a) A jet of water coming out of a tap always breaks up into a series of drops. It has been observed experimentally that these drops vibrate. Use dimensional analysis to predict how the frequency of vibration of the water drops will depend on their density, radius and surface tension. f6g b) An ideal polarising filter is oriented such that incident horizontally polarised light is transmitted 100% and vertically polarised light is all absorbed. (These directions relate to the electric field). (i) When light of amplitude E0 = 200 Vm1 and linearly polarised at 45% to the vertical is incident, what polarisation and what amplitude is transmitted? f4g (ii) When circularly polarised light of the same amplitude is incident, what polarisation and what fraction of the incident amplitude is transmitted? f3g c) An outdoor sound system comprises two identical loudspeakers distance d = 5 m apart as sketched in the diagram below. As part of a test they are both fed the same pure note, of same amplitude A0 and phase, corresponding to a sound wavelength l = 0:6 m (and same frequency w). L y d P y = 0 S1 S2 (i) The speakers S1 and S2 emit waves y1 = A0 cos (kx1wt) ; y2 = A0 cos (kx2wt) ; respectively, where k is the wavenumber and x1 and x2 are distances measured from S1 and S2. Show that the total wave at P is given by y (x; t) = 2A0 cos (kX Wt)cos(kD) ; and define X;W and D. f4g (ii) Show that for y << L the sound amplitude at point, P, in the diagram is expected to be proportional to cos pdy lL f4g: (iii) An observer stands L = 20 m away, at y = 0. They move along the positive y-direction. At what value of y(> 0) will they first hear a maximum amplitude. f4g
The objective of the coursework is to allow you a first hand appreciation of some of the key issues in measuring risk. There are two parts, one involving analysis of a portfolio having a single risk factor, the other involving analysis of a portfolio having two risk factors.
Select, and acquire historical data for, a traded financial asset. This might, for instance, be a commodity, a security, or a stock market index. Suppose that you had invested 1 million pounds in this asset at the date given by the earliest date in your data. (a) Explain your choice of sample size. (b) Using the data up to, but not including, 15th July 2021, calculate the simple daily returns for your asset [use simple returns throughout this coursework]. Examine and describe the key statistical features of your sample of returns. (c) Calculate VaR and ES for 15th July 2021 using a one day holding period, and a confidence level of 90%, using the following methods: i. Basic Historical Simulation ii. Age-weighted Historical Simulation iii. Hull-White iv. Parametric, using the Normal distribution, without volatility adjustment v. Parametric, using the Normal distribution, with volatility adjustment vi. Parametric, using what you believe to be an appropriate distribution, without volatility adjustment vii. Parametric, using what you believe to be an appropriate distribution, with volatility adjustment You should present your results in a single table, and briefly provide a commentary on the similarities and differences. (d) Explain why it would be problematic to have used log returns to calculate VaR and ES for any of the parametric methods in the previous question.
Acquire data for another asset and suppose that at at the start of the time series you invested 1 million pounds in this asset as well. You now have a portfolio which at the start of the data series was worth 2 million pounds. Suppose initially that your portfolio is not actively managed, so that your holdings of each asset remain unchanged. (a) Using the data up to, but not including, 15th July 2021, calculate the simple daily returns for each of the individual assets which constitute your portfolio. Examine and describe the key statistical features of your sample of returns. (b) Calculate VaR and ES for 15th July 2021 using a one day holding period, and a confidence level of 95%, using each of the following methods i. Basic Historical Simulation ii. Age-weighted Historical Simulation iii. Hull-White iv. Parametric, using the Normal distribution, without volatility adjustment v. Parametric, using the Normal distribution, with volatility adjustment vi. Parametric, using what you believe to be an appropriate distribution, without volatility adjustment vii. Parametric, using what you believe to be an appropriate distribution, with volatility adjustment (c) Explain how you might have been able to reduce your risk exposure for 15th July 2021 had you been able to adjust your portfolio on the 14th July 2021. Use the Basic Historical Simulation method for this part. The deadline for submission is as notified in the module outline. Please see Moodle for further discussion of useful approaches to this topic, and hints about R code.
