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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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management2
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?
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management3
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?
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management4
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management5
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management6
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%?
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management7
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)
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management8
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?
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management9
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
=
+=
+=+
===
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management10
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
∑==
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management11
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management12
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:
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management13
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management14
A $1 Investment in Different Types
of Portfolios, 1926—2005.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management15
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management16
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management17
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management18
An International Perspective: 1990-2009 vs. longer Periods
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management19
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%
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management20
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management21
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management22
• Track average returns.
• Compare performance of managed portfolios
(benchmarks).
• Base of derivatives.
• Base of ETF-s and other passive funds.
Prof. Doron Avramovדורון אברמוב ' פרופ
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)
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management24
Construction of Indexes
• How are stocks weighted in an index?
– Price weighted (DJIA).
– Market-value weighted (S&P500, NASDAQ).
– Equally weighted (Value Line Index).
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management25
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)
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management26
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%
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management27
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management28
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management29
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management30
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management31
Changing the Divisor: An Example
What would have happened to the divisor had Home
Depot shares been sold at $65.72 per share?
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management32
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%
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management33
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?
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management34
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management35
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management36
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management37
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management38
Financial Market Anomalies
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management39
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management40
• 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
Case Studies: The DFA and the AQR’s Momentum Funds
Quantitative asset management
Prof. Doron Avramov
Investment Management41
• 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.
דורון אברמוב ' פרופ
Case Studies: The DFA and the AQR’s Momentum Funds
Quantitative asset management
Prof. Doron Avramov
Investment Management42
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management43
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…).
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management44
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…
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management45
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management46
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management47
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management48
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management49
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?
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management50
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management51
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management52
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management53
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management54
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management55
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management56
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management57
:Legg Mason Value ביל מילר ב"קרן הנאמנות הטובה בארה
2006 - 1991
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management58
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!
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management59
:Legg Mason Value ביל מילר ב "קרן הנאמנות הטובה בארה
2006-2011
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management60
:Legg Mason Value ביל מילר ב "קרן הנאמנות הטובה בארה
1991-2011
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management61
And what about performance in the overall universe of U.S. equity
mutual funds? A One-Year Perspective.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management62
Here is a Longer Horizon (10 years) Perspective
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management63
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management64
What about Israel?
Quite similar… just take a look into the
figures
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management65
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 (
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management66
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
(%)
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management67
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management68
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management69
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management70
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management71
Anomaly I: The size effect
• Size effect: higher average returns on small stocks than
large stocks. Beta cannot explain the difference.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management72
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management73
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management74
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management75
The Value Effect: The International Evidence
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management76
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management77
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%.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management78
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management79
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)
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management80
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management81
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management82
Cumulative Return of 1 USD Invested 1927 - 2012
Mkt-RF SMB HML WML
$ 129.83 $ 6.97 $ 27.06 $ 322.97
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management83
Cumulative Return of 1 USD Invested Last 2 Decades (1992 – 2012)
Mkt-RF SMB HML WML
$ 2.98 $ 1.54 $ 1.69 $ 2.73
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management84
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management85
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management86
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management87
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management88
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)
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%
3 years 5 years 10 years 15 years
Trading on Down Side Risk Measures: Evidence in the US
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management89
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)
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%
3 years 5 years 10 years 15 years
Using the F-Score: Evidence in the US
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management90
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)
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%
3 years 5 years 10 years 15 years
Combining F-Score, Momentum, and Value Effects
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management91
One more Interesting Anomaly is Earnings Momentum: Cumulative Returns in Response to
Earnings Announcements
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management92
As of 31/01/2012 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)
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%
3 years 5 years 10 years 15 years
What about Israel? Strategies Involving Stocks
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management93
As of 31/01/2012 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)
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%
3 years 5 years 10 years 15 years
What about Israel? Strategies Involving Stocks
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management94
As of 31/01/2012 1 year 2 years
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management95
Portfolio Optimization and Asset Classes
96 Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management96
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?
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management97
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%.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management98
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management99
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management100
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management101
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?
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management102
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%)
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management103
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management104
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management105
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management106
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management107
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%.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management108
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management109
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
דורון אברמוב ' פרופ
( ) ( ) ABBAAABAAAp wwww ρσσσσσ −+−+= 121 22222
Prof. Doron Avramov
Investment Management110
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).
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management111
Example
•This is a diversification benefit.
•Interestingly, I did not even come up with the lowest volatility portfolio (GMVP).
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management112
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management113
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management114
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management115
Why Diversification Works?
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management116
More about Correlation
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management117
Combining Stock and Bond
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management118
Efficient Frontier for Stock and Bond
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management119
Extending to a Three-Security Portfolio
דורון אברמוב ' פרופ
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
Investment Management120
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management121
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management122
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management123
Computing Volatility – Array # 2
• 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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management124
Computing Volatility – Array # 3
• 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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management125
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!
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management126
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%.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management127
The Principle of Diversification
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management128
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!
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management129
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management130
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management131
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management132
• 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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management133
Figure 7.10 The Minimum-Variance Frontier of Risky Assets
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management134
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management135
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management136
Derivatives Securities
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management137
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management138
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management139
• 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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management140
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management141
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management142
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management143
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management144
Payoff/Profit of a Purchased Call
• Payoff = Max [0, spot price at expiration–strike price]
• Profit = Payoff – future value of option premium
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management145
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
– If index value in six months = $900
• Payoff = max [0, $900 – $1,000] = $0
• Profit = $0 – ($93.81 x 1.02) = – $95.68
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management146
Diagrams for Purchased Call
• Payoff at expiration • Profit at expiration
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management147
Payoff/Profit of a Written Call
• Payoff = – max [0, spot price at expiration–strike price]
• Profit = Payoff + future value of option premium
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management148
Payoff/Profit of a Written Call
• Example:
– 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
– If index value in six months = $900
• Payoff = – max [0, $900 – $1,000] = $0
• Profit = $0 + ($93.81 x 1.02) = $95.68
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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
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Investment Management150
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
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Put Option Examples
• Examples:
– S&R Index 6-month Put Option
• Strike price = $1,000, Premium = $74.20, 6-month risk-free rate = 2%
– If index value in six months = $1100
• Payoff = max [0, $1,000 – $1,100] = $0
• Profit = $0 – ($74.20 x 1.02) = – $75.68
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Investment Management152
Put Option Examples
• Examples:
– If index value in six months = $900
• Payoff = max [0, $1,000 – $900] = $100
• Profit = $100 – ($74.20 x 1.02) = $24.32
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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.
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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”
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Investment Management155
Options on IBM
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Investment Management156
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 $25,000.
• Let us graph the profit from this contact.
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Investment Management157
Options and Insurance
• Homeowner’s insurance as a put option.
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Investment Management158
Call Options are also Insurance
• Banks and insurance companies offer investment
products that allow investors to benefit from a rise in a
stock index and provide a guaranteed return if the
market falls.
• The equity linked CD provides a zero return if the
index falls (refund of initial investment) and a return
linked to the index if the index rises.
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Investment Management159
−×+× 17.01000,10$
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
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Investment Management160
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.
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Investment Management161
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 $450?
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Investment Management162
Much More Challenging
• In the 1980s, Bankers Trust developed index currency option notes (ICONs).
• ICONs are bonds in which the amount received by the holder at maturity varies with a foreign exchange rate. One example was its trade with the Long Term Credit Bank of Japan.
• The ICON specified that:
– If the yen-US dollar exchange rate is greater than 169 yen per dollar at maturity (in 1995), the holder of the bond will receive 1,000$.
– If it is less than 169 yen per dollar, the amount received by the holder of the bond is:
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Investment Management163
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
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Investment Management164
payoff
The Payoff Function
ST
84.5 169
1,000
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Investment Management165
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.
דורון אברמוב ' פרופ
Resolving the Complex Strategy
Prof. Doron Avramov
Investment Management166
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
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Investment Management167
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?
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Investment Management168
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.
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Investment Management169
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.
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Investment Management170
Understanding Price Quotes
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Investment Management171
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
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Investment Management172
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)
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Investment Management173
Payoff Diagram for Futures
• Long and short forward positions on the S&R 500
index.
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Investment Management174
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
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Investment Management175
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
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Investment Management176
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
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Investment Management177
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
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Investment Management178
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.
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Investment Management179
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
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Investment Management180
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
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Investment Management181
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)
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Investment Management182
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.
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Investment Management183
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
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Investment Management184
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
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Investment Management185
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.
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Investment Management186
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.
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Investment Management187
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?
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Investment Management188
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
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Investment Management189
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?
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Investment Management190
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.
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Investment Management191
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.
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Investment Management192
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.
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Investment Management193
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 )
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Investment Management194
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
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Investment Management195
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
Investment Management196
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
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Investment Management197
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
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Investment Management198
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 −=−
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Investment Management199
The Relation between Future, Strike, Call, and Put prices
• 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 −
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management200
The Relation between Future, Strike, Call, and Put prices
• 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 −=−
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Investment Management201
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 )
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Investment Management202
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?
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Investment Management203
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.
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Investment Management204
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management205
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.
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Investment Management206
60
90
30
Stock Price
C
30
0
Call Option with
K = 60
Binomial Option Pricing Example
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Investment Management207
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
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Investment Management208
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management209
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.
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Investment Management210
S
uS
dS
Stock Price
C
Cu
Cd
Call Option with
strike price = K
Binomial Option Pricing:General One-period Case
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Investment Management211
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
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Investment Management212
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
)( −
−=
−
−=∆
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management213
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(
)(
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management214
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(
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management215
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).
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management216
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).
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management217
A Two-Period Example
80
120
180
40
20
60
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Investment Management218
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
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Investment Management219
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
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Investment Management220
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
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Investment Management221
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
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Investment Management222
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
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Investment Management223
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management224
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management225
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.
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Investment Management226
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.
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Investment Management227
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)
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management228
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)
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management229
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).
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Investment Management230
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.
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Investment Management231
S
S +
S + +
S -S - -
S + -
S + + +
S + + -
S + - -
S - - -
Expanding the number of periods
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Investment Management232
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management233
The Binomial Model - summary
• 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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management234
The Binomial Model - summary
• 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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management235
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management236
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management237
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management238
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management239
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management240
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)
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management241
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).
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management242
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management243
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 .
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management244
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management245
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management246
Volatility Index –VIX
• It provides investors with market estimates of
expected volatility.
• It is computed by using near-term S&P 100 index
options.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management247
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.
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Investment Management248
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management249
Implied Volatility (cont’d)
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Investment Management250
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management251
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!
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Investment Management252
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 −
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Investment Management253
Option Payoff:
• That is, at the time of expiration it is not the price of the
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 ++=
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management254
Option Payoff:
• That is, at the time of expiration it is not the price of the
underlying stock that matters – rather it is the ratio of two
stock prices.
• The option price is then given by:
( ))(,,,,,/Pr 122
*
21 ρσσσσ −−= rTrKSSBSice
[ ]0,/max 21 KSS −
21
2
2
2
1
* 2 σρσσσσ −+=
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management255
Payoff:
• That is, at the time of expiration it is not the price of the
underlying stock that matters – rather it is the geometric
average of two stock prices.
• The option price is then given by:
−+−= )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 ++=
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management256
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).
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management257
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management258
• Dividing both sides of the equation by we
get
• Thus
0V
( )( ) ( )TT RRR
V
V+++= 111 21
0
K
( )( ) ( )TRRRR +++=+ 1111 21 K
Shortfall Probability in Long Horizon Asset Management
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management259
• Taking natural log from both sides we get
• Using properties of the normal distribution, we
get
Trrrr +++= K21
( )2,~ σµ TTNr
Shortfall Probability in Long Horizon Asset Management
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management260
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management261
Shortfall Probability and Long Horizon
• The value of such a riskfree investment is
where is the continuously compounded risk free rate.
( )T
frf RVV += 10
( )fTrV exp0=
fr
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management262
Shortfall Probability and Long Horizon
• 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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management263
Shortfall Probability and Long Horizon
• 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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management264
Shortfall Probability and long horizon
• Typically which means the probability
diminishes with increasing T.
µ<fr
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management265
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management266
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?
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management267
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management268
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
Cost of Insuring against Shortfall
• 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
Investment Management270דורון אברמוב ' פרופ
Cost of Insuring against Shortfall
• 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
Investment Management271דורון אברמוב ' פרופ
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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management272
Cost of Insuring against Shortfall
• From the B&S formula we know that
• Which becomes
( ) ( ) ( ) ( )12 expexp dNTSdNTrKPut f −−−−−= δ
−−
= TNTNPut σσ2
1
2
1
Prof. Doron Avramov
Investment Management273דורון אברמוב ' פרופ
Cost of Insuring against Shortfall
• 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
Investment Management274דורון אברמוב ' פרופ
Cost of Insuring against Shortfall
• 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
Investment Management275דורון אברמוב ' פרופ
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..
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Investment Management276
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.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management277
∆ ∆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,
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management278
Option Greeks
• The following illustrations are based on strike
price=$40, sigma=30%, riskfree rate=8%, and no
dividend payments.
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management279
Delta - Call
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management280
Gamma - call
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management281
Option Greeks (cont’d)
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management282
Option Greeks (cont’d)
Stock
price=$40
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management283
Option Greeks (cont’d)
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management284
Option Greeks (cont’d)
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management285
Option Greeks (cont’d)
CombinedOption 2Option 1
--11Wi
1.80940.97102.7804Price
0.30090.28150.5824Delta
0.00880.05630.0652Gamma
0.01060.06740.0780Vega
-0.0040-0.0134-0.0173Theta
0.02550.02570.0511Rho
Greeks for the bull spread examined in chapter 3 where
with a purchased 40-strike call and a written 45-strike call. The
column titled “combined” is a difference between column 1 and
column 2.
daysTRS 91,08.0,3.0$,40 ==== σ
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management286
Option Greeks (cont’d)
• 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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management287
Option Greeks (cont’d)
– 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
Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management288
Option Greeks (cont’d)
• 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
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Option Greeks (cont’d)
• 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
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Prof. Doron Avramovדורון אברמוב ' פרופ
Investment Management291
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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?”.
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Appendix: Value-At-Risk
• 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.
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Appendix: Value-At-Risk
• 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.
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Equity Valuation
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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.
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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.
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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
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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=
++
++
+=
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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
+×+×=+×=
+×=+
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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
=
−−
×=
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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 <
−=
−+×
=
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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 =
−×
=
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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 =
−×
=
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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.
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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
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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
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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
×=
×=
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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%
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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 <<−=
−×
=
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Example: Calculating and Using the Sustainable Growth Rate, Cont.
• 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?
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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%
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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 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
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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
−−−−++++
++++++++
++++
++++++++
−−−−−−−−
++++====
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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.
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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?
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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
=
+=
−+
+
+
+
−−−
=
−+
++
+
++
−−
+=
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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.?
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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
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Example: Using the DDM to Value a Firm Experiencing
“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
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Example: Using the DDM to Value a Firm Experiencing
“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
=
+++=
++
++
++
+=
++
++
++
+=
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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)
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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
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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.
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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.
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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.
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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.
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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.
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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
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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)
– Mid-2005 stock price = $26.50
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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)
– Mid-2005 stock price = $26.50
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