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Simple
interest: Add up all the
interest paid/payable in
a period. Divide that by
the principal at the beginning
of the period. E.g. on $100
(principal):
credit
card debt where $1/day is
charged. 1/100 = 1%/day.
corporate bond where $3
is due after six months,
and another $3 is due at
year end. (3+3)/100 = 6%/year.
certificate of deposit (GIC)
where $6 is paid at year
end. 6/100 = 6%/year.
There are three problems
with simple interest.
The
time periods used for measurement
can be different, making
comparisons wrong. You cannot
say the 1%/day credit card
interest is 'equal' to a
365%/year GIC.
The time value of money
means that $3 paid every
six months hurts more than
$6 paid only at year end.
So you cannot 'equate' the
6% bond to the 6% GIC.
When interest is due, but
not paid, it must be clear
what happens. Does it remain
'interest payable', like
the bond's $3 payment after
six months? Or does it get
added to the original principal,
like the 1%/day on the credit
card? Each time it is added
to the principal it 'compounds'.
The interest from that time
forward is calculated on
that (now larger) principal.
The more frequent the compounding,
the faster the principal
grows, and the greater the
interest.
Compound interest: In order
to solve these three problems,
there is a convention that
interest rates will be disclosed
as if the term is one year
and the compounding is yearly.
The discussion at compound
interest shows how to convert
to and from the different
measures of interest.
Real
interest: This is calculated
as (nominal interest rate)
- (inflation). It attempts
to measure the value of
the interest in units of
stable purchasing power.
See the discussion at real
interest rate.
Cumulative
interest/return: This calculation
is (FV/PV)-1. It ignores
the 'per year' convention
and assumes compounding
at every payment date. It
is usually used to compare
two long term opportunities.
Since the difference in
rates gets magnified by
time, so the speaker's point
is more clearly made.
Other
exceptions:
US
and Canadian T-Bills (short
term Government debt) have
a different convention.
Their interest is calculated
as (100-P)/P where 'P' is
the price paid. Instead
of normalizing it to a year,
the interest is prorated
by the number of days 't':
(365/t)*100. (See also:
Day count convention). The
total calculation is ((100-P)/P)*((365/t)*100)
Corporate Bonds are most
frequently payable twice
yearly. The amount of interest
paid is the simple interest
disclosed divided by two
(multiplied by the face
value of debt).
Rule of 78: Some consumer
loans calculate interest
by the "Rule of 78"
or "Sum of digits"
method. Seventy-eight is
the sum of the numbers 1
through 12, inclusive. And
the practice enabled quick
calculations of interest
in the pre-computer days.
In a loan with interest
calculated per the Rule
of 78, the total interest
over the life of the loan
is calculated as either
simple or compound interest
and amounts to the same
as either of the above methods.
Payments remain constant
over the life of the loan;
however, payments are allocated
to interest in progressively
smaller amounts. In a one-year
loan, in the first month,
12/78 of all interest owed
over the life of the loan
is due; in the second month,
11/78; progressing to the
twelfth month where only
1/78 of all interest is
due. The practical effect
of the Rule of 78 is to
make early pay-offs of term
loans more expensive. Approximately
3/4 of all interest due
on a one year loan is collected
by the sixth month, and
pay-off of the principal
then will cause the effective
interest rate to be much
higher than than the APY
used to calculate the payments.
[1]
The
United States outlawed the
use of "Rule of 78"
interest in loans over five
years in term. Certain other
jurisdictions have outlawed
application of the Rule
of 78 in certain types of
loans, particularly consumer
loans. [2]
Rule
of 72: The "Rule of
72" is a "quick
and dirty" method for
finding out how fast money
doubles for a given interest
rate. For example, if you
have an interest rate of
6%, it will take 72/6 or
12 years for your money
to double, compounding at
6%. This is an approximation
that starts to break down
above 10%.
Market interest
rates
There are markets for investments
which include the money
market, bond market, as
well as retail financial
institutions like banks,
which set interest rates.
Each specific debt takes
into account the following
factors in determining its
interest rate:
Inflation:
Since the lender is deferring
his consumption, he will
at a bare minimum, want
to recover enough to pay
the increased cost of goods
due to inflation. Because
future inflation is unknown,
there are three tactics.
Charge
X% interest 'plus inflation'.
Many governments issue 'real-return'
or 'inflation indexed' bonds.
The principal amount and
the interest payments are
continually increased by
the rate of inflations.
See the discussion at real
interest rate.
Decide on the 'expected'
inflation rate. This still
leaves both parties exposed
to the risk of 'unexpected'
inflation.
Allow the interest rate
to be periodically changed.
While a 'fixed interest
rate' remains the same throughout
the life of the debt, 'variable'
or 'floating' rates can
be reset. There are derivative
products that allow for
hedging and swaps between
the two.
Default: There is always
the risk the borrower will
become bankrupt, abscond
or otherwise default on
the loan. The risk premium
attempts to measure the
integrity of the borrower,
the risk of his enterprise
succeeding and the security
of any collaterol pledged.
Loans to developing countries
have higher risk premiums
than those to the US government.
An operating line of credit
to a business will have
a higher rate than a mortgage.
The
credit worthiness of businesses
is measured by bond rating
services and individual's
credit scores by credit
bureaus. The risks of an
individual debt may have
a large standard deviation
of possibilities. The lender
may want to cover his maximum
risk. But lenders with portfolios
of debt can lower the risk
premium to cover just the
most probable outcome.
Deferred
consumption: Charging interest
equal only to inflation
will leave the lender with
the same purchasing power,
but he would prefer his
own consumption NOW rather
than later. There will be
an interest premium of the
delay. See the discussion
at time value of money.
He may not want to consume,
but instead would invest
in another product. The
possible return he could
realize in competing investments
will determine what interest
he charges.
Length
of time: Time has two effects.
Shorter
terms have less risk of
default and inflation because
the near future is easier
to predict than events 20
year off.
Longer terms allow for investments
in larger projects with
higher eventual returns.
Contrast this to the lender's
preference for readily available
cash for contingencies.
This is why banks pay higher
interest on non-redeemable
GICs than on chequing account
balances.
Other: Borowers and lenders
may face individual tax
rates, transaction costs
and foreign exchange rate
risks. In a liquid market
they cannot exert their
personal preferences. It
is the sum total of the
participants who determine
rates. The market for financial
instruments has moved from
the local, to the national,
and is now international.
Interest rates in macroeconomics
Output and unemployment
Interest rates are the main
determinant of investment
on a macroeconomic scale.
Broadly speaking, if interest
rates increase across the
board, then investment decreases,
causing a fall in national
income. Note that if interest
rates are high, that means
the broad economy is doing
well and thus people will
be willing to borrow money
at higher interest rates.
Interest
rates are generally determined
by the market, but government
intervention - usually by
a central bank- may strongly
influence short-term interest
rates, and is used as the
main tool of monetary policy.
The central bank offers
to buy or sell money at
the desired rate and, because
of their immense size, they
are able to influence i*n.
By
altering i*n, the central
bank is able to affect the
interest rates faced by
everyone who wants to borrow
money for economic investment.
Investment can change rapidly
to changes in interest rates,
affecting national income.
Through
Okun's Law changes in output
affect unemployment.
Open Market Operations in
the United States
The effective federal funds
rate charted over fifty
yearsThe Federal Reserve
(often referred to as 'The
Fed') implements monetary
policy largely by targeting
the federal funds rate.
This is the rate that banks
charge each other for overnight
loans of federal funds,
which are the reserves held
by banks at the Fed.
Open
market operations are one
tool within monetary policy
implemented by the Federal
Reserve to steer short-term
interest rates. Using the
power to buy and sell treasury
securities, the Open Market
Desk at the Federal Reserve
Bank of New York can supply
the market with dollars
by purchasing T-notes, hence
increasing the nation's
money supply. By increasing
the money supply or Aggregate
Supply of Funding (ASF),
interest rates will fall
due to the excess of dollars
banks will end up with in
their reserves. Excess reserves
may be lent in the Fed funds
market to other banks, thus
driving down rates.
Money and inflation
Loans, bonds, and shares
have some of the characteristics
of money and are included
in the broad money supply.
By
setting i*n, the government
institution can affect the
markets to alter the total
of loans, bonds and shares
issued. Generally speaking,
a higher real interest rate
reduces the broad money
supply.
Through
the quantity theory of money,
increases in the money supply
lead to inflation. This
means that interest rates
can affect inflation in
the future.
