# Causality: interest rates and fixed income assets

### T. Fuertes

#### 19/10/2022

The blog has previously addressed interest rates in a post that splits the yield rate curve into three relevant components. This time this post tries to identify the influence of interest rates on fixed income assets by using the Granger causality test.

Interest rates obviously have a strong impact on fixed income assets because they are the base to compute their prices. However, can we find any causality effect that allows us to anticipate the movements of a fixed income asset looking at the movements of interest rates? Read on and find it out how Granger causality tests can help!

## Interest rates

The definition of an interest rate is the amount an investor will be charged for borrowing money. Generally, the interest rate is a percentage of the total amount of the loan.

There are loans of any period of time, as a consequence, there will be market rates quoted for each of those periods. The periods represent the maturity, which can be from 1 month to 30 years.

## Bond prices

So that there is market balance, bond prices only need four data to be computed:

• F: face value. This is the amount received when the bond dies,
• C: coupons. The amount of money received periodically while the bond is alive,
• n: amount of periods to receive coupons, also time to maturity, and
• r: interest rate in the period that coupons are paid.
$\begin{array}{l} &P=\sum_{i=1}^n\frac{C}{(1+r)^i}+\frac{F}{(1+r)^n} \end{array}$

As an example, let’s take a bond having a face value of $1000, carrying an annual coupon of$40 and maturing in 10 years. If the prevailing market rate of interest is 5%, and following the previous formula, the price will be $922.78. In this case, as the price is less than the face value, the bond is priced below par. As the formula reveals, there is an inverse relationship between interest rates and bond price. Therefore, if the market rate rose to 6%, a new bond with the same coupon would have a lower price, which would be$852.80. That means that the old bond would be less attractive because it is more expensive than an on the run 10 years bond. In this case, the new bond would be the cheapest to deliver, which is the bond that market participants will opt to deliver in order to settle a future contract on the 10 years bond.

On the other hand, if market rate fell to 3.5%, an on the run bond with the same coupon would be priced at \$1,041.58. In this case, as the price is higher than the face value, the bond is priced above par.

To sum up, when interest rates rise, bond prices decrease, and vice versa.

## Fixed income ETFs

Having seen the inverse relationship between interest rates and prices bonds, everything would be easier if investors could invest directly in bonds. However, in most of the cases portfolios are composed of fixed income ETFs (or mutual funds) due to capital constraints, liquidity, and expertise infrastructure of the portfolio manager. Then, investors will essentially have to anticipate how fixed income ETFs react to the interest rates’ movements.

Firstly the investor should know the holdings of the ETFs in the portfolio to know the maturity of their underlyings. For instance, let’s take the iShares 10-20 Year Treasury Bond ETF, which is an ETF that invests in USA Treasuries (state bonds directly related to zero-coupon rates) of maturities between 10 and 20 years. However, among its holdings, there is a wide range of maturities, as the next table shows:

For this ETF, most of the weight is in bonds with a maturity between 10 and 15 years. So it’s easy to think that the interest rates of these maturities will have more influence than the others on the movements of this ETF, but they won’t be the only ones.

## Causality between interest rates and fixed income ETFs

After seeing the mixture of maturities in the holding of ETFs, it is clear that the effect of interest rates in ETFs won’t be immediate. Then, to verify the actual influence of interest rates on fixed income ETFs, we run the Granger causality test. This is a hypothesis test that has the follow definition:

$\begin{cases}H_0: & X\ causes\ Y\\H_1: & X\ does\ not\ cause\ Y\end{cases}$

According to this test, run with 5% of confidence, if the p-value is less than 0.05, the null hypothesis can be refused, that is, there is causality of X on Y.

We compute the causality of USA interest rates with maturities from 1 month to 30 years on 10 fixed income ETFs. These ETFs are ultra-short, short, medium and long term. Firstly, we compute the rolling differences of interest rates in windows of 1 month, 3 months, 6 months and 1 year. Secondly, we compute the rolling returns of ETFs in windows of 1 week, 2 weeks, 1 month, 3 months, 6 months and 1 year. By using all the combinations of both groups of series, we run the Granger causality test.

In the following graph, the lighter colours represent the lowest values of p-value of each causality test. The lighter colour, the closer to threshold 0.05, which means causality. We can inference that short term interest rates cause influence on short term ETFs’ movements regardless of the window. However, medium and long term interest rates have an influence on medium and long term ETFs only in low windows, especially windows less than 3 months.

## Summing up

This post has explained how interest rates’ movements influence on bonds’ prices. However, it has also revealed that this causality is diluted when the influence is measured on fixed income ETFs, as their holdings are a mixture of bonds’ maturities.

Despite this, the Granger causality test has proved that recent movements of interest rates may cause effect on next movements of fixed income assets. This might be the start point of a fixed income strategy based on interest rates movements, don’t you think?