Interest Rate Risk

Consider again the bond pricing formula for an n period coupon-paying bond (assume only one coupon payment per year for mathematical simplicity).

Clearly for fixed C, then r, the yield to maturity, equates the present value of this stream to the price P. Note that for fixed C, changes in the price of the bond (through supply and demand shifts in the bond market) will reflect changes in r. The relative attractiveness of bonds in investors’ portfolios makes bond prices variable and, therefore, bond yields variable as well. Fiscal policy that requires deficit financing, for example, will increase the supply of bonds that the Treasury auctions and these actions, in general, will cause bond prices to fluctuate. Similarly, monetary policy executed by central banks in the form of money supply growth (Fed easing) will have implications for interest rates (and therefore discount rates). Together, market forces, policy induced or not, will cause bond yields to move around over time, which will produce compensating variations in bond prices. Investors who hold bonds in their portfolios will therefore see the value of their bond holdings, and therefore the return on bonds in their portfolios, respond to these factors. Bonds are therefore risky even if their cash flows are not. This is what we refer to as interest rate risk, also known as capital risk. It is not a risk if you hold the bond to maturity. In that case, your risk is the yield available when you reinvest coupons and face value. On the other hand, if you are holding a portfolio of bonds, then interest rate risk is very real because the value of the portfolio varies with interest rate volatility when the portfolio is periodically marked to market. Thus, if the portfolio is held as collateral, for example, and interest rates are rising, then the value of the collateral is declining—your counterparty in this case may require you to post more collateral.

The yield on bonds and the return on bonds are often misunderstood. The bond return is identical to its yield if held to maturity (with coupons reinvested at the original yield to maturity). For example, if you purchase a bond at price P, you do so because you feel that P fairly represents the discounted present value of the promised future coupon stream (plus ending face value). If you hold the bond to maturity, then the yield to maturity is identical to the return.

For example, consider the value of an n-period bond:

The price P represents the discounted present value of cash flows that are paid at various times into the future. To convert to a future value, multiply both sides by :

The left hand side is the future value of investing P today for n periods earning a return r in each period compounded over time. earns interest for n – 1 periods, for n – 2 periods, and so on while the terminal cash flows C and M earn no interest. Thus, we see that the yield to maturity (the IRR) assumes all cash flows are reinvested for the remaining time at the internal rate of return, r. Therefore, the return to the bond if held to maturity is the difference P(1 + r)ⁿ – P. The periodic return is therefore r, which is also the yield to maturity.

Many times, however, the bond is not held to maturity. For example, the bond may be sold to provide liquidity against an unexpected liability. At the time of sale, the bond's price is determined through the interaction of market supply and demand. The return on the bond is now related to the percentage change in the price of the bond from the original date of purchase, which is no longer required to be equal to the original yield. Therefore, bond yields and bond returns can be quite different. The risk is that the bond's return will be lower than its yield if the investor is forced to liquidate before maturity.

Suppose that C is a given percent of M, the face value. For example, suppose the coupon payment is 10 percent of the face value M = $100. Then the bond is a 10 percent coupon bond. If the price P is $100 and the yield to maturity is also 10 percent, then this is called a par bond. Prove this to yourselves using the Price-Yield-Relation worksheet. A bond that sells at discount (premium) sells for less (more) than $100. As you can see from this worksheet, when r > 10 percent, the bond sells for less than 100 percent, and when r < 10 percent, it sells for a premium (over 100 percent of $100). In general, when the coupon yield exceeds (is less than) the yield to maturity, the bond is selling at a premium (discount).

More precisely, the yield to maturity is a by-product of the market price P. That is, bond market participants interact to set price, which, in turn, implies a yield to maturity and not the other way around. Thus, the bond selling at a premium (discount) determines that the yield to maturity is less than (greater than) the coupon yield.

A central question for the bond portfolio manager is, “How sensitive is P to small changes in interest rates?” Specifically, we are referring to the yield to maturity when we speak of interest rates but, in general, all rates are related, as we shall see. The intuition here is that bond market participants determine bond prices depending on their individual preferences, which include their perceptions of the opportunity cost of their capital. Thus, the yield to maturity is a discounting rate that must be consistent, on average, with discounting rates on other investments of similar duration and risk.

The answer to this central question depends a lot on how long the bondholder will have her capital at risk, that is, the maturity of the bond or, more specifically, its duration. Bonds with longer maturities require waiting longer to receive the cash flows in the form of coupons. Duration is therefore a temporal measure and the longer the duration, the longer we wait for the future to unfold, and therefore the longer we wait for the possibility for bad things to happen that may affect prices and interest rates. Therefore, bonds with longer maturities are those that are less likely to be held by a single investor from date of issue to maturity. Thus, bonds with longer maturities, all other things constant, have longer durations than those of shorter maturity and, hence, longer duration means greater capital risk. This risk is a function of the time to maturity and the interest rate (the yield to maturity).

To see this, consider the several examples in the companion website's Chapter 2 Examples.xlsx (Duration Tab). Take a 10-year, a 20-year, and a 30-year 10 percent bond with M = $100. Put the yield r at 10 percent initially to see that each bond has P = $100 (that is, they are all par bonds). Now, change r to 11 percent and then to 9 percent and compare the changes in P across maturities. You should get results consistent with those in the following table.

Clearly, the prices of bonds with longer maturities, or durations as we shall see further on, are more sensitive to interest rate changes. Duration will therefore become a natural measure of interest rate risk. We derive and analyze the properties of bond duration shortly. Before we do that, however, let's explore some more of the intuition behind the example given in Table 2.4

Table 2.4 Price-Yield Relationship.

Duration measures the sensitivity of the bond's price to changes in yield, that is, duration is a slope, or derivative, . But, as we can see from this example, the change in bond price for a given change in its yield to maturity is nonlinear—a 1 percent change in yield has different implications for changes in price and depends on the duration itself. This suggests that there is some curvature in the relation between dP and dr. This curvature is called the bond's convexity and is captured through the second derivative, . Convexity measures the curvature in the price/yield trade-off. Using the three bonds in the duration worksheet, I plot several price-yield points on a graph with price as a percentage of face value on the vertical axis and yield to maturity on the horizontal axis.

Figure 2.1 Convexity

Go to the companion website for more details (see Convexity under the Chapter 2 Examples).

Notice that for each maturity, the relationship is convex to the origin and the convexity increases with maturity. What does this mean to investors? To begin with, the pricing model is highly nonlinear in the discount factor (recall, that it is a power series to the order 2n). Duration, as we illustrate further on, is a first-order approximation to risk (it is the change in price given a one unit change in yield). A first-order approximation is like a tangent line to a curve; the slope of the tangent line approximates the trade-off between price and yield as long as we don't move very far away from the point of tangency (this is the common point above at yield = 0.10). As we move away from the tangency point, duration will not do a very good job accounting for the trade-off (in this case, the risk) because it is linear and can't capture the convexity in the bond pricing relationship. That is why we specifically look at convexity; it is the second-order approximation to risk. Longer duration bonds have higher order discount rates—their pricing formulas are higher-order polynomials—and therefore more risk and more convexity. That is why the curves associated with longer maturity bonds are more convex. They are riskier. To see more clearly the differences in risk across maturities, look at Table 2.5, in which we pick two maturities and, beginning with the point at which these two convex curves intersect for their par valuations, change the yield up or down and compare the relative changes in the implied bond prices. It should be clear that the change in the price is relatively higher for a given yield change when the bond has higher maturity.

Table 2.5 Price-Yield Response.