Investment Management, Fall 2012
School of Business Administration
Hebrew University of Jerusalem
Name: Doron E. Avramov, Ph.D, Professor of Finance
Office: 5132 Mount Scopus, Jerusalem
Web: http://pluto.huji.ac.il/~davramov/
Class Hours: Sunday 15:30-18:15
Office Hour: Sunday 14:45-15:30
Room: 2113
‫דורון אברמוב; פרופסור למימון‬
Course Description:
In this course you will study the theory and practice of investment management in domestic and
global financial markets. I will cover the following topical issues:






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
Understanding the risk-return tradeoff: Essential concepts and the empirical evidence;
Constructing price weighted equity indexes (such as the Dow Jones 30 and the Nikkei 225)
as well as value weighted equity indexes (such as TA100 and the S&P500);
Understanding the concept of market efficiency and its three forms;
Analyzing performance of active investment vehicles such as mutual funds;
Examining evidence on profitable trading strategies in global and domestic markets,
especially, the size, value, and momentum effects in stock prices;
Forming optimal portfolios with and without portfolio constraints;
Investing in public and private equities, commodities, as well as nominal and indexed bonds;
Understanding derivatives securities (options and futures contracts);
Familiarity with statistics should extend through concepts of mean, standard deviation, covariance,
correlation, and regression analysis. A good grounding in Excel is essential.
Resources:

Textbook: In general, the class notes adequately cover the material to be taught. If you wish to
enhance your knowledge, however, I would recommend the following two textbooks:
o Fundamentals of Investments Valuation and Management (Fifth Edition) by Jordan &
Miller.
o Investments, Sixth Edition by Bodie, Kane, and Marcus

Instructor: I welcome students to see me during the office hour and class breaks to discuss any
aspect of the course. I welcome your feedbacks regarding any aspect of the teaching process.
Assignments (40%): There are four case studies – questions for all cases are provided below. You will
submit only one of the first two case studies (dealing with market anomalies) and one of the last two case
studies (dealing with portfolio optimization) – so overall you are responsible to submit two of the four case
studies. Each case accounts for 20% of the final grade. However, for the final exam you are responsible for
1
the content and analysis of all the four cases. I would suggest forming groups of up to four students to
submit the two cases.
Final Exam (60%): The final exam will be based on the material covered in class (in letter and spirit),
class handouts, class discussions, case studies, and assigned readings. The exam is closed books and closed
notes. However, you will be allowed to bring in one piece of paper with handwritten notes (double-sided,
A4 size). You are not allowed to use any other notes. I will allow the use of non-programmable calculators
during the exam. You will find in the end of this syllabus questions taken from past year exams. It would
be beneficial to solve those questions as a means of preparation for the final exam.
Class Participation: It is mandatory to attend all sessions. If you miss a session for good reason, make
sure you catch up on all missed material. In general, you are responsible for class lectures, case studies, as
well as any announcements, discussions, or remarks.
2
Case 1: The dimensional fund advisors (DFA)
http://hbr.org/product/dimensional-fund-advisers-2002/an/203026-PDF-ENG
In June 2002, the time when the case study was written, DFA was a $30 billion investment fund. Check out
http://www.dfaus.com/dimensional/about/ -- I believe you will be surprised to notice the phenomenal
growth in assets managed by DFA. Indeed, a number of features make DFA an unusual fund:
1.
2.
3.
4.
DFA apparently believes that the capital markets are efficient.
DFA relies on passive strategies undertaking a buy-and-hold investment approach.
DFA uses academic research to form investments and assess their performance.
DFA has specialized in the purchase of large blocks of small stocks at discount prices.
The case covers several topical subjects including efficient markets, models of capital market equilibrium
(e.g., CAPM and the Fama-French three-factor model), financial instruments, investment management, tax
management, liquidity, and stock trading.
Questions:
1. What is DFA’s business strategy? Are the DFA managers really believers in efficient markets?
2. What are the Fama-French findings? Are they theoretically based or empirically based? Should we
expect small stocks to outperform large stocks in the future? Value stocks to outperform growth
stocks? Does the Fama-French model have some value from either academic or practitioner
perspective?
3. Visit the web page http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html
Download returns on 25 size and book to market portfolios and then replicate the table in the class
notes (the one in the value anomaly section) based on:
a. the entire sample starting from 1/70
b. the 1/1/90-31/12/00 period
c. the period starting from 1/1/2001.
What do you learn?
4. Discuss DFA’s liquidity providing strategy. How does it work, and what are the costs and benefits?
5. What should be the firm strategy going forward? Does the company need to modify its basic
strategy if it wants to grow assets and/or profits?
6. Look at DFA web site – how would you explain the impressive progress DFA has made over time
in attracting new money relative to its competitors.
Guidelines
Question 1: The business strategy section is pretty much open ended. In addition, DFA has been closely
affiliated with a prominent academic from the University of Chicago (the advisor of my own advisor)
called Eugene Fama. Fama has been a strong believer that markets are efficient starting from a paper he had
written in the 60s. He has not changed his views all along. In his view, if high book-to-market small-cap
stocks earn more – these stocks are essentially more risky and moreover the famous CAPM is unable to
capture these additional risk sources. DFA claims to have adopted these beliefs as well. Do you agree that
the DFA folks are believes in market efficiency?
3
Question 2 is about describing the Fama-French findings based on past evidence and future prospects. Here
is some background. The Fama-French three-factor model (which extends the one-factor CAPM) has
become a standard in the industry and academia. Its prominence has to do with the fact that the FamaFrench three factors are available to the public at the web site of Ken French, which you are going to visit
in question 3. Of course it does not hurt that Fama and French are both prominent scholars.
Some folks dislike the Fama-French model, to say the least. What could be the objections here? First, as
noted in class, the value and size premiums appeared, disappeared, and then reappeared (you will also
examine such time variation in question 3). Second, even when the value and size premiums were there
during the sample period selected by the Fama and French study, some disagree with the notion that high
book to market small stocks are more risky.
For example, there are behavioral (non risk) explanations for the value premium. Here is one. Investors
classify stocks to growth or value categories based on past performance. These investors are willing to pay
high prices for growth stocks believing that the previously observed phenomenal growth rates will be there
also in the future. But in reality, high growth rates don’t last for long – past high growth rate firms become
low growth rate firms in the future. Therefore, growth stocks earn less in the future once investors realize
that growth firms perform poorly relative to prior expectations. Thus, the value premium reflects investors’
incorrect extrapolation of past growth rates. There are other behavioral explanations.
Question 3 is self-explanatory. Simply download the data and compute statistics similar to those displayed
in the class notes.
Question 4: DFA has established an excellent reputation in the niche in liquidity providing. They do very
nice stuff as the case describes. What do they do? Describe your own thoughts about pros and cons. If you
have anything else to say about the important concept of liquidity please feel free to detail it in your papers.
Question 5 – DFA has objectives and strategy to accomplish those objectives. Would you change their
strategy? Based on what we know today have they been able to accomplish their objectives?
Question 6 requires some thinking regarding the impressive ability of DFA to attract new investors.
4
Case 2: AQR’s Momentum Funds (2012)
Please note that here there are 2 links.
http://hbr.org/product/aqr-s-momentum-funds-a/an/211025-PDF-ENG
http://hbr.org/product/aqr-s-momentum-funds-b/an/211075-PDFENG?Ntt=AQR%27s%2520Momentum%2520Funds
This recent case study discusses the launch of several new retail mutual funds that offer investors the
exposure to ‘Momentum,’ a new investment style in the mutual funds world. While momentum strategies
has been commonplace among hedge funds, the new AQR funds would become the first retail funds to
focus on this strategy.
1. Describe in general the momentum trading strategy. What are the common explanations for the
prevalence of momentum profitability?
2. Look at one of the AQR momentum fund (at google or yahoo finance) performance since inception
(Ticker symbol: AQMOM) and at data found in the annexes - what can you say about AQR lastyears performance compared to the market? In particular,
a. Describe momentum performance over the most recent years.
b. Has the fund achieved its targets?
c. What are the pros and cons of investing in a mutual fund that chases momentum?
d. What would you advise a retail investor considering investing in one of AQR’s funds?
3. Can a retail investor easily imitate the fund? Explain in detail if and how you can build a similar
strategy at home.
4. Suppose you compare two momentum funds – one for small cap stocks (USD 100M and less) and
the other for large cap stocks (USD 1B and above). Which one you would favor and why?
5. What can we learn on market efficiency from the concept of momentum and AQR’s performance?
6. It was claimed in the case that momentum strategy performed better compared to the market in
emerging markets. Do you think that in the long run this will still be the case?
5
Case 3: The Harvard Management Company (2010)
http://hbr.org/product/harvard-management-company-2010/an/211004-PDF-ENG?Ntt=211004
The central issue in this case is to propose an asset allocation policy for Harvard Management
Company using concepts of mean variance optimization and portfolio constraints studied in class.
Let us start with the policy portfolio displayed in Exhibit 4. The policy portfolio is the long run asset
mix of Harvard. It specifies the “neutral weighting” for each asset class. HMC was given a minimum
and maximum range for each asset class within which they could trade. HMC made tactical asset
allocation bets from time to time attempting to beat the policy portfolio in anticipation for short-term
market moves. Some of the questions below involve using the Excel Solver for forming optimal
portfolios under constraints for each of the asset classes under consideration.
1. Given figures in Exhibits 4 and 17 what is the expected return and volatility of the policy
portfolio?
2. Find an efficient portfolio having the same expected return as the policy portfolio but lower
volatility based on portfolio constraints displayed in Exhibit 18.
3. Find an efficient portfolio having the same volatility as the policy portfolio but higher expected
return based on the same portfolio constraints.
4. Repeat questions 2 and 3 using the new set of constraints in Exhibit 19.
5. Plot on one space two efficient frontiers – the first one pertains to the constraints in Exhibit 18
along with the inputs in Exhibit 4; the second one pertains to the constraints displayed in
Exhibit 19 along with the inputs in Exhibit 4. Start the frontier from the Global Minimum
Variance Portfolio (GMVP); end the frontier with the maximum expected return portfolio. Pick
three other intermediate portfolios.
6. Look at the Historical Asset Mix in Exhibit 3 – how would you explain the vast change in the
asset mix for the periods 1992 through 2010 (e.g., equities versus bonds versus commodities)?
In your opinion, does the asset mix simply follow past changing market conditions or does it
predict future market conditions?
7. Look at the percentage of assets invested with internal and external managers in Exhibit 5 –
how would you explain the change in that percentage over time?
8. Explain the pros and cons of the mean variance paradigm and describe whether you would use
it in practice.
9. Would you invest in HMC?
Hints: Let me propose some guidelines for solving some of the questions. There is no need to
attach excel files. The first question is merely a computational one. The objective here is to examine
your ability to compute expected return and volatility of a portfolio accommodating multiple asset
classes. Moreover, inputs from the first question will be used to solve the three questions that
follow. In particular, the second (third) question requires displaying the weights of an efficient
portfolio dominating the policy portfolio along the volatility (expected return) dimension. Of
course, you are also required to exhibit the mean and volatility of these two dominating portfolios.
So far everything is just computations. But questions (2) and (3) suggest that the policy portfolio is
inefficient. Hence, why does HMC spend tremendous efforts on establishing the policy portfolio?
Do your best efforts to answer this important question. Question4 (4) and (5) call for some new
computations. You will see that with a tighter set of portfolio constraints the investment
opportunities appear less attractive. Then explain: why imposing constraints to begin with? The
fifth question is an open-ended question.
6
Case 4: The Harvard Management Company and Inflation Protected Bonds (2001)
http://hbr.org/product/harvard-management-co-and-inflation-protected-bond/an/201053-PDFENG?Ntt=201053-PDF-ENG
This case examines the decision of the Harvard Management Company to recommend Harvard University
to invest part of its endowment in a new asset class - Treasury Inflation Protected Bonds (TIPS). The TIPS
principal and coupons grow with the general level of prices as measured by the Consumer Price Index
(CPI).
1. When would TIPS outperform/underperform regular Treasuries (on a real basis)?
2. From a conceptual perspective, should TIPS be considered an additional asset class in Harvard’s
policy portfolio?
3. Re-form the optimal portfolios in Exhibit 5 assuming that TIPS are unavailable for investment.
4. Why did HMC ultimately recommend only 7% investment in TIPS?
5. Do TIPS have advantages or disadvantages beyond their mean-variance properties that make them
an attractive asset class for investors with long horizons such as Harvard?
6. Should the Harvard Board accept the HMC proposal? (Did the Board accept the proposal?)
Hints: TIPS have only recently been introduced in the US but such bonds have long been traded in UK
as well as other high inflation economies. Indeed TIPS are intended to protect investors from Inflation.
However, we have to be careful and distinguish between expected and unexpected inflation. In
particular, when regular bonds and TIPS are issued, inflation expectations are already accounted for in
determining the coupon paid on these bonds as well as setting the market price of the bonds. To gain
some more essential knowledge on the important differences between regular bonds and TIPS please
read the case study carefully and feel free to Google search the properties of inflation-indexed bonds.
The information in the web is invaluable. I use the web on multiple occasions. Your answer to question
1 should contain terminologies like expected and unexpected inflation as well as inflation risk premium.
Question 2 is straightforward, as we did discuss in class three qualitative indicators of an additional
asset class (the fourth one is quantitative). Do TIPS satisfy those requirements? Please be explicit.
Question 3 is self-explanatory! It would be useful, as I noted in class, to contain some quantitative
analysis here. For instance, for a given volatility what is the maximal expected return with versus
without TIPS? Then report the expected return differential. You can consider several volatility levels.
You can also compute Sharpe ratios and certainty equivalent rate of returns with versus without TIPS
(for several volatility targets). In Question 4 you will find out that TIPS account for a large fraction of
the optimal portfolio. Nevertheless the recommendation is to invest in TIPS only 7%. Why? Question 5:
There are several other aspects of investment management beyond mean variance (e.g., hedging,
skewness, liquidity, horizon effect, to name a few) that could be considered here. Question 6: Apart
from making your recommendation I would like you to tell the fraction of investment in TIPS based on
the most updated release (information is on the web).
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Past Exam Questions - Class Notes Number 1
?Dow Jones-‫ איזה מהמשפטים הבאים הינו נכון ביחס למדד ה‬.1
.‫ מניות אמריקאיות גדולות‬30 ‫ המדד הוא ממוצע משוכלל ערך של‬.‫א‬
.‫ מניות אמריקאיות גדולות‬30 ‫ המדד הוא ממוצע משוכלל מחיר של‬.‫ב‬
.‫) צריך להיות מתואם לפיצול מניות וחלוקת דיבידנד במזומן‬the divisor( ‫ מחלק המדד‬.‫ג‬
.‫ מניות אמריקאיות גדולות‬30 ‫ זהו מדד ממוצע שווה ערך של‬.‫ד‬
‫ מהו‬.‫ בהתאמה‬-01% -‫ ו‬111% ‫ הם‬8282 ‫ לאוקטובר‬82-‫ ו‬8282 ‫ לאוקטובר‬82 ‫ בתאריכים‬S&P -‫ התשואות על מדד ה‬.2
‫) של שני תשואות אלו? איזו שיטת‬geometric average( ‫) והגיאומטרי‬arithmetic average ( ‫הממוצע האריתמטי‬
?‫ממוצעים נראית יותר רלוונטית למדידת ביצועי השקעה‬
02-‫ ל‬AAA ‫ מהי תשואת אג"ח‬.2.52% ‫ הוא‬yield spread -‫ ה‬,5% ‫ שנים הינה‬02-‫ הנח כי תשואת אג"ח ממשלתי ל‬.3
?‫שנים‬
4. Which of the following is true of the S&P500 index?
A)
B)
C)
D)
E)
It is a value-weighted average of 500 stocks.
It is a price-weighted average of 500 stocks.
The divisor must be adjusted for stock splits and cash dividends.
It is an equal-weighted average of 500 stocks.
B and C.
5. What is the price-weighted index constructed with the three stocks below
Stock
Stock A
Stock B
Stock C
Price
$40
$50
$60
Number of shares outstanding
200
500
600
6. The Value Line Index is a geometric average of the return of about 1,700 firms. Following that
formulation what is the return on the index based on three stocks only with rates of return 10%, 7%, and 6%?
7. You have been given this probability distribution for the holding period return for XYZ stock:
State of the Economy
Boom
Normal growth
Recession
Probability
.30
.50
.20
HPR
8%
4%
- 5%
What is the expected holding period return for XYZ stock? What is the standard deviation for XYZ
stock?
8. You purchase a share of Clal-Israel stock for $100. One year later, after receiving a dividend of $4,
8
you sell the stock for $98. What was your holding period return?
9. The geometric average for the five year period 2006-2011 is 10% per year. The geometric average
for the three year period 2006-2007 is 11%. The geometric average for the period 2008-2009 is 8%.
What is the geometric average for the two year period 2010-2011?
10. You purchase a share of Google stock for $100. One year later, after receiving a dividend of $4,
you sell the stock for $92. What was your holding period return?
Past Exam Questions -- Class Notes Number 2
,‫ אם בכלל‬,‫ איזו צורת יעילות שוק‬.)(price momentum trading strategy ‫ הסבר מהי אסטרטגית מסחר מומנטום מחיר‬0
?‫מפרה האסטרטגיה הנ"ל‬
‫ מניבה באופן עקבי‬,‫ הנח כי אסטרטגית מסחר שבה נרכשות מניות בעלות מחיר גבוה ונמכרות מניות בעלות מחיר נמוך‬8
.‫תשואה עודפת על השוק‬
?‫ האם השווקים מקיימים את ההנחה החלשה של יעילות השוק‬.‫א‬
?‫ חזקה של יעילות השוק‬- ‫האם השווקים מקיימים את ההנחה החצי‬
.‫ב‬
?‫ האם השווקים מקיימים את ההנחה החזקה של יעילות השוק‬.‫ג‬
3. The Sharpe ratios and Jensen’s alphas of five actively managed mutual funds and the S&P 500
index are given by
4. The accruals based trading strategy means that you can outperform the market by
A. Taking long (short) position in firms with high (low) working capital
B. Taking long (short) position in firms with high (low) profitability.
C. Taking long (short) position in firms with high (low) non-cash based profits
D. Taking long (short) position in firms with high (low) cash based profits
E. None of the above
5 . Assume that a strategy that buys low volatility stocks and sells high volatility stocks consistently
outperforms the market. Does this evidence establish violation of market efficiency? If yes, what sorts of
market efficiency are being violated?
‫" מהווה הפרה של השוק היעיל? אם כן איזה צורה של שוק‬short-term reversal" ‫"? האם‬short-term reversal " ‫ מהו‬.6
?‫יעיל מופרת‬
7. Explain the relationship between the book-to-market ratio, value stocks, and growth stocks
9
Past Exam Questions - Class Notes Number 3
‫ והנכס‬01% ‫ וסטיית תקן של‬08% ‫) הינו בעל תשואה צפויה של‬A( ‫ הנכס הראשון‬.B-‫ ו‬A ‫ קיימים שני נכסים מסוכנים‬.0
A ‫ מהו משקל נכס‬.2.1 ‫ מקדם המתאם בין הנכסים הוא‬.01% ‫ וסטיית תקן של‬2% ‫) הינו בעל תשואה צפויה של‬B( ‫השני‬
)global minimum variance portfolio( ?‫בתיק בעל השונות המינימאלית‬
‫ איזה ערך של‬.2.25 ‫ ריבית חסרת הסיכון היא‬.)s=0.14( 2.01 ‫ וסטיית תקן של‬2.05 ‫ תשואה צפויה של‬,‫ לתיק השקעות‬.8
E(r) - (A/2)s2 ‫ יגרום למשקיע בעל פונקצית התועלת‬A
.‫ חישבת את תיק ההשקעות האופטימאלי כפוף לאילוצים‬,Harvard Management Company ‫ בניתוח אירוע של‬.3
.‫ הסבר מדוע‬.)commodities( ‫בפתרון מצאת כי מומלץ להשקיע שיעור גבוה מהכסף בסחורות‬
4. A portfolio has an expected rate of return of 0.12 and a standard deviation of s=0.13. The risk-free
rate is 6.085%. An investor has the following utility function: U = E(r) - (A/2)s2. Which value of
A makes this investor indifferent between the risky portfolio and the risk-free asset?
5. The Sharpe ratios and Jensen’s alphas of five actively managed mutual funds and the S&P 500
index are given by
Sharpe ratio
Jensen’s alpha
A
0.40
0.01
B
0.50
0.02
C
0.20
0.01
D
0.10
-0.02
E
0.25
-0.01
S&P 500
0.30
0.00
Fund
You apply the following decision rule: an outperforming fund is one that dominates along both the
Sharpe ratio and Jensen’s alpha criteria. Based upon that decision rule, which of the funds outperform
the S&P index
6. Your utility function is U =µ - (A/2) σ2, where A is unknown. You can invest in ONLY ONE of the
following five assets:
Asset
A
B
C
D
E
Volatility (σ)
0.15
0.17
0.20
0.25
0.32
Expected Return (µ)
0.10
0.11
0.13
0.17
0.20
Which value of A makes investment A the best investment? (There are many such values – pick only one).
10
7. You have four portfolios W (15, 36), X (12, 15), Z (5, 7), and Y (9, 21) – which one cannot lie on the
efficient frontier? Are the remaining three portfolios efficient?
8.
A portfolio consists of the following two securities
Expected return
Standard deviation
Portfolio market value
Correlation (RA, RB)
Risk-free rate
A
16%
25%
$25,000
B
10%
15%
$15,000
0.30
4%
What is the Sharpe ratio of the portfolio?
9.
Stocks offer Ex=18% and Vol=22%. Gold offers Ex=10% and Vol=30%. Would you invest in gold?
Use plot to explain!
10. Your utility function is U =µ - (A/2) σ2, where A =2. You can invest in ONLY ONE of
the
following five assets:
Asset
A
B
C
D
E
Volatility (σ)
0.15
0.17
0.20
0.25
0.32
Expected Return (µ)
0.10
0.11
0.13
0.17
0.20
Which asset is your best choice? Explain!
Questions for Class Notes Number 4
‫) לפי מודלים מקובלים לתמחור אופציות ומצאנו שמחיר האופציה בשוק גבוה‬CALL( ‫ אם חישבנו את אופציית הרכש‬.0
:‫ הסיבה לכך יכולה להיות‬,‫יותר‬
.‫ הערכת יתר של התנודתיות‬.‫א‬
.‫ אי התחשבות בדיבידנד במודל‬.‫ב‬
.‫ אי התחשבות במודל בקשר האפשרי בין שער הריבית למחיר המניה‬.‫ג‬
.‫ כל הסיבות לעיל‬.‫ד‬
.‫ אף אחת מהסיבות לעיל‬.‫ה‬
11
‫‪ .8‬להערכתך‪ ,‬שוק המניות צפוי לעלות יותר מהר משיעור הריבית חסרת הסיכון‪ .‬מנגד הינך מוכן להסתכן בהפסד של ‪32%‬‬
‫לכל היותר מערך השקעתך‪ .‬אזי‪:‬‬
‫א‪ .‬תקנה ב‪ 32%-‬מכספך מק"מ ובשאר הכסף אופציות ‪ CALL‬על מדד המעו"ף‪.‬‬
‫ב‪ .‬תקנה ב‪ 02%-‬מכספך אופציות ‪ CALL‬ובשאר הכסף אופציות ‪.PUT‬‬
‫ג‪ .‬תקנה ב‪ 32%-‬מכספך אופציות ‪ CALL‬ובשאר הכסף אופציות ‪.PUT‬‬
‫ד‪ .‬תקנה ב‪ 32%-‬מכספך אופציות ‪ CALL‬ובשאר הכסף מק"מ‪.‬‬
‫ה‪ .‬תקנה ב‪ 022%-‬מכספך מק"מ שכן זו הדרך היחידה להבטיח לא להפסיד ‪ 32%‬מהשקעתך‪.‬‬
‫‪ .3‬הנח כי רכשת דירה בבניה בניו יורק ואתה אמור לשלם בגינה בתוך ‪ 3‬חודשים‪ .‬אתה חושש מתנודתיות של שער הדולר‪.‬‬
‫כיצד תוכל להשתמש בשוק ההון כדי לגדר את הפוזיציה שלך?‬
‫‪ .4‬חוזה עתידי בשם ”‪ .“Treasury Bonds (CBT)-$100,000; pts 32nds of 100%‬המחיר העתידי (‪ (future price‬של‬
‫החוזה ביום ‪ 3020828222‬הינו ‪ .22-86‬שנה לאחר מכן ביום ‪ ,3020828222‬המחיר העתידי (‪ (future price‬של החוזה‬
‫הינו ‪ .22-32‬מהו הרווח או ההפסד שלך מהחזקת החוזה‪.‬‬
‫‪ .5‬הנח כי מחיר יצור של מוצר מסוים ביום ‪ 02228222‬הינו ‪ .$0222‬התוצרת תימכר ב‪ .02228222-‬כדי להתגונן מתנודתיות‬
‫במחיר המוצר‪ ,‬נרכשת אופציית ‪ put‬לשנה על מחיר המוצר‪ .‬מחיר האופציה הינו ‪ ,$05‬מחיר המימוש (‪)strike price‬‬
‫הינו ‪ $0022‬ושיעור הריבית הינו אפס‪ .‬הצג את גרף הרווח של החברה ביום ‪ 02228222‬כפונקציה של מחיר המוצר ביום‬
‫זה‪.‬‬
‫‪.6‬‬
‫איזה מהמשפטים הבאים הינו נכון‪:‬‬
‫א‪ .‬ה‪ call price -‬מתואם שלילית עם מחיר המימוש (‪.)strike price‬‬
‫ב‪ .‬ה‪ put price -‬מתואם שלילית עם מחיר המימוש (‪.)strike price‬‬
‫ג‪ .‬ה‪ call price -‬מתואם שלילית עם מועד הפדיון (‪.)time to maturity‬‬
‫ד‪ .‬ה‪ put price -‬מתואם חיובית עם מחיר המניה‪.‬‬
‫ה‪ .‬חוזה רכש (‪ )call contract‬הינו חוזה חסר סיכון‪.‬‬
‫‪ .0‬אתה בונה תיק השקעות משתי אופציות‪ .‬אתה קונה אופציית ‪ call‬במחיר של ‪ $0‬ומוכר אופציית ‪ put‬במחיר של ‪ .$5‬שתי‬
‫האופציות הינן על אותה המניה‪ ,‬אותה תקופת מימוש ואותו מחיר מימוש‪ .‬מייד לאחר בניית התיק‪ ,‬התנודתיות בשווקים‬
‫הפיננסים ירדה מ‪ 12%-‬ל‪ .82%-‬מחיר המניה וריבית חסרת הסיכון לא השתנו‪ .‬תאר מה קרה לערך התיק שלך?‬
‫‪ .2‬ביום ‪ ,862628202‬מחיר של אופציית ‪ call‬הינו ‪ $08‬ומחיר של אופציית ‪ put‬הינו ‪ .$02‬הנח כי שתי האופציות הינן על‬
‫אותה מניה‪ ,‬בעלות אותו מועד לפדיון (שנה) ואותו מחיר מימוש (‪ .)strike price‬מחיר המניה ‪ ,$022‬מחיר המימוש‬
‫(‪ )strike price‬הינו ‪ $002‬והריבית השנתית חסרת הסיכון הינה ‪ .02%‬הצע אסטרטגיית ארביטראז' כך שביום‬
‫‪ 862628202‬סך תזרים המזומנים הינו אפס‪ ,‬וביום ‪ 822628200‬סך תזרים המזומנים הינו חיובי ללא קשר למחיר המניה‪.‬‬
‫‪To solve the next three questions, consider the three-period Binomial Tree framework when the stock price‬‬
‫‪at time zero is 40, U=1.5, D=0.6, and R=1.1. Based upon these inputs answer the following four questions:‬‬
‫‪12‬‬
9. What is the price of a European put option with a strike price of 100?
10. What is the price of a European call option with a strike price of 100? (Hint: If you feel strongly
about your answer for the put price you can implement the put-call parity.)
11. What is the price of an American Put option with a strike price of 100?
12. Assume that the prices of the underlying stock, call, and put options are 110, 17, and 8, respectively.
In addition, the strike price is 105, the time to maturity is 0.5 years, and the annual risk free rate is
10.25%. Which of the following statements is correct?
a.
b.
c.
d.
e.
The call price must be too expensive
The put price must be too expensive
The put and call are properly priced
The leveraged equity is correctly priced
By buying the call and selling the put you make an arbitrage profit.
‫ אתה רואה שהמחיר העתידי‬. future prices -‫ במוסף המתאר את ה‬,Wall Street Journal -‫ אתה קורא את עיתון ה‬.03
‫“ הוא‬Treasury Bonds (CBT)-$100,000; pts 32nds of 100%” ‫ בשם‬future contract ‫( של‬future price(
‫ בגין‬$___________ ‫ אתה מתחייב לשלם בעתיד סכום של‬,‫ הזה‬future contract -‫ באמצעות רכישה של ה‬.22-86
.$100,000 Treasury bonds
‫ אתה חושש מתנודתיות במחיר‬.‫ חודשים‬6 ‫ הנח כי אתה מגדל פרי הדר בפלורידה ואתה מתכנן למכור את תוצרתך תוך‬.44
?‫ כיצד תוכל להשתמש בשוק ההון כדי לגדר את הפוזיציה שלך‬.‫התוצרת‬
:‫ איזה מהמשפטים הבאים אינו נכון‬.05
.)strike price( ‫ מתואם שלילית עם מחיר המימוש‬call price -‫ ה‬.‫ו‬
.)strike price( ‫ מתואם חיובית עם מחיר המימוש‬put price -‫ ה‬.‫ז‬
.)time to maturity( ‫ מתואם חיובית עם מועד הפדיון‬call price -‫ ה‬.‫ח‬
.‫ מתואם חיובית עם מחיר המניה‬put price -‫ ה‬.‫ט‬
.‫) הינו חוזה מסוכן‬call contract( ‫ חוזה רכש‬.‫י‬
‫ שתי‬.$5 ‫ במחיר של‬put ‫ ומוכר אופציית‬$0 ‫ במחיר של‬call ‫ אתה קונה אופציית‬.‫ אתה בונה תיק השקעות משתי אופציות‬.06
‫ התנודתיות בשווקים‬,‫ מייד לאחר בניית התיק‬.‫ אותה תקופת מימוש ואותו מחיר מימוש‬,‫האופציות הינן על אותה המניה‬
?‫ תאר מה קרה לערך התיק שלך‬.‫ מחיר המניה וריבית חסרת הסיכון לא השתנו‬.12%-‫ ל‬82%-‫הפיננסים עלתה מ‬
‫ בעלות‬,‫ הנח כי שתי האופציות הם על אותה מניה‬.$02 ‫ הינו‬put ‫ ומחיר של אופציית‬$08 ‫ הינו‬call ‫ מחיר של אופציית‬.00
)strike price( ‫ מחיר המימוש‬,$022 ‫ מחיר המניה כיום‬.)strike price( ‫אותו מועד לפדיון (שנה) ואותו מחיר מימוש‬
'‫ הצע אסטרטגיית ארביטראז‬.02% ‫ והריבית השנתית חסרת הסיכון הינה‬$002 ‫הינו‬
13
‫‪ .02‬אם חישבנו את אופציית הרכש (‪ )CALL‬לפי מודלים מקובלים לתמחור אופציות ומצאנו שמחיר האופציה בשוק גבוה‬
‫יותר‪ ,‬הסיבה לכך יכולה להיות‪:‬‬
‫ו‪ .‬הערכת יתר של התנודתיות‪.‬‬
‫ז‪ .‬אי התחשבות בדיבידנד במודל‪.‬‬
‫ח‪ .‬אי התחשבות במודל בקשר האפשרי בין שער הריבית למחיר המניה‪.‬‬
‫ט‪ .‬כל הסיבות לעיל‪.‬‬
‫י‪ .‬אף אחת מהסיבות לעיל‪.‬‬
‫‪14‬‬