A bank is required to pay 1,100 in one year. There are two investment options available with respect to how money can be invested now in order to provide for the 1,100 payback:

(i) a non-interest bearing cash fund, for which x will be invested, and

(ii) a two-year zero-coupon bond earning 10% per year, for which y will be invested.

Based on immunization theory, develop an asset portfolio that will minimize the risk that liability cash flows will exceed asset cash flows. Assume the effective rate of interest is equal to 10% in all calculations.

Answer: x = 500 and y = 500

I'm having trouble getting the above solutions for x and y. I'm not sure that I understand what (i) is. This is what I've been doing.

$\displaystyle p(i)=x(1.1)^{-1}+y(1.1)^{2}(1.1)^{-2}-1100(1.1)^{-1}=0$

$\displaystyle x=1100-(1.1)y$

$\displaystyle p'(i)=-x(1.1)^{-2}-2y(1.1)^{-1}+1100(1.1)^{-2}=0$

Then plug in x from the the first equation and solve the derivative for y.

$\displaystyle -1100(1.1)^{-2}+(1.1)(1.1)^{-2}y-2(1.1)^{-1}y+1100(1.1)^{-2}=0$

That leaves

$\displaystyle -0.90909y=0$

Which is obviously wrong.

Can someone point out my mistake and get me on the right direction? Thanks