ńņš. 4 |

You get $5 today, and next month you will receive a payment of (1 + Ļ )Ā·CF = 1.001 Ā· $5 = $5.005. The

13.

growing perpetuity is worth PV = CF1 /(r ā’ g) = $5.005/(0.5% ā’ 0.1%) = $1, 251.25. So, the total value is

$1,256.25.

14. $12.5 million.

15. The immediate dividend would be worth $1.5 million. In addition, you now have a growing perpetuity that

starts with a payment of $1.530 million. Therefore, the PV would be $1.500 + $1.530/12% = $14.250million.

First work out what the value would be if you stood at one month. The interest rate is (1 + 9%)1/12 ā’ 1 =

16.

0.7207% per month, and 2.1778% per quarter. Thus, in one month, you will have $5.00 plus $5.025/(2.1778%ā’

0.5%) ā $299.50. In addition, you get the $5 for a total of $304.50. Because this is your value in one month,

discount $304.50 at an 0.7207% interest rate to $302.32 today.

17. g = r ā’ E/P = 12% ā’ $5/$100 = 7% per annum.

18. Remembering this formula is not as important as remembering the other growing perpetuity formula. The

1 ā’ [1/(1 + r )]T

annuity formula is CF1 Ā· .

r

19.

1 ā’ [1/(1 + r )]T 1 ā’ [1/(1 + 0.005)]360

CF1 Ā· = $5 Ā·

r 0.005

1 ā’ 0.166

= $5 Ā· ā $833.96

0.005

20. You need to solve

$5, 000

1 ā’ 1/(1 + r )120 (3.33)

$500, 000 = .

r

The solution is r ā 0.3314% per month, or 3.8% per annum. This is the implied rate of return if you purchase

the warehouse and then rent it out. You would be better oļ¬ earning 5% elsewhere.

ļ¬le=perpetuities.tex: RP

51

Section 3Ā·4. Summary.

21. For $1,000 of mortgage, solve for CF1 in

1 ā’ [1/(1 + r )]T

PV = CF1 Ā·

r

1 ā’ [1/(1 + 0.005)]15Ā·12=180

$1, 000 = CF1 Ā·

0.005

$1, 000 = CF1 Ā· 118.504 ā CF1 ā $8.44

ā’

In other words, for every $1,000 of loan, you have to pay $8.44 per month. For other loan amounts, just

rescale the amounts.

For 1 year, the 300 bezants are worth 300/1.0212 = 236.55 bezants today. The quarterly interest rate

22.

is 1.023 ā’ 1 = 6.12%. Therefore, the 4-āquartityā is worth 75/.0612Ā·[1 ā’ 1/1.06124 ] = 300/1.06121 +

300/1.06122 + 300/1.06123 + 300/1.06124 = 259.17 bezants. The soldier would have lost 22.62 bezants,

which is 8.7% of what he was promised. (The same 8.7% loss applies to longer periods.)

23. For each ecu (e), the perpetuity is worth 1e/0.04 = 25e. The annuity is worth 1e/0.05Ā·(1ā’1/1.0541 ) = 17.29e.

Therefore, the perpetuity is better.

24. The interest rate is 5% per half-year. Be my guest if you want to add 40 terms. I prefer the annuity method.

The coupons are worth

1 ā’ [1/(1 + r )]T

PV(Coupons) = CFt+1 Ā·

r

1 ā’ [1/(1 + 0.05)]40

= $1, 500 Ā·

0.05

(3.34)

1 ā’ [1/(1 + 0.05)]40

= $1, 500 Ā·

0.05

ā .

$25, 739

The ļ¬nal payment is worth

$100, 000

PV(Principal Repayment) = ā $14, 205 . (3.35)

(1 + 0.05)40

Therefore, the bond is worth about $39,944 today.

25. For six months, (1 + 2.47%)2 ā’ 1 = 5%. Now, deļ¬ne six months to be one period. Then, for t 6-month periods,

you can simply compute an interest rate of (1 + 2.47%)t ā’ 1. For example, the 30 months interest rate is

(1 + 2.47%)5 ā’ 1 = 12.97%.

26.

(1 + 0.02)35

$4, 000/(0.08 ā’ 0.02) Ā· 1 ā’ ā $57, 649 . (3.36)

(1 + 0.08)35

(All answers should be treated as suspect. They have only been sketched, and not been checked.)

ļ¬le=perpetuities.tex: LP

52 Chapter 3. More Time Value of Money.

a. Advanced Appendix: Proofs of Perpetuity and Annuity For-

mulas

A Perpetuity The formula is

CF CF CF CF

(3.37)

+ + Ā·Ā·Ā· + + Ā·Ā·Ā· = .

t

2

1+r (1 + r ) (1 + r ) r

We want to show that this is a true statement. Divide by CF,

1 1 1 1

(3.38)

+ + Ā·Ā·Ā· + + Ā·Ā·Ā· = .

t

2

1+r (1 + r ) (1 + r ) r

Multiply (3.38) by (1 + r )

(1 + r )

1 1

(3.39)

1+ + Ā·Ā·Ā· + + Ā·Ā·Ā· = .

(1 + r )tā’1

(1 + r ) r

Subtract (3.39) from (3.38),

(1 + r ) 1

(3.40)

1= ā’

r r

which simpliļ¬es to be a true statement.

A Growing Perpetuity We know from the simple perpetuity formula that

ā ā

CF CF CF CF

= = . (3.41)

ā”

(1 + r )t ft f ā’1

r

t=1 t=1

Return to the deļ¬nition of a growing perpetuity, and pull out one (1 + g) factor from its cash ļ¬‚ows,

ā ā ā

C Ā· (1 + g)tā’1 C Ā· (1 + g)t

1 1 CF

= Ā· = Ā· . (3.42)

1+r t

(1 + r )t (1 + r )t

1+g 1+g

t=1 t=1 t=1 1+g

1+r

be f , and use the ļ¬rst formula. Then

Let

1+g

ļ£± ļ£¼ ļ£± ļ£¼

ļ£“ ļ£“

ā

ļ£² ļ£½ ļ£² ļ£½

1 CF 1 CF

Ā· = Ā· , (3.43)

1+r t

ļ£“ ļ£“ ļ£³ ā’ 1ļ£¾

1+r

1+g 1+g

ļ£³t=1 ļ£¾

1+g

1+g

and simplify this,

ļ£± ļ£¼

ļ£“ ļ£“

ļ£² ļ£½ C Ā· (1 + g)

1 CF 1 CF

= Ā· = Ā· = . (3.44)

ļ£“ ļ£“

(1+r )ā’(1+g)

1+g 1+g r ā’g r ā’g

ļ£³ ļ£¾

1+g

An Annuity Consider one perpetuity that pays $10 forever, beginning next year. Consider another perpetuity that

begins in 5 years and also pays $10, beginning in year 6, forever. If you purchase the ļ¬rst annuity and sell

the second annuity, you will receive $10 each year for ļ¬ve years, and $0 in every year thereafter.

Ā·Ā·Ā·

0 1 2 3 4 5 6 7 8

Ā·Ā·Ā·

Perpetuity 1 +$10 +$10 +$10 +$10 +$10 +$10 +$10 +$10

equivalent to +$10/r

Ā·Ā·Ā·

Perpetuity 2 ā“$10 ā“$10 ā“$10

equivalent to ā“$10/r

Net Pattern +$10 +$10 +$10 +$10 +$10

equivalent to +$10/r ā“$10/r

1 1 1 1 1

discount factor

(1 + r )1 (1 + r )2 (1 + r )3 (1 + r )4 (1 + r )5

This shows that $10, beginning next year and ending in year 5 should be worth

$10 1 $10

PV = ā’ Ā·

5

(1 + r )

r r (3.45)

C C C

1 1

= ā’ Ā· = Ā· 1ā’ ,

(1 + r )T

(1 + r )5 r

r r

which is just our annuity formula.

CHAPTER 4

Investment Horizon, The Yield Curve, and

(Treasury) Bonds

Bonds and Fixed Income

last ļ¬le change: Feb 23, 2006 (14:15h)

last major edit: Mar 2004, Nov 2004

We remain in a world of perfect foresight and perfect markets, but we now delve a little deeper

to make our world more realistic. In earlier chapters, the interest rate was the same every

periodā”if a 30-year bond oļ¬ered an interest rate of 5.6% per annum, so did a 1-year bond. But

this is usually not the case in the real world. For example, in May 2002, a 30-year U.S. Treasury

bond oļ¬ered an interest rate of 5.6% per year, while a 1-year U.S. Treasury bond oļ¬ered an

interest rate of only 2.3% per year.

The issues that interest rates that depend on the length of time (which we call horizon-

dependent) create are important not only for bond tradersā”who work with time-dependent

interest rates every dayā”but also for companies that are comparing short-term and long-term

projects or short-term and long-term ļ¬nancing costs. After all, investors can earn higher rates

of return if instead of giving money to your ļ¬rmā™s long-term projects, they invest in longer-

term Treasury bonds. Thus, if two corporate projects have diļ¬erent horizons, they should not

necessarily be discounted at the same cost of capital. In May 2002, building a 30-year power

plant probably required a higher cost of capital to entice investors than an otherwise equivalent

1-year factory. Similarly, if your corporation wants to ļ¬nance projects by borrowing, it must

pay a higher rate of return if it borrows long-term.

In this chapter, you will learn how to work with horizon-dependent rates of returns, and you

will see why rates usually diļ¬er. This chapter then takes a digressionā”it works out a number

of issues that are primarily of interest in a bond context. This digression is germane to the cor-

porate context, because almost all corporations need to borrow money. However, the material

is self-contained and you can read it after you have ļ¬nished the rest of the book.

53

ļ¬le=yieldcurve.tex: LP

54 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

4Ā·1. Time-Varying Rates of Return

We now switch direction and make the world a bit more realisticā”we allow rates of return

A second compounding

example. to diļ¬er by horizon. As promised in the previous chapter, all tools you have learned remain

applicable. In particular, compounding still works exactly the same way. For example, what is

the two-year rate of return if the interest rate is 20% in the ļ¬rst year, and 30% in the second

year? (The latter is known as a reinvestment rate.) You can determine multi-year rates of

return from one-year rates of return using the same compounding formula,

(1 + r0,2 ) = (1 + r0,1 ) Ā· (1 + r1,2 )

(4.1)

= (1 + 20%) Ā· (1 + 30%) = 1.56 .

Subtract 1, and the answer is the total two-year rate of return of 56%.

So, the compounding formula for obtaining a total rate of return from period i to period j is

The general formula for

compounding over many still the multiplicative āone-plus formulaā for each interest rate (subtracting 1 at the end). It

periods.

now can also help answer questions such as, āIf the one-year rate of return is 30% from year 1

to year 2, 40% from year 2 to year 3, and 50% from year 3 to year 4, then what is the rate of

return for investing beginning next year for three years?ā The answer is

r1,2 = 30% r2,3 = 40% r3,4 = 50%

Given :

(1 + r1,4 ) = (1 + r1,2 ) Ā· (1 + r2,3 ) Ā· (1 + r3,4 )

(4.2)

= (1 + 30%) Ā· (1 + 40%) Ā· (1 + 50%)

= (1 + 173%) .

Subtracting 1, you see that the three-year rate of return for an investment that takes money

next year (not today!) and returns money in four years, appropriately called r1,4 , is 173%. For

example, if it were midnight of December 31, 2002 right now, each dollar invested on midnight

December 31, 2003, would return an additional $1.73 on midnight December 31, 2006 for a

total return of $2.73.

Solve Now!

Q 4.1 If the ļ¬rst-year interest rate is 2% and the second year interest is 3%, what is the two-year

total interest rate?

Q 4.2 Although a promising two-year project had returned 22% in its ļ¬rst year, overall it lost

half of its value. What was the projectā™s rate of return after the ļ¬rst year?

Q 4.3 From 1991 to 2002, the stock market (speciļ¬cally, the S&P500) had the following annual

rates of return:

rS&P500 rS&P500

Ė Ė

Year Year

+0.2631 +0.3101

1991 1997

+0.0446 +0.2700

1992 1998

+0.0706 +0.1953

1993 1999

ā’0.0154 ā’0.1014

1994 2000

+0.3411 ā’0.1304

1995 2001

+0.2026 ā’0.2337

1996 2002

What was your rate of return over these 12 years? Over the ļ¬rst 6 years and over the second 6

years?

ļ¬le=yieldcurve.tex: RP

55

Section 4Ā·2. Annualized Rates of Return .

Q 4.4 A project lost one third of its value the ļ¬rst year, then gained ļ¬fty percent of its value, then

lost two thirds of its value, and ļ¬nally doubled in value. What was the overall rate of return?

4Ā·2. Annualized Rates of Return

Time-varying rates of return create a new complication, that is best explained by an analogy. Is Per-Unit Measures are

conceptual aids.

a car traveling 258,720 yards in 93 minutes fast or slow? It is not easy to say, because you are

used to thinking in āmiles per sixty minutes,ā not in āyards per ninety-three minutes.ā It makes

sense to translate speeds into miles per hour for the purpose of comparing speeds. You can

even do this for sprinters, who cannot run a whole hour. Speeds are just a standard measure

of the rate of accumulation of distance per unit of time.

The same issue applies to rates of return: a rate of return of 58.6% over 8.32 years is not A Per-Unit Standard for

Rates of Returns:

as easy to compare to other rates of return as a rate of return per year. So, most rates of

Annualization.

return are quoted as average annualized rates. Of course, when you compute such an average

annualized rate of return, you do not mean that the investment earned the same annualized rate

of return of, say, 5.7% each yearā”just as the car need not have traveled at 94.8 mph (258,720

yards in 93 minutes) each instant. The average annualized rate of return is just a convenient

unit of measurement for the rate at which money accumulates, a āsort-of-average measure of

performance.ā

So, if you were earning a total three-year holding return of 173% over the three year period, An Example of

Annualizing a

what would your average annualized rate of return be? The answer is not 173%/3 ā 57.7%,

Three-Year Total

because if you earned 57.7% per year, you would have ended up with (1 + 57.7%) Ā· (1 + 57.7%) Ā· Holding Return.

(1 + 57.7%) ā’ 1 = 287%, not 173%. This incorrect answer of 57.7% ignores the compounded

interest on the interest that you would earn after the ļ¬rst year and second year. Instead, you

need to ļ¬nd a single hypothetical rate of return which, if you received it each and every year,

would give you a three-year rate of return of 173%.

Call r3 this hypothetical annual rate which you would have to earn each year for 3 years in A Problem of ļ¬nding a

three-year annualized

order to end up with a total rate of return of 173%. To ļ¬nd r3 , solve the equation

interest rate. Solution:

Take the N-th Root of

the total return (N is

(1 + r3 ) Ā· (1 + r3 ) Ā· (1 + r3 ) = (1 + 173%)

number of years).

(4.3)

(1 + r3 ) Ā· (1 + r3 ) Ā· (1 + r3 ) = (1 + r0,3 ) ,

or, for short

(1 + r3 )3 = (1 + 173%)

(4.4)

t

(1 + rt ) = (1 + r0,t ) .

Here r3 is an unknown. Earning the same rate (r3 ) three years in a row should result in a total

holding rate of return (r0,3 ) of 173%. The correct solution for r3 is obtained by computing the

third root of the total holding rate of return (Appendix 2Ā·3 reviews powers, exponents and

logarithms):

ā

3

(1 + r3 ) = (1 + 173%)(1/3) = 1 + 173% ā 1.3976

(4.5)

(1/t) t

(1 + r0,t ) = 1 + r0,t = (1 + rt ).

Conļ¬rm with your calculator that r3 ā 39.76%,

(4.6)

(1 + 39.76%) Ā· (1 + 39.76%) Ā· (1 + 39.76%) ā (1 + 173%) .

In sum, if you invested money at a rate of 39.76% per annum for three years, you would end up

with a total three-year rate of return of 173%. As is the case here, for bonds with maturities far

from 1-year, the order of magnitude of the number would often be so diļ¬erent that you will

intuitively immediately register whether r0,3 or r3 is meant.

ļ¬le=yieldcurve.tex: LP

56 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

Important: The total holding rate of return over N years, called r0,N , is trans-

lated into an annualized rate of return , called rN , by taking the N-th root:

1 + r0,N = (1 + r0,N )1/N . (4.7)

N

(1 + rN ) =

Compounding the annualized rate of return over N years yields the total holding

period rate of return.

The need to compute annualized rates of return often arises in the context of investments. For

Translating long-term

net returns into example, what annualized rate of return would you expect from a $100 investment today that

annualized rates of

promises a return of $240 in 30 years? The ļ¬rst step is computing the total holding rate of

returns.

return. By Formula 2.2, the total 30-year rate of return is

$240 ā’ $100

r0,30 = = 140%

$100

(4.8)

CF30 ā’ CF0

r0,30 = .

CF0

The annualized rate of return is the rate r30 , which, if compounded for 30 years, oļ¬ers a 140%

rate of return,

(1 + r30 )30 = (1 + 140%)

(4.9)

t

(1 + rt ) = (1 + r0,t ) .

Solve this equation by taking the 30th root,

ā

30

(1 + r30 ) = (1 + 140%)1/30 = 1 + 140% ā 1 + 2.96%

(4.10)

1/30 30

(1 + r30 ) = (1 + r0,30 ) = 1 + r0,30 .

Thus, a return of $240 in 30 years for $100 investment is equivalent to about a 3% annualized

rate of return.

In the context of rates of return, compounding is similar to adding, while annualizing is similar

Compounding ā Adding.

Annualizing ā Averaging. to averaging. If you earn 1% twice, your compounded rate is 2.01%, similar to the rates them-

selves added (2%). Your annualized rate of return is 1%, similar to the average rate of return of

2.01%/2 = 1.005%. The diļ¬erence is the interest on the interest.

Now presume that you have an investment that doubled in value in the ļ¬rst year, and then

Compounding vs.

Averaging can lead to fell back to its original value. What would its average rate of return be? Doubling from, say,

surprising results.

$100 to $200 is a rate of return of +100%. Falling back to $100 is a rate of return of ($100 ā’

$200)/$200 = ā’50%. Therefore, the average rate of return would be [+100% + (ā’50%)]/2 =

+25%. But you have not made any money! You started with $100 and ended up with $100. If

you compound the returns, you get the answer of 0% that you were intuitively expecting:

(1 + 100%) Ā· (1 ā’ 50%) = 1 + 0%

(4.11)

(1 + r0,1 ) Ā· (1 + r1,2 ) = (1 + r0,2 ) .

Therefore, the annualized rate of return is also 0%. Conversely, an investment that produces

+20% followed by ā’20% has an average rate of return of 0%, but leaves you with

(1 + 20%) Ā· (1 ā’ 20%) = (1 ā’ 4%)

(4.12)

(1 + r0,1 ) Ā· (1 + r1,2 ) = (1 + r0,2 ) .

ļ¬le=yieldcurve.tex: RP

57

Section 4Ā·2. Annualized Rates of Return .

For every $100 of original investment, you only retain $96. The average rate of return of 0%

does not reļ¬‚ect this. The compounded and therefore annualized rate of return does:

ā

(4.13)

1 + r2 = (1 + r0,2 ) = 1 ā’ 4% = 1 ā’ 2.02% .

If you were an investment advisor and quoting your historical performance, would you rather

quote your average historical rate of return or your annualized rate of return? (Hint: The

industry standard is the average rate of return.)

Make sure to solve the following questions to gain more experience with compounding and

annualizing over diļ¬erent time horizons.

Solve Now!

Q 4.5 Assume that the two-year holding rate of return is 40%. The average rate of return is

therefore 20% per year. What is the annualized rate of return? Which is higher?

Q 4.6 Is the compounded rate of return higher or lower than the sum of the individual rates of

return? Is the annualized rate of return higher or lower than the average of the individual rates

of return? Why?

Q 4.7 Return to Question 4.3. What was the annualized rate of return on the S&P500 over these

twelve years?

Q 4.8 The following were the daily prices of an investment:

2-Jan-01 $1,283.27 4-Jan-01 $1,333.34 8-Jan-01 $1,295.86

3-Jan-01 $1,347.56 5-Jan-01 $1,298.35 9-Jan-01 $1,300.80

If returns had accumulated at the same rate over the entire 255 days of 2001, what would a

$100 investment in 2001 have turned into?

Q 4.9 If the total holding interest rate is 50% for a 5-year investment, what is the annualized

rate of return?

Q 4.10 If the per-year interest rate is 10% for each of the next 5 years, what is the annualized

total 5-year rate of return?

Q 4.11 If the annualized 5-year rate of return is 10%, what is the total 5-year holding rate of

return?

Q 4.12 If the annualized 5-year rate of return is 10%, and if the ļ¬rst yearā™s rate of return is 15%,

and if the returns in all other years are equal, what are they?

Q 4.13 There is always disagreement about what stocks are good purchases. The typical degree

of disagreement is whether a particular stock is likely to oļ¬er, say, a 10% (pessimist) or a 20%

(optimist) annualized rate of return. For a $30 stock today, what does the diļ¬erence in belief

between these two opinions mean for the expected stock price from today to tomorrow? (Assume

that there are 365 days in the year. Reļ¬‚ect on your answer for a moment, and recognize that a

$30 stock typically moves about Ā±$1 on a typical day. This is often called noise.)

ļ¬le=yieldcurve.tex: LP

58 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

4Ā·3. The Yield Curve

Let us now tackle the yield curve, which is also sometimes called the term structure of interest

The Yield Curve:

annualized interest rate rates. The yield curve is todayā™s average annualized interest (yield) that investments pay as

as a function of bond

a function of their time to maturity. If not clariļ¬ed further, the yield curve usually means

maturity.

investments in U.S. Treasuries, although it should more precisely be called the U.S. Treasuries

yield curve. Bond traders often graph other yield curves, tooā”such as the yield curve on bonds

that were issued by corporations rather than by the government.

4Ā·3.A. An Example: The Yield Curve in May 2002

Table 4.1. The Treasury Yield Curves in mid-2002

Maturity Apr 30 May 30 May 31

1 Month 1.77% 1.72% 1.72%

3 Month 1.77% 1.74% 1.73%

6 Month 1.91% 1.88% 1.89%

1 Year 2.35% 2.22% 2.26%

2 Year 3.24% 3.15% 3.20%

3 Year 3.83% 3.64% 3.65%

4 Year n/a 4.05% 4.06%

5 Year 4.53% 4.34% 4.36%

10 Year 5.11% 5.03% 5.04%

20 Year 5.74% n/a n/a

30 Year n/a% 5.60% 5.61%

The data for May 30, 2002, and May 31, 2002 were printed in the Wall Street Journal. The data for April 30, 2002,

was obtained from the U.S. Treasury website at www.ustreas.gov. As you can see, the yieldcurve changes every

dayā”though day-to-day changes are usually small.

For illustration, I am pretending that the Wall Street Journal yield curve is based on

zero bonds (which only have one ļ¬nal paymentā”these would be called Treasury STRIPS).

Although this is actually not perfectly correct (the WSJ curve is based on coupon bonds),

the differences are usually very small. This is also why the data on the Treasury website

is slightly differentā”in this example, the maximum difference is for the 10-year bond,

where it is 4 basis points.

The table in Table 4.1 shows the actual Treasury yield table on April 30, May 30, and May 31,

We will analyze the

actual yield curves at 2002. Figure 4.1 graphs the data from May 31, 2002. If you had purchased a 3-month bond

the end of May 2002.

at the end of the day on May 30, 2002, your annualized interest rate would have been 1.74%.

The following day, a 3-month bond had a yield that was one basis point lower. (In real life, the

90-day bond can also switch identity, because as bonds age, another bond may be closer to

90-days than yesterdayā™s 90-day bond.) If you had purchased a 30-year bond at the end of the

day on May 30, 2002, you would have received an annualized interest rate of 5.60% per year,

which is one basis point less than a 30-year bond would have been the following day.

Sometimes, it is necessary to determine an interest rate for a bond that is not listed. This is

You can interpolate

annualized interest rates usually done by interpolation. For example, if the 90-day Treasury note had a known interest

on the yield curve.

rate of 1.73% and the 93-day Treasury note had a known interest rate of 1.76%, a good interest

rate for an untraded 91-day Treasury note might be 1.74%.

ļ¬le=yieldcurve.tex: RP

59

Section 4Ā·3. The Yield Curve.

Figure 4.1. The Treasury Yield Curves on May 31, 2002

6

x

x

5

Annualized Rate (r), in %

x

x

4

x

x

3

x

2

x

x

1

0

0 50 100 150 200 250 300 350

Maturity in Months

This is the yieldcurve for May 31, 2002. The data is in the previous table.

Table 4.2. Relation Between Holding Returns, Annualized Returns, and Year-by-Year Re-

turns, By Formula and on May 31, 2002

Rates of Return

Maturity Total Holding Annualized Individually Compounded

1

(1 + 2.26%) = (1 + 2.26%) = (1 + 2.26%)

1 Year

= (1 + r1 )1

(1 + r0,1 ) = (1 + r0,1 )

= (1 + 3.20%)2

(1 + 6.50%) = (1 + 2.26%) Ā· (1 + 4.15%)

2 Year

2

(1 + r0,2 ) = (1 + r2 ) = (1 + r0,1 ) Ā· (1 + r1,2 )

= (1 + 3.65%)3

(1 + 11.35%) = (1 + 2.26%) Ā· (1 + 4.15%) Ā· (1 + 4.56%)

3 Year

3

(1 + r0,3 ) = (1 + r3 ) = (1 + r0,1 ) Ā· (1 + r1,2 ) Ā· (1 + r2,3 )

ļ¬le=yieldcurve.tex: LP

60 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

As notation for the annualized horizon-dependent interest rates, return to our earlier method,

The annualized interest

rate was higher for calling the two-year annualized interest rate r2 , the three-year annualized interest rate r3 , and

longer time periods.

so on. When you look at this particular yield curve, it is very clear how important it can be to

put a subscript on the annualized yields: the annualized yield varied drastically with maturity.

Just to summarizeā”we now have to be able to recognize a whole set of diļ¬erent interest rates:

holding rates of returns, such as r0,3 ; annualized rates of return, such as r3 ; and individual 1-

year annual interest rates that do not begin today, called forward rates, such as r1,2 . Table 4.2

relates the diļ¬erent types of returns, so you remember which is which. (Section 4Ā·7.A shows

you how you can construct the āindividually compoundedā column of this table from the āan-

nualizedā column that is the yield curve.) Aside, please do not forget that all the interest rates

in the yield curve themselves are just computed from the prevailing prices on corresponding

Treasury securities. It is much more intuitive to express the yield curve in this annualized

implied interest rate fashion than to give you all the Treasury security prices and let you do

the calculationsā”but the two are really one and the same.

4Ā·3.B. Compounding With The Yield Curve

On May 30, 2002, how much money did an investment of $500,000 into U.S. 2-Year notes (i.e., a

Computing the holding

period rate of return for loan to the U.S. government of $500,000) promise to return in two years? Refer to Table 4.1 on

2-Year bonds.

Page 58. Because the yield curve prints annualized rates of return, the total two-year holding

rate of return (as in Formula 4.4) is the twice compounded annualized rate of return,

r0,2 = (1 + 3.15%) Ā· (1 + 3.15%) ā’ 1 ā 6.4%

(4.14)

= (1 + r2 ) Ā· (1 + r2 ) ā’1 ,

so the $500,000 would turn into

CF2 ā (1 + 6.4%) Ā· $500, 000 ā $531, 996

(4.15)

= (1 + r0,2 ) Ā· .

CF0

(In the real world, you might have to pay a commission to arrange this transaction, so you

would end up with a little less.)

What if you invested $500,000 into 30-Year Treasuries? The 30-Year total rate of return would

Computing the holding

period rate of return for be

30-Year bonds.

(1 + r30 )30 ā’ 1

r0,30 =

(4.16)

30

= (1 + 5.60%) ā’ 1 ā 5.1276 ā’ 1 ā 412.76% .

Thus, your investment of CF0 = $500, 000 will turn into cash of CF30 ā $2, 563, 820 in 30 years.

Anecdote: Life Expectancy and Credit

Your life expectancy may be 80 years, but 30-year bonds existed even in an era when life expectancy was only

25 yearsā”at the time of Hammurabi, around 1700 B.C.E. (Hammurabi established the Kingdom of Babylon,

and is famous for the Hammurabi Code, the ļ¬rst known legal system.) Moreover, four thousand years ago,

Mesopotamians already solved interesting ļ¬nancial problems. A cuneiform clay tablet contains the oldest known

interest rate problem for prospective students of the ļ¬nancial arts. The student must ļ¬gure out how long it

takes for 1 mina of silver, growing at 20% interest per year, to reach 64 minae. Because the interest compounds

in an odd way (20% of the principal is accumulated until the interest is equal to the principal, and then it is

added back to the principal), the answer to this problem is 30 years, rather than 22.81 years. This is not an easy

problem to solveā”and it even requires knowledge of logarithms!

ļ¬le=yieldcurve.tex: RP

61

Section 4Ā·3. The Yield Curve.

4Ā·3.C. Yield Curve Shapes

What would a ļ¬‚at yield curve mean? It would mean that the interest rate was the same over A ļ¬‚at yield curve means

that the annualized

any time period. This scenario was the subject of the previous chapter. For example, at 5% per

interest rate is the same

annum and borrowing money for two years, the total (non-annualized) interest rate that would regardless of horizon.

have to be paid would be (1 + 5%) Ā· (1 + 5%) ā’ 1 = 10.25%. More generally, the interest rate over

any period can then be quickly computed as

(1 + r0,t ) = (1 + r0,1 )t . (4.17)

The yield curve is usually upward sloping. This means that longer-term interest rates are higher Yield Curves are often

upward sloping.

than shorter-term interest rates. The yield curve at the end of May 2002 was fairly steepā”

though not the steepest ever. Since 1934, the steepest yield curve (the diļ¬erence between the

long-term and the short-term Treasury rate) occurred in October 1992, when the long-term

interest rate was 7.3 percent and the short-term interest rate was 2.9 percentā”just as the

economy pulled out of the recession of 1991. Another oddity occurred in January 1940, when

the long-term interest rate was 2.3 percentā”but the short-term interest rate was practically

zero.

However, the yield curve is not always upward-sloping. If short-term rates are higher than long- They can also be

downward-sloping or

term rates, the yield curve is said to be downward sloping (or inverted). Figure 4.2 shows that

even be humped!

this was the case in December 1980 during a brief period of rapidly declining inļ¬‚ation rates

(and expansion in between two recessions). In fact, it is even possible that medium-term rates

are higher than both long-term and short-term ratesā”the yield curve is then called humped.

Inverted or humped yield curves are relatively rare.

Figure 4.2. History: One Inverted and One Humped Yield Curve

19

x x

9.5

18

Annualized Rate (r), in %

Annualized Rate (r), in %

9.4

17

9.3

16

x

x

9.2

15

x

9.1

x

x

14

9.0

x

x

x

13

x

x

8.9

x

x x

0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350

Maturity in Months Maturity in Months

December 1980 June 1979

Side Note: Economists have long wondered what they can learn from the shape of the yieldcurve. It appears

that it is a usefulā”though unreliable and noisyā”signal of where the economy is heading. Steep yield curves

often signal emergence from a recession. Inverted yield curves often signal an impending recession.

Another interesting question is what drives the demand and supply for credit, which is ultimately the determi-

nant of these interest rates. Economic research has shown that the Federal Reserve Bank has good inļ¬‚uence on

the short end of the Treasury curveā”by expanding and contracting the supply of money and short-term loans

in the economyā”but not much inļ¬‚uence on the long end of the Treasury curve. We will revisit this question

later in this chapter, and again in Chapter 6 in the context of inļ¬‚ation.

If you want to undertake your own research, you can ļ¬nd historical data at the St. Louis Federal Reserve Bank,

which maintains a database of historical interest rates at http://research.stlouisfed.org/fred. There are also the

Treasury Management Pages at http://www.tmpages.com/. Or you can look at SmartMoney.com for historical

yield curves. PiperJaļ¬ray.com has the current yield curveā”as do many other ļ¬nancial sites and newspapers.

bonds.yahoo.com/rates.html provides not only the Treasury yield curve, but also yield curves for other bonds

that will be discussed in the next section.

ļ¬le=yieldcurve.tex: LP

62 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

If you want to learn more about how to work with yield curves, donā™t forget about the optional

āforward interest ratesā section below.

Solve Now!

Q 4.14 Using information from a current newspaper or the WWW, what does an investment of

$1 in 30-year bonds yield in 30 years?

Q 4.15 Using information from a current newspaper or the WWW, what does an investment of

$1 in 1-year bonds yield in 1 year?

4Ā·4. Present Values With Time-Varying Interest Rates

In the previous chapter, you learned that present values allow you to express many future cash

The formula still looks

very similar. ļ¬‚ows in the same unit: cash today. With time-varying interest rates, nothing really changes.

The only novelty is that you can express the individual holding returns (e.g., 1 + r0,2 ) in terms

of the individual period interest rates (e.g., (1 + r0,1 ) Ā· (1 + r1,2 )). So, the Net Present Value

Formula can be rewritten as

NPV = PV( CF0 ) + PV( CF1 ) + + PV( CF3 ) + Ā· Ā· Ā·

PV( CF2 )

CF1 CF2 CF3

= + + + + Ā·Ā·Ā·

CF0

1 + r0,1 1 + r0,2 1 + r0,3

(4.18)

CF1 CF2

= + +

CF0

1 + r0,1 (1 + r0,1 ) Ā· (1 + r1,2 )

CF3

+ + Ā·Ā·Ā· .

(1 + r0,1 ) Ā· (1 + r1,2 ) Ā· (1 + r2,3 )

You must understand that the cost of capital is time-dependent. Just because a longer-term

project oļ¬ers a higher expected rate of return does not necessarily mean that it has a higher

NPV. Similarly, just because shorter-term borrowing allows ļ¬rms to pay a lower expected rate

of return does not necessarily mean that this creates value. This is also why the U.S. Treasury

is not relying exclusively on short-term borrowing. A higher expected rate of return required

for longer-term payments is (usually) a fact of life.

Important: The appropriate cost of capital depends on how long-term the pro-

ject is. The economy-wide yield curve is typically upward-sloping. Similarly, short-

term corporate projects usually have lower costs of capital than long-term projects.

And, similarly, corporations usually face lower costs of capital (expected rates of

return oļ¬ered to creditors) if they borrow short-term rather than long-term.

Let us return to our earlier example on Page 26, where you had a $10 payment in year 1 and

Present Values are alike

and thus can be added, an $8 payment in year 2, but assume that the 5-year annualized interest rate is 6% per annum

subtracted, compared,

and therefore higher than the 1-year interest rate of 5%. In this case,

etc.

$10

PV( $10 in one year ) = ā $9.52

1 + 5%

(4.19)

$8

PV( $8 in ļ¬ve years ) = ā $5.98 .

(1 + 6%)5

ļ¬le=yieldcurve.tex: RP

63

Section 4Ā·4. Present Values With Time-Varying Interest Rates.

It follows that the projectā™s total value today (time 0) would now be $15.50. If the project still

costs $12, its net present value is

$10 $8

NPV = ā’$12 + + ā $3.50

1 + 5% (1 + 6%)5

(4.20)

CF1 CF5

NPV = + + = .

CF0 NPV

1 + r0,1 1 + r0,5

You can also rework the project from Table 2.5 on Page 28, but you can now use a hypothetical Here is a typical NPV

Example.

current term structure of interest that is upward sloping. It requires an interest rate of 5% over

1 year, and 0.5% annualized interest more for every year, so it is 7% annualized for the 5-year

cash ļ¬‚ow. Table 4.3 works out the value of your project. The valuation method works the same

way as it did earlierā”you only have to use diļ¬erent interest rates now.

Table 4.3. Hypothetical Project Cash Flow Table

Project Interest Rate Discount Present

Time Cash Flow In Year Compounded Factor Value

1

t rtā’1,t r0,t PV( CFt )

CFt

1 + r0,t

Today ā“$900 any 0.0% 1.000 ā“$900.00

Year +1 +$200 5.0% 5.0% 0.952 $190.48

Year +2 +$200 5.5% 11.3% 0.898 $179.69

Year +3 +$400 6.0% 19.1% 0.840 $335.85

Year +4 +$400 6.5% 28.6% 0.778 $311.04

Year +5 ā“$100 7.0% 40.2% 0.713 ā“$71.33

Net Present Value (Sum): $45.73

Annualized interest rates apply only within this one year. They are perfectly known today.

4Ā·4.A. Valuing A Coupon Bond With A Particular Yield Curve

Let us now work a more realistic exampleā”determining the price of a coupon bond, for which

payments are 100% guaranteed, just like payments on Treasury bonds themselves. We will

recycle the 3% coupon bond example from Section 3Ā·3.B. Of course, if you wanted to value a

corporate project with risky cash ļ¬‚ows instead of a bond, it might be more diļ¬cult to determine

the appropriate inputs (cash ļ¬‚ows and discount rates), but the valuation method itself would

proceed in exactly the same way. After all, a bond is just like any other corporate projectā”an

upfront investment followed by subsequent inļ¬‚ows.

First, recall the payment pattern of your bond, which comes from the deļ¬nition of what a Step 1: Write down the

projectā™s payment

3%-level semi-annual coupon bond is.

pattern.

Due Bond Due Bond

Year Date Payment Year Date Payment

0.5 Nov 2002 $1,500 3.0 May 2005 $1,500

1.0 May 2003 $1,500 3.5 Nov 2005 $1,500

1.5 Nov 2003 $1,500 4.0 May 2006 $1,500

2.0 May 2004 $1,500 4.5 Nov 2006 $1,500

2.5 Nov 2004 $1,500 5.0 May 2007 $101,500

ļ¬le=yieldcurve.tex: LP

64 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

Second, ļ¬nd the appropriate rates of return to use for discounting. Because your bond is

Step 2: ļ¬nd the

appropriate costs of assumed default-free, it is just as good as a government bond (in our perfect world). Thus, you

capital.

can use the government yield curve to extract appropriate discount factors. Assume it is May

30, 2002, so you can use the yield curve from Table 4.1 on Page 58.

Maturity Yield Maturity Yield Maturity Yield

3 Month 1.74% 2 Year 3.15% 5 Year 4.34%

6 Month 1.88% 3 Year 3.64% 10 Year 5.03%

1 Year 2.22% 4 Year 4.05% 30 Year 5.60%

To use the PV formula to value your bond, you need to ļ¬nd the appropriate discount factors.

Begin by computing the holding rates of return from the yield curve, using the methods from

Section 4Ā·1. For example, the 6-month and 2-year holding rates of return are computed as

1 + r0,0.5 = (1 + 1.88%)0.5 ā (1 + 0.94%)

(1 + 3.15%)2 (4.21)

1 + r0,2 = ā (1 + 6.40%)

(1 + rt )t

1 + r0,t = .

The table of holding rates of return that corresponds to the yield curve is

Maturity Yield Maturity Yield Maturity Yield

3 Month 2 Year 6.40% 5 Year 23.67%

not needed

6 Month 0.94% 3 Year 11.32% 10 Year not needed

1 Year 2.22% 4 Year 17.21% 30 Year not needed

But how do you obtain a holding rate of return for the coupon that will be paid in 18 months?

You do not know the annualized 18-month interest rate, but you do know that the 1-year

annualized interest rate is 2.22% and the 2-year annualized interest rate is 3.15%. So it is

reasonable to guess that the 1.5 year annualized interest is roughly the average interest rate

of the 1-year and 2-year annualized interest ratesā”about 2.7%. Therefore, you would estimate

the 1.5 year holding rate of return to be

1 + r0,1.5 ā (1 + 2.7%)1.5 ā (1 + 4.08%)

(4.22)

t

1 + r0,t = (1 + rt ) .

You have to do similar interpolations for the coupon payments in 2.5, 3.5 and 4.5 years. Collect

this informationā”our payments, annualized interest rates, and equivalent holding interest

ratesā”into one table:

Due Bond Annual. Holding

Year Date Payment Interest Interest

0.5 Nov 2002 $1,500 1.88% 0.94%

1.0 May 2003 $1,500 2.22% 2.22%

ā2.7%

1.5 Nov 2003 $1,500 4.08%

2.0 May 2004 $1,500 3.15% 6.40%

ā3.4%

2.5 Nov 2004 $1,500 8.72%

3.0 May 2005 $1,500 3.64% 11.32%

ā3.8%

3.5 Nov 2005 $1,500 13.94%

4.0 May 2006 $1,500 4.05% 17.21%

ā4.2%

4.5 Nov 2006 $1,500 20.34%

5.0 May 2007 $101,500 4.34% 23.67%

Third, compute the discount factors, which are just 1/(1 + r0,t ), and multiply each future

Step 3: Compute the

discount factor is payment by its discount factor. This is the present value (PV) of each bond payment, and the

1/(1 + r0,t ).

overall PV of your bond.

ļ¬le=yieldcurve.tex: RP

65

Section 4Ā·5. Why is the Yield Curve not Flat?.

Due Bond Annual. Holding Discount Present

Year Date Payment Interest Interest Factor Value

0.5 Nov 2002 $1,500 1.88% 0.94% 0.991 $1,486.03

1.0 May 2003 $1,500 2.22% 2.22% 0.978 $1,467.42

ā2.7%

1.5 Nov 2003 $1,500 4.08% 0.961 $1,441.20

2.0 May 2004 $1,500 3.15% 6.40% 0.940 $1,409.77

ā3.4%

2.5 Nov 2004 $1,500 8.72% 0.920 $1,379.69

3.0 May 2005 $1,500 3.64% 11.32% 0.898 $1,347.47

ā3.8%

3.5 Nov 2005 $1,500 13.94% 0.878 $1,316.48

4.0 May 2006 $1,500 4.05% 17.21% 0.853 $1,279.75

ā4.2%

4.5 Nov 2006 $1,500 20.34% 0.831 $1,246.47

5.0 May 2007 $101,500 4.34% 23.67% 0.809 $82,073.26

Sum $94,447.55

Therefore, you would expect this 3% semi-annual level-coupon bond to be trading for $94,447.55 Common naming

conventions for this

todayā”because this is lower than the bondā™s principal repayment of $100,000, this bond is

type of bond: coupon

called a discount bond. rate is not interest rate!

4Ā·5. Why is the Yield Curve not Flat?

There is no necessary reason why capital should be equally productive at all times. For example, There is no reason why

interest rates have to be

in agrarian societies, capital could be very productive in summer (and earn a rate of return of

the same in all periods.

3%), but not in winter (and earn a rate of return of only 1%). This does not mean that investment

in summer is a better deal or a worse deal than investment in winter, because cash in winter is

not the sameā”not as valuableā”as cash in summer, so the two interest rates are not comparable.

You could not invest winter money at the 3% interest rate you will be able to invest it with 6

months later.

But although seasonal eļ¬ects do inļ¬‚uence both prices and rates of return on agricultural com- Longer-term Treasury

bonds probably have

modities, and although the season example makes it clear that capital can be diļ¬erently produc-

higher yields because

tive at diļ¬erent times, it is not likely that seasonality is the reason why 30-year Treasury bonds they are riskierā”though

in May 2002 paid 5.6% per annum, and 6-month Treasury notes paid only 1.9% per annum. So it could also have been

investment

why is it that the yield curve was so steep? There are essentially three explanations:

opportunities that are

better in the far-away

future than they are

1. The 30-year bond is a much better deal than the 1-year bond. This explanation is highly

today.

unlikely. The market for Treasury bond investments is close to perfect, in the sense that

we have used the deļ¬nition. It is very competitive and eļ¬cientā”concepts that we will

investigate more in Chapter 6. If there was a great deal to be had, thousands of traders

would have already jumped on it. So, more likely, the interest rate diļ¬erential does not

overthrow the old tried-and-true axiom: you get what you pay for. It is just a fact of life

that investments for which the interest payments are tied down for 30 years must oļ¬er

higher interest rates now.

It is important that you recognize that your cash itself is not tied down if you invest in

a 30-year bond, because you can of course sell your 30-year bond tomorrow to another

investor if you so desire.

2. Investors expect to be able to earn much higher interest rates in the future. For example, if

the interest rate r0,1 is 2% and the interest rate r1,2 is 10%, then r0,2 = (1+2%)Ā·(1+10%) ā

1 + 12%, or r2 = 5.9%. If you graph rT against T , you will ļ¬nd a steep yield curve, just as

you observed. So, higher future interest rates can cause much steeper yield curves.

However, I am cheating. This explanation is really no diļ¬erent from my āseasonsā expla-

nation, because I have given you no good explanation why investment opportunities were

expected to be much better in May 2032 than they were in May 2002. I would need to

give you an underlying reason. One particular such reason may be that investors believe

that money will be worth progressively less. That is, even though they can earn higher

interest rates over the long run, they also believe that the price inļ¬‚ation rate will increase.

Inļ¬‚ationā”a subject of Chapter 6ā”erodes the value of higher interest rates, so interest

rates may have to be higher in the future merely to compensate investors for the lesser

value of their money in the future.

ļ¬le=yieldcurve.tex: LP

66 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

However, the empirical evidence suggests that the yield curve is not a good predictor

of future interest rates, except on the very shortest horizons (a month or less). So, the

expectation of higher interest rates is not the most likely cause for the usually upward

sloping curve in the real world.

3. Long-term bonds might somehow be riskier than short-term bonds, so investors only want

to buy them if they get an extra rate of return. Although we have yet to cover uncertainty

more systematically, you can gain some intuition by considering the eļ¬ects of changes in

economy-wide interest rates on short-term bonds vs. long-term bonds. This is the plan

of the remainder of this section.

The empirical evidence indeed suggests that it is primarily compensation for taking more

risk with long-term bonds than short-term bonds that explains why long-term bonds have

higher yields than short-term bonds. That is, investors seem to earn higher expected rates

of return on average in long-term bonds, because these bonds are riskier (at least in the

interim).

4Ā·5.A. The Eļ¬ect of Interest Rate Changes on Short-Term and Long-Term Treasury Bond

Values

Why are 30-year bonds riskier than 1-year bonds? Of course, repayment is no less certain with

Our agenda is to explore

the risk of interim 30-year Treasury bonds than 1-year Treasury bonds. (This would be an issue of concern if you

interest rate changes.

were to evaluate corporate projects rather than Treasuries: long-term corporate bonds are often

riskier than short-term corporate bondsā”most ļ¬rms are unlikely to go bankrupt this week, but

fairly likely to go bankrupt over a multi-decade time horizon.) Instead of non-payment risk, the

issue here is that economy-wide bond prices (interest rates) can change in the interim, and the

eļ¬ects of interest rate changes can be much more dramatic on 30-year bonds than on 1-year

bonds.

First, the effect of a

10bp point change on

the 30-year bond.

The 30-Year Bond: Letā™s compute the value of a $1,000 30-year zero bond today at the pre-

vailing 5.60% interest rate. It is $1, 000/1.05630 ā $195.02. You already know that when

prevailing interest rates go up, the prices of outstanding bonds drop and you will have

lost money. Now, if interest rates increase by 10 basis points to 5.7%, the bond value

decreases to $1, 000/1.05730 ā $189.56. If interest rates decrease by 10 basis points to

5.5%, the bond value increases to $1, 000/1.05530 ā $200.64. Thus, the eļ¬ect of a 10

basis point increase in the prevailing 30-year yield induces an immediate percent change

(a return) in the value of your bond of

V (r30 = 5.5%) ā’ V (r30 = 5.6%) $200.64 ā’ $195.02

r= = ā +2.88%

V (r30 = 5.6%) $195.02

(4.23)

V (r30 = 5.7%) ā’ V (r30 = 5.6%) $189.56 ā’ $195.02

r= = ā ā’2.80%

V (r30 = 5.6%) $195.02

For every $1 million you invest in 30-year bonds, you expose yourself to a $29,000 risk

for a 10-basis point yield change in the economy.

Second, the effect of a

10bp point change on

The 1-Year Bond: To keep the example identical, assume that the 1-year bond also has an

the 1-year bond.

interest rate of 5.6%. In this case, the equivalent computations for the value of a 1-year

bond are $946.97 at 5.6%, $947.87 at 5.5%, and $946.07 at 5.7%. Therefore, the equivalent

change in value is

V (r1 = 5.5%) ā’ V (r1 = 5.6%) $952.38 ā’ $946.97

r= = ā +0.09%

V (r1 = 5.6%) $946.97

(4.24)

V (r1 = 5.7%) ā’ V (r1 = 5.6%) $946.07 ā’ $946.07

r= = ā ā’0.09%

V (r1 = 5.6%) $946.07

ļ¬le=yieldcurve.tex: RP

67

Section 4Ā·5. Why is the Yield Curve not Flat?.

So for every $1 million you invest in 1-year bonds, you expose yourself to a $900 risk for

a 10-basis point yield change in the economy.

It follows that the value eļ¬ect of an equal-sized change in prevailing interest rates is more Comparison

severe for longer term bonds. It follows, then, that if the bond is due tomorrow, there is very

little havoc that an interest rate change can wreak.

This brings us to an important insight: Treasury bonds are risk-free in the sense that they In the interim, T-bonds

are not risk-free!

cannot default (fail to return the promised payments). But they are risky in the sense that

interest changes can change their value. Only the most short-term Treasury bills (say, due

overnight) can truly be considered risk-freeā”virtually everything else is risky.

Important: Though āļ¬xed income,ā even a Treasury bond does not guarantee

a āļ¬xed rate of returnā over horizons shorter than the maturity: day to day, long-

term bonds are generally riskier investments than short-term bills.

But, if you really need cash only in 30 years, is this not just a paper loss? This is a cardinal āOnlyā a paper loss: A

cardinal error!

logical error many investors commit. By committing your million dollars one day earlier, you

would have lost $29,000 of your net worth in one day! Put diļ¬erently, waiting one day would

have saved you $29,000 or allowed you to buy the same item for $29,000 less. Paper money

is actual wealth. Thinking paper losses are any diļ¬erent from actual losses is a common but

capital error.

Important: āPaper lossesā are actual losses.

The primary exception to this rule is that realized gains and losses have diļ¬erent tax implica-

tions than unrealized gains and lossesā”a subject which we will discuss in Chapter 6.

Digging Deeper: I have pulled two tricks on you. First, in the real world, it could be that short-term economy-

wide interest rates typically experience yield shifts of plus or minus 100 basis points, while long-term economy-wide

interest rates never move. If this were true, long-term bonds could even be safer. But trust meā”even though the

volatility of prevailing interest rates in 20-year bonds is smaller than that of 1-year bonds, it is not that much

smaller. As a consequence, the typical annual variability in the rate of return of an investment in 20-year Treasury

bonds is higher (around 10%) than the typical variability in the rate of return of an investment in 1-month Treasury

notes (around 3%). Long-term Treasury securities are indeed riskier.

Second, when I quoted you value losses of $29,000 and $900, I ignored that between today and tomorrow, you

would also earn one dayā™s interest. On a $1,000,000 investment, this would be about $150. If you had invested

the money in 1-day Treasury bills at 1.7% instead of 30-year bonds, you would have only received about $30.

Strictly speaking, this $120 favors the long-term bond and thus should be added when comparing investment

strategiesā”but it is only about 1 basis point, and so for a quick-and-dirty calculation such as ours, ignoring it was

reasonable.

Solve Now!

Q 4.16 Using information from a current newspaper or the WWW, what is todayā™s annualized

rate of return on a 10-year bond?

Q 4.17 Using information from a current newspaper or the WWW, what is todayā™s total rate of

return on a 10-year bond over the 10-year holding period?

Q 4.18 If you invest $500,000 at todayā™s total rate of return on a 30-day Treasury note, what

will you end up with?

ļ¬le=yieldcurve.tex: LP

68 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

4Ā·6. The Yield To Maturity (YTM)

In Section 4Ā·2, you learned how to annualize rates of return, so that you could better understand

We want a āsort-of

average interest rateā the rate at which two diļ¬erent investments accumulate wealth. However, there was only one

that is implicit in future

payment involved. What do you do if each bond has many diļ¬erent payments? For example,

cash ļ¬‚ows.

what is the interest rate on a bond that costs $100,000 today, and pays oļ¬ $5,000 in 1 year,

$10,000 in 2 years, and $120,000 in 3 years? This may be an irregular coupon bond, but it is

not an illegal one. How should you even name this bondā”is there something like an āaverageā

interest rate implicit in these cash ļ¬‚ows? Is this bond intrinsically more similar to a bond

oļ¬ering a 4% rate of return or a bond oļ¬ering a 6% rate of return? Note that this has nothing

to do with the prevailing economy-wide yield curve. Our question is purely one of wanting to

characterize the cash ļ¬‚ows that are implicit to the bond itself. The answer is not obvious at

allā”until you learn it. The yield-to-maturity gives a sort of āaverage rate of returnā implicit in

many bond cash ļ¬‚ows.

Important: The Yield to Maturity is the quantity YTM, which, given a complete

set of bond cash ļ¬‚ows, solves the NPV equation set to zero,

CF1 CF2 CF3

0 = CF0 + + + + ... (4.25)

1 + YTM (1 + YTM)2 (1 + YTM)3

So, in this case, you want to solve

An example of solving

the YTM equation.

$5, 000 $10, 000 $120, 000

0 = ā’$100, 000 + + + (4.26)

.

1 + YTM (1 + YTM)2 (1 + YTM)3

In general, you solve this equation by trial and error. Start with two values, say 5% and 10%.

$5, 000 $10, 000 $120, 000

ā’$100, 000 + + + ā $17, 493 ,

1 + 5% (1 + 5%)2 (1 + 5%)3

(4.27)

$5, 000 $10, 000 $120, 000

ā’$100, 000 + + + ā $2, 968 .

1 + 10% (1 + 10%)2 (1 + 10%)3

To reach zero, you need to slide above 10%. So, try 11% and 12%,

$5, 000 $10, 000 $120, 000

ā’$100, 000 + + + ā ,

$363

1 + 11% (1 + 11%)2 (1 + 11%)3

(4.28)

$5, 000 $10, 000 $120, 000

ā’$100, 000 + + + ā ā’$2, 150 .

1 + 12% (1 + 12%)2 (1 + 12%)3

Ok, the solution is closer to 11%. Some more trial and error reveals

$5, 000 $10, 000 $120, 000

ā’$100, 000 + + + ā0. (4.29)

1 + 11.14255% (1 + 11.14255%) (1 + 11.14255%)3

2

So, the cash ļ¬‚ows of your bond with payments of $5,000 in 1 year, $10,000 in 2 years, and

$120,000 in 3 years have an embedded sort-of-average interest rateā”a yield to maturityā”that

is equal to 11.14%. There are also bonds that the corporation can call back in before maturity.

In this case, it is not uncommon to compute a YTM for such a bond assuming the ļ¬rm will do

so, then called a Yield-to-Call.

You can think of YTM as a generalization of the narrower interest rate concept. If there is only

A YTM is (usually) not an

interest rate! one cash inļ¬‚ow and one cash outļ¬‚owā”as is the case for a zero bondā”then the YTM is the same

as the annualized interest rate. However, a rate of return is deļ¬ned by exactly two cash ļ¬‚ows,

so it is meaningless to talk about it when there are multiple cash ļ¬‚ows. In contrast, the YTM

can handle multiple cash ļ¬‚ows just ļ¬ne. Although it may help your intuition to think of the

YTM as a āsort ofā average interest rate that is embedded in a bondā™s cash ļ¬‚ows, you should be

ļ¬le=yieldcurve.tex: RP

69

Section 4Ā·6. The Yield To Maturity (YTM).

clear that the YTM is not an interest rate. (An interest rate is a YTM, but not vice-versa.) Instead,

a YTM is a characteristic deļ¬ned by a cash ļ¬‚ow pattern.

Should you purchase this bond? The answer is yes if and only if this bond does not have a If the yield curve is ļ¬‚at,

YTM can substitute for

negative NPV. Fortunately, YTM can often provide the same information. If the yield curve is

NPV as a capital

uniformly below the bondā™s YTM, then the bond is a positive NPV project. So, if all prevailing budgeting tool.

economy-wide interest rates were 11%, and your bondā™s YTM is 11.14%, then this bond would

be a positive NPV project and you should buy it. If all prevailing economy-wide interest rates

were 12%, and your bondā™s YTM is 11.14%, then you should not buy this bond. Unfortunately,

when the prevailing yield curve is not uniformly above or below the YTM (e.g., if it is 11% on the

1-year horizon climbing to 12% on the 3-year horizon), YTM cannot tell you whether to purchase

the bondā”though it still gives a nice characterization of bond payments. Instead, you have to

ńņš. 4 |