Sample Exam
FM445 – Portfolio
Management
Instructions to candidates
This paper contains four questions. Answer all four questions. The exam is worth 100 points, and
the point value of each question is indicated in the question header.
Time
Allowed
Reading
Time:
10 minutes (you may not write on the answer booklet during
reading time, but you may take notes in the examination paper.)
Writing
Time:
2 hours
You are supplied with:
No additional materials.
You may also use:
No additional materials.
Calculators:
Calculators are allowed in this examination
 FM445 – Sample Exam
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Question 1: Dynamic Investment Strategies [25 Points]
A recent paper re-examines the profitability of size, value, and momentum strategies
extending the analysis to more recent sample periods. One of the goals of the paper is to
investigate the separate contribution of long and short portfolios to the profits of each
strategy. The following table shows average raw returns in excess of the one-month T-bill
rate, Sharpe ratios, and CAPM alphas of value-weighted decile portfolios formed on size,
value (book-to-market), and momentum (past 12-month returns), as well as the difference
between Deciles 10 and 1 (10-1). The returns are averaged over the period July 1926 to
December 2011. t-statistics are shown in parentheses. Returns and alphas are obtained
monthly and reported as annualized percentages.
a. [9 points] Explain how you would implement size, value, and momentum strategies to
replicate this table (assume monthly rebalancing of the portfolios).
b. [8 points] Describe the results presented in the table. Are these strategies profiting more
from long or short positions?
c. [8 points] Consider the interaction between size and momentum strategies. The following
table is based on 25 portfolios obtained from the intersection of size-momentum quintile
portfolios. The table reports profits measured in excess returns and in alphas. In particular, for
these two measures, the table reports: the return spread between high and low momentum
portfolios (5-1) for each size quintile; the profits of the long side of the strategy; the
percentage of the profits coming from the long side (including a t-statistic for whether the
profits from the long side are the same as the profits from the short side). Interpret the results
in the table.
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 FM445 – Sample Exam
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Question 2: APT and Performance Evaluation [30 Points]
[The following information applies to items (a) and (b).] The returns on assets in an
economy depend on the realization of two risk factors, denoted by and . Three welldiversified portfolios (denoted by A, B and C) are available whose returns are given by the
following expressions:
= 0.06 +
−2
= 0.05 + 2 −
= 0.05 + 3 +
As in the lecture notes, we assume that
=
= 0.
a. [15 Points] Use assets A, B and C to construct factor-replicating portfolios for factor 1,
factor 2 and for the risk-free asset in this economy. Show your step-by-step calculations.
Express the APT equation in this economy.
b. [5 Points] A fourth asset, denoted by D, is introduced in this economy. Its returns are
given by
= 0.05 +
+3
Are there any arbitrage opportunities in the economy with assets A, B, C and D?
Explain. If such an arbitrage opportunity exists, explain in detail how you would
implement it assuming that only assets A, B, C and D can be traded (that is, you cannot
trade the portfolios you constructed in item a. directly.)
c. Assume that you are currently managing a portfolio that is replicating the market index.
You identify a new portfolio P1 of stocks which, over the past five years, has produced a
monthly CAPM alpha of 0.0105, has a beta of 0.8185, and a monthly standard deviation
of excess returns of 0.0282. The market excess return over the same period is 0.0137
and its volatility is 0.0305.
i.
[5 Points] You decide to combine portfolio P1 with the market portfolio in an
optimal manner. What is the Sharpe ratio of this new portfolio of stocks? What
does this measure indicate? Explain.
ii.
[5 Points] In addition to portfolio P1, you have portfolio P2 available, with market
beta 1.1, monthly CAPM alpha of 0.0083 and monthly standard deviation of
excess returns of 0.031. When combining portfolios P1 and P2 optimally with the
market portfolio, what is the Sharpe ratio of the resulting portfolio?
 FM445 – Sample Exam
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Question 3: Dynamic Investment Strategies and Performance Attribution [25 Points]
a. [10 points] A research paper tries to understand the investment strategy that is
responsible for the remarkable performance of Warren Buffett by examining the returns
of Berkshire Hathaway. The table below shows estimates of alphas and slope
coefficients using two models: (i) the four-factor model (Carhart), and (ii) a model that
adds the Betting-Against-Beta factor (BAB). These two models are estimated for the
actual returns of Berkshire Hathaway stock, and for a replicating portfolio that uses
Berkshire’s publicly traded equities as reported in its disclosures of holdings at the end of
each quarter. Alphas are annualized and expressed in percent (t-statistics are shown in
parentheses).
b. [7 points] Define and explain the meaning of the three measures of performance attribution:
Characteristic Selectivity, Characteristic Timing, and Average Style.
c. [8 points] The following graph shows average estimated alphas for ten portfolios of
hedge funds that are sorted each month on their liquidity beta. The liquidity beta is
calculated each month from a regression of hedge fund returns on the market return and
a liquidity factor, using data over the prior 24 months.
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The bars represent the monthly alphas of the ten liquidity-beta portfolios, estimated using
the following factors: the market risk premium, SMB of Fama and French (1993), the
spread in returns between long and short-term Treasury bonds, the spread in returns
between Corporate and Treasury bonds, and three trend-following factors (returns on
lookback straddles for bonds, currencies, and commodities). The circles represent the tstatistics of the alphas. The figure also displays the alpha of the high-minus-low liquiditybeta portfolio (High-Low).
Assume that the liquidity factor used to compute liquidity betas is the Pastor-Stambaugh
factor that we studied in the course. What do these liquidity betas represent?
Interpret the results of this test.
 FM445 – Sample Exam
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Question 4: Portfolio Optimizations and Black-Litterman [20 Points]
You want to allocate your investment optimally among five equity portfolios and the risk-free
asset. The equity portfolios are: domestic equities with high past returns (winners), domestic
equities with low past returns (losers), domestic equities with high book-to-market (value),
domestic equities with low book-tomarket (growth), and a portfolio of international equities.
The following table shows the proportion of market value (w) represented by each asset.
a. [7 Points] In the context of the Black-Litterman model, you expect winners to outperform
losers by 0.8% per month, based on your estimates of a momentum strategy that you
have recently implemented. You also expect the return of growth stocks (in excess of the
risk free rate) to be 0.5% per month, based on the consensus of financial analysts.
Set up the investor optimization problem, specifying the matrix P and the vector Q for
your views. How can you specify the matrix Ω of uncertainty about your views? Assume
that you are more certain about your expectation on winners vs. losers than about your
expectation on growth stocks. Explain how you would compute the new vector of
expected returns. Finally, explain how you would obtain the optimal Black-Litterman
weights for your asset allocation.
b. [7 Points] After implementing the BL model using a coefficient of risk aversion λ=1 and a
parameter τ=0.05, you obtain the following vector of optimal BL weights:
w* = [61.5% -23.3% 18.9% 32.5% 20.8%]
Compute and discuss the differences in weights between the case in which you express
no views on the five assets and the new BL weights w* obtained using your subjective
expectations.
c. [6 Points] Assume now that, besides the views described in the table above, you expect
international stocks to outperform value stocks by 0.3% per month. How would P and Q
change? What impact do you expect this view to have on the optimal weights?
 FM445 – Sample Exam
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