<<

. 4
( 39)



>>

12. CF1 /(r ’ g). The ¬rst cash ¬‚ow occurs next period, not this period.
You get $5 today, and next month you will receive a payment of (1 + π )·CF = 1.001 · $5 = $5.005. The
13.
growing perpetuity is worth PV = CF1 /(r ’ g) = $5.005/(0.5% ’ 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 ’ 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%’
0.5%) ≈ $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 ’ E/P = 12% ’ $5/$100 = 7% per annum.


18. Remembering this formula is not as important as remembering the other growing perpetuity formula. The
1 ’ [1/(1 + r )]T
annuity formula is CF1 · .
r
19.
1 ’ [1/(1 + r )]T 1 ’ [1/(1 + 0.005)]360
CF1 · = $5 ·
r 0.005
1 ’ 0.166
= $5 · ≈ $833.96
0.005
20. You need to solve
$5, 000
1 ’ 1/(1 + r )120 (3.33)
$500, 000 = .
r
The solution is r ≈ 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 o¬ earning 5% elsewhere.
¬le=perpetuities.tex: RP
51
Section 3·4. Summary.

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

1 ’ [1/(1 + r )]T
PV = CF1 ·
r

1 ’ [1/(1 + 0.005)]15·12=180
$1, 000 = CF1 ·
0.005
$1, 000 = CF1 · 118.504 ⇐ CF1 ≈ $8.44

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 ’ 1 = 6.12%. Therefore, the 4-“quartity” is worth 75/.0612·[1 ’ 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/0.05·(1’1/1.0541 ) = 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 ’ [1/(1 + r )]T
PV(Coupons) = CFt+1 ·
r
1 ’ [1/(1 + 0.05)]40
= $1, 500 ·
0.05
(3.34)
1 ’ [1/(1 + 0.05)]40
= $1, 500 ·
0.05

≈ .
$25, 739

The ¬nal payment is worth

$100, 000
PV(Principal Repayment) = ≈ $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 ’ 1 = 5%. Now, de¬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 ’ 1. For example, the 30 months interest rate is
(1 + 2.47%)5 ’ 1 = 12.97%.
26.
(1 + 0.02)35
$4, 000/(0.08 ’ 0.02) · 1 ’ ≈ $57, 649 . (3.36)
(1 + 0.08)35




(All answers should be treated as suspect. They have only been sketched, and not been checked.)
¬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)
+ + ··· + + ··· = .
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)
+ + ··· + + ··· = .
t
2
1+r (1 + r ) (1 + r ) r
Multiply (3.38) by (1 + r )

(1 + r )
1 1
(3.39)
1+ + ··· + + ··· = .
(1 + r )t’1
(1 + r ) r
Subtract (3.39) from (3.38),
(1 + r ) 1
(3.40)
1= ’
r r
which simpli¬es to be a true statement.
A Growing Perpetuity We know from the simple perpetuity formula that
∞ ∞
CF CF CF CF
= = . (3.41)

(1 + r )t ft f ’1
r
t=1 t=1

Return to the de¬nition of a growing perpetuity, and pull out one (1 + g) factor from its cash ¬‚ows,
∞ ∞ ∞
C · (1 + g)t’1 C · (1 + g)t
1 1 CF
= · = · . (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 ¬rst formula. Then
Let
1+g
±  ± 
 

   
1 CF 1 CF
· = · , (3.43)
1+r t
   ’ 1
1+r
1+g 1+g
t=1 
1+g
1+g

and simplify this,
± 
 
  C · (1 + g)
1 CF 1 CF
= · = · = . (3.44)
 
(1+r )’(1+g)
1+g 1+g r ’g r ’g
 
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 ¬rst annuity and sell
the second annuity, you will receive $10 each year for ¬ve years, and $0 in every year thereafter.

···
0 1 2 3 4 5 6 7 8
···
Perpetuity 1 +$10 +$10 +$10 +$10 +$10 +$10 +$10 +$10
equivalent to +$10/r
···
Perpetuity 2 “$10 “$10 “$10
equivalent to “$10/r
Net Pattern +$10 +$10 +$10 +$10 +$10
equivalent to +$10/r “$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 = ’ ·
5
(1 + r )
r r (3.45)
C C C
1 1
= ’ · = · 1’ ,
(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 ¬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
period”if a 30-year bond o¬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 o¬ered an interest rate of 5.6% per year, while a 1-year U.S. Treasury bond o¬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 traders”who work with time-dependent
interest rates every day”but also for companies that are comparing short-term and long-term
projects or short-term and long-term ¬nancing costs. After all, investors can earn higher rates
of return if instead of giving money to your ¬rm™s long-term projects, they invest in longer-
term Treasury bonds. Thus, if two corporate projects have di¬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 ¬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 di¬er. This chapter then takes a digression”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 ¬nished the rest of the book.




53
¬le=yieldcurve.tex: LP
54 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

4·1. Time-Varying Rates of Return

We now switch direction and make the world a bit more realistic”we allow rates of return
A second compounding
example. to di¬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 ¬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 ) · (1 + r1,2 )
(4.1)
= (1 + 20%) · (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 “one-plus formula” for each interest rate (subtracting 1 at the end). It
periods.
now can also help answer questions such as, “If 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?” The answer is

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

(1 + r1,4 ) = (1 + r1,2 ) · (1 + r2,3 ) · (1 + r3,4 )
(4.2)
= (1 + 30%) · (1 + 40%) · (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 ¬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 ¬rst year, overall it lost
half of its value. What was the project™s rate of return after the ¬rst year?


Q 4.3 From 1991 to 2002, the stock market (speci¬cally, the S&P500) had the following annual
rates of return:

rS&P500 rS&P500
˜ ˜
Year Year
+0.2631 +0.3101
1991 1997
+0.0446 +0.2700
1992 1998
+0.0706 +0.1953
1993 1999
’0.0154 ’0.1014
1994 2000
+0.3411 ’0.1304
1995 2001
+0.2026 ’0.2337
1996 2002


What was your rate of return over these 12 years? Over the ¬rst 6 years and over the second 6
years?
¬le=yieldcurve.tex: RP
55
Section 4·2. Annualized Rates of Return .

Q 4.4 A project lost one third of its value the ¬rst year, then gained ¬fty percent of its value, then
lost two thirds of its value, and ¬nally doubled in value. What was the overall rate of return?




4·2. 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 “miles per sixty minutes,” not in “yards per ninety-three minutes.” 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 year”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 “sort-of-average measure of
performance.”
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 ≈ 57.7%,
Three-Year Total
because if you earned 57.7% per year, you would have ended up with (1 + 57.7%) · (1 + 57.7%) · Holding Return.
(1 + 57.7%) ’ 1 = 287%, not 173%. This incorrect answer of 57.7% ignores the compounded
interest on the interest that you would earn after the ¬rst year and second year. Instead, you
need to ¬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 ¬nding a
three-year annualized
order to end up with a total rate of return of 173%. To ¬nd r3 , solve the equation
interest rate. Solution:
Take the N-th Root of
the total return (N is
(1 + r3 ) · (1 + r3 ) · (1 + r3 ) = (1 + 173%)
number of years).
(4.3)
(1 + r3 ) · (1 + r3 ) · (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 2·3 reviews powers, exponents and
logarithms):

3
(1 + r3 ) = (1 + 173%)(1/3) = 1 + 173% ≈ 1.3976
(4.5)
(1/t) t
(1 + r0,t ) = 1 + r0,t = (1 + rt ).

Con¬rm with your calculator that r3 ≈ 39.76%,

(4.6)
(1 + 39.76%) · (1 + 39.76%) · (1 + 39.76%) ≈ (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 di¬erent that you will
intuitively immediately register whether r0,3 or r3 is meant.
¬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 ¬rst step is computing the total holding rate of
returns.
return. By Formula 2.2, the total 30-year rate of return is
$240 ’ $100
r0,30 = = 140%
$100
(4.8)
CF30 ’ CF0
r0,30 = .
CF0

The annualized rate of return is the rate r30 , which, if compounded for 30 years, o¬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,

30
(1 + r30 ) = (1 + 140%)1/30 = 1 + 140% ≈ 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 ≈ Adding.
Annualizing ≈ 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 di¬erence is the interest on the interest.
Now presume that you have an investment that doubled in value in the ¬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 ’
$200)/$200 = ’50%. Therefore, the average rate of return would be [+100% + (’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%) · (1 ’ 50%) = 1 + 0%
(4.11)
(1 + r0,1 ) · (1 + r1,2 ) = (1 + r0,2 ) .

Therefore, the annualized rate of return is also 0%. Conversely, an investment that produces
+20% followed by ’20% has an average rate of return of 0%, but leaves you with

(1 + 20%) · (1 ’ 20%) = (1 ’ 4%)
(4.12)
(1 + r0,1 ) · (1 + r1,2 ) = (1 + r0,2 ) .
¬le=yieldcurve.tex: RP
57
Section 4·2. Annualized Rates of Return .

For every $100 of original investment, you only retain $96. The average rate of return of 0%
does not re¬‚ect this. The compounded and therefore annualized rate of return does:

(4.13)
1 + r2 = (1 + r0,2 ) = 1 ’ 4% = 1 ’ 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 di¬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 ¬rst year™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 o¬er, say, a 10% (pessimist) or a 20%
(optimist) annualized rate of return. For a $30 stock today, what does the di¬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. Re¬‚ect on your answer for a moment, and recognize that a
$30 stock typically moves about ±$1 on a typical day. This is often called noise.)
¬le=yieldcurve.tex: LP
58 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

4·3. 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 today™s average annualized interest (yield) that investments pay as
as a function of bond
a function of their time to maturity. If not clari¬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, too”such as the yield curve on bonds
that were issued by corporations rather than by the government.


4·3.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
day”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 ¬nal payment”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 different”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 yesterday™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%.
¬le=yieldcurve.tex: RP
59
Section 4·3. 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%) · (1 + 4.15%)
2 Year
2
(1 + r0,2 ) = (1 + r2 ) = (1 + r0,1 ) · (1 + r1,2 )
= (1 + 3.65%)3
(1 + 11.35%) = (1 + 2.26%) · (1 + 4.15%) · (1 + 4.56%)
3 Year
3
(1 + r0,3 ) = (1 + r3 ) = (1 + r0,1 ) · (1 + r1,2 ) · (1 + r2,3 )
¬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 summarize”we now have to be able to recognize a whole set of di¬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 di¬erent types of returns, so you remember which is which. (Section 4·7.A shows
you how you can construct the “individually compounded” column of this table from the “an-
nualized” 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 calculations”but the two are really one and the same.


4·3.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%) · (1 + 3.15%) ’ 1 ≈ 6.4%
(4.14)
= (1 + r2 ) · (1 + r2 ) ’1 ,

so the $500,000 would turn into

CF2 ≈ (1 + 6.4%) · $500, 000 ≈ $531, 996
(4.15)
= (1 + r0,2 ) · .
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 ’ 1
r0,30 =
(4.16)
30
= (1 + 5.60%) ’ 1 ≈ 5.1276 ’ 1 ≈ 412.76% .

Thus, your investment of CF0 = $500, 000 will turn into cash of CF30 ≈ $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 years”at the time of Hammurabi, around 1700 B.C.E. (Hammurabi established the Kingdom of Babylon,
and is famous for the Hammurabi Code, the ¬rst known legal system.) Moreover, four thousand years ago,
Mesopotamians already solved interesting ¬nancial problems. A cuneiform clay tablet contains the oldest known
interest rate problem for prospective students of the ¬nancial arts. The student must ¬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 solve”and it even requires knowledge of logarithms!
¬le=yieldcurve.tex: RP
61
Section 4·3. The Yield Curve.

4·3.C. Yield Curve Shapes

What would a ¬‚at yield curve mean? It would mean that the interest rate was the same over A ¬‚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%) · (1 + 5%) ’ 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 steep”
though not the steepest ever. Since 1934, the steepest yield curve (the di¬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 percent”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 percent”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 in¬‚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 rates”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 useful”though unreliable and noisy”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 in¬‚uence on
the short end of the Treasury curve”by expanding and contracting the supply of money and short-term loans
in the economy”but not much in¬‚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 in¬‚ation.
If you want to undertake your own research, you can ¬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. PiperJa¬ray.com has the current yield curve”as do many other ¬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.
¬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, don™t forget about the optional
“forward interest rates” 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?




4·4. 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. ¬‚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 ) · (1 + r1,2 )). So, the Net Present Value
Formula can be rewritten as

NPV = PV( CF0 ) + PV( CF1 ) + + PV( CF3 ) + · · ·
PV( CF2 )

CF1 CF2 CF3
= + + + + ···
CF0
1 + r0,1 1 + r0,2 1 + r0,3
(4.18)
CF1 CF2
= + +
CF0
1 + r0,1 (1 + r0,1 ) · (1 + r1,2 )
CF3
+ + ··· .
(1 + r0,1 ) · (1 + r1,2 ) · (1 + r2,3 )

You must understand that the cost of capital is time-dependent. Just because a longer-term
project o¬ers a higher expected rate of return does not necessarily mean that it has a higher
NPV. Similarly, just because shorter-term borrowing allows ¬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 o¬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 ) = ≈ $9.52
1 + 5%
(4.19)
$8
PV( $8 in ¬ve years ) = ≈ $5.98 .
(1 + 6%)5
¬le=yieldcurve.tex: RP
63
Section 4·4. Present Values With Time-Varying Interest Rates.

It follows that the project™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 = ’$12 + + ≈ $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 ¬‚ow. Table 4.3 works out the value of your project. The valuation method works the same
way as it did earlier”you only have to use di¬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 rt’1,t r0,t PV( CFt )
CFt
1 + r0,t
Today “$900 any 0.0% 1.000 “$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 “$100 7.0% 40.2% 0.713 “$71.33
Net Present Value (Sum): $45.73


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




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

Let us now work a more realistic example”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 3·3.B. Of course, if you wanted to value a
corporate project with risky cash ¬‚ows instead of a bond, it might be more di¬cult to determine
the appropriate inputs (cash ¬‚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 project”an
upfront investment followed by subsequent in¬‚ows.
First, recall the payment pattern of your bond, which comes from the de¬nition of what a Step 1: Write down the
project™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
¬le=yieldcurve.tex: LP
64 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

Second, ¬nd the appropriate rates of return to use for discounting. Because your bond is
Step 2: ¬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 ¬nd the appropriate discount factors.
Begin by computing the holding rates of return from the yield curve, using the methods from
Section 4·1. For example, the 6-month and 2-year holding rates of return are computed as

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

(1 + 3.15%)2 (4.21)
1 + r0,2 = ≈ (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 rates”about 2.7%. Therefore, you would estimate
the 1.5 year holding rate of return to be

1 + r0,1.5 ≈ (1 + 2.7%)1.5 ≈ (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 information”our payments, annualized interest rates, and equivalent holding interest
rates”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%
≈2.7%
1.5 Nov 2003 $1,500 4.08%
2.0 May 2004 $1,500 3.15% 6.40%
≈3.4%
2.5 Nov 2004 $1,500 8.72%
3.0 May 2005 $1,500 3.64% 11.32%
≈3.8%
3.5 Nov 2005 $1,500 13.94%
4.0 May 2006 $1,500 4.05% 17.21%
≈4.2%
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.
¬le=yieldcurve.tex: RP
65
Section 4·5. 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
≈2.7%
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
≈3.4%
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
≈3.8%
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
≈4.2%
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
today”because this is lower than the bond™s principal repayment of $100,000, this bond is
type of bond: coupon
called a discount bond. rate is not interest rate!




4·5. 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 same”not as valuable”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 e¬ects do in¬‚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 di¬erently produc-
higher yields because
tive at di¬erent times, it is not likely that seasonality is the reason why 30-year Treasury bonds they are riskier”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 de¬nition. It is very competitive and e¬cient”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 di¬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 o¬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%)·(1+10%) ≈
1 + 12%, or r2 = 5.9%. If you graph rT against T , you will ¬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 di¬erent from my “seasons” 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 in¬‚ation rate will increase.
In¬‚ation”a subject of Chapter 6”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.
¬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 e¬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).



4·5.A. The E¬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 bonds”most ¬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
e¬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: Let™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 ≈ $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 ≈ $189.56. If interest rates decrease by 10 basis points to
5.5%, the bond value increases to $1, 000/1.05530 ≈ $200.64. Thus, the e¬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%) ’ V (r30 = 5.6%) $200.64 ’ $195.02
r= = ≈ +2.88%
V (r30 = 5.6%) $195.02
(4.23)
V (r30 = 5.7%) ’ V (r30 = 5.6%) $189.56 ’ $195.02
r= = ≈ ’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%) ’ V (r1 = 5.6%) $952.38 ’ $946.97
r= = ≈ +0.09%
V (r1 = 5.6%) $946.97
(4.24)
V (r1 = 5.7%) ’ V (r1 = 5.6%) $946.07 ’ $946.07
r= = ≈ ’0.09%
V (r1 = 5.6%) $946.07
¬le=yieldcurve.tex: RP
67
Section 4·5. 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 e¬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-free”virtually everything else is risky.



Important: Though “¬xed income,” even a Treasury bond does not guarantee
a “¬xed rate of return” 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 “Only” 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 di¬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 di¬erent from actual losses is a common but
capital error.



Important: “Paper losses” are actual losses.



The primary exception to this rule is that realized gains and losses have di¬erent tax implica-
tions than unrealized gains and losses”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 me”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 day™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
strategies”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 today™s annualized
rate of return on a 10-year bond?


Q 4.17 Using information from a current newspaper or the WWW, what is today™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 today™s total rate of return on a 30-day Treasury note, what
will you end up with?
¬le=yieldcurve.tex: LP
68 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

4·6. The Yield To Maturity (YTM)

In Section 4·2, you learned how to annualize rates of return, so that you could better understand
We want a “sort-of
average interest rate” the rate at which two di¬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 di¬erent payments? For example,
cash ¬‚ows.
what is the interest rate on a bond that costs $100,000 today, and pays o¬ $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 bond”is there something like an “average”
interest rate implicit in these cash ¬‚ows? Is this bond intrinsically more similar to a bond
o¬ering a 4% rate of return or a bond o¬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 ¬‚ows that are implicit to the bond itself. The answer is not obvious at
all”until you learn it. The yield-to-maturity gives a sort of “average rate of return” implicit in
many bond cash ¬‚ows.



Important: The Yield to Maturity is the quantity YTM, which, given a complete
set of bond cash ¬‚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 = ’$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
’$100, 000 + + + ≈ $17, 493 ,
1 + 5% (1 + 5%)2 (1 + 5%)3
(4.27)
$5, 000 $10, 000 $120, 000
’$100, 000 + + + ≈ $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
’$100, 000 + + + ≈ ,
$363
1 + 11% (1 + 11%)2 (1 + 11%)3
(4.28)
$5, 000 $10, 000 $120, 000
’$100, 000 + + + ≈ ’$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
’$100, 000 + + + ≈0. (4.29)
1 + 11.14255% (1 + 11.14255%) (1 + 11.14255%)3
2


So, the cash ¬‚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 rate”a yield to maturity”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 ¬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 in¬‚ow and one cash out¬‚ow”as is the case for a zero bond”then the YTM is the same
as the annualized interest rate. However, a rate of return is de¬ned by exactly two cash ¬‚ows,
so it is meaningless to talk about it when there are multiple cash ¬‚ows. In contrast, the YTM
can handle multiple cash ¬‚ows just ¬ne. Although it may help your intuition to think of the
YTM as a “sort of” average interest rate that is embedded in a bond™s cash ¬‚ows, you should be
¬le=yieldcurve.tex: RP
69
Section 4·6. 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 de¬ned by a cash ¬‚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 ¬‚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 bond™s YTM, then the bond is a positive NPV project. So, if all prevailing budgeting tool.
economy-wide interest rates were 11%, and your bond™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 bond™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 bond”though it still gives a nice characterization of bond payments. Instead, you have to

<<

. 4
( 39)



>>