Macroeconomics 1.the question is complete.

Evaluate (a) A60 and (b) A60 on the bases:

(i) A1967-70 ultimate, 4% p.a. interest;

(ii) E.L.T. No.12 - Males, 4% p.a. interest.

(i) Show that Ax = vqx + vpxAx+1

(ii) Given that p60 = 0.985, p61 = 0.98, i = 0.05 and A62 = 0.6 , evaluate A61 and A60

Consider n lives now aged x1, x2, . . . , xn respectively. Let Z be the total present value of a

contract providing the sum of Si

immediately on death of (xi), i = 1, 2, . . . , n. The n lives

are subject to the same non-select mortality table, Table B, and the interest is taken to be

fixed at rate i p.a.

(i) Show that E(Z) = S1A

x1 + + SnA

xn

on Table B with interest at rate i p.a.

(ii) Assuming further that the future lifetimes T(xi) of the lives are independent variables,

show that

Var(Z) = Xn

j=1

S

2

j

[A

xj (A

xj

)

2

]

where refers to an interest rate of 2i + i

2 p.a.

Using commutation functions or otherwise calculate the values of the following:

(i) A[40]:10 on A1967-70, 4% p.a. interest;

(ii) A

1

30:20 on A1967-70 Ultimate, 4%;

(iii) A 1

30:20 on A1967-70 Ultimate, 4%;

(iv) A

30:20 on A1967-70 Ultimate, 4%;

(v) A

30:20 on English Life Table No.12 Males, 4%.

A life aged 50 who is subject to the mortality of the A1967-70 Select table, effects a pure

endowment policy with a term of 20 years for a sum assured of 10,000.

(i) Write down the present value of the benefits under this contract, regarded as a random

variable.

(ii) Assuming an effective rate of interest of 5% per annum, calculate the mean and the

variance of the present value of the benefits available under this contract.

3.6 What are the random variables (in terms of K = curtate future lifetime of (x)) whose means

are represented by the following symbols?

(i) nEx

(ii) A1

x:n

(iii) A 1

x:n

A life aged exactly 60 wishes to arrange for a payment to be made to a charity in 10 years'

time. If he is still alive at that date the payment will be 1000. If he dies before the payment

date, the amount given will be 500. Assuming an effective interest rate of 6% per annum and

Questions 1 to 4 are based on the data set given in the table below: X 2 4 5 9 V 3.0 6.8 8.2 15.1 [x=20 _x2 =126 _xy =210.1 _y=33.1 _y2 =350.49 It is proposed to fit a simple linear regression model to these data: Y, = a+ bx; +e; where e; ~ N(0,2) Calculate the least squares estimate for the slope parameter b. Calculate an unbiased estimate of the variance parameter. Determine a 95% confidence interval for b based on the sample data. [2 (i) Show that the sample correlation coefficient is 0.9995 to 4 significant figures. (ii) Hence obtain: (a) a 95% confidence interval for p, the underlying correlation coefficient (b) the coefficient of determination, R , and comment on your result. [Total It is desired to test the value of the parameter p for a random variable that has a binomial distribution. In order to test the null hypothesis Ho : p=0.4 against the alternative hypothesis H1 : p=0.6, the following test is devised: The number of successes, X , in a sample of size 50 is determined. If X2 25, then Ho is rejected Calculate the approximate size of this test. Define the following terms: (i) a Type I error (ii) a Type ll error [1 (iii) the size of a test [1 (iv) the power of a test. [Total