MTH6113: Mathematical Tools for Asset Management Coursework 1 for Weeks 1 & 2 Spring 2023
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MTH6113: Mathematical Tools for Asset Management
Coursework 1 for Weeks 1 & 2
Spring 2023
● This Coursework consists of four parts:
I. Revision of probability theory
II. Efficient market hypothesis
III. Stochastic model of asset returns
IV. Hands-on-data homework problems using MS Excel
Challenging exercises are marked with an asterisk *.
● The exercises are there to give for your active exam preparation. Use them to prac- tice! Some exercises will be discussed during the tutorial-style lecture.
● Excel based exercises: Data is essential in finance therefore this course includes data based exercises. For this reason, the in-term assessment making up 30% of your grade will be Excel-based.
I. Revision of probability theory
A. Correlation and independence:
1) Assume playing the following game. You have two rounds of fair coin tosses. Each round you win £1 for heads, but lose £1 for tails. However if you lose in the first round, the returns for the second round will double (i.e. you can win or lose £2). Let Rl denote your profit in the first round and R2 the profit in the second round.
1) Compute the expected return _(Rl + R2 ).
2) Are Rl and R2 independent?
3) Are Rl and R2 correlated?
4) Are Rl and IR2 I correlated?
2) Let two random variables X and Y be given. The correlation coefficient given as ρx,Y = cov(X, Y)/(σx σY ) for the covariance cov(X, Y) and the standard deviations
σx and σY . We have given some scatter plots of samples of the random variables. Assign to each plot an approximate value of the corresponding correlation coefficient.
3
2
1
0
-1
-2
-3
-3 -2 -1
ρx,Y s
3
2
1
0
-1
-2
-3
-3 -2 -1
ρx,Y s
-3 -2 -1 ρx,Y s
-3 -2 -1 ρx,Y s |
3 2 1 0 -1 -2 -3 -3 -2 -1 ρx,Y s 3 2 1 0 -1 -2 -3 -3 -2 -1 ρx,Y s |
B. Basic stochastic calculations:
1) Imagine rolling dice several times. For each dice roll you receive as many pounds as the shown value. E.g. throwing 5, 4, 6, 3, 2 leaves you with £20. What is your expected gain after five rounds?
2) You play a similar game, but instead of the sum of all values, you get the average value of your dice throws. Now throwing 5, 4, 6, 3, 2 only leaves you with £4. What is the expected gain after five rounds?
Let Xi , i = 1, . . . , N be discrete random variables with expectation value µi = _(Xi ) and variance σi(2) = Var(Xi ) = _ ╱(Xi _ µi )2 ← . Define X = (Xl + . . . + XN )/N and show the following properties:
3) _(X) = (µl + . . . + µN )/N .
4) Var(X) = (σl(2) + . . . + σN(2))/N2 when Xi are mutually independent.
5) Let Xi be independent and identically distributed with µi = µ and σi = σ . Compute _(X) and _((X _ _(X))2 ). Use the result to explain the Law of Large Numbers.
II. Efficient market hypothesis
C. Predictability of asset returns.
1) Let Rt , t = 1, 2, . . . be the daily returns of a certain asset. Assume a significant corre- lation of subsequent returns, i.e. ρat+1,at > 0. How would you design an investment strategy for this asset?
2) Discuss if such market behaviour sounds reasonable.
3) Assume a strong dependency of Rt+1 and Rt , but no correlation, i.e. ρat+1,at s 0. Can you still design an investment strategy based on this dependency?
D. In the job description for a Quant position, which you think about applying for, you read the following paragraph:
“Using the latest machine learning modelling techniques, robust statistical analysis and pattern recognition, you will analyse thousands of asset price time series, extracting deep insights.”
You assume that the company hopes to gain insights on the future movement of the asset price.
Do they believe in
● the weak form of market efficiency?
● the semi-strong form of market efficiency?
● the strong form of market efficiency?
Justify your answer in 2-3 sentences.
E. Efficient markets:
You work in a bank and based on an article you read about Warren Buffet, there is a discussion emerging on market efficiency.
1. The article you read included the following comments on his investment style:
”Warren Buffet analyses companies using all information that is published. Then he invests if there is long-term prospect. We believe that he beats the market because his own trading positions are published with a delay. Since he is famous, he knows that many market participants will copy his strategy. So whatever his investment decision was, the prices of these assets will rise because he has created the interest in them.”
Which form of the efficient market hypothesis do they believe is wrong? Justify your answer
2. You explain that you do believe in the strong form of market efficiency. One colleague then tells you
“Efficient Market? No way.. I’ve heard about at least a dozen people, who made a bundle in the stock market”
How would you react?
3. Another colleague joins the discussion and says
“The random-walk theory implies that events are random, but many events are not
random. If it rains today, there’s a fair bet that it will rain again tomorrow” What do you tell them?
III. Stochastic models of asset returns
F. Log-normal model.
Let St be the simulated stock price of the lognormal model with parameters µ and σ , i.e. log(St+1/St ) ~ X(µ, σ2 ) iid and a given S0 > 0. Compute the following values for log-return and log-prices with s > t > 0:
1. _(log(St ));
2. _(Xt ), where Xt = log(St+1/St );
3. Var(log(St ));
4. cov(Xs , Xt )
G. Taken from a previous year’s sample exam: Inspect the plots of data of daily stock returns in the following figure.
Part I: What is an adequate statement related to the left graph? State the correct answer and shortly justify your choice.
A. Volatility clusters are observed. Namely, very frequently a positive return is followed by a positive return. Also on a finer scale, a large positive return is very frequently followed by a large positive return. The same is true for negative returns.
B. Volatility clustering means that large squared daily returns are likely to follow each other. We observe this in the data, in particular before 2013: A period with small absolute returns is followed by a period with frequent large absolute returns.
C. Volatility clustering means that large successive daily returns are likely to follow each other. This contradicts the hypothesis that successive daily returns are uncorrelated.
Part II: What is an adequate statement related to the right graph?
D. The graph shows the daily returns versus the daily returns of the successive days. Due to the shotgun pattern we cannot easily predict the location of the next day’s return.
E. The right graph shows for each day t the likelihood that the return of the next day Rt + 1 is positive conditioned on the return of this day Rt .
F. Volatility clusters are observed in the the shotgun pattern. Summing up all returns in the 6-ball around zero gives us the 6-intensity of the cluster.
IV. Hands-on-data homework
H. If necessary refresh your knowledge in MS Excel, e.g. based on your first-year lecture MTH4114: Computing and Data Analysis with Excel. Also check tutorials or search for solutions online, if you encounter problems.
1) Queen Mary provides a license to MS Excel. You can use it on the university com- puters or install it as a part of Microsoft Office 365 ProPlus. To install MS Excel, please follow the instructions of the IT Servicel.
2) Download the file HSBC prepared .xlsx and open it in Excel. The worksheet will include the calculated returns and plots of the stock price as well as the returns over the last 20 years:
3) Open the file AAPL.xlsx and recreate the plots for the stock price of Apple over the last year.
Use the stock price stored in column B daily return Rt = St+1/St _ 1 and generate a plot over the time.
Hint: Note that we cannot compute the return for the final date. Therefore, the vector of returns will have one item less than the stock prices.
5) Compute the average value of the daily returns. What does this mean from an investor’s point of view?
6*) Compute the correlation coefficient of subsequent daily returns. What does this
imply for investment strategies?
I. Fit and recreate the log-normal model: Use Excel to recreate the workflow discussed in Lecture 4:
1. Select a public company and download data from Yahoo Finance (use daily values and the maximal available time period)
2. Import the data in Excel and clean data (i.e. remove all columns except for the date and ‘Adj. Close’, change the header to something reasonable)
3. Compute daily log-returns using LOG()
4. Estimate parameters using AVERAGE() and STDEV.S()
5. Simulate by samples of normal random variables;
● Use NORM .INV(RAND(), mu, sigma) to create random values which are nor- mally distributed
6. Investigate the model: Plot a histogram each of the actual and the simulated returns (if histograms are not available in your excel version, use a line plot over time)
7. Discuss the model: What difference do you see in the plot of the actual and the simulated returns?
2023-05-13