Option-free bonds price/yield relationship

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 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 ... 

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. 

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

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"). 

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

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. 

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