INFLATIONINDEXED BONDS: C LINKERS AND PLINKERS
The market for inflationindexed bonds (linkers) has grown enormously in the past 30 years. PLinkers, for which the coupon rate is fixed and the principal is linked to changes in the consumer price index (ACPI), are usually issued by governments. These include U.S. Treasury TIPS (Treasury InflationProtected Securities), which have been available since 1997 and borrowed their design from linkers issued in the United Kingdom (1981), Australia (1983), and Canada (1991). CLinkers are essentially floatingrate notes for which the principal is fixed and the variable coupon rate is linked to the ACPI. They tend to be issued by commercial banks and life insurance companies.
While inflationadjusted linkers might seem to be straightforward securities, they are quite complex in theory and in practice. That's because inflation is arbitrarily measured and reported. There are many conceptual and political issues: Whose “consumption bundle” is measured for month tomonth price changes? When and how should that bundle be reweighted as new products are introduced and consumer spending patterns change?
Should the bundle be adjusted when consumers substitute one good for another because of relative price changes? How should quality improvements be handled, that is, buying a better widget that costs more and not just the same old widget at an inflated price? How is the cost of residential housing (a major household budget item) measured and included in the CPI? Does it matter that the government collects the data and calculates the official inflation rate that directly impacts the cost of its own debt? To focus on the bond math aspects of linkers, we'll neglect these issues and just assume that inflation is measured and reported accurately.
First, consider a highly stylized 2.50%, annual payment, 10year P Linker that adjusts the principal yearly based on the percentage change in the CPI. In practice, TIPS make semiannual interest payments and adjust the principal for the 2month lagged ACPI – that lag makes it difficult to model intraperiod valuation. Table 7.1 shows the nominal and real cash flows for the PLinker assuming a lowinflation scenario. The CPI starts at 100 on date 0 and rises for arbitrarily chosen values to 118.752 on date 10 for an average annual inflation rate of 1.734%, including a couple of years of deflation. Inflation is calculated as the percentage change in the CPI from one year to the next. For instance, the inflation rate of 2.863% for the tenth year is 118.752/115.4471.
TABLE 7.1 2.50%, 10Year, Annual Payment PLinker, LowInflation Scenario
Date 
CPI 
Inflation Rate 
Accrued Principal 
Nominal Cash Flow 
Real Value 
0 
100.000 
1,000.00 
1,000.00 
1,000.00 

1 
102.700 
2.700% 
1,027.00 
25.68 
25.00 
2 
105.233 
2.466% 
1,052.33 
26.31 
25.00 
3 
108.221 
2.839% 
1,082.21 
27.06 
25.00 
4 
107.988 
0.215% 
1,079.88 
27.00 
25.00 
5 
107.820 
0.156% 
1,078.20 
26.96 
25.00 
6 
108.223 
0.374% 
1,082.23 
27.06 
25.00 
7 
110.112 
1.745% 
1,101.12 
27.53 
25.00 
8 
112.338 
2.022% 
1,123.38 
28.08 
25.00 
9 
115.447 
2.768% 
1,154.47 
28.86 
25.00 
10 
118.752 
2.863% 
1,187.52 
1,217.21 
1,025.00 
Average Inflation Rate 
1.734% 
Internal Rate of Return 
4.272% 
2.500% 
The accrued principal on a PLinker is reported in Table 7.1 per $1,000 in par value. The key point is that the nominal interest payment each year is the fixed 2.50% coupon rate times the accrued principal. The $1,217.21 final payment at maturity is the accrued principal as of that date, $1,187.52, plus the interest payment, $29.69 (= 0.02 50 * $1,187.52). The real value deflates the nominal cash flow for the cumulative increase in the CPI since the base year. For the final payment, $1,217.21 is divided by 118.752/100 to get a real value of $1,025.00. The internal rates of return (IRR) are calculated for the nominal and real values assuming purchase at par value for $1,000 – the cash flow on date 0 is negative to indicate an outflow to the investor. I use the IRR financial function in Excel to get those results.
Table 7.2 repeats the exercise assuming a highinflation scenario, where the CPI rises from 100 to 240.805, resulting in an average inflation rate of 9.186%. Obviously, the nominal IRR is much higher. The salient point is that the real values of the cash flows for each date and the real IRR, overall, are the same in each scenario. That is the essence of inflation protection – to provide a predictable rate of return in real (i.e., after inflation) terms (assuming no default, of course). Note that the real yield of 2.50% is not really “locked in” because there still is coupon reinvestment risk
TABLE 7.2 2.50%, 10Year, Annual Payment PLinker, HighInflation Scenario
Date 
CPI 
Inflation Rate 
Accrued Principal 
Nominal Cash Flow 
Real Value 
0 
100.000 
1,000.00 
1,000.00 
1,000.00 

1 
104.566 
4.566% 
1,045.66 
26.14 
25.00 
2 
110.823 
5.984% 
1,108.23 
27.71 
25.00 
3 
118.398 
6.835% 
1,183.98 
29.60 
25.00 
4 
128.005 
8.114% 
1,280.05 
32.00 
25.00 
5 
139.556 
9.024% 
1,395.56 
34.89 
25.00 
6 
161.363 
15.626% 
1,613.63 
40.34 
25.00 
7 
188.857 
17.039% 
1,888.57 
47.21 
25.00 
8 
211.312 
11.890% 
2,113.12 
52.83 
25.00 
9 
228.523 
8.145% 
2,285.23 
57.13 
25.00 
10 
240.805 
5.375% 
2,408.05 
2,468.25 
1,025.00 
Average Inflation Rate 
9.186% 
Internal Rate of Return 
11.814% 
2.500% 
with respect to changes in future real rates of interest. Also, TIPS provide deflation protection as well because at maturity, the investor receives the accrued principal or par value, whichever is higher.
Second, consider a stylized annual payment CLinker paying 2.50% plus the percentage change in the CPI (i.e., the inflation rate) for the year. CLinkers in practice often make monthly interest payments. Tables 7.3 and 7.4 show the nominal and real cash flows for the annual payment CLinker purchased at par value for the same low and highinflation scenarios. On date 3 in Table 7.3, the inflation rate for the third year is determined to be 2.839%, making the coupon rate 5.339% (= 2.50% + 2.839%). The nominal interest payment is $53.39, the coupon rate times the constant principal amount of $1,000. The following year experiences deflation – in particular, an inflation rate of 0.215%. The coupon rate is 2.285% (= 2.50% – 0.215%). CLinkers offer protection in that the coupon rate will go no lower than zero if the deflation rate is more than 2.50%.
Notice that the cash flows are more frontloaded on the CLinker than on the PLinker. That is because the compensation for inflation on the C Linker is paid as it is realized, whereas it is mostly deferred until maturity on the PLinker. Three implications of this difference in the timing of cash
TABLE 7.3 2.50%, 10Year, Annual Payment CLinker LowInflation Scenario
Date 
CPI 
Inflation Rate 
Coupon Rate 
Nominal Cash Flow 
Real Value 
0 
100.000 
1,000.00 
1,000.00 

1 
102.700 
2.700% 
5.200% 
52.00 
50.63 
2 
105.233 
2.466% 
4.966% 
49.66 
47.19 
3 
108.221 
2.839% 
5.339% 
53.39 
49.34 
4 
107.988 
0.215% 
2.285% 
22.85 
21.16 
5 
107.820 
0.156% 
2.344% 
23.44 
21.74 
6 
108.223 
0.374% 
2.874% 
28.74 
26.55 
7 
110.112 
1.745% 
4.245% 
42.45 
38.56 
8 
112.338 
2.022% 
4.522% 
45.22 
40.25 
9 
115.447 
2.768% 
5.268% 
52.68 
45.63 
10 
118.752 
2.863% 
5.363% 
1,053.63 
887.25 
Average Inflation Rate 
1.734% 
Internal Rate of Return 
4.234% 
2.458% 
TABLE 7.4 2.50%, 10Year, Annual Payment CLinker HighInflation Scenario
Date 
CPI 
Inflation Rate 
Coupon Rate 
Nominal Cash Flow 
Real Value 
0 
100.000 
1,000.00 
1,000.00 

1 
104.566 
4.566% 
7.066% 
70.66 
67.57 
2 
110.823 
5.984% 
8.484% 
84.84 
76.55 
3 
118.398 
6.835% 
9.335% 
93.35 
78.85 
4 
128.005 
8.114% 
10.614% 
106.14 
82.95 
5 
139.556 
9.024% 
11.524% 
115.24 
82.58 
6 
161.363 
15.626% 
18.126% 
181.26 
112.33 
7 
188.857 
17.039% 
19.539% 
195.39 
103.46 
8 
211.312 
11.890% 
14.390% 
143.90 
68.10 
9 
228.523 
8.145% 
10.645% 
106.45 
46.58 
10 
240.805 
5.375% 
7.875% 
1,078.75 
447.97 
Average Inflation Rate 
9.186% 
Internal Rate of Return 
11.227% 
2.301% 
flows are: (1) more coupon reinvestment risk with CLinkers (more cash flow received sooner subject to fluctuation in real rates); (2) higher real rate durations with PLinkers (longer times to the receipt of cash flow); and (3) less potential credit risk on CLinkers, depending on the issuer (credit risk typically is an increasing function of time to the receipt of cash flow). These differences, along with taxation, are important when an investor has a choice between PLinkers and CLinkers.
Before turning to linker taxation, notice that the real IRR on the 2.50% CLinker is lower than on the 2.50% PLinker, even though both are purchased at par value. Given low inflation averaging 1.734%, the CLinker underperforms by 4.2 basis points. For high inflation averaging 9.186%, the difference is 19.9 basis points. The reason for this is that the typical CLinker does not provide complete inflation protection – the interest payment formula leaves out the crossproduct between the inflation rate and the real rate in setting the nominal rate. Recall from Chapter 3 the relationship among the three rates: (1 + nominal rate) = (1 + real rate) * (1 + inflation rate). Multiplying the terms, this becomes: nominal rate = real rate + inflation rate + (real rate * inflation rate). The typical CLinker only includes the first two terms.
Hopefully, investors are aware of the interest payment formula on typical CLinkers and that deficiency is reflected in the market price, especially if high inflation rates are expected. In reality, some CLinkers do include the crossproduct. The U.S. Treasury issues retailoriented Series I Savings Bonds (called IBonds). These nonmarketable, 30year CLinkers can be purchased at par value in small denominations of $25. The coupon interest rate is reset semiannually based on the ACPI, including the crossproduct. Interest accrues over the lifetime of the IBond and is paid at redemption. These, like PLinkers, have the same real IRR for all subsequent paths for inflation. But that is before taxes.