Price/discount rate relationship

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 ... 

Price/Discount Rate Relationship for an Option-Free Bond... 

As for the importance of the curvature to bond investors, let s consider what happens to bond prices in both falling and rising interest rate environments. First, what happens to bond prices as interest rates fall The answer is obvious bond prices rise. How about the rate at which they rise If the price/discount rate relationship was linear, as interest rates fell, bond prices would rise at a constant rate. However, the relationship is not linear, it is curved and curved inward. Accordingly, when interest rates fall, bond prices increase at an increasing rate. Now, let s consider what happens when interest rates rise. Of course, bond prices fall. How about the rate at which bond prices fall Once again, if the price/discount rate relationship were linear, as interest rates rose, bond prices would fall at a constant rate. Since it curved inward, when interest rates rise, bond prices decrease at a decreasing rate. In Chapter 4, we will explore more fully the implications of the curvature or convexity of the price/discount rate relationship. 

The preceding three scenarios illustrate an important general property of present value. The higher (lower) the discount rate, the lower (higher) the present value. Since the value of a security is the present value of the expected future cash flows, this property carries over to the value of a security the higher (lower) the discount rate, the lower (higher) a security s value. We can summarize the relationship between the coupon rate, the required market yield, and the bond s price relative to its par value as follows ... 

The answer to the first question is mathematical and lies in the denominator of the bond pricing formula. Since we are raising one plus the discount rate to powers greater than one, it should not be surprising that the relationship between the level of the price and the level of the discount rate is not linear. 

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. 

As discussed in chapter 1, yield to maturity is the interest rate that relates a bonds price to its future returns. More precisely, using the notation defined above, it is the rate that discounts the bond s cash flow stream C to its price P(t, T). This relationship is expressed formally in equation (3.7). 

---