Macroeconomics 1

Given that ax = 20, ax:n = 18 and ax+n = 8, find the values of nEx and ax:n .

4.3 (i) Write down an expression for ax in terms of v, px, and ax+1.

(ii) On a certain select mortality table the select period is one year. Express a[x]

in terms of

v, p[x] and ax+1. Given that q[x] = .6qx and that at 4 1

4% p.a. interest a45 = 15.719 and

a46 = 15.509, find the value of a[45] (at the same rate of interest).

4.4 Using the A1967-70 table with 4% p.a. interest find the values of

a[40]:30 , a[39]+1:30 , a40:30 , (Ia)40, 5|a[40]:25 , 5|(Ia)40:25

(i) Find the present value of a deferred annuity of 1000 p.a. to a man aged 40. Payments

commence on his 60th birthday, if he is then alive, and continue thereafter annually for

life.

Basis: A1967-70 ultimate mortality, 4 % p.a. interest.

(ii) As above, but select mortality (at entry.)

(iii) As above, but A1967-70 ultimate mortality to exact age 60 and a(55) males ultimate

mortality above exact age 60.

4.6 According to a certain mortality table, which has a select period of 1 year and is such that

q[x] = 0.6qx for each x,

a70 at 10% interest = 5.641,

A study was carried out into the effects of smoking on life expectancy. The average number ( x ) of cigarettes smoked per day from age 50 by 11 individuals was calculated and the number ( y ) of years from age 50 until their deaths was recorded. The results were as follows: X 0.0 1.1 17.3 10.6 25.1 5.2 11.8 40.0 15.6 13.8 3.6 y 42.3 30.7 26.3 36.8 8.9 25.1 10.8 10.0 25.2 17.2 29.1 For these data values: 11 11 x; =144.1, > y, =262.4 /=1 1=1 11 11 x, = 3,255.91, Ev, =7,481.26, _x,y; =2,495.43 11 /=1 1=1 Calculate Pearson's correlation coefficient. Comment on the value obtained. [2 (ii) Calculate Spearman's rank correlation coefficient and comment on the value obtained. [3 (iii) State a general advantage of using Spearman's rank correlation coefficient. [1 (iv) Carry out a test to determine if Spearman's rank correlation coefficient is significantly different from zero assuming it is appropriate to use a normal approximation and comment on this assumption. [3 (v) Calculate the Kendall's rank correlation coefficient. [3 1 17