A client deposits 100,000 in a bank, with the bank agreeing to pay 8% effective
for two years. The client indicates that half of the account balance
will be withdrawn at the end of the first year. The bank can invest in either
one year or two year zero coupon bonds. The one year bonds yield 9% and
the two year bonds yield 10%. Develop an investment program based on
immunization.
The solution goes on to solve the system of equations in x and y defined by P(i)=0 and P'(i)=0, which I'm fine with. The solution is x=$49,541.28 and y=$48,198.35, which means $2,260.37 in profit is made the day of the initial deposit.
I'm confused about why P(i) doesn't include 100,000 in assets at time 0 and -(x+y) in liabilities at time 0. It seems to me that the net cash inflow of $2,260.37 at time 0 should appear somewhere. More generally, I'm confused why it's desirable to have P(i)=0. Wouldn't the bank want to invest the extra $2,260.37 as well in order to have additional profit?