<< ńņš. 5(āńåćī 39)ŃĪÄÅŠĘĄĶČÅ >>
go back to NPV.
I must confess some small sins: First, the Treasury yield curve in Table 4.1 which was used for YTM is also called IRR in
a more general context,
illustration was not really based on zero-bonds, as I had pretended. Instead, it was based on
bonds that had some interim coupon paymentsā”and it was the YTM of these coupon bonds that built-in.
we graphed, not just the simple zero-bond annualized interest rate. (The zero-bond version
of the yield curve would be graphed based on Treasury STRIPS [see Appendix B.2Ā·6]. The
STRIP yieldcurve can diļ¬er āa littleā from the ordinary coupon-bond yieldcurve.) Second, the
concept of YTM works with or without the concept of time-varying interest rates, so it may be
misplaced in this chapter. It is about bond payments, not about the prevailing economy wide
discount rates. I just placed it here, because it allowed us to discuss how you would compare a
bondā™s YTM to the prevailing yield-curve, and how YTM becomes useless if the yield-curve is not
uniformly above or below it. Third, there is an easier method than trial-and-error yourself ā”
most computer spreadsheets oļ¬er a built-in function called āIRRā that solves the YTM equation
exactly as you just did, only without trial and error and therefore more conveniently, too. (We
will cover IRR in Chapter 8.)
Solve Now!
Q 4.19 What is the YTM of a level-coupon bond whose price is equal to the principal paid at
maturity? For example, take a 5-year bond that costs \$1,000, pays 5% coupon (\$50 per year) for
4 years, and ļ¬nally repays \$1,050 in principal and interest in year 5.

Q 4.20 What is the YTM of the following zero-bond? For example, take a 5-year bond that costs
\$1,000 and promises to pay \$1,611?

Q 4.21 Compute the yield-to-maturity of a two-year bond that costs \$25,000 today, pays \$1,000
at the end of the ļ¬rst year and at the end of the second year. At the end of the second year, it
also repays \$25,000. What is the bondā™s YTM?

Q 4.22 Let us learn how to āSTRIPā a Treasury coupon bond. (STRIP is a great acronym for
Separate Trading of Registered Interest and Principal of Securities.) Presume the 12 month
Treasury bond costs \$10,065.22 and pays a coupon of \$150 in 6 months, and interest plus coupon
of \$10,150 in 12 months. (Its payment patterns indicate that it was originally issued as a ā3-
percent semi-annual level-coupon bond.ā) Presume the 6-month Treasury bond costs \$10,103.96
and has only one remaining interest plus coupon payment of \$10,200. (It was originally issued
[and perhaps many years ago] as a ā4% semi-annual level-coupon bond.ā)

(a) What is the YTM of these two bonds?
(b) Graph a yield curve based on the maturity of these two bonds.
(c) What would be the price of a 1-year zero bond?
(d) Graph a yield curve based on zero bonds.
(e) Do the yield diļ¬erences between the 1-year zero bond and the 1-year coupon bond seem
large to you?
ļ¬le=yieldcurve.tex: LP
70 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

4Ā·7. Optional Bond Topics

There are many other ļ¬ner details of bonds that we could dive into, even though they are not
Optional for ordinary
capital budgeting, but absolutely necessary for understanding the basics of capital budgeting. This does not mean
relevant and useful!
that they are unimportantā”indeed, any CFO who wants to ļ¬nance projects by issuing bonds
will inevitably run into each of them. So, in this section, we cover a set of related issues that
are best dubbed āadvanced, but not unimportant.ā

4Ā·7.A. Extracting Forward Interest Rates

Can you lock in a 1-year interest rate beginning in 2 years? For example, you may have a project
Forward interest rates
are implied interest that will generate cash in 2 years and that you need to store for 1 year before the cash can be
rates in the future, given
used in the next project. The answer is yes, and the lock-in rate is right in the yield curve
by todayā™s yield curve.
itself. Computing and locking rates may not be important to the ordinary small investor, but
it is to bond traders and CFOs. This lock-able interest rate is the forward interest rate (or,
simply, forward rate)ā”an interest rate for an investment of cash beginning not today, but in
the future. You have already used forward rates: we called them, e.g., r2,3 , the one-year interest
rate beginning in 2 years. Still, I want to rename this to f2,3 now, both for better memorization
and (for the real world) to distinguish this forward interest rate that is known today from the
one-year interest rate that will actually come in two years, which is unknown today. (It is only
in our artiļ¬cial world of perfect certainty that the two must be identical.) In contrast to forward
rates, interest rates for investments beginning this period are called spot rates or spot interest
rates, because they are the interest that can be obtained on the spot right now.
Begin by working out the future one-year interest rates that were already computed for you in
Working out forward
rates step by step from Table 4.2 on Page 59. In Table 4.2, the formulas were
the yield curve.

Rates of Returns
Maturity Total Holding Annualized Individually Compounded
= (1 + r1 )1
(1 + r0,1 ) = (1 + r0,1 )
1 Year
= (1 + r2 )2
(1 + r0,2 ) = (1 + r0,1 ) Ā· (1 + f1,2 )
2 Year
= (1 + r3 )3
(1 + r0,3 ) = (1 + r0,1 ) Ā· (1 + f1,2 ) Ā· (1 + f2,3 )
3 Year

The Wall Street Journal yield curve gives you the annualized interest rates, i.e., the third column.
You can read them oļ¬ and insert them into your table. On May 31, 2002, these interest rates
were

Rates of Returns
Maturity Total Holding Annualized Individually Compounded
= (1 + 2.26%)1
(1 + r0,1 ) ā (1 + r0,1 )
1 Year
= (1 + 3.20%)2
(1 + r0,2 ) ā (1 + r0,1 ) Ā· (1 + f1,2 )
2 Year
= (1 + 3.65%)3
(1 + r0,3 ) ā (1 + r0,1 ) Ā· (1 + f1,2 ) Ā· (1 + f2,3 )
3 Year

The ļ¬rst step is to compute the holding rates of return in the second column:

Rates of Returns
Maturity Total Holding Annualized Individually Compounded
ā (1 + 2.26%)1
(1 + 2.26%) = (1 + r0,1 )
1 Year
ā (1 + 3.20%)2
(1 + 6.50%) = (1 + r0,1 ) Ā· (1 + f1,2 )
2 Year
ā (1 + 3.65%)3
(1 + 11.35%) = (1 + r0,1 ) Ā· (1 + f1,2 ) Ā· (1 + f2,3 )
3 Year
ļ¬le=yieldcurve.tex: RP
71
Section 4Ā·7. Optional Bond Topics.

Ultimately, you want to know what the implied future interest rates are. Work your way down.
The ļ¬rst row is easy: you know that r0,1 is 2.26%. You can also substitute this return into the
other rows:

Rates of Returns
Maturity Total Holding Annualized Individually Compounded
ā (1 + 2.26%)1
(1 + 2.26%) ā (1+2.26%)
1 Year
ā (1 + 3.20%)2
(1 + 6.50%) ā (1+2.26%) Ā· (1 + f1,2 )
2 Year
ā (1 + 3.65%)3
(1 + 11.35%) ā (1+2.26%) Ā· (1 + f1,2 ) Ā· (1 + f2,3 )
3 Year

Now you have to work on the two year row to determine f1,2 : You have one equation and one
unknown in the two year row, so you can determine the interest to be

1 + 6.50%
(4.30)
(1 + 6.50%) = (1 + 2.26%) Ā· (1 + f1,2 ) ā’ (1 + f1,2 ) = ā 1 + 4.15% .
1 + 2.26%

Substitute this solution back into the table,

Rates of Returns
Maturity Total Holding Annualized Individually Compounded
ā (1 + 2.26%)1
(1 + 2.26%) ā (1+2.26%)
1 Year
ā (1 + 3.20%)2
(1 + 6.50%) ā (1+2.26%) Ā· (1+4.15%)
2 Year
ā (1 + 3.65%)3
(1 + 11.35%) ā (1+2.26%) Ā· (1+4.15%) Ā· (1 + f2,3 )
3 Year

Now work on row 3. Again, you have one equation and one unknown in the three year row, so
you can determine the interest to be

(4.31)
(1 + 11.35%) = (1 + 2.26%) Ā· (1 + 4.15%) Ā· (1 + f2,3 )

1 + 11.35%
ā’ (1 + f2,3 ) = ā 1 + 4.56% . (4.32)
(1 + 2.26%) Ā· (1 + 4.15%)

Rates of Returns
Maturity Total Holding Annualized Individually Compounded
ā (1 + 2.26%)1
(1 + 2.26%) ā (1+2.26%)
1 Year
ā (1 + 3.20%)2
(1 + 6.50%) ā (1+2.26%) Ā· (1+4.15%)
2 Year
ā (1 + 3.65%)3
(1 + 11.35%) ā (1+2.26%) Ā· (1+4.15%) Ā· (1+4.56%)
3 Year

So, given the annualized rates of return in the yield curve, you can determine the whole set of
implied forward interest rates. For example, the implied interest rate from year 2 to year 3 is
4.56%.
Behind this arithmetic lies a pretty simple intuition: An annualized two-year interest rate is Think of the annualized
interest rate as the
āreally sort ofā an āaverageā interest rate over the interest rates from the ļ¬rst year and the
average of interest rates.
second year. (In fact, the annualized rate is called the geometric average.) If you know that
the average interest rate is 3.20%, and you know that the ļ¬rst half of this average is 2.26%, it
must be that the second half of the average must be a number around 4.2% in order to average
out to 3.20%. And, indeed, you worked out that the forward one-year interest rate was 4.15%.
It is not exactā”due to compoundingā”but it is fairly close.
ļ¬le=yieldcurve.tex: LP
72 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

Solve Now!
Q 4.23 Continuing the example, compute the one-year forward interest rate f3,4 from year 3 to
year 4, if the 4-year annualized interest rate was 4.06%.

4Ā·7.B. Shorting and Locking in Forward Interest Rates
Why are forward interest rates so interesting? The reason is that by cleverly buying and selling
Frictionless borrowing
and lending of Treasury (shorting) Treasury bonds, you can bet on future interest rates embedded in the yield curve.
bonds allow investors to
Working with and speculating on forward rates is the ābread-and-butterā of bond traders. But
lock in future interest
bond traders are not the only parties hereā”ļ¬rms often also want to ālock inā future interest
rates. How shorting
works.
rates todayā”and they can indeed lock in todayā™s forward interest rates as the future interest
rates that they will face. To understand this, assume that you can buy and sell Treasury bonds,
even if you do not own them. In eļ¬ect, you can borrow these securities, sell them, receive
the cash, buy back the bonds later, and return them to the lender. This is called a short sale
(the oppositeā”buying securitiesā”is said to be a long position). Table 4.4 explains the basic
idea behind shorting. In eļ¬ect, for Treasury bonds, short selling enables you to do what the
government doesā”āissueā a security, take in money, and return it to the lender with interest.
For example, you may sell short \$89,803.25 of a 3-year Treasury bond today with a 3.65% rate
of interest and a maturity of 3 years. This will give you \$89,803.25 cash today, but require you
to come up with \$100,000 for repayment in 3 years. In eļ¬ect, selling a bond short is a way of
borrowing money. In the real world, for professional bond traders, who can prove that they
have enough funds to make good any possible losses, this is easily possible and with extremely
small transaction costs, perhaps 1ā“2 basis points. Thus, assuming transaction costs away is a
reasonable assumption.
Holding a security (i.e., being long) speculates that the value will go up, so selling a ļ¬nancial
Shorting is the opposite
of Buying: It speculates instrument (i.e., being short) speculates that the value will go down. If the price of the bond
that the value will
tomorrow were to go down to \$50,000 (an annualized interest rate of 26%), the trader could
decline.
then purchase the government T-bill for \$50,000 to cover the \$100,000 commitment he has
made for \$89,803.25, a proļ¬t of \$39,803.25. But if the price of the bond tomorrow were to go
to \$99,000 (an annualized interest rate of 0.33%), the trader would lose \$9,196.75.
Now assume that you are able to buy a two-year bond at an annualized interest rate of 3.20%,
Future cash ļ¬‚ows from
the long leg and the and able to sell (short) a three-year bond at an annualized interest rate of 3.65%, and do so
short leg.
without transaction costs. For the three-year bond, you would have to promise to pay back
\$100 Ā· (1 + 11.35%) ā \$111.35 in three years (cash outļ¬‚ow to you) for each \$100 you are
borrowing today (cash inļ¬‚ow to you). For the two-year bond, you would invest these \$100
(cash outļ¬‚ow to you) and receive \$100 Ā· (1 + 6.50%) ā \$106.50 in two years (cash inļ¬‚ow to you).
Looking at your Payout Table 4.5, from your perspective, the simultaneous transaction in the
two bonds results in an inļ¬‚ow of \$106.50 in year two followed by a cash outļ¬‚ow of \$111.35.
Eļ¬ectively, you have committed to borrowing \$106.50 in year 2 with payback of \$111.35 in
year 3. The interest rate for this loan is
\$111.35 ā’ \$106.50
f2,3 ā ā 4.56%
\$106.50
(4.33)
CF0 Ā·(1 + r0,3 ) ā’ CF0 Ā·(1 + r0,2 )
= ,
CF0 Ā·(1 + r0,2 )

which is exactly the forward interest rate in the table.

Digging Deeper: There is an alternative way to work this. Start with the amount that you want to borrow/lend
in a future period. For example, say you want to lend \$500 in year 2 and repay however much is necessary
in year 3. Lending \$500 in year 2 requires an outļ¬‚ow, which you can only accomplish with an inļ¬‚ow today.
(Therefore, the ļ¬rst ālegā of your transaction is that you borrow, i.e., short the 2-year bond!) Speciļ¬cally, your
inļ¬‚ow today is \$500/(1+3.20%)2 ā \$469.47. Now, invest the entire \$469.47 into the 3-year bond, so that you have
zero net cash ļ¬‚ow today. (The second ālegā of your transaction is that you lend, i.e., purchase the 3-year bond.)
This will earn you an inļ¬‚ow \$469.47Ā·(1 + 3.65%)3 ā \$522.78 in 3 years. In total, your ļ¬nancial transactions
have committed you to an outļ¬‚ow of \$500 in year 2 in exchange for an inļ¬‚ow of \$522.78 in year 3ā”otherwise
known as 1-year lending in year 2 at a precommitted interest rate of 4.56%.
ļ¬le=yieldcurve.tex: RP
73
Section 4Ā·7. Optional Bond Topics.

Table 4.4. The Mechanics of an Apple Short Sale

Three Parties: Apple Lender, You, The Apple Market.

Today:

1. You borrow 1 apple from the lender in exchange for your ļ¬rm promise to the lender to
return this 1 apple next year. (You also pay the lender an extra 1 cent lending fee.)

2. You sell 1 apple into the apple market at the currently prevailing apple price. Say, 1 apple
costs \$5 today. You now have \$5 cash, which you can invest. Say, you buy bonds that
earn you a 1% interest rate.

Next year:

1. You owe the lender 1 apple. Therefore, you must purchase 1 apple from the apple market.

ā¢ If apples now cost \$6, you must purchase 1 apple from the market at \$6. You return
the apple to the lender.
Your net return on the apple is thus ā’\$1, plus the \$0.05 interest on \$5, minus the 1
cent fee to the lender. You therefore lost 96 cents.
ā¢ If apples now cost \$4, you must purchase 1 apple from the market at \$4. You return
the apple to the lender.
Your net return on the apple is thus +\$1, plus the \$0.05 interest on \$5, minus the 1
cent fee to the lender. You therefore gained \$1.04.

Net Eļ¬ects:

ā¢ The apple lender has really continued to own the apple throughout, and can sell the apple
in Year 1. There is no advantage for the lender to keep the apple in his own apple cellar
rather than to lend it to you. In addition, the lender earns 1 cent for free by lending.

ā¢ The apple market buyer purchased an apple from you today, and will never know where
it came from (i.e., from a short sale).

ā¢ The apple market seller next year will never know what you do with the apple (i.e., that
you will use it to make good on your previous yearā™s apple loan).

ā¢ You speculated that the price of an apple would decline.

ā¢ Note that you did earn the interest rate along the way. Except for the fee you paid to the
lender, you could sell the apple into the apple market today and use the proceeds to earn
interest, just like an apple grower could have.

In the real world, short-selling is arranged so that you cannot sell the apple short, receive the
\$5, and then skip town. As a short-seller, you must assure the lender that you will be able to
return the apple next year. As the short seller, you must also pay the lender for all interim
beneļ¬ts that the apple would provideā”though few apples pay dividends or coupon, the way
stocks and bonds often do.
ļ¬le=yieldcurve.tex: LP
74 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

Table 4.5. Locking in a Future Interest Rate via the Long-Short Forward Interest Rate Spread

Purchased 2-Year Shorted 3-Year
Time Bond Cash Flows Bond Cash Flows Net Cash Flow
Today ā“\$100.00 +\$100.00 \$0.00
(outļ¬‚ow) (inļ¬‚ow)

Year 1 \$0.00 \$0.00 \$0.00
Year 2 +\$106.50 \$0.00 +\$106.50
(inļ¬‚ow) (inļ¬‚ow)

Year 3 \$0.00 ā“\$111.35 ā“\$111.35
(outļ¬‚ow) (outļ¬‚ow)

This particular transaction is called a forward transaction. Indeed, this particular type of
Such forward interest
rate swaps are so forward transaction is so popular that an entire ļ¬nancial market on interest forwards has
popular that there are
developed that allows speculators to easily engage in simultaneously going long or short on
markets that make this
bonds.
even simpler.

Should you engage in this transaction? If the one-year interest rate in 2 years will be higher
You get what you pay
for: the speculation can than 4.56%, you will be able to borrow at a lower interest than what will be prevailing then. Of
end up for better or
course, if the interest rate will be lower than 4.56%, you will have committed to borrow at an
worse.
interest rate that is higher than what you could have gotten.
Solve Now!
Q 4.24 If you want to commit to saving at an interest rate of f3,4 , what would you have to do?
(Assume any amount of investment you wish, and work from there.)

Q 4.25 If you want to commit to saving \$500,000 in 3 years (i.e., you will deposit \$500,000) at
an interest rate of f3,4 (i.e., you will receive \$526,498.78), what would you have to do?

4Ā·7.C. Bond Duration

In Section 4Ā·6, you learned how to summarize or characterize the cash ļ¬‚ows promised by a bond
Maturity ignores interim
payment structure. with the YTM. But how can you characterize the āterm lengthā of a bond? The ļ¬nal payment,
i.e., the maturity, is ļ¬‚awed: zero bonds and coupon bonds may have the same maturity, but
a high coupon bond could pay out a good amount of money early on. For example, a coupon
bond could pay 99% in coupon in the ļ¬rst month, and leave 1% for a payment in 30 years. It
would count as a 30-year bond, the same as a zero-bond that pays 100% in 30 years.
To measure the payout pattern of a bond, investors often rely on both maturity and durationā”
Duration is an āaverageā
payout date. a measure of the eļ¬ective time-length of a project. The simplest duration measure computes
the time-weighted average of bond payouts, divided by the sum of all payments. For example,
a 5 Year Coupon Bond that pays \$250 for 4 years and \$1,250 in the ļ¬fth year, has a duration
of 3.89 years, because

\$250Ā·1 + \$250Ā·2 + \$250Ā·3 + \$250Ā·4 + \$1, 250Ā·5
Plain Duration = ā 3.89
\$250 + \$250 + \$250 + \$250 + \$1, 250
(4.35)
Payment at Time 1 Ā· 1 + Payment at Time 2 Ā· 2 + ... + Payment at Time T Ā· T
.
Payment at Time 1 + Payment at Time 2 + ... + Payment at Time T

(You can think of this as the āpayment-weightedā payout year.) The idea is that you now con-
sider this 5-year coupon bond to be shorter-term than a 5-year zero bond (which has a 5-year
duration)ā”and perhaps more similar to a 3.9-year zero bond.
ļ¬le=yieldcurve.tex: RP
75
Section 4Ā·7. Optional Bond Topics.

Side Note: Duration is sometimes explained through a physical analog: If all payments were weights hanging
from a (time) line, the duration is the point where the weights balance out, so that the line tilts neither right nor
left.

5-Year Equal Payments 5-Year \$250 Coupon Bond 5-Year Zero Bond

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
qq qq qq
fĀ¢
fĀ¢ fĀ¢
fĀ¢ fĀ¢
fĀ¢
a qa a a fĀ¢ a
qa qa
fĀ¢ fĀ¢

\$250
vvvv
cccc

ccv cc
v v cv v

\$1,250
c
v v
c

Duration: 3 years Duration: 3.89 years Duration: 5 years

Macaulay Duration alters plain duration by using the present value of payouts, not just nominal Macaulay duration uses
PV, and is usually a little
payouts. Thus, unlike plain duration which merely characterizes bond cash ļ¬‚ows regardless
bit less than plain
of economy-wide interest rates, Macaulay duration also depends on the prevailing yield curve. duration.
If the interest rate on all horizons is 5%, the Macaulay duration for your coupon bond is

\$238Ā·1 + \$227Ā·2 + \$216Ā·3 + \$206Ā·4 + \$979Ā·5
Macaulay Duration = ā 3.78
\$238 + \$227 + \$216 + \$206 + \$979
(4.36)
PV( Payment at Time 1 ) Ā· 1 + PV( Payment at Time 2 ) Ā· 2 + ... + PV( Payment at Time T ) Ā· T
.
PV( Payment at Time 1 ) + PV( Payment at Time 2 ) + ... + PV( Payment at Time T )

Duration Similarity
Duration can be used as a measure for the ātermā of projects other than bonds, too. However, Duration is used as an
interest exposure
duration only works if all incoming cash ļ¬‚ows are positiveā”otherwise, it may be nonsense.
measure.
Duration is important, because it helps you judge the exposure (risk) of your projects to changes
in interest rates. For example, if you have a project (or bond portfolio) that has an average
duration of 6.9 years, then it is probably more exposed to and more similar to the 7-year
Treasury bond than the 5-year or 10-year Treasury bonds.
Now presume that the yield curve is 5% for 1-year T-bonds, 10% for 2-year T-bonds, and 15% A concrete project
example.
for 3-year T-bonds. You can purchase a project that will deliver \$1,000 in 1 year, \$1,000 in 2
years, and \$1,500 in 3 years, and costs \$2,500. This bond would be a good deal, because its
present value would be \$2,765.10. The project has a YTM of 17.5%, and a Macaulay duration of
2.01 years. (We shall only work with the Macaulay duration.) But, letā™s presume you are worried
about interest rate movements. For example, if interest rates were to quadruple, the project
would not be a good one. How does the value of your project change as the yield curve moves
around?
ļ¬le=yieldcurve.tex: LP
76 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

Letā™s work out how changes in the yield curve aļ¬ect your projects and pure zero bonds, each
The effect of a constant
shift of the yield curve. promising \$1,000 at maturity. First, your project. Presume that the entire yield curve shifts
upward by 1%ā”the 5% T-bond yield becomes a 6% yield, the 10% becomes 11%, and the 15%
becomes 16%. Your project value would now be

\$1, 000 \$1, 000 \$2, 500
PV = + + ā \$2, 716.01 . (4.37)
1 + 6% (1 + 11%)2 (1 + 16%)3

This is an instant rate of return of (\$2, 716.01 ā’ \$2, 765.10)/\$2, 765.10 ā ā’1.776%.

Yield Curve Project
PV( rL ) PV( rH ) RoR

Entire yield curve shifts upward by 1%: \$2,765.10 \$2,716.01 ā“1.78%

Is this more similar to how the 1-Year zero T-bond changed, how the 2-year zero T-bond
changed, or how the 3-year zero T-bond would have changed? Of course, zero bonds are only
aļ¬ected by their speciļ¬c interest rate, so you can work out the percent change one at a time or
all simultaneously, and you would get the same answer.

Yield Curve
rL ā’ rH PV( rL ) PV( rH ) RoR
1-Year Bond 1.05%ā’1.06% \$952.38 \$943.40 ā“0.94%

2-Year Bond 1.10%ā’1.11% \$826.45 \$811.62 ā“1.79%

3-Year Bond 1.15%ā’1.16% \$640.66 \$657.52 ā“2.56%

The answer is that your projectā™s value change is most similar to the 2-year zero T-bond value
change. This is what your bondā™s duration of 2.01 year told youā”your project behaves most
similar to the 2-year bond as far as its interest rate sensitivity is concerned.

Duration Hedging
So, now you know how your project would suļ¬er from a change in the interest rate that you
A hedge matches assets
and liabilities to reduce may fear, but what can you do about it? The idea is to hedge your riskā”you try to own the
risk.
same assets long and shortā”you are matching liabilities and assetsā”so that you are ensured
against adverse changes. For example, it would be a perfect hedge if you purchased the project,
and also shorted \$1,000 in the 1-year bond, \$1,000 in the 2-year bond, and \$2,500 in the 3-year
bond. You would be totally uninterested in where interest rates would be movingā”your wealth
would not be aļ¬ected. (This is the ālaw of one priceā in action. In fact, there is absolutely no
risk to lose money, so this would be an arbitrage portfolio, explained in Section 19Ā·1.)
In the real world, perfect hedges, whereby you can match all project cash ļ¬‚ows perfectly, are
Why perfect hedges are
rare. rarely possible. First, it is more common that you know only roughly what cash ļ¬‚ows your
project will return. Fortunately, it is often easier to guess your projectā™s duration than all its
individual cash ļ¬‚ows. Second, it may also be diļ¬cult for smaller companies to short 137 zero
T-bonds to match all project cash ļ¬‚owsā”the transaction costs would simply be too high. Third,
you may not do any active matching, but you would still like to know what kind of exposure you
are carrying. After all, you may not only have this project as asset, but you may have liabilities
[e.g., debt payments] that have a duration of 2.4 yearsā”and you want to know how matched or
mismatched your assets and liabilities are. Or, you may use the newfound duration knowledge
to choose among bank or mortgage loans with diļ¬erent durations, so that your assets and
liabilities roughly match up in terms of their duration.
ļ¬le=yieldcurve.tex: RP
77
Section 4Ā·7. Optional Bond Topics.

For example, you know your project assets have a duration of 2 yearsā”what kind of loan Minimizing interest rate
risk.
would you prefer? One that has a 1-year duration, a 2-year duration or a 3-year duration? If
you want to minimize your interest rate risk, you would prefer to borrow \$2,716 of a 2-year
bondā”though the bank loan, too, may not be a zero-bond, but just some sort of loan with a
2-year duration. Would you be comfortable that interest rate would not aļ¬ect the value of your
project very much if you were short the 2-year bond and long the project? Yes and noā”you
would be comfortable that wholesale shifts of the yield curve would not aļ¬ect you. You would
however be exposed to changes in the shape of the yield curveā”if only one of the interest rates
were to shift, your project would be impacted diļ¬erently than your 2-year T-bond. In this case,
your projectā™s value would move less than the value of your 2-year bond. In the real world,
over short horizons, duration matching often works very well. Over longer horizons, however,
you will have to do constant watching and rearranging of assets and liabilities to avoid the gap
enlarging too much.

Digging Deeper: The interest-rate sensitivity of a bondā™s value is roughly its duration divided by one-plus the
bondā™s yield. Therefore, a bondā™s price change with respect to a change in interest rate is roughly

Duration
(4.38)
Bond Price Return ā Ā· Interest Rate Change .
1 + YTM
This ignores complex changes in the term structure, but it is often a useful quick-and-dirty sensitivity measure.
For example, take our project, and consider a change in interest rates of 10 basis points. That is, the 1-year
interest rate moves to 1.051%, the 2-year to 1.101%, and the 3-year to 1.151%. The value of your project would
change from \$2,765.10 to \$2,760.12, an immediate percent change of 18 basis points.

2.01
ā Ā· 10bp .
18bp
1 + 17.5% (4.39)
Duration
Bond Price Return ā Ā· Interest Rate Change .
1 + YTM

Solve Now!
Q 4.26 Compute the duration of a two-year bond that costs \$25,000 today, pays \$1,000 at the
end of the ļ¬rst year and at the end of the second year. At the end of the second year, it also
repays \$25,000.

Q 4.27 If the yield curve is a ļ¬‚at 3%, compute the Macaulay duration for this two-year bond.

Q 4.28 If the yield curve is a ļ¬‚at 10%, compute the Macaulay duration for this two-year bond.

Q 4.29 Compute the yield-to-maturity of a 25-year bond that costs \$25,000 today, and pays
\$1,000 at year-end for the following 25 years. In the ļ¬nal year (t = 25), it also pays \$25,000.
What is the YTM?

Q 4.30 Compute the plain duration of this 25-year bond.

Q 4.31 If the yield curve is a ļ¬‚at 3%, compute the Macaulay duration for this 25-year bond.

Q 4.32 If the yield curve is a ļ¬‚at 10%, compute the Macaulay duration for this 25-year bond.
ļ¬le=yieldcurve.tex: LP
78 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

4Ā·7.D. Continuous Compounding

A subject of some interest to Wall Street traders, i.e., the people who trade bonds or options for
Continuously
compounded interest a living, is the concept of a continuously compounded interest rate. This is easiest to explain
rates are āas if interest
by example.
is paid every instant.ā

Assume that you receive \$120 next year for an investment of \$100 today. You already know
Progressively more
frequently paid interest that this represents a simple rate of return of 20%. What would the interest be if it were paid
payments converge to
twice per year, not once per year, the interest rate remained constant, and the \$100 would still
the continuously
come out to be \$120 at the end of the year. You have done this before:
compounded interest
rate.

(4.40)
(1 + rsemi-annual ) Ā· (1 + rsemi-annual ) = (1 + 20%) ā’ r ā 9.54% .

If you multiply this semiannual interest rate by two, you get 19.08%. What if you received
interest twelve times a year?

(1 + rmonthly )12 = (1 + 20%) (4.41)
ā’ r ā 1.53% .

Multiply this monthly (m) interest rate by 12 and you get 18.36%. What if you received interest
365 times a year?
(1 + rdaily )365 = (1 + 20%) (4.42)
ā’ r ā 0.05% .

The 20% was called an āeļ¬ective annual rateā in Table 2.3. Multiply this daily (d) interest rate
by 365 and you get 18.25% (the annual quote). Now, what would this number be if you were to
receive interest every single moment in timeā”the annual rate, compounded every instant?
The answer is, you guessed it, the continuously compounded interest rate and it can be com-
The limit: Use logs and
exponents to translate puted by taking the natural logarithm (abbreviated ālnā on your calculator and below) of one
simple interest rates to
plus the simple interest rate
continuously
compounded interest
rates.
rcontinuously compounded = ln(1 + 20%) ā 18.23%
(4.43)
rcontinuously compounded = ln(1 + rsimple ) .

(Appendix 2Ā·3 reviews powers, exponents and logarithms.)
You must never directly apply a continuously compounded interest rate to a cash ļ¬‚ow to com-
Warning: Never, ever
apply cc rates of return pute your return. In this example, investing \$100 would not leave you with \$118.23 after one
to a cash ļ¬‚ow!
year. Indeed, if someone quoted you a continuously compounded interest rate, to determine
how much money you will end up with, you would ļ¬rst have to convert the continuously com-
pounded return into a simple interest rate

rsimple = ercontinuously compounded ā’ 1 ā e18.23% ā’ 1 ā 20% , (4.44)

and then apply this interest rate to the cash ļ¬‚ow. Alternatively, you can multiply the cash ļ¬‚ow
not by one plus the simple interest rate, but by erc c .
Continuously compounded rates have two nice features: First, if the continuously compounded
To obtain multi-period
interest returns, rate in period 1 is 10% and in period 2 is 20%, the total two-period continuously compounded
continuously
rate is 30%ā”yes, continuously compounded interest rates can be added, so no more multiplying
compounded interest
one-pluses! This additivity is not a big advantage. Second, they are more āsymmetric.ā See, an
rates are never
ordinary rate of return lies between ā’100% and +ā, while the continuously compounded rate
of return lies between ā’ā and +ā. (This can be an advantage in statistical work, as can be the
fact that the logarithm helps āpull inā large outliers.) However, the main need for continuously
compounded interest rates arises in other formulas (such as the Black-Scholes option formula,
the subject of the Web Chapter on Options and Derivatives).
ļ¬le=yieldcurve.tex: RP
79
Section 4Ā·8. Summary.

Solve Now!
Q 4.33 A bond pays \$150 for every \$100 invested. What is its continuously compounded interest
rate?

Q 4.34 Show my claim that you can add continuously compounded interest rates. That is, a
bond pays a continuously compounded interest rate of 10%. Upon maturity, the money can be
reinvested at a continuously compounded interest rate of 20%. If you invest \$100 today, how
much money will you end up with? What is the simple and continuously compounded interest
rate over the two periods?

4Ā·8. Summary

The chapter covered the following major points:

ā¢ Compounding works just as well for time-varying interest rates.

ā¢ A holding rate of return can be annualized for easier interpretation.

ā¢ Diļ¬erent interest rates apply to diļ¬erent horizon investments. The graph of interest rates
as a function of horizon is called the āterm structure of interestā or āyield curve.ā

ā¢ Net present value works just as well for time-varying interest rates. You merely need to
use the appropriate opportunity cost of capital as the interest rate in the denominator.

ā¢ Diļ¬erent horizon interest rates carry diļ¬erent risks, not because one is a better deal than
the other. Instead, it is either that future interest rates are expected to be diļ¬erent, or
that longer-term investments carry more interim risk.
For Treasury bonds, the risk from interim interest rate changes seems to be the primary
reason why the yield curve is usually upward sloping.

ā¢ More often than not, āpaper lossesā are no diļ¬erent from real losses.

ā¢ The yield curve is usually upward sloping, but can be downward sloping (inverted),
humped, or ļ¬‚at.

ā¢ The Yield-to-Maturity is a āsort-of-averageā interest rate characterizing the payoļ¬s of a
bond. It does not depend on economy-wide interest rates (the yield curve).

If you covered the optional bond topics section, you also learned the following.

ā¢ The information in the set of annualized rates of return, individual holding rates of return,
and total holding rates of return is identical. Therefore, you can translate them into one
another. For example, you can extract all forward interest rates from the prevailing yield
curve.

ā¢ How shorting transactions work.

ā¢ If you can both buy and short bonds, then you can lock in forward interest rates today.

ā¢ Bond duration is a characterization of when bond payments typically come in.

ā¢ The continuously compounded interest rate is ln(1 + r ), where r is the simple interest
rate.
ļ¬le=yieldcurve.tex: LP
80 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

Solutions and Exercises

1. r0,2 = (1 + r0,1 ) Ā· (1 + r1,2 ) ā’ 1 = (1 + 2%) Ā· (1 + 3%) ā’ 1 = 5.06%.
2. Solve (1 + 22%) Ā· (1 + x) = (1 ā’ 50%), so the project had a rate of return of ā’59%.
3. 166.4%. For checking, 124.3% followed by 18.8%.
4. The returns were (ā’33%, +50%, ā’67%, +100%), so the overall rate of return was ā’33.3%.

5. The annualized rate of return is 18.3%. It is therefore lower than the average rate of return.
6. The compounded rate of return is always higher, because you earn interest on interest. The annualized rate of
return is lower than the average rate of return, again because you earn interest on the interest. For example, an
investment of \$100 that turns into an investment of \$200 in two years has a total holding period rate of return
of 100%ā”which is an average rate of return of 50% and an annualized rate of return of (1 + 100%) ā’ 1 = 41%.
Investing \$100 at 41%/annum would yield \$200, which is higher than 50% per annum.
ā
12
7. r12 = 1 + 166.4% = 8.5%.
8. Your rate of return over the six days here was r0,6 = \$1, 300.80/\$1, 283.27ā’1 = 1.366%. You can compound
this over the 255/6 = 42.5 time periods to obtain 78% per annum. (Alternatively, the 1.366% is a daily rate of
0.2264%, which can compound over 255 days.) So, your \$100 would have turned into \$178. (Actually, these
were the index values of the S&P500; the rest of 2001 was a lot bleaker for investors, and they lost 13% in
2001.)
(1 + r5 )5 = (1 + 50%) r5 = (1 + 50%)1/5 ā’ 1 = 8.45%.
9. r0,5 = 50% ā’
10. The same.
(1 + r0,5 ) = (1 + r5 )5 = (1 + 61.05%).
11. r5 = 10%
12. The basic formula is (1 + r5 )5 = (1 + r0,1 ) Ā· (1 + r1,2 ) Ā· (1 + r2,3 ) Ā· (1 + r3,4 ) Ā· (1 + r4,5 ).
Substituting what you know, (1 + 10%)5 = (1 + 15%) Ā· (1 + x) Ā· (1 + x) Ā· (1 + x) Ā· (1 + x).
Calculating, 1.61051 = 1.15 Ā· (1 + x)4 , (1 + x)4 = 1.40, , and (1 + x) = 1.401/4 = 1.0878. So, the return in
all other years would have to be 8.78% per year.
13. The daily interest rate is either (1 + 10%)1/365 ā’ 1 ā 0.026% or (1 + 20%)1/365 ā’ 1 ā 0.05% per day. Thus, the
pessimist expects a stock price of \$30.008 tomorrow; the optimist expects a stock price of \$30.015 tomorrow.
Note that the 1 cent or so expected increase is dwarfed by the typical day-to-day noise in stock prices.

14. Do it!
15. Do it!

16. Do it!
17. Take the above number, call it r10 , and compute (1 + r10 )10 .
18. Do it.

19. 5%, because

\$50 \$50 \$50 \$50 \$1, 050
ā’\$1, 000 + + + + + =0 (4.45)
1 + 5% (1 + 5%)2 (1 + 5%)3 (1 + 5%)4 (1 + 5%)5
The YTM of such a bond is just the coupon itself.
20. The YTM is 10%, because
\$1, 611
ā’\$1, 000 + =0 (4.46)
(1 + 10%)5

21. You are seeking the solution to
2
\$1, 000 \$25, 000
(4.47)
ā’\$25, 000 + + =0.
(1 + r )t (1 + r )2
t=1

The correct solution is 4%.
ļ¬le=yieldcurve.tex: RP
81
Section 4Ā·8. Summary.

22.
(a)
\$150 \$10, 150
(4.48)
ā’\$10, 065.22 + 0.5 + =0,
YTM1
YTM
so YTM = 2.35%. The YTM of the six-month bond is

\$10, 200
(4.49)
ā’\$10, 103.96 + =0,
YTM0.5
so YTM = 1.91%. Okayā”I admit I chose the equivalent of the yield curve that we plotted in the text.
(b) Do it.
(c) The \$150 coupon is worth \$150/1.01910.5 ā \$1, 48.59. Therefore, the one-year zero bond with one
payment of \$10,150 due in one year costs \$10, 065.22 ā’ \$148.59 ā \$9, 916.63. This means that the
1-year zero bond with payoļ¬ of \$10,150 has a YTM of \$10, 150/\$9, 916.63 ā’ 1 ā 2.3533%.
(d) Do it.
(e) The diļ¬erence between the YTM of the coupon and the zero bond is only 0.3 basis pointsā”very small,
even though the yield curve here is fairly steep. The reason is that the early 6-month coupon earns a
lower interest makes little diļ¬erence because the coupon payment is only \$150, and most of the YTM
comes from the ļ¬nal payment. The coupon eļ¬ect can become larger on very long horizons when the
yield curve is steep, but it is very rarely more than 10-20 basis points.

23. The 4-year holding rate of return is r0,4 ā (1 + 4.06%)4 ā 17.26%. Therefore, the 1-year forward rate from
(1 + r0,4 ) (1 + 17.27%)
year 3 to year 4 is f3,4 ā ā’1ā ā’ 1 ā 5.30%.
(1 + r0,3 ) (1 + 11.35%)
24. Buy \$1,000 of a 4-year zero bond (4.06%/year) and short \$1,000 of a 3-year zero bond (3.65%/year). Today,
you receive and pay \$1,000, so the transaction does not cost you anything. In 3-years, you need to pay the
3-year bond, i.e., you need to pay in \$1,113.55. In 4-years, you receive from the 4-year bond \$1,172.56. This
is the equivalent of saving at an interest rate of 5.30%.
25. You can do this from ļ¬rst principles, as before. An alternative is to rely on the previous solution, where you
were saving \$1,113.50. So, you now have to do this transaction at a scale that is \$500, 000/\$1113.5 ā 449.03
times as much. Therefore, instead of buying \$1,000 of the 4-year bond, you must buy 449.035 Ā· \$1, 000 ā
\$449, 035 of the 4-year bond, and short the same amount of the 3-year bond.
26. 2
\$1, 000 Ā· t + \$25, 000 Ā· 2 53, 000
t=1
Plain Duration ā = ā 1.96296 . (4.50)
2 27, 000
\$1, 000 + \$25, 000
t=1

The units here are years, because we quoted the multiplication factors ā1ā and ā2ā are in years.
27. 2 \$1,000Ā·t \$25,000Ā·2
+
t=1 (1+3%)t (1+3%)2
Macaulay Duration at 3% = 2 \$1,000 \$25,000
+ (1+3%)2
t=1 (1+3%)t
\$1,000Ā·1 \$1,000Ā·2 \$25,000Ā·2
+ (1+3%)2 + (1+3%)2
(1+3%)1 (4.51)
= \$1,000 \$1,000 \$25,000
+ (1+3%)2 + (1+3%)2
(1+3%)1

49, 986
= ā 1.96189 .
25, 478
The units here are years, because we quoted the multiplication factors ā1ā and ā2ā are in years.
28. 2 \$1,000Ā·t \$25,000Ā·2
+
t=1 (1+10%)t (1+10%)2
Macaulay Duration at 10% = 2 \$1,000 \$25,000
+ (1+10%)2
t=1 (1+10%)t
\$1,000Ā·1 \$1,000Ā·2 \$25,000Ā·2
+ (1+10%)2 + (1+10%)2
(1+10%)1 (4.52)
= \$1,000 \$1,000 \$25,000
+ (1+10%)2 + (1+10%)2
(1+10%)1

43, 884.3
= ā 1.95941 .
22, 396.7

29. You are seeking the solution to
25
\$1, 000 \$25, 000
(4.53)
ā’\$25, 000 + + =0.
(1 + r )t (1 + r )25
t=1

25
1 ā’ [1/(1 + r )25 ]
\$1, 000/(1 + r )t = \$1, 000 Ā·
The middle piece is an annuity, so . The correct solution is
r
t=1
4%.
ļ¬le=yieldcurve.tex: LP
82 Chapter 4. Investment Horizon, The Yield Curve, and (Treasury) Bonds.

30. 25
\$1, 000 Ā· t + \$25, 000 Ā· 25
t=1 (4.54)
Plain Duration = ā 19.62
25
\$1, 000 + \$25, 000
t=1

31. 25 \$1,000Ā·t \$25,000Ā·25
+
t=1 (1+3%)t (1+3%)25
(4.55)
Macaulay Duration at 3% = ā 16.98 .
25 \$1,000 \$25,000
+ (1+3%)25
t=1 (1+3%)t

32. 25 \$1,000Ā·t \$25,000Ā·25
+
t=1 (1+10%)t (1+10%)25
(4.56)
Macaulay Duration at 10% = ā 11.81 .
25 \$1,000 \$25,000
+
t=1 (1+10%)t (1+10%)25

33. The simple interest rate is 50%. The cc interest rate is log(1 + 50%) ā 40.55%.
34. A 10% cc interest rate is a simple interest rate of r0,1 ā e0.10 ā’ 1 ā 10.52%, so you would have \$110.52 after
one year. A 20% cc interest rate is a simple interest rate of f1,2 ā e0.20 ā’ 1 ā 22.14%. This means that your
\$110.52 investment would turn into (1 + 22.14%) Ā· \$110.52 ā \$134.99. This means that the simple interest
rate is r0,2 ā 34.99%. Thus, the cc interest rate is ln(1 + r0,2 ) ā ln(1.3499) ā 30%. Of course, you could have
computed this faster: Vt = e0.10 Ā· e0.20 Ā· V0 = e0.10+0.20 Ā· V0 = e0.30 Ā· \$100 ā 1.3499 Ā· \$100 ā \$134.99.

(All answers should be treated as suspect. They have only been sketched, and not been checked.)
CHAPTER 5
Uncertainty, Default, and Risk

Promised vs. Expected Returns; Debt vs. Equity
last ļ¬le change: Feb 23, 2006 (14:17h)

last major edit: Mar 2004, Nov 2004

We now enter the world of uncertaintyā”though we shall still pretend that we live in a perfect
world of no taxes, no transaction costs, no diļ¬erences of opinion, and inļ¬nitely many investors
and ļ¬rms.
Net present value still rules, but you will now have to face the sad fact that it is not easy to use
in the real world. It is not the NPV concept that is diļ¬cultā”in fact, you already āalmostā know
it. Instead, it is the present value formula inputsā”the expected cash ļ¬‚ows and appropriate
costs of capital under uncertaintyā”that can be so very diļ¬cult to estimate in the real world.
What does uncertainty really do? There will be scenarios in which you will get more than you
expected and scenarios in which you will get less than you expected. This is the case for almost
all corporate projects. The single-most important insight under uncertainty is that you must
always draw a sharp diļ¬erence between promised (or quoted) and expected returns. Because
ļ¬rms can default on payments or go bankrupt in the future, promised returns are higher than
expected returns.
After setting forth the necessary statistical background, our chapter will cover two important
topics: First, we need to determine how lenders should charge borrowers if there is the possi-
bility of default. Second, once we know about how to handle uncertainty, we can really discuss
the diļ¬erences between the two important building blocks of ļ¬nanceā”debt and equity.

Anecdote: The Ruin of the First Financial System
The earliest known example of widespread ļ¬nancial default occurred in the year of 1788 B.C.E., when King
Rim-Sin of Uruk (Mesopotamia) repealed all loan repayments. The royal edict eļ¬ectively destroyed a system of
ļ¬‚ourishing commerce and ļ¬nance, which was already many thousands of years old! It is not known why he did
so.

83
ļ¬le=uncertainty.tex: LP
84 Chapter 5. Uncertainty, Default, and Risk.

5Ā·1. An Introduction to Statistics

Statistics has a reputation of being the most painful of the foundation sciences for ļ¬nanceā”but
to bet in an uncertain we absolutely need it to describe an uncertain future. Fortunately, although statistics can be
world.
a diļ¬cult subject, if you have ever placed a bet in the past, chances are that you already have
a good intuitive grasp of what you need. In fact, I had already sneaked the term āexpectedā
into previous chapters, even though we only now ļ¬rm up your knowledge of this important
statistical concept.

5Ā·1.A. Random Variables and Expected Values

The most important statistical concept is the expected value, which is most often just a fancy
A āfairā bet means that
both sides break even if phrase for mean or average. The only necessary clariļ¬cation is that we use āmeansā and
the bet is repeated
āaveragesā for past outcomes and āexpected valueā for future outcomes.
inļ¬nitely many times.

Important: The expected value is just a mean (or average) that is computed
over future outcomes if hypothetical scenarios are repeated (inļ¬nitely) often.

For example, say you toss a coin, which can come up with either heads or tails and with equal
The āExpected Valueā is
the average outcome. probability. You receive \$1 if the coin comes up heads and \$2 if the coin comes up tails. Because
An expected value can be
you know that there is a 50% chance of \$1 and a 50% chance of \$2, the expected value of each
an impossible realization.
coin toss is \$1.50ā”repeated inļ¬nitely often, the mean will be exactly \$1.50. Of course, exactly
\$1.50 will never come upā”the expected value does not need to be a possible realization of a
single coin toss.
Statisticians have invented the concept of random variables to make it easier to work with
A random variable is a
number whose uncertainty. A random variable is a variable whose value (i.e., outcome) has not yet been de-
realization is not yet
termined. In the coin toss example, we can deļ¬ne a random variable named c (for ācoin toss
known.
outcomeā) that takes the value \$1 with 50% probability and the value \$2 with 50% probability.
The expected value of c is \$1.50. To distinguish a random variable from an ordinary non-
random variable, we use a tilde over the variable. To denote the expected value, we use the
notation E. So, in this bet,

E (Ė) = 50% Ā· \$1 + 50% Ā· \$2 = \$1.50
c
(5.1)
Expected Value(of Coin Toss) = Prob( Heads ) Ā· \$1 + Prob( Tails ) Ā· \$2 .

After the coin has been tossed, the actual outcome c could, e.g., be

(5.2)
c = \$2 ,

and c is no longer a random variable. Also, if you are certain about the outcome, perhaps
because there is only one possible outcome, then the actual realization and the expected value
are the same. The random variable is then really just an ordinary non-random variable. Is the
expected outcome of the coin toss a random variable? No: we know the expected outcome is
\$1.50 even before we toss the coin. The expected value is known, the uncertain outcome is not.
The expected value is an ordinary non-random variable; the outcome is a random variable. Is
the outcome of the coin throw after it has come down heads a random variable? No: we know
what it is (heads), so it is not a random variable.
ļ¬le=uncertainty.tex: RP
85
Section 5Ā·1. An Introduction to Statistics.

Figure 5.1. A Histogram For a Random Variable With Two Equally Likely Outcomes, \$1 and
\$2.

100%
T
Probability

50%

E
\$0 \$1 \$2
Outcome

A random variable is deļ¬ned by the probability distribution of its possible outcomes. The Computing expected
values from
coin throw distribution is simple: the value \$1 with 50% probability and the value \$2 with 50%
distributions.
probability. This is sometimes graphed in a histogram, as depicted in Figure 5.1.
For practice, let us make this a bit trickier. Assume that there is a 16.7% chance that you will An example with 3
possible outcomes.
get \$4, a 33.3% chance that you will get \$10, and a 50% chance that you will get \$20. You can
getting Ā  if it comes up Ā¢ q Ā”, \$10 if it
Ā£Ā
simulate this Ā payoļ¬ Ā  structure by throwing a dieĀ and Ā  \$4
Ā£q Ā£q Ā£q q Ā£q q Ā£q q
comes up Ā¢ q Ā” or Ā¢ q q Ā”, and \$20 if it comes up Ā¢q q Ā”, Ā¢q q q Ā”, or Ā¢q q Ā”.
qq
Now, a fair bet is a bet that costs its expected value. If repeated inļ¬nitely often, both the person If it costs its expected
value to buy a bet, the
oļ¬ering the bet and the person taking the bet would expect to end up even. What is a fair price
bet is fair. An example
Ė Ė
for our die bet? Call the uncertain payoļ¬ D. First, you must determine E(D). It is of a fair bet.

Ė
E (D) = Ā·
16.7% \$4

+ Ā·
33.3% \$10

+ Ā· = \$14
50.0% \$20
(5.3)
Prob( Ā¢ q Ā” (Payout if Ā¢ q Ā”)
Ā£Ā  Ā£Ā
Ė
E (D) = Ā·
)
Ā£q Ā  Ā£q Ā  Ā£q Ā  Ā£q Ā
Prob( Ā¢ q Ā” or Ā¢ q q Ā” (Payout if Ā¢ q Ā” or Ā¢ q q Ā”)
+ Ā·
)
Ā£q q Ā  Ā£q q Ā  Ā£q qq Ā  Ā£q q Ā  Ā£q q Ā  Ā£q qq Ā
+ Prob( Ā¢q q Ā” or Ā¢q q q Ā”or Ā¢q q Ā” Ā· (Payout if Ā¢q q Ā” or Ā¢q q q Ā” or Ā¢q
q q q Ā”) .
)

Therefore, if you repeat this bet a million times, you would expect to earn \$14 million. (On
average, each bet would earn \$14, although some sampling variation in actual trials would
make this a little more or less.) If it costs \$14 to buy each bet, it would be a fair bet.
Generally, you compute expected values by multiplying each outcome by its probability and The expected value is the
Ė probability weighted
adding up all these products. If X is a random variable with N possible outcomes, named X1
sum of all possible
through XN , then you would compute outcomes.

Ė Ė Ė Ė (5.4)
E (X) = Prob( X = X1 ) Ā· X1 + Prob( X = X2 ) Ā· X2 + Ā· Ā· Ā· + Prob( X = XN ) Ā· XN .

This is the formula that you used above,

Ė
E (D) = 16.7% Ā· \$4 + 33.3% Ā· \$10 + 50% Ā· \$20 = \$14 .
(5.5)
Ė Ė Ė
= Prob( D = D1 ) Ā· D1 + Prob( D = D2 ) Ā· D2 + Prob( D = D3 ) Ā· D3 .
ļ¬le=uncertainty.tex: LP
86 Chapter 5. Uncertainty, Default, and Risk.

Note that N could be a trillion possible diļ¬erent outcomes, and many of them could be impos-
sible, i.e., have probabilities of zero.

Important: You must understand

1. the diļ¬erence between an ordinary variable and a random variable;

2. the diļ¬erence between a realization and an expectation;

3. how to compute an expected value, given probabilities and outcomes;

4. what a fair bet is.

We will consider the expected value the measure of the (average) reward that we expect to
There are also summary measures of spread, often used as measures of risk. They will play
standard deviation is the starring roles in Part III, where I will explain them in great detail. For now, if you are curious,
most common measure
think of them as measures of the variability of outcomes around your expected mean. The
most common measure is the standard deviation, which takes the square-root of the sum of
squared deviations from the meanā”a mouthful. Letā™s just do it once,

Ė
Sdv(D) = 16.7% Ā· (\$4 ā’ \$14)2 + 33.3% Ā· (\$10 ā’ \$14)2 + 50% Ā· (\$20 ā’ \$14)2 ā \$6.3 .

Ė Ė Ė Ė Ė Ė
= Prob( D = D1 ) Ā· [D1 ā’ E (D)]2 + Prob( D = D2 ) Ā· [D2 ā’ E (D)]2 + Prob( D = D3 ) Ā· [D3 ā’ E (D)]2
(5.6)
Losely speaking, this adds to your summary that describes that you expect to earn \$14 from a
single die throw that you expect to earn \$14 plus or minus \$6.3.
Solve Now!
Q 5.1 Is the expected value of a die throw a random variable?

Q 5.2 Could it be that the expected value of a bet is a random variable?

q qq
Q 5.3 An ordinary die came up with a Ā¢q q Ā” yesterday. What was its expected outcome before the
Ā£Ā
q
throw? What was its realization?

Q 5.4 A stock that has the following probability distribution (outcome P+1 ) costs \$50. Is an
investment in this stock a fair bet?

Prob Prob Prob Prob
P+1 P+1 P+1 P+1
5% \$41 20% \$45 20% \$58 5% \$75
10% \$42 30% \$48 10% \$70
ļ¬le=uncertainty.tex: RP
87
Section 5Ā·1. An Introduction to Statistics.

5Ā·1.B. Risk Neutrality (and Risk Aversion Preview)

Fortunately, the expected value is all that we need to learn about statistics until we will get Choosing investments
on the basis of expected
to Part III (investments). This is because we are assumingā”only for learning purposesā”that
values is assuming
everyone is risk-neutral. Essentially, this means that investors are willing to write or take any risk-neutrality.
fair bet. For example, you would be indiļ¬erent between getting \$1 for sure, and getting either
\$0 or \$2 with 50% probability. And you would be indiļ¬erent between earning 10% from a risk-
free bond, and earning either 0% or 20% from a risky bond. You have no preference between
investments with equal expected values, no matter how safe or uncertain they may be.
If, instead, you were risk-averseā”which you probably are in the real worldā”you would not Risk aversion means you
would prefer the safe
want to invest in the more risky alternative if both the risky and safe alternative oļ¬ered the
project, so you would
same expected rate of return. You would prefer the safe \$1 to the unsafe \$0 or \$2 investment. demand an extra kicker
You would prefer the 10% risk-free Treasury bond to the unsafe corporate bond that would pay to take the riskier
project.
either 0% or 20%. In this case, if I wanted to sell you a risky project or bond, I would have to
oļ¬er you a higher rate of return as risk compensation. I might have to pay you, say, 5 cents to
get you to be willing to accept the project that pays oļ¬ \$0 or \$2 if you can instead earn \$1. Or,
I would have to lower the price of my corporate bond, so that it oļ¬ers you a higher expected
rate of return, say, 1% and 21% instead of 0% and 20%.
Would you really worry about a bet for either +\$1 or ā’\$1? Probably not. For small bets, you Where risk-aversion
matters and where it
are probably close to risk-neutralā”I may not have to oļ¬er even one cent to take this bet. But
does not matter.
what about a bet for plus or minus \$100? Or for plus and minus \$10,000? My guess is that you
would be fairly reluctant to accept the latter bet without getting extra compensation. For large
bets, you are probably fairly risk-averseā”I would have to oļ¬er you several hundred dollars
to take this bet. However, your own personal risk aversion is not what matters in ļ¬nancial
markets. Instead, it is an aggregate risk aversion. For example, if you could share the \$10,000
bet with 100 other students in your class, the bet would be only \$100 for you. And some of
your colleagues may be willing to accept even more risk for less extra moneyā”they may have
healthier bank accounts or wealthier parents. If you could lay bets across many investors, the
eļ¬ective risk aversion would therefore be less. And this is exactly how ļ¬nancial markets work:
the aggregate risk absorption capability is considerably higher than that of any individual. In
eļ¬ect, ļ¬nancial markets are less risk averse than individuals.
We will study risk aversion in the investments part of the book. There, we will also need to The tools we shall learn
now will remain
deļ¬ne good measures of risk, a subject we can avoid here. (Appendix 2Ā·4 provides more ad-
applicable under
vanced statistical background, if you are interested.) But, as always, all tools we learn under risk-aversion.
the simpler scenario (risk-neutrality) will remain applicable under the more complex scenario
(risk-aversion). And, in any case, in the real world, the diļ¬erence between promised and ex-
pected returns that we shall discuss in this chapter is often more important (in terms of value)
than the extra compensation for risk that we shall ignore in this chapter.
ļ¬le=uncertainty.tex: LP
88 Chapter 5. Uncertainty, Default, and Risk.

5Ā·2. Interest Rates and Credit Risk (Default Risk)

Most loans in the real world are not risk-free, because the borrower may not fully pay back
what was promised. So, how do we compute appropriate rates of returns for risky bonds?

5Ā·2.A. Risk-Neutral Investors Demand Higher Promised Rates

Put yourself into the position of a banker. Assume a one-year Treasury bond oļ¬ers a safe
interest rate from an annual rate of return of 10%. You are about to lend \$1 million to a risky company. The loan
investor as from the
maturity is one year. You must decide what interest rate to charge on the loan. If you are 100%
Treasury if payment is
certain that the lender will fully pay the agreed-upon amount, you can earn as much charging
certain.
a 10% interest rate from the borrower as you can from buying the Treasury bond. Both provide
\$1,100,000 in repayment.
However, in the real world, there are few if any borrowers for whom you can be 100% certain
If you quote a risky
borrower a particular that they will fully repay a loan. Letā™s take an extreme example. Assume you think there is a
interest rate, you must
50% chance that the company will default (fail to pay all that it has promised), and the borrower
expect to earn a lower
can only pay back \$750,000. There is also a 50% chance that the company will pay back the
interest rate.
principal plus interest (in which case the company is called solvent). In this case, if you charge
a 10% interest rate, your expected payout would be

Ā· + Ā· = \$925, 000 .
50% \$750, 000 50% \$1, 100, 000
(5.7)
Prob(Default) Ā· Payment if Default + Prob(Solvent) Ā· Payment if Solvent

So your expected return would not be \$1,100,000, but only \$925,000. Your expected rate of
return would not be +10%, but only \$925, 000/\$1, 000, 000 ā’ 1 = ā’7.5%. Extending such a
loan would not beā”pardon the punā”in your best interest: you can do better by putting your
\$1,000,000 into government Treasury bills.
So, as a banker, you must demand a higher interest rate from risky borrowers, even if you just
higher promised want to ābreak evenā (i.e., earn the same \$1,100,000 that you could earn in Treasury bonds). If
you solve
in good timesā”in order
to make up for default
risk.
50% Ā· + 50% Ā· (promised repayment) =
\$750, 000 \$1, 100, 000
(5.8)
Prob Ā· Payment if Default + Prob Ā· = Treasury Payment
Payment if Solvent

for the desired promised repayment, you ļ¬nd that you must ask the borrower for \$1,450,000.
The promised interest rate is therefore \$1, 450, 000/\$1, 000, 000 ā’ 1 = 45%. Of this 45%, 10%
is the time premium. Therefore, we can call the remaining 35% the default premiumā”the
diļ¬erence between the promised rate and the expected rate that allows the lender to break
even.
We rarely observe expected rates of return directly. Newspaper and ļ¬nancial documents al-
Common terminology
most always provide only the promised interest rate, which is therefore also called the quoted
interest rate or the stated interest rate. (The Yield-to-Maturity, YTM, also is usually merely a
promised rate, not an expected rate.) On Wall Street, the default premium is often called the
credit premium, and default risk is often called credit risk.
Solve Now!
Q 5.5 For what kind of bonds are expected and promised interest rates the same?
ļ¬le=uncertainty.tex: RP
89
Section 5Ā·2. Interest Rates and Credit Risk (Default Risk).

5Ā·2.B. A More Elaborate Example With Probability Ranges

Now work a similar but more involved example in which we assume that you want to lend Borrowers may
sometimes not be able
money to me. Assume I am often not able to pay you back, despite my best intentions and
to repay.
promises. You believe I will pay you with 98% probability what I promise, with 1% probability
half of what I borrowed, and with 1% probability nothing. I want to borrow \$200 from you, and
you could alternatively invest the \$200 into a government bond promising a 5% interest rate,
so you would receive \$210 for certain. What interest rate would you ask of me?

Table 5.1. Risky Loan Payoļ¬ Table: 5% Promised Interest Rate

How Likely
Flow CFt=1 return of (Probability)
\$210 5.0% 98% of the time
\$100 ā“52.4% 1% of the time
\$0 ā“100.0% 1% of the time

Table 5.1 shows that if you ask me for a 5% interest rate, your expected cash return will be If you ask me to pay the
risk-free interest rate,
you will on average earn
Ė
E (Ct=1 ) = Ā·
98% \$210 less than the risk-free
interest rate.
+ Ā·
1% \$100

+ Ā· = \$206.80
1% \$0
(5.10)
Ė
E (Ct=1 ) = Prob CFt=1 will be case 1 Ā· CFt=1 cash ļ¬‚ow in case 1
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