# How are the notionals on proceeds-weighted bond butterflies calculated?

Most LDI (Liability-Driven Investment) accounts construct bond butterfly (fly) trades by weighting them according to proceeds. This creates two constraints:

1. The fly is duration-neutral (the usual constraint for a fly trade), and

2. The fly is notional-neutral (the proceeds-weighted constraint).

I've been taking several bonds (Gilts) and have been trying to compute the notionals necessary to obtain a proceeds-weighted fly. However, I seem to be getting strange results that don't make much sense.

I'm going to outline how I approached it and would appreciate an input as to what I'm doing wrong.

The maths

First of all, we can set up the two constraints mathematically. Everything with subscript $$2$$ will refer to the bond in the belly of the fly, and those with subscript $$1$$ and $$3$$ will be the wings.

To obtain duration-neutrality, we need

$$D_2-(D_1+D_3)=0,$$

and for notional-neutrality we need

$$N_2-(N_1+N_3)=0.$$

If we let $$P$$ be the PVBP of a bond (known values), then the first constraint becomes

$$100N_2P_2-(100N_1P_1+100N_3P_3)=0$$

$$N_2P_2-(N_1P_1+N_3P_3)=0,$$

and also

$$N_1P_1=N_3P_3$$

$$N_2P_2-2N_1P_1=0$$

In the first formula I use the fact that $$D=100NP$$, which uses N in millions. E.g. if $$P=10$$ and $$N=100m$$, then the duration is $$D=100\times 100\times 10=100,000$$, a duration of 100k per basis point.

Assuming that we will use a belly notional of $$N_2=100\text{m}$$, we now have a system of two equations with two unknowns:

$$N_1+N_3=N_2$$

$$N_2P_2-2N_1P_1=0.$$

If we let $$N_1=N_2-N_3$$ and substitute it into the second formula then we get:

$$N_2P_2-2(N_2-N_3)P_1=0$$

$$2N_3P_1=2N_2P_1-N_2P_2$$

$$N_3=\frac{N_2(2P_1-P_2)}{2P_1}$$

and subsequently we have $$N_1=N_2-N_3=100-N_3.$$

A real example

Consider now the following PVBPs:

$$P_1=10.8$$

$$P_2=17.2$$

$$P_3=18.7,$$

and $$N_2=100$$. The notional required to satisfy the two constraints for a proceeds-neutral fly should be

$$N_3=\frac{N_2(2P_1-P_2)}{2P_1}=\frac{100\times ((2\times 10.8)-17.2)}{2\times 10.8}=20.4\text{m}.$$

And hence $$N_1=100-20.4=79.6\text{m}.$$

We can check the notionals get the appropriate durations, by using $$D=100NP$$

$$100\times 20.4\times 18.7=34,000$$

$$100\times 79.6\times 10.8=85,000.$$

For the bond in the belly we have

$$100\times 100\times 17.2=172,000.$$

The problem

Clearly something is very wrong. I have used what seem to be to be valid constraints, yet I have ended up with non-similar durations for the wings, and durations on the wings that don't add up to the duration on the belly.

I've tried various other constraints, but end up with similar issues. I will just state the other set of formulas that I have used.

First is:

$$N_2=N_1+N_3$$

$$N_1P_1-N_3P_3=0$$

And I've also tried using

$$N_2=N_1+N_3$$

$$N_1P_1+N_3P_3=N_2P_2$$

Thanks for getting this far, and I hope you can help!

I believe the duration constraint and the proceeds constraint are not self consistent. You cannot satisfy both. The duration constraint alone fixes $$N_1/N_2$$ and $$N_3/N_2$$, so you cannot also satisfy the proceeds constraint.