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The Investment FAQ (part 3 of 20)

( Part1 - Part2 - Part3 - Part4 - Part5 - Part6 - Part7 - Part8 - Part9 - Part10 - Part11 - Part12 - Part13 - Part14 - Part15 - Part16 - Part17 - Part18 - Part19 - Part20 )
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Archive-name: investment-faq/general/part3
Version: $Id: part03,v 1.61 2003/03/17 02:44:30 lott Exp lott $
Compiler: Christopher Lott

See reader questions & answers on this topic! - Help others by sharing your knowledge
The Investment FAQ is a collection of frequently asked questions and
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Rules, regulations, laws, conditions, rates, and such information
discussed in this FAQ all change quite rapidly.  Information given
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Please send comments and new submissions to the compiler.

--------------------Check for updates------------------

Subject: Analysis - Internal Rate of Return (IRR)

Last-Revised: 25 June 1999
Contributed-By: Christopher Yost (cpy at, Rich Carreiro
(rlcarr at

If you have an investment that requires and produces a number of cash
flows over time, the internal rate of return is defined to be the
discount rate that makes the net present value of those cash flows equal
to zero.  This article discusses computing the internal rate of return
on periodic payments, which might be regular payments into a portfolio
or other savings program, or payments against a loan.  Both scenarios
are discussed in some detail. 

We'll begin with a savings program.  Assume that a sum "P" has been
invested into some mutual fund or like account and that additional
deposits "p" are made to the account each month for "n" months.  Assume
further that investments are made at the beginning of each month,
implying that interest accrues for a full "n" months on the first
payment and for one month on the last payment.  Given all this data, how
can we compute the future value of the account at any month? Or if we
know the value, what was the rate of return?

The relevant formula that will help answer these questions is:
  F = -P(1+i)^n - [p(1+i)((1+i)^n - 1)/i]
In this formula, "F" is the future value of your investment (i.e., the
value after "n" months or "n" weeks or "n" years--whatever the period
over which the investments are made), "P" is the present value of your
investment (i.e., the amount of money you have already invested), "p" is
the payment each period, "n" is the number of periods you are interested
in, and "i" is the interest rate per period.  Note that the symbol '^'
is used to denote exponentiation (2 ^ 3 = 8). 

Very important! The values "P" and "p" should be negative .  This
formula and the ones below are devised to accord with the standard
practice of representing cash paid out as negative and cash received (as
in the case of a loan) as positive.  This may not be very intuitive, but
it is a convention that seems to be employed by most financial programs
and spreadsheet functions. 

The formula used to compute loan payments is very similar, but as is
appropriate for a loan, it assumes that all payments "p" are made at the
end of each period:
          F = -P(1+i)^n - [p((1+i)^n - 1)/i]
Note that this formula can also be used for investments if you need to
assume that they are made at the end of each period.  With respect to
loans, the formula isn't very useful in this form, but by setting "F" to
zero, the future value (one hopes) of the loan, it can be manipulated to
yield some more useful information. 

To find what size payments are needed to pay-off a loan of the amount
"P" in "n" periods, the formula becomes this:
          p =  ------------
               (1+i)^n - 1
If you want to find the number of periods that will be required to
pay-off a loan use this formula:
               log(-p) - log(-Pi - p)
          n =  ----------------------

Keep in mind that the "i" in all these formula is the interest rate per
period .  If you have been given an annual rate to work with, you can
find the monthly rate by adding 1 to annual rate, taking the 12th root
of that number, and then subtracting 1.  The formula is:
          i = ( r + 1 ) ^ 1/12 - 1
where "r" is the rate. 

Conversely, if you are working with a monthly rate--or any periodic
rate--you may need to compound it to obtain a number you can compare
apples-to-apples with other rates.  For example, a 1 year CD paying 12%
in simple interest is not as good an investment as an investment paying
1% compounded per month.  If you put $1000 into each, you'll have $1120
in the CD at the end of the year but $1000*(1.01)^12 = $1126.82 in the
other investment due to compounding.  In this way, interest rates of any
kind can be converted to a "simple 1-year CD equivalent" for the
purposes of comparison.  (See the article "Computing Compound Return"
for more information.)

You cannot manipulate these formulas to get a formula for "i," but that
rate can be found using any financial calculator, spreadsheet, or
program capable of calculating Internal Rate of Return or IRR. 

Technically, IRR is a discount rate: the rate at which the present value
of a series of investments is equal to the present value of the returns
on those investments.  As such, it can be found not only for equal,
periodic investments such as those considered here but for any series of
investments and returns.  For example, if you have made a number of
irregular purchases and sales of a particular stock, the IRR on your
transactions will give you a picture of your overall rate of return. 
For the matter at hand, however, the important thing to remember is that
since IRR involves calculations of present value (and therefore the
time-value of money), the sequence of investments and returns is

Here's an example.  Let's say you buy some shares of Wild Thing
Conservative Growth Fund, then buy some more shares, sell some, have
some dividends reinvested, even take a cash distribution.  Here's how to
compute the IRR. 

You first have to define the sign of the cash flows.  Pick positive for
flows into the portfolio, and negative for flows out of the portfolio
(you could pick the opposite convention, but in this article we'll use
positive for flows in, and negative for flows out). 

Remember that the only thing that counts are flows between your wallet
and the portfolio.  For example, dividends do NOT result in cash flow
unless they are withdrawn from the portfolio.  If they remain in the
portfolio, be they reinvested or allowed to sit there as free cash, they
do NOT represent a flow. 

There are also two special flows to define.  The first flow is positive
and is the value of the portfolio at the start of the period over which
IRR is being computed.  The last flow is negative and is the value of
the portfolio at the end of the period over which IRR is being computed. 

The IRR that you compute is the rate of return per whatever time unit
you are using.  If you use years, you get an annualized rate.  If you
use (say) months, you get a monthly rate which you'll then have to
annualize in the usual way, and so forth. 

On to actually calculating it... 

We first have the net present value or NPV:

NPV(C, t, d) = Sum C[i]/(1+d)^t[i]
     C[i] is the i-th cash flow (C[0] is the first, C[N] is the
     d is the assumed discount rate. 
     t[i] is the time between the first cash flow and the i-th. 
     Obviously, t[0]=0 and t[N]=the length of time under
     consideration.  Pick whatever units of time you like, but
     remember that IRR will end up being rate of return per chosen
     time unit. 

Given that definition, IRR is defined by the equation: NPV(C, t, IRR) =

In other words, the IRR is the discount rate which sets the NPV of the
given cash flows made at the given times to zero. 

In general there is no closed-form solution for IRR.  One must find it
iteratively.  In other words, pick a value for IRR.  Plug it into the
NPV calculation.  See how close to zero the NPV is.  Based on that, pick
a different IRR value and repeat until the NPV is as close to zero as
you care. 

Note that in the case of a single initial investment and no further
investments made, the calculation collapses into:
     (Initial Value) - (Final Value)/(1+IRR)^T = 0 or
     (Initial Value)*(1+IRR)^T - (Final Value) = 0
     Initial*(1+IRR)^T = Final
     (1+IRR)^T = Final/Initial
     And finally the quite familiar:
     IRR = (Final/Inital)^(1/T) - 1

A program named 'irr' that calculates IRR is available.  See the article
Software - Archive of Investment-Related Programs in this FAQ for more

--------------------Check for updates------------------

Subject: Analysis - Paying Debts Early versus Making Investments

Last-Revised: 14 July 2000
Contributed-By: Gary Snyder, Thomas Price (tprice at,
Chris Lott ( contact me ), John A.  Weeks III (john at

This article analyzes the question of whether you should apply any extra
cash you might have lying around to making extra payments on a debt, or
whether you should instead leave the debt on its regular payment
schedule and invest the cash instead.  An equivalent question is whether
you should cash out an existing investment to pay down debt, or just let
it ride.  We'll focus on the example of a first mortgage on a house, but
the analysis works (with some changes) for a car loan, credit-card debt,

Before we compare debts with investments, it's important to frame the
debate.  A bit of financial planning is appropriate here; there are
several articles in the FAQ about that.  To start with, an individual
should have an emergency fund of 3-6 months of living expenses. 
Emergency funds need to be readily available (when was the last
emergency that you could plan for), like in a bank, credit union, or
maybe a money market fund.  And most people would not consider these
investments.  So the first thing to do with cash is arguably to
establish this sort of rainy-day fund.  If you have to cash out a stock
to get this fund, that's ok; remember, emergencies rarely happen at
market tops. 

Before we run numbers, I'd like to point out two important issues here. 
The most important issue to remember is risk.  Making early payments to
a loan exposes you to relatively few risks (once the loan is paid, it
stays paid), but two notable risks are liquidity and opportunity.  The
liquidity risk is that you might not have cash when you need it (but see
above for the mitigation strategy of a rainy-day fund).  The opportunity
risk is the possibility that a better opportunity might present itself
and you would be unable to take advantage of it since you gave the bank
your extra cash.  And when you invest money, you generally expose
yourself to market risk (the investment's price might fall) as well as
other risks that might cause you to lose money.  Of course the other
important issue (you probably guessed) is taxes.  The interest paid on
home mortgages is deductable, so that acts to reduce the cost of the
loan below the official interest rate on the loan.  Not true for
credit-card debt, etc.  Also, monies earned from an investment are
taxed, so that acts to reduce the return on the investment. 

One more caveat.  If you simply cannot save; i.e., you would cash out
the investments darned quick, then paying down debt may be a good
choice! And owning a home gives you a place to live, especially if you
plan to live in it on a modest income. 

Finally, all you can do in advance is estimate, guess, and hope.  No one
will never know the answer to "what is best" until long after it is too
late to take that best course of action.  You have to take your shot
today, and see where it lands tomorrow. 

Now we'll run some numbers.  If you have debt as well as cash that you
will invest, then maintaining the debt (instead of paying it) costs you
whatever the interest rate on the loan is minus whatever you make from
the investment.  So to justify your choice of investing the cash,
basically you're trying to determine whether you can achieve a return on
your investment that is better than the interest rate on the debt.  For
example, you might have a mortgage that has an after-tax rate of 6%, but
you find a very safe investment with a guaranteed, after-tax return of
9% (I should be so lucky).  In this case, you almost certainly should
invest the money.  But the analysis is never this easy -- it invariably
depends on knowing what the investments will yield in the future. 

But don't give up hope.  Although it is impossible to predict with
certainty what an investment will return, you can still estimate two
things, the likely return and the level of risk.  Since paying down any
debt entails much lower risk than making an investment, you need to get
a higher level of return to assume the market risk (just to name one) of
an investment.  In other words, the investment has to pay you to assume
the risk to justify the investment.  It would be foolish to turn down a
risk-free 10% (i.e., to pay off a debt with an after-tax interest rate
of 10%) to try to get an after-tax rate of 10.5% from an investment in
the stock market, but it might make very good sense to turn down a
risk-free 6.5%.  It is a matter of personal taste how big the difference
between the return on the investment and the risk-free return has to be
(it's called the risk premium), but thinking like this at least lets you
frame the question. 

Next we'll characterize some investments and their associated risks. 
Note that characterizing risk is difficult, and we'll only do a
relatively superficial job it.  The purpose of this article is to get
you thinking about the options, not to take each to the last decimal

Above we mentioned that paying the debt is a low-risk alternative.  When
it comes to selecting investments that potentially will yield more than
paying down the debt, you have many options.  The option you choose
should be the one that maximizes your return subject to a given level of
risk (from one point of view).  Paying off the loan generates a
rock-solid guaranteed return.  The best option you have at approximately
this level of risk is to invest in a short-term, high-grade corporate
bond fund.  The key market risk in this investment is that interest
rates will go up by more than 1%; another risk of a bond fund is that
companies like AT&T will start to default on their loans.  Not quite
rock-solid guaranteed, but close.  Anyway, these funds have yielded
about 6% historically. 

Next in the scale of risk is longer-term bonds, or lower rated bonds. 
Investing in a high-yield (junk) bond fund is actually quite safe,
although riskier than the short-term, high grade bond fund described
above.  This investment should generate 7-8% pre-tax (off the top of my
head), but could also lose a significant amount of money over short
periods.  This happened in the junk bond market during the summer of
1998, so it's by no means a remote possibility. 

The last investment I'll mention here are US stock investments. 
Historically these investments have earned about 10-11%/year over long
periods of time, but losing money is a serious possibility over periods
of time less than three years, and a return of 8%/year for an investment
held 20 years is not unlikely.  Conservatively, I'd expect about an 8-9%
return going forward.  I'd hope for much more, but that's all I'd count
on.  Stated another way, I'd choose a stock investment over a CD paying
6%, but not a CD paying 10%. 

Don't overlook the fact that the analysis basically attempted to answer
the question of whether you should put all your extra cash into the
market versus your mortgage.  I think the right answer is somewhere in
between.  Of course it's nice to be debt free, but paying down your
debts to the point that you have no available cash could really hurt you
if your car suddenly dies, etc.  You should have some savings to cushion
you against emergencies.  And of course it's nice to have lots of
long-term investments, but don't neglect the guaranteed rate of return
that is assured by paying down debt versus the completely unguaranteed
rate of return to be found in the markets. 

The best thing to do is ask yourself what you are the most comfortable
with, and ignore trying to optimize variables that you cannot control. 
If debt makes you nervous, then pay off the house.  If you don't worry
about debt, then keep the mortgage, and keep your money invested.  If
you don't mind the ups and downs of the market, then keep invested in
stocks (they will go up over the long term).  If the market has you
nervous, pull out some or all of it, and ladder it into corporate bonds. 
In short, each person needs to find the right balance for his or her

--------------------Check for updates------------------

Subject: Analysis - Price-Earnings (P/E) Ratio

Last-Revised: 27 Jan 1998
Contributed-By: E.  Green, Aaron Schindler, Thomas Busillo, Chris Lott (
contact me )

P/E is shorthand for the ratio of a company's share price to its
per-share earnings.  For example, a P/E ratio of 10 means that the
company has $1 of annual, per-share earnings for every $10 in share
price.  Earnings by definition are after all taxes etc. 

A company's P/E ratio is computed by dividing the current market price
of one share of a company's stock by that company's per-share earnings. 
A company's per-share earnings are simply the company's after-tax profit
divided by number of outstanding shares.  For example, a company that
earned $5M last year, with a million shares outstanding, had earnings
per share of $5.  If that company's stock currently sells for $50/share,
it has a P/E of 10.  Stated differently, at this price, investors are
willing to pay $10 for every $1 of last year's earnings. 

P/Es are traditionally computed with trailing earnings (earnings from
the past 12 months, called a trailing P/E) but are sometimes computed
with leading earnings (earnings projected for the upcoming 12-month
period, called a leading P/E).  Some analysts will exclude one-time
gains or losses from a quarterly earnings report when computing this
figure, others will include it.  Adding to the confusion is the
possibility of a late earnings report from a company; computation of a
trailing P/E based on incomplete data is rather tricky.  (I'm being
polite; it's misleading, but that doesn't stop the brokerage houses from
reporting something.) Even worse, some methods use so-called negative
earnings (i.e., losses) to compute a negative P/E, while other methods
define the P/E of a loss-making company to be zero.  The many ways to
compute a P/E may lead to wide variation in the reporting of a figure
such as the "P/E for the S&P whatever." Worst of all, it's usually next
to impossible to discover the method used to generate a particular P/E
figure, chart, or report. 

Like other indicators, P/E is best viewed over time, looking for a
trend.  A company with a steadily increasing P/E is being viewed by the
investment community as becoming more and more speculative.  And of
course a company's P/E ratio changes every day as the stock price

The price/earnings ratio is commonly used as a tool for determining the
value the market has placed on a common stock.  A lot can be said about
this little number, but in short, companies expected to grow and have
higher earnings in the future should have a higher P/E than companies in
decline.  For example, if Amgen has a lot of products in the pipeline, I
wouldn't mind paying a large multiple of its current earnings to buy the
stock.  It will have a large P/E.  I am expecting it to grow quickly. 

PE is a much better comparison of the value of a stock than the price. 
A $10 stock with a PE of 40 is much more "expensive" than a $100 stock
with a PE of 6.  You are paying more for the $10 stock's future earnings
stream.  The $10 stock is probably a small company with an exciting
product with few competitors.  The $100 stock is probably pretty staid -
maybe a buggy whip manufacturer. 

It's difficult to say whether a particular P/E is high or low, but there
are a number of factors you should consider.  First, a common rule of
thumb for evaluating a company's share price is that a company's P/E
ratio should be comparable to that company's growth rate.  If the ratio
is much higher, then the stock price is high compared to history; if
much lower, then the stock price is low compared to history.  Second,
it's useful to look at the forward and historical earnings growth rate. 
For example, if a company has been growing at 10% per year over the past
five years but has a P/E ratio of 75, then conventional wisdom would say
that the shares are expensive.  Third, it's important to consider the
P/E ratio for the industry sector.  For example, consumer products
companies will probably have very different P/E ratios than internet
service providers.  Finally, a stock could have a high trailing-year P/E
ratio, but if the earnings rise, at the end of the year it will have a
low P/E after the new earnings report is released.  Thus a stock with a
low P/E ratio can accurately be said to be cheap only if the
future-earnings P/E is low.  If the trailing P/E is low, investors may
be running from the stock and driving its price down, which only makes
the stock look cheap. 

--------------------Check for updates------------------

Subject: Analysis - Percentage Rates

Last-Revised: 15 Feb 2003
Contributed-By: Chris Lott ( contact me )

This article discusses various percentage rates that you may want to
understand when you are trying to choose a savings account or understand
the amount you are paying on a loan. 

Annual percentage rate (APR)
     In a savings account or other account that pays you interest, the
     annual percentage rate is the nominal rate paid on deposits.  This
     may also be known as just the rate.  Most financial institutions
     compute and pay out interest many times during the year, like every
     month on a savings account.  Because you can earn a tiny bit of
     interest late in the year on the money paid out as interest early
     in the year, to understand the actual net increase in account
     value, you have to use the annual percentage yield (APY), discussed
     In a loan or other arrangement where you pay interest to some
     financial institution, you will also encounter annual percentage
     rates.  Every loan has a rate associated with it, for example a 6%
     rate paid on a home mortgage.  Federal lending laws (Truth in
     Lending) require lenders to compute and disclose an annual
     percentage rate for a loan as means to report the true cost of the
     loan.  This just means that the lender is supposed to include all
     fees and other charges with the note rate to report a single
     number, the APR.  This sounds great, but it doesn't actually work
     so well in practice because there do not appear to be clear
     guidelines for lenders on what fees must be included and which can
     be omitted.  Some fees that are usually included are points, a loan
     processing fee, private mortgage insurance, etc.  Fees that are
     usually omitted include title insurance, etc.  So the APR of a loan
     is a useful piece of data but not the only thing you should
     consider when shopping for a loan. 
Annual percentage yield (APY)
     The annual percentage yield of an account that pays interest is the
     actual percentage increase in the value of an account after a
     1-year period when the interest is compounded at some regular
     interval.  This is sometimes called the effective annual rate.  You
     can use APY to compare compound interest rates.  The formula is:
             APY = (1 + r / n ) ^ n - 1
     where 'r' is the interest rate (e.g., r=.05 for a 5% rate) and 'n'
     is the number of times that the interest is compounded over the
     course of a year (e.g., n=12 for monthly compounding).  The symbol
     '^' means exponentiation; e.g., 2^3=8. 
     For example, if an account pays 5% compounded monthly, then the
     annual percentage yield will be just a bit greater than 5%:
             APY = (1 + .05 / 12 ) ^ 12 - 1
                 = 1.0042 ^ 12 - 1
                 = 1.0512 - 1
            = .0512 (or 5.12%)
     If interest is compounded just once during the year (i.e.,
     annually), then the APY is the same as the APR.  If interest is
     compounded continuously, the formula is
             APY = e ^ n - 1
     where 'e' is Euler's constant (approximately 2.7183). 

--------------------Check for updates------------------

Subject: Analysis - Risks of Investments

Last-Revised: 15 Aug 1999
Contributed-By: Chris Lott ( contact me ), Eugene Kononov (eugenek at

Risk, in general, is the possibility of sustaining damage, injury, or
loss.  This is true in the world of investments also, of course. 
Investments that are termed "high risk" have a significant possibility
that their value will drop to zero. 

You might say that risk is a measure of whether a surprise will occur. 
But in the world of investments, positive as well as negative surprises
happen.  Sometimes a company's revenue and profits explode suddenly and
the stock price zooms upward, a very pleasant and positive surprise for
the stockholders.  Sometimes a company implodes, and the stock crashes,
a not very pleasant and decidedly negative surprise for the

Because investments can rise or fall unexpectedly, the primary risk
associated with an investment (the market risk) is characterized by the
variability of returns produced by that investment.  For example, an
investment with a low variability of return is a savings account with a
bank (low market risk).  The bank pays a highly predictable interest
rate.  That interest rate also happens to be quite low.  An internet
stock is an investment with a high variability of return; it might
quintuple, and it might fall 50% (high market risk). 

The standard way to calculate the market risk of investing in a
particular security is to calculate the standard deviation of its past
prices.  So, the academic definition is:

market risk = volatility = StdDev(price history)

However, it has long been noticed that the standard deviation may not be
appropriate to use in many instances.  Consider a hypothetical asset
that always goes up in price, in very small and very large increments. 
The standard deviation of the prices (and returns) for that asset may be
large, but where is the market risk?

For practical purposes (trading and system evaluation), a much better
measure of market risk is the distribution of the drawdowns.  Given the
history of the prices, and assuming some investment strategy (be it
buy-and-hold or market timing), what is the maximum loss that would have
been suffered? How frequent are the losses? What is the longest
uninterrupted string of losses? What is the average gain/loss ratio?

Other risks in the investment world are the risk of losing purchasing
power due to inflation (possibly by making only risk-free investments),
and the risk of underperforming the market (of special concern to mutual
fund mangers).  Occasionally you may see "liquidity risk" which
basically means that you might need your money at a time when an
investment is not liquid; i.e., not easily convertible to cash.  The
best example is a certificate of deposit (CD) which is payable in full
when it matures but if you need the money before then, you will pay a

Bond holders face several risks unique to bonds, the most prominent
being interest rate risk.  Because the price of bonds drops as the
prevailing interest rates rise, bond holders tend to worry about rising
interest rates.  Other risks more-or-less unique to bonds are the risk
of default (i.e., the company that issued the bond decides it cannot pay
the obligation), as well as call (or prepayment) risk.  What's that last
one? Well, in a nutshell, a bond issuer can call (prepay) the bond
before the bond matures, depending of course whether the terms and
conditions associated with the bond allow it.  A bond that can be repaid
before the maturity date is called "callable" and a bond that cannot is
called "non callable" (see the basics of bonds article elsewhere in this
FAQ for more details).  Hmm, you might be saying to yourself, the bond
holder got the money back, where's the risk? Because the investor will
have to reinvest the money at some random time, the risk is that the
investor might not be able to find as good of a deal as the old bond. 

Market risk has additional components for investments outside your home
country.  To the usual volatility of the markets you have to add the
volatility of the currency markets.  You might have great gains, but
lose them when you swap the foreign currency for your own.  Other risks
(especially in emerging markets) are problems in the economy or
government (that might lead to severe market declines) and the risk of
illiquidity (no one is buying when you want to sell). 

This seems like a good place to discuss the classic risk-reward
tradeoff.  If we use volatility as our risk measure, then it's clear an
investor will obtain only modest returns from low-volatility (low-risk)
investments.  An investor must put his or her money into volatile (i.e.,
risky) investments if he or she hopes to experience returns on
investment that are greater than the risk-free rate of return. 

Different individuals will have very different tolerances for risk, and
their tolerance for risk will change during their lifetimes.  In
general, if an investor will need cash within a short period of time
(and will be forced to sell investments to raise that cash), the
investor should not put money into high-volatility (i.e., high-risk)
vehicles.  Those investments might not be worth very much when the
investor needs to sell.  On the other hand, if an investor has a very
long time horizon, such as a young person investing 401(k) monies, he or
she should seriously consider choosing investments that offer the best
possibility of good returns (i.e., investments with significant
historical volatility).  The long period of time before that person
needs the money offers an unparalleled chance to allow the investment to
grow; the occasional downturn will most likely be offset by other gains. 
All things being equal, it's reasonable to expect that a young worker
will tolerate more risk than a retired person. 

A commonly accepted quantification of market risk is beta, which is
explained in another article in this FAQ. 

--------------------Check for updates------------------

Subject: Analysis - Return on Equity versus Return on Capital

Last-Revised: 7 June 1999
Contributed-By: John Price (johnp at

This article analyzes the question of whether return on equity (ROI) or
return on capital (ROC) is the better guide to performance of an

We'll start with an example.  Two brothers, Abe and Zac, both inherited
$10,000 and each decided to start a photocopy business.  After one year,
Apple, the company started by Abe, had an after-tax profit of $4,000. 
The profit from Zebra, Zac's company, was only $3,000.  Who was the
better manager? I.e., who provided a better return? For simplicity,
suppose that at the end of the year, the equity in the companies had not
changed.  This means that the return on equity for Apple was 40% while
for Zebra it was 30%.  Clearly Abe did better? Or did he?

There is a little more to the story.  When they started their companies,
Abe took out a long-term loan of $10,000 and Zac took out a similar loan
for $2,000.  Since capital is defined as equity plus long-term debt, the
capital for the two companies is calculated as $20,000 and $12,000. 
Calculating the return on capital for Apple and Zebra gives 20% (= 4,000
/ 20,000) for the first company and 25% (= 3,000 / 12,000) for the
second company. 

So for this measure of management, Zac did better than Abe.  Who would
you invest with?

Perhaps neither.  But suppose that the same benefactor who left money to
Abe and Zac, also left you $100 with the stipulation that you had to
invest in the company belonging to one or other of the brothers.  Who
would it be?

Most analysts, once they have finished talking about earnings per share,
move to return on equity.  For public companies, it is usually stated
along the lines that equity is what is left on the balance sheet after
all the liabilities have been taken care of.  As a shareholder, equity
represents your money and so it makes good sense to know how well
management is doing with it.  To know this, the argument goes, look at
return on equity. 

Let's have a look at your $100.  If you loan it to Abe, then his capital
is now $20,100.  He now has $20,100 to use for his business.  Assuming
that he can continue to get the same return, he will make 20% on your
$100.  On the other hand, if you loan it to Zac, he will make 25% on
your money.  From this perspective, Zac is the better manager since he
can generate 25% on each extra dollar whereas Abe can only generate 20%. 

The bottom line is that both ratios are important and tell you slightly
different things.  One way to think about them is that return on equity
indicates how well a company is doing with the money it has now, whereas
return on capital indicates how well it will do with further capital. 

But, just as you had to choose between investing with Abe or Zac, if I
had to choose between knowing return on equity or return on capital, I
would choose the latter.  As I said, it gives you a better idea of what
a company can achieve with its profits and how fast its earnings are
likely to grow.  Of course, if long-term debt is small, then there is
little difference between the two ratios. 

Warren Buffett (the famous investor) is well known for achieving an
average annual return of almost 30 percent over the past 45 years. 
Books and articles about him all say that he places great reliance on
return on equity.  In fact, I have never seen anyone even mention that
he uses return on capital.  Nevertheless, a scrutiny of a book The
Essays of Warren Buffett and Buffett's Letters to Shareholders in the
annual reports of his company, Berkshire Hathaway, convinces me that he
relies primarily on return on capital.  For example, in one annual
report he wrote,"To evaluate [economic performance], we must know how
much total capital—debt and equity—was needed to produce these
earnings." When he mentions return on equity, generally it is with the
proviso that debt is minimal. 

If your data source does not give you return on capital for a company,
then it is easy enough to calculate it from return on equity.  The two
basic ways that long-term debt is expressed are as long-term debt to
equity DTE and as long-term debt to capital DTC.  (DTC is also referred
to as the capitalization ratio.) In the first case, return on capital
ROC is calculated from return on equity ROE by

ROC = ROE / (1 + DTE),

and in the second case by:

ROC = ROE * (1 - DTC)

For example, in the case of Abe, we saw DTE = 10,000 / 10,000 = 1 and
ROE = 40% so that, according to the first formula, ROC = 40% / ( 1 + 1)
= 20%.  Similarly, DTC = 10,000 / 20,000 = 0.5 so that by the second
formula, ROC = 40% (1 – 0.5) = 20%.  You might like to check your
understanding of this by repeating the calculations with the results for
Zac's company. 

If you compare return on equity against return on capital for a company
like General Motors with that of a company like Gillette, you'll see one
of the reasons why Buffett includes the latter company in his portfolio
and not the former. 

For more articles, analyses, and insights into today's financial markets
from John Price, visit his web site.

--------------------Check for updates------------------

Subject: Analysis - Rule of 72

Last-Revised: 19 Feb 1998
Contributed-By: Chuck Cilek (ccilek at, Chris Lott (
contact me ), Richard Alpert

The "Rule of 72" is a rule of thumb that can help you compute when your
money will double at a given interest rate.  It's called the rule of 72
because at 10%, money will double every 7.2 years. 

To use this simple rule, you just divide the annual interest into 72. 
For example, if you get 6% on an investment and that rate stays
constant, your money will double in 72 / 6 = 12 years.  Of course you
can also compute an interest rate if you are told that your money will
double in so-and-so many years.  For example, if your money has to
double in two years so that you can buy your significant other that
Mazda Miata, you'll need 72 / 2 = 36% rate of return on your stash. 

Like any rule of thumb, this rule is only good for approximations.  Next
we give a derivation of the exact number for the case of an interest
rate of 10%.  We want to know how long it takes a given principal P to
double given either the interest rate r (in percent per year) or the
number of years n.  So, we are solving this equation:
     P * (1 + r/100) ** n = 2P

Note that the symbol '**' is used to denote exponentiation (2 ** 3 = 8). 
Since we said we'll try the case of r = 10%, we're solving this:
     P * (1 + 10/100) ** n = 2P

We cancel the P's to get:
     (1 + r/100) ** n = 2


     (1 + 10/100) ** n = 2
     1.1 ** n = 2

From calculus we know that the natural logarithm ("ln") has the
following property:
     ln (a ** b) = b * ln ( a )

So we'll use this as follows:
     n * ln(1.1) = ln(2)
     n * (0.09531) = 0.693147

Finally leaving us with:

     n = 7.2725527

Which means that at 10%, your money doubles in about 7.3 years.  So the
rule of 72 is pretty darned close. 

You can solve the equation for other values of r to see how rough of an
approximation this rule provides.  Here's a table that shows the actual
number of years required to double your money based on different
interest rates, along with the number that the rule of 72 gives you. 

% Rate Actual Rule 72
1 69.66 72
2 35.00 36
3 23.45 24
4 17.67 18
5 14.21 14.4
6 11.90 12
7 10.24 10.29
8 9.01 9
9 8.04 8
10 7.27 7.2
..  ..  .. 
15 4.96 4.8
20 3.80 3.6
25 3.11 2.88
30 2.64 2.4 (note: 10pct error)
40 2.06 1.8
50 1.71 1.44 (note: 19pct error)
75 1.24 0.96
100 1.00 0.72 (note: 38pct error)

--------------------Check for updates------------------

Subject: Analysis - Same-Store Sales

Last-Revised: 9 Jan 1996
Contributed-By: Steve Mack

When earnings for retail outlets like KMart, Walmart, Best Buy, etc. 
are reported, we see two figures, namely total sales and same-store
sales.  Same-store comparisons measure the growth in sales, excluding
the impact of newly opened stores.  Generally, sales from new stores are
not reflected in same-store comparisons until those stores have been
open for fifty three weeks.  With these comparisons, analysts can
measure sales performance against other retailers that may not be as
aggresive in opening new locations during the evaluated period. 

--------------------Check for updates------------------

Subject: Bonds - Basics

Last-Revised: 5 Jul 1998
Contributed-By: Art Kamlet (artkamlet at, Chris Lott ( contact
me )

A bond is just an organization's IOU; i.e., a promise to repay a sum of
money at a certain interest rate and over a certain period of time.  In
other words, a bond is a debt instrument.  Other common terms for these
debt instruments are notes and debentures.  Most bonds pay a fixed rate
of interest (variable rate bonds are slowly coming into more use though)
for a fixed period of time. 

Why do organizations issue bonds? Let's say a corporation needs to build
a new office building, or needs to purchase manufacturing equipment, or
needs to purchase aircraft.  Or maybe a city government needs to
construct a new school, repair streets, or renovate the sewers. 
Whatever the need, a large sum of money will be needed to get the job

One way is to arrange for banks or others to lend the money.  But a
generally less expensive way is to issue (sell) bonds.  The organization
will agree to pay some interest rate on the bonds and further agree to
redeem the bonds (i.e., buy them back) at some time in the future (the
redemption date). 

Corporate bonds are issued by companies of all sizes.  Bondholders are
not owners of the corporation.  But if the company gets in financial
trouble and needs to dissolve, bondholders must be paid off in full
before stockholders get anything.  If the corporation defaults on any
bond payment, any bondholder can go into bankruptcy court and request
the corporation be placed in bankruptcy. 

Municipal bonds are issued by cities, states, and other local agencies
and may or may not be as safe as corporate bonds.  Some municipal bonds
are backed by the taxing authority of the state or town, while others
rely on earning income to pay the bond interest and principal. 
Municipal bonds are not taxable by the federal government (some might be
subject to AMT) and so don't have to pay as much interest as equivalent
corporate bonds. 

U.S.  Bonds are issued by the Treasury Department and other government
agencies and are considered to be safer than corporate bonds, so they
pay less interest than similar term corporate bonds.  Treasury bonds are
not taxable by the state and some states do not tax bonds of other
government agencies.  Shorter term Treasury bonds are called notes and
much shorter term bonds (a year or less) are called bills, and these
have different minimum purchase amounts (see the article elsewhere in
this FAQ for more details about US Treasury instruments.)

In the U.S., corporate bonds are often issued in units of $1,000.  When
municipalities issue bonds, they are usually in units of $5,000. 
Interest payments are usually made every 6 months. 

A bond with a maturity of less than two years is generally considered a
short-term instrument (also known as a short-term note).  A medium-term
note is a bond with a maturity between two and ten years.  And of
course, a long-term note would be one with a maturity longer than ten

The price of a bond is a function of prevailing interest rates.  As
rates go up, the price of the bond goes down, because that particular
bond becomes less attractive (i.e., pays less interest) when compared to
current offerings.  As rates go down, the price of the bond goes up,
because that particular bond becomes more attractive (i.e., pays more
interest) when compared to current offerings.  The price also fluctuates
in response to the risk perceived for the debt of the particular
organization.  For example, if a company is in bankruptcy, the price of
that company's bonds will be low because there may be considerable doubt
that the company will ever be able to redeem the bonds.  When you buy a
bond, you may pay a premium.  In other words, you may pay more than the
face value (also called the "par" value).  For example, a bond with a
face value of $1,000 might sell for $1050, meaning at a $50 premium. 
Or, depending on the markets and such, you might buy a bond for less
than face value, which means you bought it at a discount. 

On the redemption date, bonds are usually redeemed at "par", meaning the
company pays back exactly the face value of the bond.  Most bonds also
allow the bond issuer to redeem the bonds at any time before the
redemption date, usually at par but sometimes at a higher price.  This
is known as "calling" the bonds and frequently happens when interest
rates fall, because the company can sell new bonds at a lower interest
rate (also called the "coupon") and pay off the older, more expensive
bonds with the proceeds of the new sale.  By doing so the company may be
able to lower their cost of funds considerably. 

A bearer bond is a bond with no owner information upon it; presumably
the bearer is the owner.  As you might guess, they're almost as liquid
and transferable as cash.  Bearer bonds were made illegal in the U.S. 
in 1982, so they are not especially common any more.  Bearer bonds
included coupons which were used by the bondholder to receive the
interest due on the bond; this is why you will frequently read about the
"coupon" of a bond (meaning the interest rate paid). 

Another type of bond is a convertible bond .  This security can be
converted into shares of the company that issues the bond if the
bondholder chooses.  Of course, the conversion price is usually chosen
so as to make the conversion interesting only if the stock has a pretty
good rise.  In other words, when the bond is issued, the conversion
price is set at about a 15--30% premium to the price of the stock when
the bond was issued.  There are many terms that you need to understand
to talk about convertible bonds.  The bond value is an estimate of the
price of the bond (i.e., based on the interest rate paid) if there were
no conversion option.  The conversion premium is calculated as ((price -
parity) / parity) where parity is just the price of the shares into
which the bond can be converted.  Just one more - the conversion ratio
specifies how many shares the bond can be converted into.  For example,
a $1,000 bond with a conversion price of $50 would have a conversion
ratio of 20. 

Who buys bonds? Many individuals buy bonds.  Banks buy bonds.  Money
market funds often need short term cash equivalents, so they buy bonds
expiring in a short time.  People who are very adverse to risk might buy
US Treasuries, as they are the standard for safeness.  Foreign
governments whose own economy is very shaky often buy Treasuries. 

In general, bonds pay a bit more interest than federally insured
instruments such as CDs because the bond buyer is taking on more risk as
compared to buying a CD.  Many rating services (Moody's is probably the
largest) help bond buyers assess the riskiness of any bond issue by
rating them.  See the FAQ article on bond ratings for more information. 

Listed below are some additional resources for information about bonds. 
   * The Bond Market Association runs an information site.

--------------------Check for updates------------------

Subject: Bonds - Amortizing Premium

Last-Revised: 12 Jul 2001
Contributed-By: Chris Lott ( contact me )

The IRS requires investors who purchase certain bonds at a premium
(i.e., above par, which means above face value) to amortize that premium
over the life of the bond.  The reason is fairly straightforward.  If
you bought a bond at 101 and were redeemed at 100, that sounds like a
capital loss -- but of course it really isn't, since it's a bond (not a
stock).  So the IRS prevents you from buying lots and lots of bonds
above par, taking the interest and a phony loss that could offset a bit
of other income. 

Here's a bit more discussion, excerpted from a page at the IRS.  If you
pay a premium to buy a bond, the premium is part of your cost basis in
the bond.  If the bond yields taxable interest, you can choose to
amortize the premium.  This generally means that each year, over the
life of the bond, you use a part of the premium that you paid to reduce
the amount of interest that counts as income.  If you make this choice,
you must reduce your basis in the bond by the amortization for the year. 
If the bond yields tax-exempt interest, you must amortize the premium. 
This amortized amount is not deductible in determining taxable income. 
However, each year you must reduce your basis in the bond by the
amortization for the year. 

To compute one year's worth of amortization for a bond issued after 27
September 1985 (don't you just love the IRS?), you must amortize the
premium using a constant yield method.  This takes into account the
basis of the bond's yield to maturity, determined by using the bond's
basis and compounding at the close of each accrual period.  Note that
your broker's computer system just might do this for you automatically. 

--------------------Check for updates------------------

Subject: Bonds - Duration Measure

Last-Revised: 19 Feb 1998
Contributed-By: Rich Carreiro (rlcarr at

This article provides a brief introduction to the duration measure for
bonds.  The duration measure for bonds is a invention that allows bonds
of different maturities and coupon rates to be compared directly. 

Everyone knows that the maturity of a bond is the amount of time left
until it matures.  Most people also know that the price of a bond swings
more violently with interest rates the longer the maturity of the bond
is.  What many people don't know is that maturity is actually not that
great a measure of the lifetime of a bond.  Enter duration. 

The reason why maturity isn't that great a measure is that it does not
account for the differences in bond coupons.  A 10-year bond with a 5%
coupon will be more sensitive to interest rate changes than a 10-year
bond with an 8% coupon.  A 5-year zero-coupon bond may well be more
sensitive than a 7-year 6% bond, and so forth. 

Faced with the inadequacy of maturity, the investment gurus came up with
a measure that takes both maturity and coupon rate into account in order
to make apples-to-apples comparisons.  The measure is called duration. 

There are different ways to compute duration.  I will use one of the
common definitions, namely:
     Duration is a weighted average of the times that interest
     payments and the final return of principal are received.  The
     weights are the amounts of the payments discounted by the
     yield-to-maturity of the bond. 

The final sentence may be alternatively stated:
     The weights are the present values of the payments, using the
     bond's yield-to-maturity as the discount rate. 

Duration gives one an immediate rule of thumb -- the percentage change
in the price of a bond is the duration multiplied by the change in
interest rates.  So if a bond has a duration of 10 years and
intermediate-term interest rates fall from 8% to 6% (a drop of 2
percentage points), the bond's price will rise by approximately 20%. 

In the examples and formulas that follow, I make the simplifying
assumptions that:
  1. Interest payments occur annually (they actually occur every 6
     months for most bonds). 
  2. The final interest payment occurs on the date of maturity. 
  3. It is always one year from now to the first interest payment. 

It turns out that (especially for intermediate- and long-term bonds)
these simplifications don't affect the final numbers that much --
duration is well less than a year different from its "true" value, even
for something as short as a duration of 5 years. 

Example 1:
Bond has a $10,000 face value and a 7% coupon.  The yield-to-maturity
(YTM) is 5% and it matures in 5 years.  The bond thus pays $700 a year
from now, $700 in 2 years, $700 in 3 years, $700 in 4 years, $700 in 5
years and the $10,000 return of principal also in 5 years. 

As you may recall, to compute the weighted average of a set of numbers,
you multiply the numbers by the weights and add those products up.  You
then add all the weights up and divide the former by the latter.  In
this case the weights are $700/1.05, $700/1.05^2, $700/1.05^3,
$700/1.05^4, $700/1.05^5, and $10,000/1.05^5, or $666.67, $634.92,
$604.69, $575.89, $548.47, and $7,835.26.  The numbers being average are
the times the payments are received, or 1 year, 2 years, 3 years, 4
years, 5 years, and 5 years.  So the duration is:
    1*$667.67 + 2*$634.92 + 3*$604.69 + 4*$575.89 + 5*$548.47 + 
D = 
       $667.67 + $634.92 + $604.69 + $575.89 + $548.47 + $7,835.26
D = 4.37 years

Example 2:
Bond has a face value of $P, coupon of c, YTM of y, maturity of M years. 
    1Pc/(1+y) + 2Pc/(1+y)^2 + 3Pc/(1+y)^3 + ...  + MPc/(1+y)^M + 
D = 
           Pc/(1+y) + Pc/(1+y)^2 + Pc/(1+y)^3 + ...  + Pc/(1+y)^M + 
We can use summations to condense this equation:
    Pc*Sum i/(1+y)^i + MP/(1+y)^M
D = ------------------------------
      Pc*Sum 1/(1+y)^i + P/(1+y)^M
We can cancel out the face value of P, leaving a function only of
coupon, YTM and time to maturity:
    c*Sum i/(1+y)^i + M/(1+y)^M
D = -----------------------------------
    c*Sum 1/(1+y)^i + 1/(1+y)^M
It is trivial to write a computer program to carry out the calculation. 
And those of you who remember how to find a closed-form expression for
Sum{i=1 to M}(x^i) and Sum{i=1 to M}(ix^i) can grind through the
resulting algebra and get a closed-form expression for duration that
doesn't involve summation loops :-)

Note that any bond with a non-zero coupon will have a duration shorter
than its maturity.  For example, a 30 year bond with a 7% coupon and a
6% YTM has a duration of only 14.2 years.  However, a zero will have a
duration exactly equal to its maturity.  A 30 year zero has a duration
of 30 years.  Keeping in mind the rule of thumb that the percentage
price change of a bond roughly equals its duration times the change in
interest rates, one can begin to see how much more volatile a zero can
be than a coupon bond. 

--------------------Check for updates------------------

Subject: Bonds - Moody Bond Ratings

Last-Revised: 12 Nov 2002
Contributed-By: Bill Rini (bill at, Mike Tinnemeier

Moody's Bond Ratings are intended to characterize the risk of holding a
bond.  These ratings, or risk assessments, in part determine the
interest that an issuer must pay to attract purchasers to the bonds. 
The ratings are expressed as a series of letters and digits.  Here's how
to decode those sequences. 

Rating "Aaa"
     Bonds which are rated Aaa are judged to be of the best quality. 
     They carry the smallest degree of investment risk and are generally
     referred to as "gilt edged." Interest payments are protected by a
     large or an exceptionally stable margin and principal is secure. 
     While the various protective elements are likely to change, such
     changes as can be visualized are most unlikely to impair the
     fundamentally strong position of such issues. 
Rating "Aa"
     Bonds which are rated Aa are judged to be of high quality by all
     standards.  Together with the Aaa group they comprise what are
     generally known as high grade bonds.  They are rated lower than the
     best bonds because margins of protection may not be as large as in
     Aaa securities or fluctuation of protective elements may be of
     greater amplitude or there may be other elements present which make
     the long-term risk appear somewhat larger than the Aaa securities. 
Rating "A"
     Bonds which are rated A possess many favorable investment
     attributes and are considered as upper-medium-grade obligations. 
     Factors giving security to principal and interest are considered
     adequate, but elements may be present which suggest a
     susceptibility to impairment some time in the future. 
Rating "Baa"
     Bonds which are rated Baa are considered as medium-grade
     obligations (i.e., they are neither highly protected not poorly
     secured).  Interest payments and principal security appear adequate
     for the present but certain protective elements may be lacking or
     may be characteristically unreliable over any great length of time. 
     Such bonds lack outstanding investment characteristics and in fact
     have speculative characteristics as well. 
Rating "Ba"
     Bonds which are rated Ba are judged to have speculative elements;
     their future cannot be considered as well-assured.  Often the
     protection of interest and principal payments may be very moderate,
     and thereby not well safeguarded during both good and bad times
     over the future.  Uncertainty of position characterizes bonds in
     this class. 
Rating "B"
     Bonds which are rated B generally lack characteristics of the
     desirable investment.  Assurance of interest and principal payments
     of of maintenance of other terms of the contract over any long
     period of time may be small. 
Rating "Caa"
     Bonds which are rated Caa are of poor standing.  Such issues may be
     in default or there may be present elements of danger with respect
     to principal or interest. 
Rating "Ca"
     Bonds which are rated Ca represent obligations which are
     speculative in a high degree.  Such issues are often in default or
     have other marked shortcomings. 
Rating "C"
     Bonds which are rated C are the lowest rated class of bonds, and
     issues so rated can be regarded as having extremely poor prospects
     of ever attaining any real investment standing. 

A Moody rating may have digits following the letters, for example "A2"
or "Aa3".  According to Fidelity, the digits in the Moody ratings are in
fact sub-levels within each grade, with "1" being the highest and "3"
the lowest.  So here are the ratings from high to low: Aaa, Aa1, Aa2,
Aa3, A1, A2, A3, Baa1, Baa2, Baa3, and so on. 

Most of this information was obtained from Moody's Bond Record. 
Portions of this article are copyright 1995 by Bill Rini. 

--------------------Check for updates------------------

Compilation Copyright (c) 2003 by Christopher Lott.

User Contributions:

Gerri Pisciotta
My employer accidentally advised the company handling the 401k investment that I had been terminated, when in fact I had not. As a result, withdrawals discontinued from my pay and I missed a couple years of contributions. Since I never withdrew from the plan, is my employer liable for making up these contributions? If I made a lump sum catchup contribution,could they do the same?
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