Interest rate risk measurement

Note that our calculation for duration of 17.636 agrees (within rounding error) with Bloomberg s calculation in Exhibit 4.1. Bloomberg s interest rate risk measures are located in a box titled Sensitivity Analysis in the lower left-hand corner of the screen. The duration measure we just calculated is labeled Adj/Mod Duration which stands for adjusted/ modified duration. 

Duration measures the sensitivity of a bond s price to a given change in yield. The traditional formulation is derived under the assumption that the reference yield curve is flat and moves in parallel shifts. Simply put, all bond yields are the same regardless of when the cash flows are delivered across time and changes in yields are perfectly correlated. Several recent attempts have been made to address this inadequacy and develop interest rate risk measures that allow for more realistic changes in the yield curve s shape. ... 

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. 

Bonds differ in their exposure to interest rate risk so investors want to know the sensitivity of a bond to change in interest rates. This sensitivity is first approximated by a bond s duration. There are various measures of duration (e.g., Macaulay, modified, effective, etc.) that will be... 

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. 

Some managers use another measure of the price volatility of a bond to quantify interest rate risk—the price value of a basis point (PVBP). This mea-... 

Thomas S. Y. Ho, Key Rate Durations Measures of Interest Rate Risk, Journal of Fixed Income (September 1992), pp. 29-44. 

Measuring Interest Rate Risk Frank J. Fabozzi and Steven V. Mann... 

Although newcomers to the market commonly consider duration, much as Macaulay did, a proxy for a bond s time to maturity, this interpretation misses the main point of duration, which is to measure price volatility, or interest rate risk. Using the Macaulay duration can derive a measure of a bond s interest rate price sensitivity, i.e., how sensitive a bond s price is to changes in its yield. This measure is obtained by applying a mathematical property known as a Taylor expansion to the basic equation. 

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 represents an attempt to remedy the drawbacks of duration. A second-order measure of interest rate risk uses second-order derivatives. It measures the curvature of the price-yield graph and the degree to which this diverges from the straight-line estimation. Convex-... 

The modified duration of a bond measures its price sensitivity to a change in yield. It is essentially a snapshot of one point in time. It assumes that no change in expected cash flows will result from a change in market interest rates and is thus inappropriate as a measure of the interest rate risk borne by a mortgage-backed bond, whose cash flows are affected by rate changes because of the prepayment effect. 

Cost of Capital The value of the interest rate of return used in calculating the net present value (NPV) of a project is usually referred to as the cost of capital. It is not a constant value since it depends on the financial structure of the company, the policy of the company toward a particular project, the local method of assessing taxation, and, in some cases, the measure of risk associated with the particular projec t. The last-named fac tor is best dealt with by calculating the entrepreneurs risk allowance inherent in the project i from Eq. (9-108), written in the form... 

Banks are not in business to take risks. They rent money and do everything they can to insure the return of their principal as well as the interest. Elaborate rating systems have been developed to measure each company s ability to repay its loans. One criterion is the debt to equity ratio. The higher the debt the more risk in a loan, and the higher the interest rate. 

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