I am studying for the Society of Actuaries - Exam FM and encountered the following problem:
Let $x$ be the face amount of the 5-year bond and let $y$ be the face amount of the 10-year bond.
Since the coupon rate of the 5-year bond is the same as the yield rate, the bond is purchased at face value.
The first condition of immunization requires that:
PV(Assets) = PV(Liabilities) or
$x + \frac{y}{(1.03)^{20}} = \frac{5000}{(1.03)^{20}} = 3115.83.$
The second condition of immunization requires that
MacD(Assets) = MacD(Liabilities).
The duration of a 5-year coupon bond with semi-annual coupons that sells at its par value is $a_{10}\cdot(1.03) = 8.7862$ half-years, or 4.3931 years. ($a_{10}$ is a 10-year annuity immediate).
The duration of the 10-year zero-coupon bond is simply 10 years. The duration of the liability is 8 years.
Now here is where I am confused. Is duration linear, in the sense that the "total" duration of the assets is the sum of the duration of the individual assets?
Would we have MacD(Assets) = MacD(x) + MacD(y)?
To meet the second condition, it seems like we have 10 + 4.3931 = 8, which I don't know what to make sense of.