Price/yield convexity

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) ... 

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. 

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... 

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. 

EXHBIT 4.17 Negative Convexity Region of the Price/Yield Relationship for a Callable Bond... 

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. 

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. 

Convexity is a measure of curvature of a bond s price-yield function. One can also think of convexity as the rate of change of duration with respect to yield. Convexity is discussed in more detail in Chapter 4. 

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. 

Equation (2.17) shows that convexity is the rate at which price sensitivity to yield changes as yield changes. That is, it describes how much a bond s modified duration changes in response to changes in yield. Formula (2.18) expresses this relationship formally. The convexity term can be seen as an adjustment for the error made by duration in approximating the price-yield curve. 

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 Negative and Positive Convexity ... 

From Figure 7.3, we see fliat to price a very Iraig-dated bond off the yield of the 30-year government bond would lead to errors. The unbiased expectations hypothesis suggests that 100-year bond yields are essentially identical to 30-year yields however, this is in fact incorrect. The theoretical 100-year yield in fact will be approximately 20-25 basis points lower. This reflects the convexity bias in longer dated yields. In our illustration, we used a hypothetical scenario where only three possible interest-rate states were permitted. Dybvig and Marshall showed that in a more realistic environment, with forward rates calculated using a Monte Carlo simulation, similar observations would result. Therefore, the observations have a practical relevance. 

Phoa (1998) presents an approximation of the convexity bias as follows. Consider a conventional fixed coupon bond, which has a yield at a future time t of r and a price at this time of P r). The convexity bias is estimated using... 

Exhibit 3.1 depicts this inverse relationship between an option-free bond s price and its discount rate (i.e., required yield). There are two things to infer from the price/discount rate relationship depicted in the exhibit. First, the relationship is downward sloping. This is simply the inverse relationship between present values and discount rates at work. Second, the relationship is represented as a curve rather than a straight line. In fact, the shape of the curve in Exhibit 3.1 is referred to as convex. By convex, it simply means the curve is bowed in relative to the origin. This second observation raises two questions about the convex or curved shape of the price/discount rate relationship. First, why is it curved Second, what is the import of the curvature ... 

We will discuss two approaches for assessing the interest rate risk exposure of a bond or a portfolio. The first approach is the full valuation approach that involves selecting possible interest rate scenarios for how interest rates and yield spreads may change and revaluing the bond position. The second approach entails the computation of measures that approximate how a bond s price or the portfolio s value will change when interest rates change. The most commonly used measures are duration and convexity. We will discuss duration/convexity measures for bonds and bond portfolios. Finally, we discuss measures of yield curve risk. 

The price volatility characteristic of a callable bond is important to understand. The characteristic of a callable bond—that its price appreciation is less than its price decline when rates change by a large number of basis points—is referred to as negative convexity. But notice from Exhibit 4.16 that callable bonds do not exhibit this characteristic at every yield level. When yields are high (relative to the issue s coupon... 

As can be seen from the exhibits, when a bond exhibits negative convexity, the bond compresses in price as rates decline. That is, at a certain yield level there is very little price appreciation when rates decline. When a bond enters this region, the bond is said to exhibit price compression. ... 

EXHIBIT 4.23 Estimating the New Price for a Large Yield Change for Bonds with Different Convexities... 

Mathematically, any function can be estimated by a series of approximations referred to a Taylor series expansion. Each approximation or term of the Taylor series is based on a corresponding derivative. For a bond, duration is the first-term approximation of the price change and is related to the first derivative of the bond s price with respect to a change in the required yield. The convexity measure is the second approximation and related to the second derivative of the bond s price. 

For example, for the UK gilt principal strip, the convexity adjustment to the percentage price change based on duration if the yield increases from 4.435% to 5.435% is... 

The approximate percentage price change based on duration and the convexity adjustment is found by summing the two estimates. So, for example, if yields change from 4.435% to 5.435%, the estimated percentage price change would be... 

The prices used in equation (4.4) to calculate convexity can be obtained by either assuming that when the yield changes the expected cash flows either do not change or they do change. In the former case, the resulting convexity is referred to as modified convexity. (Actually, in the industry, convexity is not qualified by the adjective modified. ) In contrast, effective convexity assumes that the cash flows do change when yields change. This is the same distinction made for duration. 

Convexity is a positive number for most normal bonds. However, for bonds with embedded call options such as mortgage-backed securities, it is always negative. Intuitively, it is obvious that if interest rates fall, the bond prices rise and the option to call the bond turns in the money and is often exercised, which shortens the duration of the bond and hence the rate of change of duration with respect to change in yields is negative. 

Convexity comes into play when yield curve changes are moderate to large and serves to increase the value of the bond irrespective of whether the yield rises or falls. In other words, if yields rise, then bonds with positive convexity fall less than expected from duration alone, and when yields fall, bond prices rise more than expected. To put it bluntly, convexity is good for a bond portfolio, but it is exceptionally hard to actively manage a credit portfolio and maximise convexity at the same time. 

The first two chapters of this section discuss bond pricing and yields, moving on to an explanation of such traditional interest rate risk measures as modified duration and convexity. Chapter 3 looks at spot and forward rates, the derivation of such rates from market yields, and the yield curve. Yield-curve analysis and the modeling of the term structure of interest rates are among the most heavily researched areas of financial economics. The treatment here has been kept as concise as possible. The References section at the end of the book directs interested readers to accessible and readable resources that provide more detail. 

It is clear from the bond price formula that a bonds yield and its price are closely related. Specifically, the price moves in the opposite direction from the yield. This is because a bonds price is the net present value of its cash flows if the discount rate—that is, the yield required by investors— increases, the present values of the cash flows decrease. In the same way, if the required yield decreases, the price of the bond rises. The relationship between a bond s price and any required yield level is illustrated by the graph in FIGURE 1.5, which plots the yield against the corresponding price to form a convex curve. 

Duration is a first-order measure of interest rate risk, using first-order derivatives. If the relationship between price and yield is plotted on a graph, it forms a curve. Duration indicates the slope of the tangent at any point on this curve. A tangent, however, is a line and, as such, is only an approximation of the actual curve—an approximation that becomes less accurate the farther the bond yield moves from the original point. The magnitude of the error, moreover, depends on the curvature, or convexity, of the curve. This is a serious drawback, and one that applies to modified as well as to Macaulay duration. 

Convexity is a second-order approximation of the change in price resulting from a change in yield. This relationship is expressed formally in (2.21). 

Note that the value for convexity given by the expressions above will always be positive—that is, the approximate price change due to convexity is positive for both yield increases and decreases, except for certain bonds such as callable bonds. 

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