THE UNIVERSITY OF HONG KONG
DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE
STAT3609 THE STATISTICS OF INVESTMENT RISK
December 13, 2019 Time: 8:30 p.m. - 10:30 p.m.
Only approved calculators as announced by the Examinations Secretary can be used in
this examination. It is candidates' responsibility to ensure that their calculator operates
satisfactorily, and candidates must record the name and type of the calculator used on the
front page of the examination script.
• This examination has a total of 100 points.
• The points for each question are indicated at the beginning of the question.
S&AS: STAT3609 The Statistics of Investment Risk
l. (Total: 7 points)
Write down TRUE or FALSE (not just a single letter T or F) for each of the following
statements.
( a) It is possible for a portfolio to have a larger Sharpe ratio but a smaller Treynor
ratio when comparing to another portfolio.
(b) CAPM and APT have the same set of assumptions.
(c) If in a stock market there are always positive returns on Fridays, then it is
likely that the efficient market hypothesis (EMH) is violated.
( d) Different utility functions will end up with different capital market lines, given
all other settings being the same.
(e) If Jensen's alpha of a portfolio (read from the security market line) is positive,
that portfolio performs better than the market.
(f) The Fama and French model is special case of the single index market model.
(g) Based on a moving average trading rule, when a short term moving average
line drops from above to below a long term moving average line, a buy signal
will be generated.
2. (Total: 9 points)
Explain the difference between a risk-averse investor and a risk-neutral investor in
terms of
(a) (3 points) verbal definition,
(b) ( 3 points) the utility function,
( c) ( 3 points) the indifference curve.
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S&AS: STAT3609 The Statistics of Investment Risk
3. (Total: 14 points)
Consider a price-weighted index, a value-weighted index, and an equally weighted
index. (all with base value 10000) are calculated based on the closing price of 5
constituent stocks. Suppose the number of shares of each stock remains unchanged.
Stock Shares
Closing Price (\$)
Base Day Day 1 Day 2
A 1000 40 50 80
B 2000 55 60 70
C 4000 50 75 65
D 2500 80 85 70
E 1500 65 70 60
(a) (4 points) Calculate the price-weighted indices on Day 1 and Day 2.
(b) (4 points) Calculate the value-weighted indices on Day 1 and Day 2.
(c) (2 points) Using the results in parts (a) and (b), describe the change of the
two indices from Day 1 to Day 2.
(d) (i) (2 points) Calculate the equally weighted index on Day 1.
(ii) (2 points) Consider a equally weighted portfolio of the 5 stocks created on
the Base Day. What is the ratio of the number of shares of the 5 stocks in
the portfolio? Simplify the ratio.
4. (Total: 10 points)
Given the following information for a universe of available risky securities:
• Returns are identically distributed with an expected return of 10% and standard
deviation of 50%.
• Securities share a common correlation of 0.45.
(a) Consider an equally-weighted portfolio of 12 stocks.
(i) (3 points) Calculate the portfolio variance.
(ii) (3 points) Show that when the number of stocks in the portfolio increases,
the portfolio variance approaches a limit of 0.1125.
(b) ( 4 points) Calculate the minimum number of stocks necessary so that the
portfolio volatility (standard deviation) is less than 0.5% plus the theoretical
minimum volatility of a well-diversified portfolio.
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S&AS: STAT3609 The Statistics of Investment Risk
5. (Total: 8 points)
(a) (4 points) Explain the details and the significance of the two-fund theorem.
(b) (4 points) Explain the details and the significance of the one-fund theorem.
6. (Total: 12 points)
Suppose SIMM and CAPM are valid. The risk-free interest rate is 8% and the
expected rate of return on the market index is 18%. All the rates are quoted on an
annual basis. Assume that the market index is an accurate proxy for the market
portfolio. Stock A is selling at \$100 per share now, and it will pay \$9 per share at
the end of the year. It is also known that the expected share price of stock A at the
end of the year is \$111.
(a) (5 points) Show that the beta of stock A is 1.2.
(b) The variance-covariance matrix of the returns (in percentage) on stock A and
the market index is given as
[ 70 X ] ,
42 y
where x and y are constants.
(i) (3 points) Find x and y.
(ii) (4 points) Find the residual variance of stock A.
7. (Total: 10 points)
Consider purchasing 10,000 shares of stock at \$10 per share, with initial margin
40% and maintenance margin 25%. Suppose the future price is \$P where P rv
LogN(3.6, 0.9) , and the loan interest up to that future date is 8% of the borrowed
amount.
(a) (2 points) What is the expected future price?
(b) (3 points) What is the standard deviation of the future price?
(c) (3 points) What is the probability of receiving a margin call on that future
date? (Ignore the interest effect in this part.)
(d) (2 points) What is the probability of having a profit on the future date?
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S&AS: STAT3609 The Statistics of Investment Risk
8. (Total: 10 points)
Give a one-page summary of Markowitz mean-variance portfolio model.
(Note: There is no specific format as long as the presentation is clear and readable.
You may combine your descriptions with mathematical formulae with well-defined
symbols, diagrams, tables, etc.)
9. (Total: 20 points)
Consider a portfolio of two risky assets B and C. Denote µB, µc , aB and ac as the
expected returns and volatilities of B and C, respectively. Also denote aBc as the
covariance between the returns on B and C. Suppose the returns of B and C are not
perfectly correlated.
(a) (4 points) Show that the weighting of asset B for the mm1mum variance
portfolio is
ac - aBc
WE = 2 2 aB + ac - 2aBc
(b) (4 points) Show that the weighting of asset B for the tangency portfolio (with
risk-free rate R1) is
(Hint: You are not required to derive the result from first principle.)
(c) Suppose R1 = 4%, µB = 10%, µc = 15%, aB = 20%, ac = 30%. It is further
given that the correlation between the returns on B and C is 0.5, and the betas
of assets B and C are f3B = 0.8 and /3c = 1.25, respectively. Find the Sharpe
ratios and the Treynor ratios of the following:
(i) (3 points) Asset B.
(ii) (3 points) Asset C.
(iii) (3 points) The minimum variance portfolio of assets B and C.
(iv) (3 points) The tangency portfolio of assets B and C.
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