Examiners’ commentaries 2016
Examiners’ commentaries 2016
FN3023 Investment management
Important note
This commentary reflects the examination and assessment arrangements for this course in the
academic year 2015–16. The format and structure of the examination may change in future years,
and any such changes will be publicised on the virtual learning environment (VLE).
Information about the subject guide and the Essential reading
references
Unless otherwise stated, all cross-references will be to the latest version of the subject guide (2011).
You should always attempt to use the most recent edition of any Essential reading textbook, even if
the commentary and/or online reading list and/or subject guide refer to an earlier edition. If
different editions of Essential reading are listed, please check the VLE for reading supplements – if
none are available, please use the contents list and index of the new edition to find the relevant
section.
General remarks
Learning outcomes
At the end of this course, and having completed the Essential reading and activities, you should be
able to:
•
list given types of financial instruments and explain how they work in detail
•
contrast key characteristics of given financial instruments
•
briefly recall important historical trends in the innovation of markets, trading and financial
instruments
•
name key facts related to the historical return and risk of bond and equity markets
•
relate key facts of the managed fund industry
•
define market microstructure and evaluate its importance to investors
•
explain the fundamental drivers of diversification as an investment strategy for investors
•
aptly define immunisation strategies and highlight their main applications in detail
•
discuss measures of portfolio risk-adjusted performance in detail and critically analyse the
key challenges in employing them
•
competently identify established risk management techniques used by individual investors
and corporations.
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FN3023 Investment management
Format of the examination paper
The examination paper consists of eight questions of which you have to answer any four. The
questions are a mixture of three types. The first type is a question that asks for a numerical problem
to be solved. The second type of question asks for institutional knowledge, for instance candidates
are asked to explain what instruments are traded in the money market or how a limit order market
works. The third type of question asks for an essay-style answer about a conceptual issue in finance,
for instance what implications the efficient market hypothesis have on investment returns and
factors that may explain why the efficient market hypothesis may not hold in practice.
What are the examiners looking for?
With numerical questions, it is important that answers and steps are carefully and clearly explained.
A very good answer would specify what knowledge is used. For instance, when the CAPM model is
used as a basis for a cost of capital calculation, it is important that this is outlined in the answer.
When the question asks for an outline of institutional details, an ideal answer is brief and concise,
with a clear emphasis on relevant facts. For instance, if you explain what instruments are traded in
the money market, you need to focus on the distinguishing features of these instruments – that they
are fixed income instruments of short maturity, often of large denominations, and issued by the
government, banks or corporations. When the question asks for a critical evaluation of a conceptual
issue, it is important that you address all aspects of the question and structure your argument
carefully so that it is clear to the examiners what level of understanding you have.
Key steps to improvement
The key test of how much you understand about this subject is whether you can transfer knowledge
about one type of problem in finance to other problems.
The typical pattern that the examiners find when marking the papers for this course is that
questions that may appear difficult (in the sense they are technically demanding, for instance)
achieve higher scores than questions that may appear to be easy, if the difficult question is closer to
material that candidates have studied beforehand.
In other words, the examiners find that candidates tend to find it difficult to transfer their
knowledge into new areas. Therefore, problem-solving practice is probably the most valuable
preparation for the examination, and it is important that you attempt to solve problems that go
outside what you encounter in the subject guide.
Examination revision strategy
Many candidates are disappointed to find that their examination performance is poorer than they
expected. This may be due to a number of reasons. The Examiners’ commentaries suggest ways of
addressing common problems and improving your performance. One particular failing is ‘question
spotting’, that is, confining your examination preparation to a few questions and/or topics which
have come up in past papers for the course. This can have serious consequences.
We recognise that candidates may not cover all topics in the syllabus in the same depth, but you
need to be aware that the examiners are free to set questions on any aspect of the syllabus. This
means that you need to study enough of the syllabus to enable you to answer the required number of
examination questions.
The syllabus can be found in the Course information sheet in the section of the VLE dedicated to
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Examiners’ commentaries 2016
each course. You should read the syllabus carefully and ensure that you cover sufficient material in
preparation for the examination. Examiners will vary the topics and questions from year to year and
may well set questions that have not appeared in past papers. Examination papers may legitimately
include questions on any topic in the syllabus. So, although past papers can be helpful during your
revision, you cannot assume that topics or specific questions that have come up in past examinations
will occur again.
If you rely on a question-spotting strategy, it is likely you will find yourself in difficulties
when you sit the examination. We strongly advise you not to adopt this strategy.
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FN3023 Investment management
Examiners’ commentaries 2016
FN3023 Investment management
Important note
This commentary reflects the examination and assessment arrangements for this course in the
academic year 2015–16. The format and structure of the examination may change in future years,
and any such changes will be publicised on the virtual learning environment (VLE).
Information about the subject guide and the Essential reading
references
Unless otherwise stated, all cross-references will be to the latest version of the subject guide (2016).
You should always attempt to use the most recent edition of any Essential reading textbook, even if
the commentary and/or online reading list and/or subject guide refer to an earlier edition. If
different editions of Essential reading are listed, please check the VLE for reading supplements – if
none are available, please use the contents list and index of the new edition to find the relevant
section.
Comments on specific questions – Zone A
Candidates should answer FOUR of the following EIGHT questions. All questions carry equal
marks.
Question 1
(a) Explain what we mean by efficient markets. List empirical findings that
challenge the notion of efficient markets.
(7 marks)
(b) A stock is trading at a price of $15 in year 0. In year 1 it pays a dividend of $1
and the price (ex dividend) has increased to $16. In year 2 it also pays a
dividend of $1, and the price has gone down to $13. In year 0 you short 100
shares which you buy back in year 2, and you trade on a margin where you are
required to hold 150% of the value of your short position (you should assume
the initial margin and the maintenance margin are both 150% of the value of
the short position) in a margin account paying zero interest. What is the return
on your short position?
(9 marks)
(c) A non-dividend paying stock has price that follows a geometric Brownian
motion with drift (instantaneous return) equal to 10% and volatility 20%.
Derive the 1%, 30-day VaR of the stock (you should assume there are 20
trading days over the 30-day period). Your answer should be given in terms of
log-returns, and also recall that if z is a standard normally distributed random
variable, then Pr(z -2.33) = 1%.
(9 marks)
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Examiners’ commentaries 2016
Reading for this question
Chapter 2 in the subject guide and Chapter 17 in the Elton/Gruber/Brown/Goetzmann text
provide the background readings for this question.
Approaching the question
(a) The examiners awarded 4 marks for explanations of what we mean by efficient markets, and
3 marks for challenging evidence.
(b) Year 0: sell 100 shares at a price of 15, which generates a cash in-flow of 1,500; year 1: you
pay a dividend of 1 per share which generates a cash out-flow of 100; year 2: you pay
another dividend of 1 per share (cash out-flow of 100) and buy back 100 shares at a price of
13 (cash out-flow of 1,300), which generates a net cash out-flow of 1,400. The underlying
cash-flows are therefore (+1,500, −100, −1,400). The initial margin is 150% of 1,500, which
generates a cash out-flow of 2,250. The portfolio value in year 1 is 100 shares times a price
of 16 which is 1,600. Therefore, the increase in margin deposit is 150% of (1600 − 1500)
which generates a cash out-flow of 150. The deposit is released in year 2, which generates a
cash in-flow of 2,400. Therefore, the margin cash-flows are (−2,250, −150, +2,400). The net
cash-flow is therefore (−750, −250, +1,000), which is equivalent to an IRR of 0%.
The examiners awarded 3 marks for the underlying cash-flows, 3 marks for the margin
cash-flows, and 3 marks for the IRR.
(c) The log return is:
x = ln
vt
v0
∼N
(0.2)2
0.1 −
2
30
30
, (0.2)2
365
365
.
The normalised 30-day log return with mean 0 and variance 1 is therefore:
2
30
x − 0.1 − (0.2)
2
365
q
z=
.
30
0.2 365
The probability P (Z ≤ y) = N (y) = 0.01 implies y = −2.33. The associated x is therefore:
2
30
x − 0.1 − (0.2)
2
365
q
= −2.33
30
0.2 365
which corresponds to x = −0.196, or a log-return of −19.6%.
9 marks in total were awarded for this calculation. The subject guide
p states that in practice
the standard deviation in the denominator is often adjusted to 0.2 20/365 because there
are 20 trading days during the period. This would be fine.
Question 2
(a) Explain why an optimal portfolio for investors who are variance averse consists
of only two assets. What are the two assets? How is variance aversion different
from risk aversion?
(7 marks)
(b) A 5-year bond with face value $1,000 and coupon rate 5% is trading at par
value. What is the yield-to-maturity of the bond? What is the duration of the
bond?
(9 marks)
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FN3023 Investment management
(c) What is the convexity of the bond in (b)? Suppose the yield-curve is flat and
the yield-to-maturity increases by 0.1% from its current level (i.e. 10 basis
points). Based on your answers in (b) and (c), what is the approximate change
in the $-value of the bond?
(9 marks)
Reading for this question
Chapter 6 in the subject guide and Chapters 13 and 20 in the Elton/Gruber/Brown/Goetzmann
text provide background readings for this question.
Approaching the question
(a) The examiners awarded 4 marks for explaining the two-fund separation property of the
optimal portfolios, 2 marks for the two assets involved, and 1 mark for explaining how
variance aversion is a special form of risk aversion.
(b) The value is 1,000 since the bond is trading at par, which implies the yield-to-maturity is
equal to the coupon rate (i.e. 5%). The duration is:
50
1.05
+
2×50
(1.05)2
+ ··· +
5×1050
(1.05)5
1000
= 4.546.
The examiners awarded 2 marks for the yield-to-maturity, and 7 marks for the duration.
(c) The convexity is:
1×2×50
(1.05)3
+
2×3×50
(1.05)4
+ ··· +
5×6×1050
(1.05)7
1000
= 23.936.
The change in yield is 0.001. The approximate %-change in the value of the bond is:
∆P
Duration
1
=−
× 0.001 + × Convexity × (0.001)2 = −0.00432
P
1.05
2
so that the $-change in the bond is 1000 times the %-change, or −$4.32.
The examiners awarded 5 marks for the convexity and 4 marks for working out the
approximate change in the $-value of the bond.
Question 3
(a) Explain how corporations approach the issue of managing risk. What is the aim
of corporate risk management, and how do corporations create value through
risk management?
(7 marks)
(b) You are given the following data on 6-month put and call values on a stock. The
stock is currently trading at a value of £10.
Exercise price
£9.5
£10
£10.5
Call price
£0.948
£0.706
£0.467
Put price
£0.253
£0.510
£0.771
Can you identify an arbitrage opportunity? Explain.
(9 marks)
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Examiners’ commentaries 2016
(c) Suppose you have invested in 100,000 shares of a stock currently trading at £10
per share. The 6-month risk free interest rate is 2%, the 6-month call option on
the stock with exercise price £10 is trading at £0.706, and the corresponding
put option is trading at £0.510. Construct a new portfolio which consists of the
stock, a risk free investment, and put options on the stock such that: (i) the
initial value of the portfolio is the same as your current portfolio; (ii) the value
in 6 months’ time will never fall below the initial value of the portfolio; and (iii)
the portfolio retains as much as possible of the up-side return of the stock in
(b). Specify the exact amounts invested in the three assets.
(9 marks)
Reading for this question
Chapter 9 in the subject guide, and Chapter 3 in the Stultz text provide background readings for
this question.
Approaching the question
(a) The examiners awarded 4 marks for explaining the objective of risk management in
mitigating the effects of shortfall of funds, and 3 marks for explaining how shortfall can lead
to deadweight costs for the firm. Avoiding such costs creates value through risk
management.
(b) Using put-call parity, we know that P V (X) = S + P − C, and we also know that
P V (X) = X/(1 + r) where r is the 6-month rate. Therefore:
P V (9.5) = 10 + 0.253 − 0.948
which implies a risk-free rate r = 2.0956 over the 6-month period. Doing the same with
P V (10) and P V (10.5), which yield risk-free rates of r = 1.9992 and r = 1.9022, respectively,
we find arbitrage opportunities by borrowing at the cheapest rate (best available rate is
1.9022) and lending at the most expensive rate (best available rate is 2.0956). This holds
regardless of what the actual risk-free rate is – if it is less than the best cheapest rate or
greater than the best expensive rate the arbitrage transaction would become more attractive
by including the risk-free asset. The arbitrage profit increases by the amount of capital
used, and decreases by the transaction costs incurred.
The examiners awarded 6 marks for identifying the implied risk-free rates, and 3 marks for
specifying the arbitrage transaction.
(c) The portfolio has value 100,000 shares times the price 10 per share, which equals 1,000,000.
To generate the floor of the investment strategy, invest P V (10) per share in the risk-free
asset. This leaves 10 − P V (10) = 10 − 10/1.02 to be invested in calls with exercise price 10
which provides the upside potential of the investment. This implies:
10 − P V (10)
= 0.2777
0.706
call options per share. By put-call parity, each call can be replicated by an investment in the
stock, a risk-free investment, and a put option: C = S + P − P V (X). Therefore, for each
share, invest P V (10) in the risk-free investment, and invest 0.2777 shares in the stock,
0.2777 puts, and borrow 0.2777 × P V (10) risk-free. In total, therefore, for each share there
is (1 − 0.2777) invested risk-free in P V (10), 0.2777 shares of the stock, and 0.2777 puts.
Multiplying by 100,000 shares, the total position is 708,106 invested risk-free, 277,731
invested in stocks, and 14,164 invested in puts.
The examiners awarded 5 marks for the basic investment strategy involving the risk-free
investment plus the correct investment in calls, and 4 marks for converting the call
investment into an investment involving stocks, puts and risk-free borrowing.
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FN3023 Investment management
Question 4
(a) What do certificates and deposits have in common with commercial papers? In
what ways are they different?
(7 marks)
(b) Suppose a bond is currently quoted at a price of $1,025. The face value is
$1,000 and the coupon rate 3%. The number of days since the last coupon is 49,
and the number of days between coupons is 365. If you purchase this bond,
what is the actual transaction price?
(9 marks)
(c) An investor has preferences over portfolios that can be expressed as a function
of expected return (m) and portfolio variance (s2 ) only: U (m, s2 ) = m − 0.8s2 .
The risk free rate of return is 3% and the average return on the market index is
8%. The standard deviation of the market index is 30%. What is the optimal
portfolio for the investor?
(9 marks)
Reading for this question
Chapter 2 in the subject guide, and Chapter 2 in the Elton/Gruber/Brown/Goetzmann text
provide background readings for this question.
Approaching the question
(a) The examiners awarded 4 marks for explaining the similarity of these instruments to other
money market instruments, and 3 marks for explaining the key difference in issuing body for
the two instruments.
(b) Accrued interest is:
1000 × 0.03 × 49
= 4.027.
365
The relationship between the quoted price and the transaction price is that the transaction
price (dirty price) = quoted price (clean price) + accrued interest. Therefore, the
transaction price is 1025 + 4.03 = 1029.03.
The examiners awarded 5 marks for working out the accrued interest and 4 marks for the
transaction price.
(c) First, the investor is variance averse which implies that the optimal portfolio consists of a
mixture of the risk-free asset and the market portfolio. We can then exclude all other
portfolios from consideration. The expected return on the mixture portfolio is:
m = 0.03(1 − x) + 0.08x = 0.03 + 0.05x.
The variance of the mixture portfolio is:
s2 = (0.3)2 x2 .
The weight on the risk-free asset is 1 − x and the weight on the market portfolio is x. The
expected utility is therefore:
U = 0.03 + 0.05x − 0.8(0.09)x2 .
Differentiating the expected utility with respect to x to maximise the expression, we find the
first-order condition is:
0.05 − 2(0.8)0.09x = 0
8
or
x=
0.05
= 0.35.
2(0.8)0.09
Examiners’ commentaries 2016
Therefore, the optimal portfolio has 65% invested risk-free and 35% invested in the market
portfolio.
The examiners awarded 4 marks for the argument about two-fund separation and the
derivation of the expression for expected utility, and 5 marks for the derivation of the
optimal weights.
Question 5
(a) Explain why it is difficult to estimate correct Sharpe ratios for funds when the
fund manager engages in market timing where he moves the fund’s capital in
and out of risky assets often?
(7 marks)
(b) You are given the following data.
Fund
Risk free asset
Market index
Portfolio A
Portfolio B
Average return
3%
8%
12%
9%
Standard deviation
0
30%
50%
30%
Beta
0
1
1.4
1.2
Are the data in the table reasonable? What is the Sharpe ratio and the Treynor
ratio for A and B? Would you recommend these two portfolios to investors?
(9 marks)
(c) You are given the following data on 4 stocks A, B, C, and D.
Stock
A
B
C
D
Beta
1.2
1.2
1.2
1.2
Idiosyncratic variance
0.12
0.12
0.12
0.2
The total variance of the market portfolio is 0.09. Assume the idiosyncratic risk
is independent across the four stocks. What portfolio consisting of the four
stocks has the smallest total variance? What is the idiosyncratic risk of this
portfolio?
(9 marks)
Reading for this question
Chapter 8 in the subject guide provides the background readings for this question.
Approaching the question
(a) The examiners awarded 2 marks each for the explanation that the Sharpe ratio aggregates
the excess return of fund performance linearly, and the standard deviation of fund
performance non-linearly. They awarded a further 3 marks for recognising that the high-risk
periods will be given higher weight in the standard deviation estimate than the low-risk
periods, and therefore the denominator in the Sharpe ratio is overestimated, and the Sharpe
ratio itself underestimated.
(b) We notice that portfolio B has the same standard deviation as the market index, but with
beta greater than 1. This implies negative idiosyncratic risk which is unreasonable – hence
it is likely there is some estimation error in the table where the standard deviation of B is
greater than the estimate of 30% or the beta is less than the estimate of 1.2.
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FN3023 Investment management
Using the data available, Sharpe(A) = (0.12 − 0.03)/0.5 = 0.18, Sharpe(B) =
(0.09 − 0.03)/0.3 = 0.2, and Sharpe(Market) = (0.08 − 0.03)/0.3 = 0.167. Similarly,
Treynor(A) = (0.12 − 0.03)/1.4 = 0.064, Treynor(B) = (0.09 − 0.03)/1.2 = 0.05, and
Treynor(Market) = 0.08 − 0.03 = 0.05.
A is favourable in terms of Sharpe and Treynor. B is problematic since the superior Sharpe
ratio may be due to measurement error – the true Sharpe ratio is likely to be less. B is likely
to be favourable in terms of Treynor as the beta estimate of 1.2 is likely to be greater than
the true value – hence it matches the market Treynor ratio or is better than the market. For
investors that are well-diversified you may recommend both A and B, but for investors that
are looking for a stand-alone investment you can really only be sure about A.
The examiners awarded 2 marks for identifying the problem with the data, 4 marks for
working out the Sharpe and Treynor ratios of A and B, and 3 marks for making the
recommendations.
(c) (Note that the original examination paper had a typo: the column market ‘Idiosyncratic
variance’ was incorrectly labelled ‘Total variance’ which implies negative idiosyncratic risk
for portfolios A, B and C.)
Since portfolios A, B and C are identical in terms of risk decomposition we put weights x in
each, and the remaining 1 − 3x in D. Regardless of how the portfolio is mixed, the beta of
the mixed portfolio is 1.2, therefore the market risk of the mixed portfolio is
(1.2)2 × 0.09 = 0.1296. The idiosyncratic risk is:
x2 × 0.12 + x2 × 0.12 + x2 × 0.12 + (1 − 3x)2 × 0.2
and the objective is to choose x such that this expression is minimised. The first-order
condition is:
3 × 2 × x × 0.12 − 2 × 3 × (1 − 3x) × 0.2 = 0
which implies x = 0.278 which is equivalent to holding 27.8% in each of A, B and C and
16.7% in D. The total risk of this portfolio is:
0.1296 + 3 × (0.278)2 × 0.12 + (0.167)2 × 0.2 = 0.163.
The examiners awarded 6 marks for identifying the correct weighting scheme and 3 marks
for working out the total variance.
Question 6
(a) Explain how credit default swap instruments work. What primary use do credit
default swaps have?
(7 marks)
(b) Suppose the price of an asset is 110 with probability 0.5 and 90 with probability
0.5. There are uninformed noise traders present who buy or sell one unit of the
asset with equal probability, and informed traders who buy one unit if they
know the price is 110 and sell one unit if they know the price is 90. There are 9
uninformed traders for every informed trader. The market is cleared by a risk
neutral competitive market maker quoting prices at which the trading takes
place. What are the initial bid and ask quotes?
(9 marks)
(c) Suppose the mid-price in the market for a stock follows a random walk
mt = mt+1 + ut ,
where mt is the mid-price for transaction t and ut is a random error term which
is distributed iid. Suppose transaction prices are
pt = mt + qt c,
10
Examiners’ commentaries 2016
where pt is the transaction price and qt is an iid binary +1/ − 1 variable.
Outcome +1 represents a transaction at the ask and outcome −1 a transaction
at the bid. The two outcomes are equally likely. The constant c is the
half-spread. Demonstrate that the half-spread c equals the square root of the
negative of the covariance between two successive price changes ∆pt = pt − pt−1
and ∆pt−1 = pt−1 − pt−2 .
(9 marks)
Reading for this question
Chapter 3 in the subject guide, and Chapter 18 in the Stultz text provide background readings
for this question.
Approaching the question
(a) The examiners awarded 4 marks for an explanation of how CDS instruments work, and a
further 3 marks for explaining their primary use.
(b) The conditional probability of buy given high value is:
90% × 50% + 10% × 100% = 55%
since the market-maker expects the noise traders to buy and sell with equal probability and
the informed traders to buy for sure in this case. Therefore, the conditional probability of
sell given high value is 45%. The conditional probability we are looking for at the bid is
therefore:
P (buy and high value)
.
P (high value | buy) =
P (buy)
We know that:
P (buy)
= P (buy | high value) P (high value) + P (buy | low value) P (low value)
=
55% × 50% + 45% × 50%
=
50%.
Therefore:
55% × 50%
= 55%.
50%
Similarly, the conditional probability of sell given low value is 55% (using the symmetrical
argument above), and the conditional probability of buy given low value is 45%. We know
that:
P (sell) = 45% × 50% + 55% × 50% = 50%
P (high value | buy) =
hence:
55% × 50%
= 55%.
50%
The ask price (at which the market-maker sells) is therefore 55% × 110 + 45% × 90 = 101
and the bid price (at which the market-maker buys) is 55% × 90 + 45% × 110 = 99.
P (low value | sell) =
The examiners awarded 3 marks for applying Bayes formula and 6 marks for working out
the bid and the ask prices.
(c) The expression for the relevant covariance is:
Cov(∆pt , ∆pt−1 )
=
Cov(pt − pt−1 , pt−1 − pt−2 )
=
Cov(ut + (qt − qt−1 )c, ut−1 + (qt−1 − qt−2 )c)
= c2 Cov(qt − qt−1 , qt−1 − qt−2 )
=
−c2 Cov(qt−1 , qt−1 )
=
−c2 Var(qt−1 )
=
−c2
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FN3023 Investment management
since m and q are iid and uncorrelated with any other variable, and since qt is uncorrelated
over time. Therefore, the half-spread is equal to the square root of the negative of the
covariance of successive price changes.
The examiners awarded 3 marks for recognising the expression for the price changes, 3
marks for applying the covariance formula correctly, and 3 marks for finally demonstrating
the answer.
Question 7
(a) Explain how risk capital allocation is carried out in bank systems that have
adopted the Basel framework of bank regulation. How can capital requirements
make the banking system less fragile?
(7 marks)
(b) Suppose the stocks follow a 2-factor structure. You are given the following data:
Asset
Risk free asset
Fund A
Fund B
Fund C
Avg return
3%
10%
12%
?
Factor-1 beta
0
1.2
0.8
?
Factor-2 beta
0
0.5
1.3
0.9
Fund A and B are well diversified. Fund C consists of a linear combination of A
and B only. What is your best estimate of the missing values for fund C?
(9 marks)
(c) Suppose the term structure of interest rates can be estimated correctly from the
following 1-year, 2-year and 3-year bond data given in the table (the coupons are
paid annually and the yield-to-maturity is expressed with annual compounding).
Bond
1-year bond
2-year bond
3-year bond
Yield-to-maturity
5%
5%
4.9%
Coupon rate
5%
5%
0%
Suppose you can enter an agreement now to borrow 1,000 at the beginning of
year 3 and repay 1,050 at the end of year 3. Would you take this borrowing
opportunity?
(9 marks)
Reading for this question
Chapter 9 in the subject guide, and Chapter 4 in the Stultz text provide the background
readings for this question.
Approaching the question
(a) Chapter 9 in the subject guide, and Chapter 4 in the Stultz text provide the background
readings for this question.
(b) Since C is a linear combination of A and B we can identify the weights from factor-2 beta,
where factor-2(C) = x factor-2(A) + (1 − x) factor-2(B), or 0.9 = 0.5x + 1.3(1 − x), which
implies x = 0.5. Using the data in the table we find, therefore, the average return for C as
0.5(10%) + 0.5(12%) = 11% and factor-1 beta as 0.5(1.2) + 0.5(0.8) = 1.0.
The examiners awarded 3 marks for identifying the weights and 3 marks each for working
out the average return and factor-1 beta for C.
12
Examiners’ commentaries 2016
(c) The one-year spot rate is 5%, the yield-to-maturity of the 2-year bond is 5%, and removing
the one-year spot rate we find that the 2-year spot must also be 5%. The 3-year spot is
simply the yield-to-maturity since the 3-year bond is a zero-coupon bond. Therefore, the
implied one-year forward rate between year 2 and 3 is:
(1 + r2 )2 (1 + f ) = (1 + r3 )3
and the implied borrowing rate 1050/1000 − 1 = 5% must be compared to f . In this case,
since r2 = 5% > r3 = 4.9%, it must be the case that f < 5% and the implied borrowing
opportunities in the market are cheaper than the borrowing opportunity on offer. The offer
should therefore be declined.
An alternative way is to use net present value. The net present value of the borrowing
opportunity is:
1050
1000
−
<0
NPV =
(1 + r2 )2
(1 + r3 )3
when using 5% for the 2-year spot and 4.9% for the 3-year spot. Since the net present value
is negative the offer should be declined.
The examiners awarded 2 marks each for identifying the 2-year and 3-year spot rates, 4
marks for the net present value calculation (or its equivalent argument), and 1 mark for
interpreting the result correctly.
Question 8
(a) Explain the problem of hedging non-linear risk. Why does hedging over short
time interval using the current hedge ratio reduce this problem?
(7 marks)
(b) The Treynor–Black model dictates that it is optimal to mix an active portfolio
(with Sharpe ratio exceeding that of the market portfolio) with the passive
market portfolio with weight w and 1 − w, respectively, where
w=
αA
(eA )
αA (1 − βA ) + (ErP − rF ) Var
σ2
P
Here, αA is Jensen’s alpha of the active portfolio, βA is the beta of the active
portfolio, ErP is the expected return on the passive portfolio, rF is the risk free
2
rate, Var(eA ) is the idiosyncratic risk of the active portfolio, and σP
is the
variance of the passive portfolio. Suppose the expected return of the active
portfolio is 10%, total variance is 0.10, and the beta is 0.8. Also suppose the
risk free rate is 3%, the expected return on the passive portfolio is 8%, and the
variance of the portfolio is 0.09. What is the optimal portfolio according to the
Treynor–Black model?
(9 marks)
(c) A company has a pension liability under which payments of $900,000 need to be
made at the end of years 6, 7, 8, 9 and 10. The term structure of interest rate is
flat at 5%. The company seeks to immunize the liability with respect to small
changes in the interest rates. Explain how much the firm should buy or sell of a
fixed interest portfolio, and at what duration, to achieve this objective.
(9 marks)
Reading for this question
Chapter 9 in the subject guide, and Chapter 13 in the Stultz text provide background to this
question.
13
FN3023 Investment management
Approaching the question
(a) The examiners awarded 7 marks for an explanation of hedging non-linear risk.
(b) The missing ingredients in the formula are the alpha and the variance of the idiosyncratic
risk term for the active portfolio. The alpha is:
αA = 0.10 − (0.03 + 0.8(0.08 − 0.03)) = 0.03
and the variance term is:
Var(eA ) = 0.10 − (0.8)2 0.09 = 0.0424.
Putting these into the formula we find:
w=
0.03
= 1.02
0.03(1 − 0.8) + (0.08 − 0.03) 0.0424
0.09
which implies an investment of 102% of wealth in the active portfolio and shorting of 2% of
the passive portfolio.
The examiners awarded 3 marks each for the alpha calculation and the missing variance
term, and 3 marks for the final answer plus its interpretation.
(c) The value of the liability is:
(1.05)−5
900000
0.05
1−
1
(1.05)5
= 3053032
and the duration is:
6(900000)
(1.05)6
+
7(900000)
(1.05)7
+ ··· +
3053032
10(900000)
(1.05)10
= 7.90.
This implies that the firm should buy bonds of value x with duration y such that
xy = 3053032 × 7.90, for instance if the duration of the bond portfolio is 10, then the firm
needs to invest in x = (3053032 × 7.90)/10 in order to immunise the liability against interest
rate movements. Any combination will however do.
The examiners awarded 3 marks for working out the value of the liability, 3 marks for
working out the duration, and 3 marks for outlining the details of a suitable hedging
strategy.
14
Examiners’ commentaries 2016
Examiners’ commentaries 2016
FN3023 Investment management
Important note
This commentary reflects the examination and assessment arrangements for this course in the
academic year 2015–16. The format and structure of the examination may change in future years,
and any such changes will be publicised on the virtual learning environment (VLE).
Information about the subject guide and the Essential reading
references
Unless otherwise stated, all cross-references will be to the latest version of the subject guide (2016).
You should always attempt to use the most recent edition of any Essential reading textbook, even if
the commentary and/or online reading list and/or subject guide refer to an earlier edition. If
different editions of Essential reading are listed, please check the VLE for reading supplements – if
none are available, please use the contents list and index of the new edition to find the relevant
section.
Comments on specific questions – Zone B
Candidates should answer FOUR of the following EIGHT questions. All questions carry equal
marks.
Question 1
(a) Explain why an optimal portfolio for investors who are variance averse consists
of only two assets. What are the two assets? How is variance aversion different
from risk aversion?
(7 marks)
(b) A 5-year bond with face value $1,000 and coupon rate 5% is trading at par
value. What is the yield-to-maturity of the bond? What is the duration of the
bond?
(9 marks)
(c) What is the convexity of the bond in (b)? Suppose the yield-curve is flat and
the yield-to-maturity increases by 0.1% from its current level (i.e. 10 basis
points). Based on your answers in (b) and (c), what is the approximate change
in the $-value of the bond?
(9 marks)
Reading for this question
Chapter 6 in the subject guide and Chapters 13 and 20 in the Elton/Gruber/Brown/Goetzmann
text provide background readings for this question.
15
FN3023 Investment management
Approaching the question
(a) The examiners awarded 4 marks for explaining the two-fund separation property of the
optimal portfolios, 2 marks for the two assets involved, and 1 mark for explaining how
variance aversion is a special form of risk aversion.
(b) The value is 1,000 since the bond is trading at par, which implies the yield-to-maturity is
equal to the coupon rate (i.e. 5%). The duration is:
50
1.05
+
2×50
(1.05)2
+ ··· +
5×1050
(1.05)5
1000
= 4.546.
The examiners awarded 2 marks for the yield-to-maturity, and 7 marks for the duration.
(c) The convexity is:
1×2×50
(1.05)3
+
2×3×50
(1.05)4
+ ··· +
5×6×1050
(1.05)7
= 23.936.
1000
The change in yield is 0.001. The approximate %-change in the value of the bond is:
∆P
Duration
1
=−
× 0.001 + × Convexity × (0.001)2 = −0.00432
P
1.05
2
so that the $-change in the bond is 1000 times the %-change, or −$4.32.
The examiners awarded 5 marks for the convexity and 4 marks for working out the
approximate change in the $-value of the bond.
Question 2
(a) Explain how corporations approach the issue of managing risk. What is the aim
of corporate risk management, and how do corporations create value through
risk management?
(7 marks)
(b) You are given the following data on 6-month put and call values on a stock. The
stock is currently trading at a value of £10.
Exercise price
£9.5
£10
£10.5
Call price
£0.948
£0.706
£0.467
Put price
£0.253
£0.510
£0.771
Can you identify an arbitrage opportunity? Explain.
(9 marks)
(c) Suppose you have invested in 100,000 shares of a stock currently trading at £10
per share. The 6-month risk free interest rate is 2%, the 6-month call option on
the stock with exercise price £10 is trading at £0.706, and the corresponding
put option is trading at £0.510. Construct a new portfolio which consists of the
stock, a risk free investment, and put options on the stock such that: (i) the
initial value of the portfolio is the same as your current portfolio; (ii) the value
in 6 months’ time will never fall below the initial value of the portfolio; and (iii)
the portfolio retains as much as possible of the up-side return of the stock in
(b). Specify the exact amounts invested in the three assets.
(9 marks)
Reading for this question
Chapter 9 in the subject guide, and Chapter 3 in the Stultz text provide background readings for
this question.
16
Examiners’ commentaries 2016
Approaching the question
(a) The examiners awarded 4 marks for explaining the objective of risk management in
mitigating the effects of shortfall of funds, and 3 marks for explaining how shortfall can lead
to deadweight costs for the firm. Avoiding such costs creates value through risk
management.
(b) Using put-call parity, we know that P V (X) = S + P − C, and we also know that
P V (X) = X/(1 + r) where r is the 6-month rate. Therefore:
P V (9.5) = 10 + 0.253 − 0.948
which implies a risk-free rate r = 2.0956 over the 6-month period. Doing the same with
P V (10) and P V (10.5), which yield risk-free rates of r = 1.9992 and r = 1.9022, respectively,
we find arbitrage opportunities by borrowing at the cheapest rate (best available rate is
1.9022) and lending at the most expensive rate (best available rate is 2.0956). This holds
regardless of what the actual risk-free rate is – if it is less than the best cheapest rate or
greater than the best expensive rate the arbitrage transaction would become more attractive
by including the risk-free asset. The arbitrage profit increases by the amount of capital
used, and decreases by the transaction costs incurred.
The examiners awarded 6 marks for identifying the implied risk-free rates, and 3 marks for
specifying the arbitrage transaction.
(c) The portfolio has value 100,000 shares times the price 10 per share, which equals 1,000,000.
To generate the floor of the investment strategy, invest P V (10) per share in the risk-free
asset. This leaves 10 − P V (10) = 10 − 10/1.02 to be invested in calls with exercise price 10
which provides the upside potential of the investment. This implies:
10 − P V (10)
= 0.2777
0.706
call options per share. By put-call parity, each call can be replicated by an investment in the
stock, a risk-free investment, and a put option: C = S + P − P V (X). Therefore, for each
share, invest P V (10) in the risk-free investment, and invest 0.2777 shares in the stock,
0.2777 puts, and borrow 0.2777 × P V (10) risk-free. In total, therefore, for each share there
is (1 − 0.2777) invested risk-free in P V (10), 0.2777 shares of the stock, and 0.2777 puts.
Multiplying by 100,000 shares, the total position is 708,106 invested risk-free, 277,731
invested in stocks, and 14,164 invested in puts.
The examiners awarded 5 marks for the basic investment strategy involving the risk-free
investment plus the correct investment in calls, and 4 marks for converting the call
investment into an investment involving stocks, puts and risk-free borrowing.
Question 3
(a) What do certificates and deposits have in common with commercial papers? In
what ways are they different?
(7 marks)
(b) Suppose a bond is currently quoted at a price of $1,025. The face value is
$1,000 and the coupon rate 3%. The number of days since the last coupon is 49,
and the number of days between coupons is 365. If you purchase this bond,
what is the actual transaction price?
(9 marks)
(c) An investor has preferences over portfolios that can be expressed as a function
of expected return (m) and portfolio variance (s2 ) only: U (m, s2 ) = m − 0.8s2 .
The risk free rate of return is 3% and the average return on the market index is
8%. The standard deviation of the market index is 30%. What is the optimal
portfolio for the investor?
(9 marks)
17
FN3023 Investment management
Reading for this question
Chapter 2 in the subject guide, and Chapter 2 in the Elton/Gruber/Brown/Goetzmann text
provide background readings for this question.
Approaching the question
(a) The examiners awarded 4 marks for explaining the similarity of these instruments to other
money market instruments, and 3 marks for explaining the key difference in issuing body for
the two instruments.
(b) Accrued interest is:
1000 × 0.03 × 49
= 4.027.
365
The relationship between the quoted price and the transaction price is that the transaction
price (dirty price) = quoted price (clean price) + accrued interest. Therefore, the
transaction price is 1025 + 4.03 = 1029.03.
The examiners awarded 5 marks for working out the accrued interest and 4 marks for the
transaction price.
(c) First, the investor is variance averse which implies that the optimal portfolio consists of a
mixture of the risk-free asset and the market portfolio. We can then exclude all other
portfolios from consideration. The expected return on the mixture portfolio is:
m = 0.03(1 − x) + 0.08x = 0.03 + 0.05x.
The variance of the mixture portfolio is:
s2 = (0.3)2 x2 .
The weight on the risk-free asset is 1 − x and the weight on the market portfolio is x. The
expected utility is therefore:
U = 0.03 + 0.05x − 0.8(0.09)x2 .
Differentiating the expected utility with respect to x to maximise the expression, we find the
first-order condition is:
0.05 − 2(0.8)0.09x = 0
or
x=
0.05
= 0.35.
2(0.8)0.09
Therefore, the optimal portfolio has 65% invested risk-free and 35% invested in the market
portfolio.
The examiners awarded 4 marks for the argument about two-fund separation and the
derivation of the expression for expected utility, and 5 marks for the derivation of the
optimal weights.
Question 4
(a) Explain why it is difficult to estimate correct Sharpe ratios for funds when the
fund manager engages in market timing where he moves the fund’s capital in
and out of risky assets often?
(7 marks)
(b) You are given the following data.
Fund
Risk free asset
Market index
Portfolio A
Portfolio B
18
Average return
3%
8%
12%
9%
Standard deviation
0
30%
50%
30%
Beta
0
1
1.4
1.2
Examiners’ commentaries 2016
Are the data in the table reasonable? What is the Sharpe ratio and the Treynor
ratio for A and B? Would you recommend these two portfolios to investors?
(9 marks)
(c) You are given the following data on 4 stocks A, B, C, and D.
Stock
A
B
C
D
Beta
1.2
1.2
1.2
1.2
Idiosyncratic variance
0.12
0.12
0.12
0.2
The total variance of the market portfolio is 0.09. Assume the idiosyncratic risk
is independent across the four stocks. What portfolio consisting of the four
stocks has the smallest total variance? What is the idiosyncratic risk of this
portfolio?
(9 marks)
Reading for this question
Chapter 8 in the subject guide provides the background readings for this question.
Approaching the question
(a) The examiners awarded 2 marks each for the explanation that the Sharpe ratio aggregates
the excess return of fund performance linearly, and the standard deviation of fund
performance non-linearly. They awarded a further 3 marks for recognising that the high-risk
periods will be given higher weight in the standard deviation estimate than the low-risk
periods, and therefore the denominator in the Sharpe ratio is overestimated, and the Sharpe
ratio itself underestimated.
(b) We notice that portfolio B has the same standard deviation as the market index, but with
beta greater than 1. This implies negative idiosyncratic risk which is unreasonable – hence
it is likely there is some estimation error in the table where the standard deviation of B is
greater than the estimate of 30% or the beta is less than the estimate of 1.2.
Using the data available, Sharpe(A) = (0.12 − 0.03)/0.5 = 0.18, Sharpe(B) =
(0.09 − 0.03)/0.3 = 0.2, and Sharpe(Market) = (0.08 − 0.03)/0.3 = 0.167. Similarly,
Treynor(A) = (0.12 − 0.03)/1.4 = 0.064, Treynor(B) = (0.09 − 0.03)/1.2 = 0.05, and
Treynor(Market) = 0.08 − 0.03 = 0.05.
A is favourable in terms of Sharpe and Treynor. B is problematic since the superior Sharpe
ratio may be due to measurement error – the true Sharpe ratio is likely to be less. B is likely
to be favourable in terms of Treynor as the beta estimate of 1.2 is likely to be greater than
the true value – hence it matches the market Treynor ratio or is better than the market. For
investors that are well-diversified you may recommend both A and B, but for investors that
are looking for a stand-alone investment you can really only be sure about A.
The examiners awarded 2 marks for identifying the problem with the data, 4 marks for
working out the Sharpe and Treynor ratios of A and B, and 3 marks for making the
recommendations.
(c) (Note that the original examination paper had a typo: the column market ‘Idiosyncratic
variance’ was incorrectly labelled ‘Total variance’ which implies negative idiosyncratic risk
for portfolios A, B and C.)
Since portfolios A, B and C are identical in terms of risk decomposition we put weights x in
each, and the remaining 1 − 3x in D. Regardless of how the portfolio is mixed, the beta of
the mixed portfolio is 1.2, therefore the market risk of the mixed portfolio is
(1.2)2 × 0.09 = 0.1296. The idiosyncratic risk is:
x2 × 0.12 + x2 × 0.12 + x2 × 0.12 + (1 − 3x)2 × 0.2
19
FN3023 Investment management
and the objective is to choose x such that this expression is minimised. The first-order
condition is:
3 × 2 × x × 0.12 − 2 × 3 × (1 − 3x) × 0.2 = 0
which implies x = 0.278 which is equivalent to holding 27.8% in each of A, B and C and
16.7% in D. The total risk of this portfolio is:
0.1296 + 3 × (0.278)2 × 0.12 + (0.167)2 × 0.2 = 0.163.
The examiners awarded 6 marks for identifying the correct weighting scheme and 3 marks
for working out the total variance.
Question 5
(a) What is a hedge fund? What is a consensus trade, and why can consensus
trades create problems in financial markets?
(7 marks)
(b) According to the Treynor–Black model the optimal mix of the passive market
portfolio P and an active superior portfolio A is given by the following formula:
w=
αA
(eA )
αA (1 − βA ) + (ErP − rF ) Var
σ2
,
P
where αA is Jensen’s alpha and βA is the beta of the active portfolio,
respectively; ErP and rF are the expected return on the passive market
portfolio and the risk free asset, respectively; Var(eA ) is the idiosyncratic risk of
2
the active portfolio and σP
is the variance of the market portfolio. You have the
following data: rF = 3%; ErP = 8%, the expected return of the active portfolio
2
ErA = 10%, βA = 0.9, σP
= 0.09 and the variance of the active portfolio
2
σA = 0.10. Work out the optimal mix of the active and the passive portfolio,
and explain what this result means.
(9 marks)
(c) The term structure of interest rates is 5% for all maturities. A firm has a
pension liability under which it needs to pay $10m each year from year 11
through year 15. What is the duration of this liability? The firm seeks to
reduce the duration of this liability to 5 by trading bonds with duration 10.
What position should the firm take to achieve this target?
(9 marks)
Reading for this question
Chapter 4 in the subject guide, Chapter 1 in the Lo text and the MacKenzie article from the
London Review of Books provide background readings for this question.
Approaching the question
(a) The examiners awarded 3 marks for explaining the nature of hedge funds, and 2 marks each
for explaining what a consensus trade is and how such a trade can cause problems.
(b) The missing variables in the formula are the alpha of the active portfolio and the variance of
the idiosyncratic term of the active portfolio. We find:
αA = 0.10 − (0.03 + 0.9(0.08 − 0.03)) = 0.025
and:
Var(eA ) = 0.10 − (0.9)2 0.09 = 0.0271.
20
Examiners’ commentaries 2016
Plugging these into the formula we find:
w=
0.025
= 1.424.
0.025(1 − 0.9) + (0.08 − 0.03) 0.0271
0.09
This implies the investor holds 142.4% of their wealth in the active portfolio and shorts
42.4% of the passive portfolio. This portfolio will then be mixed with the risk-free portfolio
depending on the investor’s risk aversion.
The examiners awarded 3 marks each for the missing variables in the formula, 2 marks for
working out the weights and 1 mark for explaining what the number means.
(c) The value of the liability is:
(1.05)−10
and the duration is:
11×10m
(1.05)11
+
10m
(1 − (1.05)−5 ) = 26.6m
0.05
12×10m
(1.05)12
+ ··· +
15×10m
(1.05)15
= 12.9.
26.6m
The asset side needs to be balanced by an investment of x in a bond portfolio with duration
10. The liability side consists of 26.6m in pension liability with duration 12.9, and equity
with value x − 26.6m and desired duration 5. Therefore, to achieve the target we solve
10x = 12.9 × 26.6m + 5(x − 26.6m), which has solution x = 42.03.
The examiners awarded 3 marks each for working out the value of the liability and the
duration, and 3 marks for finding the optimal investment in the bond portfolio.
Question 6
(a) Extreme risk deals with so called extreme events. What characterises extreme
events? Why do we sometimes care more about extreme events than with the
everyday risk we are exposed to?
(7 marks)
(b) Demonstrate that a portfolio of x calls and y puts on a stock S has value V
where
V = S (xN (d1 ) − yN (−d1 )) − P V (X) (xN (d2 ) − yN (−d2 )) ,
where the call value C and the put value P are given by
C = SN (d1 ) − P V (X)N (d2 )
P = P V (X)(1 − N (d2 )) − S(1 − N (d1 ))
and where P V (X) is the present value of the exercise price X, N () is the
standard normal distribution function, and d1 , d2 are parameters that depend
on the time to maturity, the risk free rate of return, and the volatility of the
stock. Explain how you can make use of such a portfolio to form a hedge against
changes in the volatility of the stock.
(9 marks)
(c) You are given the following returns on 5 stocks over the most recent 3-month
period.
Stock
3-month return
A
2%
B
−4%
C
5%
D
4%
E
−2%
Suppose you wish to construct a zero-investment position in these 5 stocks
exploiting momentum effects. According to a scheme outlined in the study
guide, what are the weights in your position?
(9 marks)
21
FN3023 Investment management
Reading for this question
Chapter 9 in the subject guide, and the ‘Reader Guidelines’ of the
Embrechts/Kluppelberg/Mikosch text provide background readings for this question.
Approaching the question
(a) The examiners awarded 6 marks for the characterisations of extreme events and 1 mark for
why we should worry about such risk.
(b) The portfolio is:
xC + yP
= x (SN (d1 ) − P V (X)N (d2 )) + y (P V (X)(1 − N (d2 ))) − S (1 − N (d1 ))
= S (xN (d1 ) − y(1 − N (d1 ))) − P V (X) (xN (d2 ) − y(1 − N (d2 )))
= S (xN (d1 ) − yN (−d1 )) − P V (X) (xN (d2 ) − yN (−d2 ))
(the last equality follows from the fact that the standard normal distribution function is
symmetrical around 0). A volatility hedge should ideally depend on the volatility but not on
other variables, which motivates creating a portfolio such that the first term vanishes.
Therefore, set:
N (−d1 )
x
=
.
xN (d1 ) = yN (−d1 ),
⇒
y
N (d1 )
This portfolio is insensitive to small changes in S but will respond positively (for x, y
positive) to increasing volatility and negatively to decreasing volatility.
The examiners awarded 3 marks for the derivation of the formula, 3 marks for explaining
the optimal volatility hedge, and 3 marks for explaining how the value responds to changes
in volatility.
(c) The equally-weighted index in the stocks would have earned:
1
(2% − 4% + 5% + 4% − 2%) = 1%
5
so the weights should reflect the difference between return and the return on the index. For
A, we get 1/5 × (2 − 1) = 1/5; for B, 1/5 × (−4 − 1) = −1; for C, 1/5 × (5 − 1) = 4/5; for D,
1/5 × (4 − 1) = 3/5; and for E, 1/5 × (−2 − 1) = −3/5. The long positions are
1/5 + 4/5 + 3/5 = 8/5 and the short positions −1 − 3/5 = −8/5 so the position is
self-financing. Any scaling of this scheme is permitted.
The examiners awarded 2 marks for the index return, 5 marks for working out the weights,
and 2 marks for noting the self-financing nature of the scheme.
Question 7
(a) What is market microstructure? Why does market microstructure matter to
investors?
(7 marks)
(b) Year 0 value of a stock is $12. In year 1 the stock pays a dividend of $0.5 per
share, and the value (ex-dividend) increases to $13. In year 2 the stock pays
another dividend of $0.5 per share, and the value (ex-dividend) goes back to
$12. In year 0 you buy 1,000 shares in the stock on a margin loan of 40% (you
should assume the initial margin and the maintenance margin are both 40%).
You increase your holding of the stock to 1,500 in year 1, and sell the entire
holding in year 2. What is the return of your investment?
(9 marks)
22
Examiners’ commentaries 2016
(c) A portfolio has annual log-return which is normally distributed with mean 10%
and standard deviation 30%. What is the 30-day, 1% value-at-risk for the
portfolio? You should give your answer in terms of log-returns, and assume the
30-day period consists of 20 trading days. Hint: For a standard normally
distributed random variable z, P r(z − 2.33) = 1%.
(9 marks)
Reading for this question
Chapter 5 in the subject guide, and Chapter 1 of the Lo text provide background readings for
this question.
Approaching the question
(a) The examiners awarded 4 marks for a characterisation of market microstructure and 3
marks for why it matters to investors.
(b) Year 0: buy 1,000 shares at $12 to an outlay of $12,000. Year 1 you receive dividends of $0.5
per share on your holding of 1,000 shares, which generates a cash in-flow of $500. You also
buy an additional 500 shares at a price of $13 to an outlay of $6,500. The net cash out-flow
in year 1 is $6,000. In year 2 you receive dividends of $0.5 per share on your holding of
1,500, which generates a cash in-flow of $750. You sell your holding of 1,500 shares at a
price of $12, which generates an additional cash-flow of $18,000. The total cash in-flow is
$18,750. The underlying cash-flows are (−12,000, −6,000, +18,750). The margin is 40%. In
year 0 the margin loan is $4,800; in year 1 your margin loan increases by $3,000, and the
margin loan is repaid in year 2. The margin cash-flows are (+4,800, +3,000, −7,800). The
return is the IRR of the cash-flow (−7,200, −3,000, +10,950), which is 4.2%.
The examiners awarded 3 marks each for the underlying cash-flows, the margin cash-flows,
and the IRR calculation.
q
30
30
and standard deviation 30% 365
(you can also use 20
(c) The log-return y has mean 10% 365
days in the numerator for the standard deviation as it sometimes is customary in practice to
assume the volatility comes on trading days only). The normalised log-return is therefore:
z=
30
y − 10% 365
q
30
30% 365
with mean 0 and variance 1, and the standard normal distribution function N (−2.33) = 1%
can be applied. Therefore, we need to solve:
−2.33 =
30
y − 10% 365
q
30
30% 365
which implies y = −19.2%. The probability of a greater loss than 19.2% of the portfolio’s
value is therefore 1%, so VaR = 19.2%.
The examiners awarded 9 marks for this argument.
Question 8
(a) Explain how the strategy of selling deep out-of-the-money options can distort
the measurement of investment performance for hedge funds. Why does this
strategy not work in the long run?
(7 marks)
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FN3023 Investment management
(b) Suppose an asset has value 110 with probability 0.5 and 90 with probability 0.5.
Noise traders buy or sell x units of the asset with equal probability, and an
informed trader with perfect knowledge of the asset value decides to buy y units
of the asset if the value is 110 and sell z units of the asset if the value is 90. A
risk neutral competitive market maker observes x and y or z, and sets a price p
which clears the market. Derive the expected profits for the informed trader as
a function of the absolute value of the noise traders’ trading volume |x|. Explain
why the profits are increasing in |x|.
(9 marks)
(c) Suppose the covariance between successive price changes between transactions
of a stock over a trading day is −0.015. Using the Roll model as basis for your
answer, explain why you would expect this number to be negative rather than
positive. Suppose the average price over the trading day is $18. What is your
best estimate, again based on Roll, of the spread in this market as a percent of
the price?
(9 marks)
Reading for this question
Chapter 4 in the subject guide and Chapter 1 in the Lo text provide background readings for this
question.
Approaching the question
(a) The examiners awarded 5 marks for explaining the distortions caused by such option selling,
and 2 marks for why they cannot pull this trick off in the long run.
(b) The insider must trade in the same quantities as the noise traders otherwise the
market-maker can infer the insider’s information from the insider’s trade. Therefore, if the
insider has good information there is a 50% chance the insider and the noise traders trade in
the same direction and the aggregate trade is 2|x|. In this case the market-maker infers
perfectly the insider’s information and sets price 110. The insider makes zero profits. There
is also a 50% chance the insider and the noise traders trade in opposite directions and the
aggregate trade is 0. In this case the market-maker cannot infer any information as it is
equally likely the insider is buying as selling. The market-maker sets the price as the
average of 90 and 110, i.e. 100. The insider makes profit of (110 − 100) = 10 per unit traded,
i.e. 10|x| profit in total, and this happens with probability 50%. Conditional on good
information, the expected profit is 5|x|. If the insider has bad information the same
argument applies, and the insider makes expected profits of 5|x|. Therefore, the
unconditional expected profit is 5|x|. The noise traders’ activity |x| can be interpreted as
‘liquidity’ so the insider makes greater profits in markets that are more liquid, which makes
intuitive sense.
The examiners awarded 2 marks each for the market-maker’s price setting strategy and the
insider’s trading behavior, 3 marks for expected trading profits, and 2 marks for explaining
why the insider makes greater profits in the more liquid markets.
(c) The covariance is negative because the transaction prices bounce between the ask and the
bid prices, which generates a risk with negative autocovariance. The fundamental price
movements we expect have zero autocovariance, so the magnitude of the negative covariance
can be used to predict the bid-ask spread. We know from Roll’s model that the half-spread:
p
c = −Cov(∆pt , ∆pt−1 )
√
so that with the data given c = 0.015 = 0.122. Therefore, the spread is twice this quantity,
or $0.245, which is approximately 1.36% of $18.
The examiners awarded 4 marks for the explanation of negative covariance, 4 marks for
applying the Roll equation, and 1 mark for expressing the spread as a percentage of the
average price.
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