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Sign in. The yield curve is a line that plots the various interest rates of bonds with equal credit quality and different maturities. Government bonds are said to have negligible default risk, as the government can simply borrow more money to finance their repayments.

The discount function, which determines the value of all future nominal payments, is the most basic building block of finance and is usually inferred from the Treasury yield curve. It is therefore surprising that researchers and practitioners do not have available to them a long history of high-frequency yield curve estimates. This paper fills that void by making public the Treasury yield curve estimates of the Federal Reserve Board at a daily frequency from to the present. We use a well-known and simple smoothing method that is shown to fit the data very well. The resulting estimates can be used to compute yields or forward rates for any horizon.

The U.S. Treasury Yield Curve: 1961 to the Present

Sign in. The yield curve is a line that plots the various interest rates of bonds with equal credit quality and different maturities. Government bonds are said to have negligible default risk, as the government can simply borrow more money to finance their repayments.

According to the expectations theory of interest rates, the yield curve is made up of two aspects:. An average of market expectations concerning future short-term interest rates. The term premium — the extr a compensation an investor receives for holding a longer-term bond.

Therefore, for a fixed-income investment to be worth the extra time the investor must part with their cash, the bond issuer must pay the investor some extra amount.

We can model these aspects of the yield curve using principal components decomposition. Data has two main properties: noise and signal. Principal components analysis aims to extract the signal and reduce the dimensionality of a dataset; by finding the least amount of variables that explain the largest proportion of the data.

For this analysis I will use various UK government bond spot rates from 0. When finding the principal components of the yield curve, the main theory held by econometricians is that:.

After importing the relevant modules needed, and importing the spot curve data from the Bank of England, we can see in the DataFrame that there are some nan values due to missing data. The next step is to standardize the data into z-scores, assuming a mean of 0 and a variance of 1.

We can do this by using the formula:. We then form a covariance matrix from the standardized data using numpy. Although correlation matrices are more commonly used in finance, due to our data being standardized, a correlation matrix and covariance matrix will actually yield the same result. This is because a correlation matrix is simply a standardized covariance matrix.

We can then use the linalg. This performs eigendecomposition on our standardized data. Eigenvalues are scalars of the linear transformation that has been applied in np. We can use eigenvalues to find the proportion of the total variance that each principal component explains using this formula:. Eigenvectors are the coefficients of these linear transformations, leaving the direction unchanged. This diagram and equation should help to explain the concept visually:.

To form a time series for the principal components, we simply need to calculate the dot product between the eigenvectors and the standardized data. When we plot the first principal component, we can see that it looks very similar to the actual year yield curve. The second principal component represents the slope — this should have a correlation with the slope of the actual yield curve. One way we can calculate the slope is the year spot minus the 2-year spot rate. Simply from visual inspection, we can see that the slope looks almost identical to our second principal component.

When running the correlation between the second principal component and the 10Y-2M slope of the yield curve, the high correlation of 0. Thanks for reading! Please feel free to leave any comments for any insights you may have. The full Jupyter Notebook which contains the source code I used to do this project can be found on my Github repository.

Disclaimer: All views expressed in this article are my own, and are not in any way associated with Vanguard or any other financial entity. I am not a trader and am not making any money from the methods used in this article. This is not financial advice.

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A Medium publication sharing concepts, ideas and codes. Get started. Open in app. Editors' Picks Features Explore Contribute. Applying PCA to the yield curve — the hard way. Learn how to apply one of the most popular applications of principal components analysis using current financial data in python. Nathan Thomas. References:  Alexander, Carol, Sign up for The Variable. Get this newsletter. More from Towards Data Science Follow. Read more from Towards Data Science.

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Applying PCA to the yield curve — the hard way

In finance , the yield curve is a curve showing several yields to maturity or interest rates across different contract lengths 2 month, 2 year, 20 year, etc. The curve shows the relation between the level of the interest rate or cost of borrowing and the time to maturity , known as the "term", of the debt for a given borrower in a given currency. The U. Treasury securities for various maturities are closely watched by many traders, and are commonly plotted on a graph such as the one on the right, which is informally called "the yield curve". Yield curves are usually upward sloping asymptotically : the longer the maturity, the higher the yield, with diminishing marginal increases that is, as one moves to the right, the curve flattens out.

A Recession is coming in months; 2. The Fed is too tight, and needs to cut rates; 3. Investors are rotating out of risk assets and into safe US Treasuries bonds;.

Nathan Thomas

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