FINE-452: Applied Quantitative Finance -- Asset Management
Fall, 2020
Dr. Anisha Ghosh
Suggested Solution to Quiz 1
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1. Is each of the following statement true or false? Explain your answer.
a. (5) In a CAPM world, every investor, regardless of her risk preferences,
invests in the market portfolio.
Answer: True, except an investor who is infinitely risk averse (such an
investor puts all her wealth in the risk free asset). This is because, under
the CAPM assumptions, all investors forms the same beliefs about
expected return, the variance of the return, and the covariances between
the returns of each pair of assets. Therefore, every investor faces the same
minimum variance frontier. Also, under the CAPM assumptions, every
investor faces the same risk free rate. Therefore, every investor identifies
the same tangency portfolio as having the maximum Sharpe ratio. Since
every investor invests in the same risky portfolio (the tangency portfolio),
the tangency portfolio is the market portfolio in equilibrium. Thus, every
investor (except the infinitely risk averse) invests in the market portfolio.
They just invest different proportions in the market portfolio based on
their risk aversion.
b. (5) According to the CAPM, an asset that covaries positively with the
market earns a higher risk premium than an asset that covaries negatively
with the market.
Answer: True, because such an asset is risky – it pays off in good times
when the market is doing well and poorly in bad times when the market
crashes. The latter are times when investors need the money the most.
Therefore, investors view such an asset as risky and demand a high
average return to hold it.
c. (5) If the CAPM is true, then the relevant measure of risk of an asset (e.g.,
a stock) is the variance of the return on the asset.
Answer: False. Part of the variance is idiosyncratic risk, which can be
diversified away by increasing the number of stocks in the portfolio.
Therefore, investors do not demand compensation for this component of
risk. Only the component of variance that represents systematic risk (the
beta), that cannot be diversified away, commands a risk premium.
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Therefore, the relevant measure of risk of an asset is the systematic risk –
the beta of the asset with respect to the market in the context of the
CAPM.
d. (5) According to the CAPM, bad times correspond to periods of low return
on the market portfolio.
True. Each risk factor identifies its own set of bad times. The CAPM has
only one risk factor – the return on the overall market portfolio. The bad
times are characterized by periods of low return on the market portfolio.
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2. (8) During the 2008 financial crisis, several (seemingly very different) asset
classes simultaneously performed very poorly. The asset classes included the
US large market capitalization equity, US small market capitalization equity,
international equity, emerging markets equity, public real estate, private real
estate, equity hedge funds, fixed income hedge funds, and commodities. The
return on each of these asset classes was large in magnitude and negative in
2008. However, the magnitude of the negative return was different for the
different asset classes, varying from -16.9% for private real estate to -53.2%
for emerging markets equity. Can this observation be rationalized in the
context of an underlying factor model? Explain your answer.
Answer: The simultaneous poor performance of the mentioned asset classes
is consistent with the existence of a small number of common underlying risk
factor(s) for these assets. The observation that different asset classes went
down by differing amounts suggests that they have different exposures
(betas) to the risk factors.
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3. A regression of the monthly excess returns on hedge fund ABC on the
monthly excess returns on the market portfolio delivered the following
result:
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𝑟𝑖,𝑡 − 𝑟𝑓,𝑡 = 0.0001 + 1.30(𝑟𝑀,𝑡 − 𝑟𝑓,𝑡 ) + 𝜀𝑖,𝑡 ;
𝑅 2 = 0.74
The standard error of the intercept is 0.001 and that of the slope is 0.50.
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a. (2+2+2+2) What is the estimated monthly alpha of the Fund? What is the
annualized alpha of the Fund? Is the alpha statistically significant?
Interpret the alpha.
Answer: Monthly alpha is 0.0001.
The annualized alpha is 0.0001*12=0.0012.
The t statistic for alpha is 0.0001/0.001=0.1. Thus, the alpha is not
statistically significant.
The hedge fund does not deliver any average returns in excess of the
benchmark return.
b. (2+2+2) What is the market beta of the Fund? Is the beta statistically
significant? Interpret the beta.
Answer: the market beta is 1.3.
The t statistic for beta is 1.3/0.5=2.6. Thus, the beta is strongly statistically
significant.
Exposure to market risk explains well the returns on the hedge fund.
c. (2+2) Is the Fund riskier than the market portfolio? Explain your answer.
Answer: The beta of the fund is higher than that of the market (1.3 versus
1). Therefore, the fund has higher systematic risk than the market.
d. (4) Interpret the 𝑅 2 of the above regression.
Answer: Variation in the market return explains 74% (about threequarters) of the variation in the returns on the fund.
e. (4+2) Write down the formula for the Tracking Error of the Fund in the
context of the above regression equation. Explain the rationale for
imposing tracking error constraints on a fund manager.
Answer: The tracking error is standard deviation of 𝜀𝑖,𝑡 .
Tracking error constraints try to ensure that the manager does not stray
too far from the benchmark (because, if he/she does, then the benchmark
may no longer be the relevant benchmark with respect to which to
measure alpha).
f. (4+2) Write down the formula for the Information Ratio of the Fund in the
context of the above regression equation. Explain the rationale for using
the Information Ratio as a measure of performance.
𝛼
0.0001
Answer: The Information ratio is 𝑇𝑟𝑎𝑐𝑘𝑖𝑛𝑔 𝐸𝑟𝑟𝑜𝑟 =
.
√𝑉𝑎𝑟(𝜀𝑖,𝑡 )
The Information Ratio, unlike the alpha, adjusts for risk. So, it is a measure
of risk-adjusted performance.
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4. (8) Martingale Asset Management’s low volatility strategy produces an alpha
of 3.44% per year relative to the CAPM benchmark. Warren Buffet’s
Berkshire Hathaway, on the other hand, yields an annualized alpha of 8.64%.
Based on this observation, an investor concludes that Berkshire Hathaway is
a much more attractive investment vehicle than Martingale Asset
Management’s low volatility strategy. Is the investor correct in her
conclusion? What would your advice be to such an investor?
Answer: Alpha, as a measure of performance, ignores (non-market) risk.
Therefore, a manager may deliver a large alpha, but at the cost of exposing
the investor to a large amount of risk. Therefore, the advice to such an
investor would be to compare a risk-adjusted measure of performance, e.g.
the information ratio, for the above two investment strategies, rather than
focusing solely on alpha.
5. Suppose that the monthly earnings-price ratio predicts the next month’s
return on the market portfolio with an 𝑅 2 = 0.38%, and the T-bill rate
predicts the next month’s return on the market portfolio with an 𝑅 2 =
0.57%. Suppose further that the average monthly return on the market
portfolio is 0.9%, the volatility of the monthly market return is 5.53%, and
the average monthly risk free rate is 0.3%.
An investor with mean-variance preferences can increase her average
monthly returns by an amount given by
𝑅2
𝛾
by investing in an active portfolio
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constructed by forecasting the monthly market return using a predictor
variable versus not using the predictor variable to forecast market returns
(and, instead) using the historical mean of the market return as the best
predictor of the market return in the next month).
The percentage increase in average monthly returns achieved by moving
from forecasting using the historical mean to forecasting using a predictor
𝑅2
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variable is given by 𝑆 2 . Note that 𝑆 denotes the Sharpe ratio of the market
and 𝛾 denotes the risk aversion coefficient of the investor.
a. (2+2) What is the monthly Sharpe ratio of the market? What is the
annualized Sharpe ratio of the market?
Answer: The monthly Sharpe ratio is (0.009-0.003)/0.0553=0.108.
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The annualized Sharpe ratio is 0.108 × √12 = 0.374.
b. (5) Suppose an investor has mean-variance preferences with a risk
aversion coefficient of 2. What is the absolute increase in average
monthly returns that she can achieve by investing in an active portfolio
constructed by forecasting the monthly market return using the earningsprice ratio as the predictor variable versus not using the earnings price
ratio to forecast market returns and using the historical mean of the
market return as the best predictor of the market return in the next
month?
Answer: The average returns can be increased by an absolute amount of
𝑅2
𝛾
=
0.0038
2
= 0.0019 per month, or by 0.0019*12=0.0228=2.28% per
𝑅2
𝛾
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year.
c. (5) How does the answer to (b) change if the investor is more risk averse
and has a risk aversion coefficient of 5?
Answer: The average returns can be increased by an absolute amount of
=
0.0038
5
= 0.00076 per month, or by 0.00076*12=0.00912=0.912%
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per year.
d. (5) Explain the difference between the results in (b) and (c).
Answer: In (b), the investor is more risk averse and, therefore, takes less
extreme positions in the market (to benefit from the forecasts) and so her
average returns increase by less than a less risk averse investor who takes
more extreme positions in the market in response to the forecasts.
e. (5) Suppose an investor has mean-variance preferences with a risk
aversion coefficient of 2. What is the percentage increase in average
monthly returns that she can achieve by investing in an active portfolio
constructed by forecasting the monthly market return using the earningsprice ratio as the predictor variable versus not using the earnings price
ratio to forecast market returns and using the historical mean of the
market return as the best predictor of the market return in the next
month?
𝑅2
0.0038
sh
Answer: The average returns can be increased 𝑆 2 = 0.108×0.108 = 0.326 =
32.6% per month. Note this is the percentage increase in average returns,
while (b) is the absolute increase in average returns.
f. (3) How does the answer to (e) change if the investor is more risk averse
and has a risk aversion coefficient of 5?
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𝑅2
0.0038
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Answer: The average returns can be increased 𝑆 2 = 0.108×0.108 = 0.326 =
32.6% per month, i.e. does not change with the level of risk aversion.
g. (3) Explain the difference between the results in (e) and (f).
Answer: The answers in (e) and (f) are identical because the percentage
increase does not depend on the risk aversion.
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