### Yield Measures for Fixed-rate Bonds

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

## Street Convention

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

## True-Yield

True yield is the internal rate of return that is calculated using a calendar, including weekends and bank holidays. The true yield is always less than street convention yield since the weekends and the 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 it does not consider capital gain or loss. It is calculated as the sum of 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 priced at 98 per 100 of the par values. Calculate the current yield of the bond.

Solution

We know that current yield is the sum of 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 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

An example of such one is the callable bond where the issuer of the bond has the right to repurchase back the bond from the investor at a specified price on predetermined date, which usually coincide 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 years of protection from the one being called. After the lapse of the protection period, the issuer might exercise the call option if the interest rates decrease or the credit quality of the issuer improves. If the call option is exercised, then the issuer pays a premium above the bar.

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

A 5-year callable bond pays 10% coupon payments and 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 in 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 trial and error method (financial calculator might be quicker):
Starting at $$r_1=5\%$$
$$p_1=105=\frac{10}{(1.05)^1}+\frac{102+10}{(1.05)^2}=102.0408$$
At $$r_2=3\%$$
$$p_2=105=\frac{10}{(1.03)^1}+\frac{102+10}{(1.03)^2}=105.8535$$

Using the interpolation method:
$$\frac{p-p_1}{p_2-p_1}≈\frac{r-r_1}{r_2-r_1}=\frac{105-102.0408}{105.8535-102.0408}≈\frac{r-0.05}{0.03-0.05}$$

$$r=0.05+(\frac{105-102.0408}{105.8535-102.0408})(0.03-0.05)=0.034477≈3.44 \%$$

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 are using 100 per 100 of par value.

The yield to worst is the lowest potential yield that can be received on a bond without the issuer defaulting. It is the smallest possible values of sequence of yield-to-call and the yield to maturity. For example, our example above, if the yield-to-second call is 9.312% and the yield to maturity is higher than yield-second-call, then the yield to worst is 3.44% (yield-t-first call). 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 embedded call option and flat prices 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 and hence the name “floating.” FRN offers an investor a security that has relatively low market price risk as compare o fixed-rate bonds. This is true because price volatility affects the fixed-rate bonds extensively since the interest payments are constant over the payment period. On the other hand, interest volatility affects future payments as well.

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

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

As we have 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 margin compensates the investor for the difference in credit risk of the issuer, and that 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 at par value and that it pays a 3-moth LIBOR and added 0.25%. Therefore, the quoted margin is 25 pbs (0.25×100) so 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 be either 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 pbs, the floater which initially had a quoted margin of 25 pbs, pays a “deficient” interest since the floater will be priced at a discount below par value. However, if the quoted margins decrease from 25 pbs to 15 pbs, the FRN will now be priced at a premium.

In the case that 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 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.

Some points about this model are that the N is evenly spaced; it assumes a 30/360-day count convention (the periodicity is an integer) and that the reference rate also called 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 six-month Libor plus 0.5%. The current six-month Libor is 1.25%. Moreover, the yield spread required by the investors is 50 pbs over the reference rate. Calculate the price of the FRN per 100 of the par values.

Solution

Now using the formula:

$$Index=1.25=0.0125$$

$$QM=0.25%=0.0025$$

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

$$FV=100$$

$$DM=50 pbs=0.50%=0.0050$$

$$N=2×2=4$$

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:

A. 0.0840%

B. 0.0863%

C. 0.0686%

Solution

Using the information given and the formula provided, then:

$$\frac{(Index+QM)×FV}{m}=\frac{(0.015+0.005)×100}{4}=0.25$$

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

$$99=\frac{0.25}{(1+\frac{0.015+DM}{4})^1} +\frac{0.25}{(1+\frac{0.015+DM}{4})^2} +⋯+\frac{0.25}{(1+\frac{0.015+DM}{4})^8}$$

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

$$99=\frac{0.25}{1+r)^1} +\frac{0.25}{1+r)^2} +⋯+\frac{0.25}{1+r)^8}$$

Using financial calculator:

$$r=0.003771$$

$$\Rightarrow \frac{0.015+DM}{4}=0.003771$$

$$\therefore DM=0.000084=0.0084 \%$$

Reading 44 LOS 44g:calculate and interpret yield measures for fixed-rate bonds and floating-rate notes;

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