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Company XYZ has a liability of $6,000 that is due in 3 years. The company could invest in zero-coupon bonds to be immunized against the liability from future large interest rate changes. Bond X is a 1-year zero coupon and Bond Y is a 5-year zero coupon bond. Company XYZ plans to invest in Bond X and Bond Y. How much should Company XYZ invest in Bond X, assuming an effective interest rate of 5%?

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Answer:

Explanation:

From the given information:

The present value of the cash flow of assets and liabilities is:

[tex]P(i) = x(1.05)^1*(1+i)^{-1}+y(1.05)^5 *(1+i)^{-5} -60000*(1+i)^{-3}[/tex]

Thus; [tex]P'(i) =\Big [(-1) (1.05)^1*(1+i)^{-2} \Big] -\Big [y(1.05)^5 *(1+i)^{-6} \Big] + \Big [(3)*60000*(1+i)^{-4}\Big][/tex]

Solving for x&y in the equations P(0.05) = 0 & P'(0.05) = 0

Then;

x+y = (60000) * (1.05)⁻³

x + y = 51830.26

and

x + 5y = 180000* (1.05)⁻³

x + 5y = 155490.7677

x + y = 51830.26

x + 5y = 155490.7677

0  +  4y = 103660.51  

4y = 103660.51

y = 103660.51/4

y = 25915.1275

Since y = 25915.1275 and x + y = 51830.26

Then x = 51830.26 - y

x =  51830.26 - 25915.1275

x = $25915.13

Thus, Company XYZ should invest the sum of $25915.13 into Bond X.

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