Ill give it a shot:
Step 1) Find the terminal value of your currne bonds position, using the minimim acceptable return
Step 2) Discount this value back to today, using the immunized return given. Thios value subtracted from the terminal value gives us our dollar safety margin.
Now, if rates change, we discount the Terminal value found in step one by the new rates
Step 3) Calculate the value PV of the bond. If this value is less than your step 2 calculation with the new rates, yuo must switch out of contingent immunization.
So the example…
Step 1) We have a $500M portfokio, and min acceptable return is 3%. The immunization period is 3 years. So, N=6 and Interest is 3/2 = 1.5
Terminal Value = 500M(1+.015)^6 = $546.72M
Step 2) It can immunize at 4.75%, so the PV of the Terminal value is:
546.72/(1+(.0475/2))^6 = $474.89M. Note, this step may not be needed if we are asked to see what happens with interest rate changes. But its good to do it to practice and see yuor original safety net margin.
NOW, rates drop to 3.75%, or 3.75/2 = .01875 per period. Now, the PV of our Terminal value becomes:
546.72/(1+.01875)^6 = 489.0567 (notice the dollar safety margin has grown - since rates dropped, our bond position will increase in value).
Step 3) we find the PV of the bond the manager invests in with new rates:
FV = -500M
N = 10 years, so 20
PMT = we knwo its trading at par (given) so coupon rate = Original YTM of 4,75. Payment is .0475/2*500M -= -11.875M
IY = new rate of .0375/2 = .01875 or 1.875
Solve for PB. we get 541.36. Since this value is greater than step 2 value, we can stick with Contingent
Now, if rates go up to 5.8, the PV of the Terminal value is:
=546.72/(1.029^6) = 460.5
the value of the bond the manage rinvests in is:
FV = -500
PMT = -11.875
IY = 2.9
PV = 460.58
The net saftey margin is pretty much 0 at this point. I think differences are due to rounding
Does that help?