Unfortunately, we are now encountering a new hindrance to progress in ¬nance. Financial

institutions have come to consider their data to be their proprietary competitive advantage.

Fear of legal liability is further limiting the data that becomes available for public study”and

given the litigiousness of U.S. society, justly so. Sadly, many of the most interesting questions

in ¬nance therefore may no longer be researchable or answerable.

The fact that we do not have all the answers is good news and bad news. The bad news is

that we will never fully understand ¬nancial markets and individuals. The good news is that

our knowledge will continue to improve, and that there is plenty of space for new and exciting

research in ¬nance. For me, this means ¬nance is still intellectually challenging enough to

remain “fun.”

¬le=epilogue.tex: RP

779

Section A·3. Finance Research.

A·3. Finance Research

Finance research is not just for aspiring academics: consulting ¬rms are basically research ¬rms.

Academics and consultants may have di¬erent audiences, production speeds, team systems,

and evaluation processes, but they both research issues of interest to business and do so using

similar methodologies. There is also much cross-fertilization: many professors work regularly

with major consulting ¬rms”and some have even quit Academia altogether and departed for

higher paying jobs in consulting.

A·3.A. Accomplishments of Finance

Rather than taking up space here, let me just refer you to my paper called The Top Achieve-

ments, Challenges, and Failures of Finance, available for free download at the book™s website

or the Social Science Research Network (www.ssrn.com).

A·3.B. Interesting Current Academic Research

Fortunately, ¬nance is by nature a very applied discipline. If you have read this book, you

already understand the main questions and problems in ¬nance and ¬nancial research today.

You do not need a higher ¬nance degree. Unfortunately, academic ¬nance journals (and many

academics) love obscure jargon and algebra. It may or may not require some extra training in

“language” for you to follow the writeups of academic papers in academic journals. But, in the

end, with just a little bit of extra jargon, you should be able to pick up the important journals

and understand the most cutting-edge and interesting research ideas in ¬nance today.

A·3.C. Getting Involved in Academic Research

My own recommendation to an aspiring student of ¬nance is ¬rst to learn what the top profes-

sors (and especially the younger professors) in your own school are working on. Then, browse

SSRN for current working papers. Finally, you should work for a professor in your ¬nance or

economics department, even if it is unpaid”though you should pick a professor who does not

have too many assistants already. You will learn more in this one-on-one contact than you will

learn from taking many classes.

A·3.D. Finance Degrees

The most common ¬nance degree in many of the top schools is the M.B.A. with a specialization

in ¬nance. But increasingly, many universities, such as UC/Berkeley, Princeton, and Wharton,

are o¬ering undergraduate degrees in ¬nance. The Harvard economics department may well

feature the best ¬nance department in the world right now, and it teaches only undergradu-

ates and Ph.D. students. Similarly, universities like Brown and the University of Chicago are

just beginning to expand ¬nancial economics curricula into undergraduate education. Finance

de¬nitely quali¬es as a subject with no less intellectual rigor than economics, and no more of

a specialization/vocational education component than, say, pre-med or biochemistry.

There are also some other programs that o¬er masters programs in ¬nance, e.g., the N.Y.U. pro-

gram in mathematical ¬nance, o¬ered by the Courant Institute. Typically, these programs have

a bent towards ¬nancial engineering. Their graduates tend to come from speci¬c backgrounds

(usually some other engineering discipline), and their graduates tend to work in speci¬c types

of jobs (typically in derivatives and ¬xed income modeling). Finally, there is the Ph.D. track,

discussed next.

¬le=epilogue.tex: LP

780 Chapter A. Epilogue.

A·3.E. Academic Careers in Finance and Economics: A Ph.D.?

Finance is a sub¬eld of economics. About one-third of its professors have an economics Ph.D.

instead of a ¬nance Ph.D. Either degree is su¬cient”although it is imperative for the future

academic to have solid grounding in both disciplines.

The typical Ph.D. program in ¬nance takes between 4 and 8 years. Unlike most degree programs,

success is not guaranteed. About one-third of accepted students drop out, typically after 2

to 4 years”not a cheap outcome. Although qualifying exams, usually taken in the ¬rst two

years of the program, are very challenging, the biggest hurdle for almost every Ph.D. student

to overcome is the transition from classroom work to academic research. This is a Gordian

knot, and success is di¬cult to predict. Although intelligence and smarts are necessary, it

is not mathematical sophistication that determines success. Very little of ¬nance uses more

than plain algebra”although it does use lots of it. Instead, the successful Ph.D. student must

develop a problem-relevant intuition and creativity. If I only knew how to translate this skill

into a recipe!

Although the ¬rst 4 years in Ph.D. programs are usually paid for by full stipends by the uni-

versity, the opportunity costs and the uncertainty of ultimate success mean that only the most

intellectually interested students will ¬nd a Ph.D. program to be a rewarding endeavor. For the

successful graduate, job opportunities tend to be plenty and lucrative. Even academic careers

are not exactly a vow of poverty. In 2003, the typical ¬rst year Assistant Professor in a top

business school earned somewhere between $130,000 and $180,000 per year. Industry jobs

in ¬nancial or consulting institutions sometimes pay more even in the ¬rst year, but their big

advantage is that salaries tend to escalate far more rapidly than those in Academia in subse-

quent years. Finally, many economics and ¬nance Ph.D.s pursue governmental careers, e.g., at

the International Monetary Fund or the World Bank.

It is very encouraging that many universities and institutions today conduct terri¬c academic

research in ¬nance and economics. Thirty years ago, only a handful of schools were able to

produce great papers, but this time of exclusivity has passed. This does not mean that there

are no di¬erences in average academic quality. I will volunteer here my personal impression

of the rank order of academic ¬nance departments today, which is based on the tendency of

departments to successfully attract faculty from other departments. In my opinion, the top

academic department today is the University of Chicago. It is followed closely by “Cambridge,”

which is really the combination of Harvard (economics and ¬nance) and M.I.T. (economics and

¬nance). A large number of schools vie for the ranking spots right after. Among them, but

not exclusively, are (in alphabetical order) Columbia, Duke, N.Y.U., Northwestern, Stanford,

U.C./Berkeley, U.C.L.A., Wharton, and Yale. These schools each have their unique advantages

and disadvantages, and regularly succeed in stealing faculty from one another.1 There are also

a large number of excellent schools, many of which have individual faculty who are every bit as

good as some faculty at, say, Chicago, but which typically do not have the same overall average

academic quality or resources.

The average quality of a ¬nance or economics department can make an important di¬erence

for Ph.D. students, however. They bene¬t greatly from the variety of interaction. Therefore, a

Ph.D. from any top academic institution would make an excellent springboard into a top-notch

academic economics or ¬nance department, or into a very high-quality investment or consulting

career.

1

I almost surely have omitted some schools by mistake.

¬le=epilogue.tex: RP

781

Section A·3. Finance Research.

A·3.F. Being a Professor ” A Dream Job for the Lazy?

So, what does a professor do? Multiply the number of classes per year by the hours per class,

and you arrive at a number of 120“180 hours per year. Is being a ¬nance professor the ultimate

dream job for the lazy?

Sorry to disappoint you”the opposite is the case. The classroom hours during which you see

your instructor are just a small part of the job”most comparable perhaps to the small number

of hours in which a litigation lawyer is in the courtroom. The rule of thumb is that every hour of

teaching of a new course requires about ten hours of preparation. This includes topic selection,

comparative evaluations of various textbooks, reading of the relevant literature, preparation

of slides and homeworks, and so on. Many ¬nance professors do not teach exactly what is

in any one textbook, but inform themselves about what they should teach, how their material

¬ts together in one coherent set, what relevant papers have recently appeared in the literature,

what relevance their courses and subjects have to current events and their own locale and

audience, where they think the textbooks are wrong, how their ¬nance courses relate to other

academic areas, and so on. (Fortunately, once prepared, a course would take only about two

hours of preparation for each hour of teaching.) Add this all up, and the 150 hours have already

increased to about 600“800 hours. In addition to course preparation and lecturing, there are

class handling tasks, o¬ce hours, teaching assistance coordination, and grading. This easily

adds another 100 hours per year. Finally, many ¬nance professors get roped into holding

speeches at school events, and giving lectures not within the context of their regular classes.

So, a typical ¬nance professor may spend about 800 to 1,000 hours per year on teaching related

issues.

Is this it? Of course not! Tenure-track ¬nance professors are promoted based on their research.

Where do you believe the insights in this book have originally come from? Yes, most ¬nancial

concepts are now heavily used in practice, so even practitioners know them”especially if they

were taught concepts in their own academic training decades ago”but it is the academic pub-

lished research that is responsible for 99% of what you have read in this book. After all, if

smart practitioners invent something useful, they do not teach it”they keep it secret and try

to sell it. So how much time do professors spend on creating knowledge? Writing an academic

paper can take anywhere from 100 hours to 1,000 hours. I know this from painful experience,

having written papers that fall into both extremes of this spectrum. Moreover, a good amount

of research ¬‚ops and thus never ends up in a published paper. After all, this is why it is called

research and not development! In total, a research-active professor will publish one or two pa-

pers per year spending about 500 to 1,500 hours per year on research. Attending conferences

and seminars that are necessary to keep up with the profession and publicize one™s work may

require another 100 hours.

Is this it? Sorry, still no. There is service. Students need advising”undergraduate students,

masters students, and Ph.D. students. Universities are governed by the faculty and run by com-

mittees that need to be sta¬ed. Alumni and potential donors need to be charmed. Depending

on the particular university and one™s particular role, this can be anything from 2 hours per

week to 10 hours per week. In-school service therefore sums to another 100-500 hours per

year.

For all of the aforementioned tasks, you may be able to catch your professor in the act”that

is, you are the direct bene¬ciary or may be present when he or she is spending time working

thereon. However, an important part of a professor™s job is service to the profession over-

all. Academia lives by peer evaluation. This applies both to papers and careers. Journals

need referees to judge papers. Schools need outsiders to write academic letters for promotion.

Refereeing a paper or writing a reference for another professor at another university (should)

take at least a day (10 hours). Di¬erent professors get di¬erent number of external evaluation

requests”my own number sits at about 30 per year, consuming about 300 hours per year of

my time. There is very little direct reward for doing a good, conscientious job on refereeing and

referencing, but it is necessary to make Academia work. As an external referee or evaluator,

you are also literally making or breaking someone else™s career. It is every professor™s duty to

take these tasks very seriously.

¬le=epilogue.tex: LP

782 Chapter A. Epilogue.

Putting this all together, my typical year has about 2,500-3,000 hours of work per year. On

an hourly basis, my compensation would probably be ¬ve times higher if I worked for a top

consulting ¬rm or investment bank. (If you read my chapter on ethics, in which I more or

less describe economics as the science of “pro¬t maximization” with little concern for others, I

hope you will see the irony.) So, why do I work for a university? Simple”I love my work. I love

teaching students, I love writing research papers, and I love having the relative independence

to do what I want to do that only an academic job can provide. Yes, not every single task is

enjoyable, but overall, it is the best job for me.

Now I must admit that telling you all about what I do in a typical year had a second hidden

agenda. I want you to understand the di¬erence between a full-time professor and a part-

time professor. Understanding the full scope of professorial obligations will hopefully make

you appreciate why you need “the real deal.” Yes, both faculty and students can bene¬t from

some lecturers who know practice well, who are only teaching what they themselves learned in

their programs (often decades ago, though supplemented with their practical experience), and

who do not participate in academic research and in the running of the university and of the

academic profession. In fact, many lecturers are very valuable, both to the research faculty and

to the students. They can complement our academic knowledge with some practical experience.

And a small number start out as lecturers and over time turn into full faculty and excellent

researchers. But it is the regular faculty that remains the backbone of ¬nancial economics”

who provide you with new knowledge to navigate the broad continent of ¬nance over the next

few years.

A·3.G. Top Finance Journals

The top academic journals in ¬nance today are The Journal of Finance, the Journal of Financial

Economics, the Journal of Financial and Quantitative Analysis, the Review of Financial Studies,

and the Journal of Business. However, there are also many other good outlets for academic

research. For example, economics journals have published some of the most in¬‚uential work

in ¬nance. Other journals are written with more of a practitioner audience in mind, such as

Financial Analysts Journal.

Although numbers do not tell the whole story (it is impact that counts!), the tenure standards

for professors range from about 8“10 papers in the top journals for a Chicago professor, to 5“7

papers for a school ranking at around #10, to 3“5 for a school ranking at around #30. The top

journals have rejection rates of about 90%. A successful academic will write about 2“3 papers

per year, but publish only one of them in a top journal.

¬le=epilogue.tex: RP

783

Section A·4. Bon Voyage.

A·4. Bon Voyage

Our book has covered the principles of ¬nance in some depth and breadth. You can trust me

when I say that if you have read and understood these chapters, you are very well prepared for

the next steps in your ¬nance/business education. (You can choose your next courses á la carte:

investments, derivatives, corporate ¬nance, ¬xed income, ¬nancial institutions, international

¬nance, or something else. If you are still curious to learn more, visit the book™s web site at

http://welch.econ.brown.edu/book.)

But even more important to me than teaching you ¬nance has been teaching you how to ap-

proach problems: when you need to solve a new problem, think in terms of the easiest numer-

ical example that you can come up with, and only then translate whatever you have learned

from your simple example into something more complex”be it a formula or a more complex

scenario. So, if you are facing a new problem, even if you do not know or remember any of

our formulas, given time, you should now be able to “reinvent” them. When you encounter a

complex new problem in your company, do not despair, but gradually work your way up from

the simplest versions.

I have enjoyed writing this book in the same way that I enjoy writing my academic research

papers, and pretty much for the same reason: it has been like solving an intriguing puzzle

that no one else has ¬gured out in quite the same way”a particular way to see and explain

¬nance. Of course, writing it has taken me far longer than I had anticipated”four years and

still counting just for the ¬rst edition.

But it will all have been worth it if you have learned from my presentation. If you have studied

the book, you should now know about 90% of what I know about ¬nance. Interestingly, there

were a number of topics that I thought I had understood, but had not”and it was only my

having to explain it that made this clear to myself. And this brings me to a key point that I

want to leave you with”never be afraid to ask questions, even about ¬rst principles. To do

so is not a sign of stupidity”on the contrary, it is often a sign of deepening awareness and

understanding.

I have no illusions that you will remember all the ¬ne details in this book as time passes”nor

will I. But more than the details, I hope that I will have left you with an appreciation for the big

ideas, an arsenal of tools, a method to approach novel problems, and a new perspective. You

can now think like a ¬nancier.

Ivo Welch

¬le=epilogue.tex: LP

784 Chapter A. Epilogue.

APPENDIX B

More Resources

An NPV Checklist, Some Data URL Links, Algebra, Statistics, and Portfolios

last ¬le change: Feb 19, 2006 (12:33h)

last major edit: n/a

785

¬le=appendix.tex: LP

786 Chapter B. More Resources.

2·1. An NPV Checklist

The NPV formula is easy. For most projects, its application is hard. It is usually very di¬cult to

Here is an abbreviated

list of issues to worry estimate future cash ¬‚ows (and even their appropriate interest rates), especially for far-in-the-

about when using NPV.

future returns. It is usually more important and more di¬cult to avoid errors for the expected

cash ¬‚ow (the NPV numerator) than it is for the cost of capital (the NPV denominator). The NPV

formula is less robust to cash ¬‚ow errors than it is to cost of capital (r ) errors, and it is “easier”

to commit dramatic errors in the cash ¬‚ow estimation than in the cost of capital estimation.

Here is an abbreviated checklist of items to consider when working out NPV estimates.

Real After-Tax Dollars (Page 138, Page 141, Page 141):

Have all relevant inputs and outputs been quoted in what-is-relevant-to-you after-tax

dollars? This applies to both expected cash ¬‚ows and to appropriate discount rates.

Has in¬‚ation been properly included? Preferably, have all computations used nomi-

nal expected future cash ¬‚ows and nominal costs of capital, with in¬‚ation used only

to gross up nominal cash ¬‚ows appropriately?

Interactions (Page 154, Page 441):

Have all projects been properly credited with their contributions, positive or negative,

to the values of other projects (externalities)?

Have all projects been judged “on the margin,” i.e., without charging them for unal-

terable or previously made choices, such as sunk costs, overhead, etc.?

Has the cost of capital applicable to each project component, respectively, been used,

and not the (incorrect) overall average cost of capital? (Note: some errors and sim-

pli¬cations here are unavoidable in the real world, because it is impossible to put a

di¬erent cost of capital on each paper clip.)

Conditionals (Strategic Options) (Page 165, and in the Web Chapter on Options and Deriva-

tives):

Have all possible future options been considered (using scenario analyses) in order

to ¬nd the correct expected cash ¬‚ows, e.g.:

The ability to leverage a product into future markets?

The ability to ¬nd product spino¬s?

The ability to learn about (how to do) future products?

The ability to stop the project if conditions are bad.

The ability to delay the project if conditions are bad.

The ability to mothball the project if conditions are bad and restart the project

if conditions improve.

The ability to accelerate the project if conditions are good.

The ability to expand the project if conditions are good.

¬le=appendix.tex: RP

787

Section 2·1. An NPV Checklist.

Accuracy (Page 104, Page 175, Page 176, Page 211, Page 449):

How accurate are the estimated project cash ¬‚ows?

If project success and project cash ¬‚ows were estimated by someone else, what are

the motives of the estimator? Does the estimator want the project taken or rejected?

Can the cash ¬‚ow estimates be improved by doing more research?

Is it possible to get another independent evaluation/audit of the project estimates?

Given unavoidable simpli¬cations, assumptions, and errors, how sensitive/robust is

the NPV computation to changes?

Correct Inputs (Page 163):

Are the cash ¬‚ows expected, rather than just promised? Are the interest rates ex-

pected, rather than promised? (Recall: expected interest rates are below promised

interest rates due to default premia, not just due to risk premia.)

Are the expected cash ¬‚ows the “average outcome” (correct!), and not the “most likely

outcome”?

Do the expected cash ¬‚ow estimates include the correct weighted probabilities of

low-probability events, especially for negative outcomes?

If money needs to be borrowed to execute the project, is the used cost of capital r

the borrowing rate? If capital is already available, is the used cost of capital r the

lending (investments) rate?

Corporate Income Taxes (Page 552):

For use of WACC and APV, is the numerator in the NPV calculation the expected cash

¬‚ows “as if all equity ¬nanced”? (This means that the company bears the full brunt

of its corporate income tax load.)

“ In the weighted cost of capital, is the debt cost of capital the expected (not the

promised!) interest rate on debt? Is the numerator the expected cash ¬‚ow, not the

promised cash ¬‚ow?

A ¬nal warning: although many of these issues seem obvious in isolation, they are much harder

to spot and take care of in complex real-world situations than in our highlighted expositions.

Watch out! The most common error is worth its own box:

Important: The most common NPV method is to estimate cash ¬‚ows for the

numerator, and to use an expected rate of return (cost of capital) from the CAPM

formula (see Chapter 17).

The default risk is handled only in the numerator, i.e., in the computation of

expected cash ¬‚ows.

The time-premium and risk-premium are handled only in the denominator.

The CAPM formula provides an expected rate of return, which contains only

these two components.

Do not try to adjust the numerator for the time premium or the risk premium.

Do not try to add a default-premium to the rate of return in the denominator.

(This would yield a promised, not an expected rate of return on capital.) Do

not believe that by using the CAPM expected rate of return, you have taken

care of the default risk.

Q B.1 Recall as many items from the NPV checklist as you can remember. Which are you most

likely to forget?

¬le=appendix.tex: LP

788 Chapter B. More Resources.

2·2. Prominently Used Data Websites

The following data and information websites have been prominently used in this book. (If you are reading this on

the Acrobat reader , you can click on the links!) Please note that the list is not complete, and that the links may have

changed by the time you read this.

Overall Market Information

Marketgauge.com Various market gauges (incl. S&P500 dividend http://tal.marketgauge.com/dvmgpro/gauges/dvplast.htm

and earnings yields).

Yahoo!Finance Stock and Index Quotes, Current and Historical http://quote.yahoo.com

Yahoo!Finance Current Interest Rates http://bonds.yahoo.com/rates.html

CNN Money General Information and Quotes http://money.cnn.com

Federal Reserve Data Historical Interest Rates http://www.federalreserve.gov/releases/h15/data.htm

Fred (Federal Reserve) U.S. economic time series, macroeconomic and http://research.stlouisfed.org/fred

¬nancial.

SmartMoney Animated Yield Curve http://www.smartmoney.com/onebond/index.cfm?story=yieldcurve

Treasury Direct In¬‚ation Protected Interest Rates http://www.publicdebt.treas.gov/gsr/gsrlist.htm

Treasury “ Debt O¬ce of Public Debt http://www.publicdebt.treas.gov/

Bloomberg Index rates (incl. muni bonds) http://www.bloomberg.com/markets/rates/index.html

R. Shiller™s Website Very long-run indexes http://aida.econ.yale.edu/∼shiller/

Yahoo!Finance Foreign ¬nancial market websites e.g., Germany: http://de.¬nance.yahoo.com/

Individual Stock and Fund Information

Yahoo!Biz Firm-speci¬c corporate pro¬les (earnings, http://biz.yahoo.com/p/i/ibm.html

sales, etc.). Here, IBM.

Edgar All public corporate SEC ¬lings http://www.edgar.sec.gov

PWC Price-Waterhouse-Coopers™ Edgarscan http://edgarscan.pwcglobal.com/EdgarScan/

Vanguard Funds Information http://www.vanguard.com

PepsiCo Investor Information http://www.pepsico.com/investors/

PepsiCo Annual Reports http://www.pepsico.com/investors/annual-reports/

PepsiCo 2000 10-K Filing http://www.pepsico.com/¬lings/200010k.shtml

Other Information

Ivo Welch General Website http://welch.econ.brown.edu/

SSRN Finance Working Papers http://www.ssrn.com

AFA The American Finance Association http://www.afajof.org

AEA The American Economics Association http://www.aeaweb.org

Moody™s Monthly bond default reports. http://riskcalc.moodysrms.com/us/research/mdr.asp

Moody™s Extended report on default rates, 1992. http://riskcalc.moodysrms.com/us/research/defrate/0085.pdf

BankruptcyFinger Bankruptcy Related Information. http://bankruptcy¬nder.com/

BLS Bureau of Labor Statistics (In¬‚ation). http://www.bls.gov/

SEC Securities Exchange Commission http://www.sec.gov

Governance Web Chapter

CalPers Corporate Governance Focus http://www.calpers-governance.org/alert/focus/

International Web Chapter

Barchart Currencies http://www2.barchart.com/mktcom.asp?section=currencies

PACIFIC Exchange Rate Related Information and Data http://paci¬c.commerce.ubc.ca/

Bloomberg Market Indices http://www.bloomberg.com/markets/rates/index.html

The Economist The “Big-Mac” Price Index http://www.economist.com/markets/Bigmac/Index.cfm

¬le=appendix.tex: RP

789

Section 2·3. Necessary Algebraic Background .

2·3. Necessary Algebraic Background

• Finding a base:

32 = 9 ” 3 = 91/2

(B.1)

a 1/a

=b x=b

x .

”

• Finding an exponent:

ln(9)

32 = 9 ”2=

ln(3)

(B.2)

ln(b)

ax = b ”x= .

ln(a)

• Summation Notation:

N

f (i) = f (1) + f (2) + · · · + f (N) . (B.3)

i=1

This should be read as the “sum over all i from 1 to N.” There are N terms in this sum.

i is not a real variable: it is simply a dummy counter to abbreviate the notation. When 1

and N are omitted, it usually means “over all possible i.”

• Summation Rules:

N

[a · f (i) + b] = a · f (1) + b + a · f (2) + b + · · · + a · f (N) + b

®

i=1

(B.4)

N

a·° f (i)» + N · b

= .

i=1

Here is an illustration:

3

5 · ii + 2 5 · 11 + 2 + 5 · 22 + 2 + 5 · 33 + 2

= = 7 + 22 + 137 = 166 . (B.5)

i=1

• The following is not necessary but interesting. A function L(·) is called a linear function,

if and only if L(a + b · x) = a + L(b · x) = a + b · L(x), where a and b are constants.

Here is an illustration. (Weighted) averaging is a linear function. For example, start with

(5,10,15) as a data series. The average is 10. Pick an a = 2 and a b = 3. For averaging to

be a linear function, it must be that

(B.6)

Average(2 + 3 · Data) = 2 + 3 · Average(Data)

Let™s try this”the LHS would become the average of 17, 32, 47, which is 32. The RHS would

become 2+3·10 = 32. So, in our example, averaging indeed behaves like a linear function.

√ √

In contrast, the square-root is not a linear function, because ’2 + 3·9 ≠ ’2 + 3· 9. The

LHS is 5, the RHS is 7.

“ Similar to averaging, expected values are linear functions. This is what has permitted

us to interchange expectations and linear functions:

˜ ˜ (B.7)

E (a + b · X) = a + b · E (X) .

This will be explained in the next section.

“ The rate of return on a portfolio is also a linear function of the investment weights.

For example, a portfolio rate of return may be r (x) = 20% · rx + 80% · ry , where rx

¬le=appendix.tex: LP

790 Chapter B. More Resources.

is the rate of return on the component into which you invested $20. For r (x) be a

linear function, we need

2 + 3 · r (x) = r (2 + 3 · x)

(B.8)

a + b · r (x) = r (a + b · x)

Substitute in

(B.9)

2 + 3 · [20% · rx + 80% · ry ] = 20% · (2 + 3 · rx ) + 80% · (2 + 3 · ry )

Both sides simplify to 2 + 60% · rx + 240% · ry , so our statement is true and a portfolio

return is indeed a linear function.

However, not all functions are linear. The variance is not a linear function, because

˜ ˜ (B.10)

Var(a + b · X) ≠ a + b · Var(X) .

This will also be explained in the next section.

Solve Now!

Q B.2 If (1 + x)10 = (1 + 50%), what is x?

Q B.3 If (1 + 10%)x = (1 + 50%), what is x?

N N

Q B.4 Are xi and xs the same?

s=1

i=1

3

(3 + 5 · x). Is x a variable or just a placeholder to write the

Q B.5 Write out and compute

x=1

expression more conveniently?

« «

3 3

Q B.6 Write out and compute 3 + 5 · y . Compare the result to the previous expres-

y=1 x=1

sion.

« «

3 3 3

(i · i) the same as · i?

Q B.7 Is

i=1 i=1 i=1

¬le=appendix.tex: RP

791

Section 2·4. Laws of Probability, Portfolios, and Expectations.

2·4. Laws of Probability, Portfolios, and Expectations

This section describes some of the algebra that we are using in our investments chapters. The

material is exposited in a more mathematical fashion that in the chapters, which you may ¬nd

easier or harder depending on your background.

2·4.A. Single Random Variables

The Laws of Expectations for single random variables (illustration will follow):

• De¬nition of Expectation

N

˜ ˜

E (X) = Prob(i) · [X = X(i)] (B.11)

i=1

• The expected value of a linear transformation (a and b are known constants):

˜ ˜ (B.12)

E (a · X + b) = a · E (X) + b .

˜

This works because expectation is a linear operator. Similarly, you could rename X as

˜

f (X), so

˜ ˜ (B.13)

E [a · f (X) + b] = a · E [f (X)] + b .

˜ ˜

However, you cannot always “pull” expectations in, so E(f (X)) is not always f (E(X))

For example, if f (x) = x 2 , it is the case that

˜˜ ˜ ˜ (B.14)

E (X · X) ≠ ·E (X) · E (X) .

To see this, consider a fair coin that can be either 0 or 1. E(X 2 ) = 0.5 · 02 + 0.5 · 12 = 0.5,

˜

2 2

˜

but E(X) = (0.5 · 0 + 0.5 · 1) = 0.25.

• De¬nition of Variance:

2

˜ ˜ ˜ (B.15)

Var(X) = E X ’ E (X) .

˜2

˜ ˜

It is sometimes easier to manipulate this formula Var(X) = E X ’ E(X) .

• De¬nition of a Standard Deviation:

˜ ˜ (B.16)

Standard Deviation(X) = Var(X) .

• The variance of a linear transformation (a and b are known constants):

˜ ˜

Var(a · X + b) = a2 · Var(X) . (B.17)

˜

Here is an extended illustration. A coin, outcome called X, with 4 and 8 written on the two

sides. These two outcomes can be written as 4 · i where i is either 1 or 2. So, the expected

˜

value of X is

2

˜ ˜

E (X) = Prob X = (4 · i) · (4 · i)

i=1

(B.18)

˜ ˜

= Prob X = 4 · (4) + Prob X = 8 · (8)

= 50% · 4 + 50% · 8 =6.

¬le=appendix.tex: LP

792 Chapter B. More Resources.

2

2

˜ ˜

Var(X) = Prob X = (4 · i) · [(4 · i) ’ 6]

i=1

(B.19)

˜ ˜

= Prob X = 4 · (4 ’ 6)2 + Prob X = 8 · (8 ’ 6)2

= 50% · 4 + 50% · 4 =4.

The standard deviation is the square root of the variance, here 2.

As we noted earlier, E(X 2 ) is of course not the same as [E(X)]2 = [3]2 = 9, because

˜ ˜

2

˜ ˜

E (X 2 ) = Prob X = (2 · i) · (2 · i)2

i=1

(B.20)

˜ ˜

= Prob X = 2 · (22 ) + Prob X = 4 · (42 )

= 50% · 4 + 50% · 16 = 10 .

˜ ˜

Now we work with a linear transformation of the X, say Z = $2.5 · X + $10. (In ¬nance, the rate

of return on portfolios are such linear transformation; for example, if you own 25% in A and

75% in B you will earn 0.25 · rA + 0.75 · rB + 0.) Thus,

˜ ˜

˜ ˜

Prob X Z

Coin

1/2 Heads 4 $20

1/2 Tail 8 $30

˜ ˜

We want to convince ourselves that the expected value of Z, de¬ned as is $2.5 · X + $10, is

˜ ˜

$2.5 · E(X) + $10 = $25. So, we hand-compute the expected value the long way from Z,

2

˜ ˜ ˜

E (Z) = Prob X = (4 · i) same as Z = $2.5 · X + $10 · (Zi )

i=1

˜ ˜

= Prob X = 4 same as Z = $20 · ($20)

(B.21)

˜ ˜

+ Prob X = 8 same as Z = $30 · ($30)

= 50% · $20 + 50% · $30 = $25 .

Unlike the mean (the expected value), the variance is not a linear function, so the variance of

˜ ˜ ˜ ˜

Z = $2.5 · X + $10 is not $2.5 · Var(X) + $10 = $2.5 · 4 + $10 = $20. Instead, Var(Z) =

Var(a · X + c) = a2 · Var(X) = ($2.5)2 · Var(X) = $2 · 2.52 · 4 = $2 25. We can con¬rm this

˜ ˜ ˜

˜

working with Z directly:

2

2

˜ ˜ ˜ ˜

Var(Z) = Prob X = (4 · i) · (Zi ) ’ E (Z)

i=1

˜ ˜

Prob X = 4 same as Z = $20 · ($20 ’ $25)2

=

(B.22)

˜ ˜

+ Prob X = 8 same as Z = $30 · ($30 ’ $25)2

50% · ($5)2 + 50% · ($5)2 = $2 25

= .

˜ $2 · 25 = $5.

The standard deviation of Z is therefore

˜

You should con¬rm Formula B.12: the expected value of Z should be ($5) times the expected

˜ ˜ ˜

value of X plus $10. Con¬rm Formula B.17: the variance of Z should be the variance of X

multiplied by ($5) squared.

¬le=appendix.tex: RP

793

Section 2·4. Laws of Probability, Portfolios, and Expectations.

Solve Now!

Q B.8 What is the expected value and standard deviation of a bet B that pays o¬ the number of

points on a fair die, squared? For example, if the die comes down 3, you receive $9.

Q B.9 Assume that you have to pay $30, but you receive twice the outcome of the previous bet

˜ ˜

˜ ˜

B. This is a new bet, called C. That is, your payo¬ is C = ’$30 + 2 · B. What is the expected

payo¬ and risk of your position? Make your life easy!

2·4.B. Portfolios

Portfolios are de¬ned as follows (illustration will follow):

rP ≡ wi · ri ,

˜ ˜ (B.23)

i

where wi is the known investment weights in security i and ri is the security return on security i.

˜

Unlike the above, simpler de¬nitions, portfolios are the weighted sum of multiple random

variables.

• Portfolio Expectations «

E wi · ri = wi · E (˜i ) .

r

˜ (B.24)

i i

Although the weights are ¬xed and known constants, they cannot be pulled out of the

summation, because they are indexed by i (each could be di¬erent from the others).

• ±

«

N

N

Var wi · ri = wi · wj · Cov(˜i , rj )

r˜

˜

j=1

i i=1

(B.25)

NN

= wi · wj · Cov(˜i , rj )

r˜

i=1 j=1

˜

Here is an illustration. A coin toss outcome is a random variable, T , and it will return either $2

(head) or $4 (tail). You have to pay $2 to receive this outcome. This looks like a great bet: The

mean rate of return on each coin toss, E(˜T ) is 50%, the variance on each coin toss is

r

Var(˜T ) = 1/2 · (0% ’ 50%)2 + 1/2 · (100% ’ 50%)2 = 0.50 . (B.26)

r

Therefore, the standard deviation of each coin toss is $0.707.

Now, bet on two independent such coin toss outcomes. You have $10 invested on the ¬rst bet

and $20 on the second bet. In other words, your overall actual and unknown rates of return

are 2

r= wi · ri ,

i=1

(B.27)

2

r= wi · ri .

˜ ˜

i=1

(The second equation is in random variable terms.) Now, your investment portfolio consists of

the following investments

$10

w1 = = 0.33 ,

$30

(B.28)

$20

= = 0.67 .

w2

$30

¬le=appendix.tex: LP

794 Chapter B. More Resources.

We can now use the formulas to compute your expected rate of return (E(˜)) and risk (Sdv(˜)).

r r

To compute your expected rate of return, use

2

E (˜) = wi · E (˜i ) = w1 · E (˜1 ) + w2 · E (˜2 )

r r r r

(B.29)

i=1

= 1/3 · (50%) + 2/3 · (50%) = 50% .

To compute your variance, use

2 2

Var(˜) = wi · wj · Cov(˜i , rj )

r r˜

i=1 j=1

= w1 · w1 · Cov(˜1 , r1 ) + w1 · w2 · Cov(˜1 , r2 )

r˜ r˜

+w2 · w1 · Cov(˜2 , r1 ) + w2 · w2 · Cov(˜2 , r2 )

r˜ r˜

2

= w1 · Cov(˜1 , r1 ) + 2 · w1 · w2 · Cov(˜1 , r2 )

r˜ r˜

2

+w2 · Cov(˜2 , r2 )

r˜ (B.30)

2

= w1 · Var(˜1 ) + 2 · w1 · w2 · Cov(˜1 , r2 )

r r˜

2

+w2 · Var(˜2 )

r

= (1/3)2 · Var(˜1 ) + 2 · w1 · w2 · 0 + (2/3)2 · Var(˜2 )

r r

(1/9) · Var(˜1 ) + (4/9) · Var(˜2 )

= r r

(1/9) · 0.5 + (4/9) · 0.5

= = 0.278 .

√

0.278 = 52.7%. This is lower than the 70.7% that a single

The standard deviation is therefore

coin toss would provide you with.

Solve Now!

Q B.10 Repeat the example, but assume that you invest $15 into each coin toss, rather than $10

and $20 respectively. Would you expect the risk to be higher or lower? (Hint: What happens if

you choose a portfolio that invests more and more into just one of the two bets.)

¬le=appendix.tex: RP

795

Section 2·5. Cumulative Normal Distribution Table.

2·5. Cumulative Normal Distribution Table

We used the cumulative normal distribution in Subsection 13·6.D. It is also used in the famous

Black-Scholes formula, explained in the Web Chapter on Options and Derivatives.

¬le=appendix.tex: LP

796 Chapter B. More Resources.

Table B.1. Cumulative Normal Distribution Table

N(z) N(z) N(z) N(z) N(z) N(z)

z z z z z z

-4.0 0.00003

-3.5 0.00023

-3.0 0.0013 -2.0 0.0228 -1.0 0.1587 0.0 0.5000 1.0 0.8413 2.0 0.9772

-2.9 0.0019 -1.9 0.0287 -0.9 0.1841 0.1 0.5398 1.1 0.8643 2.1 0.9821

-2.8 0.0026 -1.8 0.0359 -0.8 0.2119 0.2 0.5793 1.2 0.8849 2.2 0.9861

-2.7 0.0035 -1.7 0.0446 -0.7 0.2420 0.3 0.6179 1.3 0.9032 2.3 0.9893

-2.6 0.0047 -1.6 0.0548 -0.6 0.2743 0.4 0.6554 1.4 0.9192 2.4 0.9918

-2.5 0.0062 -1.5 0.0668 -0.5 0.3085 0.5 0.6915 1.5 0.9332 2.5 0.9938

-2.4 0.0082 -1.4 0.0808 -0.4 0.3446 0.6 0.7257 1.6 0.9452 2.6 0.9953

-2.3 0.0107 -1.3 0.0968 -0.3 0.3821 0.7 0.7580 1.7 0.9554 2.7 0.9965

-2.2 0.0139 -1.2 0.1151 -0.2 0.4207 0.8 0.7881 1.8 0.9641 2.8 0.9974

-2.1 0.0179 -1.1 0.1357 -0.1 0.4602 0.9 0.8159 1.9 0.9713 2.9 0.9981

3.5 0.99977

4.0 0.99997

Normal Score (z) vs. standardized Normal Cumulative Distribution Probability N (z) Table: This table allows

determining the probability that an outcome X will be less than a prespeci¬ed value x, when standardized into the

score z. For example, if the mean is 15 and the standard deviation is 5, an outcome of X = 10 is one standard

deviation below the mean. This standardized score can be obtained by computing z(x) = [x ’ E (x)]/Sdv (x) =

˜

(x ’ 15)/5 = (10 ’ 15)/5 = (’1). This table then indicates that the probability that the outcome of X (drawn from

this distribution with mean 15 and standard deviation 5) will be less than 10 (or less than its score of z = ’1) is

15.87%.

Normal Distribution Cumulative Normal Distribution

0.4

1.0

0.8

0.3

0.6

N(z)

n(z)

0.2

0.4

0.1

0.2

Area

=15.87%

0.0

0.0

’3 ’2 ’1 0 1 2 3 ’3 ’2 ’1 0 1 2 3

z z

These two ¬gures show what the table represents. The left-side is the classical bell curve. Recall that at z = ’1, the

table gave N(z = ’1) = 15.87%. This 15.87% is the area under the left curve up to an including z = ’1. The right

¬gure just plots the values in the table itself, i.e., the area under the graph to the left of each value from the left-side

¬gure.

If you need to approximate the cumulative normal distribution, you can use the formula

2

e’z /2

b1 ·kz + b2 ·k2 + b3 ·k3 + b4 ·k4 + b5 ·k5

N(z) ≈ 1 ’ √ · z z z z

2π (B.31)

1

≡

kz .

1 + a · |z|

where a = 0.2316419, b1 = 0.319381530, b2 = (’0.356563782), b3 = 1.781477937, b4 = (’1.821255978), b5 =

1.330274429, and π = 3.141592654.

¬le=appendix.tex: RP

797

Section 2·6. A Short Glossary of Some Bonds and Rates.

2·6. A Short Glossary of Some Bonds and Rates

This appendix brie¬‚y describes a plethora of di¬erent interest rates and bonds that you may

encounter. More complete ¬nance glossaries are at www.investopedia.com and the New York

Times and Campbell Harvey™s Dictionary of Money and Investing (also online at www.duke.edu/ char-

vey/Classes/wpg/glossary.htm).

In the real world, there are many di¬erent interest rates. Every borrower and every lender Other interest rate

information sources.

may pay a slightly di¬erent interest rate, depending on the bond™s default risk, risk premium,

liquidity, maturity, identity, convenience, etc. It is impossible to describe every common bond

or rate. The C-section of the Wall Street Journal describes daily interest rates on many common

and important interest instruments: the C-1 Page Markets Diary; the C-2 Page Interest Rates

and Bonds section; some boxes on a later page describing the interest rates paid on individual

government and government agency bonds (headlines Treasury Bonds, Notes and Bills and

Government Agency & Similar Issues); the Credit Markets page thereafter, which includes the

important Yield Comparisons and Money Rates boxes, as well as some Corporate Bonds; and

what is often the ¬nal page, which contains the Bond Market Data Bank. In addition, futures

on interest rates (similar to forward rates) are listed in the B-section.

Here are short descriptions of some of the interest rates printed in the Wall Street Journal on Some real-world interest

rates explained.

a daily basis, as well as some other bond subclasses.

Agency Bonds Issued by quasi-governmental companies, such as FannieMae, FreddieMac, and

SallieMae (all described below). These agencies were originally set up by the U.S. govern-

ment to facilitate loans for a particular purpose, then bundle them, and sell them to the

¬nancial markets. These companies are huge. Sometimes they are thought to be implicitly

backed by the U.S. government, though no explicit guarantees may exist.

APR (Annual Percentage Rate) A measure of interest due on a mortgage loan that accounts for

upfront costs and payments. Unfortunately, there are no clear rules of how to compute

APR, so the APR computation can vary across companies.

ARM Rate (Adjustable Rate Mortgage) A mortgage with an interest rate that is usually reset

once per year according to a then prevailing interest rate, pre-speci¬ed by a formula, but

subject to some upper limit (called a cap). Repayable by the borrower.

Bankers Acceptances Loans by banks to importers, used to pay the exporting ¬rm. Backed by

the issuing bank if the importer defaults. Usual maturities are 30 to 180 days.

Certi¬cate of Deposit (CD) Rate paid by banks to bank retail customers willing to commit

funds for a short-term or medium-term period. Unlike ordinary savings accounts, CDs

are not insured by the government if the bank fails.

Callable Bonds Bonds that the issuer can redeem. We will discuss these in Chapter 27.

CMO (Collateralized Mortgage Obligation) A security backed by a pool of real estate mort-

gages, with speci¬ed claims to interest and principal payments. For example, there are

Interest Only (IO) bonds and Principal Only (PO) bonds, which entitle bond holders to

only the interest or principal that the pool of mortgages receives.

Collateralized Trust Bonds Often issued by corporations, these bonds pledge as collateral the

securities owned by a subsidiary.

Commercial Paper Short-term bonds issued by corporations to the public markets. Often

backed by bank guarantees. Because commercial paper is short-term and often backed

by assets, it is usually very low risk.

Consumer Credit Rates The Wall Street Journal lists typical credit-card rates and car loan

rates.

Convertible Bonds Bonds that the holder can convert into common equity”we will discuss

these in Chapter 27.

¬le=appendix.tex: LP

798 Chapter B. More Resources.

Debenture Unsecured general obligation bond.

Discount Rate The interest rate that the Federal Reserve charges banks for short-term loans

of reserves.

Equipment Obligations Unlike debentures, these corporate bonds usually pledge speci¬c equip-

ment as collateral.

Eurobond Bonds issued by the U.S. government outside the domain of the Securities Exchange

Commission (e.g., in Europe) and purchased by foreign investors. Eurobonds need not be

denominated in dollars.

Federal Funds Rate Banks must hold ¬nancial reserves at the Federal Reserve Bank. If they

have more reserves than they legally need, they can lend them to other banks. The rate

at which they lend to one another overnight is the Federal Funds Rate. It is this interest

rate which is the interest rate primarily under the control of the Board of Governors of

the Federal Reserve.

FannieMae, originally the Federal National Mortgage Association (or FNMA), a corporation set

up by the government to help facilitate mortgage lending. It holds mortgages as assets.

FannieMae and FreddieMac together hold most U.S. mortgages, though they sell o¬ claims

against these mortgage bundles into the ¬nancial markets. The FNMA bonds are them-

selves collateralized (backed) by the mortgages, but, despite common perception, not by

the U.S. government.

FreddieMac, originally the Federal Home Loan Mortgage Corporation (FHLMC). An agency sim-

ilar to FNMA.

GICs (Guaranteed Investment Contracts) Usually issued by insurance companies and purchased

by retirement plans. The interest rate is guaranteed, but the principal is not.

G.O. Bonds (General Obligation Bonds) Bonds whose repayment is not guaranteed by a spe-

ci¬c revenue stream. See also Revenue Bonds.

High-Yield Bonds Sometimes also called Junk Bonds, high-yield bonds are bonds (usually of

corporations) that have credit ratings of BB and lower. This will be discussed in the next

chapter.

Home Equity Loan Rate The rate for loans secured by a home. Usually second mortgages, i.e.,

taken after another mortgage is already in place.

Investment Grade Bonds Bonds that have a rating higher than BBB. This is a common classi-

¬cation for corporate bonds, discussed in the next chapter.

Jumbo Mortgage Rate Like the N-year mortgage rate (see below), but for loans which exceed

the FNMA limit on mortgage size.

LIBOR London Interbank O¬er Rate Typical rate at which large London banks lend dollars to

one another.

Money-Market Rate Rate paid to cash sitting in a brokerage account and not invested in other

assets.

Mortgage Bonds Bonds secured by a particular real-estate property. In case of default, the

creditor can foreclose the secured property. If still not satis¬ed, the remainder of the

creditor™s claim becomes a general obligation.

Municipal Bond Bonds issued by a municipality. Often tax-exempt.

N-year Mortgage Rate A ¬xed-rate loan, secured by a house, with standard coupon payments.

The rate is that paid by the borrower. Usually limited to an amount determined by FNMA.

Prime Rate An interest rate charged by the average bank to their best customers for short-term

loans. (This rate is used less and less. It is being replaced by the LIBOR rate, at least in

most commercial usage.)

¬le=appendix.tex: RP

799

Section 2·6. A Short Glossary of Some Bonds and Rates.

Repo Rate A Repo is a repurchase agreement, in which a seller of a bond agrees to repurchase

the bond, usually within 30 to 90 days, but also sometimes overnight. (Repos for more

than 30 days are called Term Repos.) This allows the bond holder to obtain actual cash

to make additional purchases, while still being fully exposed to (speculate on) the bond.

Revenue Bond A bond secured by a speci¬c revenue stream. See also G.O. bond.

SallieMae, originally Student Loan Marketing Association (SLMA). Like FannieMae, an agency

(corporation) set up by the U.S. government. It facilitates student loans.

Savings Bonds Issued by the U.S. Treasury, Savings Bonds can only be purchased from or sold

to agents authorized by the Treasury Department. They must be registered in the name

of the holder. Series E Bonds are zero bonds; Series H Bonds are semi-annual coupon

payers and often have a variable interest feature. In contrast to Savings Bonds, other

bonds are typically bearer bonds, which do not record the name of the owner and are

therefore easy to resell (or steal).

Tax-Exempt Bonds Typically bonds issued by municipalities. Their interest is usually exempt

from some or all income taxes. The designation G.O. Bond means General Obligation

Bond, i.e., a Bond that was not issued to ¬nance a particular obligation. In contrast, a

Revenue Bond is a Bond backed by speci¬c municipal revenues”but it may or may not

be tax-exempt.

Treasury Security See Section 2·1.C.

Treasury STRIPS , or Separate Trading of Registered Interest and Principal of Securities. Fi-

nancial institutions can convert each coupon payment and principal payment of ordinary

Treasury coupon bonds into individual zero bonds. We brie¬‚y described these in the

previous chapter. See also www.publicdebt.treas.gov/of/ofstrips.htm. for a detailed ex-

planation.

Yankee Bonds U.S. Dollar denominated and SEC-registered bonds by foreign issuers.

Note: mortgage (and many other) bonds can be paid o¬ by the borrower before maturity. Re- Prepayment.

payment is common, especially if interest rates are dropping.

¬le=appendix.tex: LP

800 Chapter B. More Resources.

Solutions and Exercises

1. See text for list. Your personal propensity to forget is probably unique to yourself.

2. x ≈ 4.138%. Check: (1 + 4.138%)10 ≈ 1.5.

3. x ≈ 4.254. Check: 1.14.254 ≈ 1.5.

4. Yes! i and s are not variables, but notation!

5. The expression is

3

(3 + 5 · x) = (3 + 5 · 1) + (3 + 5 · 2) + (3 + 5 · 3) = 8 + 13 + 18 = 39 . (B.32)

x=1

x is not an unknown. It is simply a counter dummy used for writing convenience. It is not a part of the

expression itself.

6. The expression is

« «

3 3

3 + 5 · y = (3 + 3 + 3) + 5 · (1 + 2 + 3) = 39 . (B.33)

y=1 y=1

«

a+b·x = a + b · x.

The result is the same. This is an example why

i i i

7.

3

(i · i) = 1+4+9 = 14 .

« «

i=1

(B.34)

3 3

i · i = (1 + 2 + 3) · (1 + 2 + 3) = 36 .

i=1 i=1

The two are not the same. So, be careful not to try to pull out multiplying i™s! You can only pull out constants,

not counters. Incidentally, is also why E (X 2 ) ≠ E (X)2 , as stated in the next section.

˜ ˜

8. The expected value is

E (B) = (1/6) · $1 + (1/6) · $4 + (1/6) · $9 + (1/6) · $16 + (1/6) · $25 + (1/6) · $36

˜

(B.35)

= $15.17

The variance is

(1/6) · ($1 ’ $15.17)2 + (1/6)($4 ’ $15.17)2 + (1/6) · ($9 ’ $15.17)2

ar ˜

V (B) =

+ (1/6) · ($16 ’ $15.17)2 + (1/6) · ($25 ’ $15.17)2 + (1/6) · ($36 ’ $15.17)2 (B.36)

$2 149.14

=

The standard deviation is therefore

√

˜ ˜ (B.37)

Sdv(B) = Var(B) = 149.14 = 12.21

9. You expect to receive

˜ ˜

E (C) = ’$30 + 2 · E (B) = ’$30 + 2 · $15.17 = ,

$0.34

˜ ˜

22 · E (B) = 4 · $149.14 (B.38)

Var(C) = = $595.56 ,

˜ ˜

Sdv(C) = Var(B) = $24.42 .

¬le=appendix.tex: RP

801

Section 2·6. A Short Glossary of Some Bonds and Rates.

Your investment weights are now w1 = w2 = 0.5. The mean rate of return remains the same 50%. The

10.

variance of the rate of return is computed similarly to the example in the text,

Var(˜) = (1/4) · 0.5 + (1/4) · 0.5 = 0.25 . (B.39)

r

Therefore, the risk (standard deviation) is 50%. This is lower than where you put more weight on one of the

coin tosses. This makes sense: as you put more and more into one of the two coin tosses, you lose the bene¬t

of diversi¬cation.!

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

¬le=appendix.tex: LP

802 Chapter B. More Resources.

APPENDIX C

Sample Exams

Applied Torture!

last ¬le change: Jan 18, 2006 (13:22h)

last major edit: n/a

The following are the midterm and ¬nal exams that I gave in my introductory ¬nance course in Spring 2005. The

exams did not cover all subjects that were covered in the course, but students did not know which subjects would

be on the exam and which subjects would be omitted. The exam answers follow.

The student instructions common to both exams were

• This is a closed-book, closed-notes exam. You are allowed to use your prepared 3*5 index card, and a calcu-

lator. No Internet connections are allowed, either.

• The ¬nal answer must be in the right units, so make sure to distinguish between raw numbers and percent,

between dollars and dollars-squared, etc.

• We will try to give partial credit, so show your work.

• You have enough time to write clearly: we will mercilessly penalize hard-to-read and hard-to-comprehend

answers. It is your task to make it clear to us that you know the answer, not our task to decipher what you

mean. Be concise.

• If you believe a question is ambiguous, please make reasonable assumptions, and spell them out in your

answer.

• We will liberally subtract points for wrong answers”in particular, we do not like the idea of 3 di¬erent

answers, one of which is correct, two of which are incorrect. So, if you show us two di¬erent solutions, you

can at best only get half credit, unless you clearly outline assumptions that you have to make because my

question is ambiguous.

• Assume a perfect market, unless otherwise indicated.

803

¬le=exam-sample.tex: LP

804 Chapter C. Sample Exams.

3·1. A Sample Midterm

Students were told that the midterm was 80 minutes for 12 questions, and that each question was worth 10 points,

regardless of di¬culty or time required to solve.

Q C.1 Market Perfection Questions:

(a) What are the four conditions that make a market “perfect”?

(b) What kind of ambiguity happens if the market is not perfect? (You do not need to spell it out for each reason

why the market can be imperfect. You need to tell us what breaks generally.)

Q C.2 The interest rate (at a zero-tax rate) is 12 basis points per week, A year is always 52 weeks.

(a) What is the payo¬ on a $200 investment in 5 years?

(b) If the in¬‚ation rate is 5 basis points per week, what is the PV of your answer?

(c) Now introduce an imperfect market. Your tax rate is now 20%, and due immediately each Jan 1. What will

your cash ¬‚ow in 5 years be? What is this worth in real terms (in 2005 dollars), i.e., adjusted for purchasing

power using the in¬‚ation rate?

Q C.3 If it takes 9 years for you to triple your investment, what is your annualized rate of return?

Q C.4 Risk-free Treasury bonds earn holding rates of return (not annualized) of 10% over 1 year, 25% over 2 years,

and 40% over 3 years.

(a) Draw the yield curve and provide the appropriate table that you use to draw your yield curve. (Use the same

abbreviations that we have been using in class.)

(b) What are the two forward rates?

Use at least 4 signi¬cant digits in your calculation, so we know you are computing the right thing.

Q C.5 What is the IRR of a project that costs $100 today, earns $100 next year, and costs $50 the year after?

Q C.6 What is the monthly payment on a ¬xed 30-year 8% home mortgage for $500,000? (Interpret the 8% quote the

same way a normal mortgage company or bank interprets it.)

Q C.7 Tomorrow, a project will be worth either $200 million (60% probability), or $10 million liquidation value (40%

probability). Today, the project is worth and can be bought for $100 million. You only have $80 million, so you borrow

$20 million today from a bank.

(a) If the world is risk-neutral, what interest rate do you have to promise the bank?

(b) If the world is not risk-neutral, but you know that in equilibrium the bank asks for a 50% promised rate of

return, what would you as residual equity holder demand as your expected rate of return?

Q C.8 A project reports the following:

Year 1 Year 2 Year 3 Beyond

Sales = Income $200 $300 $500 $0

A/R $100 $100 $50 $0

What are the cash ¬‚ows?

Q C.9 Some accounting questions:

(a) What is the main di¬erence between how an accountant thinks of cash ¬‚ows (not earnings!) and how a ¬nancier

thinks of the same?

¬le=exam-sample.tex: RP

805

Section 3·2. A Sample Final.