Reading 52
Fixed-Income Bond Valuation: Prices and Yields

Calculating the Value of an Annual Coupon Bond

The value of a coupon bond can be calculated by summing the present values of all of the bond's promised cash flows. The market discount rate appropriate for discounting a bond's cash flows is called the bond's yield to maturity (YTM). If we know a bond's YTM, we can calculate its value, and if we know its value (market price), we can calculate its YTM.

Consider a newly issued 5-year, 10% coupon, annual-pay bond. For $100 of par value, the coupon payments will be $10 at the end of each year, and the $100 par value will be paid at the end of Year 5. First, let's value this bond assuming the appropriate discount rate (also called "yield") is 10%. The present value of the bond's cash flows discounted at 10% is:

$$\frac{10}{(1.10)^1}+\frac{10}{(1.10)^2}+\frac{10}{(1.10)^3}+\frac{10}{(1.10)^4}+\frac{110}{(1.10)^5}=100$$

The calculator solution is as follows:

• N = number of years
• PMT = the annual coupon payment
• I/Y = the annual discount rate (YTM)
• FV = the par value or face value of the bond received at maturity

This calculation shows that when the coupon of a bond is equal to its yield, the bond's price is equal to par.

Now let's value the same bond with a discount rate of 8%:

$$\frac{10}{(1.08)^1}+\frac{10}{(1.08)^2}+\frac{10}{(1.08)^3}+\frac{10}{(1.08)^4}+\frac{110}{(1.08)^5}$$

If the market discount rate for this bond were 8%, it would sell at a premium of $7.99 above its par value. When bond yields decrease, the present value of a bond's payments — its market value — increases. Here, we also see that a bond with a coupon greater than its yield will be trading above par.

If we discount the bond's cash flows at 12%:

$$\frac{10}{(1.12)^1}+\frac{10}{(1.12)^2}+\frac{10}{(1.12)^3}+\frac{10}{(1.12)^4}+\frac{110}{(1.12)^5}$$

If the market discount rate for this bond were 12%, it would sell at a discount of $100 − $92.79 = $7.21 to its par value. When bond yields increase, the market value decreases. Here, a bond with a coupon less than its yield will be trading below par.

Calculating the Value of a Semiannual Coupon Bond

Let's calculate the value of the same bond with semiannual payments. Rather than $10 per year, the security will pay $5 every six months. Assuming an annual YTM of 8%, we discount the coupon payments at 4% per period:

$$\frac{5}{(1.04)^1}+\frac{5}{(1.04)^2}+\cdots+\frac{5}{(1.04)^9}+\frac{105}{(1.04)^{10}}$$

The stated annualized YTM equals the periodic return of the bond multiplied by the number of periods in the year. In this case, the bond actually earns 4% every six months, hence the stated annualized YTM = 4% × 2 = 8%.

Calculating Yield to Maturity

Now let's calculate the stated yield of the same bond after market conditions have caused its price to move to 105. (This price is stated as a percentage of par — investors pay 105% of par. The easiest interpretation is the price of $100 of par.) We solve for the semiannual return:

This is the true semiannual return. To quote a YTM, multiply by two: stated YTM = 4.37% × 2 = 8.74%. Note the negative sign on PV — we must respect cash-flow direction (negative outflow today, positive inflows later) or the calculator cannot solve for the rate of return.

To actually earn the YTM over the life of a bond, the investor must (1) hold the bond to maturity, (2) the issuer must make all promised payments, and (3) the investor must be able to reinvest the periodic cash flows at the same YTM.

Accrued Interest, Flat Price, and Full Price

So far we have been calculating bond values on a coupon payment date, as the present value of the remaining coupons. For most actual bond trades, the settlement date (when cash is exchanged for the bond) falls between coupon payment dates.

Bond pricing must account for the fact that the next coupon will be paid to the buyer, but a portion of it (the accrued interest) is owed to the seller. Accrued interest since the last payment date is calculated as:

$$\text{accrued interest}=\text{coupon payment}\times\frac{\text{days from last coupon to settlement}}{\text{days in coupon period}}$$

Financial markets use a variety of day count methods:

• actual/actual — uses the actual number of days between coupon payments and the actual number of days between the last coupon date and settlement. Typical for government bonds.
• 30/360 — assumes each month has 30 days and a year has 360 days. Typical for corporate bonds.

Bond prices are quoted without accrued interest. If accrued interest were included in the quoted price, it would appear to drift up during a coupon period and drop suddenly on the coupon date. The quoted price is called the flat price (or clean price). The full price (also called invoice price or dirty price) is the sum of the flat price and the accrued interest. We cannot simply calculate a flat price and add accrued interest to it; instead we calculate the full price first and derive the flat price:

$$\text{flat price}=\text{full price}-\text{accrued interest}$$

The full price is calculated as follows:

1. Step 1:Calculate the value of the bond on the last coupon date.
2. Step 2:Compound this value at the YTM per period, over the number of days since the last coupon payment:

$$\text{full price}=PV_{\text{last coupon}}\times\left(1+\frac{YTM}{\text{periods per year}}\right)^{\frac{\text{days since last coupon}}{\text{days in coupon period}}}$$

We can summarize the relationships between price and specific bond features as follows:

1. At a point in time, a decrease (increase) in a bond's YTM will increase (decrease) its price. There is an inverse relationship between yield and price.
2. Other things equal, the price of a bond with a lower coupon rate is more sensitive to a change in yield than the price of a bond with a higher coupon rate.
3. Other things equal, the price of a bond with a longer maturity is more sensitive to a change in yield than a shorter maturity.
4. The percentage decrease in value when YTM increases by a given amount is smaller than the increase in value when YTM decreases by the same amount (the price-yield relationship is convex).

Relationship Between Price and Maturity

Before maturity, a bond can be selling at a significant discount or premium to par value. However, regardless of its required yield, the price will converge to par value as maturity approaches. Consider a bond with a 3-year life paying 6% semiannual coupons. The bond values corresponding to required yields of 3%, 6%, and 12% as the bond approaches maturity are presented in Figure 52.2.

The convergence to par at maturity ("pull to par") is known as the constant-yield price trajectory. It shows how a bond's price would change over time if its YTM remained constant. Figure 52.3 illustrates the same idea graphically: the premium bond (YTM < coupon) drifts down toward par; the discount bond (YTM > coupon) drifts up toward par; the par bond stays at par.

Matrix pricing is a method of estimating the required YTM (or price) of bonds that are currently not traded, or infrequently traded. The procedure is to use the YTMs of traded bonds with credit quality very close to that of the nontraded bond and similar in maturity and coupon, to estimate the required YTM.

In Step 2 we used simple linear interpolation. Because the maturity of the nontraded bond is three years, we estimate its YTM as the 2-year YTM plus one-third of the difference between the 2-year YTM and the average 5-year YTM. (The maturity gap from 2 to 3 is one year; from 2 to 5 is three years.)

A variation of matrix pricing used for new bond issues focuses on spreads. The required yield spread to a benchmark for a new issue can be estimated by observing spreads on existing similar securities.