2
3
4
400
400
400
400
400
1.1
1.1
1.1
1.1
1,667.95
PV

58
4.4
Perpetuity
Perpetuity is a special case of an annuity in which the number of equal cash flows is
infinite
.
The formula for the present value of a perpetuity is:
Present Value of a Perpetuity
C
r
Example 4.10
In the early 1900's the Canadian Government issued $100 par value 2% Consol bonds.
The
holder of these bonds is entitled to receive a coupon (or interest) payment of $2 per year
forever.
If the current appropriate discount rate is 5% p.a. and the next coupon is due one
year from now, how much is one of the Consols worth?
2
0.05
40
C
PV
r
4.5
Comparing Rates
Suppose a bank offers you two deals: (1) pays you 10% interest per year or (2) pays you 5%
interest compounded every six months.
Which deal would you prefer?
If you invest $1, then after a year,
Option (1) will give you:
$1
1.1
$1.1
Option (2) will give you:
2
$1
1.05
$1.1025
Obviously, option 2 is better as you can enjoy the
interest on interest
.
As the example
illustrates, 10% compounded semiannually is actually equivalent to 10.25% per year.

59
4.5.1
Effective Annual Rate (EAR)
In the example, the 10% is called the
quoted interest rate
.
The 10.25%, which is actually the
rate that you can earn, is called the
effective annual rate
(EAR).
If you want to compare two
alternative investments with different compounding periods, you need to compute the EAR
and use that for comparison.
To get the effective annual rate,
Quoted Rate
1
1
m
EAR
m
Where
m
is the number of times the interest is compounded during the year.
Example 4.11
Suppose a bank offers a nominal interest rate of 5% on your time deposit.
Compare the
different EARs with various times the interest is compounded each year.
Compounding
Formula
Effective Annual Rate
Annually
1
0.05
1
1
1
r
5.0000%
Semiannually
2
0.05
1
1
2
r
5.0625%
Quarterly
4
0.05
1
1
4
r
5.0945%
Monthly
12
0.05
1
1
12
r
5.1162%
Weekly
52
0.05
1
1
52
r
5.1246%
Daily
365
0.05
1
1
365
r
5.1267%
Hourly
8760
0.05
1
1
8760
r
5.1271%
Continuously
0.05
1
r
e
5.1271%
You will always prefer more compounding periods to less.

60
Example 4.12
You are looking at two savings accounts.
HSBC pays you 5.25%, with daily compounding.
BOC pays 5.3% with semiannual compounding.
Which account should you use?
HSBC:
365
0.0525
1
1
365
5.3899%
EAR
BOC:
2
0.053
1
1
2
5.3702%
EAR
4.5.2
Annual Percentage Rate (APR)
Another rate we often calculate is the
annual percentage rate
(APR).
APR is the interest rate
charged per period multiplied by the number of periods per year.
Since the law requires that
lenders disclose an APR on all loans, this rate must be displayed on a loan document in an
unambiguous way.
Example 4.13
What is the APR if (1) the monthly rate is 0.5%; (2) the semiannual rate is 0.5%?
For (1):
0.5% 12
6%
APR
For (2):
0.5%
2
1%
APR
Remember, APR is only an
annual rate
that is quoted by law.
In order to figure out the
actual rate, you need to compute the EAR.

61
The relationship between EAR and APR:
1
1
m
APR
EAR
m
If you have an effective rate, you can compute the APR.

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