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Interest Rates (Ch. 4)
04/17/06
Compounding interest
• Thus far, we have assumed that interest is compounded (or
paid/earned) once during the period.
• However, often interest is compounded more frequently than once a
period.
• This means that interest is earned (but not necessarily paid) more
than once during the period.
• We examine here what the effect of the frequency of compounding
has on interest rates.
Quoting conventions
• Annual Percentage Rate (APR):
– a.k.a. nominal, stated or quoted rate
– Rate required to be disclosed in lending
agreements (Truth-in-lending laws)
– Does not reflect the actual interest
earned/paid
APR = periodic interest rate * compounding periods per year (C/Y)
Quoting conventions
• Effective Annual Rate (EAR)
– The annual rate of interest actually paid or earned.
– Incorporates the effect of compounding
– The EAR is equal to the annual percentage yield
(APY) which is the rate required to be disclosed in
savings products (Truth-in-savings laws)
m
APR 

EAR  1 
 1
m 

where m is the compounding frequency (or C/Y)
– For continuously compounded interest,
EAR  e APR  1
Employing compounded interest in
PV/FV calculations (rules to follow)
• For lending (savings) products, assume that the
interest rate stated is the APR (APY) unless
otherwise specified.
• Ensure that the frequency of cash flows and
interest rate used is consistent.
– If APR is compounded and the compounding
frequency (C/Y) is the same as the cash flow
frequency, use APR/ (C/Y) for the interest rate.
Employing compounded interest in
PV/FV calculations (rules to follow)
• Ensure that the frequency of cash flows and
interest rate used is consistent. (contd.)
– If cash flows are annual, use EAR regardless of the
compounding frequency
– For simple (single cash flow) PV/FV problems, use
EAR
– If you are provided with an interest rate over the
same period as the cash flow period, no adjustments
need to be made
Nominal and Real Interest Rates
• APR and Periodic Rates are nominal rates
• Nominal Rates have two components
– Real Rate
– Expected Inflation Rate
• Real Rate is the reward for saving
• Expected Inflation is the rising price of a good
Nominal and Real Interest Rates
• Fisher Effect
– Relationship between real rate, expected
inflation, and nominal rate
(1+r) = (1+r*) x (1+h)
where r is the nominal rate, r* is the real rate, and h
is expected inflation
- We can get an approximate value for r:
r ≈ r* + h
Risk-free Rate and Premiums
• Nominal interest rates (of return)
associated with a particular investment or
asset are based on four components.
– Risk-free rate
– Default Risk
– Maturity
– Liquidity
Risk-free Rate and Premiums
• Risk-free rate (rf)
– a “guaranteed” rate available to investors
– 3-month U.S. Treasury Bill rate
• Default Risk
– Different Investments have different default risk
based on the issuers ability to meet future promised
payments
– Credit ratings (by Standard and Poors, etc.) evaluate
the default risk of public companies.
Risk-free Rate and Premiums
• Maturity Premium
– Investors demand more compensation for investing in
longer-maturity investments
– The term structure of interest rates and yield curve
reflects the difference in rates as the borrowing time
increases and provides an estimate of the maturity
premium
• Liquidity Premium – Different investments can
be converted back to cash at different speeds
and ease
Risk-free Rate and Premiums
• Summary of Interest Rates
– The nominal interest rate can be summarized
as follows:
r ≈ rf* + h + dp + mp + lp
where dp, mp and lp represent the premiums
required by investors for default risk, maturity
and liquidity of the investment.