You want to make a one-year investment in a fixed-income security. You have three investment choices. The first is a Treasury security with an annual (nominal) interest rate of 4.5%, the second is a municipal security with a (nominal) rate of 3.5%, and the third is a Treasury inflation-protected security (TIPS), which promises a real rate of 3%. (Assume that the coupon is paid only once at the end of the year, that the securities trade currently at par and that they will pay back the face value at the end of the year. TIPS adjust the principal according to the inflation rate.)

Answer explained below

Explanation:

We have

Treasury Security giving 4.5% nominal rate of return

Municipal Security giving 3.5$ nominal rate of return

and TIPS giving 3% real rate of return

In case of 0 inflation (50% probability)

Treasury Security giving 4.5% nominal rate of return

Municipal Security giving 3.5$ nominal rate of return

and TIPS giving 3% nominal rate of return

In case of 2.5% inflation (50% probability)

Treasury Security giving 4.5% nominal rate of return

Municipal Security giving 3.5$ nominal rate of return

and TIPS giving 3% real rate of return i.e. return of 1.03*1.025 -1 =0.0575 = 5.75% nominal rate of return

So, the expected nominal returns(before tax) of Treasury security = 0.5*4.5% +0.5*4.5% = 4.5%

The expected nominal returns(before tax) of Municipal security = 0.5*3.5% +0.5*3.5% = 3.5%

and The expected nominal returns(before tax) of TIPS = 0.5*3% +0.5*5.575% = 4.2875%

For low income individual (marginal tax rate of 20%)

The expected nominal returns(after tax) of Treasury security = 4.5%*(1-0.2)= 3.6%

The expected nominal returns(after tax) of Municipal security = 3.5% (as they are tax exempt)

The expected nominal returns(after tax) of TIPS = 4.2875% * (1-0.2) =3.43%

For high income individual (marginal tax rate of 20%)

The expected nominal returns(after tax) of Treasury security = 4.5%*(1-0.4)= 2.7%

The expected nominal returns(after tax) of Municipal security = 3.5% (as they are tax exempt)

The expected nominal returns(after tax) of TIPS = 4.2875% * (1-0.4) =2.5725%

Expected Inflation Rate = 0%*0.5+2.5% *0.5 =1.25%

The Expected after tax Real Return of Low Income Individual for Treasury security = 1.036/1.0125 - 1 =0.02321 = 2.32%

The Expected after tax Real Return of Low Income Individual for Municipal security = 1.035/1.0125 - 1 = 0.02222 = 2.22%

The Expected after tax Real Return of Low Income Individual for TIPS=1.0343/1.0125 -1 =0.021531 =2.15%

The Expected after tax Real Return of High Income Individual for Treasury security = 1.027/1.0125 - 1 =0.014321 = 1.43%

The Expected after tax Real Return of High Income Individual for Municipal security = 1.035/1.0125 - 1 = 0.02222 = 2.22%

The Expected after tax Real Return of High Income Individual for TIPS=1.025725/1.0125 -1 =0.013062 =1.31%

We can clearly see that TIPS are riskier in real terms as they offer lower real rates which remain nominal in case of no inflation scenario and hence the overall expected returns decline. Further, these are taxed, so they are risky even in real terms.