PDF Comparing the After Tax Return on TIPS and I-Bonds Floyd Vest ...

Comparing the After Tax Return on TIPS and I-Bonds Floyd Vest, Feb. 2012 (Preliminary Version)

In the Jan. 23, 2012 issue of Time magazine, the well known TV financial advisor, Suze Orman, advised people "to put their money in TIPS ? Treasury Inflation Protected Securities." She says that interest rates and inflation rates will rise. The 29 year bond bull market will come to an end. As usual, for financial forecasting, only time will tell.

For TIPS (Treasury Inflation Protected Securities), the principal in the bond is adjusted by changes in the Consumer Price Index (CPI). TIPS pay interest at a fixed rate of the adjusted principal every six months. Both the coupons and increase in principal are taxed each year.

For I-Bonds, there is a base rate with which the inflation rate is combiuned and the interest can be accumulated in the bond at the combined rate each period. When the bond is cashed, the profit is taxed at the marginal income tax rate.

Example 1. TIPS. In , there is a graph entitled "Cumulative

Inflation

and

Annual

Interest

on

the

7 3%

Treasury

10-year

TIPS

due

1/15/2009

(Par

8

Value = $1000) which displays the accumulated amount of growth in dollars for the

principal and the annual coupons (interest payments) beginning 1/15/1999. A footnote

indicates that cumulative interest payments for 10 years = $447.43. Examination of the

bar graph for the accumulated principal gives an estimate of a $308 increase. Thus the

before tax value of the TIPS principal at maturity is approximately $1,308. (The author

apologizes for the approximate data.)

The interest does not accumulate in the bond. For the TIPS, solving for the average inflation rate gives 1000(1+I )10 = 1308 with I = .0272 = 2.72%, which means that

the principal increased at an annual average rate of 2.72%. Using for the period from the end of 1998 to the end of 2008, gives 20(1+I )10 =26.42, with I = 2.82%. (See the article in this course "Using .")

Continuing to look at the inflation rate, consider that the semiannual coupon

37% is 8

of the accumulated principal .

For simplicity we will use

7 3%

= .03875 =

2

8

3.875% per year. We will assume that the yearly coupon is 3.875% of the principal and

the principal increases yearly at the rate of inflation, the coupon also increases at the rate

of inflation. This gives a formula for the sum of (cumulative) interest payments.

(1)

.03875(1000)(1+I)

(1

I )10 I

1

= 447.43.

Solving for I gives 2.6% as another

estimate for the inflation rate. (See the Side Bar Notes for derivation of Formula 1.)

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Since we are attempting an after tax comparison, for TIPS we will calculate with an average inflation rate of 2.72% and a marginal income tax rate of 25%. Someone with exact historical data and a spreadsheet could do much better.

After tax gain on the principal. Considering taxes paid each year on the increase

in principal at 25%, and an average inflation rate of 2.72%, the after tax value of the

principal for ten years is

(2)

1000

+

1000(.0272)(1-.25)

(1

.0272)10 .0272

1

= $1230.87 . (See the Side Bar

Notes for the derivation.)

Total coupons paid. The footnote said that the total coupons paid is $447.43, which implies that under our assumptions, that interest is paid at the rate 3.875% on increasing values of the bond. To approximate this, we will calculate that total interest is S = 1000(1+.0272)(.03875)+1000(1+.0272 )2 (.03875) + ... + 1000(1+.0272 )10 (.03875)

=1000(.03875)(1+.0272)

(1

.0272)10 .0272

1

=

$450.47

which

approximates

the

footnote

$447.43.

After tax return on the coupons. For the coupons, we will assume that they are

reinvested each year at the rate I=2.72% since for some periods, rates on six month CDs

track the inflation rate, for example from 1970 to 1982. (From Jan. 1999 to Jan 2009,

the Vanguard Prime Money Market Fund averaged 3.48% compounded.) For the after

tax accumulation of reinvestment of coupons we will consider a coupon rate r, annual

income taxes on coupons at the rate t, income taxes on interest earned by coupons at the

rate t, and the inflation rate I, and interest rate on coupons of I. (See the Side Bar Notes

for derivation of Formulas 3 and 4 which are used.) Calculation gives the after tax and

reinvestment value of coupons of $368.96. The after tax value of the TIPS bond

investment is $368.96 + $1230.87 = $1599.83, as calculated below.

(3)

D=

(1 I )10 [1 I (1 t)]10

.

(1 .0272)10 1 .0272(1 .25)10

D

= 12.359292

It

.25(.0272)

(4) The after reinvestment and tax value for coupons = rP(1+I)(1-t)D. Calculating

gives .03875(1000)(1+.0272)(1-.25)(12.359292) = $368.96 . This makes the after tax

value of the TIPS bond at maturity = 1230.87 + 368.96 = $1599.83 .

Rate of return. For the TIPS, the before tax rate of return was 5.79%, not considering reinvestment. The after tax and reinvestment rate of return was 4.81%. What was the before tax rate of return with reinvestment?

Example 2. Comparing the after tax value of the I-Bond. Acquiring the after tax value of an I-Bond is much simpler. Go to , Tools, Savings Bond Calculator, and put in an I-Bond of $1000, purchased 1/1999, Value at 1/2009, and get the report, Value = $1827.60. We will assume for simplicity that taxes are paid at 10 years at the 25% rate giving 1827.60-.25(827.60) = $1620.70. The I-Bond can remain invested for many 30 years. The reader can see the comparison of the after tax value of

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the TIPS bond of $1599.83 and the I-Bond of $1620.70 and make their own assumptions and do their own calculations. They can compare the many advantages and disadvantages of the two types of inflation protected bonds. For the I bond, the after tax and reinvestment rate of return was 4.95%. The before tax yield was 6.22%. The reinvestment rate was the base rate combined with inflation rate which could be different from that assumed for the TIPS. We can estimate the base rate to be 6.22% - 2.72% = 3.5%. According to , I-Bonds sold in 1999 and 2000 had fixed rates of 3.30% to 3.60%. Everybody should have bought these. The fixed rates haven't been that high since. For the formula for combining the two rates, see indiv/research. Fixed rate on May 1, 2013 was 0.0%.

Side Bar Notes:

Derivation of the after tax and reinvestment value of coupons (Formulas 3 and 4). Let the after tax value of the first coupon at the end of the first year be rP(1+I)(1-t) where r is the annual coupon rate, P is the principal, and t is the marginal income tax rate, and I is the inflation rate and rate of return on reinvested coupons. This value is reinvested at the rate I giving rP(1+I)(1-t)[1+I(1-t) ]9 at maturity. This concept

gives the after tax and reinvestment value of all of the coupons to be

rP(1+I)(1-t) [1 I (1 t)]9 (1 I )[1 I (1 t)]8 ... (1 I )8[1 I (1 t)] +(1+I )9

Let D = [1 I (1 t)]9 (1 I)[1 I(1 t)]8 ... (1 I) 8[1 I(1 t)] +(1+I )9 . Consider

1 I D (1 I )[1 I (1 t)]8 (1 I )2[1 I (1 t)]7 ... (1 I )9 + (1 I )10

[1 I (1 t)]

[1 I (1 t)]

Subtracting gives

[1

1 I I (1

t

)]

1

D

=

(1 I )10 [1 I (1 t)]9 . [1 I (1 t)]

Solving for D and

simplifying gives

(1 I )10 [1 I (1 t)]10

(3) D =

.

It

(4) The after reinvestment and tax value for coupons = rP(1+I)(1-t)D. Calculating

gives .03875(1000)(1+.0272)(1-.25)(12.245545) = $368.96 . This makes the after tax

value of the TIPS bond = 1230.87 + 368.96 = $1599.83.

Derivation of Formula 1.

(5) The sum of the annual coupon payments = rP(1+I)+...+rP(1+I )10 = 447.43.

Simplifying gives

(6)

rP(1+I)[1+...+(1+I

)9

]

=

rP(1+I)

(1

I )10 I

1

= 447.43.

Solving for I by the TI83

gives I = 2.6% using r = 3.875%.

3

(8) Derivation of Formula 2, After tax gain of principal. For principal, end of year after tax gain: Year 1: P(I)(1 t)

Year 2: P(1 I )(I )(1 t)

Year 3: P(1 I )2(I )(1 t) . . . Year 10: P(1 I )9(I )(1 t) This gives the sum

S = P(I)(1-t) 1 (1 I ) (1 I )2 ... (1 I )9 so

S

=

P(I)(1-t)

(1

I )10 I

1

which is the sum of yearly taxed increases in principal.

The after tax value of principal will be 1000 + S.

The results of recent Treasury auctions of TIPS. 10 year TIP, 1-31-2012, 1-15-2022, Interest rate .125%, Yield -.046%, Price per $100, $101.6661834. What does this tell you about investors' expectations?

Recent fixed rates on I-Bonds. Fixed rate Nov. 1, 2011, 0.0%. How would you have liked to buy an I-Bond in May 2000 with a fixed rate of 3.60%? (See indiv/research/indepth/ibonds/res_ibonds_iratesandterms.htm.)

More Information on TIPS and I-Bonds can be found at , , , .

Financial Education is a Priority. From : "For years, teachers have used Money Math: Lessons for Life in core mathematics classes, .... It is important that we make financial education a priority, and provide or kids with knowledge they need to manage their money, ..., and save for retirement." Lesson 1 ? The Secret of Becoming a Millionaire, for Grades 7 and 8.

For a free course in financial mathematics with emphasis on personal finance, see . Click on the box for the free financial mathematics course, register, and COMAP will e-mail you a password. Simply click on an article in the annotated bibliography, download it, and teach it, or study it. Unit 1: The Basics of Mathematics of Finance Unit 2: Managing Your Money Unit 3: Long-Term Financial Planning Unit 4: Investing in Bonds and Stocks Unit 5: Investing in Real Estate Unit 6: Solving Financial Formulas for i.

Current money market rates. (1-27-2012) : The national average rate on a bank money market fund was .20% APY. The top reported was .90% and lowest .15%. Make up problems for long term comparisons. Bank money market funds were paying "much" more that mutual fund money market funds.

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Looking for the right calculator? See calculators.aspx for personal financial calculators.

A year by year record of inflation. See research/index.cfm, It gives annual U.S. Inflation from 1946. Also see .

How does the work? It reports that for 1999 to 2009, the average inflation rate is 2.56%. Is this an eleven year average or a ten year average, if so which year is not included, the first, or the last? Also when does a year begin? Perhaps the easiest way to check is to use one year. For example, for 1999 to 2000, the gives, "If in 1999 an item costs $20, then in 2000, the same item would cost $20.67. The annual rate of inflation is 3.40%. Using the inflation table from or the Cleveland Fed, for 1999 inflation is 2.186%, and for 2000 inflation is 3.376%. Maybe 1999 to 2000 means from Jan. 1, 2000 to Dec.31, 2000, inclusive. See the article in this course "Using ." Lets try again, with 2000 to 2001. The usinflationcalculator gives 2.8%. We can check 20(1+I) = 20.56, I = .028 = 2.8%. The Cleveland Fed gives inflation for 2000 is 3.376% and for 2001 it is 2.832%. The calculator must mean by 2000 to 2001, the period from the end of 2000 to the end of 2001. For our original figures, 1999 to 2009 must mean from the end of 1999 to the end of 2009. The TIPS bond in Example 1 was from Jan. 15, 1999 to Jan. 15, 2009. We should redo our usinflationcalculator for from 1998 to 2008 which is from the beginning of 1999 through 2008. This gives 2.823% which doesn't agree with the other inflation figures we calculated.

How inflation is applied to TIPS. A research question for a serious TIPS investor: The question is, for a year in the TIPS, what inflation figure is applied? : For a TIPS bond, inflation index ratios are periodically published for each day of the year and are used to adjust the principal. The owner of a TIPS is instructed to locate the inflation index ratio that corresponds to the interest payment date, multiply the par amount of the bond by the applicable index ratio, and use the fixed coupon rate to calculate the coupon (, TIPS: FAQs) .

Envisioning finances in your old age. Social Security benefits amount to 90 percent or more of income for more than a third of retirees. Social Security amounts to 50 percent or more for two-thirds. Interviews report that these people worry daily about "not enough money."

For the financially independent (somewhat wealthy), they report a happy realization that they are very unlikely to run out of money. Their sense of ease is palpable. They are awash in an ambience of raw, joyous life, and productivity.

There is mounting economic evidence that affluence adds years to your life, and being poor shortens it. In the U.S., the most affluent live 4.5 years longer than the least. The same trend had been observed in England among government employees.

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