5.   (a) An investor has a liability payable on the 1st of January 2021. The present value of the liability on the 1st of January 2019 is 2X. He matches it with two zero coupon bonds payable on the 1st of January 2020 and on the 1st of January 2022. The present value of each bond on the 1st of January 2019 is X. Show that according to Redington’s theory the position is fully immunised against interest rate changes. [5 marks]

(b) Explain the flaw in Redington’s theory. You may use the example in (a). [5 marks]

6. A restaurant business opens a new outlet. The initial costs are 400000. There are also outgoings of 2000 per month payable in advance. The outgoings will increase by 2% at the beginning of each year. There is a continuous income at a rate of 30000 per annum for 0  < t < 5 and 60000 per annum thereafter. The business will be sold at a price of

400000 after 10 years. Calculate the net present value of the project using an effective rate of interest of 5% per annum. Should the project go ahead if the effective interest rate is higher than 5%? [9+2=11 marks]

7. A bond having nominal value of 10000 is repayable in ten equal annual instalments at par at the end of each year. It is also subject to a coupon of 5% of the nominal amount outstanding per annum payable at the end of each year. An investor is subject to income tax at 40% and capital gains tax at 30%. However, capital gains that occur more than 5 years after the purchase are not liable to tax. Capital gains are calculated assuming each instalment has an equal purchase price. Find the price the investor should pay to realise a yield of 4%. [10 marks]

8. The price of a t-year zero coupon bond is given by

exp( −0.02t) + exp( −0.04t)

2

(a) Calculate the 5-year spot rate. [3 marks]

(b) Calculate the 3-year forward rate in 2 years’ time. [3 marks]

(c) Calculate the present value of a 10-year continuous annuity certain payable at a rate of 1 per annum during the first 5 years and 2 per annum thereafter. [8 marks]

(d) Using the R output at the end of the paper, calculate A¯ 40:20 . [8 marks]

9. A combined assurance and pension policy issued for a 40 year old provides the following benefits

· A continuous life annuity of o20000 per annum commencing at time t = 20.

· The sum of o50000 payable at time t = 20 or immediately upon death if death occurs before time t = 20.

The policy is financed by a continuous premium payable till time t = 20. Using the R output at the end of the paper and a force of interest 0.04 per annum calculate the following.

(a) The annual premium [8 marks]

(b) The reserve at time t = 10 [10 marks]

(c) The reserve at time t = 30. Why does the issuing office need to set up reserves after time 20? [5+2=7 marks]

This is the R output

#The function p is the probability of survival#

p=function(x ,t)

{

a1=exp (0.0875*(x+t))

a2=exp (0.0875*x)

b1=0.00045*t+0.00006*(a1-a2)/0.0875

return(exp (-b1))

}

#For life annuities, we use a large value such as n=90#

ann=function(x ,n ,r)

{

p1=function(t)

{p (x,t)*exp(-r*t)}

integrate(p1,0,n)$value

}

Here are some calculations. p(x, n) represents the probability of survival of a life aged x for n years and a(x, 90, r) the value of a continuous life annuity for a life aged x calculated at a force of interest r. You might not need all of them.

p (40 ,20)

##  [1]  0 .8896152

p (50 ,10)

##  [1]  0 .9224695

p (60 ,10)

##  [1]  0 .8291977

ann (40 ,90 ,0.02)

##  [1]  25 .49443

ann (50 ,90 ,0.02)

##  [1]  20 .9831

ann (60 ,90 ,0.02)

##  [1]  16 .17504

ann (70 ,90 ,0.02)

##  [1]  11 .45234

ann (40 ,90 ,0.04)

##  [1]  18 .52201

ann (50 ,90 ,0.04)

##  [1]  16 .08722

ann (60 ,90 ,0.04)

##  [1]  13 .10672

ann (70 ,90 ,0.04)

##  [1]  9 .79586