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# INTEREST RATE RISK FIN 6547

UF

GPA 3.77

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This 9 page Class Notes was uploaded by Roosevelt Romaguera on Friday September 18, 2015. The Class Notes belongs to FIN 6547 at University of Florida taught by David Brown in Fall. Since its upload, it has received 18 views. For similar materials see /class/206602/fin-6547-university-of-florida in Finance at University of Florida.

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Date Created: 09/18/15

INFLATION PROTECTED SECURITIES AND FUNDING REAL LIABILITIES David T Brown University of Florida p A Overview This class will cover four topics First we will discuss the mechanics of how Treasury In ation Protected Securities TIPs work The second topic is calculating the break even in ation rate between quotUPS and ordinary Treasury Bonds Third we will discuss evaluating the duration of quotUPS Fourth we will the issues associated with funding a real liability and the unique role of TIPs in funding a real liability N The Mechanics ofTreasury In ation Protected Securities Treasury In ation Protected Securities were rst issued by the United States Treasury in 1997 In ation protected securities have been around awhile in other countries Israel UK and Canada The cash ows both principal and interest to the holder of an in ation protected are indexed to in ation based on the CPI The following Table walks through an example of the cash ows for a ten year TIP with a 4 annual real coupon issued on 11599 The IRR from holding a TIP until maturity is 1R1I 7 1 Z R 1 where R is the real coupon rate and I is the actual in ation rate over the holding period TIPs provide in ation 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 203 206 103100 709 0 There is some risk that the de nition of the CPI changes 3 Break Even In ation 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 in ation turns out to be high low The following shows the break even in ation rates for the veyear ten year and thirtyyear sectors based on 111699 prices The breakeven in ation rates are based on the difference between the yield on the TIP and the yield on a comparable Treasury Bond as report by Bloomberg P Many investors view TIPs as cheap relative to Treasury Bonds given the current break even in ation rates The difference between the break even in ation and an investor s expectation about future in ation is the price of in ation protection Suppose that you think that in ation is going to be 150 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 in ation protection is 45 per year The price of in ation 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 Duration ofInflation Protected Securities Break Interest Rate Changes Down into the In ation and Real Rate Component The nominal interest rate is approximately the sum of the real rate of interest and the expected in ation rate r R Real Rate Duration and In ation Rate Duration The real rate duration of an in ation rate bond is equal to its cash ow duration A change in the real rate of interest will have the same effect on the price of a TIP and Treasury Bond The in ation rate duration of an in ation protected security is zero A change in expected future in ation 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 in ation rate duration of a Treasury Bond is equal to its cash ow 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 in ation or a change in real rates Effective Duration of an In ation Protected Bond In order to assess the effective duration of an in ation protected security you have to make a statement about what percent of subsequent interest rate changes will be due to expected in ation rate changes versus real rate changes If you think that X of the future changes in in ation rate are due to changes in real rates obviously lX must be due to changes in in ation expectations then the effective duration of an in ation protected bond is XD 1XDI 1 Where DH is the real rate duration and D1r is the in ation rate duration Given the discussion above 1 reduces to X Cash Flow Duration Comments on the Relative Volatili of In ation and Real Rates Historically interest rate changes have been largely driven by changes in eXpected in ation rates It is argued that in the current environment where the Fed has actively managed in ation the relative volatility of in ation rates When the Federal reserve sought to anticipate in ation pressures and prevent them from being imbedded in the economy it was forced to make real interest rates more volatile Thus in ation 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 in ation 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 in ation sensitivity of the liability is explicitly recognized Nuclear Decommission Trusts NDTs Environmental Clean Up Trusts PampC Insurance Pension Bene ts 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 in ation rate duration of the liability is zero 6 Funding Real Liabilities Advanced Issues In certain situations a liability may not be completely indexed to in ation The following example walks through some work I did for a PampC 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 in ation In creating this example I rst de ne a liability in ation parameter that describes the correlation between a given liability and in ation If the parameter is 1 then the liability is a real liability and the rate of in ation of the item being insured is equal to the general rate of in ation 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 in ation is equal to the economy wide rate of in ation If the parameter is 0 then the insurer has agreed to replace a totaled car with a xed amount of money Interest rate changes result from changes in the real rate of interest or changes in expectations about future in ation 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 in ation I calculate the following duration measures for the liability cash ow duration real rate duration in ation 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 modi ed duration of the expected cash ows today In the example I generated the cash ow duration is 286 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 in ation constant The real rate duration always equals the cash ow duration 3 The in ation rate duration of the liability is the change in the present value of the liability resulting from a change in the in ation rate holding the real rate constant The in ation rate duration is large small when 1 you have large small existing claims relative to anticipated claims andor 2 the liability in ation parameter is small large For example if the in ation sensitivity parameter is 0 the in ation 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 in ation The following two tables summarize the ndings In ation Rate Duration of Liability In ation Duration of Liability In ation The following shows how the effective duration of the liability va1ies with the liability in ation parameter and the relative volatility parameter Effective Duration of Liability Liability Liability Liability Liability Liability In ation In ation In ation In ation In ation Parameter 0 Parameter Parameter Parameter Parameter 1 Relative 286 286 286 286 286 volatility 0 Relative volatility 286 270 252 235 218 25 Relative volatility 286 252 218 184 150 Relative volatility 286 235 184 133 83 75 Relative 286 218 150 83 15 volatility 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 de nite role for in ation indexed bonds for funding insurance liabilities Consider funding an insurance liability with l 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 De ne the following notation DRL real rate duration of the liability DR cash ow duration of real bonds in a portfolio DE in ation rate duration of liability DN cash ow 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 in ation parameter to obtain the in ation rate duration of the liability The in ation rate duration of the real bond is zero and the in ation rate duration of the nominal bond is its cash ow duration The real rate durations of the real and nominal bonds equal their respective cash ow durations To eliminate interest rate risk you choose N R DR and DN subject to RDR NDN DRL 1 NDN DIL 2 N R 1 3 Ngt o 4 Rgt o 5 Equations 1 and 2 ensure that your bond portfolio has the same real rate duration and the same in ation rate duration as the liability You can always satisfy 1 2 and 3 Constraints 4 and 5 require that DRL gt DIL which is always true The second approach is to fund the liability using only nominal bonds Since 1 the in ation rate duration and the real rate duration of nominal bonds are the same and 2 the in ation and real rate duration of the liability are different you need to make a judgement about the relative volatility of in ation 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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