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 P-Linker 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 P-Linker. 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 fixed-rate 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 C-Linker, 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 cross-product term. The projected cash flows
again are discounted using the nominal rate. In closed-form, this reduces to equation 7.17.
(7.17)
The real rate Macaulay duration for the C-Linker (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 P-Linker (and to equation 6.15 for the traditional fixed-rate bond). The key point is that, unlike the P-Linker, the real rate duration of the C-Linker 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 P-Linkers than with C-Linkers.
The inflation Macaulay duration for the C-Linker (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 long-term P-Linkers and C-Linkers. Perhaps this is to allow individuals to build low-risk retirement portfolios protected from inflation and deflation and hold them as either tax-deferred 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 20-year, annual payment P-Linkers and C-Linkers 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 25-year or 30-year securities at par value and since then the real rates have been volatile. The 2.50%, 20-year linkers are the newly issued on-the-run offerings.
Table 7.7 shows the prices and Macaulay durations for these 20-year 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 P-Linker are the same for both inflation rates. The Macaulay real rate durations are high, given the 20-year time to maturity, and are inversely related to the fixed coupon rate (as is a traditional fixed-rate bond). The modified real rate durations are easily calculated. For instance, the modified duration for the 1%, 20-year P-Linker is 17.464 (= 17.901/1.025). The inflation durations are all zero.
You might be questioning these prices for the P-Linker, thinking that if expected future inflation were to jump suddenly from 1.00% to 4.00%, the prices of P-Linkers such as TIPS surely would react. The demand for inflation-protected securities would go up, driving their prices up and the real rate down. So, if a change in inflation impacts the price of the P-Linker, 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%, 20-year P-Linker 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 20-Year P-Linkers and C-Linkers
Inflation 1.00%, Real Rate 2.50%, Nominal Rate 3.525% |
|||
Fixed Coupon Rate |
1.00% |
2.50% |
4.00% |
P-Linker |
|||
Price (% of Par Value) |
76.616 |
100.000 |
123.384 |
Real Rate Duration |
17.901 |
15.979 |
14.786 |
Inflation Duration |
0 |
0 |
0 |
C-Linker |
|||
Price (% of Par Value) |
78.375 |
99.645 |
120.916 |
Real Rate Duration |
16.149 |
14.699 |
13.759 |
Inflation Duration |
-2.125 |
0.326 |
1.915 |
Inflation 4.00%, Real Rate 2.50%, Nominal Rate 6.60% |
|||
Fixed Coupon Rate |
1.00% |
2.50% |
4.00% |
P-Linker |
|||
Price (% of Par Value) |
76.616 |
100.000 |
123.384 |
Real Rate Duration |
17.901 |
15.979 |
14.786 |
Inflation Duration |
0 |
0 |
0 |
C-Linker |
|||
Price (% of Par Value) |
82.510 |
98.907 |
115.304 |
Real Rate Duration |
12.336 |
11.689 |
11.225 |
Inflation Duration |
-1.443 |
0.194 |
1.365 |
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% P-Linker 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% C-Linker are less than par value even though the real rate is also 2.50%. That's because of the absence of the cross-product term. This effect is heightened when the inflation rate is higher. The real rate durations for the C-Linkers are high but still are lower than the corresponding P-Linker because, by design, the compensation for inflation is more front-loaded. That reduces the weighted-average time to the receipt of cash flow. Importantly, we now see that negative inflation duration occurs when the C-Linker is trading at a price sufficiently below par value.
The reason why discount C-Linkers have negative inflation duration and why premium C-Linkers have positive inflation duration is that they fundamentally are floating-rate 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 C-Linker, it's in the fixed coupon rate along with compensation for credit and liquidity risk.
When the real rate goes up, the C-Linker 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 C-Linker. In sum, a lower inflation rate reduces the price – that's negative inflation duration.
Not understanding the inflation duration of C-Linkers 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 C-Linkers, however, it definitely does matter – the trader would want to hold low-coupon C-Linkers trading at a discount and having negative inflation duration. If the view is toward lower-than-expected inflation, the trader prefers high-coupon, 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.
CONCLUSION
Floating-rate notes and inflation-indexed bonds demonstrate how we can employ a toolkit of bond math techniques to go beyond basic fixed-rate and zero-coupon 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 fixed-income portfolios of assets or liabilities – interest rate swaps.