Solving for a Zero-Duration Bond Portfolio with an Investment Constraint

Summary

The document considers how to combine a coupon bond and a perpetuity to create a portfolio with zero duration. It identifies an error in multiplying weighted durations and replaces it with a linear duration condition: the sum of each bond’s weight times its duration must equal zero.

That condition alone does not determine both weights. A second equation is needed to specify the portfolio’s total invested value, using each bond’s price and weight. Substituting the duration condition into the investment constraint then yields a solution for the weights. The example supplies bond terms but does not work through their prices, durations, or numerical weights. The method also depends on the chosen definition of duration and the specified investment amount; the document does not address transaction costs, financing, or how duration neutrality behaves as yields change.

Key ideas

• Portfolio duration is the sum of each asset’s weight multiplied by its duration.
• Zero duration requires the weighted durations to sum to zero.
• A separate investment-value constraint is needed to determine both bond weights.
• The duration equation can be substituted into the value constraint to solve for the weights.

Tags

# Duration of portfolio equals to zero


# Duration of portfolio equals to zero












I am solving the following problem:

> Consider a 2000 dollars bond with maturity of 5 years and a half-year coupon of 25 dollars at a nominal interest rate of 8% p.a and a consolidation bond (eternal annuity) with a half-year coupon of 50 dollars and the same nominal interest rate. Create a portfolio of these two bonds, which will have a zero duration.

So I calculated the price of the first and the second bond and I know that the formula is: $0=w_{1}D_{1}*w_{2}D_{2}$

But how should I calculate weights so that it would be equal to zero?

## Answer by Dom (score 2, accepted)

https://quant.stackexchange.com/a/50419

The equation to be solved should be $w_1 D_1 + w_2 D_2=0$ where $D_1$ and $D_2$ are the respective durations of the two bonds. However you need an investment constraint to fix the values of $w_1$ and $w_2$. Hence you also need $w_1 P_1 + w_2 P_2 = \Pi$ where $\Pi$ is the amount invested.

You can then subsitute $w_1 = - w_2 D_2 /D_1$ into the second equation to have a solution for both $w_1$ and $w_2$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.