Modified duration

Barbituric Acid Generie Name Trade Name R5 R Other Modifieation Duration of Aetion (h) Onset of Aetion (h) Average Adult Hypnotie Dose (g)... 

The inflation adjustment can be made on both, Macaulay and Modified duration. 

Index duration is usually equal to the time until the next reset date, whereas spread duration is equal to a modified duration of a bond paying fixed coupon, with same coupon payments and time to maturity. Therefore, conventionally floaters have lower index duration and higher spread duration. 

The duration shows the bond s price sensitivity to its yield to maturity. The change in bond s price is plotted in a curve in which the duration represents the slope of the tangent at any point of the curve. Conversely, the effective duration, or also known as curve duration, shows the price sensitivity to the change of the benchmark yield curve or market yield curve. This duration is more suitable than Macaulay or modified duration for bonds with embedded options because the latter ones have not a well-defined yield to maturity. The effective duration is given by (11.1) ... 

As one might expect the yields on bonds are correlated, in most cases very closely positively correlated. This enables us to analyse interest-rate risk in a portfolio for example, but also to model the term structure in a systematic way. Much of the traditional approach to bond portfolio management assumed a parallel shift in the yield curve, so that if the 5-year bond yield moved upwards by 10 basis points, then the 30-year bond yield would also move up by 10 basis points. This underpins traditional duration and modified duration analysis, and the concept of immunisation. To analyse bonds in this way, we assume therefore that bond yield volatilities are identical and correlations are perfectly positive. Although both types of analysis are still common, it is clear that bond yields do not move in this fashion, and so we must enhance our approach in order to perform more accurate analysis. 

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. 

Bloomberg s Risk measure is simply the PVBP x 100. For bonds that are trading close to par, risk should be close to modified duration. 

It is worth comparing the relationship between modified duration to another duration measure. Modified duration can also be written as ... 

The expression in the brackets of the modified duration formula given by equation (4.3) is a measure formulated in 1938 by Frederick Macaulay. This measure is popularly referred to as Macaulay duration. 

The general formulation for duration as given by equation (4.1) provides a short-cut procedure for determining a bond s modified duration. Because it is easier to calculate the modified duration using the shortcut procedure, most vendors of analytical software will use equation (4.1) rather than equation (4.3) to reduce computation time. 

However, it must be clearly understood that modified duration is a flawed measure of a bond s price sensitivity to interest rate changes for a bond with an embedded option and therefore so is Macaulay duration. 

More specifically, this is the formula for the modified duration of a bond on a coupon anniversary date. 

The reason it is only approximate is because modified duration assumes a flat yield curve whereas key rate duration takes the spot curve as given. 

How does an investor measure the modified duration of linkers It sounds like a straightforward question and there is an easy answer, but it is sadly not the answer that people generally want. The easy answer is that a linker s modified duration is the (normalised) first derivative of price with respect to real yield, just as a conventional bond s modified duration is that with respect to nominal yield. This answer is a flippant one, because what people really want to know is some empirical rule about the sensitivity of a linker s price with respect to nominal yields, either for hedging purposes or in order to calculate aggregate duration statistics for portfolios holding both nominal and real bonds. 

Hedging the underlying portfolio hence requires the relative volatility that exists between the cash market bond(s) held in the portfolio and the futures contract. Dollar duration can assist in this search since modified duration can help predict the effect of small changes in yield on the price of a bond ... 

Consider the case where the CTD bond is priced at 104.28 and has a modified duration of 6.874 while the bond held in the portfolio is priced at 104.32 and has a modified duration of 7.491. Since bond to be hedged is not the CTD, then an adjustment needs to be made to the number of contracts shorted. This is the case where the relationship between the CTD and the bond to be hedged plays an important role. 

From equation (16.6) it is evident that to be able to perform the necessary calculations the modified durations of the two bonds need to be established... 

Modified duration of the portfolio Target duration Futures quote... 

The bond portfolio has the following characteristics Modified duration is 7.4908 yield is 4.595%, current valuation is 300,000,000. The BPV of the Euro-Bund future is 72.03. 

EXHIBIT 25.1 iBoxx Euro Index—Modified Duration 31 January 2002 to January 31, 2003... 

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. 

If the expression for Macaulay durarion, (2.5), is substituted into equation (2.4), which calculates the approximate percentage change in price, (2.7) is obtained. This is the definition of modified duration. 

Modified duration can be used to demonstrate that small changes in yield produce inverse changes in bond price. This relationship is expressed formally in (2.7), repeated as (2.9). 

Taking the first derivative of (2.10) and dividing the result by the current bond price, P, produces an alternative formulation for modified duration, shown as (2.11). 

Plugging these values into the modified-duration equation (2.11) gives... 

To obtain the bond s Macaulay duration, this modified duration is multiplied by (1 + r), or 1.07634, for a value of 7.28354 years. 

---