STK3505/4505 Mandatory assigment 1 of 1
October 5th, 2022
1 Submission deadline Thursday 20th 2022, 14:30 in Canvas
2 Instructions
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You can choose between scanning handwritten notes or typing the solution directly on a computer (for instance with LATEX). The assignment must be submitted as a single PDF le. Scanned pages must be clearly legible. The submission must contain your name, course and assignment number. It is expected that you give a clear presentation with all necessary explanations. Remember to include all relevant plots and gures. Students who fail the assignment, but have made a genuine eort at solving the exercises, are given a second attempt at revising their answers. All aids, including collaboration, are allowed, but the submission must be written by you and reect your understanding of the subject. If we doubt that you have understood the content you have handed in, we may request that you give an oral account.
In exercises where you are asked to write a computer program, you need to hand in the code along with the rest of the assignment. It is important that the submitted program contains a trial run, so that it is easy to see the result of the code.
3 Application for postponed delivery
If you need to apply for a postponement of the submission deadline due to illness or other reasons, you have to contact the Student Administration at the Department of Mathematics (e-mail: well before the deadline.
All mandatory assignments in this course must be approved in the same semester, before you are allowed to take the nal examination.
4 Complete guidelines about delivery of mandatory assignments:
https://www.uio.no/english/studies/examinations/compulsory-activities/mn- math-mandatory.html
Good luck!!!
5 Assignments
5.1 Assignment 1 – Time value of money (30 p)
A xed-coupond bond is a nancial instrument characterised with maturity T-years, coupon c (which is assumed to be paid out annually) and principal P. For such a bond, a bond holder gets cP amound of money at the end of each year before maturity, and at the maturity they get cP + P . The value of the bond at time t = 0 is equal to the present value of the underlying cashows.
1. Assume that the interest rate at time t = 0 is at and equal r. Write down the formula for calculating the value of the bond at time t = 0 given the parameters r,T,c,P.
2. Assume that r = c. Prove that the value of the bond is equal to P regardless c, P, T .
3. Write a function which calculates the value of the bond, given pa- rameters r,c,P,T. Run the function for r = 5%, c = 4%, T = 10 and P = 1000. Run the function again changing dierent parame- ters and plot the results (paramater agains the value of the bond), rst with the interest rate (r = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10%), then coupon (c = 1,2,3,4,5,6,7,8,9,10%) and at last, the maturity of the bond
(T = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20). What does the results tell you about the relation between dierent level of parameters and the va- lue of the bond?
4. Calculatethedurationofthebondforr=5%,c=4%,T =10and P = 1000 and give an interpretation to it. Assume that the interest rate has increased from r = 5% to r = 6%. How much would the price of the bond change, as prescribed by the duration? What is the real change in the value of the bond? Why those two value are dierent?
5.2 Assignment 2 – Monte Carlo and parameter estimation (30 p)
We assume that the claims in the non-life insurance are log-normally distri- buted with parameters μ = 4 and σ = 2
1. Derive maximum likelihood estimators of μ and σ and write a function which calculates the estimates given a sample X. Function should be general enough to handle dierent lengths of X.
2. Assume that our sample has n = 100 observations. Generate such a sample nb = 1000 times. For each such a sample calculate the maximum likelihood estimators and create a plot of estimated μ and σ agains each
other for all the nb samples. The plot should therefore has nb = 1000 points.
3. Increase n in point be to n = 1000 and rerun the plot. Compare the results. Which parameter was more uncertain?
5.3 Assignment 3 – Non-life insurance (30 p)
We assume the compound Poisson model for J = 1000 policies. In this model, the claim number from all policies is modelled as a random variable N ∼ Poiss(λ),(λ = Jμ , μ = 0.1) and the claim size Zi is distributed according to Weibull distribution with parameters α = 2, β = 3. Then the sum of all claims can be dened as X = Ni=1 Zi.
1. Compute the expectation and the standard deviation of X using the closed-form formula. Calculate the pure premium for one policy and compare it with the expected value of X. What is the relation?
2. Write a program which simulates the sum of claims X and simulate n = 10000 simulations. Compute the mean and standard deviation from the simulation and compare it with point 1.
3. Assume excess of loss reinsurance, where the reinsurer covers the loss between 2.5 and 5 (i.e. we have a layer 2.5xs2.5). Write a program which would simulate reinsurance recoverables as well as sum of net claims in this case.
4. Calculate the pure reinsurance premium in this case as well as mean of the sum of net claims.
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