Tutorial Session 1
Time Value of Money (TVM Part I)
Fall 2024
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The problems solved in the tutorials would be
related to the most important concepts
introduced by your Professors.
 Financial Calculator is your best friend in this
course. Start learning how to use it today!
 We will use “BA II Plus Texas Instruments.”

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Agenda

Problem solving related to:
1)
2)
3)
4)
5)
Simple Interest Rate
Compound Interest Rate
Simple vs. Compound Interest Rates
Compounding vs. Discounting
Annuities
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Time Value of Money
Example:
 What is the difference between option A and
B below?
Having $100,000 today
B. Having $100,000 after 10 years
A.
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Time Value of Money
Example:
What is the difference between options A and B below:
A. Having $100,000 today
B. Having $150,000 after 10 years

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Simple Interest
Example :
You invest $500 today for a 1-year term and
receive 8% simple interest annually on your
investment. How much will you have after 1
year?
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Simple Interest (Cont’d)
Example:
You invest $500 today for a 5-year term and
receive 8% simple interest annually on your
investment. How much will you have after five
years?
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Compound Interest (Cont’d)
Example:
We invest $500 today and receive 8%
annual compound interest. What would be
the FV at years 1,2,3,4,5?
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Compound Interest (Cont’d)
We notice from the above, that the basic compounding
equation can be reduced to:
CVIF:
Compound
 FVn = PV0 x ( 1 + k)n

Now let us continue with our example
Value
Interest
Factor
FV3 =
 FV4 =
 FV5 =

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Compound Interest (Cont’d)
What is the additional amount of interest you
receive due to compounding in year 5?
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Compound Interest (Cont’d)
Example:
 Suppose you invest $100 in a savings account
with a compounding annual interest rate of 4%
for five years. How much will you have at the
end of the
First year?
B. Second year?
C. Fifth year?
A.
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Compounding vs. Discounting
Example:
Given that you just won a special kind of lottery
that will either pay you $650,000 today or $1
million dollars in five years. Given an interest
rate of 10%, which option would you choose?
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Compounding vs. Discounting
Example
On January 1, 2011, Nick signs a contract to
play for a baseball team. He will receive:
$500,000 for 2011
 $600,000 for 2012
 $700,000 for 2013
 $800,000 for 2014

All payments are made at the end of the year.
If you assume a 10% interest rate per year,
what is the present value of Nick’s contract?
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Other variables we can look for:
Example (Number of years):
You have $1,000 now. You need $1,500 to
go on a trip. The available IR is 4% which is
compounding annually. For how long do you
need to invest your money to be able to go
on this trip?
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Other variables we can look for:
Example (Interest rate):
You have $1,000 now. You need $1,500 to
go on a trip in three years. At what rate you
need to invest your money to be able to go
on this trip?
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Problem
Suppose that you want to purchase a house in
two years. You expect that you will need to
have $20,000 at that time to use as a down
payment. If a savings account pays 5% per
year, how much will you need to invest in the
account today in order to meet your goal?
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Problem
Which investment is worth more today at an annual
4% interest rate: $1,000 to be paid in 8 years or
$800 to be paid in 4 years?
Which investment is worth more if the interest rate is
6%?
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Problem
You have $10,000 now. You need to purchase a Car
that costs $14,000. The current available IR is 5%
annually. How many years do you need to be able to
purchase your car?
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Winter 2018, Q2
Su Mei deposits $10,000 today and is promised
a return of $18,700 in 5 years. What is the
implied annual rate of return?
A) 13.34%
B) 11.50%
C) 14.63%
D) 13.81%
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Annuities
•
Definition:
•
•
An annuity is a series of cash flows with the same amount and occurring over
the same interval.
The interval could be a year (as in the example below), a month, a week ...
$500K
$500K
$500K
$500K
period
•
0
1
2
3
4
Jan 1,
2009
Dec 31,
2009
Dec 31,
2010
Dec 31,
2011
Dec 31,
2012
T
Our objective is to bring all these cash flows to
the present or the future
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Annuities
•
Equations related to Normal Annuity:
1+𝑘 𝑛 − 1
𝐹𝑉𝑛 = 𝑃𝑀𝑇 [
]
𝑘
𝑃𝑉0 = 𝑃𝑀𝑇
1
1−
1+𝑘 𝑛
𝑘
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PVAF:
Present Value
Annuity
Formula
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Example of Annuity
 We are now Jan 1, 2009. Suppose that you plan
to invest $500 at the end of each year starting
from 2009 and for the coming 4 years and you
expect to earn 10% per year.
A.
B.
How much will you have after 4 years?
How much you need to deposit today to get the
same results? (What’s the present value of your
investment)
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GOLDEN RULE !!!!!!!
 Whenever you have PMTs , N
and I/Y Follow PMT.
 So if your PMT is Monthly, N
should be in Months and I/Y
should be Monthly.
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Very Important
Whenever you compute PV, it is one
Period before the 1st PMT.
So if you are computing the PV of an annuity
starting at year 4. the PV you get is at t=3
 If you’re dealing with months, the PV you get
is one month before the first pmt. So if you
receive the 1st pmt in month 11 the PV u get is
at t= 10 months.

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Annuity Due
 Definition:
•
Annuity due is an annuity whose cash flows occurs at
the beginning of the period rather than at the end.
 Equations related to Annuity Due:
𝐹𝑉𝑛 = 𝑃𝑀𝑇
𝑃𝑉0 = 𝑃𝑀𝑇
1+𝑘 𝑛 − 1
(1 + 𝑘)
𝑘
1
1−
1+𝑘 𝑛
(1 + 𝑘)
𝑘
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Annuity Due (example)
 Example:

We are now Jan 1, 2009. Suppose that you plan to invest
$500 at the beginning of each year starting from 2009
and for the coming 4 years and you expect to earn 10%
per year.
A.
How much will you have after 4 years?
How much you need to deposit today to get the same
results?
B.
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Fall 2014, Q7
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Note: Annuity is covered in this session. We will
cover “Perpetuity” and “Growing Annuities &
Perpetuities” next session.
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Thank you
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