FINANCE 525: FIXED INCOME SECURITIES



INFLATION PROTECTED SECURITIES AND FUNDING REAL LIABILITIES

David T. Brown

University of Florida

1. Overview.

This class will cover four topics. First, we will discuss the mechanics of how Treasury Inflation Protected Securities (TIPs) work. The second topic is calculating the break even inflation rate between TIPs and ordinary Treasury Bonds. Third, we will discuss evaluating the duration of TIPs. Fourth, we will the issues associated with funding a real liability and the unique role of TIPs in funding a real liability.

2. The Mechanics of Treasury Inflation Protected Securities.

• Treasury Inflation Protected Securities were first issued by the United States Treasury in 1997. Inflation protected securities have been around awhile in other countries: Israel, U.K. and Canada.

• The cash flows (both principal and interest) to the holder of an inflation protected are indexed to inflation based on the CPI. The following Table walks through an example of the cash flows for a ten year TIP with a 4% annual real coupon issued on 1/15/99.

|Date |Reference |Inflation Adjusted |Coupon |Total Cash |

| |CPI |Principal | |Flow |

|1/15/99 |159.0 |100 | | |

|7/15/99 |161.4 |100*(161.4/159) = 101.5 |2%*101.5 = 2.03 |2.03 |

|1/15/00 |163.8 |100*(163.8/159) = 103.0 |2%*103.0 = 2.06 |2.06 |

| | | | | |

|1/15/09 |213.7 |134.4 |2.69 |2.69 + 134.4 = 137.10 |

• The IRR from holding a TIP until maturity is (1+R)(1+I) – 1 ≈ R + I, where R is the real coupon rate and I is the actual inflation rate over the holding period.

• TIPs provide inflation protection in that the real return on the investment is known with certainty when the bond is purchased.

• Both the coupon and the appreciation of the principal are taxable in the year they occur. Thus, the taxable income in 1999 would be 2.03 + 2.06 + (103-100) = $7.09.

• There is some risk that the definition of the CPI changes.

3. Break Even Inflation Rates.

The credit risk associated with TIPs and conventional Treasury Bonds is obviously both tiny and identical. Therefore, an investor will have a higher (lower) return owning TIP if inflation turns out to be high (low). The following shows the break even inflation rates for the five-year, ten-year and thirty-year sectors based on 11/16/99 prices.

|Coupon Rate |Maturity |Yield |Break Even Inflation Rate* |

|3.625% |7/02 |3.85% |1.95% |

|3.875% |1/09 |4.08% |1.84% |

|3.875% |4/29 |4.08% |1.95% |

*The break-even inflation rates are based on the difference between the yield on the TIP and the yield on a comparable Treasury Bond as report by Bloomberg.

• Many investors view TIPs as “cheap” relative to Treasury Bonds given the current break even inflation rates.

• The difference between the break even inflation and an investor’s expectation about future inflation is the price of inflation protection. Suppose that you think that inflation is going to be 1.50% per year for the next thirty years. If you buy a TIP you are expecting to under perform Treasury Bonds by .45% and thus the price of inflation protection is .45% per year.

• The price of inflation protection appears to be negative. There is currently limited interest in these securities among institutional money managers and the liquidity is low compared to Treasuries.

4. Duration of Inflation Protected Securities.

Break Interest Rate Changes Down into the Inflation and Real Rate Component.

The nominal interest rate is approximately the sum of the real rate of interest and the expected inflation rate: r = R + E(I).

Real Rate Duration and Inflation Rate Duration.

The real rate duration of an inflation rate bond is equal to its cash flow duration. A change in the real rate of interest will have the same effect on the price of a TIP and Treasury Bond.

The inflation rate duration of an inflation protected security is zero. A change in expected future inflation rates will have no effect on the value of a TIP. The numerator and denominator of the present value formula change by the same amount.

The inflation rate duration of a Treasury Bond is equal to its cash flow duration: the effect of an interest rate change has the same effect on a “nominal” bond regardless of whether the interest rate change comes from a change in expected inflation or a change in real rates.

Effective Duration of an Inflation Protected Bond.

In order to assess the effective duration of an inflation protected security you have to make a statement about what percent of subsequent interest rate changes will be due to expected inflation rate changes versus real rate changes. If you think that x% of the future changes in inflation rate are due to changes in real rates (obviously (1-x%) must be due to changes in inflation expectations), then the effective duration of an inflation protected bond is

X%*Drr + (1-X%)DIr (1)

Where Drr is the “real rate duration” and DIr is the inflation rate duration. Given the discussion above, (1) reduces to

X% * Cash Flow Duration.

Comments on the Relative Volatility of Inflation and Real Rates.

Historically, interest rate changes have been largely driven by changes in expected inflation rates. It is argued that in the current environment where the Fed has actively managed inflation, the relative volatility of inflation rates. When the Federal reserve sought to anticipate inflation pressures and prevent them from being imbedded in the economy, it was forced to make real interest rates more volatile. Thus, inflation rate volatility was transformed into real interest rate volatility.

(The graphs on the following page show the yield history for the ten year Treasury Bond and ten year TIP)

5. Funding Real Liabilities.

Obviously, individual investors are concerned about real returns and hence include inflation protection in their security selection. Individual investors are in effect saving to fund a real liability: retirement consumption or college tuition.

There are some situations where the inflation sensitivity of the liability is explicitly recognized:

Nuclear Decommission Trusts (NDTs)

Environmental Clean Up Trusts

P&C Insurance

Pension Benefits with COLAs

TIPS are ideally suited for liabilities like NDTs and Environmental Clean Up Trusts where the real rate duration is the weighted average maturity of the real liability and the inflation rate duration of the liability is zero.

6. Funding Real Liabilities: Advanced Issues.

In certain situations a liability may not be completely indexed to inflation. The following example walks through some work I did for a P&C Insurance company.

I created an insurance liability example where an insurer has (1) known claims that lead to a known nominal liability stream and (2) anticipated losses from its current book of policies. The anticipated losses from its current book of policies are actuarily derived and presented in nominal dollars. Anticipated losses from its current policies vary with inflation. In creating this example, I first define a “liability inflation” parameter that describes the correlation between a given liability and inflation. If the parameter is 1, then the liability is a real liability and the rate of inflation of the item being insured is equal to the general rate of inflation. In simple terms, if the parameter is 1, then the insurer has agreed to replace a totaled car with a new car and the car price rate of inflation is equal to the economy wide rate of inflation. If the parameter is 0, then the insurer has agreed to replace a totaled car with a fixed amount of money.

Interest rate changes result from changes in the real rate of interest or changes in expectations about future inflation. I use a “relative volatility” parameter that varies between 0 and 1 that describes interest rate changes. If the parameter is 0, then the interest rate change is all due to changes in the real rate. If the parameter is 1, then the interest rate change is all due to changes in expected inflation.

I calculate the following duration measures for the liability: cash flow duration, real rate duration, inflation rate duration and effective duration. The values of these duration measures depends on the relative volatility and liability sensitivity parameters described above.

1. The cash flow duration of the liability is the traditional modified duration of the expected cash flows today. In the example I generated, the cash flow duration is 2.86 years.

2. The real rate duration of the liability is the change in the present value of the liability resulting from a change in the real rate of interest, holding inflation constant. The real rate duration always equals the cash flow duration.

3. The inflation rate duration of the liability is the change in the present value of the liability resulting from a change in the inflation rate, holding the real rate constant. The inflation rate duration is large (small) when (1) you have large (small) existing claims relative to anticipated claims, and/or (2) the liability inflation parameter is small (large). For example, if the inflation sensitivity parameter is 0, the inflation duration equals the real rate duration.

4. The effective duration of the liability is the change in the present value of the liability resulting from a change in interest rates. The effective duration depends a lot on the extent to which the change in rates is due to a change in real rates versus a change in inflation.

The following two tables summarize the findings.

Inflation Rate Duration of Liability

|Inflation Duration of |Liability Inflation |

|Liability |Parameter |

|.15 |1 |

|.42 |.9 |

|.69 |.8 |

|.96 |.7 |

|1.23 |.6 |

|1.50 |.5 |

|1.77 |.4 |

|2.05 |.3 |

|2.32 |.2 |

|2.59 |.1 |

The following shows how the effective duration of the liability varies with the liability inflation parameter and the relative volatility parameter.

Effective Duration of Liability

| |Liability Inflation |Liability Inflation |Liability Inflation |Liability Inflation |Liability Inflation|

| |Parameter = 0 |Parameter = .25 |Parameter = .50 |Parameter = .75 |Parameter = 1 |

|Relative volatility|2.86 |2.86 |2.86 |2.86 |2.86 |

|= 0 | | | | | |

|Relative volatility| | | | | |

|= .25 |2.86 |2.70 |2.52 |2.35 |2.18 |

|Relative volatility| | | | | |

|= .50 |2.86 |2.52 |2.18 |1.84 |1.50 |

|Relative volatility| | | | | |

|= .75 |2.86 |2.35 |1.84 |1.33 |.83 |

|Relative volatility|2.86 |2.18 |1.50 |.83 |.15 |

|= 1 | | | | | |

If you want to play with the model the effective duration (blue) varies as you change the parameters or the liability (red).

There is a definite role for inflation indexed bonds for funding insurance liabilities. Consider funding an insurance liability with (1) a combination of real bonds and nominal bonds or (2) only nominal bonds. The following shows that (1) either approach requires judgement and (2) using a combination of real bonds and nominal bonds requires less judgement than using only nominal bonds.

Define the following notation:

DRL = real rate duration of the liability

DR = cash flow duration of real bonds in a portfolio

DIL = inflation rate duration of liability

DN = cash flow duration of nominal bonds in a portfolio

N = % of nominal bonds in portfolio

R = % of real bonds in a portfolio

Notes: You need to make a judgement about the liability inflation parameter to obtain the inflation rate duration of the liability.

The inflation rate duration of the real bond is zero and the inflation rate duration of the nominal bond is its cash flow duration.

The real rate durations of the real and nominal bonds equal their respective cash flow durations.

To eliminate interest rate risk, you choose N, R, DR, and DN subject to:

R*DR + N*DN = DRL (1)

N*DN = DIL (2)

N + R = 1 (3)

N > 0 (4)

R > 0 (5)

Equations (1) and (2) ensure that your bond portfolio has the same real rate duration and the same inflation rate duration as the liability. You can always satisfy (1), (2) and (3). Constraints (4) and (5) require that DRL > DIL, which is always true.

The second approach is to fund the liability using only nominal bonds. Since (1) the inflation rate duration and the real rate duration of nominal bonds are the same and (2) the inflation and real rate duration of the liability are different, you need to make a judgement about the relative volatility of inflation and real rates to determine the effective duration of the liability. You then choose your nominal bond portfolio to have the same duration as the effective duration of the liability. By using a combination of real and nominal bonds you can eliminate the need to make the relative volatility judgement that is required when using only nominal bonds.

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