# Ways to Compare the Yields of Different Bonds

Investors who purchase T-bills do not receive interest payments. Instead, they earn a return based on the difference between the purchase price and the face value of the bill at maturity. However, determining this rate can be complicated as it is based on a hypothetical year of 360 days.

On the other hand, when investing in CDs, the annual percentage rate (APR) may not accurately reflect the actual return. This is because the better figure to consider is the annual percentage yield (APY), which considers compounding and provides a more accurate representation of the return on investment.

## Basics

Bond yield comparisons pose a challenge due to diverse coupon payment frequencies and the utilization of various yield conventions in fixed-income investments. Converting yields to a standardized basis becomes imperative for accurate comparisons among distinct bonds.

Individually, these conversions present a clear process. However, the complexity escalates when confronted with both compounding period and day-count conversions, making the attainment of an accurate solution more challenging.

## Analyzing Bond Yields: A Different Perspective

When it comes to U.S. Treasury bills (T-bills) and corporate commercial paper investments, market quoting and trading follow a discount model. In this scenario, investors do not receive coupon interest payments. Profit is earned from the difference between the current purchase price and the face value at maturity, which is a form of implicit interest payment.

Expressed as a percentage of the face value, the discount amount is then annualized over a 360-day year. However, relying on rates quoted on a discount basis presents inherent challenges. Firstly, such rates underestimate the true rate of return over the term to maturity due to the percentage being based on face value.

A more rational approach involves considering the rate of return as the interest earned divided by the current price, not the face value. Given that T-bills are acquired at a value lower than their face value, the denominator is disproportionately high, resulting in an understated discount rate. The second challenge lies in the fact that the rate is anchored in a hypothetical year with only 360 days, introducing further complexities.

### Bank CD Yields

Historically, bank certificates of deposit (CDs) have been traditionally quoted on a 360-day year, with some persisting in this practice. However, due to the slightly elevated rate derived from a 365-day year, the majority of retail CDs are now presented using the latter basis.

These returns showcase their annual percentage yield, distinct from the annual percentage rate commonly associated with mortgages. Unlike APR, which simply multiplies interest rates by the number of periods in a year without accounting for compounding, APY incorporates the effects of compounding. For instance, a six-month CD offering 3% interest may have an APR of 6%, but its APY is calculated at 6.09%, factoring in compounding effects:

In contrast, yields on Treasury notes and bonds, corporate bonds, and municipal bonds follow a semi-annual bond basis (SABB), aligning with their semi-annual coupon payments. Compounding in this context occurs twice a year, using a 365-day year.

## Bond Yield Transformations

Navigating the landscape of fixed-income investments requires a unified approach to yield calculations. An initial conversion involves moving from a 360-day yield to a 365-day yield, achieved by increasing the 360-day yield with the factor 365/360. For instance, an 8% 360-day yield equates to an 8.11% 365-day yield.

Delving into discount rates, commonly applied to T-bills, necessitates a shift to bond-equivalent yield (BEY). For "short-dated" bills maturing in 182 days or fewer, the conversion formula is:

**where:**

*BEY = the bond-equivalent yield**DR = the discount rate (expressed as a decimal)**N = # of days between settlement and maturity*

In contrast, "long-dated" T-bills with a maturity exceeding 182 days involve a more intricate formula, considering compounding effects:

For short-dated T-bills, the implicit compounding period for BEY aligns with the days between settlement and maturity. However, for long-dated T-bills, no well-defined compounding assumption exists, adding complexity to interpretation.

BEYs consistently trail annualized yields for semi-annual compounding. The frequency of compounding inversely correlates with the rate, leading to lower BEYs for more frequent compounding periods.

Caution is warranted when comparing BEYs reported by the Federal Reserve and financial institutions to yields on longer-maturity bonds. While BEYs compare T-bills, T-notes, and T-bonds maturing on the same date, a more accurate comparison with longer-maturity bonds involves converting discount rates to a semi-annual bond basis (SABB). The SABB, computed using the APY formula with semi-annual compounding, allows for direct comparison with yields based on SABB. The conversion formula for a discount rate (DR) on an N-day T-bill to SABB is:

## Conclusion

Fixed-income investments bring challenges to yield calculations. T-bill investors face complexity with a hypothetical 360-day year, while CD investments highlight the need to consider the annual percentage yield for accuracy. Diverse bond yield comparisons demand a standardized basis, with caution urged in comparing bond-equivalent yields to longer-maturity bonds. Emphasizing a semi-annual bond basis remains crucial for precise evaluations.