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Stochastic Applications of Actuarial Models with R coding

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Introduction

Part I

Introduction

Credit risk refers to the probability that the counterparty may not meet its contractual obligations. Bond rating is a good indicator of credit risk of a sovereign issuing the bonds. Thus it is important to analyse the possible future rating movements of a sovereign due to credit events.

1. Probability of Default Rating

To model the credit risks of bonds over time, a continuous time Markov model is implemented to calculate the probability of credit migration. In using a continuous time Markov model, the Kolmogorov forward equations are used:

image00.png

where Pij(t) is the transition probability from rating i to rating j at time t, and rkj refers to the transition rates from rating k to rating j based on the annual transition rates matrix estimated. The above equation can be rewritten in matrix formimage01.png, where P(t) is the matrix of transition probabilities over time t, and R is the annual transition rate matrix estimated.

A possible solution to the above equation isimage05.png, where P(0) is the matrix of initial transition probabilities. It is assumed that the sovereign will remain in their current rating initially (at t=0), so for all ratings the probability of transition from one rating to another will be zero at time 0, while probability of remaining in the same rating is 1 (implying that P(0) is the identity matrix).

To observe the probability of default over time for sovereign rated AAA, AA, A and B, the probability transition matrix will be calculated for 100 equally spaced time frames over a 10 year period (starting from beginning of year 1995 to end of year 2004), The probability transition matrix

...read more.

Middle

0.03489511

It can be observed that the probability of a sovereign initially rated BB or above being rated default increases over time, due to greater time frame for credit events and hence rating migration to occur. Also the increase in transition rate matrix means that there will be more transitions between credit ratings, meaning sovereigns starting from a rating close to default are more likely to move upwards than usual.


Part II

1. a) Consider the stock price modelled by the geometric Brownian motion process:

image08.png

For each of the 2 methods suggested, we simulated S(t) for t=150000 with increments of 1/12 for time and then calculated ln(S(t))/t for each value of t.

It can be noted from the above results, both methods will cause the function ln(S(t))/t will converge to the limit which is approximately 0.0558077, which is close to the true value of p of 0.055. The true value can be found based on the property of the probability distribution of ln(S(t)).image09.pngimage10.png

We note that: image11.png as S(0)=1, image12.png,  image13.png and image14.png

Using the Delta Method, we can deduce the distribution of ln(S(t))/t as:

image15.png

Taking limits as t goes towards infinity, we note that:

image16.png

So as t goes to infinity, the variance will reduce to 0, meaning ln(S(t))/t will approach the mean which is 0.055. Thus the true value of p is 0.055 and verified the results of the simulation.

b) To find the value of t to obtain 2 digit accuracy with 95% probability, we find t such that:

image17.png

This is equivalent to finding:

image18.png

Rearranging gives

image19.png

Noting that from part a) that image20.png, where p=v, we can deduce that image22.png. This implies that image23.png, and noting that image13.png and image24.png, we deduce that image25.png.

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Conclusion

Plots function for 1(a) Method 2

plot(Q2, type="l", lwd=3, xlab="Time", ylab="(ln[S(t)])/t", main="Result from Method 1")

>Q1[150000] = 0.0558077   ←Used as an estimate of the limit

Plots function for 1(c) Method 1

plot(P1, type="l", lwd=3, xlab="Time", ylab="((ln[S(t)]-p)^2)/t", main="Result from Method 1")

Plots function for 1(c) Method 2

plot(P2, type="l", lwd=3, xlab="Time", ylab="((ln[S(t)]-p)^2)/t", main="Result from Method 2")

2.1.2 – Derivation of Gamma distribution of the function

Consider image32.png and hence image44.png. So:

Let image45.png, so then:

image46.png

image47.png

image48.png

image49.png        by noting the cumulative distribution function of standard normal

Thus the probability density function of image51.pngcan be found by differentiating image52.png with respect to x. Using the Fundamental theorem of calculus, this gives:

image53.png where image54.png, which is in the form of image55.png distribution.


2.2 Part II Question 2

Code is in Blue

Results are in Red

Comments are in Green

Collects the exchange rate data from a CSV file

exchange <-  read.csv("C:/Users/Chung-Yu/Documents/Uni/Actuarial Studies/ACTL2003/Assignment/exchange.csv",header=T)

exchange<-ts(exchange[,1],start=1980,freq=365/7)

Plots time series of data as well as sample ACF and PACF

par(mfrow=c(2,2))

plot(exchange)

acf(exchange)

acf(exchange, type="partial")

Plots the differenced data

par(mfrow=c(2,2))

plot(diff(exchange))

acf(diff(exchange))

acf(diff(exchange), type="partial")

fit=arima(exchange, order=c(0,1,0)) fit ARIMA(0,1,0) model

tsdiag(fit) Shows residual plots for fitted ARIMA(0,1,0) model, used for residual analysis

> fit=arima(exchange, order=c(0,1,0))

> fit

Shows AIC for ARIMA(0,1,0) model

Call:

arima(x = exchange, order = c(0, 1, 0))

sigma^2 estimated as 0.0007593:  log likelihood = 1018.97,  aic = -2035.93

The next 2 ARIMA models are fitted to compare the AIC and to see that the AIC for ARIMA(0,1,0) is optimal.

> fit=arima(exchange, order=c(0,1,1))

> fit

Call:

arima(x = exchange, order = c(0, 1, 1))

Coefficients:

          ma1

      -0.0610

s.e.   0.0449

sigma^2 estimated as 0.0007563:  log likelihood = 1019.87,  aic = -2035.74

> fit=arima(exchange, order=c(1,1,0))

> fit

Call:

arima(x = exchange, order = c(1, 1, 0))

Coefficients:

          ar1

      -0.0629

s.e.   0.0461

sigma^2 estimated as 0.0007563:  log likelihood = 1019.89,  aic = -2035.79

...read more.

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