Rebalancing Duration-Immunized Portfolios Over Time
Summary
The document considers funding a liability due in the future with a portfolio of four-year zero-coupon bonds and a perpetual annuity, using duration matching. It writes one duration equation at the initial date and another one year later, when the bond’s remaining maturity and the liability’s horizon have each shortened. The annuity’s duration is calculated from the interest rate, and the post includes an attempted calculation of the change in its portfolio weight.
The author notes that the computed change does not match an expected answer and asks where the reasoning fails. The material is therefore an unresolved exercise rather than a validated rebalancing solution. It also highlights a key modeling issue: portfolio weights are based on present values, so they evolve with asset prices and liability value; matching duration equations alone does not determine the new weights without applying those valuation relationships consistently.
Key ideas
- Duration immunization matches the asset portfolio’s weighted duration to the liability horizon.
- A zero-coupon bond’s duration falls as it approaches maturity.
- A level perpetual annuity’s duration depends on the assumed interest rate.
- Portfolio weights are present-value weights and can change as asset and liability values evolve.
- The proposed rebalancing calculation is left unresolved and should not be treated as a confirmed result.
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Full text
# Portfolio immunization in time
# Portfolio immunization in time
The company will have to pay out an amount of liabilities $13594$ at the moment $t=9$. In $t=0$ they want to cover it with 4-years zero-coupon bonds and yearly perpetual annuity that is due in arrears using immunization strategy (match the duration of portfolio assets with the duration of future liabilities). How much (in %) has to change share of perpetual annuity in portfolio in $t=1$ to keep portfolio immuziation? Interest rate r=8%.
My approach: At the momment $t=0$:
$$p_1\underbrace{Duration_{0}(bonds)}_{4}+p_2Duration_{0}(annuity)=\underbrace{Duration_{0}(liabillities)}_{9} $$
At the momment $t=1$:
$$ p_1^{'}\underbrace{Duration_{1}(bonds)}_{3}+p_2^{'}Duration_{1}(annuity)= \underbrace{Duration_{1}(liabillities)}_{8} $$
Where $$ p_1=\frac{PV_{0}(bonds)}{PV_{0}(bond)+PV_{0}(annuity)}=\frac{PV_{0}(bonds)}{13594v^9}, \\ p_2=\frac{PV_{0}(annuity)}{PV_{0}(bond)+PV_{0}(annuity)}=\frac{PV_{0}(annuity)}{13594v^9}$$ $$ p_1^{'}=\frac{PV_{1}(bonds)}{PV_{1}(bond)+PV_{1}(annuity)}=\frac{PV_{1}(bonds)}{13594v^8}, \\ p_2^{'}=\frac{PV_{1}(annuity)}{PV_{1}(bond)+PV_{1}(annuity)}=\frac{PV_{1}(annuity)}{13594v^8}$$
For perpetual annuity $Duration_{0}(annuity)=Duration_{1}(annuity)=:D$. So we have simultaneous equations
$$ (\star) \begin{cases} 4p_1+Dp_2=9 \\ 3p_1^{'}+Dp_2^{'}=8 \\ p_1+p_2=1 \\ p_1^{'}+p_2^{'}=1 \end{cases} $$
Thus
$$ \begin{cases} 4\frac{PV_{0}(bonds)}{13594v^9}+D\frac{PV_{0}(annuity)}{13594v^9}=9 \\ 3\frac{PV_{1}(bonds)}{13594v^8}+D\frac{PV_{1}(annuity)}{13594v^8}=8 \\ \frac{PV_{0}(bonds)}{13594v^9}+\frac{PV_{0}(annuity)}{13594v^9}=1 \\ \frac{PV_{1}(bonds)}{13594v^8}+\frac{PV_{1}(annuity)}{13594v^8}=1 \end{cases} $$
And I cannot solve it and generally don't know if the approche is right. I'd be really appreciate it if somebody could help me out with that exercise.
Edit:
Duration for perpetual annuity is dependent only on interest rate:
$D=\frac{\sum_{k=1}^{\infty} kv^kCF}{\sum_{k=1}^{\infty} v^kCF}=\frac{Ia{\infty}}{a_{\infty}}=\frac{\frac{1}{r}+\frac{1}{r^2}}{\frac{1}{r}}=1+\frac{1}{r}=\frac{1.08}{0.08}=13.5$
If we insert it to $(\star)$, then we get $p_2=\frac{5}{9.5}=0.526, p_2^{'}=\frac{5}{10.5}=0.476 $. So $p_2^{'}-p_2=-0.05$ and the right answer is $-0.0228$.
Maybe somebody detects what is not ok here? In this solution I don't use information that the amount of liabillities is $13594$ so probably there is some other (right) solution.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.