Lesson 3
Fair Market Value
Objectives
At the end of this lesson, the learner should be able to
● accurately define fair market value;
● accurately calculate the fair market value of a cash
flow stream; and
● correctly solve word problems involving fair market
value.
Essential Questions
● What is a fair market value and why is it important?
● How is fair market value applied in real-life situations?
Guide Questions
● Suppose you are a seller, what factors should you consider
in setting the price of a property being sold?
● If you are a buyer, what conditions must be satisfied in
establishing a deal with the seller?
● Give an example of a real-life situation that illustrates the
concept of fair market value.
Learn about It!
Fair market value (FMV)
1 the price that two parties are willing to pay for an asset or liability, given the following
conditions:
• Both parties are well informed about the condition of the asset or liability.
• Neither party is under undue pressure to buy or sell the item.
• There is no time pressure to complete the deal.
Example:
Suppose you want to sell a car for ₱500 000. A buyer offers to
purchase the car at a lower price of ₱420 000. If you and the
buyer agreed on ₱450 000 after negotiating the price, then the
fair market value of the car is ₱450 000.
Try It!
Example 1: A house is for sale at ₱25 500 000. Mr. Guavis
offers the seller ₱20 350 000 for the property. The two
discussed the price and agreed on ₱23 500 000. What is the
fair market value of the property?
Try It!
Example 1: A house is for sale at ₱25 500 000. Mr. Guavis
offers the seller ₱20 350 000 for the property. The two
discussed the price and agreed on ₱23 500 000. What is the
fair market value of the property?
Solution:
Since Mr. Guavis and the seller agreed to the same price, the
fair market value of the house and lot is ₱23 500 000.
Try It!
Example 2: Mrs. Monceda received two offers on a
beachfront house that she wants to sell.
Option A: Mr. Medina has offered ₱1 500 000 down payment, plus
₱4 000 000 lump sum payment six years from now.
Option B: Ms. Arnan has offered ₱700 000 down payment, plus ₱200 000
payment every end of the quarter for six years.
Compare the economic values of the two offers if money can
earn 6% compounded annually.
Try It!
Solution:
To compare their economic values, we must get the
present value of the two offers and add their respective
down payments.
Try It!
Solution:
1. Determine the present value for Mr. Medina’s offer.
Use the formula for finding the present value involving
compound interest.
𝐹𝑉
𝑃𝑉 =
1+𝑟 𝑛
4 000 000
𝑃𝑉 =
1 + 0.06 6
4 000 000
𝑃𝑉 =
1.06 6
Try It!
Solution:
1. Determine the present value for Mr. Medina’s offer.
Use the formula for finding the present value involving
compound interest.
4 000 000
𝑃𝑉 =
1.06 6
4 000 000
𝑃𝑉 =
1.418519112
𝑷𝑽 = 𝟐 𝟖𝟏𝟗 𝟖𝟒𝟐. 𝟏𝟔
Try It!
Solution:
2. Find the economic value for Mr. Medina’s offer.
We add the down payment to the computed present
value of the lump sum payment to determine the
economic value. That is,
𝐸𝑉 = 𝐷𝑃 + 𝑃𝑉
𝐸𝑉 = 1 500 000 + 2 819 842.16
𝑬𝑽 = 𝟒 𝟑𝟏𝟗 𝟖𝟒𝟐. 𝟏𝟔
Try It!
Solution:
3. Determine the present value for Ms. Arnan’s offer.
Here, we will use the formula for finding the present
value of ordinary general annuity.
First, find the value of 𝒄.
𝑚 𝟏
𝒄= =
𝑝 𝟒
Try It!
Solution:
3. Determine the present value for Ms. Arnan’s offer.
Next, find the value of 𝒊𝟐 .
𝑐
𝑖2 = 1 + 𝑖 −1
1
𝑖2 = 1 + 0.06 4 −1
1
𝑖2 = 1.06 − 14
𝑖2 = 1.014673846 − 1
𝒊𝟐 = 𝟎. 𝟎𝟏𝟒𝟔𝟕𝟑𝟖𝟒𝟔
Try It!
Solution:
3. Determine the present value for Ms. Arnan’s offer.
Then, solve for the present value using the formula for
the present value of a general ordinary annuity.
Try It!
Solution:
3. Determine the present value for Ms. Arnan’s offer.
1 − 1 + 𝑖2 −𝑛
𝑃𝑉𝑂𝐺𝐴 = 𝑅
𝑖2
1 − 1 + 0.014673846 −24
𝑃𝑉𝑂𝐺𝐴 = 200 000
0.014673846
1 − 1.014673846 −24
𝑃𝑉𝑂𝐺𝐴 = 200 000
0.014673846
1 − 0.7049605433
𝑃𝑉𝑂𝐺𝐴 = 200 000
0.014673846
Try It!
Solution:
3. Determine the present value for Ms. Arnan’s offer.
1 − 0.7049605433
𝑃𝑉𝑂𝐺𝐴 = 200 000
0.014673846
0.2950394567
𝑃𝑉𝑂𝐺𝐴 = 200 000
0.014673846
𝑃𝑉𝑂𝐺𝐴 = 200 000 20.10648447
𝑃𝑉𝑂𝐺𝐴 = 4 021 296.89
Try It!
Solution:
4. Find the economic value for Ms. Arnan’s offer.
Let us add the down payment to the present value of
the general ordinary annuity to determine the economic
value.
𝐸𝑉 = 𝐷𝑃 + 𝑃𝑉
𝐸𝑉 = 700 000 + 4 021 296.89
𝑬𝑽 = 𝟒 𝟕𝟐𝟏 𝟐𝟗𝟔. 𝟖𝟗
Try It!
Solution:
5. Compare the results.
It is better to choose Ms. Arnan’s offer since it is higher
than Mr. Medina’s offer.
Let’s Practice!
Individual Practice:
1. A piece of land in a commercial area is for sale at
₱2.4 million. A prospective buyer wants to negotiate since
the price is 25% higher than the market value and offers
to buy the land at ₱2 million instead. After a month-long
negotiation, the two parties agreed at the middle price.
How much is the fair market value of the land?
Let’s Practice!
Individual Practice:
2. Given that the money grows by 5% compounded
quarterly for five years, which has a greater future value
for a high-end camera: (a) a payment of ₱6 000 at the end
of every six months or (b) a payment of ₱2 800 at the end
of every quarter?
Key Points
Fair market value (FMV)
1 the price that two parties are willing to pay for an asset or liability, given the following
conditions:
• Both parties are well informed about the condition of the asset or liability.
• Neither party is under undue pressure to buy or sell the item.
• There is no time pressure to complete the deal.
Synthesis
● What is the fair market value?
● How can you apply the concept of fair market value in
your daily life as a student?
● Can the first payment interval of an annuity be delayed?
Lesson 4
Deferred Annuity
Objectives
At the end of this lesson, the learner should be able to
● accurately define deferred annuity and period of deferral;
● correctly calculate the period of deferral of a deferred annuity;
● correctly solve the present value of a deferred annuity; and
● correctly find the number of periodic payments given the
present value of a deferred annuity.
Essential Questions
● What is a deferred annuity?
● How does a deferred annuity compare with an ordinary
annuity?
● How can you find the present value and period of deferral
of a deferred annuity?
Warm Up!
Suppose you are considering an investment that will allow you to claim
₱20 000 every year for 5 years beginning on 2022 with a 5% interest
rate compounded annually. How much should you pay at present year
2019 for this investment?
The associated time diagram is
Learn about It!
Deferred annuity
1 an annuity in which the first payment interval is delayed or deferred for a period
of time
In a deferred annuity, the time interval to the beginning of the first payment
interval is called period of deferral.
Example:
‘A loan with an interest rate of 7% and a quarterly payment of
₱5 000 for 3 years starting at the end of 1 year’ is an example of a
deferred annuity since the phrase starting at the end of 1 year
indicates that the payment started on a later date.
Learn about It!
Deferred annuity
1 an annuity in which the first payment interval is delayed or deferred for a period
of time
In a deferred annuity, the time interval to the beginning of the first payment
interval is called period of deferral
Example:
At the end of 1 year will be at time 4 if one quarter is considered as one
period. Hence, the period of deferral is from time 0 to time 3 which is
equivalent to 3 quarters.
Learn about It!
Present value of a deferred annuity (𝑃𝑉def )
2 the sum of the present value of the payments given by the following formula:
−𝑑
1+𝑖 − (1 + 𝑖)−(𝑝+𝑑)
𝑃𝑉def = 𝑅
𝑖
Learn about It!
Example: To find the present value of annuity using the
same problem in the previous example,
1. Determine the given information.
• 𝑑=3
• 𝑅 = 5000
• 𝑟 = 0.07
• 𝑚=4
• 𝑝 = 4 ∙ 3 = 12
0.07
• 𝑖= = 0.0175
4
Learn about It!
2. Substitute the given values in the formula.
−𝑑
1+𝑖 − 1 + 𝑖 − 𝑝+𝑑
𝑃𝑉def =𝑅
𝑖
1 + 0.0175 −3 − 1 + 0.0175 − 12+3
𝑃𝑉def = 5 000
0.0175
1.0175 −3 − 1.0175 −15
𝑃𝑉def = 5 000
0.0175
0.9492852794 − 0.7708745919
𝑃𝑉def = 5 000
0.0175
Learn about It!
2. Substitute the given values in the formula.
0.1784106875
𝑃𝑉def = 5 000
0.0175
𝑃𝑉def = 5 000 10.19489643
𝑃𝑉𝑑𝑒𝑓 = 50 974.48
Thus, the present value of the deferred annuity is
₱50 974.48.
Try It!
Example 1: Find the present value of a deferred annuity of
₱600 every three months for 6 years that is deferred for 4
years, if money is worth 8% compounded quarterly.
Try It!
Example 1: Find the present value of a deferred annuity of ₱600 every three months for 6
years that is deferred for 4 years, if money is worth 8% compounded quarterly.
Solution:
1. Determine the number of deferred payments 𝒅.
To do so, multiply the number of deferred years by 𝒎.
That is,
𝒅 = 𝟒 ∙ 𝟒 = 𝟏𝟔
Try It!
Example 1: Find the present value of a deferred annuity of ₱600 every three months for 6
years that is deferred for 4 years, if money is worth 8% compounded quarterly.
Solution:
2. Determine the other given information.
• 𝑅 = 600
• 𝑟 = 0.08
• 𝑚=4
• 𝑝 = 4 ∙ 6 = 24
0.08
• 𝑖= = 0.02
4
Try It!
Example 1: Find the present value of a deferred annuity of ₱600 every three months for 6
years that is deferred for 4 years, if money is worth 8% compounded quarterly.
Solution:
3. Substitute the given values in the formula.
1 + 𝑖 −𝑑 − 1 + 𝑖 − 𝑝+𝑑
𝑃𝑉def = 𝑅
𝑖
1 + 0.02 −16 − 1 + 0.02 − 24+16
𝑃𝑉def = 600
0.02
1.02 −16 − 1.02 −40
𝑃𝑉def = 600
0.02
Try It!
Example 1: Find the present value of a deferred annuity of ₱600 every three months for 6
years that is deferred for 4 years, if money is worth 8% compounded quarterly.
Solution:
3. Substitute the given values in the formula.
0.7284458137 − 0.4528904152
𝑃𝑉def = 600
0.02
0.2755553985
𝑃𝑉def = 600
0.02
𝑃𝑉def = 600 13.77776993
𝑃𝑉𝑑𝑒𝑓 = 8 266.66
Try It!
Example 1: Find the present value of a deferred annuity of ₱600 every three months for 6
years that is deferred for 4 years, if money is worth 8% compounded quarterly.
Solution:
Thus, the present value of the deferred annuity is
₱8 266.66.
Try It!
Example 2: Find the present value of a deferred annuity of
₱6 300 every six months for 10 years, if the first payment is
made in 5 years, and money is worth 10% compounded
semiannually.
Try It!
Example 2: Find the present value of a deferred annuity of ₱6 300 every six months for 10
years, if the first payment is made in 5 years, and money is worth 10% compounded
semiannually.
Solution:
1. Determine the number of deferred payments 𝒅.
To do so, multiply the number of deferred years by 𝒎
and subtract 1 since the first payment immediately starts
at the end of the period of deferral.
𝒅= 𝟓∙𝟐 −𝟏=𝟗
Try It!
Example 2: Find the present value of a deferred annuity of ₱6 300 every six months for 10
years, if the first payment is made in 5 years, and money is worth 10% compounded
semiannually.
Solution:
2. Determine the other given information.
• 𝑅 = 6300
• 𝑟 = 0.1
• 𝑚=2
• 𝑝 = 2 ∙ 10 = 20
0.1
• 𝑖= = 0.05
2
Try It!
Example 2: Find the present value of a deferred annuity of ₱6 300 every six months for 10
years, if the first payment is made in 5 years, and money is worth 10% compounded
semiannually.
Solution:
3. Substitute the given values in the formula.
1 + 𝑖 −𝑑 − 1 + 𝑖 − 𝑝+𝑑
𝑃𝑉def = 𝑅
𝑖
1 + 0.05 −9 − 1 + 0.05 − 20+9
𝑃𝑉def = 6300
0.05
1.05 −9 − 1.05 −29
𝑃𝑉def = 6 300
0.05
Try It!
Example 2: Find the present value of a deferred annuity of ₱6 300 every six months for 10
years, if the first payment is made in 5 years, and money is worth 10% compounded
semiannually.
Solution:
3. Substitute the given values in the formula.
0.6446089162 − 0.2429463211
𝑃𝑉def = 6 300
0.05
0.4016625951
𝑃𝑉def = 6 300
0.05
𝑃𝑉def = 6 300 8.033251902
𝑃𝑉𝑑𝑒𝑓 = 50 609.49
Try It!
Example 2: Find the present value of a deferred annuity of ₱6 300 every six months for 10
years, if the first payment is made in 5 years, and money is worth 10% compounded
semiannually.
Solution:
Thus, the present value of the deferred annuity is
₱50 609.49.
Let’s Practice!
Individual Practice:
1. If money is worth 12% compounded quarterly, find the present
value of a deferred annuity of ₱20 000 every three months for
4 years that is deferred for 2 years.
2. Find the present value of a deferred annuity of ₱42 500 every
six months for 8 years, if the first payment is made after 3
years, and money is worth 10% compounded semiannually.
Key Points
Deferred annuity
1 an annuity in which the first payment interval is delayed or deferred for a period
of time
In a deferred annuity, the time interval to the beginning of the first payment
interval is called period of deferral.
Key Points
Present value of a deferred annuity (𝑃𝑉def )
2 the sum of the present value of the payments given by the following formula:
−𝑑
1+𝑖 − (1 + 𝑖)−(𝑝+𝑑)
𝑃𝑉def = 𝑅
𝑖
Synthesis
● How are you going to find the present value and period of
deferral of a deferred annuity?
● What difficulties did you encounter while solving problems
involving deferred annuity? How did you overcome them?
● How do annuities compare with other types of
investments such as stocks and bonds?
