prof. Doron Avramov


Cost of insuring against shortfall

• This is a European put option expiring in T years with

a. S=1

b. .

c. Riskfree rate given by .

d. Volatility given by

e. Dividend yield given by

( )frTK ⋅= exp

fr

σ

0=δ




• From the B&S formula we know that

• Which becomes

( ) ( ) ( ) ( )12 expexp dNTSdNTrKPut f −−−−−= δ

−−

= TNTNPut σσ2

1

2

1

Prof. Doron Avramov



• The B&S option-pricing model gives the current put

price P as

where

and is

( ) ( )21 dNdNPut −=

12

12

dd

Td

−=

=σ

( )dN dzprob <

Prof. Doron Avramov



• For (per year)

• The cost of the insurance increases in T, even

though the probability of needing it decreases in

T (if ).

2.0=σ

r>µ

PT (years)

0.081

0.185

0.2510

0.3520

0.4230

0.5250

Prof. Doron Avramov


Open questions

• Do you believe in time diversification?

• What are the difficulties with using shortfall

probability as a risk measure and tool for asset

allocation?

• We finish up this option section with the so-called

option Greeks..



Option Greeks

• What happens to option price when one input changes?

– Delta (∆): change in option price when stock price increases by $1.

– Gamma (Γ): change in delta when the stock price increases by $1.

– Vega: change in option price when volatility increases by 1%.

– Theta (θ): change in option price when time to maturity decreases by 1 day.



∆ ∆portfolio i ii

n

==∑ω

1

Option Greeks

– Rho (ρ): change in option price when interest rate

increases by 1%

• Greek measures for portfolios

– The Greek measure of a portfolio is weighted

average of Greeks of individual portfolio

components. For example,



Option Greeks

• The following illustrations are based on strike

price=$40, sigma=30%, riskfree rate=8%, and no

dividend payments.



Delta - Call



Gamma - call



Option Greeks (cont’d)




Stock

price=$40




CombinedOption 2Option 1

--11Wi

1.80940.97102.7804Price

0.30090.28150.5824Delta

0.00880.05630.0652Gamma

0.01060.06740.0780Vega

-0.0040-0.0134-0.0173Theta

0.02550.02570.0511Rho

Greeks for the bull spread examined in chapter 3 where

with a purchased 40-strike call and a written 45-strike call. The

column titled “combined” is a difference between column 1 and

column 2.

daysTRS 91,08.0,3.0$,40 ==== σ




• Option elasticity (Ω):

– Ω describes the risk of the option relative to the

risk of the stock in percentage terms: If stock price

(S) changes by 1%, what is the percent change in

the value of the option (C)?

Ω

∆∆

≡ = =

×

=%

%

C

S

C

C S

S

S

C S

S

S

C

change in

change in

change in

change in

change in

change in




– Example 12.8: S = $41, K = $40, σ = 0.30, r = 0.08, T = 1,

δ = 0

• Elasticity for call: Ω=S∆/C = $41 x 0.6911 / $6.961 =

4.071

• Elasticity for put: Ω=S∆/C = $41 x (– 0.3089 / $2.886) =

– 4.389




• Option elasticity (Ω) (cont’d)

– The volatility of an option

– The risk premium of an option

– The Sharpe ratio of an option.

σ σoption stock= × Ω

γ α− = − ×r r( ) Ω




• Where | . | is the absolute value, g: required return on

option, a: expected return on stock, and r: risk-free

rate.

.r

ckio for stoSharpe ratlio for calSharpe rat =−

=σ

α



Appendix: Value-At-Risk

• The shortfall probability is related to a widely applied

topic in risk management called value at risk.

• Value-At-Risk (VaR) answers the question, “How

much can the value of a portfolio decline with a given

probability in a given time period?”.




• The most common assumption is that returns follow a

normal distribution. One of the properties of the

normal distribution is that 95 percent of all

observations occur within 1.96 standard deviations

from the mean. This means that the probability that an

observation will fall 1.96 standard deviations below

the mean is only 2.5 percent. For the purposes of

calculating VAR we are interested only in losses, not

gains, so this is the relevant probability.




• Example: XYZ Fund has an (arithmetic) average

monthly return of 2.03 percent and a standard

deviation of 3.27 percent. Thus, its monthly VAR at

the 2.5 percent probability level is 2.03%-1.96*3.27=-

4.38%, or $43.80 for a $1,000 investment, meaning

that the probability of losing more than this is 2.5

percent.



Equity Valuation



Common Stock Valuation

• Our goal in this chapter is to examine the methods

commonly used by financial analysts to assess the

economic value of common stocks.

• These methods are grouped into three categories:

– Dividend discount models.

– Price ratio models.



Security Analysis: Be Careful Out There

• Fundamental analysis is a term for studying a

company’s accounting statements and other financial

and economic information to estimate the economic

value of a company’s stock.

• The basic idea is to identify “undervalued” stocks to buy and “overvalued” stocks to sell.

• In practice however, such stocks may in fact be correctly priced for reasons not immediately apparent to the analyst.



The Dividend Discount Model

• The Dividend Discount Model (DDM) is a method to

estimate the value of a share of stock by discounting

all expected future dividend payments. The basic

DDM equation is:

• In the DDM equation:

– V(0) = the present value of all future dividends.

– D(t) = the dividend to be paid t years from now.

– k = the appropriate risk-adjusted discount rate.

( ) ( ) ( ) ( )T32k1

D(T)

k1

D(3)

k1

D(2)

k1

D(1)V(0)

++

++

++

+= L



Example: The Dividend Discount Model

• Suppose that a stock will pay three annual dividends

of $200 per year, and the appropriate risk-adjusted

discount rate, k, is 8%.

• In this case, what is the value of the stock today?

( ) ( ) ( )32k1

D(3)

k1

D(2)

k1

D(1)V(0)

++

++

+=

( ) ( ) ( )$515.42

0.081

$200

0.081

$200

0.081

$200V(0)

32=

++

++

+=



The Dividend Discount Model:the Constant Growth Rate Model

• Assume that the dividends will grow at a constant growth rate g. The dividend next period (t + 1) is:

• For constant dividend growth, the DDM formula becomes:

g k if D(0) T V(0)

g k if k1

g11

gk

g)D(0)(1V(0)

T

=×=

≠

++

−−

+=

( ) ( ) ( )

g)(1g)(1D(0) g)(1 D(1) D(2) So,

g1tD1tD

+×+×=+×=

+×=+



Example: The Constant Growth Rate Model

• Suppose the current dividend is $10, the dividend

growth rate is 10%, there will be 20 yearly dividends,

and the appropriate discount rate is 8%.

• What is the value of the stock, based on the constant

growth rate model?

g k if k1

g11

gk

g)D(0)(1V(0)

T

≠

++

−−

+=

( ) ( )$243.86

1.08

1.101

.10.08

1.10$100V

20

=

−−

×=



The Dividend Discount Model:the Constant Perpetual Growth

Model

• Assuming that the dividends will grow forever at a constant growth rate g.

• For constant perpetual dividend growth, the DDM formula becomes:

( ) ( ) ( ) ( )k)g :(Important

gk

1D

gk

g10D0V <

−=

−+×

=



Example: Constant Perpetual Growth Model

• Think about the electric utility industry.

• In mid-2005, the dividend paid by the utility company,

American Electric Power (AEP), was $1.40.

• Using D(0)=$1.40, k = 7.3%, and g = 1.5%, calculate

an estimated value for DTE.

Note: the actual mid-2005 stock price of AEP was

$38.80.

What are the possible explanations for the difference?

( ) ( )$24.50

.015.073

1.015$1.400V =

−×

=



Example: Constant Perpetual Growth Model

• Think about the electric utility industry.

• In 2007, the dividend paid by the utility company,

DTE Energy Co. (DTE), was $2.12.

• Using D0 =$2.12, k = 6.7%, and g = 2%, calculate an

estimated value for DTE.

Note: the actual mid-2007 stock price of DTE was

$47.81. Quite close!

( )$46.01

.02.067

1.02$2.12P0 =

−×

=



The Dividend Discount Model:Estimating the Growth Rate

• The growth rate in dividends (g) can be estimated in a

number of ways:

– Using the company’s historical average growth

rate.

– Using an industry median or average growth rate.

– Using the sustainable growth rate.



The Historical Average Growth Rate

• Suppose the Kiwi Company paid the following

dividends:

– 2000: $1.50 2003: $1.80

– 2001: $1.70 2004: $2.00

– 2002: $1.75 2005: $2.20



The Historical Average Growth Rate

• The spreadsheet below shows how to estimate historical

average growth rates, using arithmetic and geometric

averages.

2005 $2.20 10.00%

2004 $2.00 11.11%

2003 $1.80 2.86%

2002 $1.75 2.94% Year: 7.96%:

2001 $1.70 13.33% 2000 $1.50

2000 $1.50 2001 $1.62

2002 $1.75

8.05% 2003 $1.89

2004 $2.04

7.95% 2005 $2.20

Arithmetic Average:

Geometric Average:

Grown at



The Sustainable Growth Rate

• Return on Equity (ROE) = Net Income / Equity

• Payout Ratio = Proportion of earnings paid out as

dividends.

• Retention Ratio = Proportion of earnings retained for

investment.

Ratio) Payout - (1 ROE

Ratio Retention ROE Rate Growth eSustainabl

×=

×=



Example: Calculating and Using the Sustainable Growth Rate

• In 2005, American Electric Power (AEP) had an ROE of 14.59%, projected earnings per share of $2.94, and a per-share dividend of $1.40. What was AEP’s:

– Retention rate?

– Sustainable growth rate?

• Payout ratio = $1.40 / $2.94 = .476• So, retention ratio = 1 – .476 = .524 or 52.4%

• Therefore, AEP’s sustainable growth rate = .1459 × 52.4% = 7.645%



Example: Calculating and Using the Sustainable Growth Rate, Cont.

• What is the value of AEP stock, using the perpetual

growth model, and a discount rate of 7.3%?

• Recall the actual mid-2005 stock price of AEP was

$38.80.

( ) ( )$38.80 $436.82

.07645.073

1.07645$1.400V <<−=

−×

=




• Clearly, there is something wrong because we have a

negative price.

• What causes this negative price?

• Suppose the discount rate is appropriate. What can we

say about g?



Example: Calculating and Using the Sustainable Growth Rate

• In 2007, AEP had an ROE of 10.17%, projected earnings per share of $2.25, and a per-share dividend of $1.56. What was AEP’s:

– Retention rate?

– Sustainable growth rate?

• Payout ratio = $1.56 / $2.25 = .693• So, retention ratio = 1 – .693 = .307 or 30.7%

• Therefore, AEP’s sustainable growth rate = .1017 × .307 = .03122, or 3.122%




• What is the value of AEP stock, using the perpetual

growth model, and a discount rate of 6.7%?

• The actual mid-2007 stock price of AEP was $45.41.

• In this case, using the sustainable growth rate to value

the stock gives a reasonably accurate estimate.

• What can we say about g and k in this example?

( )$44.96

.03122.067

1.03122$1.56P =

−×

=0



The Two-Stage Dividend Growth Model

• The two-stage dividend growth model assumes that a

firm will initially grow at a rate g1 for T years, and

thereafter grow at a rate g2 < k during a perpetual

second stage of growth.

• The Two-Stage Dividend Growth Model formula is:

2

2

T

1

T

1

1

1

gk

)g1)(0(D

k1

g1

k1

g11

gk

)g1)(0(D)0(V

−−−−++++

++++++++

++++

++++++++

−−−−−−−−

++++====



Using the Two-StageDividend Growth Model, I.

• Although the formula looks complicated, think of it as

two parts:

– Part 1 is the present value of the first T dividends

(it is the same formula we used for the constant

growth model).

– Part 2 is the present value of all subsequent

dividends.



Using the Two-StageDividend Growth Model, I.

• So, suppose MissMolly.com has a current dividend of D(0)

= $5, which is expected to “shrink” at the rate g1 - -10% for

5 years, but grow at the rate g2 = 4% forever.

• With a discount rate of k = 10%, what is the present value

of the stock?



Using the Two-StageDividend Growth Model, II.

• The total value of $46.03 is the sum of a $14.25 present value of the first five dividends, plus a $31.78 present value of all subsequent dividends.

$46.03.

$31.78 $14.25

0.040.10

0.04)$5.00(1

0.101

0.90

0.101

0.901

0.10)(0.10

)$5.00(0.90V(0)

gk

)gD(0)(1

k1

g1

k1

g11

gk

)gD(0)(1V(0)

55

2

2

T

1

T

1

1

1

=

+=

−+

+

+

+

−−−

=

−+

++

+

++

−−

+=



Example: Using the DDM to Value a Firm Experiencing

“Supernormal” Growth, I.

• Chain Reaction, Inc., has been growing at a phenomenal rate of 30% per year.

• You believe that this rate will last for only three more years.

• Then, you think the rate will drop to 10% per year.

• Total dividends just paid were $5 million.

• The required rate of return is 20%.

• What is the total value of Chain Reaction, Inc.?



Example: Using the DDM to Value a Firm Experiencing “Supernormal” Growth, II.

• First, calculate the total dividends over the “supernormal”

growth period:

Year Total Dividend: (in $millions)

1 $5.00 x 1.30 = $6.50

2 $6.50 x 1.30 = $8.45

3 $8.45 x 1.30 = $10.985




“Supernormal” Growth, II.

• Using the long run growth rate, g, the value of all the

shares at Time 3 can be calculated as:

V(3) = [D(3) x (1 + g)] / (k – g)

V(3) = [$10.985 x 1.10] / (0.20 – 0.10) = $120.835




“Supernormal” Growth, III.

• Therefore, to determine the present value of the firm

today, we need the present value of $120.835 and the

present value of the dividends paid in the first 3 years:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

million. $87.58

$69.93$6.36$5.87$5.42

0.201

$120.835

0.201

$10.985

0.201

$8.45

0.201

$6.50V(0)

k1

V(3)

k1

D(3)

k1

D(2)

k1

D(1)V(0)

332

332

=

+++=

++

++

++

+=

++

++

++

+=



Discount Rates for Dividend Discount Models

• The discount rate for a stock can be estimated using

the capital asset pricing model (CAPM ).

• We will discuss the CAPM in a later chapter.

• However, we can estimate the discount rate for a stock

using this formula:

Discount rate = time value of money + risk premium

= U.S. T-bill rate + (stock beta x stock market risk premium)



Discount Rates for Dividend Discount Models

T-bill rate return on 90-day U.S. T-

bills

Stock Beta risk relative to an

average stock

Stock Market Risk Premium:

risk premium for an

average stock



Observations on Dividend Discount Models, I.

Constant Perpetual Growth Model:

• Simple to compute

• Not usable for firms that do not pay dividends

• Not usable when g > k

• Is sensitive to the choice of g and k

• k and g may be difficult to estimate accurately.

• Constant perpetual growth is often an unrealistic

assumption.



Observations on Dividend Discount Models, II.

Two-Stage Dividend Growth Model:

• More realistic in that it accounts for two stages of growth

• Usable when g > k in the first stage

• Not usable for firms that do not pay dividends

• Is sensitive to the choice of g and k

• k and g may be difficult to estimate accurately.



Price Ratio Analysis, I.

• Price-earnings ratio (P/E ratio)

– Current stock price divided by annual earnings per

share (EPS).

• Earnings yield

– Inverse of the P/E ratio: earnings divided by price

(E/P).

• High-P/E stocks are often referred to as growth stocks,

while low-P/E stocks are often referred to as value

stocks.



Price Ratio Analysis, II.• Price-cash flow ratio (P/CF ratio)

– Current stock price divided by current cash flow per share.

– In this context, cash flow is usually taken to be net income plus depreciation.

• Most analysts agree that in examining a company’s financial performance, cash flow can be more informative than net income..

• Earnings and cash flows that are far from each other may be a signal of poor quality earnings.



Price Ratio Analysis, III.

• Price-sales ratio (P/S ratio)

– Current stock price divided by annual sales per share.

– A high P/S ratio suggests high sales growth, while a

low P/S ratio suggests sluggish sales growth.

• Price-book ratio (P/B ratio)

– Market value of a company’s common stock divided

by its book (accounting) value of equity.

– A ratio bigger than 1.0 indicates that the firm is

creating value for its stockholders.



Price/Earnings Analysis, Intel Corp.

• Intel Corp (INTC) - Earnings (P/E) Analysis.

– 5 year average P/E ratio 37.30.

– Current EPS $1.16.

– EPS growth rate 17.5%.

• Expected stock price = historical P/E ratio × projected EPS.

– $50.84 = 37.30 × ($1.16 × 1.175)

– Mid-2005 stock price = $26.50



Price/Cash Flow Analysis, Intel Corp.

• Intel Corp (INTC) - Cash Flow (P/CF) Analysis.

– 5 year average P/CF ratio 19.75.

– Current CFPS $1.94.

– CFPS growth rate 13.50%.

• Expected stock price = historical P/CF ratio × projected CFPS.

– $43.49 = 19.75 × ($1.94 × 1.135)




Price/Sales Analysis, Intel Corp.

• Intel Corp (INTC) - Sales (P/S) Analysis.

– 5-year average P/S ratio 6.77.

– Current SPS $5.47.

– SPS growth rate 10.50%.

• Expected stock price = historical P/S ratio×projected SPS

– $40.92 = 6.77 × ($5.47 × 1.105)




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