
The Effective Annual Rate (EAR)
◦ Indicates the total amount of interest that will be earned at the end of one year
◦ The EAR considers the effect of compounding
 Also referred to as the effective annual yield (EAY) or annual percentage yield (APY)


Adjusting the Discount Rate to Different
Time Periods
◦ If 5% is the effective annual rate what is the 6month rate?
◦ Earning a 5% return annually is not the same as earning 2.5% every six months.
n

(1
 r n
)

1
 (1.05) 0.5
– 1= 1.0247 – 1 = .0247 = 2.47%
 Note: n = 0.5 since we are solving for the six month
(or 1/2 year) rate


Problem
◦ Suppose your bank account pays interest monthly with an effective annual rate of 5%. What is the interest you will earn in one month? What is the interest you will earn in two years?
◦ We know from above that a 5% EAR is equivalent to earning (1.05) 1/12 -1 = .4074% per month.
◦ We can use the same formula to find the two year rate (1.05) 2 – 1 = 10.25% (in particular, not 10%).


The annual percentage rate (APR), indicates the amount of simple interest earned in one year.
◦ Simple interest is the amount of interest earned without the effect of compounding.
◦ The APR is typically less than the effective annual rate (EAR).
◦ APR is simply a communication device.


◦ The APR with k compounding periods each year is simply a standardized way of quoting the actual interest earned each compounding period:
Interest Rate per Compounding Period

APR k periods / year


Converting an APR to an EAR
1

EAR


1

APR k

 k
◦ The EAR increases with the frequency of compounding.
 Continuous compounding is compounding every instant.

◦ If the APR is 6% what are the EARs for different compounding intervals?
 Annual compounding: (1 + 0.06/1) 1 – 1 = 6%
 Semiannual compounding: (1 + 0.06/2) 2 – 1 = 6.09%
 Monthly compounding: (1 + 0.06/12) 12 – 1 = 6.1678%
 Daily compounding: (1 + 0.06/365) 365 – 1 = 6.1831%
 A 6% APR with continuous compounding results in an
EAR of approximately 6.1837%. This is an abstraction we will not make much use of in this course.




You are offered a chance to buy a perpetuity paying a $100 per year (beginning in one year) for $1,650 today. You are able to borrow and lend money at a 6% APR with monthly compounding. Should you buy the perpetuity?
First we must find the EAR since the perpetuity makes annual payments. The effective annual rate is (1+0.06/12) 12 – 1 = 6.16778%.
The value of the perpetuity is $100/0.0616778
= $1,621.33.


Inflation and Real Versus Nominal Rates
◦ Nominal Interest Rate: The rates quoted by financial institutions and used for discounting or compounding cash flows
◦ Real Interest Rate: The rate of growth of your purchasing power, after adjusting for inflation

Growth in Purchasing Power

1
 r r

1
 r
1
 i

Growth of Money
Growth of Prices

The Real Interest Rate r r

1

1
 r i r
 i
1
 i
 r
 i


Problem
◦ In the year 2006, the average 1-year Treasury rate was about 4.93% and the rate of inflation was about 2.58%.
◦ What was the real interest rate in 2006?

Solution
◦ Using the equation given above, the real interest rate in
2006 was:
 (4.93% − 2.58%) ÷ (1.0258) = 2.29%
 Which is approximately equal to the difference between the nominal rate and inflation: 4.93% – 2.58% = 2.35%


An increase in interest rates will typically reduce the NPV of an investment.
◦ Consider an investment that requires an initial investment of $3 million and generates a cash flow of $1 million per year for four years. If the interest rate is 4%, the investment has an NPV of:
NPV 3
1

1

1

1
2 3
(1.04) (1.04) (1.04) (1.04)
4

$0.629 million
◦ Recall the number $0.629 million means something very specific. What is it?


If the interest rate jumps to 15%, the NPV of this investment becomes negative and we would not undertake such a project.
NPV 3
1

1

1

1
2 3
(1.15) (1.15) (1.15) (1.15)
4
 
$0.145 million


Term Structure: The relationship between the investment term and the interest rate

Yield Curve: A graph of the term structure


When the yield curve is not flat we must discount future cash flows at the appropriate rates.
◦ Compute the present value of a risk-free three-year annuity of $500 per year, given the following yield curve:
January-07
Term (Years)
1
Rate
5.06%
2
3
4.88%
4.79%


Solution
◦ Each cash flow must be discounted by the corresponding interest rate:
PV

$500

$500

$500
2
1.0506
1.0488
1.0479
3
PV

$475.92

$454.55

434.52

$1, 364.99
◦ It is very important to note that even though this is a
3-year annuity, we cannot use the annuity formula to find its present value.


The shape of the yield curve is influenced by interest rate expectations.
◦ An inverted yield curve indicates that interest rates are expected to decline in the future.
 Because interest rates tend to fall in response to an economic slowdown, an inverted yield curve is often interpreted as a negative forecast for economic growth.
 Each of the last six recessions in the United States was preceded by a period in which the yield curve was inverted.


The shape of the yield curve is influenced by interest rate expectations.
◦ A steep yield curve generally indicates that interest rates are expected to rise in the future.
 The yield curve tends to be sharply increasing as the economy comes out of a recession and interest rates are expected to rise.


Risk and Interest Rates
◦ U.S. Treasury securities are considered “risk-free.”
All other borrowers have some risk of default, so investors require a higher rate of return.
◦ The yield curve is commonly presented in terms of risk free yields but one can also examine a AAA or
AA yield curve at any point in time.

When computing the present value of future cash flows that are risky we adjust the discount rate to account for the risk.


Taxes reduce the amount of interest an investor can keep, and we refer to this reduced amount as the after-tax interest rate.
r
 r (1
 
)


Opportunity Cost of Capital: The best available expected return offered in the capital market on an investment of comparable risk and term (investment horizon) to the cash flow being discounted
◦ Also referred to as Cost of Capital