Price/yield relationship

The convexity is a more correct measure of the price sensitivity. It measures the curvature of the price-yield relationship and the degree in which it diverges from the straight-line estimation. Like the duration, the standard measure of convexity does not consider the changes of market interest rates on bond s prices. Therefore, the conventional measure of price sensitivity used for bonds with embedded options is the effective convexity. It is given by (11.2) ... 

A fundamental property is that an upward change in a bond s price results in a downward move in the yield and vice versa. This result makes sense because the bond s price is the present value of the expected future cash flows. As the required yield decreases, the present value of the bond s cash flows will increase. The price/yield relationship for an option-free bond is depicted in Exhibit 1.9. This inverse relationship embodies the major risk faced by investors in fixed-income securities—interest rate risk. Interest rate risk is the possibility that the value of a bond or bond portfolio will decline due to an adverse movement in interest rates. 

EXHIBIT 4.8 Price/Yield Relationship for Four Hypothetical Option-Free Bonds ... 

When the price/yield relationship for any hypothetical option-free bond is graphed, it exhibits the basic shape shown in Exhibit 4.9. Notice that as the required yield decreases, the price of an option-free bond increases. Conversely, as the required yield decreases, the price of an option-free bond increases. In other words, the price/yield relationship is negatively sloped. In addition, the price/yield relationship is not linear (i.e., not a straight line). The shape of the price/yield relationship for any option-free bond is referred to as convex. The price/yield relationship is for an instantaneous change in the required yield. 

Exhibit 4.10 shows the price/yield relationship for the UK gilt principal strip that is shown in Exhibit 4.1. Recall, using a settlement date of... 

EIHBIT 4.9 Price/Yield Relationship for a Hypothetical Option-Free Bond... 

EXHIBIT 4.10 Price/Yield Relationship for a UK Gilt Principal Strip... 

May 30, 2003, the yield is 4.435%. To construct the graph, the gilt strip was repriced using increments and decrements of 10 basis points from 2.435% to 6.435%. Exhibit 4.11 shows the two price/yield relationships for the 4% coupon, 2-year Italian government and 5.75% coupon, 30-year Italian government shown in Exhibits 4.3 and 4.4, respectively. 

EXHIBIT 4.11 Price/Yield Relationship for a 2-Year and a 30-Year Italian Government Bonds... 

Note the 30-year security s price/yield relationship is more steeply sloped and more curved than the price/yield relationship for the 2-year security. The reasons for these differences will be discussed shortly. 

An explanation for these two properties of bond price volatility lies in the convex shape of the price/yield relationship. Exhibit 4.13 illustrates this. The following notation is used in the exhibit... 

To see how the convexity of the price/yield relationship impacts Property 4, look at Exhibits 4.14 and 4.15. Exhibit 4.14 shows a less convex price/yield relationship than Exhibit 4.13. That is, the price/ yield relationship in Exhibit 4.14 is less bowed than the price/yield relationship in Exhibit 4.13. Because of the difference in the convexities, look at what happens when the yield increases and decreases by the same number of basis points and the yield change is a large number of basis points. We use the same notation in Exhibits 4.14 and 4.15 as in Exhibit 4.13. Notice that while the price gain when the required yield decreases is greater than the price decline when the required yield increases, the gain is not much greater than the loss. In contrast. Exhibit... 

The two most common types of embedded options are call (or prepay) options and put options. As interest rates in the market decline, the issuer may call or prepay the debt obligation prior to the scheduled principal repayment date. The other type of option is a put option. This option gives the investor the right to require the issuer to purchase the bond at a specified price. Below we will examine the price/yield relationship for bonds with both types of embedded options (calls and puts) and implications for price volatility. 

In the discussion below, we will refer to a bond that may be called or is prepayable as a callable bond. Exhibit 4.16 shows the price/yield relationship for an option-free bond and a callable bond. The convex curve given by a-a" is the price/yield relationship for an option-free bond. The unusual shaped curve denoted by a-b in the exhibit is the price/yield relationship for the callable bond. 

The reason for the price/yield relationship for a callable bond is as follows. When the prevailing market yield for comparable bonds is higher than the coupon rate on the callable bond, it is unlikely that the issuer will call the issue. For example, if the coupon rate on a bond is 7% and the prevailing market yield on comparable bonds is 12%, it is highly unlikely that the issuer will call a 7% coupon bond so that it can issue a 12% coupon bond. Since the bond is unlikely to be called, the callable bond will have a similar price/yield relationship as an otherwise comparable option-free bond. Consequently, the callable bond is going to be valued as if it is an option-free bond. However, since there is still... 

Let s look at the difference in the price volatility properties relative to an option-free bond given the price/yield relationship for a callable bond shown in Exhibit 4.16. Exhibit 4.17 blows up the portion of the... 

The value of a putable bond is equal to the value of an option-free bond plus the value of the put option. Thus, the difference between the value of a putable bond and the value of an otherwise comparable option-free bond is the value of the embedded put option. This can be seen in Exhibit 4.19 which shows the price/yield relationship for a putable bond (the curve a-b) and an option-free bond (the curve a-a"). 

This result should come as no surprise to careful readers of the last section on price volatility characteristics of bonds. Specifically equation (4.2) is somewhat at odds with the properties of the price/yield relationship. We are using a linear approximation for a price/yield relationship that is convex. 

EXHBIT 4.21 Price/Yield Relationship for an Option-Free Bond with a Tangent Line... 

Earlier we used the graph of the price/yield relationship to demonstrate the price volatility properties of bonds. We can use graphs to illustrate what we observed in our examples about how duration estimates the percentage price change, as well as some other noteworthy points. 

Also note that regardless of the magnitude of the yield change, the tangent line always underestimates what the new price will be for an option-free bond because the tangent line is below the price/yield relationship. This explains why we found in our illustration that when using duration we underestimated what the actual price will be. 

The reason for this result is that duration is in fact a first (linear) approximation for a small change in yield.The approximation can be improved by using a second approximation. This approximation is referred to as convexity. The use of this term in the industry is unfortunate since the term convexity is also used to describe the shape or curvature of the price/yield relationship. The convexity measure of a security can be used to approximate the change in price that is not explained by duration. 

Exhibit 16.5 shows very clearly the price responses for three 20-year bonds offering 10%, 6% annual coupon, and zero-coupon payments. One feature worth noting is that the curvilinear, price/yield relationship is not constant. It varies at different points on the curve for each type of... 

Note that the price duration figure, calculated from the modified duration measurement, underestimates the change in price resulting from a fall in yields but overestimates the change from a rise in yields. This reflects the convexity of the bond s price-yield relationship, a concept that will be explained in the next section. 

As explained in chapter 1, the curve representing a plain vanilla bond s price-yield relationship is essentially convex. The price-yield curve for a bond with an embedded option changes shape as the bond s price approaches par, at which point the bond is said to exhibit negative convexity. This means that its price will rise by a smaller amount for a decline in yield than it will fall for a rise in yield of the same magnitude. FIGURE 11.13 summarizes the price-yield relationships for both negatively and positively convex bonds. 

FIGURE 11.13 Price-Yield Relationships Associated with... 

As noted earlier, indexation lags prevent indexed bonds returns from being completely inflation-proof According to Deacon and Derry (1998), this suggests that an indexed bond can be regarded as a combination of a true indexed instrument (with no lag) and an nonindexed bond. Equation (12.13) expresses the price-yield relationship for a bond whose indexation lag is exactly one coupon period. 

In this situation, the final cash flows are not indexed, and the price-yield relationship is identical to that for a conventional bond. This, then, represents the nonindexed component of the indexed bond. Its yield can be compared with those of conventional bonds, making it possible to quantify the indexation element. This implies a true real yield measure for the indexed bond. 

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