Why Bond Portfolio IRR Moves with a Parallel Yield Shift

# Parallel shift in spot yield curve moves the IRR of a bond portfolio in the same direction: Analytical Proof


# Parallel shift in spot yield curve moves the IRR of a bond portfolio in the same direction: Analytical Proof












I am trying to prove that a parallel shift in the spot yield curve will as its effect have the IRR of a bond portfolio move in the same direction and by the same amount.

I have tested this on few simple portfolios and the statement seems to hold up empirically, but I would like to prove it mathematically (if it's even valid and not just me misinterpreting something).

Just to start off with a manageable case, which should hopefully reveal how a more complex one should be dealt with, let's assume we have a portfolio of one zero-coupon bond maturing in 1-year with par 100 (denoted $ZCB_1$), and one zero-coupon bond maturing in 2 years with par 100 (denoted $ZCB_2$).

Let's also assume that from the spot curve we observe that:

$$\text{Spot rate for 1 year maturity is } y_1 = 2\%$$ and $$\text{Spot rate for 2 year maturity is } y_2 = 4 \%.$$

The above assumptions imply that the present values of the zero coupon bonds in our portfolio are

$$PV(ZCB_1)=\frac{100}{1.02}=98.0392$$ $$PV(ZCB_2)=\frac{100}{(1.04)^{2}}=92.4556.$$

That means that we can now calculate the IRR of our portfolio by solving the following equation for IRR $$98.0392+92.4556=\frac{100}{1+IRR}+\frac{100}{(1+IRR)^{2}}.$$

If I plug this into Mathematica, I get $$IRR = 0.033$$

If we now assume a parallel +1% shift in the spot yield curve, we would have

$$\text{Spot rate for 1 year maturity after this parallel shift is } \overline{y}_1 = 3\%$$ and $$\text{Spot rate for 2 year maturity after this parallel shift is } \overline{y}_2 = 5 \%.$$

We could repeat the same process as above and end up with after the shift $IRR$ being equal to $$\overline{IRR}=0.043.$$

Thus, our IRR increased by 1% after the 1% positive parallel shift in the yield curve.

I am trying to prove this relation analytically for this very simple case of two bonds maturing in 1 and 2 years. The thing is that using my brute force approach, this very quickly gets quite tedious and I am thinking that there might be a much simpler way of doing this?

This is my attempt:

$$\frac{100}{1+y_1}+\frac{100}{(1+y_2)^2}=\frac{100}{1+IRR}+\frac{100}{(1+IRR)^2}$$ $$\text{If we introduce } t = 1+IRR, \text{then we have}$$ $$\frac{100}{1+y_1}+\frac{100}{(1+y_2)^2}=\frac{100}{t}+\frac{100}{t^2}$$ $$t^2\left( \frac{100}{1+y_1}+\frac{100}{(1+y_2)^2}\right)-100t-100=0$$ $$t_{1,2}=\frac{100 \pm \sqrt{10000+400\left( \frac{100}{1+y_1}+\frac{100}{(1+y_2)^2}\right)}}{2\left(\frac{100}{1+y_1}+\frac{100}{(1+y_2)^2} \right)}$$ $$IRR_1=\frac{100 + \sqrt{10000+400\left( \frac{100}{1+y_1}+\frac{100}{(1+y_2)^2}\right)}}{2\left(\frac{100}{1+y_1}+\frac{100}{(1+y_2)^2} \right)}-1$$

Now if I assume a positive parallel shift of $\delta$, in order to prove what I am after, I would need to prove that $\overline{IRR} = IRR_1 + \delta$, i.e. that the following equation holds:

$$\frac{100}{1+y_1+\delta}+\frac{100}{(1+y_2+\delta)^2}=\frac{100}{1+IRR_1+\delta}+\frac{100}{(1+IRR_1+\delta)^2}.$$

This is approximately where I lose all my faith and confidence that I am doing the right thing because it gets so messy.

## Answer by Attack68 (score 2, accepted)

https://quant.stackexchange.com/a/79577

First, for a parallel shift, $x$, in the spot curve, show that for any bond, $i$, $$\frac{\partial P_i}{\partial x} < 0, \implies \frac{\partial P}{\partial x} = \sum_i \frac{\partial P_i}{\partial x} < 0$$ That is, as rates go up, prices of bonds go down. Since every bond goes down in price the whole portfolio goes down in price.

Now,

$$\frac{\partial P}{\partial x} = \frac{\partial P}{\partial Irr} \frac{\partial Irr}{\partial x} < 0 $$

Thus, either;

- $\frac{\partial P}{\partial Irr}$ is negative and $\frac{\partial Irr}{\partial x}$ is positive, or,

- $\frac{\partial P}{\partial Irr}$ is positive and $\frac{\partial Irr}{\partial x}$ is negative.

In very simple cases, i.e. for one bond portfolio, it is easy to show that $\frac{\partial P}{\partial Irr} < 0$. You could generalise this, possibly by induction that, or looking at the sum of each individual bond. Your requirement is then shown that:

$$\frac{\partial Irr}{\partial x} > 0$$

I.e. both the IRR and the spread move in the same direction. As expected.