Bond Valuation and Term Structure of Interest Rates

A bond is an obligation by the bond issuer to pay money to the bondholder according to certain rules specified at the time of the bond issuance. Generally, a bond is composed of timed cash flows over determined periods. A bond’s final payments period is called maturity time. A bondholder receives the bond’s face value or, equivalently, its par value at the maturity time. The holder also may acquire periodic
coupon payments.

Zero coupon bonds: Zero coupon bonds have no coupon payments and the principal payment is received at the maturity time T = Tm. The zero coupon bond’s cash flow stream is {FT } = {F }.

A bond’s yield is the interest rate implied by its payment structure. Specifically, it is the interest rate at which the sum of the present values of the stream of payments (consisting of the coupon payments and the final face-value redemption) is exactly equal to the current price. This value is termed more properly the yield to maturity (YTM) γ. Yields are always quoted on an annual basis.

The yield to maturity of the coupon bond with m - periods discrete compounding over T years, γ, is determined by:
 

The following relationships between the price of a bond, its coupon rate, its current yield, and its yield to maturity hold:

Par Bond: Coupon rate = Current yield = Yield to maturity
Discount Bond: Coupon rate < Current yield < Yield to maturity
Premium Bond: Coupon rate > Current yield > Yield to maturity

A par bond’s trading price is equal to its face value. On the other hand, a discount bond’s price is less than its face value, and a premium bond’s price is higher than its face value.

When the yield rate decreases (γ ↓) the price of the bond goes up (PBond,0↑) at an increasing rate because of convexity: when the yield rate is low, less than ŷ for example, a decrease in the yield rate might cause a big increase in the price. At very low yield rates a decrease in the yield rate might lead to a very big increase in the price.


A bond with a very low yield rate indicates that the bond’s future payments are highly attractive  and the demand for the bond is high, leading to the high bond price. Such bonds are premium bonds. On the other hand, when the yield rate increases (γ ↑) the price goes down (PBond,0↓) at a decreasing rate. This means, that at high yield rates, higher than ŷ for example, the price’s yield rate sensitivity is low: an increase in the yield rate may result in negligible changes in the price. Moreover, the price of the bond approaches to zero when yield rates reach to very high levels causing heavy discounting rates (limγ→∞PBond,0 = 0).

The price of the bond approaches to zero when yield rates reach to very high levels causing heavy discounting rates (limγ→∞PBond,0 = 0). This phenomenon may appear when investing in a particular bond is unprofitable and alternative assets bring much higher rates of returns. These types of bonds are discount bonds.

At very low yield rates a decrease in the yield rate might lead to a very big increase in the price. A bond with a very low yield rate indicates that the bond’s future payments are highly attractive and the demand for the bond is high, leading to the high bond price. Such bonds are premium bonds

Basically, a bond’s duration measure is formulated as the weighted average of the periods in which the bond’s future cash flows will accrue. Generally, the weight of each period is assigned as the ratio of that period’s discounted cash flow to the cost of the bond, which is its price.

The most common measures are Macaulay and modified durations.

Macaulay duration measures the number of periods required to recover the cost of a bond, considering present values of all future coupon and principal payments.

Modified duration adjusts Macaulay duration to measure the responsiveness of a bond’s price to interest changes.

Bonds of different maturities and coupon rates can be compared with measures of their durations.

Contributors to the duration measures are

• coupon rates (settling periodic cash flows’ sizes),
• yield rates (determining present values of future cash flows), and
• cash flows’ payment times (weighting each of future cash flows) and time to maturity

For any given bond, all else being constant, the bond’s price is a decreasing convex function of the yield rate. This indicates the limitation of the duration’s linear approximation in measuring price-interest rate sensitivity. Taking into account convexity in combination with duration provides a more accurate approximation of a percentage price change resulting from a specified change in a bond’s yield than using duration alone. That approximation becomes a quadratic approximation.

The relationship between interest rates of different maturities is referred to as the term structure of interest rates. The fact that similar loans borrowed at relatively higher maturities bear comparatively higher interest rates forms the basis of the term structure theory. The term structure theory concentrates on pure interest rates instead of yield rates.

A graphical representation of the term structure displaying the relationship between interest rates of different maturities is known as the yield curve. Figure 8.7 displays some possible shapes of yield curves. Generally, yield curves slope upward to reflect the fact that the longer maturity instruments carry higher interest rates. The positive rates of change in spot rates are higher in shorter maturities than the longer maturities with upward-sloping yield curves.

Liquidity preference: Investments in long-term assets are more sensitive to interest rate changes. Hence, investors may prefer more liquid short-term investments rather than less liquid long-term ones. That may explain why long-term investments are motivated by high rates of returns (or high-interest rates).

Market segmentation: Markets for securities are segmented by their maturity times and they are subject to different demand and supply characteristics. Then forward rates may not provide informationabout future spot rates.

Preferred habitat hypothesis: Markets may be segmented but they are not independent. High enough differences in spot rates for different maturities may motivate investors to shift their investments from short rates to long rates or the other way around.

The Pure Expectations Hypothesis: The pure expectations hypothesis claims that forward interest rates are unbiased predictors of corresponding future spot interest rates under the assumption that different maturities are perfect substitutes.