## Qualitative Nature of Price Yield Curves

would not buy a bond with a yield of 6% when bank CDs are offering 10%, The general interest rate environment exerts a force on every bond, urging its yield to conform to that of other bonds. However, the only way that the yield of a bond can change is for the bond's price to change. So as yields move, prices move correspondingly But the price change required to match a yield change varies with the structure of the bond (its coupon rate and its maturity). So as the yields of various bonds move more or less in harmony, their prices move by different amounts, To understand bonds, it is important to understand this relation between the price and the yield For a given bond, this relationship is shown pictorially by the price-yield curve.

Examples of price-yield curves are shown in Figure .3,3. Here the price, as a percentage of par, is shown as a function of YTM expressed in percentage terms, Let us focus on the bond labeled 10%. This bond has a 10% coupon (which means 10% of the face value is paid each year, or 5% every 6 months), and it has .30 years to maturity The price-yield curve shows how yield and price are related

The first obvious feature of the curve is that it has negative slope; that is, price and yield have an inverse relation. If yield goes up, price goes down If I am to obtain a higher yield on a fixed stream of received payments, the price I pay for this stream must be lower. This is a fundamental feature of bond markets., When people say "the bond market went down," they mean that interest rates went up

Some points on the curve can be calculated by inspection First, suppose that YTM = 0. This means that the bond is priced as if it offered no interest, Within the framework of this bond, money in the future is not discounted. In that case, the present value of the bond is just equal to the sum of all payments: here coupon payments of 10 points each year for 30 years, giving 300, plus the 100% of par value received at maturity, for a total of 400. This is the value of the bond at zero yield. Second, suppose that YTM — 10%. Then the value of the bond is equal to the par value The reason for this is that each year the coupon payment just equals the 10% yield expected on the

Price

Price FIGURE 3.3 Price-yield curves and coupon rate. All bonds shown have a maturity of 30 years and the coupon rates indicated on the respective curves Prices are expressed as a percentage of par

investment. The value remains at 100 every year The bond is like a loan where the interest on the principal is paid each year and hence the principal remains constant. In this situation, where the yield is exactly equal to the coupon rate, the bond is termed a par bond. In addition to these two specific points on the price-yield curve, we can deduce that the price of the bond must tend toward zero as the yield increases—large yields imply heavy discounting, so even the nearest coupon payment has little present value Overall, the shape of the curve is convex since it bends toward the origin and out toward the horizontal axis. lust given the two points and this rough knowledge of shape, it is possible to sketch a reasonable approximation to the true curve

Let us briefly examine another one of the curves, say, the 15% bond The price at YTM = 0 is 15 x 30+ 100 = 550, and the par point of 100 is at 15% We see that with a fixed maturity date, the price-yield curve rises as the coupon rate increases,

Now let us consider the influence of the time to maturity, Figure 3.4 shows the price-yield curves for three different bonds. Each of these bonds has a 10% coupon rate, but they have different maturities: 30 years, 10 years, and 3 years All of these bonds are at par when the yield is 10%; hence the three curves all pass through the common par point However, the curves pivot upward around that point by various amounts, depending on the maturity. The values at YTM — 0 can be found easily, as before, by simply summing the total payments The main feature is that as the maturity is increased, the price-yield curve becomes steeper, essentially pivoting about the par point. This increased steepness is an indication that longer maturities imply greater-sensitivity of price to yield

The price-yield curve is important because it describes the interest rate risk associated with a bond For example, suppose that you purchased the 10% bond illustrated in Figure 3 3 at par (when the yield was 10%) It is likely that all bonds of maturity approximately equal to 30 years would have yields of 10%, even though some might not be at par Then 10% would represent the market rate for such bonds

Price

Price All bonds have a 10% coupon

TABLE 3 5

Prices of 9% Coupon Bonds

Yield

TABLE 3 5

Prices of 9% Coupon Bonds

Yield

 Time to maturity 5% 8% 9% 10% 15% 1 year 103 85 100 94 100 00 99 07 94 61 5 years 117 50 104 06 100.00 96.14 79 41 10 years t3i 18 106 80 100 00 93 77 69 42 20 years 150.21 109 90 100 00 91.42 62 22 .30 years 161.82 11! 31 100.00 90 54 60 52

The prices of long-maturity bonds are more sensitive to yield changes than are the prices of bonds of short maturity

The prices of long-maturity bonds are more sensitive to yield changes than are the prices of bonds of short maturity

Now suppose that market conditions change and the yield on your bond increases to 11% The price of your bond will drop to 91.28. This represents an 8.7.2% change in the value of your bond. It is good to consider the possibility of such a change when purchasing this bond. For example, with a ,3~year 10% par bond, if the yield rose to 11 %, the price would drop only to 97 50, and hence the interest rate risk is lower with this bond. Of course if yields decreased, you would profit by similar amounts

Bond holders are subject to yield risk in the sense described: if yields change, bond prices also change This is an immediate risk, affecting the near-term value of the bond You may, of course, continue to hold the bond and thereby continue to receive the promised coupon payments and the face value at maturity This cash flow stream is not affected by interest rates. (That is after all why the bond is classified as a fixed-income security.) But if you plan to sell the bond before maturity, the price will be governed by the price-yield curve.

Table 3.5 displays the price-yield relation in tabular form for bonds with a 9% coupon rate, It is easy to see that the bond with .30-year maturity is much more sensitive to yield changes than the bond with I-year maturity.

It is the quantification of this risk that underlies the importance of the price-yield relation Our rough qualitative understanding is important. The next sections develop additional tools for studying this risk 