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BENG0091 Coursework 1

Things to do

1.    We will first do some preliminary work that will enable us to simulate stochastic realisations of the prices of the stocks of interest.

(a) Derive an explicit mathematical formula for the cumulative distribution function of the Laplace distribution.                   [2]

(b)    Describe a simple approach for the generation of random deviates from the Laplace distribution.                                                                                                                                             [4]

(c)    Write a program that generates 100000 samples from the Laplace distribution with μ = 1.5 and b = 0.8 (use the approach you developed in the previous question; do not use built-in functions for this). To verify that your method works correctly, show a histogram of these samples and a line-graph of the probability density function (both in the same plot).           [6]

2.    We are now able to perform the stochastic simulations of the prices of the three stocks of

interest. We will assume that each year contains 260 days of trading (roughly 52 weeks of 5

working days each), and we will simulate stock prices for a 5-year window,i.e.  nmax   = 1300 days.

(a)    Write a program that simulates the stock pricesequencies  Sn(EIF) , Sn(CES) , Sn(Q)NT , n = 0,1,2,..., nmax  .

To do this, first generate three random streams, one for  Xn , n = 1,2,..., nmax , one for  Yn    and   similarly, one for  Zn , using the method you developed in Question1. Thus, you are allowed to use only a uniform random number generator available for the programming language of your choice (MATLAB, Python etc.), initialised with a seed value of your choice.

Subsequently, use these sequences in the application of the equations that give the stock

price per day, e.g. for CES, you will evaluate the expression  Sn(CES) =  (1 + 0.3Xn  + 0.8Yn ) . S1(S) recursively. Present graphs of the stock prices per day in the same plot.                                [12]

(b)    Repeat the procedures of Question2(a)with two more different random seeds and

produce plots of the stock prices ped day. You will present two separate plots, one for each random seed. Each of these plots will contain three line-graphs, one for each stock (EIF,

CES, and QNT). Comment on the observed behaviour in the three “scenarios” that you

simulated and plotted in Questions2(a)and2(b). For instance, how does the magnitude of the fluctuations (volatility) compare with the overall “average” behaviour of a stock (drift) in the three different scenarios? Which stock(s) would you pick for your portfolio if you were an investor?                             [10]

3.    We will continue with the calculation of certain quantities that will allow us to apply the efficient frontier approach to the data we generated for the prices of the three stocks overtime.

(a)    Pick one of the three scenarios you simulated (whichever you want), and for each of the stock pricesequencies, calculate the daily returns as follows:

DRn(ABC) =  −1              n = 1,2,...,nmax

where ABC will be each one of the stocks in discussion (EIF, CES, QNT). Present, in three    separate plots, one for each stock, histograms of the daily returns you calculated. Discuss movements you plotted earlier.                         [10]

(b)   Calculate and report the following quantities to 3 significant figures:

•    Annualised mean returns:

ARABC  = (1 +DRABC )260  −1

where: DR     = nmax  DRn

Report one number for each stock, ABC = EIF, CES, QNT.

•    Annualised (co)variances of the daily returns. The following formula is used to calculate variances when ABC = DEF, but also covariances when ABC ≠ DEF (where ABC and DEF    can be any of the stock symbols: EIF, CES, QNT):

 ABC   DEF                                          ABC   DEF−−

ACV          =260 .DCV

where: DCVABC −DEF  =  DRn(ABC) − DRABC )(DRn(DEF) − DRDEF )

Thus, report 6 numbers in total: 3 variances, one for each stock, and 3 covariances, one   for each pair: EIF-CES, EIF-QNT, CES-QNT.                                                                                  [9]

4.    We now have the data necessary to apply efficient frontier theory. To this end, we will consider an ensemble (collection) of portfolios, all of which invest in the three stocks under

consideration. However, each portfolio will have different weighting factors per stock,e.g. in    one portfolio the allocation may be 40% EIF, 20% CES and 40% QNT; for another portfolio the   allocation maybe 30% EIF, 35% CES and 35% QNT. Note that the weighting factors will have to sum up to 1 (i.e. 100%).

(a)    For the scenario you picked in Question3(a), create an ensemble of 2000 portfolios, each      with different allocation. Thus, for each portfolio you will generate 3 weights that sum up to 1 and represent the relative numbers of shares bought for each of the 3 assets (EIF, CES,

QNT). Then calculate the expected (annualised) return and variance of each portfolio according to the following formulas:

 ABC

Return  =         Σ     wABC AR

ABC E {EIF,  CES,  QNT}

Variance  =         Σ          Σ     wABC  wDEF  ACVABC −DEF

ABC E {EIF,  CES,  QNT}  DEF E {EIF,  CES,  QNT}

where wABC  is the weight of the stock of ABC. Present a scatter plot of the return in they-

axis with respect to the standard deviation, i.e. Variance½  in the x-axis. Find the portfolio

with the minimum variance, highlight it on your plot (e.g. with a marker with different shape or colour) and report its allocation (i.e. the weights).                                                      [15]

(b)    Discuss the results you have obtained. Compare the weights of the minimum-variance

portfolio, with the weights of “riskier” portfolios that may have better or worse returns

than the minimum-variance one. How does the above plot guide you in choosing an allocation for your portfolio?        [12]

5.    Finally, present your Matlab Python (or other) codes as Appendixes to your report or preferably upload them to Datalore. If presenting your codes in an Appendix make sure these are copy-

pasted as text, not as figures, and that they don’t contain the code’s line numbers (see the guidelines in the first page for more detailed instructions) .                              [20]