MATH377: Financial and Actuarial Modelling in R Tutorial 3
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MATH377: Financial and Actuarial Modelling in R
Tutorial 3
Exercise 1. The file “snow_data.csv” contains ten observations taken during the years 1970-79 on October’s snow cover for Eurasia (snow cover is in millions of square kilometers):
a) Enter the data into R.
b) Plot year versus snow.cover.
c) Add a new column to your data frame with the logarithms of snow.cover and save the data into a new CSV file.
Exercise 2. The “temp_data.xlsx” file contains data on damages that had occurred in space shuttle launches before the Challenger space shuttle launch of Jan 28, 1986.
a) Enter the temperature data into R.
b) Plot temperature against the number of incidents. Do you see any relationship?
Exercise 3. Consider the data set “EuStockMarkets”in R corresponding to the “Daily Closing Prices of Major European Stock Indices, 1991-1998”.
a) Write an R program to plot a histogram of“DAX”. Note: to access“DAX” use EuStockMarkets[, 1].
b) Compute the log-returns of“DAX”. Recall that the log-returns are computed as
log(Pt /Pt − 1 ) ,
where Pt are the closing prices.
c) Compute the mean and standard deviation of the log-returns.
d) Plot a histogram of the log-returns.
e) In your plot in d), add the density function of a normal distributed random variable with mean and standard deviation the values computed in c).
f) Plot “DAX” versus “SMI” and compute their correlation.
Exercise 4. Let X be a normal random variable with mean 2 and variance 9, that is, X ∼ N(2, 9).
Compute:
a) P(|X| ≤ 2.4).
b) The 95% quantile of the given distribution.
Exercise 5. Let X ∼ LN (µ, σ2 ). Then, the moments of X are given by
E [Xn] = exp (nµ + 
) , n ∈ N .
a) Plot the density of a lognomal distribution with mean 10 and Coefficient of Variation (CV) of 0.3. Recall that CV(X ) =“var (X )/E [X].
b) Generate 100 and 1000 observations from the distribution in a). Then draw histograms for the generated samples.
c) What are the mean, standard deviation, and CV of each sample?
Exercise 6. The F distribution with d1 and d2 degrees of freedom is the distribution of
S1 /d1
X =
S2 /d2 ,
where S1 and S2 are independent random variables with chi-square distributions with respective degrees of freedom d1 and d2 .
a) Simulate 1000 observation from a F distribution with d1 = 5 and d2 = 10 degrees of freedom without using rf(). Hint: Use rchisq() to simulate two independent chi-square distributed random variables.
b) Verify that your simulated sample comes from an F distribution by:
i. Plotting the histogram of the sample versus the theoretical density. Use df() for the theoretical density.
ii. A QQ-lot. Use qf() to compute the quantiles of the theoretical distribution.
Exercise 7. Consider X a random variable with distribution a mixture of two normal distributions, that is, the density function of X is given by
fX (x) = pf1 (x) + (1 − p)f2 (x) ,
where p ∈ [0, 1] and fi are the density functions of N (µi , σi(2)) random variables, i = 1, 2.
a) Write an R function to compute the density above.
b) Plot your density in a) for the parameter values p = 0.5, µ 1 = 1, µ2 = 5, σ 1 = 1, σ2 = 2.
c) Note that X can be expressed as
X = IY1 + (1 − I)Y2 ,
where Yi ∼ N (µi , σi(2)) and I ∼ Bernoulli (p), independent. Using this representation, write an R
code to simulate data from X . Test your code by generating 1000 observations with the same
parameters as b).
d) Convince yourself that your simulated data comes from the model with density in b). Exercise 8. Let Y be exponentially distributed with mean 4 and consider X defined as
X = Y0 .5 .
a) Write an R function to compute the density function of X . Evaluate your function at x = 1.
b) Simulate a sample of size 1000 from X .
c) Approximate E[X4 ] using your simulated sample in b).
d) Write an R program to compute E[X4 ] via numerical integration.
2023-03-08