2019 Exam commentary

Examiners’ commentaries 2019
Examiners’ commentaries 2019
FN3023 Investment management
Important note
This commentary reflects the examination and assessment arrangements for this course in the
academic year 2018–19. 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.
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 2019
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 2019
FN3023 Investment management
Important note
This commentary reflects the examination and assessment arrangements for this course in the
academic year 2018–19. 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) What are stop-loss orders and stop-buy orders? What benefits do such orders
offer to investors?
(7 marks)
(b) A 5-year bond with annual coupons 4% of the face value is trading at
yield-to-maturity 5%. What is the duration of the bond?
(9 marks)
(c) Consider the Kyle framework for market microstructure. A market for a stock
where a competitive risk-neutral market maker clears incoming buy and sell
orders in a sequence. Each order is for one share, and the incoming orders are
drawn randomly from a pool consisting of uninformed traders, making up 90%
of the pool, and informed traders, making up 10% of the pool. The uninformed
traders buy or sell with equal probability, whereas the informed traders buy for
sure when the value is high and sell for sure when the value is low. The high
value is $25 per share and the low value is $15 per share. The market maker
believes initially that the probability of a high value is 40% and the probability
of a low value is 60%. What bid and ask prices does the market maker quote in
anticipation of the first order?
(9 marks)
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Examiners’ commentaries 2019
Reading for this question
For (a), you should read Chapter 2 in the subject guide. For (b), you should read Chapter 7. For
(c), you should read Chapter 5.
Approaching the question
(a) A good answer should contain the following. Stop-loss are orders to sell a given quantity as
long as the price remains below a given limit; stop-buy are orders to buy a given quantity as
long as the price remains above a given limit. They provide quick exits from loss-making
long and short positions, respectively, limiting the investor’s loss.
(b) A good answer should contain the following. You can assume an arbitrary face value, say
$1, since the duration depends on yields, coupon rates and maturity and not the face value
of the bond. Then we find that the bond price is:
0.04
× (1 − 1.05−5 ) + 1.05−5 = $0.9567.
0.05
Duration is then:
1
× (1 × 0.04 × 1.05−1 + 2 × 0.04 × 1.05−2 + · · · + 5 × 1.04 × 1.05−5 ) = 4.62.
0.9567
(c) A good answer should contain the following. The market maker revises their beliefs about
the likelihood of high (H) and low (L) values depending on the order flow. Using Bayes’
theorem, we find that the conditional probability of H given a buy order is:
P (H | buy) =
(0.9 × 0.5 + 0.1) × 0.4
P (buy | H) P (H)
=
= 0.4490.
P (buy)
(0.9 × 0.5 + 0.1) × 0.4 + (0.9 × 0.5) × 0.6
Simlarly, the conditional probability of L given a sell order is:
P (L | sell) =
P (sell | L) P (L)
(0.9 × 0.5 + 0.1) × 0.6
=
= 0.6471.
P (sell)
(0.9 × 0.5 + 0.1) × 0.6 + (0.9 × 0.5) × 0.4
Therefore, the ask price is designed to meet the buy order, which is:
0.4490 × $25 + (1 − 0.4490) × $15 = $19.50
and the bid price is designed to meet the sell order, which is:
(1 − 0.6471) × $25 + 0.6471 × $15 = $18.53.
Question 2
(a) What are zero-coupon bonds? What benefits do such bonds offer to investors?
(7 marks)
(b) The following table shows data on a portfolio, the market index, and the risk
free interest rate.
Average return of the portfolio
Correlation coefficient of the portfolio with
the market index
Standard deviation of the portfolio
Average return of the market index
Standard deviation of the market index
Observed risk free interest rate
10%
0.6
50%
8%
30%
3%
Work out the M 2 -measure and Jensen’s alpha for the portfolio.
(9 marks)
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FN3023 Investment management
(c) The Treynor–Black model gives the weights w and (1 − w) in an ‘active’
portfolio A and a ‘passive’ market index P , respectively, by the formula:
w=
αA
2
αA (1 − βA ) + (ErP − rF ) Var(εA )/σP
where αA is Jensen’s alpha for A, βA is the beta of A, ErP is the expected
return of P , rF is the risk-free interest rate, Var(εA ) is the unsystematic
2
variance of A, and σP
is the variance of P . The risk-free interest rate is 3%, and
the following table shows relevant information on A and P :
Expected return
Cov(ri , rP )
Total variance
A
9%
0.07
0.15
P
8%
—
0.09
Work out the portfolio weight w.
(9 marks)
Reading for this question
For (a), you should read Chapter 3 in the subject guide. For (b), you should read Chapter 8. For
(c), you should read Chapter 6.
Approaching the question
(a) A good answer should contain the following. Zero-coupon bonds are bonds which pay only
the face value at maturity (coupon payments are stripped out). Therefore, they are very
flexible investment instruments which can be targeted towards giving investment returns at
specific dates, with no reinvestment risk.
(b) A good answer should contain the following. Using the definition of the measure given in
the subject guide, we find that:
σM σM
0.3
0.3
M2 = 1 −
rF +
E(r) − E(rM ) = 1 −
× 0.03 +
× 0.10 − 0.08 = −0.8%.
σ
σ
0.5
0.5
The beta is:
0.6 × 0.5
=1
0.3
so, using the definition of Jensen’s alpha we find:
Jensen’s alpha = 0.1 − 0.03 − 1 × (0.08 − 0.03) = 2%.
(c) A good answer should contain the following. First, we find that the beta of A is
0.07/0.09 = 0.7778. Using the definition of Jensen’s alpha, we find that Jensen’s alpha for A:
0.09 − 0.03 − 0.7778 × (0.08 − 0.03) = 0.0211.
The idiosyncratic risk of A is:
0.15 − (0.778)2 × 0.09 = 0.0956.
Hence applying these numbers in the formula given in the question, we find that the weight
is:
0.0211
= 0.37
w=
(0.0211 × (1 − 0.7778) + (0.08 − 0.03) × (0.0956/0.09))
that is, the optimal mixture portfolio consists of 37% in A and 63% in P .
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Examiners’ commentaries 2019
Question 3
(a) What is an inverted shape of the term structure of interest rate? Is this shape
consistent with the money substitute hypothesis for the term structure?
(7 marks)
(b) The value of a portfolio over a 10-day period is log-normally distributed, so that
the 10-day log-return has mean 0.2% and variance 0.001. For a standard,
normally distributed, random variable with mean 0 and variance 1, the
distribution function is N (−2.33) = 1%. What is the 10-day, 1% VaR of the
portfolio log-return?
(9 marks)
(c) Suppose you hold an equally weighted portfolio of 10 stocks. Each stock has
covariance 0.05 with the market index and total variance 0.08. The market
index has variance 0.09. Assume the unsystematic risk is independent between
any two stocks in your portfolio. What is the total risk of your portfolio?
(9 marks)
Reading for this question
For (a), you should read Chapter 7 in the subject guide. For (b), you should read Chapter 9. For
(c), you should read Chapter 6.
Approaching the question
(a) A good answer should contain the following. An inverted term structure is a
downward-sloping term structure, which makes longer maturities cheaper than shorter ones.
The money substitute hypothesis asserts that shorter-termed bonds are a closer substitute
to cash than longer ones. Since money has zero interest rates, this hypothesis is unlikely to
explain inverted shapes of the term structure.
(b) A good answer should contain the following. The first approach is to standardise the
log-return so that it has zero mean and unit variance. Standardising the 10-day log-return
yields:
x − 0.002
(0.001)0.5
and we are looking to solve for x such that the standard log-return value is equal to −2.33:
x = −2.33 × (0.001)0.5 + 0.002 = −7.2%.
Therefore, the portfolio will lose at most 7.2% of its value (measured by log-returns) over a
10-day period for 99% of the time.
(c) A good answer should contain the following. First, we work out the beta of each stock,
which is 0.05/0.09 = 0.556. Therefore, the systematic risk of each stock is
(0.556)2 × 0.09 = 0.028 (which is also the systematic risk of the portfolio since the portfolio
has the same beta as each individual stock), and the unsystematic risk is
0.08 − 0.028 = 0.052. The unsystematic risk of the portfolio will be diversified away in part.
The unsystematic risk is for the portfolio is 0.052/10 = 0.0052. Therefore, the total risk of
the portfolio is 0.028 + 0.0052 = 0.033.
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FN3023 Investment management
Question 4
(a) What is accrued interest (in bond markets)? Explain the quoting convention for
bond prices.
(7 marks)
(b) You have purchased and sold shares in a company using an interest free margin
loan over a 2-year period. The initial (and maintenance) margin is 70%. The
company pays dividends, and the following table shows the dividends and
ex-dividend prices over the period.
Trading activity
Ex-div price
Dividend
Year 0
Buy 1,000
shares
$25
—
Year 1
Sell 500
shares
$30
$2
Year 2
Sell 500
shares
$23
$2
What are the cash flows of your trading activity before and after taking into
account the margin loan?
(9 marks)
(c) Consider the following information about a given portfolio A, the market index
M , and the risk-free interest rate F :
Average return of A
Standard deviation of A
Average return of M
Standard deviation of M
Correlation between A and M
Risk-free interest rate
12%
45%
8%
30%
0.6
3%
Work out the information ratio of portfolio A.
(9 marks)
Reading for this question
For (a) and (b), you should read Chapter 2 in the subject guide. For (c), you should read
Chapter 8.
Approaching the question
(a) A good answer should contain the following. Bond prices go up and down depending on how
close the next coupon payment is. To correct for these movements, the convention is to
quote bond prices net of accrued interest. The accrued interest is the coupon payment ×
days since the last coupon payment/days between coupon payments. The transaction (or
dirty) price minus accrued interest is the quoted (or clean) price.
(b) A good answer should contain the following. The cash flows associated with the underlying
transactions, and margin transactions, are:
Year
Cash flow
Margin Loan
Net cash flow
8
0
−1,000 × 25 = −25,000
Total −25,000
30% × 25,000 = 7,500
Cash flow +7,500
−17,500
1
+500 × 30 = 15,000
+1,000 × 2 = 2,000
Total +17,000
30% × 15,000 = 4,500
Cash flow −3,000
+14,000
2
+500 × 23 = 11,500
+500 × 2 = 1,000
Total +12,500
0
Cash flow −4,500
+8,000
Examiners’ commentaries 2019
(c) A good answer should contain the following. We need to find the beta of A, which is
0.6 × 0.45/0.3 = 0.9. Using the definition of Jensen’s alpha, we find that the Jensen’s alpha
of A is:
0.12 − 0.03 − 0.9 × (0.08 − 0.03) = 4.5%.
Therefore, the systematic risk of A is:
(0.9)2 × (0.3)2 = 0.0729.
The unsystematic risk of A is the residual:
(0.45)2 − 0.0729 = 0.1296.
Using the definition of the information ratio, we find that it is equal to 0.045/0.1296 = 0.35.
Question 5
(a) What is the difference between direct and indirect bankruptcy costs? Explain
why bankruptcy costs are an important reason why corporations engage in risk
management.
(7 marks)
(b) The stock market has a 2-factor structure. The risk free rate is 3%. The
following table gives the factor betas and returns on three well-diversified
portfolios.
Average
return
Factor-1 beta
Factor-2 beta
Portfolio
A
9%
Portfolio
B
9%
Portfolio
C
?
1.2
0.3
0.8
0.9
0.7
0.6
What is the average return on portfolio C?
(9 marks)
(c) You have the following information about a portfolio, the market index, and the
risk-free interest rate:
Jensen’s alpha
Beta
Unsystematic variance
Average return of the market
index
Variance of the market index
Risk free interest rate
1.5%
1.1
0.02
8%
0.09
3%
What are the Sharpe ratios of the portfolio and the market?
(9 marks)
Reading for this question
For (a), you should read Chapter 9 in the subject guide. For (b), you should read Chapter 6. For
(c), you should read Chapter 8.
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FN3023 Investment management
Approaching the question
(a) A good answer should contain the following. Direct bankruptcy costs relates to the advisory
costs, legal costs, court costs, management time spent with creditors – the indirect costs
relate to the loss of customers, the costs associated with dealing with suppliers, the costs of
poor management decisions as a consequence of the financial distress. Risk management can
reduce the costs of financial distress and therefore the direct and indirect bankruptcy costs.
Since these costs are borne by the debt holders, risk management can help secure cheaper
borrowing.
(b) A good answer should contain the following. We know that there is a risk premium which is
associated with each factor, and that these risk premia add to the average return linearly,
weighted by the factor beta. Therefore, the risk premia can be extracted using A and B:
0.09 = 0.03 + λ1 × 1.2 + λ2 × 0.3
and:
0.09 = 0.03 + λ1 × 0.8 + λ2 × 0.9.
This yields λ1 = 4.29% and λ2 = 2.86%. Therefore, the average return on C should be 7.7%.
(c) A good answer should contain the following. The average return on the portfolio is:
1.5% + 3% + 1.1(8% − 3%) = 10%.
Total risk (variance) of the portfolio is:
(1.1)2 × 0.09 + 0.02 = 0.1289.
Therefore, the Sharpe ratio of the portfolio is:
0.10 − 0.03
= 0.195.
(0.1289)0.5
The Sharpe ratio of the market is, in comparison:
0.08 − 0.03
= 0.167.
(0.09)0.5
Question 6
(a) Explain how market risk and liquidity risk differ. How can liquidity risk cause a
problem for the management of market risk?
(7 marks)
(b) You want to make a 6-month put-protected investment consisting of the stock
market index, 6-month put options on the index, and a risk-free investment.
Your objective is to achieve the maximum return when the index has positive
return, and zero return when the index has negative return. The following table
sets out the relevant information.
Current index level
Current 6-month call price
Current 6-month put price
Annual risk free rate (continuously
compounded)
6,956
636.43
532.87
3%
The options have exercise price equal to the current index level, so the payoffs of
the call and put options at maturity are max(0, index level in 6 months − 6,956)
and max(0, 6,956 − index level in 6 months), respectively. Your total investment
is $5m. Derive the breakdown of your investment into the three assets: the
index, put options, and risk-free investment.
(9 marks)
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Examiners’ commentaries 2019
(c) The auto-covariance of successive transaction price changes of a stock is −0.015.
The average stock price is $25. Use Roll’s model to estimate the average
%-spread of the stock.
(9 marks)
Reading for this question
For (a), you should read Chapter 8 in the subject guide. For (b), you should read Chapter 9. For
(c), you should read Chapter 5.
Approaching the question
(a) A good answer should contain the following. Market risk deals with the risk of changes in
the market values of assets and securities, whereas liquidity risk deals with the risk of the
costs of rebalancing portfolios. Since the management of market risk often involves making
rebalancing transactions of portfolios, an increase in liquidity risk can expose the investor to
extra market risk.
(b) A good answer should contain the following. The idea is to invest part of the portfolio risk
free and the other part in call options. This is done by investing risk free:
6,956 × exp(−0.5 × 3%)
which yields a payoff of 6,956 at maturity, and by investing:
6,956 × (1 − exp(−0.5 × 3%)) = 103.5613
in calls, which equals 103.5613/636.43 = 0.1627 units of the call. Next, using put-call parity,
we convert the call into puts, risk free investment and index:
0.1627 calls = 0.1627 × puts + 0.1627 × index − 0.1627 × 6,956 × exp(−0.5 × 3%).
This yields the following final position: a risk free investment of:
(1 − 0.1627) × 6,956 × exp(−0.5 × 3%) = 5,737.39
an index investment of 0.1627 × 6,956 = 1,139.90, and a put investment of
0.1627 × 532.87 = 86.71, a total of 6,956. Since we want to scale up the investment to $5m,
you multiply by 5m/6956 to get a risk-free investment of $4.124m, and index investment of
$0.814m, and a put option investment of $0.062m.
(c) A good answer should contain the following. The Roll model predicts that the half-spread
squared equals the negative of the auto-covariance of transaction prices, or that the half
spread (in $-terms) is the square root of 0.015, equal to $0.1225, or a $-spread of $0.245.
This represents a %-spread of 0.0098, or 0.98%.
Question 7
(a) What is convexity (in bond markets)? Explain how convexity is used to improve
predictions of bond price changes given changes in the yield-to-maturity.
(7 marks)
(b) Assume a stock market where all investors are variance-averse. The average
return on the market index is 8% and the variance is 0.09. The risk free return
is 3%. Suppose an investor has utility function U (µ, σ 2 ) = µ − 0.2σ 2 . What is
the investor’s optimal portfolio?
(9 marks)
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FN3023 Investment management
(c) A pension liability consists of annual payments according to the schedule in the
table below:
Year
1
2
3
4
5
Payments
$10m
$11m
$12m
$12m
$12m
The interest rate is 5% (assume flat term structure). You seek to hedge the
liability against changes in the flat term structure, taking positions in a bond
portfolio with duration 10 years. How much do you need to invest in these
bonds to hedge the liabilities?
(9 marks)
Reading for this question
For (a) and (c), you should read Chapter 7 in the subject guide. For (b), you should read
Chapter 6.
Approaching the question
(a) A good answer should contain the following. Convexity refers to the convex curved shape of
the bond price when plotted against the yield-to-maturity. The first order effect on the bond
price is duration, which yields a linear approximation. If convexity is taken into account you
will also capture the second order effects due to the curvature of the shape.
(b) A good answer should contain the following. We know that two-fund separation obtains
when we consider portfolio choice with variance-averse investors. Therefore, the optimal
portfolio consists of the market index and the risk free asset, in proportions x and (1 − x),
respectively. The expected return of the portfolio is:
rF + x(rM − rF )
and the variance is:
x2 × Var(rM )
so the problem for the investor is to maximise:
rF + x(rM − rF ) − 0.2 × (x2 × Var(rM ))
which yields a first-order condition:
rM − rF = 0.4 × x × Var(rM )
equal to 0.08 − 0.03 = 0.4 × 0.09 × x. Therefore, the optimal portfolio is x = 1.38, which
represents 138% invested in the market index and borrowing 38% risk free.
(c) A good answer should contain the following. Before we work out the duration we need the
value of the liability. The present value of the pension liability is:
11
12
10
+
+ ··· +
= 49.14.
2
1.05 (1.05)
(1.05)5
Using the definition of duration, we find it equals:
1
10
11 × 2
12 × 5
+
·
·
·
+
= 2.99.
×
+
49.14
1.05 (1.05)2
(1.05)5
Next, we need to balance the duration by investing x in bonds. Therefore,
x × 10 = 49.14 × 2.99, or x = 14.71. Therefore, the ‘immunising’ investment in the bond
portfolio is $14.71.
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Examiners’ commentaries 2019
Question 8
(a) What is a certificate of deposit? How does the certificate of deposit differ from
a US T-bill instrument?
(7 marks)
(b) A 5-year bond with annual coupons 5% of the face value is trading at
yield-to-maturity 5%. What is the convexity of the bond?
(9 marks)
(c) Consider the Glosten–Milgrom framework for market microstructure. A
risk-neutral competitive market maker clears the market for trading in a stock
after observing the incoming orders from a noise trader and an informed trader.
The noise trader buys and sells 1,000 shares of the stock with equal probability,
whereas the informed trader buys the stock for sure if the value is high and sells
the stock for sure if the value is low. The high value is $60, and the low value is
$40, and the market maker believes initially that there is an equal probability of
high and low value. What profits can the informed trader expect to make in
this market?
(9 marks)
Reading for this question
For (a), you should read Chapter 2 in the subject guide. For (b), you should read Chapter 7. For
(c), you should read Chapter 5.
Approaching the question
(a) A good answer should contain the following. A certificate of deposit is a money market
instrument, which is typically a discount security with no coupon payments (the price is
simply a discounted face value) and maturity less than one year. The instrument
distinguishes itself from US T-bills mainly by the issuing entity: US T-bills are issued by the
US government whereas certificates of deposits are issued by commercial banks
(corporations’ money market version of this instrument is called a commercial paper).
(b) A good answer should contain the following. We know that the bond is trading at par
(which we assume equals $1) since the coupon rate equals the yield-to-maturity. Therefore,
using the convexity formula we find it equals:
5 × 6 × 1.05
1 × 2 × 0.05 2 × 3 × 0.05
+
+
·
·
·
+
= 15.13.
0.5 ×
(1.05)3
(1.05)4
(1.05)7
(c) A good answer should contain the following. The informed trader’s order is fully revealed
unless the order has the same quantity as the noise trader, and goes in the opposite
direction. Trading 1,000 shares in the opposite direction to the noise trader’s order happens
with probability 50%, and the market clearing price is in this case equal to
0.5 × 60 + 0.5 × 40 = $50. The informed trader makes expected profits of:
0.5 × (60 − 50) × 1,000 = $5,000
if the value is high, and:
0.5 × (50 − 40) × 1,000 = $5,000
if the value is low. Therefore, the unconditional expected profit is $5,000.
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FN3023 Investment management
Examiners’ commentaries 2019
FN3023 Investment management
Important note
This commentary reflects the examination and assessment arrangements for this course in the
academic year 2018–19. 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) What are limit buy and limit sell orders? What benefits do such orders offer to
investors?
(7 marks)
(b) A 5-year bond with annual coupons 4% of the face value is trading at
yield-to-maturity 4%. What is the convexity of the bond?
(9 marks)
(c) If investors are variance-averse, then the expected return on a given stock is
such that Eri − rF = γ Cov(ri , rM ), where Eri is the expected return on stock i,
rF is the risk-free interest rate, γ is a constant (the same for all stocks), and
Cov(ri , rM ) is the covariance between the return on stock i and the market
index. Suppose the stock market consists of only two stocks, A and B, with
E(rA ) = 10%, E(rB ) = 6%, rF = 3%, the variance of stock A is Var(rA ) = 0.4,
the variance of stock B is Var(rB ) = 0.2, and the covariance between A and B is
Cov(rA , rB ) = 0.11. What is the composition of the market index? (Hint: if x,
y and z are random variables and a and b are constants, then Cov(x, ay + bz) =
a Cov(x, y) + b Cov(x, z).)
(9 marks)
14
Examiners’ commentaries 2019
Reading for this question
For (a), you should read Chapter 2 in the subject guide. For (b), you should read Chapter 7. For
(c), you should read Chapter 6.
Approaching the question
(a) A good answer should contain the following. Limit buy orders are orders to buy a given
quantity as long as the price is below a certain limit, and limit sell orders are orders to sell a
given quantity as long as the price is above a certain limit. These orders are helpful in
ensuring that buy and sell transactions are carried out with minimal execution costs due to
illiquid markets.
(b) A good answer should contain the following. Since convexity does not depend on the face
value directly but on the yield, the coupon rate, and the maturity, we can choose the face
value arbitrarily, say $1. Using the formula for convexity we find it is equal to:
5 × 6 × 1.04
1 × 2 × 0.04 2 × 3 × 0.04
+
+ ··· +
= 25.01.
0.5 ×
(1.04)3
(1.04)4
(1.04)7
(c) A good answer should contain the following. Since the investors are variance-averse,
two-fund separation obtains. Therefore, the market portfolio is xr(A) + (1 − x)r(B) for
some x. Hence the covariance of r(A) with the market portfolio is:
x × Var(r(A)) + (1 − x) × Cov(r(A), r(B)).
Similarly, the covariance of r(B) with the market portfolio is:
(1 − x) × Var(r(B)) + x × Cov(r(A), r(B)).
Therefore, we get a system with:
E(r(A)) − r(F ) = γ × x × Var(r(A)) + γ × (1 − x) × Cov(r(A), r(B))
and:
E(r(B)) − r(F ) = γ × (1 − x) × Var(r(B)) + γ × x × Cov(r(A), r(B))
or:
and:
0.10 − 0.03
= 0.11 + x(0.4 − 0.11)
γ
0.06 − 0.03
= 0.2 + x × (0.11 − 0.2).
γ
Simplifying, 0.07 × (1/γ) − 0.29 × x = 0.11 and 0.03 × (1/γ) + 0.09 × x = 0.2, and solving we
find x = 0.71, so the market index consists of 71% in A and 29% in B.
Question 2
(a) What is a normal shape for the term structure of interest rates? How does the
liquidity preference theory explain this shape?
(7 marks)
(b) A stock has correlation 0.6 with the market index and total variance 0.15. The
market index has variance 0.09. What are the stock’s systematic and
unsystematic variances?
(9 marks)
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FN3023 Investment management
(c) Consider the Kyle framework for market microstructure. A risk-neutral
competitive market maker clears the market for trading in a stock after
observing the incoming orders from a noise trader and an informed trader (who
knows perfectly the true value of the stock). The noise trader buys and sells 100
shares of the stock with equal probability, whereas the informed trader buys the
stock if the value is high and sells the stock if the value is low. The high value is
$12, and the low value is $6, and the market maker believes initially there is a
60% probability of high value and a 40% probability of low value. What profits
can the informed trader expect to make in this market?
(9 marks)
Reading for this question
For (a), you should read Chapter 7 in the subject guide. For (b), you should read Chapter 6. For
(c), you should read Chapter 5.
Approaching the question
(a) A good answer should contain the following. A gently upward-sloping term structure is
considered normal. Liquidity preference theory predicts that the longer yields reflect greater
uncertainty, and hence they offer higher compensation for investors.
(b) A good answer should contain the following. First we need the beta of the stock, which is:
0.6 × (0.15)0.5
= 0.775.
(0.09)0.5
Therefore, the systematic variance is, (0.775)2 × 0.09 = 0.054. The unsystematic risk is
simply the residual between the total variance and the systematic variance, or
0.15 − 0.054 = 0.096.
(c) A good answer should contain the following. First, we make the observation that the
informed trader reveals his type by trading in different quantities to the noise trader, so will
also need to trade 100 shares per transaction. The market maker sets price equal to
0.6 × 12 + 0.4 × 6 = $9.6 if the two orders go in opposite directions, which happens with
probability 0.5 (otherwise, the price is fully revealing). Therefore, the informed trader
makes a gain of 0.5 × (12 − 9.6) = $1.20 if the value is high and 0.5 × (9.6 − 6) = $1.80 if the
value is low. Therefore, the unconditional expected gain is 0.6 × $1.20 + 0.4 × $1.80 = $1.44.
Note that whereas the conditional expected profits are different, their weights in the
unconditional expected profit, 0.6 × 1.20 = 0.72 and 0.4 × 1.80 = 0.72, are the same. Hence
the informed trader can expect to make high gains with low probability or low gains with
high probability, but the product of the two are the same.
Question 3
(a) What is accrued interest (in bond markets)? Explain the quoting convention for
bond prices.
(7 marks)
(b) You have purchased and sold shares in a company using an interest free margin
loan over a 2-year period. The initial (and maintenance) margin is 70%. The
company pays dividends, and the following table shows the dividends and
ex-dividend prices over the period.
Trading activity
Ex-div price
Dividend
16
Year 0
Buy 1,000
shares
$25
—
Year 1
Sell 500
shares
$30
$2
Year 2
Sell 500
shares
$23
$2
Examiners’ commentaries 2019
What are the cash flows of your trading activity before and after taking into
account the margin loan?
(9 marks)
(c) Consider the following information about a given portfolio A, the market index
M , and the risk-free interest rate F :
Average return of A
Standard deviation of A
Average return of M
Standard deviation of M
Correlation between A and M
Risk-free interest rate
12%
45%
8%
30%
0.6
3%
Work out the information ratio of portfolio A.
(9 marks)
Reading for this question
For (a) and (b), you should read Chapter 2 in the subject guide. For (c), you should read
Chapter 8.
Approaching the question
(a) A good answer should contain the following. Bond prices go up and down depending on how
close the next coupon payment is. To correct for these movements, the convention is to
quote bond prices net of accrued interest. The accrued interest is the coupon payment ×
days since the last coupon payment/days between coupon payments. The transaction (or
dirty) price minus accrued interest is the quoted (or clean) price.
(b) A good answer should contain the following. The cash flows associated with the underlying
transactions, and margin transactions, are:
Year
Cash flow
Margin Loan
Net cash flow
0
−1,000 × 25 = −25,000
Total −25,000
30% × 25,000 = 7,500
Cash flow +7,500
−17,500
1
+500 × 30 = 15,000
+1,000 × 2 = 2,000
Total +17,000
30% × 15,000 = 4,500
Cash flow −3,000
+14,000
2
+500 × 23 = 11,500
+500 × 2 = 1,000
Total +12,500
0
Cash flow −4,500
+8,000
(c) A good answer should contain the following. We need to find the beta of A, which is
0.6 × 0.45/0.3 = 0.9. Using the definition of Jensen’s alpha, we find that the Jensen’s alpha
of A is:
0.12 − 0.03 − 0.9 × (0.08 − 0.03) = 4.5%.
Therefore, the systematic risk of A is:
(0.9)2 × (0.3)2 = 0.0729.
The unsystematic risk of A is the residual:
(0.45)2 − 0.0729 = 0.1296.
Using the definition of the information ratio, we find that it is equal to 0.045/0.1296 = 0.35.
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FN3023 Investment management
Question 4
(a) What is the difference between direct and indirect bankruptcy costs? Explain
why bankruptcy costs are an important reason why corporations engage in risk
management.
(7 marks)
(b) The stock market has a 2-factor structure. The risk free rate is 3%. The
following table gives the factor betas and returns on three well-diversified
portfolios.
Average
return
Factor-1 beta
Factor-2 beta
Portfolio
A
9%
Portfolio
B
9%
Portfolio
C
?
1.2
0.3
0.8
0.9
0.7
0.6
What is the average return on portfolio C?
(9 marks)
(c) You have the following information about a portfolio, the market index, and the
risk-free interest rate:
Jensen’s alpha
Beta
Unsystematic variance
Average return of the market
index
Variance of the market index
Risk free interest rate
1.5%
1.1
0.02
8%
0.09
3%
What are the Sharpe ratios of the portfolio and the market?
(9 marks)
Reading for this question
For (a), you should read Chapter 9 in the subject guide. For (b), you should read Chapter 6. For
(c), you should read Chapter 8.
Approaching the question
(a) A good answer should contain the following. Direct bankruptcy costs relates to the advisory
costs, legal costs, court costs, management time spent with creditors – the indirect costs
relate to the loss of customers, the costs associated with dealing with suppliers, the costs of
poor management decisions as a consequence of the financial distress. Risk management can
reduce the costs of financial distress and therefore the direct and indirect bankruptcy costs.
Since these costs are borne by the debt holders, risk management can help secure cheaper
borrowing.
(b) A good answer should contain the following. We know that there is a risk premium which is
associated with each factor, and that these risk premia add to the average return linearly,
weighted by the factor beta. Therefore, the risk premia can be extracted using A and B:
0.09 = 0.03 + λ1 × 1.2 + λ2 × 0.3
and:
0.09 = 0.03 + λ1 × 0.8 + λ2 × 0.9.
This yields λ1 = 4.29% and λ2 = 2.86%. Therefore, the average return on C should be 7.7%.
18
Examiners’ commentaries 2019
(c) A good answer should contain the following. The average return on the portfolio is:
1.5% + 3% + 1.1(8% − 3%) = 10%.
Total risk (variance) of the portfolio is:
(1.1)2 × 0.09 + 0.02 = 0.1289.
Therefore, the Sharpe ratio of the portfolio is:
0.10 − 0.03
= 0.195.
(0.1289)0.5
The Sharpe ratio of the market is, in comparison:
0.08 − 0.03
= 0.167.
(0.09)0.5
Question 5
(a) Explain how market risk and liquidity risk differ. How can liquidity risk cause a
problem for the management of market risk?
(7 marks)
(b) You want to make a 6-month put-protected investment consisting of the stock
market index, 6-month put options on the index, and a risk-free investment.
Your objective is to achieve the maximum return when the index has positive
return, and zero return when the index has negative return. The following table
sets out the relevant information.
Current index level
Current 6-month call price
Current 6-month put price
Annual risk free rate (continuously
compounded)
6,956
636.43
532.87
3%
The options have exercise price equal to the current index level, so the payoffs of
the call and put options at maturity are max(0, index level in 6 months − 6,956)
and max(0, 6,956 − index level in 6 months), respectively. Your total investment
is $5m. Derive the breakdown of your investment into the three assets: the
index, put options, and risk-free investment.
(9 marks)
(c) The auto-covariance of successive transaction price changes of a stock is −0.015.
The average stock price is $25. Use Roll’s model to estimate the average
%-spread of the stock.
(9 marks)
Reading for this question
For (a), you should read Chapter 8 in the subject guide. For (b), you should read Chapter 9. For
(c), you should read Chapter 5.
Approaching the question
(a) A good answer should contain the following. Market risk deals with the risk of changes in
the market values of assets and securities, whereas liquidity risk deals with the risk of the
costs of rebalancing portfolios. Since the management of market risk often involves making
rebalancing transactions of portfolios, an increase in liquidity risk can expose the investor to
extra market risk.
19
FN3023 Investment management
(b) A good answer should contain the following. The idea is to invest part of the portfolio risk
free and the other part in call options. This is done by investing risk free:
6,956 × exp(−0.5 × 3%)
which yields a payoff of 6,956 at maturity, and by investing:
6,956 × (1 − exp(−0.5 × 3%)) = 103.5613
in calls, which equals 103.5613/636.43 = 0.1627 units of the call. Next, using put-call parity,
we convert the call into puts, risk free investment and index:
0.1627 calls = 0.1627 × puts + 0.1627 × index − 0.1627 × 6,956 × exp(−0.5 × 3%).
This yields the following final position: a risk free investment of:
(1 − 0.1627) × 6,956 × exp(−0.5 × 3%) = 5,737.39
an index investment of 0.1627 × 6,956 = 1,139.90, and a put investment of
0.1627 × 532.87 = 86.71, a total of 6,956. Since we want to scale up the investment to $5m,
you multiply by 5m/6956 to get a risk-free investment of $4.124m, and index investment of
$0.814m, and a put option investment of $0.062m.
(c) A good answer should contain the following. The Roll model predicts that the half-spread
squared equals the negative of the auto-covariance of transaction prices, or that the half
spread (in $-terms) is the square root of 0.015, equal to $0.1225, or a $-spread of $0.245.
This represents a %-spread of 0.0098, or 0.98%.
Question 6
(a) What is convexity (in bond markets)? Explain how convexity is used to improve
predictions of bond price changes given changes in the yield-to-maturity.
(7 marks)
(b) Assume a stock market where all investors are variance-averse. The average
return on the market index is 8% and the variance is 0.09. The risk free return
is 3%. Suppose an investor has utility function U (µ, σ 2 ) = µ − 0.2σ 2 . What is
the investor’s optimal portfolio?
(9 marks)
(c) A pension liability consists of annual payments according to the schedule in the
table below:
Year
1
2
3
4
5
Payments
$10m
$11m
$12m
$12m
$12m
The interest rate is 5% (assume flat term structure). You seek to hedge the
liability against changes in the flat term structure, taking positions in a bond
portfolio with duration 10 years. How much do you need to invest in these
bonds to hedge the liabilities?
(9 marks)
Reading for this question
For (a) and (c), you should read Chapter 7 in the subject guide. For (b), you should read
Chapter 6.
20
Examiners’ commentaries 2019
Approaching the question
(a) A good answer should contain the following. Convexity refers to the convex curved shape of
the bond price when plotted against the yield-to-maturity. The first order effect on the bond
price is duration, which yields a linear approximation. If convexity is taken into account you
will also capture the second order effects due to the curvature of the shape.
(b) A good answer should contain the following. We know that two-fund separation obtains
when we consider portfolio choice with variance-averse investors. Therefore, the optimal
portfolio consists of the market index and the risk free asset, in proportions x and (1 − x),
respectively. The expected return of the portfolio is:
rF + x(rM − rF )
and the variance is:
x2 × Var(rM )
so the problem for the investor is to maximise:
rF + x(rM − rF ) − 0.2 × (x2 × Var(rM ))
which yields a first-order condition:
rM − rF = 0.4 × x × Var(rM )
equal to 0.08 − 0.03 = 0.4 × 0.09 × x. Therefore, the optimal portfolio is x = 1.38, which
represents 138% invested in the market index and borrowing 38% risk free.
(c) A good answer should contain the following. Before we work out the duration we need the
value of the liability. The present value of the pension liability is:
11
12
10
+
+ ··· +
= 49.14.
2
1.05 (1.05)
(1.05)5
Using the definition of duration, we find it equals:
1
10
11 × 2
12 × 5
×
+
+
·
·
·
+
= 2.99.
49.14
1.05 (1.05)2
(1.05)5
Next, we need to balance the duration by investing x in bonds. Therefore,
x × 10 = 49.14 × 2.99, or x = 14.71. Therefore, the ‘immunising’ investment in the bond
portfolio is $14.71.
Question 7
(a) What is a commercial paper? How is a commercial paper different from a US
T-bill instrument?
(7 marks)
(b) A 5-year bond with annual coupons 5% of the face value is trading at
yield-to-maturity 4%. What is the duration of the bond?
(9 marks)
(c) Consider the Glosten–Milgrom framework for market microstructure. A
competitive risk-neutral market maker clears incoming buy and sell orders in a
sequence. Each order is for one share, and the incoming orders are drawn
randomly from a pool consisting of uninformed traders, making up 80% of the
pool, and informed traders, making up 20% of the pool. The uninformed
traders buy or sell with equal probability, whereas the informed traders buy for
sure when the value is high and sell for sure when the value is low. The high
value is $50 per share, and the low value is $30 per share. The market maker
believes initially that the probability of a high value is 60% and the probability
of a low value is 40%. What bid and ask prices does the market maker quote in
anticipation of the first order?
(9 marks)
21
FN3023 Investment management
Reading for this question
For (a), you should read Chapter 2 in the subject guide. For (b), you should read Chapter 7. For
(c), you should read Chapter 5.
Approaching the question
(a) A good answer should contain the following. Commercial paper is a money market
instrument, which is typically a discount security with no coupon payments (the price is
simply a discounted face value) and maturity less than one year. The commercial paper
distinguishes itself from US T-bills mainly by the issuing entity: US T-bills are issued by the
US government whereas commercial paper is issued by corporations (non-bank corporations,
that is, banks’ version is called a certificate of deposit).
(b) A good answer should contain the following. Duration does not depend directly on the face
value, but rather on the yield, the coupon rate, and the maturity of the bond. Therefore, we
can choose a $1 face value. The price of the bond is, per $1 face value:
0.05
× (1 − (1.04)−5 ) + (1.04)−5 = 1.0445.
0.04
Using the definition of duration we find it equals:
0.05 0.05 × 2
1.05
1
×
+
+
·
·
·
+
= 4.56.
1.0445
1.04
(1.04)2
(1.04)5
(c) A good answer should contain the following. Conditional on an incoming buy order, the
market maker revises their beliefs according to Bayes’ theorem:
P (H | buy) =
(0.8 × 0.5 + 0.2) × 0.6
P (buy | H) P (H)
=
= 0.6923.
P (buy)
(0.8 × 0.5 + 0.2) × 0.6 + (0.8 × 0.5) × 0.4
Similarly, conditional on an incoming sell order, the revised beliefs are:
P (L | sell) =
(0.8 × 0.5 + 0.2) × 0.4
P (sell | L) P (L)
=
= 0.5.
P (sell)
(0.8 × 0.5 + 0.2) × 0.4 + (0.8 × 0.5) × 0.6
Therefore, the ask price is designed to meet the information contained in an incoming buy
order:
0.6923 × 50 + (1 − 0.6923 × 30 = $43.85
and the bid price is designed to meet the information contained in an incoming sell order:
(1 − 0.5) × 50 + 0.5 × 30 = $40.00.
Question 8
(a) What is a credit default swap? What benefits does such an instrument offer to
investors?
(7 marks)
(b) The value of a portfolio over a 30-day period is log-normally distributed, so that
the 30-day log-return has mean 0.4% and variance 0.03. For a standard,
normally distributed, random variable with mean 0 and variance 1, the
distribution function is N (−2.33) = 1%. What is the 30-day, 1% VaR of the
portfolio log-return?
(9 marks)
22
Examiners’ commentaries 2019
(c) Consider the following information about a given portfolio A, the market index
M , and the risk-free interest rate F :
Average return of A
Standard deviation of A
Average return of M
Standard deviation of M
Correlation between A and
M
Risk-free interest rate F
8.5%
40%
8%
30%
0.6
3%
Work out the M 2 -measure and Jensen’s alpha for A.
(9 marks)
Reading for this question
For (a), you should read Chapter 3 in the subject guide. For (b), you should read Chapter 9. For
(c), you should read Chapter 8.
Approaching the question
(a) A good answer should contain the following. Credit default swaps are instruments which
have payoffs for credit events, such as default or a downgrading of the reference bond’s
rating. They are useful to investors because they offer a targeted hedge against credit risk.
(b) A good answer should contain the following. The first approach is to standardise the
log-return so that it has zero mean and unit variance. Standardising the 30-day log-return
yields:
x − 0.004
(0.03)0.5
and we are looking to solve for x such that the standard log-return value is equal to −2.33:
x = −2.33 × (0.03)0.5 + 0.004 = −39.96%.
Therefore, the portfolio will lose at most 39.96% of its value (measured by log-returns) over
a 30-day period for 99% of the time.
(c) A good answer should contain the following. Using the definition of the M 2 measure we find
it equals:
0.3
0.3
1−
× 0.03 +
× 0.085 − 0.08 = −0.875%.
0.4
0.4
The beta of the portfolio is 0.6 × 0.4/0.3 = 0.8. Using the definition of Jensen’s alpha we
find it equals:
0.085 − 0.03 − 0.8 × (0.08 − 0.03) = 1.5%.
23