A question on immunization and Macaulay duration

Question

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.

Answer 1

Duration is not linear. It is the weighted average of the duration of the underlyings with the weightings being their values. To get a linear system multiply the durations by the associated pvs and match that quantity instead.

Answer 2

The correct answer is D.1111

We have a liability of 5000 $\$$ due to 8 years. The discount rate is 6%. We have two different bonds: Bond1: 5 years to maturity with semiannual payments, 6% annualy.
Bond2: 10years to maturity, zero coupon bond Both bonds have a face value equal to 100 and their market price is $ B1_t , B2_t= 100, \forall t \ to \hspace{0.2cm} maturity$.

The Macaulay duration of the liability is equal to 8. Hence $D_{liability}=8$. The zero has $D_2 =10$ by definition. The duration of the 5-year bond is:

$D_1(semiannualy)= \frac{\sum_{i=1}^{10} \frac{3i}{1.03^i}+\frac{10 *100}{1,03^{10}}}{100}=8,786$

We convert the above duration into annually $D_1=\frac{D_1(semiannualy)}{2}=4,393$

We construct a portfolio whose assets are bond1 and bond2 with weights $w_1,w_2$.

The duration of the portfolio should equal to the duration of the liability We have to solve the following $2*2$ linear equation system:

$w_1 *D_1+ w_2*D_2=D_{liability} \implies w_1*4,393+w_2*10=8$ $w_1+w_2=1$.

The solutions are: $w_1=0,3566 , w_2=0,6434$.

Assuming that rate remains constant across time. We should invest $x$ amount of the 5-year bond:

$x=\frac{5000}{1,03^{16}}*0,3566 =1111,11 \approx 1111 \$$