About immunization! Please help!! URGENT!!

[ActuaryFIA]

Quite rusty but will have a go…….

Wendy706 wrote:
A company has to pay $25,000 in 6 years and $35,000 in 8 years. It sets up an investment funds consisting of 5-year and 10-year zero-coupon bonds with a face value of $100 in order to meet these requirements. The current interest rate ,continuously compounded, is 4%.
1) How many 5-year and 10-year bonds should the company buy based on immunisation?

Create a potfolio of the two assets such that:

• PV of Asset Portfolio (PVA) = PV Liabilities (PVL)

PVL can be immediately calculated from the data given. For PVA we can calculate the current price of each of the 5 & 10 - year ZCBs, and then assume X units of the 5-Year ZCB & Y units of the 10-Year ZCB are bought we can form a simple linear equation that equates to the PVL (which we know).

• Duration of the Asset Porfolio = Duration of Liabilities

Similarly, as with the PVs, we can do the same for the Durations and get a second linear equation.

Then we can solve for X & Y based on the two linear equations.

I did this quickly in a spreadsheet and got the following (by no mean claiming this is correct becasue did it really quickly):
PVL = $45,081
Price of 5-Year ZCB= $81.9
Price of 10-Year ZCB = $67.0
X = 316
Y = 286

Wendy706 wrote:
2) instead of 5-year and 10-year bonds, the company uses 5-year and n-year bonds for some value of n. Construct a portfolio of 5-year and n-year bonds such that the present value and duration of the portfolio matches the present value and duration of the liabilities.

I would suppose this is a similar idea as above but:

• We have another unknown, i.e. “n”.
• So since no values are being asked for I would think its more of a qualitative question to explain the process and importantly exlain Redington Immunization conditions ……….. and setting up systems of linear equations (3 in this case)…….The third equation comes in because of the next bullet point.
• ………the condition that: Convexity of Asset Portfolio (CA) > Convexity of Liabilities (CL).

There may be more ……..?

Wendy706 wrote:
3) for which values of n can the conditions of Redington immunisation be satisfied?

I think this is solving the system of three equations to calculate 3 unknowns X, Y, n ? ….and more?

Hope I am making sense and you find it useful.