Income Taxes on Zero Coupon Bonds (Preliminary Version)

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Income Taxes on Zero Coupon Bonds (Preliminary Version)

Floyd Vest, Nov. 2014

A person interested in a tax free zero coupon bond should compare its yield to the after tax return on a taxable zero coupon bond. (See the Exercises and Side Bar Notes.) Comparing the tax free rate to the after tax rate of return is a common comparison in evaluating taxable and tax free investments.

A zero coupon bond pays only at maturity and doesn't pay periodic interest. For a taxable zero coupon bond, the IRS requires that income taxes be paid annually as OID securities which are sold at a discount from the value at maturity resulting in phantom annual coupon payments. First the yield needs to be calculated. We will present a simplified three year zero coupon bond as an example.

The taxable zero coupon bond is purchased for P = $900 with a value at maturity

M = $1000 in three years. Let y = yield so that

1

(1)

900(1 + y )3

= 1000,

y

=

1000 3 900

- 1 = .0357442 = 3.57442%

.

Next the (phantom

value) Implied Values V1,V2,V3 will be calculated for the end of each year. This sequence of

Implied Values is the basis for the Implied Taxable Interest, I1, I2 , I3 .

The Implied Value for year 1 is V1 P(1 y) = 900(1+.0357442) = 932.165975. The Implied Taxable Interest I1 V1 P = 932.16978 ? 900 = $32.17.

The Implied Value for year 2 is V2 = V1(1 y) = 932.17(1 + .0357442) = 965.55. The Implied Taxable Interest I2 V2 V1 = 965.55 ? 932.17 = $33.38 .

For year 3, Implied Value V3 = V2(1 y) = 965.55(1+.0357442) = $1000 . Implied Taxable Interest I3 V3 V2 = 1000 ? 965.55 = $34.45 . The bond owner pays income taxes each year on Implied Taxable Interest. (See Sharpe in the References.)

The above bond in general.

I1 P(1 y) P P(1 y) 1 P( y).

I2 V1(1 y) P(1 y) P(1 y)2 P(1 y) P(1 y)( y) I3 V2 (1 y) V1(1 y) P(1 y)3 P(1 y)2 P(1 y)2( y)

A n-year bond in general. In P(1 y)n1( y) and M P(1 y)n

An after tax rate of return for the above three year bond. Let t = income tax rate = .25 . Consider the following cash flow time line.

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M=P(1+y )3 = 900(1+y )3 = 1000

0

1

2

3

__|_________________|__________________|___________________|_____

P

I1 (t )

I2 (t)

I3 (t)

-900

-32.17(.25)

-33.38(.25)

-34.45(.25)

Let the after tax rate of return = i = irr.

(2) 900 = -32.17(.25)(1+i )1 - 33.38(.25)(1+i )2 - 34.45(.25)(1+i )3 + 1000(1+i )3

(3) 900 = -8.0425(1+i )1 - 8.34(1+i )2 + 991.39(1+i )3 We will solve for i = irr by the irr(

function on the TI83/84. First on the home screen, {-8.0425, -8.345, 991.39} L1. You will find irr( in 2nd Finance 8. Code and commentary: 2nd Finance 8. You see irr( . Write in

-900,L1) Alpha Solve. You see i = irr = 2.68089% as the after tax rate of return on the above three year zero coupon bond.

A general formula for the after tax rate of return i. Consider the following time line for a n-year taxable zero coupon bond with price P, with yield y, tax rate t on Implied taxable interest Ik , and value at maturity M. For year k, tax owed is Pyt(1+y )k1 = Ik (t) .

M

0

1

2

3

. . .

n-1

n

__|________|_______|__________|__________________________|__________|____________

P

I1t

I2t

I3t . . .

I n1t

Int

Let i = irr which discounts cash flows to initial equity P:

(4) P = I1t(1 i)1 I2t(1 i)2 I3t(1 i)3 ... In1t(1 i)(n1) Int(1 i) n M(1 i) n

P = - Pyt(1+i )1 - Pyt(1 y)(1 i)2 Pyt(1 y)2(1 i)3 ... Pyt(1 y)n1(1 i)n M(1 i)n

P = -Pyt(1+i )1 1 (1 y)(1 i)1 (1 y)2(1 i)2 ... (1 y)n1(1 i)(n1) +M(1+i )n Let S = 1 (1 y)(1 i)1 (1 y)2(1 i)2 ... (1 y) n1(1 i) (n1) . Consider

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(1 y)(1 i)1S (1 y)(1 i)1 (1 y)2(1 i)2 ... (1 y)n1(1 i)(n1) (1 y)n(1 i)n

(1 y)(1 i)1 1 S (1 y)n (1 i)n 1 (1 y)n (1 i)n 1

(5) S = (1 y)(1 i)1 1

(6)

P

=

Pyt

(1

i)1

(1 y)n (1 i)n 1 (1 y)(1 i)1 1

P(1

y)n (1 i)n

(7)

1

=

yt

(1

i)1

(1 y)n (1 i)n 1

(1

y)(1

i)1

1

(1

y)n (1 i)n

Given y, t, n, use Formula 7 and

a general Solver to solve for i = after tax rate of return. You could let x = 1 + i and solve for x

hopefully between 1 and 2.

Checking Formula 7 for the above three year bond.

We use y = .0357442. We put 1+ y = 1.0357442 in Sto 1. Let t = .25 . From the above calculation

irr = i = .0268089. We put 1.0268089 in Sto 0. Calculation and substitution gives

1

=

(-.0357442)(.25)

(1

i)1

.0263363 .0087028

+1.0263363 which gives

1 = -.0262601 + 1.0263363 = 1.0000762. Formula 7 checks.

See the Side Bar Notes for a short cut for calculating the after tax rate of return i from y and t. But this short cut doesn't explain everything.

Comparing municipal bonds which are zero coupon and semiannual pay bonds. Most municipal bonds provide semiannual interest payments and value at maturity. An example is a municipal bond with a face value of $20,000 maturing in 20 years with a 5.5% coupon. Each six months, the investor is paid an interest dividend of .055 (20,000) = $550, federal income tax

2 free, and at maturity receives $20,000. A comparable zero coupon bond paying $20,000 at maturity in 20 years might be purchased for $6757. For the yield y, we solve

6757(1 + y )20 = 20,000 to get y = .0557566 = 5.57566%. Exercise: Calculate the effective yield on the semiannual pay bond.

Bonds in a portfolio with stocks. William Baldwin gives the formula

R = 5S + 2B ? E . R is the expected real return (after inflation). S is the fraction of the portfolio that is in stocks, for example 60%. B is the fraction that is in bonds, for example 40%. E is the annual expense ratio for the portfolio, for example 0.5% of assets in the portfolio. According to Baldwin, for a 60/40 portfolio, the expected annual real rate of return =

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R = 5(.60) + 2(.40) - .005 = 3.3 percentage points, 3.3% real return per year. The numbers 5 and 2 are Baldwin's constants based on historical stock and bond returns over long periods. On

Dec. 2013, when this was written, Baldwin says that for bonds to earn 2% real return is optimistic and requires a willingness to invest in corporate bonds. What does Baldwin's formula assume about the real return on bonds? Using Baldwin's formula, what is the expected real return for stocks? John Bogle, the founder of Vanguard Mutual Funds, said to expect a 7% nominal return on stocks less 2% inflation. Using the formula R N I for real return R,

1 I inflation rate I, and nominal return N, what real return on stocks does Bogle expect? What rate of inflation does Baldwin seem to have in mind? With a real rate of return of 5% for stocks with 2% inflation, what is the nominal rate of return for stocks?

Baldwin says that 3.3% real return converts one dollar into $4.04 by age 68. At what age did the investment start? What is meant by the $4.04? What is the inflated nominal value of the $4.04?

Derivation of Baldwin's formula: Assume a 60%,40% portfolio with $600 in stocks earning a real rate of return of 5% and $400 in bonds earning a real return of 2%. For the total portfolio, Real return in percent = 5(600) 2(400) - E = 5(.60) + 2(.40) ? E = R . Expense

1000 ratios E as a percent can range from .1% to 2% of total value of portfolio for mutual funds.

Baldwin said "If you are young and can stand volatility, aim for a stock heavy portfolio."

Try a portfolio of 80% stocks and 20% bonds. If $100,000 is invested, how do the two portfolios compare in 43 years in real terms, in nominal terms? What expense ratio E did you use?

See "Joint Life Expectancy and Sustainability of Retirement Funds' (Preliminary Version), Nov. 2013 in this course for the historical separate earning of stocks and bonds in a 60/40 portfolio. If you study this article, you can write your own formula to replace Baldwin's formula.

For the 60/40 portfolio, what does increased expense ratio E do to R? Some mutual funds have an expense ratio of 1.5%. After 43 years, with E = 1.5%, what is the real and nominal value of $100,000 invested, with 2% inflation. Compare this to the E = .5%.

Although both Baldwin and Bogle seem to be thinking in terms of 2% inflation, a long term average rate of inflation is 3.2%. See and check this number. Under these conditions what would be the real rate of return for stocks? Would bonds keep up with inflation? (For Baldwin's formula, see "Nine Formulas for Wealth Building, " Forbes, Dec. 2013, pp 144-150. )

Target date retirement savings funds and college savings funds reduce their exposure to stocks as the target date approached by including a greater proportion of bonds.

Can bonds lose money? When long-term interest rates rose 1.3 percentage points from May to September, 2013, the average bond fund slumped 4%. The 30-year Treasury lost 11% from Jan. 2013 to Dec. 2013. With interest rates at a historic low, they are likely to rise. The Fed says in 2015. One kind of bond fund has a hedge against a loss in NAV due to a rise in

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interest rates. Floating-rate bond funds can raise their rates when Fed Funds rates rise. One such fund returned 4.2% in 2013 (Kiplinger's Personal Finance, 1/2014, p. 44). See the References in this course: "The Effect of Interest Rate Increase on a Bond Investment in a Low Interest Rate Environment" (Preliminary Version) Jan. 2013. "High Dividend Yields and Low Interest Rates on CDs and Bonds" (Preliminary Version) Jan. 2013.

Side Bar Notes:

Real and UnReal rates. In 1981, the 30 year zero-coupon Treasury bond yielded 15.21%. If you owned $100,000, what was it worth in 30 years? Currently 30 year Treasury zeroes yield around 3%. In 30 years, what is this worth? An UnReal example: Suppose that in 1981, the zero coupon Treasury rate on the remaining 30 years suddenly drops to a market rate of 3%, what is the immediate benefit of owning the 15.21% bond? Forbes says "Since 1981 the zero-coupon Treasury bonds have risen six time more than the S&P 500 total return," whatever this means. (Forbes, Nov. 24, 2014, page 76). What is the income tax on zero-coupon bonds? How do the rates of return on zero-coupon STRIPs compare to the comparable semiannual pay bond? How is a contract that pays only the coupons taxed?

The world's largest company. On Nov. 25, 2014, the market capitalization of Apple, Inc. (APPL) was $300 billion, making it larger than Exxon Mobile Corp. (XOM), the former largest company by this measure.

Who buys zero coupon bonds? People who need a specific amount at a future date might buy a zero coupon bond. An example might be to fund a child's college education beginning at a specific date. If a zero coupon bond is purchased for $1000 , the gift giver will have used only $1000 of the annual gift tax exclusion, but the recipient will in the future receive a much larger amount. A zero coupon owner does not have to manage the reinvestment of periodic dividends. Is it true that a zero coupon bond yield is equivalent to reinvestment of coupons at the bond rate? It is often inconvenient to reinvest dividends at the bond rate. Why? What type of tax payer might buy a taxable zero coupon bond? What type would prefer a municipal zero coupon bond?

Taxable zero coupon bonds are often used in tax-deferred accounts.

Who sells zero coupon bonds? They are sold by the federal government, corporations, brokers, banks, municipalities, state or local governments. Some zero coupon bonds issued by local government entities may be triple tax free, with no federal, state, or local income tax. They can be bought and sold on the secondary market. Some zero coupon bonds may be callable. Some zero coupon corporate bonds can be converted into stocks in the corporation.

Duration of a zero coupon bond is equal to the length in years to the value at maturity. See if you can prove this. What does this imply about interest rate risk of a long term bond? Give an example. Discuss.

Pricing zero coupon bonds. Quotations can be relative to a face amount of $100. That is if the asked price is 93.12, this means the quote is 93 + 12/32 for a face amount of $100. Calculating gives 93.12 = $(93+12/32) = $93.375 .

Coupon payments in an annuity. Assume annual payments of $1200 for fifteen years and priced at 5%. The Price is $12,455.59. There is no tax on the payments until you get all of your

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$12,455.59 back. 12,455.59 1200 = 10.38 years. There is no tax through the tenth year. The $1200 for the eleventh year is partially taxable with .38(1200) = $456 is taxable. Payments for years 12 through 15 are taxable.

Coupons and value at maturity which make STRIPS can make several zero coupon bonds each of a coupon at maturity, and one made of the value of the original bond at maturity. For taxable zeros, each zero coupon bond is taxed as indicated in the above derivation. We have not dealt with zero coupon bonds purchased between coupon dates. (See .)

Bond prices. Search Google, Wall Street Journal, bond prices. On Nov. 26, 2014: Treasury, 30-year bond; Price 0/32; Yield 2.955 .

U. S. Treasury Strips, Maturity 2014, Aug. 15; Bid 40.044; Ask 40.161; Chg 0.327; Asked Yield 3.09 .

Bonds and retirement. "A 65 year old who wanted to pay for retirement with annuities tied to bonds needed 24% more wealth in 2013 than in 2005, according to the National Bureau of Economic Recovery." (Kiplinger's Personal Finance, 01/2015, page 6). What does this tell you about bond earnings?

Derailed retirement. In a survey by Ameriprise Financial, 63% said that low interest rates had derailed their retirement plans because assets were growing so slowly (Kiplinger's Personal Finance, 01/2015, page 6).

Stock dividend rates are often higher than 30 year bond rates. Lockheed Martin (LMT), dividend yield 3.1%.

Rising interest rates by the Federal Reserve are expected in 2015. How do returns of stocks follow from rising interest rates? For nine raises in interest rates from 1971 to 2004, the average S&P 500 index return over the next six months was 3.8% ranging from as low as

-17.6%. (Kiplinger's Personal Finance, 01/2015, page 30)

Can you explain this? From Kiplinger's Personal Finance, 01/2015, page 38.

Over the past year, the yield of the benchmark 10-year Treasury bond dropped from 2.6% to 2.3%. Government debt returned 13.1% over the past year.

Making money in stocks. The price of Biogen Idec (BIIB) climbed from $70 to $322. What was the percent increase? Biogen developed an effective treatment for multiple sclerosis.

Wide ranging returns. Among top performing stock ETFs, three year annualized returns ranged from 97.9% to -14.9%. A S&P 500 ETF earned 19.7%. (Kiplinger's Personal Finance, 01/2015, page 45).

Quality of bonds. Bonds are rated by Fitch, Moody's, Standard and Poor's, and others. See , bond ratings. The rating companies have been wrong, particularly in rating mortgage-backed securities (MBSs) and collateralized debt obligations (CDOs) which set off the international mortgage crisis when many institutions lost money on the bonds. Several banks that took the advice of their financial mathematicians didn't by these toxic investments. See for percentages of default by bonds with various ratings. Bonds rated lower by

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S&P have significant default percentages. New York lost three Fortune 500 companines due to the financial crisis: Merrill Lynch, Bear Stearns, and Lehman Bros.

Taxably equivalent bond interest rates. What taxable interest rates is equivalent to 2% tax free? It depends on the marginal income tax rate which varies according to taxable income. For 2012, marginal rates varied from 0%, to 15%, 25%, 28%, 33%, to 35%, Consider a 33% bracket, and r the taxable equivalent rate to 2% tax free. To answer, we calculate r(1-.33) = .02, r = .0298507% = 3%. Also there is consideration of state and local income taxes in some areas. Qualified dividends may be taxed at 15%. Some municipal bonds pay 5% tax free.

So r(1-.33) = .05 gives r = .0746 = 7.46% taxable. See , Tax Rate Schedule. Will this formula replace the derivations and formulas for taxable zero coupon bonds? Consider the example leading up to Formula 2 in the above article? Let y = tax free rate, i = taxable rate, and t = marginal income tax rate, and consider i(1 ? t) = y. List ten things which could be learned from the above formulas and derivations?

For CD interest rates, see and compare top rates and average rates and the long term accumulation difference in five year CD earnings. For some people, up to 85% of their Social Security is taxed. If they can reduce their modified adjusted income, they might escape some of this tax. In a sense, Social Security is double taxed.

Exercises: Show your work. Name variables, numbers, and answers. Discuss in complete sentences. Give the name of your grapher or calculator and some of the code.

#1. (a) A six year zero coupon bond has a face value of $100 and a yield of 6%. What is its price? (b) After one year the market rate drops to 5%. At this rate, what is the market price of the remaining five years of the bond?

#2. A four year taxable zero coupon bond had a face value of $100 and a yield of 6%.

(a) Calculate the yield y. (b) Give the Implied Values for years 1 through 4. (c) Give the Implied Taxable Interest for each year. (d) Calculate with the irr function the annual after tax rate of return i if the marginal tax rate is 28%. (e) Check this answer in Formula 7. (f) Use Formula 7 in a general Solver to solve for the after tax rate of return i.

#3. A 20-year municipal zero coupon bond, matures Jan. 1, 2028. On Jan. 1, 2008, it sells for $6757.04 . Original coupon 5.5%. (a) Calculate the yield y. (b) Graph accrete value for 2008 to 2028. Use Trace to build a five year table. (c) Calculate the yield for the 5.5% coupon rate for the semiannual pay bond. How does the effective yield of the semiannual pay bond compare to the yield of the zero?

#4. One should figure 10-year Treasuries will be yielding between 2.25% and 3.25% over the coming year (Kiplinger's Personal Finance, 01/2015, page 38). What is a ten year $100 zero worth with a yield of 2.25%? What is it worth at 3.25%? What is the percent loss? How does this compare to the duration of the bond?

#5. Name different kinds of zero coupon bonds or near zero coupon bonds not mentioned in this article. Explain each.

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#6. (a) In Jan. 1987, Sam bought a 20 year 10% zero coupon bond which at maturity pays $100,000. What did he pay for the bond? (b) Five years later he needed to sell the bond. The current interest rate on 15 year bonds was 12%. What did he sell the bond for? What rate of return did he make on his original investment? (c) What rate of interest would make the new selling price equal to what he paid originally for the bond? (d) If the rate for a 15 year bond is 10%, what would he be paid for the bond? Build a table of interest rates for 15 year bonds and prices, and discuss.

References:

For a free course in financial mathematics, with emphasis on personal finance, for upper high school and undergraduate college, see . Register and they will e-mail you a password. Simply click on an article in the annotated bibliography, download it, and teach it. Unit 1: The Basics of Mathematics of Finance, Unit 2: Managing Your Money, Unit 3: LongTerm Financial Planning, Unit 4: Investing in Bonds and Stocks, Unit 5: Investing in Real Estate, Unit 6: Solving Financial Formulas for Interest Rate, For about thirteen more advance or technical articles, see the UMAP Journal at COMAP. The last section is Additional Articles on Financial Mathematics or Related to Personal Finance. In all, there are about eighty articles.

There are at least twelve articles on bonds in Unit 4 of this course.

To get a great amount of information about municipal zero coupon bonds and taxable zero coupon bonds, search the internet.

Sharpe, William F. and Gordon Alexander, Investments, Fourth Edition, Prentice-Hall,

page 331.

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