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MATH 377
MOCK FINAL EXAM - I
Financial and Actuarial Modelling in R
1. An investor invests in three stocks, A, B, and C. The stocks A, B, and C have expected rates of return E[RA] = 9%, E[RB ] = 12%, and E[RC ] = 15%. The risk-free rate is 3%. The covariance matrix of the three stocks is:
|
|
RA |
RB |
RC |
|
RA |
0.08 |
0.045 |
0.01 |
|
RB |
0.045 |
0.25 |
0.07 |
|
RC |
0.01 |
0.07 |
0.06 |
(a) Find the mean and standard deviation of a portfolio such that the weight of stock A is twice that of stock B and twice that of stock C. [3 marks]
(b) Write an R program that plots the opportunity set available to any investor. [6 marks]
(c) Write an R program to plot the capital market line. [3 marks]
(d) Write an R program to find the minimum variance portfolio. What are the weights of stocks A, B, and C in this portfolio? [3 marks]
2. A stock price is currently 30. Over each of the next two 3-week periods, it is expected to increase by 10% or decrease by 8%. The risk-free interest rate is 6% per annum with monthly compounding during the first 3-week period and 4% per annum with weekly compounding during the second 3-week period.
(a) Write an R program that gives the binomial tree evolution of the stock price. [3 marks]
(b) Write an R program to find the initial price of a 6-week derivative that pays off max(4/1 (900 − ST/2), 0). [5 marks]
3.
(a) Consider a 4-month European call option on a stock with a current price of 30. The exercise price is 29, the risk free interest rate with continuous compounding is 5% per annum, and the volatility is 25% per annum. What is the option ’s price? [4 marks]
(b) Write an R code to plot 100 simulated trajectories of a geometric Brownian motion with an initial value of 30, a drift of 0.01875 (0.05 − 0.252 /2), and volatility of 0.25 up to time 1/3. [6 marks]
(c) Generate 5000 simulations of the process in (b) to approximate the price of the option described in (a). [4 marks]
4.
(a) Consider the following two vectors: height <− c(151 , 174 , 138 , 186 , 128 , 136 , 179 , 163 , 152 , 131) weight <− c(63 , 81 , 56 , 91 , 47 , 57 , 76 , 72 , 62 , 48)
(i) Fit a linear regression model to explain the dependent variable “weight” in terms of the independent (explanatory) variable “height.” [4 marks]
(ii) Create a scatter plot of “height” vs “weight” along with the regression line. [3 marks]
(iii) Predict “weight” for a value of “height” equal to 170. [3 marks]
(b) Consider a collective risk model S where the distribution of the frequency is Poisson with parameter 2, and the severities are exponentially distributed with mean 1/3. Write an R program to find the approximation of the CDF of S (use the rounding method to discretize the severity distribution over the interval (0, 20) with a step size of 0.1) and find the mean of S. [5 marks]
5.
(a) Consider the classical risk model
N(t)
U(t) = u + ct − 
Xi , U(0) = u > 0 ,
i=1
where N(t) is a Poisson process with intensity λ and X1 , X2 , . . . are i.i.d. random variables independent of N(t).
The intensity of the Poisson process is λ = 2, the density of the claim amounts is given by
fX (x) = e −2x + 
e−x , x ≥ 0 ,
and the premium received per unit of time is c = 2. For an initial surplus of u = 5:
|
(i) Find an upper bound for the ruin probability. |
[5 marks] |
|
(ii) Calculate the exact ruin probability. |
[5 marks] |
(b) Consider the above classical risk model under proportional insurance. Sup- pose that the relative security loading of the company in a reinsurance-free environment is θ = 10% and that the relative security loading under pro- portional reinsurance is θh = 20%. Write an R program to plot the risk adjustment coefficient as a function of the proportion a. [8 marks]