EFFECTIVE RATE
𝑟
𝑖= 𝑎𝑛𝑑 𝑛 = 𝑚𝑡
𝑚
𝑟 𝑛
𝐸𝑅 = (1 + ) −1
𝑚
EFFECTIVE RATE
NOMINAL RATE- basic annual rate of interest
EFFECTIVE RATE OF INTEREST- actual or exact rate of interest earned
on the principal during a one-year period
EX: A principal is invested at 5% compounded quarterly.
NOMINAL RATE = 5%
EFFECTIVE RATE = greater that 5% because of the compounding
which occurs 4x a year.
SAMPLE PROBLEM
What is the effective rate
corresponding to 18%
compounded daily? Take 1 year =
360 days.
SAMPLE PROBLEM
𝐸𝑅 = (1 + 𝑖)𝑛 −1
0.18 360
𝐸𝑅 = (1 + ) −1
360
𝑬𝑹 = 𝟏𝟗. 𝟕𝟐%
SAMPLE PROBLEM
What is the corresponding
effective rate of 18%
compounded semi-quarterly?
Take 1 year = 360 days.
SAMPLE PROBLEM
𝐸𝑅 = (1 + 𝑖)𝑛 −1
0.18 8
𝐸𝑅 = (1 + ) −1
8
𝑬𝑹 = 𝟏𝟗. 𝟒𝟖%
SAMPLE PROBLEM
Compute the equivalent rate of
6% compounded semi-annually to
a rate compounded quarterly.
SAMPLE PROBLEM
𝐸𝑅𝑞𝑢𝑎𝑟𝑡𝑒𝑟𝑙𝑦 = 𝐸𝑅𝑠𝑒𝑚𝑖−𝑎𝑛𝑛𝑢𝑎𝑙𝑙𝑦
𝑖 4 0.06 2
(1 + ) −1= (1 + ) −1
4 2
𝑖 = 0.0596 = 𝟓. 𝟗𝟔%
ANNUITY
ANNUITY
SIMPLE ANNUITY
ORDINARY ANNUITY DEFERRED
PERPETUITY
ANNUITY DUE ANNUITY
ANNUITY
ORDINARY ANNUITY
𝐴 (1 + 𝑖)𝑛 − 1
𝐹=
𝑖
𝑛
𝐹 𝐴 (1 + 𝑖) − 1
𝑃= =
(1 + 𝑖)𝑛 (1 + 𝑖)𝑛 𝑖
SAMPLE PROBLEM
Find the annual payment to
extinguish a debt of ₱10,000
payable for 6 years at 12%
interest annually.
SAMPLE PROBLEM
𝑃 = 𝐷𝑒𝑏𝑡 = 10,000
𝐹 𝐴 (1 + 0.12)6 − 1
10000 = =
(1 + 𝑖)𝑛 (1 + 0.12)6 0.12
𝑨 = 𝟐, 𝟒𝟑𝟐. 𝟐𝟔 Debt
1 2 3 4 5 6
0
A A A A A A
P
SAMPLE PROBLEM
What annuity is required over 12
years to equate with a future
amount of ₱20,000? Assume i=6%
annually.
SAMPLE PROBLEM
𝐴 (1 + 𝑖)𝑛 − 1
𝐹=
𝑖 0
1 2 3… 12
𝐴 (1 + 0.06)12 − 1
20000 = A A A A A A
0.06 F = 20,000
𝑨 = 𝟏, 𝟏𝟖𝟓. 𝟓𝟒
ANNUITY
ANNUITY DUE
𝐴 (1 + 𝑖)𝑛 − 1
𝐹 = 𝐹1 1 + 𝑖 = (1 + 𝑖)
𝑖
𝐹1 𝐴 (1 + 𝑖)𝑛 − 1
𝑃= 𝑛
= 𝑛
(1 + 𝑖)
(1 + 𝑖) (1 + 𝑖) 𝑖
SAMPLE PROBLEM
Suppose that a court settlement results
in a ₱750,000 award. If this is invested
at 9% compounded semiannually,
how much will it provide at the
beginning of each half-year for a period
of 7 years?
SAMPLE PROBLEM
𝐹1 𝐴 (1 + 𝑖)𝑛 − 1
𝑃= 𝑛
= 𝑛
(1 + 𝑖)
(1 + 𝑖) (1 + 𝑖) 𝑖
0.09 2 7
𝐴 (1 + ) −1 0.09
750000 = 2 1+
0.09 2 7 0.09 2
(1 + ) ( )
2 2
𝑨 = 𝟕𝟎, 𝟐𝟎𝟓. 𝟗𝟕
SAMPLE PROBLEM
Betty plans to spend 2 months in a rented
house in Palawan in April to May 2018. The
monthly rent is ₱5 000, payable in advance.
The landlady can earn interest at 6%
compounded monthly. How much should
Betty pay on April 1 for both months’ rent
paid in advance?
SAMPLE PROBLEM
𝐹1 𝐴 (1 + 𝑖)𝑛 − 1
𝑃= 𝑛
= 𝑛
(1 + 𝑖)
(1 + 𝑖) (1 + 𝑖) 𝑖
0.06 2
5000 (1 + ) −1 0.06
𝑃= 12 1+
0.06 2 0.06 12
(1 + ) ( )
12 12
𝑷 = 𝟗, 𝟗𝟕𝟓. 𝟏𝟐
ANNUITY
DEFERRED ANNUITY
𝑛
𝐴 (1 + 𝑖) − 1
𝐹=
𝑖
𝐹 𝐴 (1 + 𝑖)𝑛 − 1
𝑃= =
(1 + 𝑖)𝐾+𝑛 (1 + 𝑖)𝑛 𝑖
SAMPLE PROBLEM
A house and lot can be acquired by a
downpayment of ₱500,000 and a yearly
payment of ₱100,000 at the end of each
year for a period of 10 years, starting at the
end of 5 years from the date of purchase. If
money is worth 14% compounded
annually, what is the cash price of the
property?
1… 4 5 6… 15
0
SAMPLE PROBLEM 500,000
A A A A
P1
𝐴 (1 + 𝑖)𝑛 − 1 P2
𝑃1 = 𝑃1
(1 + 𝑖)𝑛 𝑖
𝑃2 =
100000 (1 + 0.14)10 − 1 (1 + 𝑖)𝑛
𝑃1 = 521,611.56
(1 + 0.14)10 (0.14) 𝑃2 =
(1 + 0.14)4
𝑷𝟏 = 𝟓𝟐𝟏, 𝟔𝟏𝟏. 𝟓𝟔 𝑃2 = 308,835.92
𝐶𝑎𝑠ℎ 𝑃𝑟𝑖𝑐𝑒 = 500,000 + 308,835.92
𝑪𝒂𝒔𝒉 𝑷𝒓𝒊𝒄𝒆 = 𝟖𝟎𝟖, 𝟖𝟑𝟓. 𝟗𝟐
ANNUITY
PERPETUITY
When ninfinity,
𝐴
𝑃=
𝑖
SAMPLE PROBLEM
You are valuing a firm that is
expected to earn cash flows of
₱10m per year in perpetuity. You
estimate a discount rate of 11%.
What is the present value of these
cash flows?
SAMPLE PROBLEM
𝐴
𝑃=
𝑖
10,000,000
𝑃=
0.11
𝑃 = 90,909,090.91
PRACTICE PROBLEMS
1. What interest rate, compounded monthly, is
equivalent to a 10% effective rate?
2. How much must you invest today in order to
withdraw ₱2,000 annually for 10 years if the interest
rate is 9%?
3. What uniform annual amount should be deposited
each year in order to accumulate ₱100,000 at the end
of 5th annual deposit if money earns 10% interest?
