The IR-exempt title is not always better; see a simplified way to compare between exempt and taxed fixed income – 09/10/2023 – From Grão to Grão

When deciding to invest in fixed income, we must consider at least three elements: term, risk and return. Comparing deadlines is simpler. However, the comparison of return and risk is not trivial, especially when we analyze corporate bonds exempt from IR and taxed. Right now, I will focus on comparing returns.

After the article from the day before yesterday, when I showed a table equating income exempt from IR and its gross IR equivalents, I received a few e-mails asking me how the calculation would have been performed.

Understanding how to compare tax-exempt bonds is essential to choose the one that will provide the best result for the portfolio.

Unfortunately, the precise account demands that the calculation be carried out more carefully and it depends on the interest rate and the term of the security.

However, there is a pocket formula that simplifies the comparison of exempt bonds with their taxed peers. It must be remembered that every simplification carries an error.

In the case of post-fixed titles, that is, those referenced to the CDI, I wrote how to do the calculation at this link. At that moment, I warned that the simplification carries an error that increases with the term of the title.

But, the simplification is reasonably accepted. Let’s see an example applied to a prefixed title.

In a simplified form, to calculate the equivalent gross IR yield of an exempt prefixed security, simply divide the exempt security by 1 minus the IR rate.

For example, consider an IR-exempt bond with a yield of 12% per year maturing in 2 years. The gross IR rate equivalent to it is 14.12% per year ( = 0.12/(1-0.15)).

As I explained, this account is not accurate, as the taxed bond that has a return of 14.12% per year in two years is equivalent, more precisely, to an exempt bond with a return of 12.12% per year. In five years, the income tax-free return on this taxed bond will be 12.41% per year.

As I mentioned, realize that the error grows over time, but the pocket formula approximation is good enough not to be discarded.

The complexity of the title referenced to the IPCA goes a little beyond the deadline.

A security referenced to the IPCA is made up of two income portions: the IPCA and the real rate, which is pre-fixed.

Therefore, in these bonds, it is not enough to simply divide the real interest portion by 1 minus the income tax rate, as we did above.

In this case, it will be necessary to make an assumption about future inflation, as IR affects both real interest and inflation.

Therefore, see in the table below that the higher the IPCA in the future, the greater the advantage of the exempt title over the taxed one.

Real interest rate (txR, % pa)	IPCA expectation (% pa)	IR gross equivalent real rate (% pa)
6.50%	3.00%	8.16%
6.50%	4.00%	8.33%
6.50%	5.00%	8.49%
6.50%	6.00%	8.65%
6.50%	7.00%	8.80%
6.50%	8.00%	8.95%

In the table above, I considered an IR-exempt bond with a yield of IPCA+6.5% per year. Note that for each IPCA expectation, we have a different equivalent gross real rate.

For the calculation performed in the table, the IPCA yield must be combined with the real rate, calculate the gross equivalence of IR and then extract the real rate, dividing by the IPCA again:

(((1 + IPCA)*(1 + txR) – 1)/ (1 – IR) + 1)/ (1 + IPCA) -1,
where:
IPCA is the inflation expectation
txR is the real interest rate of the bond
IR is the IR rate

I remember that the account above is the pocket formula. Therefore, there is a small error in the estimate, but it is compensated by the benefit of simplicity of calculation.

With this simplified formula, it will be possible to properly compare exempt with taxed bonds and understand which of the two will promote better results in your portfolio.

Michael Viriato is an investment advisor and founding partner of Investor House.

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