MTH6113: Mathematical Tools for Asset Management Coursework 3 for Weeks 5 & 6 Spring 2023
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MTH6113: Mathematical Tools for Asset Management
Coursework 3 for Weeks 5 & 6
Spring 2023
● This Coursework consists of four parts:
I. Risk and Return
II. Mean-Variance Portfolio Theory
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. Risk and Return
A. Assume the following lottery L, which costs S0 = 1 (pound sterling) to participate. A random number within the interval (0.5, 1.5) (with a uniform distribution) is generated and determines your payoff S1 in £ . E.g. if the random number is 1.3 your effective gain is £0.30.
1. Compute the variance and the semi-variance _ ((min(0, R0 _ _(R0 ))2 ) of your return R0 = S1 /S0 _ 1.
2. What is your 10% shortfall probability SF(0.1, R0 )?
3. What is the Value at Risk VaR0.99 ?
B. Monotonicity of the risk measures.
Let the returns be given by a real random variable X .
1. Show that for b1 > b2 we have SF(b1 , X) < SF(b2 , X).
2. Assume that the distribution function of X , FX is strictly monotonic increasing. Show that for b1 > b2 we have SF(b1 , X)< SF(b2 , X).
3. Assume that the distribution function of X , FX is continuous and strictly monotonic
increasing. For α 1 > α2 show that VaRα 1 (X) > VaRα2 (X).
Hint: Use that VaRα (X) = _FX(−)1 (1 _ α)
4.* For a general distribution X , α 1 > α2 show that VaRα 1 (X) > VaRα2 (X).
Note: Do not assume continuity/strict monotonicity of FX .
II. Mean-Variance Portfolio Theory
C. Preparation for Portfolio Theory
In your new job, you mange a portfolio, which was initiated by your colleague. One year ago, the client gave your company £100 000 to manage. Your colleague invested them as follows
● 4 000 stocks of HSBC trading for £7.93 each;
● 800 stocks of Shell trading for £25.67 each;
● the remaining money was invested in a risk-free deposit with 2% interest compoun- ded continuously.
You look at the current rates and notice that HSBC lost some value and now trades for £6.47, while Shell trades for £23.61. Thus you start to analyse the current state of the portfolio as follows. (You may round the value of the portfolio to the nearest £ . Also we assume that no dividends are paid.)
1. How much money was invested in the risk-free deposit?
2. What is the current value of the portfolio?
3. At the current time, what is the proportion of the portfolio invested in HSBC and Shell (in terms of the value in £).
4. The investor wished to always have not less then 45% but not more than 55% of the portfolio’s value to be risk-free. Do you need to adjust your portfolio now?
D. Portfolio analysis
Assume you have a portfolio consisting of two stocks of company 1 (S1 ) and one stock of company 2 (S2 ). Today each of the stocks is worth £1, 000.
The probabilities of the stock prices tomorrow are given in the following table:
|
p=1/4 |
p=1/4 |
p=1/4 |
p=1/4 |
|
750 |
950 |
1050 |
1300 |
|
950 |
750 |
1300 |
1050 |
1. Use Excel to compute the expectation and variance of the return for each well as their correlation.
2. State the random variable for the value of your portfolio at t = 1 and compute its expectation and variance of the return.
E. Minimal Variance Portfolio
Compute the weights associated to the MVP in the case of two assets and a general ρ . Let σ 1 > σ2 > 0 be the standard deviation for both assets and ρ e (_1, 1) their correlation.
1. Show that the MVP without short-selling is given for
w 1(M)VP = max ╱ 
, 0、.
Hint: compare the calculation from class for ρ = 0 .
2. Show that w 1(M)VP > 0 iff ρ < σ2 /σ 1 .
F. Attainable portfolios
Assume that µ 1 = 10%, µ2 = 20%, σ 1 = 0.1, σ2 = 0.3 and ρ = 0.1.
1. Find a portfolio with σP < σ 1 .
2. Is it possible to construct a portfolio with expected return equal to 30% without short-selling.
III. Hands-on-data homework
G. Download data from the Excel file CW3 .xlsx. Column A contains daily returns, which were used to fit a lognormal model with the parameters µ and σ given in cells D2 and D3, respectively. Under the lognormal model, our return is given by the normally distributed random variable X ~ N(µ, σ2 ).
1. Evaluate the following Value at Risk:
● VaR90% (X);
● VaR99% (X).
You may use Excel’s function NORM .INV to evaluate the inverse of the distribution function.
2. For comparison, evaluate the empirical Value at Risk VaR90% (X) and VaR99% (X) using the 5, 000 real-life returns provided in column A.
Hint: Use PERCENTILE or modify column A to sort the data. With 5000 data points in total, 10% correspond to 500 data points, while 1% corresponds to 50 data points. Note that you might get slightly different results with both methods.
3. Interpret the difference in the VaR99% (X) computed using the empirical returns and using the lognormal model.
2023-05-13