## Yield Measures for Fixed-rate Bonds

There are many ways to calculate the rate of return on a fixed rate bond. However, investors want a yield measure that is standardized for comparison between bonds with different time-to-maturities, and therefore yield measures are generally annualized.

In general, an annualized and compounded yield on a fixed rate bond depends on the number of periods in a year, called the periodicity of the annual rate, which typically matches the frequency of coupon payments.

For instance, the annual yield-to-maturity on a 3-year zero-coupon bond priced at 85 per 100 of par value could be calculated as the following:

• For annual compounding, annual yield-to-maturity is 5.567%. Periodicity is 1.

$$\frac { 100 }{ (1+r)^{ 1 } } =85.00;\quad r=5.567\% \quad(per \quad year)$$

• For semiannual compounding, annual yield-to-maturity is 5.491\%. Periodicity is 2.

$$\frac { 1000 }{ (1+r)^{ 6 } } =85.00;\quad r=2.746\% \quad (per \quad 6 \quad months)×2=5.491\% \quad(per \quad year)$$

• For quarterly compounding, annual yield-to-maturity is 5.454\%. Periodicity is 4.

$$\frac { 100 }{ (1+r)^{ 12 } } =85.00;\quad r=13=1.364\% \quad (per \quad 6 \quad quarter)×4=5.454\%\quad(per \quad year)$$

• For monthly compounding, annual yield-to-maturity is 5.430\%. Periodicity is 12.

$$\frac { 100 }{ (1+r)^{ 36 } } =85.00;\quad r=0.452\%\quad(per \quad month)×12=5.43\%quad(per \quad year)$$

## A few Definitions…

The effective annual rate has a periodicity of 1 as there is only 1 compounding period.

The semiannual bond basis yield (or semi-annual bond equivalent yield) is the annual rate with a periodicity of 2.

The street convention assumes that payments are made on scheduled dates, excluding weekends and holidays.

The true-yield is calculated using a calendar including weekends and holidays.

The current yield relates the annual dollar coupon interest to the market price. This is the simplest yield measure and it does not consider capital gain or loss.

The yield to worst is the lowest potential yield that can be received on a bond without the issuer defaulting.

The option-adjusted yield is the yield-to-maturity also including the theoretical value of the call option.

## Yield Measures for Floating-rate Notes

As we’ve seen previously, the interest rate volatility affects the price of a fixed-rate bond. However, a floating-rate note (FRN) results in a more stable price due to the flexibility of interest rates.

The quoted margin is the yield spread over or under the reference rate such as LIBOR. This compensates the investor for the difference in credit risk of the issuer. The discount margin (or required margin) is the spread required by investors and the quoted margin must be set for FRNs to trade at par value. Changes in discount rates are directly correlated to issuer’s credit risk.

## Yield Measures for Money Market Instruments

Money market instruments are short-term debt securities. They different from longer-term fixed-rate bonds in the following ways:

• Bond yields-to-maturitiy are annualized and compounded. However, yield measures in money market are annualized but not compounded. The rate of return is stated on a simple interest basis.
• Bond yields-to-maturitiy are calculated using standard time value of money whereas money market instruments are often quoted using nonstandard interest rates such as discount rates or add-on rates.
• Money market instruments with different times-to-maturities have different periodicities.

The bond equivalent yield states the money market rate on a 365-day add-on rate basis. The formulas shown below help us calculate the discount rates and add-on rates on money market instruments:

#### Formula for the Discount Rate (DR)

$$PV=FV×(1-\frac { Days }{ Year } ×DR)$$

$$DR=\frac { Year }{ Days } ×\frac { (FV-PV) }{ PV }$$

#### Formula for the Add-on Rate (AOR)

$$PV=\frac { FV }{ 1+\frac { Days }{ Year } \times AOR }$$

$$AOR=\frac { Year }{ Days } ×\frac { (FV-PV) }{ PV }$$

## Question

A 180-day US Treasury Bill with a face value of 100 has a quoted discount rate of 4.5%. Its bond equivalent yield is closest to:

A. 3.33%

B. 4.25%

C. 4.67%

Solution

$$PV=100×(1-\frac { 180 }{ 360 } ×4.5\% ) =97.750$$

And the bond-equivalent yield for a 365-day year is 4.67%.

$$AOR=\frac { 365 }{ 180 } ×\frac { (100-97.75) }{ 97.75 } =4.67\%$$

Calculate and interpret yield measures for fixed-rate bonds, floating-rate notes, and money market instruments

Share:

#### Related Posts

##### Compare, Calculate, and Interpret Yield Spread Measures

Yield spread (measured in basis points) is the difference between any two bond...

##### Term Structure of Yield Volatility and Interest Rate Risk

Time-horizon is a very important aspect in understanding interest rate risk and the...