MATH 377 Financial and Actuarial Modelling in R 2022
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MATH 377
Financial and Actuarial Modelling in R
2022
1. An investor invests his wealth 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 of 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 minimal 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(
(900 − S
), 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) Consider a collective risk model S where the distribution of the frequency is
Poisson with parameter 2, and the severities are exponential 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 (0,20))
and find the mean of S . [5 marks]
4.
(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 time is c = 2. For an initial surplus of u = 5:
i) Find an upper bond 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]
2022-05-09