A company faces liabilities of e$2$ million and e$5$ million due in three and

six years, respectively. The company plans to invest its assets in two zerocoupon bonds: one paying e$2$ million in one year and the other paying R in

n years.

$(1)$ If n $=7$ determine the value of R needed to match the company$$s

liability position with the two bonds at an effective interest rate of

$6\%$ per annum.

$(2)$ Calculate the volatility of the company$$s assets and liabilities, using

the values of n and R as above, at an effective interest rate of $6\%$

per annum.

$(3)$ Assuming the annual effective interest rate increases to $6\mathrm{.}5\%$ per

annum, discuss, without further calculation, whether the company$$s

surplus position will be positive or negative with the given values of

n and R$.$

$(4)$ Find the values of R and n such that the present value of the company$$s assets equals its liabilities, and the volatility of the company$$s

assets equals that of its liabilities, both at an effective interest rate

of $6\%$ per annum.