First-Order Bond Portfolio Immunization Against Yield Curve Factors
Summary
The document develops a first-order approach to immunizing a bond portfolio against nonparallel interest-rate shifts. It approximates each bond’s price change by differentiating its discounted cash flows with respect to rates at each maturity, producing key-rate sensitivities. If the liability is represented by its own discounted cash flows, portfolio quantities can be chosen to reduce the rate-weighted difference between asset and liability cash flows.
The discussion also suggests representing yield curve movements with factors such as level, slope, and curvature, or fitting a Nelson–Siegel or Svensson curve. The cited replies report that a small number of curve factors can explain most observed variation, but give no dataset or estimation details. The method is a local, first-order approximation, so it is intended for small rate changes; it does not establish protection against larger moves, nonlinear price effects, or errors in the chosen factor model. The optimization criterion is sketched rather than fully specified for implementation.
Key ideas
- For small rate changes, bond price changes can be approximated from discounted cash flow sensitivities to rates at each maturity.
- These maturity-specific sensitivities correspond to a key-rate duration view of curve exposure.
- Liability cash flows can be included to choose bond holdings that reduce asset–liability rate mismatch.
- Yield curve changes can be represented with level, slope, and curvature factors or parametric curve models.
- The proposed immunization is local and depends on the accuracy of the factor representation.
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Full text
# Factor immunization for bond portfolio
# Factor immunization for bond portfolio
I'm trying to figure out some kind of immunization using a factor model I developed for interest rates. Here is the basic problem. Let's say that we have a bond portfolio containing $N$ bonds with weights $x_i$ and one liability. Call the present value of the liability $P_L$. I'll suppose that the bond prices are given by $$P_i=\Sigma_{t=1}^T F_{it}e^{-r_tt}$$ where $F_{it}$ is the future payoff of bond $i$ and $r_t$ is the interest rate at time $t$. Now the factor model for the term structure of the interest rates can be written $$\Delta r_t=\Sigma_{j=1}^k \beta_{jt} \Delta f_j +\epsilon_t$$ for $k$ independent factors and standard normal error term $\epsilon_t$. I'm assuming I have small but not necessarily parallel-shifts in the term structure, I want a first order condition for factor immunization. Can someone help?
## Answer by Richi Wa (score 1, accepted)
https://quant.stackexchange.com/a/16234
The present value is $$ P_i= \sum_{t=1}^T F_{i,t} \exp(-r_t t), $$ what happens if rates change to $r_t + \Delta r_t$ then the new price is $$ P_i^{new} = \sum_{t=1}^T F_{i,t} \exp(-(r_t+\Delta r_t) t). $$ by the exponential series $\exp(x)\approx 1 + x$ we can write $$ P_i^{new} - P_i =: \Delta P_i \approx -\sum_{t=1}^T F_{i,t} \Delta r_t t. $$ Observing the shifts in these rates we have some kind of key-rate duration setting.
If you can decompose your liablity $P_L$ in the is way too (denote the cash flows by $F_t$, then you can try to find quantities $Q_i$ such that $$ |\sum_{t=1}^T (Q_i F_{i,t} - F_t) \Delta r_t t| -> Min $$
## Answer by Tulio Carnelossi (score 0)
https://quant.stackexchange.com/a/16217
Usually the moviments in the yield curve are decomposed by level, slope and curvature or parallel, steepness and bends in the term structure. These 3 factors can explain over 95% 98% of the total variance. Sometimes a second curvature point is used and the explanation reaches 99%. Instead of approximate it numerically why won't you fit the Svensson or Nelson Siegel model and use it as your factors?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.