MTH6113: Mathematical Tools for Asset Management Coursework 2 for Weeks 3 & 4 Spring 2023
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MTH6113: Mathematical Tools for Asset Management
Coursework 2 for Weeks 3 & 4
Spring 2023
● This Coursework consists of four parts:
I. Stochastic model of asset returns
II. Risk and Return
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. Stochastic model of asset returns
A. Last year’s exam question
One of the following figures shows the empirical returns of a stock since 1993. The other figure shows simulated returns using the lognormal model with parameters fitted to the empirical data.
Figure A:
Figure B:
Complete the following statements, such that they are true.
The daily return shown in Figure A
1. show a significant underestimation of the largest total loss.
2. were generated using the autoregressive AR(1) model.
3. are not independent and identically distributed as we observe volatility clustering.
The daily return shown in Figure B
1. are not independent and identically distributed as we observe volatility clustering.
2. show a significant underestimation of the largest total loss.
3. were generated using the autoregressive AR(1) model.
II. Risk and Return
B. Order the following assets (current price s0 = 1. 000) by their pairwise dominance and find the efficient subset. To do this consider the following table, which states s1 as well as the expectation and variance of the return R0 = s1 /s0 - 1:
Asset proba 50% |
proba 50% |
E(R0 ) |
Var(R0 ) |
A1 |
1,000 |
1,200 |
0.1 |
0.01 |
A2 900 |
1,300 |
0.1 |
0.04 |
|
A3 950 |
1,350 |
0.15 |
0.04 |
|
A4 |
1,350 |
850 |
0.1 |
0.0625 |
A5 100 |
2,000 |
0.05 |
0.9025 |
C. Assume that we have two assets. The first one has an expected return of μ 1 = 10% and standard deviation of return equal to u1 = 0725. The second asset has the current value s and the future value s . Assume that the future price of the second asset will have E(s) = 100 and the standard deviation of the return is equal to u2 = 072. For the following two cases, find the valid range of the current price s, such that the following conditions are satisfied:
1. The second asset dominates the first asset.
2. No asset is dominated by the other asset.
Hint: First evaluate μ2 = E(R) for the return R = s/s - 1 in dependence of s .
III. Hands-on-data homework
D. Let our return be given by the normally distributed random variable X ~ N(0701. 0712 ).
1. Evaluate the shortfall probabilities
● SF(10%. X);
● SF(50%. X).
Represent them in terms of the distribution function, and evaluate the term, e.g. using the Excel function NORM .DIST.
Hint: Use Excel’s formula builder for help with the use of NORM.DIST. For the cum- mulative distribution function use the option TRUE for cummulative .
2* Now create 1 000 samples of this random variable by executing
NORM7INV(RAND(). 0701. 071)
in the cells A2:A1001 Then the 1 000 samples are stored in the vector empirical. Compute the empirical shortfall probabilities and the Value at Risk:
● SFe (10%),
● SFe (50%)
and compare them to the theoretically computed values.
Hint: Use COUNTIF to count the number of values fulfilling a certain condition. For example COUNTIF(A1:A1000, "<0 .0") counts the number of negative entries in the cell range.
E. Download the Excel file CW2 E .xlsx from QMplus.
1. Watch the corresponding video on QMplus to see its construction. We consider a stochastic volatility AR(1) model. The parameters have already been fitted and the modelled returns are computed.
2. Continue the calculation of the modelled asset price in column L using the modelled returns.
3. Create a line plot of the empirical return and the modelled return.
4. Interpret the line plot of the returns with respect to volatility clusters and spikes.
2023-05-13