Please solve the question below precisely

(1) S State the principle of correspondence as it relates to mortality investigations. [1] Two small countries conduct population censuses on an annual basis. Country A records its population on 1 February every year based on an age definition of age lost birthday. Country B records its population on every 1 August using a definition of age nearest birthday. Each country records deaths as they happen based on age next birthday. Below are some data from the last few years. Country A Age last Population Population Population birthday ! February 20]] ] February 2012 ] February 2013 44 382,000 394,000 401,000 45 374,000 381,000 385,000 46 354,000 372,000 375,000 Country B Age nearest Population Population Population birthday I August 201 1 I August 2012 I August 2013 44 382,000 394,000 401,000 45 374,000 381,000 385,000 46 354,000 372,000 375,000 In the combined lands of Countries A and B in the calendar year 2012 there were 4,800 deaths of those aged 46 next birthday and 4,500 deaths of those aged 45 next birthday. The two countries decide to form an economic union, after which it will be mandatory to offer the same rates for life insurance to residents of each country. (ii) Estimate the death rate at age 45 years last birthday for the two countries combined. [6] (iii) Explain the exact age to which your estimate relates. [1] [Total 8]Let R; denote the return on security i given by the following multifactor model R; = a;+ bill + bialy+ ... + billy + c; a; and c are the constant and random parts respectively of the component of the return unique to security i. I .. It are the changes in a set of L indices. bak is the sensitivity of security i to factor k. (i) State the category of the above model where: (a) index 1 is a price index index 2 is the yield on government bonds index 3 is the annual rate of economic growth (b) index 1 is the level of Research and Development expenditure index 2 is the price earnings ratio [2] (ii) Determine the number of parameters to be estimated in a single index model and in a multifactor model. [4] [Total 6] The following unusual model has been proposed for the (real-world) stochastic behaviour of the short term interest rate: dr, = ur, di + dZ,, where u > 0 and o are fixed parameters and Z is a standard Brownian motion under the proposed real-world measure P. Under the same measure P, a (zero coupon) bond with maturity T has price at time f B(t, T) = exp(-(T-D)r, + 6 (T-);6). (a) Derive the SDE satisfied by B(t. 7). (b) Determine the market price of risk and deduce the corresponding SDE for r, under the risk neutral measure @. [7]