Examiners’ commentaries 2015
Examiners’ commentaries 2015
FN3023 Investment management
Important note
This commentary reflects the examination and assessment arrangements
for this course in the academic year 2013–14. 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 refers 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.
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
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FN3023 Investment management
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.
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Examiners’ commentaries 2015
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 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 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 2015
FN3023 Investment management – Zone A
Important note
This commentary reflects the examination and assessment arrangements
for this course in the academic year 2014–15. 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 refers 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
Candidates should answer FOUR of the following EIGHT questions. All
questions carry equal marks.
A calculator may be used when answering questions on this paper and
it must comply in all respects with the specification given with your
admission notice. The make and type of machine must be clearly stated on
the front cover of the answer book.
Question 1
a. What is the difference between arithmetic average and geometric average
returns? Suppose you seek to compare the returns of various funds over a
period of time. Explain when you would make use of arithmetic averages
and when you would make use of geometric averages when making this
comparison. (7 marks)
Reading for this question
Appendix 1 in the subject guide, sections Averages: geometric vs
arithmetic, and Investment returns.
Approaching the question
Arithmetic average return is the ‘normal’ average, where we aggregate
the return over each period and divide by the number of periods, so for
n periods the arithmetic average is (1/n)(r1 + r2 + … + rn). Geometric
average return over n periods is the nth-root of the product of one plus
the period return minus one, so ((1+r1)(1+r2)…(1+rn))^(1/n) – 1. The
difference in economic terms is that the arithmetic average return is the
average return over the periods if you invest 1 at the start of each period,
and the geometric average return assumes you invest 1 at the start of
period 1, 1+r1 at the start of period 2, (1+r1)(1+r2) at the start of period
3, etc, so that what you have earned in the previous periods is aggregated
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Examiners’ commentaries 2015
into what you invest in the current period. If your main concern is what
an investor with one-period investment horizon will earn on average,
therefore, the arithmetic average is the most appropriate. If you main
concern is what an investor with n-period buy-and-hold strategy will earn,
the geometric return is the most appropriate. For cross-sectional averages,
or index returns, arithmetic return is the most appropriate.
Allocation of marks: A total of 4 marks were awarded for an
explanation of the two types (2 marks) and the difference between the
approaches (2 marks), and 3 marks for when geometric and arithmetic is
the most appropriate.
b. Suppose the value of a stock is either 110 or 90 with equal probabilities. A
market maker (who is risk neutral) is willing to trade at bid and ask prices
such that the expected trading profit of the market maker is zero whether
the next trade is at the bid or at the ask. The market maker assumes there
is 10% probability that the next trade is made by an informed trader who
knows perfectly the value of the stock, and that there is 90% probability that
the next trade is made by a noise trader who randomly buys and sells the
stock regardless of the value of the stock. The noise traders buy the stock
with probability 55% and sell with probability 45%. What are the bid and ask
prices set by the market maker? (9 marks)
Reading for this question
Chapter 5 in the subject guide, in the section called Glosten-Milgrom.
Approaching the question
The Glosten-Milgrom model outlines an approach for solving problems
of this kind. We need to work out the beliefs of the market maker,
conditional on observing the event that an investor wants to buy the
stock. Using Bayes’ rule, we find P(110|Investor Buys) = (P(Investor
Buys|110)P(110))/P(Investor Buys). Since P(Investor Buys|110) = 0.1
+ 0.9(0.55), i.e. the probability that the investor is informed (10%) plus
the probability the investor is uninformed and buys (90% times 55%),
the unconditional probability can be found using the law of iterated
conditional probabilities: P(Investor Buys) = P(Investor Buys|110)P(110)
+ P(Investor Buys|90)P(90) = (0.1 + 0.9(0.55))0.5 + (0.9(0.55))0.5.
Putting it all together we find P(110|Investor Buys) = 0.5459 and
therefore P(90|Investor Buys) = 1 – 0.5459 = 0.4541. The ask price
at which the investor buys is, therefore, 0.5459(110) + 0.4541(90) =
100.918. Similarly, P(110|Investor Sells) = (0.9(0.45)0.5)/(0.9(0.45)0.5
+ (0.1 + 0.9(0.45))0.5) = 0.4451, and P(90|Investor Sells) = 1 –
0.4451 = 0.5549. The bid price at which the investor sells is, therefore,
0.4451(110) + 0.5549(90) = 98.902.
Allocation of marks: The examiners awarded 3 marks for the correct
application of Bayes’ rule, 3 marks for the correct derivation of the ask
price, and 3 marks for the correct derivation of the bid price.
c. You are given the following data on two funds (A and B) and the market
portfolio (M). The risk free rate is 5%.
Fund
Average Return
Beta
Standard Deviation
A
10%
0.8
0.2
B
12%
0.9
0.35
M
12%
1.0
0.3
Work out the Sharpe ratio and Jensen’s alpha for the three funds. If you were
an investment manager advising a client about his stock market investments,
what would your advice be on the basis of your results? (9 marks)
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FN3023 Investment management
Reading for this question
Chapter 8 in the subject guide, in the sections called The Sharpe ratio and
More portfolio performance measures.
Approaching the question
Sharpe ratio of A is (0.10 – 0.05)/0.2 = 0.25. Sharpe ratio for B is (0.12 –
0.05)/0.35 = 0.2. Sharpe ratio for the market index is (0.12 – 0.05)/0.3 =
0.235. Jensen’s alpha for A is 0.10 – (0.05 + 0.8(0.12 – 0.05)) = -0.006.
Jensen’s alpha for B is 0.12 – (0.05 + 0.9(0.12 – 0.05)) = 0.007.
A appears a good investment opportunity as a stand-alone investment
but not as part of a well-diversified portfolio. This is inconsistent so there
is reason to doubt the estimates (all portfolios above the capital market
line should also lie above the security market line – here A is above
the capital market line but below the security market line). You should
therefore make further investigations into portfolio A before making
recommendations to clients. Portfolio B appears a good investment as part
of a well-diversified portfolio but not as a stand-alone investment.
Allocation of marks: There were 3 marks for the correct calculation of
the Sharpe ratio for A, B and the market, 3 marks for the corresponding
numbers for Jensen’s alpha, and 3 marks for the recommendation.
Question 2
a. Explain what we mean by futures trading, and why this form of trading is a
very efficient way for investors to transfer risk to and from other investors.
(7 marks)
Reading for this question
Chapter 3 in the subject guide, in the section called Recent financial
innovations, in the subsection called Futures trading.
Approaching the question
A futures contract is a standardised forward contract traded on a futures
exchange market (forward is an agreement to buy or sell the underlying
asset at a specific time in the future at a specific price). The contract is
marked-to-market each day, and it will involve very little up-front capital –
the buyer or seller must maintain margin payments to the clearing house.
The clearing house (futures exchange) will use the margin system to limit
counterparty risk. Therefore, a futures contract enables the buyers and
sellers to transfer risk cheaply and with minimal counterparty risk.
Allocation of marks: There were 4 marks for the description of the
contract, 1 mark for recognising it is marked-to-market, 1 mark each for a
description of the margin system and the limited counterparty risk.
b. Consider a 5-year bond with face value 1,000 and annual coupon rate
5% paid annually. The term structure is flat at 5% per annum annually
compounded, and the first coupon is due exactly one year from now. What is
the Macaulay duration of the bond?
(9 marks)
Reading for this question
Chapter 7 in the subject guide, in the section called Duration.
Approaching the question
The bond is trading at par because the yield-to-maturity is the same as
the coupon rate. Therefore the Macaulay duration is (1/1000)(50/1.05 +
100/1.052 + 150/1.053 + 200/1.054 + 5(1050)/1.055) = 4.546.
Allocation of marks: 9 marks were allocated.
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Examiners’ commentaries 2015
c. What is the convexity of the bond in b)? (9 marks)
Reading for this question
Chapter 7 in the subject guide, in the section called Convexity.
Approaching the question
Convexity is (1/2)(1/1000)(1(2)(50)/1.053 + 2(3)(50)/1.054 + 3(4)
(50)/1.0545 + 4(5)(50)/1.056 + 5(6)(1050)/1.057) = 11.968.
Allocation of marks: 9 marks were allocated.
Question 3
a. What do we mean by the term structure of interest rates? Why the term
structure is not flat at certain times? (7 marks)
Reading for this question
Chapter 7 in the subject guide, in the section called The term structure.
Approaching the question
The term structure of interest rates is the collection of spot rates for
various maturities. There are several hypotheses explaining the shape
of the term structure. The various spot rates represent the ‘time value of
money’ for various horizons, and the hypotheses explain why these are
different: The expectation hypothesis states that the forward rates implied
by the term structure are expected future spot rates; the liquidity premium
and money substitute hypothesis states that the short rates are formed on
the basis that the short spot rates represent the yield-to-maturity of bonds
that are very liquid or close substitutes to money, whereas the longer spot
rates are formed by other factors; and the segmentation hypothesis states
that the yield-to-maturity of bonds of various maturities are traded by
different segments of investors, and reflect the time value of money of
these varying segments.
Allocation of marks: 2 marks were allocated for the definition of term
structure, and 2 marks each for two descriptions of the hypotheses above,
and 1 mark for the third.
b. You consider investing 10% of your wealth in each of 8 stocks, and 20%
in the 9th stock. Each stock has a beta of 0.8, and the variance of the
idiosyncratic risk of each stock, which is independent across the stocks and
the market portfolio, is 0.1. The standard deviation of the market portfolio is
30%. What is the total variance of your portfolio? (9 marks)
Reading for this question
Chapter 6 in the subject guide, in the section called Diversification: The
single index model.
Approaching the question
The portfolio has beta 0.8. The idiosyncratic risk is var(0.1 e1 + 0.1 e2 +
0.1 e3 + … + 0.1 e8 + 0.2 e9) = (8(0.12) + 0.22)0.1 = 1.2%, where ei is
the idiosyncratic risk of stock i. Total variance is therefore 0.82(0.32) +
0.012 = 5.808%.
Allocation of marks: 5 marks were allocated for working out the
idiosyncratic risk of the portfolio, 2 marks for working out the systematic
risk, and 2 marks for applying the variance of a sum formula correctly.
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FN3023 Investment management
c. The one-year log-return of a portfolio is normally distributed with mean 10%
and standard deviation 20%. If x is a normal random variable with mean 0 and
standard deviation 1, Pr(x ≤ –2.33) = 1%. Work out the one-year 1% Value-atRisk for the portfolio. You should ignore discounting effects. Also note that
with a standard calculator you will not be able to work out the numerical
value, so your answer should be in implied form using the exponential
function. (9 marks)
Reading for this question
Chapter 9 in the subject guide, in the section called Put protection vs VaR.
Approaching the question
We know that ln(r) is normal with mean 0.1 and variance 0.22. This implies
that a normalised random variable y = (ln(r) – 0.10)/0.2 is normal with
mean 0 and variance 1. Therefore, we are looking for a value of y for
which N(y) = 0.01 or y = –2.33. This corresponds to a log-return level
–2.33 = (ln(r) – 0.10)/0.2 which implies ln(r) = –0.366. This corresponds
to a return r = e–0.366, so the one-year 1% VaR is 1 – e–0.366, approximately
30.65%.
Allocation of marks: 3 marks were awarded by recognising the
normalised variable y, 3 marks for working out the critical return threshold,
and 3 marks for the correct VaR expression.
Question 4
a. Hedge funds often take derivatives positions to boost their performance.
Explain how performance statistics for hedge funds can look better than what
their actual holdings should justify. (7 marks)
Reading for this question
Chapter 4 in the subject guide, in the section called Performance of hedge
funds.
Approaching the question
The practice the examiners were looking to having described is the practice
of selling deep out-of-the-money put options. This practice raises a steady
income stream through the sale price of the options, but since the option
are rarely exercised the liability is very infrequent. When looking at
performance data, therefore, the hedge fund’s income stream looks low risk
because the options are typically not exercised, but also high return because
the hidden risk of exercise is priced into the options. This practice can mask
the actual performance.
Allocation of marks: 4 marks were allocated for a description of the
practice, and 3 marks for the explanation of why this practice can mask the
underlying performance of the fund.
b. You buy 1,000 shares in a stock at a price of 100 per share. You hold the
portfolio until the end of the year when you collect a dividend payment of 5
per share. The stock price is at this point 95. You invest also at this point in
another 500 shares, which you hold for a further year. At the end of the second
year you collect another dividend payment of 6 per share. The stock price is at
this point 100, and you sell your entire holding. You trade on a margin account
that allows you to borrow up to 40% of the value of your portfolio at zero
interest. The maintenance margin is also 40% and you borrow maximally at all
times. What is the 2-year return on your investment? (9 marks)
Reading for this question
Chapter 2 in the subject guide, in the section called Working out the
profitability of margin trades.
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Examiners’ commentaries 2015
Approaching the question
The underlying cash flow is –100,000 in year 0 (buy 1,000 shares at a
price of 100 each); –42,500 in year 1 (receive dividend of 1,000 times
dividend per share of 5, but buy 500 more shares at a price of 95 each);
+159,000 in year 2 (receive dividend of 1,500 shares times dividend per
share of 6, and sell 1,500 shares at a price of 100 each).
The margin cash flow is +40,000 in year 0 (borrow 40% of the initial
purchase), +17,000 in year 1, (to get up to 40% of the total value of
portfolio of 142,500); and –57,000 in year 2 (repaying total loan).
The net cash flow is –60,000 in year 0; –25,500 in year 1; and +102,000
in year 2. The return is the IRR of this cash flow, which solves –60,000 –
25,500/(1+IRR) + 102,000/(1+IRR)2, which is 10.85% approximately.
Allocation of marks: 3 marks for identifying the underlying cash flow;
3 marks for working out the margin cash flow; and 3 marks for working
out the IRR.
c. You observe the following European option price data for 2-year call and put
options on a certain stock, currently valued at 100.
Exercise price
90
100
110
Calls
26.39
20.41
14.42
Puts
8.02
11.11
14.41
Can you find arbitrage opportunities in this market? Assume that you can
only buy or sell positions up to 100,000 in value, and pay a transaction cost
of 0.01% of the value of each position. (9 marks)
Reading for this question
The essential tool is put-call parity, which is derived in Chapter 9 in the
subject guide, in the section called Portfolio insurance with calls.
Approaching the question
Since we are not given the actual risk free rate we need to use put-call
parity to create a synthetic risk free investment (PV(X) = S + P – C). The
discount factor (1/(1+r)) is identified as the solution to PV(X) = X*(1/
(1+r)) and can be found for each case respectively as: PV(90) = 100 +
8.02 – 26.35 = 81.63 implies (1/(1+r)) = 0.907; PV(100) = 100 + 11.11
– 20.41 = 90.7 implies (1/(1+r)) = 0.907; and PV(110) = 100 + 14.41 –
14.42 = 99.99 implies (1/(1+r)) = 0.909.
The obvious strategy is therefore to borrow the maximum 100,000 at the
discount factor 0.909, and to invest at the discount factor 0.907. This
yields a cash flow x next year where 100,000 = 0.909x, which implies x
= 110,011. The necessary investment y is such that y = 110,011(0.907)
= 99,779.98. The difference is the gross profit 220.02. The transaction
costs are 0.0001(100,000) + 0.0001(99,779.98) = 19.98. The net profit is
therefore 220.02 – 19.98 which is positive so arbitrage profits exist.
Allocation of marks: 6 marks were awarded for identifying the
difference in the synthetic risk free rate, 2 marks for working out the gross
profits, and 1 mark for the net profits.
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Question 5
a. What do we mean by credit default swaps? Explain why these financial
instruments can be useful for investors. (7 marks)
Reading for this question
Chapter 3 in the subject guide, in the section called Recent financial
innovation, in the subsection called Credit default swaps.
Approaching the question
CDS are financial instruments in which the buyer pays a rate (which is
determined by market conditions) until a defined credit-event occurs.
Following this event the seller of the CDS must make payments to the
buyer according to the rules set out in the CDS contract. The credit-event
can range from a default on certain debt payments, a down-grade in
credit ratings, to a full blown bankruptcy. The CDS are useful as insurance
against losses of default on debt. A lender can, for instance, buy CDS
insurance to meet future credit related losses.
Allocation of marks: 5 marks are allocated for a description of the CDS
instrument, and 2 marks for an explanation of their usefulness.
b. Suppose you observe the following returns of 6 stocks over a 3-month
period.
Stock:
A
B
C
D
E
F
Return:
2.3%
-4.5%
5.2%
15.1%
-8.8%
7.8%
Suppose you wish to place a returns-based strategy aimed at capturing
momentum effects in stock returns. Using a scheme outlined in the subject
guide, what is the optimal position in the 6 stocks? (9 marks)
Reading for this question
Chapter 4, in the section called Return based trading strategies.
Approaching the question
We need to work out the 3-month return of an equally weighted index of
the 6 stocks: (1/6)(2.3 – 4.5 + 5.2 + 15.1 – 8.8 + 7.8)% = 2.85%. The
weight on A is therefore (1/6)(2.3 – 2.85) = –0.09167; on B (1/6)(–4.5 –
2.85) = –1.225; on C 0.39167; on D 2.04167; on E –1.94167; and on
F 0.825.
Allocation of marks: 3 marks allocated to working out the index
benchmark, and 6 marks to calculating the weights.
c. Suppose you have the following data on 3-month put and call options on a
stock market index, currently valued at 100. The option values are the BlackScholes values (BS values), which are given by Call value = S N(d1) – PV(X)
N(d2) and Put value = PV(X) N(-d2) – S N(-d1), where S is the current index
level and PV(X) is the 3-month discounted value of the exercise price X. N(d1)
and N(d2) are parameters that enter the valuation formula.
Option type:
Call
Put
Exercise price:
100
100
BS value:
6.583
5.341
N(d1):
0.562902928
0.562902928
N(d2):
0.503324481
0.503324481
Using these data, construct a volatility hedge consisting of the options in the
table that yields a positive payoff if there is an increase in the volatility of
the index at the same time as being insensitive to small changes in the value
of the index. (9 marks)
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Examiners’ commentaries 2015
Reading for this question
Chapter 9 in the subject guide, in the section called Hedging volatility.
Approaching the question
Since the portfolio should increase in value you need to buy options. The
portfolio consists of x units of the call and y units of the put, which equals
xSN(d1) – xPV(X)N(d2) + y(PV(X) + C – S) = xSN(d1) – xPV(X)N(d2)
+ yPV(X) + ySN(d2) – yPV(X)N(d2) – yS. To make the portfolio delta
neutral, we set the derivative with respect to S equal to zero, which yields
xN(d1) + yN(d2) – y = 0, or xN(d1) = y(1 – N(d2)). This implies that a
delta neutral portfolio must include (x/y) in the ratio (1 – N(d2))/N(d1).
This implies x/y = (1-0.503324481)/0.56290298 = 0.7765, (i.e. you buy
0.7765 calls for each put).
Allocation of marks: 4 marks allocated to working out the delta
neutral portfolio formula, 4 marks for the numbers, and 1 mark for the
interpretation of the result.
Question 6
a. Explain why the Sharpe ratio can give a misleading picture of performance
for a fund with time varying risk strategy, switching between high and low
risk in its portfolio. (7 marks)
Reading for this question
Chapter 8 in the subject guide, in the section called Changing risk.
Approaching the question
If there are both high risk and low risk periods in the evaluation period,
the average return for the fund is aggregated linearly so the numerator
in the Sharpe ratio is a linear average of the different periods, but the
denominator is aggregated non-linearly with the high risk periods
receiving disproportionately high weights. Therefore, the Sharpe ratio may
compare favourably with the market’s Sharpe ratio for each sub-period in
the evaluation period, but the aggregate Sharpe ratio may not. This is just
an artefact of the way standard deviation is calculated.
Allocation of marks: 4 marks allocated to the explanation of the
aggregation problem, and 3 marks for recognising how this affects the
Sharpe ratio.
b. For this question you should assume the term structure is flat at 5% per
annum annually compounded. You manage a pension fund for your company’s
employees where the current assets consist of a bond portfolio with
duration 10 years. The liabilities of the fund consist of expected withdrawals
of 100,000 annually in each of the next 10 years. The fund is currently in
deficit with the value of the current assets only half the value of the fund’s
liabilities. You need to increase the contribution to the fund’s assets in such
a way that the value of the assets matches the value of the liabilities and
the fund is immunized against changes in the discount rate. Work out the
duration and value of your investment of new contributions. (9 marks)
Reading for this question
Chapter 7 in the subject guide in the section called Duration.
Approaching the question
The value of the liability is (100,000/0.05)(1 – (1/1.05)10) = 772,173.
Duration is (1/772,173)(100,000/1.05 + 200,000/1.052 + … +
1,000,000/1.0510) = 5.1. The value of the existing bond is 0.5(772,173)
with duration 10. To bridge the duration gap, therefore, we need 0.5x =
0.5(10) = 5.1, which yields x = 1.98.
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Allocation of marks: 3 marks allocated to working out the value of the
liability, 2 marks for the duration, 2 marks for the duration gap equation,
and 2 marks for the duration of the new investment.
c. The stock market has a 2-factor structure where the risk of the two factors
is independent. The factor betas for factor 1 of two portfolios A and B are
known, but the factor betas for factor 2 are not. You have the following data:
Fund:
Beta factor 1
Beta factor 2
Total variance
A
1.2
?
10%
B
0.5
?
11%
The variance of factor 1 is 5% and the variance of factor 2 is 7% and both A
and B are well diversified. On the basis of the information given, what is your
guess of the factor 2 betas for the two funds? State clearly the assumptions
made. (9 marks)
Reading for this question
Chapter 6 in the subject guide, in the section called Factor models.
Approaching the question
We start out with the assumption that the portfolios are diversified to the
extent that they have zero idiosyncratic risk. This implies that var(A) =
10% = 1.22 var(f1) + x2 var(f2) where x is the factor beta on factor 2 for
portfolio A. Solving we find x = 0.63. Repeating for B we find var(B) =
11% = 0.52 var(f1) + y2 var(f2), where y is the factor beta on factor 2 for
B. Solving we find y = 1.18. If the portfolios have idiosyncratic risk then the
estimates are lower because the idiosyncratic risk feeds into the right hand
sides but not the left hand sides of the equations above, so the numbers we
have found are the upper bound for the factor betas on factor 2.
Allocation of marks: 2 marks allocated for specifying the critical
assumption that the idiosyncratic risk is zero, 3 marks each for identifying
the upper bonds, and 1 mark for commenting on the direction of the betas
if the idiosyncratic risk grows.
Question 7
a. Explain why Value-at-Risk is an appropriate risk management tool for banks. (7 marks)
Reading for this question
Chapter 9 in the subject guide, in the section called Risk management for
banks.
Approaching the question
Normally risk management is aimed at protecting a firm from a shortfall
in its ability to fund its investments/operations, because this is where the
firm incurs deadweight costs which cannot be recovered. Therefore, risk
management is typically aimed at managing the risk in the operational
cash flow of the firm, since it is shortfalls in cash flow that lead to a
liquidity crisis for the firm. For a firm where most of the assets are
liquid financial assets rather than illiquid real assets, such as banks, the
problem is much more related to protecting the value of the assets than
the operational cash flow. This is why risk management is focused on
values rather than cash flow, and VaR is designed precisely to measure the
likelihood of a shortfall in value of portfolios.
Allocation of marks: 4 marks allocated to recognising the primary
role of risk management in protecting the firm from liquidity crises, and
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Examiners’ commentaries 2015
3 marks for explaining why banks are different from firms which hold
illiquid real assets.
b. You have the following data on an investor:
Wealth of the Risk aversion
investor
coefficient
Risk free
rate
Expected return Variance of the
on the market
market index
index
2,000,000
5%
12%
2
10%
You should assume the investor has CARA utility function and the stock
market index has near normally distributed returns. What capital market
investments do you recommend for this investor? (9 marks)
Reading for this question
Chapter 6 in the subject guide, in the subsection called CARA utility and
normal returns.
Approaching the question
The utility function can be written as E(r) – (k/2) var(r), where r is the
return on the investor’s portfolio and k is the risk aversion coefficient.
Second, with the information given the optimal portfolio for the investor
has the 2-fund separation property: it will be a linear combination of the
risk free asset and the market portfolio. Therefore, the investor maximises
(1-x)rF + xErM – x2 var(rM) with respect to x (which is the fraction of
wealth invested in the market portfolio). The first order condition is x
= (1/2)(ErM – rF)/var(rM) = 0.35. Therefore, the investor invests 65%
(1,300,000) in the risk free asset and 35% (700,000) in the market index.
Allocation of marks: 1 mark allocating to recognising the 2-fund
separation property, 3 marks for deriving the optimal weight formula, 4
marks for working out the actual portfolio, and 1 mark for interpreting the
result.
c. The market index has expected return 12% and variance 10%. We call
this portfolio the passive portfolio. The risk free rate is 5%. There exists a
portfolio with expected return 13%, beta equal to 1, and variance 11%. We
call this portfolio the active portfolio. According to the Treynor-Black model,
the optimal weight in the active portfolio is given by w:
w = αA/(αA + (ErP – rF) Var(eA)/σP2)
where αA is Jensen’s alpha of the active portfolio, ErP is the expected return
of the passive portfolio, rF is the risk free rate, Var(eA) is the idiosyncratic risk
of the passive portfolio, and σP2 is the variance of the passive portfolio. If
you seek to invest 50% risk free and 50% in the active and passive portfolio,
what is your optimal investment in these portfolios? (9 marks)
Reading for this question
Chapter 6 in the subject guide, in the section called The Treynor-Black
model.
Approaching the question
Note: the denominator in the formula above should read (αA (1–βA) +
(ErP – rF) Var(eA)/σP2). All the numbers in the formula are given except
Var(eA) = 0.11 – 12 (0.10), so with this information we find w = 1.425.
This implies that the investor must replace a ‘normal’ market investment
with a portfolio which consists of 1.425 times A and (–0.425) times P.
Therefore, the investment of the investor is 0.5 times wealth in the risk
free asset, 0.713 times wealth in the active portfolio, and short 0.213 times
wealth in the passive portfolio.
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FN3023 Investment management
Allocation of marks: 3 marks allocated to working out the idiosyncratic
risk of the active portfolio, 4 marks for applying the formula to obtain the
optimal combination of the active and passive portfolio, and 2 marks for
interpreting the results into an optimal portfolio holding.
Question 8
a. Explain what we mean by a Treasury bill, a certificate of deposit and
commercial paper. What is the essential difference between the three classes
of securities?
(7 marks)
Reading for this question
Chapter 2 in the subject guide, in the section called Money market
instruments.
Approaching the question
All of these are money market instruments, or short term debt claims, and
the only way they differ is in terms of the issuing body. A T-Bill is issued by
the US government and has a credit risk that reflects the US government’s
ability to honor its debt; a certificate of deposit is issued by a private bank
and has a credit risk reflecting the bank’s ability to honour its debt; and
finally a commercial paper is issued by a corporation and has a credit risk
reflecting the corporation’s ability to honour its debt.
Allocation of marks: 1 mark awarded for a general recognition that
these are all money market instruments; 2 marks each for listing the three
claims’ issuing body.
b. Demonstrate, using Roll’s model, that the half-spread in a market c can
be written as c = (–cov(Δpt-1,Δpt))0.5, where Δpt is the market price change
between time t–1 and time t, and Δpt–1 is the market price change between
time t–2 and t-1. Explain the intuition for this result. (9 marks)
Reading for this question
Chapter 5 in the subject guide, in the section called Bid-ask bounce: The
Roll model.
Approaching the question
Fundamental prices are m, and the transaction price is p = m + qc where
the first term is the fundamental price and the second term is the spread
where q = +1 if the transaction is at the ask and q = –1 if the transaction
is at the bid. The derivation is cov(Δpt, Δpt–1) = cov(Δmt + Δqtc, Δmt–1 +
Δqt-1c) (using the decomposition of price changes into fundamental price
changes and price changes linked to the bid-ask bounce) = cov(Δqtc,
Δqt–1c) (assuming that fundamental price changes has zero serial
covariance) = c2 cov(Δqt, Δqt–1) (using the linearity of the covariance
operator) = c2 cov(qt – qt–1, qt–1 – qt–2) (using the definition of the change
in q) = c2 cov(–qt–1, qt-1) (assuming that q is serially independent) =
–c2 var(qt) (using the linearity of the covariance operator and the
definition of variance) = –c2 (0.5(1)2 + 0.5(–1)2) (calculating the
variance) = –c2. Hence c2 = –cov(Δpt, Δpt–1) and taking square roots on
both sides yields the result. The intuition is that the prices tend to bounce
up and down because of the bid-ask spread, and an up-movement is then
likely to be followed by a down-movement and vice versa. This generates
negative serial correlation, which is linked to the magnitude of the spread.
Allocation of marks: 3 marks were awarded for the decomposition
of transaction prices into fundamental changes and bid-ask bounces, 3
marks for the covariance calculus, and 3 marks for an explanation of the
intuition.
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Examiners’ commentaries 2015
c. The quoted price for a bond is 1,010 per 1,000 in face value. The coupon is
5% annual. It is 45 days since the last coupon payment. If you were to buy or
sell the bond, what is the transaction price of your deal (ignoring spreads)? (9 marks)
Reading for this question
Chapter 2 in the subject guide, in the section called Bond market
instruments.
Approaching the question
The quoted price (clean price) equals the transaction price (dirty price)
minus the accrued interest. Accrued interest is 0.05(1,000)(45/365) =
6.164, so the transaction price is 1,010 + 6.164 = 1,016.64.
Allocation of marks: 3 marks were awarded for recognising the
relationship between the quoted price and the transaction price, 3 marks
for the accrued interest calculation, and 3 marks for getting the final
numbers correct.
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FN3023 Investment management
Examiners’ commentaries 2015
FN3023 Investment management – Zone B
Important note
This commentary reflects the examination and assessment arrangements
for this course in the academic year 2014–15. 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 refers 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
Candidates should answer FOUR of the following EIGHT questions. All
questions carry equal marks.
A calculator may be used when answering questions on this paper and
it must comply in all respects with the specification given with your
admission notice. The make and type of machine must be clearly stated on
the front cover of the answer book.
Question 1
a. Explain what we mean by futures trading, and why this form of trading is a
very efficient way for investors to transfer risk to and from other investors. (7 marks)
Reading for this question
Chapter 3 in the subject guide, in the section called Recent financial
innovations, in the subsection called Futures trading.
Approaching the question
A futures contract is a standardised forward contract traded on a futures
exchange market (forward is an agreement to buy or sell the underlying
asset at a specific time in the future at a specific price). The contract is
marked-to-market each day, and it will involve very little up-front capital –
the buyer or seller must maintain margin payments to the clearing house.
The clearing house (futures exchange) will use the margin system to limit
counterparty risk. Therefore, a futures contract enables the buyers and
sellers to transfer risk cheaply and with minimal counterparty risk.
Allocation of marks: There were 4 marks for the description of the
contract, 1 mark for recognising it is marked-to-market, 1 mark each for a
description of the margin system and the limited counterparty risk.
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Examiners’ commentaries 2015
b. Consider a 5-year bond with face value 1,000 and annual coupon rate 5%.
The yield-to-maturity is also 5%, and the first coupon is due exactly one year
from now. What is the Macaulay duration of the bond? (9 marks)
Reading for this question
Chapter 7 in the subject guide, in the section called Duration.
Approaching the question
The bond is trading at par because the yield-to-maturity is the same as
the coupon rate. Therefore the Macaulay duration is (1/1000)(50/1.05 +
100/1.052 + 150/1.053 + 200/1.054 + 5(1050)/1.055) = 4.546.
Allocation of marks: 9 marks were allocated.
c. What is the convexity of the bond in b)? (9 marks)
Reading for this question
Chapter 7 in the subject guide, in the section called Convexity.
Approaching the question
Convexity is (1/2)(1/1000)(1(2)(50)/1.053 + 2(3)(50)/1.054 + 3(4)
(50)/1.0545 + 4(5)(50)/1.056 + 5(6)(1050)/1.057) = 11.968.
Allocation of marks: 9 marks were allocated.
Question 2
a. Hedge funds often take derivatives positions to boost their performance.
Explain how performance statistics for hedge funds can look better than
what their actual holdings should justify. (7 marks)
Reading for this question
Chapter 4 in the subject guide, in the section called Performance of hedge
funds.
Approaching the question
The practice the examiners were looking to having described is the
practice of selling deep out-of-the-money put options. This practice raises
a steady income stream through the sale price of the options, but since the
option are rarely exercised the liability is very infrequent. When looking
at performance data, therefore, the hedge fund’s income stream looks low
risk because the options are typically not exercised, but also high return
because the hidden risk of exercise is priced into the options. This practice
can mask the actual performance.
Allocation of marks: 4 marks were allocated for a description of the
practice, and 3 marks for the explanation of why this practice can mask
the underlying performance of the fund.
b. You buy 1,000 shares in a stock at a price of 100 per share. You hold the
portfolio until the end of the year when you collect a dividend payment of
5 per share. The stock price is at this point 95. You invest also at this point
in another 500 shares, which you hold for a further year. At the end of the
second year you collect another dividend payment of 6 per share. The stock
price is at this point 100, and you sell your entire holding. You trade on a
margin account that allows you to borrow up to 40% of the value of your
portfolio at zero interest. The maintenance margin is also 40% and you
borrow maximally at all times. What is the 2-year return on your investment? (9 marks)
Reading for this question
Chapter 2 in the subject guide, in the section called Working out the
profitability of margin trades.
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FN3023 Investment management
Approaching the question
The underlying cash flow is –100,000 in year 0 (buy 1,000 shares at a
price of 100 each); –42,500 in year 1 (receive dividend of 1,000 times
dividend per share of 5, but buy 500 more shares at a price of 95 each);
+159,000 in year 2 (receive dividend of 1,500 shares times dividend per
share of 6, and sell 1,500 shares at a price of 100 each).
The margin cash flow is +40,000 in year 0 (borrow 40% of the initial
purchase), +17,000 in year 1, (to get up to 40% of the total value of
portfolio of 142,500); and –57,000 in year 2 (repaying total loan).
The net cash flow is –60,000 in year 0; –25,500 in year 1; and +102,000
in year 2. The return is the IRR of this cash flow, which solves –60,000 –
25,500/(1+IRR) + 102,000/(1+IRR)2, which is 10.85% approximately.
Allocation of marks: 3 marks for identifying the underlying cash flow;
3 marks for working out the margin cash flow; and 3 marks for working
out the IRR.
c. You observe the following option price data for 2-year call and put options
on a certain stock, currently valued at 100.
Exercise price
90
100
110
Calls
26.39
20.41
14.42
Puts
8.02
11.11
14.41
Can you find arbitrage opportunities in this market? Assume that you can
only buy or sell positions up to 100,000 in value, and pay a transaction cost
of 0.01% of the value of each position. (9 marks)
Reading for this question
The essential tool is put-call parity, which is derived in Chapter 9 in the
subject guide, in the section called Portfolio insurance with calls.
Approaching the question
Since we are not given the actual risk free rate we need to use put-call
parity to create a synthetic risk free investment (PV(X) = S + P – C). The
discount factor (1/(1 + r)) is identified as the solution to PV(X) = X*(1/
(1 + r)) and can be found for each case respectively as: PV(90) = 100
+ 8.02 – 26.35 = 81.63 implies (1/(1 + r)) = 0.907; PV(100) = 100 +
11.11 – 20.41 = 90.7 implies (1/(1 + r)) = 0.907; and PV(110) = 100 +
14.41 – 14.42 = 99.99 implies (1/(1 + r)) = 0.909.
The obvious strategy is therefore to borrow the maximum 100,000 at the
discount factor 0.909, and to invest at the discount factor 0.907. This
yields a cash flow x next year where 100,000 = 0.909x, which implies x
= 110,011. The necessary investment y is such that y = 110,011(0.907)
= 99,779.98. The difference is the gross profit 220.02. The transaction
costs are 0.0001(100,000) + 0.0001(99,779.98) = 19.98. The net profit is
therefore 220.02 – 19.98, which is positive so arbitrage profits exist.
Allocation of marks: 6 marks were awarded for identifying the
difference in the synthetic risk free rate, 2 marks for working out the gross
profits, and 1 mark for the net profits.
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Examiners’ commentaries 2015
Question 3
a. Explain why the Sharpe ratio can give a misleading picture of performance
for a fund with time varying risk strategy, switching between high and low
risk in its portfolio. (7 marks)
Reading for this question
Chapter 8 in the subject guide, in the section called Changing risk.
Approaching the question
If there are both high risk and low risk periods in the evaluation period,
the average return for the fund is aggregated linearly so the numerator
in the Sharpe ratio is a linear average of the different periods, but the
denominator is aggregated non-linearly with the high risk periods
receiving disproportionately high weights. Therefore, the Sharpe ratio may
compare favourably with the market’s Sharpe ratio for each sub-period in
the evaluation period, but the aggregate Sharpe ratio may not. This is just
an artefact of the way standard deviation is calculated.
Allocation of marks: 4 marks allocated to the explanation of the
aggregation problem, and 3 marks for recognising how this affects the
Sharpe ratio.
b. For this question you should assume the term structure is flat 5% annually
compounded. You manage a pension fund for your company’s employees
where the current assets consist of a bond portfolio with duration 10 years.
The liabilities of the fund consist of expected withdrawals of 100,000
annually in each of the next 10 years. The fund is currently in deficit with
the value of the current assets only half the value of the fund’s liabilities.
You need to increase the contribution to the fund’s assets in such a way
that the value of the assets matches the value of the liabilities and the fund
is immunized against changes in the discount rate. Work out the Macaulay
duration and value of your investment of new contributions. (9 marks)
Reading for this question
Chapter 7 in the subject guide in the section called Duration.
Approaching the question
The value of the liability is (100,000/0.05)(1 – (1/1.05)10) = 772,173.
Duration is (1/772,173)(100,000/1.05 + 200,000/1.052 + … +
1,000,000/1.0510) = 5.1. The value of the existing bond is 0.5(772,173)
with duration 10. To bridge the duration gap, therefore, we need
0.5x = 0.5(10) = 5.1, which yields x = 1.98.
Allocation of marks: 3 marks allocated to working out the value of the
liability, 2 marks for the duration, 2 marks for the duration gap equation,
and 2 marks for the duration of the new investment.
c. The stock market has a 2-factor structure where the risk of the two factors
is independent. The factor betas for factor 1 of two portfolios A and B are
known, but the factor betas for factor 2 are not. You have the following data:
Fund:
Beta factor 1
Beta factor 2
Total variance
A
1.2
?
10%
B
0.5
?
11%
The variance of factor 1 is 5% and the variance of factor 2 is 7% and both A
and B are well diversified. On the basis of the information given, what is your
guess of the factor 2 betas for the two funds? State clearly the assumptions
made. (9 marks)
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FN3023 Investment management
Reading for this question
Chapter 6 in the subject guide, in the section called Factor models.
Approaching the question
We start out with the assumption that the portfolios are diversified to the
extent that they have zero idiosyncratic risk. This implies that var(A) =
10% = 1.22 var(f1) + x2 var(f2) where x is the factor beta on factor 2 for
portfolio A. Solving we find x = 0.63. Repeating for B we find var(B) =
11% = 0.52 var(f1) + y2 var(f2), where y is the factor beta on factor 2 for
B. Solving we find y = 1.18. If the portfolios have idiosyncratic risk then the
estimates are lower because the idiosyncratic risk feeds into the right hand
sides but not the left hand sides of the equations above, so the numbers we
have found are the upper bound for the factor betas on factor 2.
Allocation of marks: 2 marks allocated for specifying the critical
assumption that the idiosyncratic risk is zero, 3 marks each for identifying
the upper bonds, and 1 mark for commenting on the direction of the betas
if the idiosyncratic risk grows.
Question 4
a. Explain what we mean by a Treasury bill, a certificate of deposit and a
commercial paper. What is the essential difference between the three classes
of securities? (7 marks)
Reading for this question
Chapter 2 in the subject guide, in the section called Money market
instruments.
Approaching the question
All of these are money market instruments, or short term debt claims, and
the only way they differ is in terms of the issuing body. A T-Bill is issued by
the US government and has a credit risk that reflects the US government’s
ability to honour its debt; a certificate of deposit is issued by a private
bank and has a credit risk reflecting the bank’s ability to honour its debt;
and finally a commercial paper is issued by a corporation and has a credit
risk reflecting the corporation’s ability to honour its debt.
Allocation of marks: 1 mark awarded for a general recognition that
these are all money market instruments; 2 marks each for listing the three
claims’ issuing body.
b. Demonstrate, using Roll’s model, that the half-spread in a market c can
be written as c = (–cov(Δpt-1,Δpt))0.5, where Δpt is the market price change
between time t–1 and time t, and Δpt–1 is the market price change between
time t–2 and t–1. Explain the intuition for this result. (9 marks)
Readings
Chapter 5 in the subject guide, in the section called Bid-ask bounce: The
Roll model.
Approaching the question
Fundamental prices are m, and the transaction price is p = m + qc where
the first term is the fundamental price and the second term is the spread
where q = +1 if the transaction is at the ask and q = –1 if the transaction
is at the bid. The derivation is cov(Δpt, Δpt–1) = cov(Δmt + Δqtc, Δmt–1 +
Δqt–1c) (using the decomposition of price changes into fundamental price
changes and price changes linked to the bid-ask bounce) = cov(Δqtc,
Δqt–1c) (assuming that fundamental price changes has zero serial
covariance) = c2 cov(Δqt, Δqt–1) (using the linearity of the covariance
20
Examiners’ commentaries 2015
operator) = c2 cov(qt – qt–1, qt–1 – qt–2) (using the definition of the change
in q) = c2 cov(–qt–1, qt–1) (assuming that q is serially independent) = –c2
var(qt) (using the linearity of the covariance operator and the definition
of variance) = -c2 (0.5(1)2 + 0.5(–1)2) (calculating the variance) = –c2.
Hence c2 = –cov(Δpt, Δpt–1) and taking square roots on both sides yields
the result. The intuition is that the prices tend to bounce up and down
because of the bid-ask spread, and an up-movement is then likely to be
followed by a down-movement and vice versa. This generates negative
serial correlation, which is linked to the magnitude of the spread.
Allocation of marks: 3 marks were awarded for the decomposition
of transaction prices into fundamental changes and bid-ask bounces, 3
marks for the covariance calculus, and 3 marks for an explanation of the
intuition.
c. The quoted price for a bond is 1,010 per 1,000 in face value. The coupon is
5% annual. It is 45 days since the last coupon payment. If you were to buy or
sell the bond, what is the transaction price of your deal (ignoring spreads)? (9 marks)
Reading for this question
Chapter 2 in the subject guide, in the section called Bond market
instruments.
Approaching the question
The quoted price (clean price) equals the transaction price (dirty price)
minus the accrued interest. Accrued interest is 0.05(1,000)(45/365) =
6.164, so the transaction price is 1,010 + 6.164 = 1,016.64.
Allocation of marks: 3 marks were awarded for recognising the
relationship between the quoted price and the transaction price, 3 marks
for the accrued interest calculation, and 3 marks for getting the final
numbers correct.
Question 5
a. “Put protection” is a form of portfolio insurance strategy. The firm can
alternatively hold risk capital (determined for instance by a Value-at-Risk
model) against future potential losses. Explain the difference between the
two risk management strategies, and why both can be useful in practice. (7 marks)
Reading for this question
Chapter 9 in the subject guide, in the section called Put protection vs VaR.
Approaching the question
A put protected hedge is a targeted hedge that provides protection when
the value of a portfolio drops below a certain threshold, but can be illiquid
or there may simply not be put options traded that are a suitable hedge
for the portfolio in question. A risk capital approach is in this sense more
robust as it is always possible to hold a cash-cushion to dampen the effects
of shortfall of capital.
Allocation of marks: 4 marks were awarded for explaining the
differences in the way the two strategies work, and 3 marks for explaining
the relative advantages and disadvantages.
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FN3023 Investment management
b. Suppose you hold 100,000 in a stock market index, and you want to make
sure your portfolio does not lose value over the year to come if the market
index goes down at the same time as securing as much as possible of the
gains made if the market index goes up. Construct a portfolio that achieves
this, based on the data in the following table. The table shows one-year
option prices with exercise price equal to the current index level.
Current
index value
1 year Call option on the index
(per 1,000 capital invested)
1 year Put option on the index
(per 1,000 capital invested)
6,650
84.126
35.355
How much of the upside risk of the index can you keep when eliminating the
downside risk? (9 marks)
Reading for this question
Chapter 9 in the subject guide, in the section called Portfolio insurance
with calls.
Approaching the question
We can work out the present value of 1,000 at maturity from the put-call
parity formula: PV(1,000) = Capital invested in index + Put – Call =
1,000 + 35.355 – 84.126 = 951.229. This investment provides a floor of
1,000 at maturity and the difference 1,000 – 951.229 = 48.771 can be
invested in calls. If we could invest 84.126 in calls we would keep 100%
of the upside, in this case we invest less so keep only 48.771/84.126 =
57.97% of the upside. Therefore, final portfolio is 95,122.9 invested risk
free and 100,000 – 95,122.9 = 4,877.1 in call options.
Allocation of marks: 3 marks allocated for working out the amount
to invest risk free, 3 marks for explaining the investment in calls, and 3
marks for working out the amount of upside risk retained.
c. Suppose a risk neutral competitive market maker clears incoming orders
from a noise trader, who trades x units of the asset, and an informed trader,
who trades y units of the asset. The market maker observes the order flows
x and y but cannot identify the traders submitting them. The true asset value
is –1 or 1. Assume the market maker initially puts a value of 0 on the asset,
believing the asset is worth 1 with probability 50% and –1 with probability
50%. The noise traders either buy or sell one unit with equal probability
regardless of the value of the asset. The informed trader trades the optimal
quantity y, which maximizes the expected trading profit, conditional on
the true asset value, which is observable to the informed trader. Work out
the equilibrium trading quantity y and the expected trading profit of the
informed trader. (9 marks)
Reading for this question
Chapter 5 in the subject guide, in the section called Discrete version of the
Kyle model.
Approaching the question
The informed trader must trade in the same quantities as the noise trader
otherwise they are identifiable to the market maker. Therefore trading
quantity is |y| = |x| = 1. The market maker observes x and y which
individually are indistinguishable but the sum x + y is either 2, 0, or -2.
If 2 then the informed trader buys and the price is 1, if –2 the informed
trader sells and the price is –1, and if 0 the market maker learns no new
information and the price remains 0. If the true value is high the informed
trader buys and x + y is either 2 or 0, each equally likely, and the market
price is 1 or 0, respectively. Therefore the trading profits are 0.5(1–1) +
0.5(1–0) = 0.5. If the true value is low the informed trader sells and x +
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Examiners’ commentaries 2015
y is either –2 or 0, each equally likely, and using the same argument the
trading profits are 0.5. Expected profits are 0.5.
Allocation of marks: 3 marks awarded for working out the optimal
trading quantity, 3 marks for deducing the market maker’s price setting,
and 3 marks for working out the trading profits.
Question 6
a. Explain the “pairs trading” strategy. Explain how this strategy will work in
both bull and bear markets. (7 marks)
Reading for this question
Chapter 4 in the subject guide, in the section called Algorithmic or
program trading (statistical arbitrage).
Approaching the question
The pairs trading strategy is essentially a bet that two portfolios (A and
B) which historically have similar price patterns will converge again after
a brief divergence in prices. Assume PA > PB. Then you short A (the
expensive one) and invest in B (the cheap one) such that the capital outlay
is zero, that is to say xA PA – xB PB = 0, or xA = (PB/PA) xB. At a future
point where the prices become identical again so that both portfolios are
worth the same price P, you buy back xA units of A and sell xB units in B
at the same price P to clear your position, so your profit is (xB – xA)P = (1
– PB/PA) xB P which is positive since PB < PA, regardless of whether the
prices converge to a level higher than PA or lower than PB (the essential
point is they converge).
Allocation of marks: 3 marks allocated for explaining the pairs trading
idea, and 4 marks for working out how the profits emerge (an example is
acceptable).
b. You have the following data on the performance of a managed fund.
Average return
Total variance
Beta
Fund
15%
20%
1.2
Market
12%
10%
1
Risk free asset
5%
-
-
What is the Sharpe ratio and Jensen’s alpha for the fund? If you were to
advise investors on this fund, what would your recommendations be? (9 marks)
Reading for this question
Chapter 8 in the subject guide, in the sections called The Sharpe ratio and
More portfolio performance measures.
Approaching the question
Sharpe ratio for the fund is (0.15 – 0.05)/0.20.5 = 0.224, Sharpe ratio for
market is (0.12 – 0.05)/0.10.5 = 0.221. Jensen’s alpha is 0.15 – (0.05 +
1.2(0.12 – 0.05)) = 0.016. Your recommendation is that this fund looks
like an attractive investment regardless of whether it is part of a diversified
portfolio (positive alpha) or as a stand alone investment (Sharpe greater
than market). You may recommend the investor mixes the portfolio with
the market portfolio as prescribed by the Treynor-Black formula.
Allocation of marks: 4 marks allocated to the Sharpe ratio, 3 marks for
Jensen’s alpha, and 2 marks for making the recommendation.
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FN3023 Investment management
c. The utility of an investor can be described by the following utility function,
which depends on the expected return Er, and the variance v2, of the
investor’s portfolio: U(Er,v2) = Er – 1.5 v2. Suppose all investments available
to the investor are located on the security market line which cuts through the
risk free return rF = 5% and the market portfolio with expected return
ErM = 10%. The variance of the market portfolio is 10%. What portfolio would
you recommend to this investor? (9 marks)
Reading for this question
Chapter 6 in the subject guide, in the section called CARA utility and
normal returns.
Approaching the question
The investor is variance averse, therefore the optimal portfolio is a linear
combination of the risk free asset and the market portfolio (2-fund
separation). The investor maximises (1–x) rF + x ErM – 1.5 x2 var(rM)
with respect to x, which yields the first order condition (ErM – rF) =
3x var(rM), which implies x = 0.167. The optimal portfolio is therefore
16.7% of wealth in the market index and 83.3% of wealth risk free.
Allocation of marks: 2 marks allocated for recognising the 2-fund
separation property, 5 marks for working out the optimal portfolio, and 2
marks for interpreting the result.
Question 7
a. What do we mean by collateralized debt/loan obligations? Explain why these
instruments became popular in the run-up to, and why they may also have
contributed to, the 2007-08 financial crisis in the US? (7 marks)
Reading for this question
Chapter 3 in the subject guide, in the section called Recent financial
innovations, in the subsection called Collateralised debt/loan obligations.
Approaching the question
Collateralised debt/loan obligations is a financial instrument whose
payoffs are derived from the payoffs generated by a debt or loan portfolio,
a so-called asset backed security. Normally they are issued in tranches with
varying seniority where the most senior tranche is relatively risk free and
the least senior tranche very risky. They became popular vehicles for offloading loans from banks’ balance sheets, which enabled the bank to make
more loans without raising more risk capital. The problems associated
with these instruments were (i) that the quality of the underlying loan
portfolio became difficult to assess for investors and even for credit rating
agencies, and (ii) they encouraged the banks to make new loans of poorer
quality (subprime). In the end the risk associated with these instruments
led to the markets drying up and banks unable to offload very risky loans,
and ultimately to bank bailouts and a crisis in the financial system.
Allocation of marks: 2 marks were allocated to an explanation of
the instruments, 2 marks to explain their popularity, and 3 marks for
explaining the path to the crisis.
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Examiners’ commentaries 2015
b. You are given the following data on two funds (A and B), the market portfolio
(M), and the risk free asset (R):
Fund
Expected return
Beta
Standard
deviation
A
15%
1.2
50%
B
13%
1.2
40%
M
12%
1
30%
5%
-
-
R
Work out the M measure and Jensen’s alpha for A and B. If you were
to advise investors who have no current stock market exposure in their
investments, what recommendations would you make? (9 marks)
2
Approaching the question
The M2 for portfolio P is ((1 – x)rF + xErP) – ErM, where x = Std.
dev(rM)/Std.dev(rP), so x = 0.3/0.5 for A and x = 0.3/0.4 for B. The M2
is then –0.01 for A and –0.01 for B. Jensen’s alpha is ErP – (rF + βP(ErM
– rF)) and equal to 0.016 for A and –0.004 for B. The relevant measure is
M2, and neither fund is attractive as a stand-alone investment.
Reading for this question
Chapter 8 in the subject guide, in the section called More portfolio
performance measures.
Allocation of marks: 4 marks allocated for working out the M2
measures, 4 marks for Jensen’s alpha, and 1 mark for the interpretation.
c. The utility function of an individual is given by u(w) = ln(w) where w is
wealth. Demonstrate that this utility function has the property of CRRA
(constant relative risk aversion). Myopic asset allocation is the practice
of making portfolio choice optimal only for the next period even if the
investor’s investment horizon is longer than that. Explain why log-utility can
justify myopic asset allocation. (9 marks)
Reading for this question
Chapter 6, in the section called Asset allocation over longer time horizons.
Approaching the question
The absolute risk aversion coefficient is –u’’(w)/u’(w) = (1/w2)/(1/w) =
(1/w). The relative risk aversion coefficient is w times the absolute risk
aversion coefficient, or 1, which is constant.
With constant relative risk aversion coefficient you make constant relative
asset allocation choices (i.e. you hold the same fractions of risky to risk
free investments), and therefore it does not matter that you make myopic
asset allocation choices.
Allocation of marks: 6 marks for working out the absolute risk
aversion coefficient, for working out the derivatives correctly and getting
the relative risk aversion coefficient correct, and 3 marks for explaining
the argument about myopic asset allocation choices.
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FN3023 Investment management
Question 8
a. Explain how stock markets are regulated, and for what reason this regulation
is in place. (7 marks)
Reading for this question
Chapter 2 in the subject guide, in the section called Regulation of financial
markets.
Approaching the question
Regulation is in the form of self-regulation where the institutions that
facilitate trade impose their own rules on the users, and in the form of
government regulation. The main reasons for regulation is (i) to provide
protection to market participants, particularly the small investors who
are likely to be ill-informed and vulnerable to abuse by others, and (ii) to
regulate the information flow from the firms whose securities are traded.
This involves both the accuracy of the information and the manner in
which the information is released.
Allocation of marks: 3 marks allocated for explaining how the markets
are regulated, and 4 marks for the reason they are regulated.
b. The stock market has a two-factor structure, where the risk free return is 5%,
the risk premium of factor 1 is 6%, and the risk premium of factor 2 is 4%.
You observe the following data on 3 portfolios:
Portfolio
Average return
Factor 1 beta
Factor 2 beta
A
10%
0.6
0.8
B
12%
1.1
0.2
C
15%
0.3
1.5
Outline a strategy for trading these portfolios (and the risk free asset) to
make positive expected returns with zero capital invested and zero exposure
to factor 1 and 2. (9 marks)
Reading for this question
Chapter 6 in the subject guide, in the section called Factor models.
Approaching the question
There are several ways of doing this. The method here is to combine A
and B such that we eliminate factor 2 beta. If x is in A and (1 – x) in B
we need 0.8x + 0.2(1–x) = 0, or x = –1/3. The average return on this
portfolio is –(1/3)0.10 + (4/3)0.12 = 0.1267 and the loading on factor
1 risk is –(1/3)0.6 + (4/3)1.1 = 1.267. Next, we combine A and C such
that we also eliminate factor 2 risk. If y is in A and (1 – y) in B we need
08y + 1.5(1 – y) = 0, or y = 15/7. The average return on this portfolio
is (15/7)0.10 – (8/7)0.15 = 0.0428 and the loading on factor 1 risk
is (15/7)0.6 – (8/7)0.3 = 0.6571. Combining the two new portfolios
to eliminate factor 1 risk (invest z in the first and (1–z) in the second)
we need 1.267z + 0.6571(1–z) = 0, or z = 1.0781. The return on this
portfolio is 1.0781(0.1267) – 0.0781(0.0428) = 0.1333, or 13.33%. This
portfolio should earn the risk free rate which is 5%, so you can borrow at
5% and invest in the portfolio at 13.33% to make expected trading profits.
Allocation of marks: 5 marks were awarded for the formation of an
arbitrage portfolio and 4 marks for working out the specific arbitrage
profits.
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Examiners’ commentaries 2015
c. The Kyle model derives the pricing rule of a risk neutral market maker and
the optimal quantity traded by a risk neutral insider who knows perfectly the
value of the asset. The pricing rule is p(y) = p0 + 0.5(σ/σ)y and the optimal
quantity traded is x = p0(σ/Σ) + (σ/Σ)v where σ is the standard deviation of
the asset value v, with initial price p0, and σ is the standard deviation of the
trading quantity of the noise traders, u. The aggregate trade is y = x + u, and
assume the initial price is zero, p0 = 0. Work out the expected trading profit
for the insider, E(v–p(y))x, conditional on the true asset value v. Explain why
the conditional profit is increasing in the standard deviation of the noise
traders, σ. (9 marks)
Reading for this question
Chapter 5 in the subject guide, in the section called Kyle.
Approaching the question
The expected trading profits are E(v – p(y))x, and substituting for p(y)
we find E(v – 0.5(Σ/σ)y)x, and substituting further for y we find E(v 0.5(Σ/σ)(x + u))x = (v – 0.5(Σ/σ)x)x since Eu = 0. Finally, substitute for
x and find the profits are v(σ/Σ)v – 0.5(Σ/σ) (σ/Σ)2v2 = 0.5 (σ/Σ)v2. The
intuition the profits are increasing in the standard deviation of the noise
traders trading activity is that this makes it easier to “hide” the informed
trading (effectively the market becomes more liquid).
Allocation of marks: 6 marks for working out the expected trading
profits and 3 marks for explaining the intuition.
27