Exact match and bond immunization

Question

I am currently studying a risk management course, and there is one question from past year final exam about exact-match that I don't quite understand.

The problem is:

Consider an investor that needs to cover a liability of 1000 dollars in two years from now. He can invest in a one-year pure discount bond and a three-year pure discount bond both having a face value of 100. Assume that the yield to maturity is 10%

Q: How many units of the three-year pure discount bond do investors need to purchase to be able to cover the liability of 1000 dollars for sure even if the yield to maturity changes in the second year? The answer is 5.5

so first of all, I think I should use exact match method but I have no idea about how to set up the equations to solve for the investment proportion for each asset.

Can this problem be solved by Immunization or it can only be solved by exact match?

Any help would be greatly appreciated!

Thank you so much

Answer 1

Well, here is an approach that justifies the answer of 5.5. I'm not really up on the terminology. Given that, I think the exact match approach is not possible because there is not a bond that pays off in two periods so interest rate changes may preclude an exact match.

For immunization, we want to form a portfolio of bonds that delivers $\$1000$ in period two, and whose value is not very sensitive to changes in interest rates. Let $x$ be the number of one-period bonds we buy, and $y$ be the number of three-period bonds we buy (each with face value 100).
The value of this portfolio in period two is $$V=x(1+R_{12})100 +\frac{y}{(1+R_{12})}100.$$ Here $R_{12}$ is the interest rate between period 1 period 2 (today is 0, just to be clear). That is, you buy a one period bond today, it gives you 100 tomorrow, and you roll it over from 1 to 2, and it gives you $100(1+R_{12})$ though as of today we don't know what that will be. The value of the three-period bond in period 2 should be described as $\frac{y}{(1+R_{23})}100$, where $R_{23}$ is the interest rate between 2 and 3, but, and this is a crucial point, in these immunization problems, we assume parallel shifts in the yield curve, that is all rates change by the same amount, and since we initially had a flat yield curve (since the yield to maturity of both bonds is 0.10) we assume the yield curve remains flat implying $R_{23}=R_{12}$.

Now, we want to minimize the change in $V$ with respect to changes in $R_{12}$. Take the derivate and set to 0 to minimize, to get $$\frac{d V}{d R}= -x +\frac{y}{(1+R_{12})}=0.$$ This reduces to $$x=\frac{y}{(1+R_{12})^2}$$ Now, we don't know $R_{12}$, but we are thinking about small changes around the current interest rate, which is $10\%$, to that becomes $x=\frac{y}{(1.10)^2}$.

But we also know the value in period 2 has to be 1000, so using the formula for $V$ and substituting in the answer above for $x$ we know that $$\frac{y}{(1.10)^2} (1.10) + \frac{y}{1.10} =10$$ where I've divided out the 100 face value. That simplifies to $2y=11$ and the answer follows.