Using PCA to Shock an On-the-Run Yield Curve and Reprice Bonds

# Shocking an OTR curve to price an OFF THE RUN


# Shocking an OTR curve to price an OFF THE RUN












I would like to shock an OTR curve in a certain way and reprice an off-the-run bond after that shock.

I have a yield curve that I want to shock. This yield curve is the YTM of the OTR bonds. The way I want to shock it is as follow: I want to be able to shock the YTM of the OTR at the 2 year point, 5year point or 30 year point. I'll call these shocks $s_2, s_5, s_{30}$. For example if I think the $30$y point is going to increase by $1$bp I'll have: $s_2 = 0, s_5 = 0, s_{30} = 1$. If I think the $2y$ will decrease by $2$bp, the $5$y increase by $3$bp and the $30$y increase by $1$ I'll get: $s_2 = -2, s_5 = 3, s_{30} = 1$.

Since all points on the curve are correlated if I shock the $5$year point it should also shock the $7, 20, ...$ and all the other point on the curve (except the $2, 30$ since these are the points I shock myself => the points I have a view on)

The first way I thought about was to use PCA. I do PCA on the difference in yield of that OTR curve. I thus get $PC_1, PC_2, PC_3$. where $PC_i(t)$ is the coordinate of $PC_i$ at $t$ years.

Now if I introduce a shock I solve the system of equations:

$$s_2 = \lambda_1 \cdot PC_1(2) + \lambda_2 \cdot PC_2(2) + \lambda_3 \cdot PC_3(2) \\ s_5 = \lambda_1 \cdot PC_1(5) + \lambda_2 \cdot PC_2(5) + \lambda_3 \cdot PC_3(5)\\ s_{30} = \lambda_1 \cdot PC_1(30) + \lambda_2 \cdot PC_2(30) + \lambda_3 \cdot PC_3(30)$$

Hence after solving these equations I get the new yield curve: $y_{today} + \lambda_1 \cdot PC_1 + \lambda_2 \cdot PC_2 + \lambda_{30} \cdot PC_{30}$ where $y_{today}$is the yield curve today.

Hence my first question is:

Q1: Is that a correct way of doing it? I don't feel super confident because this system of equation don't take into account the explained variance of each $PC$. Also the problem is how to perform PCA correctly since OTR changes, so what to do when I don't have enough historical data for an OTR?

Now that I have shocked my yield curve I want to see how I can reprice an off the run bon in this new environment. Can I just assume a constant Z-spread and use the point on my new OTR curve to reprice the bond?

The problem is that the number of points is very small ($2, 3, 5, 7, 10, 20, 30$), so using only these points to reprice my curve is probably not ideal.

Hence my second question:

Q2: Can I just assume a constant Z-spread and use the new yield curve to reprice my off-the-run? or is there a better way of doing it?

Many thanks.