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LINKER DURATION(7.12) This equation is simplified using the standard relationship among the nominal, real, and inflation rates: (1 + y) = (1 + r) * (1 + i). (7.13) This can be written more compactly as shown in equation 7.14. (7.14) The inflation rate drops out in equations 7.13 and 7.14, so the inflation duration for the stylized PLinker is zero. That's because the first derivative of the pricing equation with respect to a change in the inflation rate is zero. In practice, the time between measuring the ACPI and changing the accrued principal (i.e., the indexation lag) matters, and the inflation duration for TIPS is close to but not exactly zero. In equation 7.14, only changes in the real rate impact the market value of the PLinker. Its Macaulay duration, derived in the Technical Appendix and denoted RealMacDurPLINK in equation 7.15, is going to look familiar to you. (7.15) This is the same as equation 6.15 for the Macaulay duration of a standard fixedrate bond on a coupon date (when t/T = 0). Real rate durations on TIPS are relatively high compared to Treasury notes and bonds for the same maturity because their fixed coupon rates and the real yields are relatively low. In general, the Macaulay yield duration statistic is inversely related to both the coupon rate and yield to maturity. The present value of the CLinker, PVCLINK, assuming a constant inflation rate, is shown in equation 7.16. (7.16) The principal is fixed and the inflation rate simply is added to the fixed coupon rate without the crossproduct term. The projected cash flows again are discounted using the nominal rate. In closedform, this reduces to equation 7.17. (7.17) The real rate Macaulay duration for the CLinker (RealMacDurCLINK) is derived in the Technical Appendix. It entails taking the first derivative of equation 7.17 with respect to changes in r, which is contained in y, and doing some algebraic manipulation. (7.18) This equation has a similar structure to equation 7.15 for the PLinker (and to equation 6.15 for the traditional fixedrate bond). The key point is that, unlike the PLinker, the real rate duration of the CLinker is a function of the inflation rate, which enters the equation directly as i and indirectly in y. This will matter in Chapter 10 when we get to strategies, in particular, an immunization strategy that rests on matching the duration of the bond portfolio to some target. It will be much easier to implement such a strategy using PLinkers than with CLinkers. The inflation Macaulay duration for the CLinker (InflationMacDurCLINK) will turn out to be low but not equal to zero, even for these stylized securities. Equation 7.19 is derived in the Technical Appendix. (7.19) Notice that the first two terms are the same as RealMacDurCLINK. The third term reduces InflationMacDurCLINK – sometimes all the way into negative territory. Some numerical examples will establish the circumstances when negative inflation duration occurs. It's not obvious from looking at equation 7.19. Suppose that someday an inspired government offers a full array of longterm PLinkers and CLinkers. Perhaps this is to allow individuals to build lowrisk retirement portfolios protected from inflation and deflation and hold them as either taxdeferred or currently taxable investments. Or perhaps the government seeks to assure holders of its traditional fixed income debt that it has no intention of “inflating” its way out of its obligations. Assume that at present there are 20year, annual payment PLinkers and CLinkers having coupon rates of 1.00%, 2.50%, and 4.00%. Investors require a real rate of 2.50%, so the 1.00% linkers trade at substantial discounts and the 4.00% linkers at substantial premiums. Assume that they originally were issued as 25year or 30year securities at par value and since then the real rates have been volatile. The 2.50%, 20year linkers are the newly issued ontherun offerings. Table 7.7 shows the prices and Macaulay durations for these 20year linkers given two inflation rates, 1.00% and 4.00%. For the real rate of 2.50%, the nominal rates are 3.525% (= 1.0250 * 1.01 – 1) and 6.60% (= 1.0250 * 1.04 1). The prices and durations are calculated by substituting N = 20, FV = 100, r = 0.0250, / = 0.01 or 0.04, y = 0.03525 or 0.0660, and c = 0.01, 0.0250, or 0.04 into equations 7.12, 7.13, 7.15, 7.16, and 7.17 (using a spreadsheet). There is a lot going on these numerical examples, so it is worthwhile to examine them closely. First, notice that the prices and durations for the PLinker are the same for both inflation rates. The Macaulay real rate durations are high, given the 20year time to maturity, and are inversely related to the fixed coupon rate (as is a traditional fixedrate bond). The modified real rate durations are easily calculated. For instance, the modified duration for the 1%, 20year PLinker is 17.464 (= 17.901/1.025). The inflation durations are all zero. You might be questioning these prices for the PLinker, thinking that if expected future inflation were to jump suddenly from 1.00% to 4.00%, the prices of PLinkers such as TIPS surely would react. The demand for inflationprotected securities would go up, driving their prices up and the real rate down. So, if a change in inflation impacts the price of the PLinker, how could its inflation duration be zero? Good point. Table 7.7 implicitly assumes that the real rate remains the same whereas your analysis has the real rate changing. However, how much so is estimated by the real rate duration statistic, not the inflation duration. Suppose you figure that the heightened demand for linkers will reduce the real rate by 10 basis points from 2.50% to 2.40%. The price of the 4%, 20year PLinker would go up from 123.384 by approximately 1.780, estimated by the modified real rate duration times the price times the change in the real rate. The key point is that these real rate and inflation duration statistics are what we call comparative static properties of the security – we assume other variables are held constant when we change one in particular. In reality, when the nominal interest rate goes up or down, both inflation and TABLE 7.7 Prices and Real Rate and Duration Macaulay Duration Statistics on 20Year PLinkers and CLinkers
real rates change as well, but not necessarily by the same amount or in the same direction or in any consistent manner. One way of dealing with this is to calculate the duration of the linker with respect to the nominal rate and just assume some breakdown between the two component rates. That breakdown sometimes is called the “yield beta.” For example, if you assume a yield beta of 0.50, you would get a nominal Macaulay duration for the 4% PLinker of 7.393 (= 14.786/2). That means you assume that a 1.00% change in the nominal rate is half due to inflation and that the other half is attributable to the real rate. In my opinion, this is a rather ad hoc approach. I think it is more insightful to work with the underlying real rate and inflation duration statistics. Notice that the prices on the 2.50% CLinker are less than par value even though the real rate is also 2.50%. That's because of the absence of the crossproduct term. This effect is heightened when the inflation rate is higher. The real rate durations for the CLinkers are high but still are lower than the corresponding PLinker because, by design, the compensation for inflation is more frontloaded. That reduces the weightedaverage time to the receipt of cash flow. Importantly, we now see that negative inflation duration occurs when the CLinker is trading at a price sufficiently below par value. The reason why discount CLinkers have negative inflation duration and why premium CLinkers have positive inflation duration is that they fundamentally are floatingrate notes. Their real rate durations are like the credit duration on a traditional floater; their inflation durations are like rate duration. The key difference is the location of the real rate. With a floater, it's in the nominal money market reference rate. With a CLinker, it's in the fixed coupon rate along with compensation for credit and liquidity risk. When the real rate goes up, the CLinker trades at a discount. The amount of the discount is the present value of the annuity representing the deficiency in the fixed coupon. If there is no further change in the real rate, a drop in the inflation rate lowers the nominal rate used to get that present value. A lower discount rate increases the present value of the annuity, increasing the amount of the discount and lowering the price of the CLinker. In sum, a lower inflation rate reduces the price – that's negative inflation duration. Not understanding the inflation duration of CLinkers could lead to real surprises for aggressive traders positioning a portfolio based on an expected level of inflation. Suppose that the trader's view is that inflation will go up by more than is generally expected by other market participants. Normally the trader is not particularly concerned whether bonds are trading at a discount or a premium. With CLinkers, however, it definitely does matter – the trader would want to hold lowcoupon CLinkers trading at a discount and having negative inflation duration. If the view is toward lowerthanexpected inflation, the trader prefers highcoupon, premium C Linkers having positive inflation duration. The trader in each case also has to factor in the likely impact on real rates and use the real rate duration to assess the additional price change. CONCLUSIONFloatingrate notes and inflationindexed bonds demonstrate how we can employ a toolkit of bond math techniques to go beyond basic fixedrate and zerocoupon securities. We have to extend our duration analysis to assess why the required rate of return changes and how that impacts market value. For floaters, changes in credit spreads and benchmark yields impact market value differently. For linkers, changes in the real rate and inflation have different impacts. Best of all, we can understand why and when floaters and linkers have negative duration. Now we can use the toolkit to delve into derivatives and the most commonly used product to manage fixedincome portfolios of assets or liabilities – interest rate swaps. 
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