MTH6113: Mathematical Tools for Asset Management Coursework 4 for Weeks 7 & 8 Spring 2023
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MTH6113: Mathematical Tools for Asset Management
Coursework 4 for Weeks 7 & 8
Spring 2023
● This Coursework consists of three parts:
I. Mean-Variance Portfolio Theory
II. Factor Models of Asset Returns
III. 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. Mean-Variance Portfolio Theory
A. Portfolio analysis and optimisation
Assume you have a portfolio consisting of 12 stocks S1 of company 1 and two stocks S2 of company 2.
The current stock prices are S1 (0) = £25 and S2 (0) = £100. From your data analysis you know that the expected returns are µ 1 = 匝(R1 (1)) = 0.08 and µ2 = 匝(R2 (1)) = 0.1, the variance is the same in both cases, with σ 1 = ^Var(R1 (1)) = σ2 = ^Var(R2 (1)) = 0.1.
Furthermore you know that both stocks are positively correlated with cov(R1 (1), R2 (1)) = 0.005.
1. What is the current value of the portfolio P (0)?
2. What are the weights of both investments w1 , w2 ?
3. What is the expected return of the portfolio 匝(RP )?
3. What is the variance of the portfolio Var(RP )?
4. Consider the three investment choices ”Stock 1”, ”Stock 2” and ”Portfolio” in terms of their expectation and variance. Find the efficient subset.
5. Find the minimum variance portfolio consisting of Stock 1 and 2 without using the general formula shown in I-B-2.
B. Attainable set and efficient subsets
Let us consider two assets (σ1 , µ 1 ) = (0.2, 0.2) and (σ2 , µ2 ) = (0.2, 0.1), correlated with a coefficient ρ = 0.75. Risk-free interest is provided with a return of µ0 = 7.5%. Investigate the attainable set as follows:
1. Compute the minimal variance portfolio based on asset 1 and 2 only. Use it to sketch the attainable set of portfolios consisting of assets 1 and 2 with no short positions.
2. Now include the risk-free asset and compute the market portfolio. Use it to sketch the capital market line as well as the attainable set for portfolios consisting of all three assets with the possibility of short positions.
3. Sketch the efficient frontier for the two cases with and without short-selling.
C Portfolios in the σ-µ-plane. Consider two assets (σ1 , µ 1 ) and (σ2 , µ2 ), the attainable set (of these two assets only) and a risk-free asset as shown in the following graph:
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0 0.1 0.2 0.3 0.4
Note that we are looking for a graphical solution so the precise values of σ and µ are not given. Sketch the following portfolios in the σ-µ-plane.
1. With an initial wealth of £1 000:
● Buy £500 of asset 1;
● Buy £500 of asset 2;
2. With an initial wealth of £1 000:
● Borrow £1 000 from the bank;
● Buy £1 000 of asset 1;
● Buy £1 000 of asset 2;
3. With an initial wealth of £500:
● Buy £1 000 of asset 1;
● Short-sell £500 of asset 2;
D. We consider the same situation as in part B and optimise our portfolio according to given restrictions. Sketch the position of the following portfolios in the graph.
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0 0.1 0.2 0.3 0.4
You wish to invest in a portfolio with the same risk as asset 1. Of course your investment shall be an efficient portfolio. Compute the portfolio for the following three cases and explain the investment.
1. with no limits on short-selling;
2. without short-selling;
3. without short-selling of the assets, but the possibility to borrow money at the rate µ0 .
E. Attainable portfolios
Reconsider question F from CW3, i.e. assume that µ 1 = 10%, µ2 = 20%, σ 1 = 0.1, σ2 = 0.3 and ρ = 0.1. In the previous coursework, you have seen that short selling is required to construct a portfolio with expected return equal to 30%.
With short selling being allowed, construct the portfolio with expected return equal to 30%.
F. Previous year’s exam question 1
Note: The following two questions are excerpts from last year’s final exam and take a form suitable for a QMplus exam. In this case, they are multiple choice and fill-the- blank questions. Solve them on paper as usual and then select the correct answer among the possibilities.
Assume that we have four assets. The first one has expected return µ 1 = 20% and standard deviation of return equal to σ 1 = 10%. The second has expected return µ2 = 40% and standard deviation of return equal to σ2 = 20%.
● Assume that the third asset has expected return µ3 = 10%. What is the range of the standard deviation σ3 of the third asset so that the three assets form an efficient
set?
Select one:
1. The standard deviation of the third asset needs to be below 10%.
2. The range is the empty set, as it is not possible that all three assets are efficient in this case.
3. The standard deviation of the third asset needs to be above 10%.
4. The standard deviation of the third asset needs to be below 20%.
5. The standard deviation of the third asset needs to be above 20%.
6. The standard deviation of the third asset needs to be between 10% and 20%.
● Fill the two gaps [Gap A] and [Gap B] in the following text among the possibilities below:
Next we want to determine the range of µ4 and σ4 such that Asset 4 dominates Asset 2, but does not dominate Asset 1. After careful calculation and checking our result by drawing a graph (please take this as a hint how to work to obtain the solution), we know that Asset 4 dominates Asset 2 and not Asset 1 if and only if
the expected return of Asset 4 is [Gap A] and the standard deviation of the return of Asset 4 is [Gap B] and the pair (µ4 , σ4 ) is not equal to the pair (µ2 , σ2 ). Possible choices for Gap A:
1. larger than or equal to the expected return of Asset 2.
2. smaller than or equal to the expected return of Asset 2.
3. smaller than the expected return of Asset 2.
4. larger than the expected return of Asset 2.
5. larger than that of Asset 1 and smaller than that of Asset 2.
6. larger than or equal to that of Asset 1 and smaller or equal to that of Asset 2.
7. larger than the expected return of Asset 1.
Possible choices for Gap B:
1. larger than or equal to that of Asset 1 and smaller or equal to that of Asset 2.
2. larger than that of Asset 1.
3. larger than or equal to that of Asset 1.
4. smaller than or equal to that of Asset 2.
5. smaller than that of Asset 2.
6. is larger than that of Asset 1 and smaller than or equal to that of Asset 2.
7. smaller than or equal to that of Asset 1.
8. smaller than that of Asset 1.
9. larger than that of Asset 2.
10. larger than or equal to that of Asset 2.
II. Factor Models of Asset Returns
G Portfolios in Sharpe’s Single Index Model
Let the returns of d stocks be given: Ri = 1, . . . , d, which follow Sharpe’s Single Index
Model:
Ri 一 µ0 = αi + βi ╱ RMP 一 µ0、+ εi ,
with the risk-free rate µ0 , αi , βi ← R and εi ↓ 衬(0, σε(2)i ) mutually independent for i = 1, . . . , d.
We consider a portfolio built by these stocks with weights w1 , . . . , wd , such that w1 +...+ wd = 1. The portfolio’s return is given as
d
RP = wi Ri .
i=1
Show that the portfolio’s return also follows Sharpe’s single index model, i.e.,
RP 一 µ0 = αP + βP ╱ RMP 一 µ0、+ εP ,
with αP = i(d)=1 wi αi , βP = i(d)=1 wi βi and εP ↓ 衬(0, σP(2)), where σP(2) = i(d)=1 wi(2)σε(2)i .
H Factor loadings.
Let Z1 , Z2 , Z3 ↓ 衬(0, 1) be iid. We build a factor model for the dependent random variables
Y1 = 4Z1 + 2Z2 + Z3 ,
Y2 = 4Z1 一 4Z2 ,
Y3 = 4Z1 + 2Z2 一 Z3 .
We have identified the single fundamental factor F = Z1 + Z2 + Z3 and the factor model
Yi = bi F + εi .
Find the best fit of the factor loadings bi , i.e. values of bi such that the idiosyncratic risk εi has mean zero and minimal variance.
III. Hands-on-data homework
J. Numerical portfolio optimisation
Consider the two stocks (GSK and AZN) given in CW4J .xlsx. Evaluate the attainable set and evaluate the MVP as follows:
1. Estimate the mean, standard deviation and the correlation.
Hint: Revisit CW1 and 2 to recap how to estimate the parameters of stock returns. Use caRREL to evaluate the empirical correlation between the two assets.
2. Create a list of weights for w1 from 0 to 1 in steps of 0.01. For each value, calculate the expected return and the standard deviation of a portfolio with weights w1 and w2 = 1 一 w1 .
3. Create a scatter plot with lines to plot the attainable portfolios in the σ-µ-plane. Try to make the plot look as nice as possible, e.g., by including a marker for the two assets, labels and legends.
4. Calculate the MVP (minimal variance portfolio). Add it as a data point to the plot to validate your results.
5. Provide a short economic interpretation of your result
6.* Now choose your own stocks. Try to find two UK-based companies with a large
correlation of the stock price; Download the data from Yahoo Finance and again use Excel to evaluate the attainable set and evaluate the MVP.
Hint: When downloading data make sure to select equal time ranges for both assets, so you can merge the data properly.
K Parameter estimation
We consider the situation as in problem H, but instead of an explicit formula for Y and F, we have empirical data. Download the Excel file CW4K.xlsx for the data.
Find the best fit of the factor loadings bi and the standard deviation of the idiosyncratic risk σi in the factor model
Yi = bi Fi + εi ,
with Fi , εi ↓ 衬(0, σi(2)) pairwise independent. We will use the empirical mean, variance and correlation together with their theoretical calculations
Cov(Yi , F) = bi Var(F). (3)
a) We evaluate the empirical mean of X and F to see that they are both close to zero and (1) is fulfilled.
b) Equation 3 yields a formula for bi only depending on values which we can estimate empirically:
bi = Cov(Yi , F)/ Var(F).
We evaluate the empirical covariance and variance to get an empirical estimate for bi .
Note: Use cav4RI4NcE .s to evaluate the empirical covariance between the two data series
c) With bi known (2) now yields a formula to evaluate σi .
d) After solving Question I: Compare the empirical factor loadings to the ones we computed by hand. How good was the estimate?
2023-05-13