I have a question regarding immunization portfolios that are continuously compounded. Suppose we have the following three bonds:

  • Bond 1: one year zero coupon with principal of $100

  • Bond 2: two year zero coupon with principal of $100

  • Bond 3: continuous constant cash flow with total one year payment of $100.

Let $P_i(y)$ bet the price of the $i$th bond as a function of $y \equiv$ annual yield, compounded continuously.

$P_1(y) = 100e^{-y}$, $P_2(y) = 100e^{-2y}$ and $P_3 = \int_0^1 100e^{-yt}dt = \frac{100}{y}\left(1-e^{-y} \right)$

I want to determine an immunization portfolio that includes bonds 2 and 3 to replicate the value and duration of bond 1 when the annual yield is $50\%$, but I'm not sure if the following is done correctly:

Trivially the modified duration $\left(D_i(y) \equiv -\frac{1}{P_i(y)}P_i'(y) \right)$ of bond $1$ and $2$ are $1$ and $2$ respectively, while the duration of bond 3 is given by $$\frac{y}{100(1-e^{-y})} \left(\frac{100}{y^2}(1-e^{-y}) - \frac{100}{y}e^{-y}\right) = \frac{1}{y} - \frac{1}{e^{y}-1} = D_3(y)$$

We solve the following two equations for $V_2$ and $V_3$ the value of bonds 2 and 3 we need to generate the immunization portfolio:

$$\begin{aligned} V_2 + V_3 = P_1(0.5) = 100e^{-0.5}\quad \quad \quad (1) \\ D_2V_2 + D_3V_3 = 2V_2 + \left(2 - \frac{1}{e^{0.5}-1}\right)V_3 = D_1P_1(0.5) = 100e^{-0.5} \quad \quad \quad (2)\end{aligned}$$

from which we obtain $$\boxed{V_2 = 21.31 \quad \text{and} \quad V_3 = 39.35}$$

Is this correct?


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