 # Measures for Fixed-Rate Bonds and Floating-Rate Notes

## Yield Measures for Fixed-rate Bonds

Fixed-rate bonds are those that pay the same amount of interest throughout their specified term. Measurement for fixed-rate bonds depends on the timing of the bond’s cash flows. The following are the measures of fixed-rate bonds.

## Street Convention

Street convention assumes that payments are made on scheduled dates, excluding weekends and holidays. It is commonly used because it simplifies bond price and yield calculations.

## True-Yield

True yield is the internal rate of return calculated using a calendar, including weekends and bank holidays. The true yield is always less than the street convention yield since the weekends and holidays delay the payment of the bond’s proceeds.

## Current Yield

Current yield (also called income or running yield) relates the annual dollar coupon interest to the market price. This is the most straightforward yield measure and does not consider capital gain or loss. It is calculated as the sum of the bond’s coupon payments over one year divided by the flat price of the bond.

The current yield is a crude measure of the investor’s return since it does not include the frequency of coupon payments and the accrued interest.

#### Example: Calculating the Current Yield

A five-year bond pays annual coupons of 4% and is priced at 98 per 100 of the par values. Calculate the current yield of the bond.

Solution

We know that the current yield is the sum of the bond’s coupon payments over one year divided by the flat price of the bond. So, in this case:
$$\text{Current Yield}=\frac{4}{98}=0.04082=4.082\%$$

## Simple Yield

Simple yield is defined as the sum of the coupon payments and the straight-line proportion of the gain or loss divided by the flat price of the bond.

### Yield Measures in Fixed-Rate Bond Containing Embedded Option

A good example here is the callable bond, where the issuer of the bond has the right to recall and buy off the bond from the investor at a specified price on a predetermined date. The date usually coincides with the call protection period (a period where the issuer of the bond is prohibited from exercising the call option).

Depending on the date of the exercise, the yield of the callable option can be termed as yield-to-first call, yield-to-second call, and so forth. For instance, if a bond is first callable after two years, then the investor has two-year protection. After the lapse of the protection period, the issuer might exercise the call option if the interest rates decrease or the issuer’s credit quality improves. If the call option is exercised, then the issuer pays a premium above par.

#### Example: Calculating Measures in a Bond Containing Embedded Option

A 5-year callable bond pays 10% coupon payments and is currently priced at 105. The bond is first callable at 102 per 100 of par value after 2 years and 101 per 100 of par value after four years. Calculate the yield-to-first call.

Solution

The yield-to-first call is calculated from the following equation:

$$p=105=\frac{10}{(1+r)^1}+\frac{102+10}{(1+r)^2}$$

Using a financial calculator: $$r=8.15%$$

At this point, you can easily calculate the yield-to-second call (and, of course, the yield to maturity).

Note that in our calculation of yield to the first call, we are using the call price as FV. This is quite different from the yield to maturity, where we use 100 per 100 of par value.

The yield-to-worst is the lowest potential yield on a callable bond. It is the lowest between the yield-to-maturity and the yields at each call date (yield-to-call). For example, if the yield-to-second call is 9.312%, then the yield-to-second call is greater than the yield-to-first call. The lower of the two calculated yields is the yield-to-worst (equal to 8.15%). The yield-to-worst gives the investor the most stable assumption rate of return.

Option-Adjusted Yield is the yield-to-maturity, also including the theoretical value of the call option. That is, it is a measure calculated from the option-adjusted price. The option-adjusted price is the sum of the value of the embedded call option and the flat price of a bond. Option-adjusted yield is a market discount rate where the price of the bond is adjusted to accommodate the embedded option.

### Yield Measures for Floating-Rate Notes (FRN)

Floating-Rate Notes are bonds whose interest payments vary from period to period depending on the level of the reference interest rate. In other words, the interest payments on the floating-rate notes can fall or rise, hence the name “floating”. FRN offers an investor security that has relatively low market price risk when compared to fixed-rate bonds. This is true because price volatility affects fixed-rate bonds extensively since the interest payments are constant over the payment period. It is equally noteworthy that interest volatility affects future payments as well.

The reference rate on a floating note is a short-term money market rate (such as a 3-month LIBOR). It is determined at the start of each period, and the interest payment is made at the end of the period (in arrears) while maintaining the actual/360-day convention.

We either subtract or add a stated yield spread to the reference rata to obtain these floaters. This specified yield spread is termed the quoted margin on FRN.

As we have seen previously, 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 margin compensates the investor for the difference in the issuer’s credit risk and the risk suggested by the reference rate.

The required margin (or Discount) is the spread above or below the reference rate that makes the FRN priced at par value on the date the rate is reset. For instance, if the issued floater is at par value, pays a 3-month LIBOR, and adds 0.25%, the quoted margin will be 25 bps (0.25×100). This implies that at each quarterly reset, the floater will be priced at par value. Moreover, in between the coupon date, the flat price of the floater will either be at a premium or discount to par value, depending on the fluctuation of the LIBOR. For instance, in the example, the yield spread rises to 50 bps, and the floater, which initially had a quoted margin of 25 bps, pays a “deficient” interest since the floater will be priced at a discount below par value. However, if the quoted margins decrease from 25 bps to 15 bps, the FRN will now be priced at a premium.

If the required margin and the quoted margin continue to be equal, the flat price is pulled to par when the reset date reaches.

### Valuation of FRN

The pricing model for FRN is given by:
$$PV=\frac{\frac{(Index+QM)×FV}{m}}{(1+\frac{Index+DM}{m})^1} +\frac{\frac{(Index+QM)×FV}{m}}{(1+\frac{Index+DM}{m})^2} +⋯+\frac{\frac{(Index+QM)×FV}{m}}{(1+\frac{Index+DM}{m})^N}$$

$$PV$$ =  price of the floating-rate note (as a present value)
$$Index$$ = annualized reference rate
$$QM$$ = annualized quoted margin
$$FV$$ = future value paid at maturity, or the par value of the bond
$$m$$ = periodicity of the floating-rate note (the number of payment periods per year)
$$DM$$ = annualized discount margin (required margin)
$$N$$ = number of evenly spaced periods to maturity

Note that the above formula is similar to that of the fixed-rate bond only that here the PMT is equivalent to the annual rate (Index +QM)  multiplied by the par value (FV) and divided by the periodicity. Moreover, the market discount rate per period is equivalent to the sum of the reference rate and the discount margin (Index +DM) divided by the periodicity.

It is imperative to note that the N is evenly spaced; it assumes a 30/360-day count convention (the periodicity is an integer). The reference rate, also called the index, is used in all periods in both the numerators and denominators.

#### Example: Calculating the Price of a Floating-Rate Note

The 2-year  FRN pays a six-month Libor plus 0.25%. The current six-month Libor is 1.25%. Moreover, the yield spread required by the investors is 50 bps over the reference rate. Calculate the price of the FRN per 100 of the par values.

Solution

Now using the formula:

$$Index=1.25=0.0125$$

$$QM=0.25%=0.0025$$

$$m=2 \text{(the payment is semiannual)}$$

$$FV=100$$

$$DM=50 bps =0.50%=0.0050$$

$$N=2×2=4$$

So:
$$\frac{(Index+QM)×FV}{m}=\frac{(0.0125+0.0025)×100}{2}=0.75$$

and
$$\frac{Index+DM}{m}=\frac{0.0125+0.0050}{2}=0.00875$$

Therefore, the price of the floating rate note is given by:

$$PV=\frac{0.75}{(1+0.00875)^1} +\frac{0.75}{(1+0.00875)^2} +⋯+\frac{0.75+100}{1+0.00875)^4} =99.511$$

So, the price of the floating rate is 99.511 per 100 of par value.

The FRN is priced at a discount since the quoted margin (0.0025) is less than the discount margin (0.0050).

Using the formula, you can be asked to compute the discount margin when you are provided with the price of the FRN. Check out the question below.

### Question

A two-year  British floating rate note pays a regular 3-month of 1.5% Euribor plus 0.5%. The floater is priced at 99 per 100 of par value. Assuming the 30/360-day count convention and that the periods are evenly spaced,  the discount margin for the floater is closest to:

1. 0.0840%
2. 0.0863%
3. 0.0686%

Solution

Using the information given and the formula provided, then:

$$\frac{(Index+QM)×FV}{m}=\frac{(0.015+0.005)×100}{4}=0.50$$

We can  calculate the DM if we solve the following equation:

$$99=\frac{0.50}{(1+\frac{0.015+DM}{4})^1} +\frac{0.50}{(1+\frac{0.015+DM}{4})^2} +⋯+\frac{0.50}{(1+\frac{0.015+DM}{4})^8}$$

Now, Let $$r=\frac{0.015+DM}{4}$$ so that:

$$99=\frac{0.5}{1+r)^1} +\frac{0.5}{1+r)^2} +⋯+\frac{0.5}{1+r)^8}$$

Using financial calculator:

$$r=0.0062856$$

$$\Rightarrow \frac{0.015+DM}{4}=0.0062856$$

$$\therefore DM=0.01014=1.014 \%$$

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