Time Value of Money

Gitman, chapter 5
 Content
The Role of Time Value in Finance

FV, PV of a single amount

FV, PV of an annuity

FV, PV of Mixed Streams

Compounding Interest More Frequently Than Annually

 The Role of Time Value in Finance
Introduction
You're given a choice: receive $1,000 today or $1,000 a year from now. Which one
do you prefer?

Time value of money recognizes that a dollar today has more purchasing power
than a dollar tomorrow. Why?

Why we need to understand time value of money?

Can we compare different cash flows from different periods? How?

 The Role of Time Value in Finance
Time Lines

100$

Time Period

Used to depict an investment’s cash flows

 The Role of Time Value in Finance
Simple vs Compound Interest
Simple Interest:

P = 1300

Compound Interest:
 The Role of Time Value in Finance
Compounding vs Discounting
Compounding: used to find the future value of each cash
flow in the future

Discounting: used to find the find the present value of

each future cash flow at time zero
 The Role of Time Value in Finance
Calculation approaches
Different approaches to resolve time-value-of-money
related problems:
• Step-by-Step Approach

• Formula Approach

• Spreadsheets/Excel Approach
 The Role of Time Value in Finance
Future vs Present Values
Future Values - FV: The value at a given future date of an
amount placed on deposit today and earning interest at
a specified rate.

Present Values - PV: The amount of money that would

have to be invested today at a given interest rate over a
specified period to equal the future amount
 FV, PV of a single amount
Future value of a single amount
General equation : FVn = PV × (1 + r)n

FVn: future value at the end of period n

PV : initial principal or present value
r: annual rate of interest paid
n: number of periods (typically years)
(In case of simple interest: FVn = PV × (1 + n × r))
 FV, PV of a single amount
Example
Mr.A places $100 in a savings account paying 5% interest
compounded annually. He wants to know how much money will
be in the account at the end of 3 years?

FV3 = PV × (1 + 0.05)3 = $100(1.05)3=$115.76

=FV(rate, n, 0, PV)
 FV, PV of a single amount
Application
You have $1,500 to invest today at 7% interest
compounded annually. Find how much you will have
accumulated in the account at the end of (1) 3 years, (2)
6 years, and (3) 9 years
How much is 1000 VNĐ worth, with the growth rate of
5% per year, after 100 years? 10%? 15%?
FV, PV of a single amount
 FV, PV of a single amount
Present value of a single amount

FVn
General equation: PV =
(1 + r)n
 FV, PV of a single amount
Example
Mr. B wishes to find the present value of $115.76 that he will
receive 3 years from now. Its opportunity cost is 5%

FVn $115.76
PV = = = $100
(1 + r) n (1.05) 3

=PV(rate, n, 0, FV)
 FV, PV of a single amount
Application
You want to have 40 billion VNĐ after 40 years for your
retirement life. How much money do you need to
deposit in your bank account to achieve this goal?
Suppose that the bank’s deposit interest rate is 8%

What is the present value of 1 billion VNĐ in 100 years

from now if the interest rate is 5%, 10%, 20% per year?
FV, PV of a single amount
 FV, PV of a single amount
Special Applications of Time Value

• Find number of periods (N):

=NPER(I,PMT,PV,FV)

• Find interest rate (r):

=RATE (N,PMT,PV,FV)
 FV, PV of a single amount
Application
You have $100 to invest. If you can earn 12% interest, about how long does it
take for your $100 investment to grow to $200?

In 1995, O. G. McClain of Houston, Texas, mailed a $100 check to a descendant

of Texas independence hero Sam Houston to repay a $100 debt of McClain’s
great great-grandfather, who died in 1835, to Sam Houston. A bank estimated
the interest on the loan to be $420 million for the 160 years it was due. Find the
interest rate the bank was using, assuming interest is compounded annually.
Source: The New York Times.
 FV, PV of an annuity
Introduction
Definition: A stream of equal periodic cash flows over a specified
period.

These cash flows can be inflows of returns earned on investments or

outflows of funds invested to earn future returns

Two types of annuities:

 FV, PV of an annuity
Ordinary annuity: the cash flow occurs at the end of each period

Annuity due: the cash flow occurs at the beginning of each period
FV, PV of an annuity
 FV, PV of an annuity
Future value of an ordinary annuity
General equation:
1+𝑟 𝑛−1
𝐹𝑉𝑛 = 𝐶𝐹 ×
𝑟

FVn: future value at period n

CF : annual cash flow
r: annual rate of interest paid
n: number of periods
 FV, PV of an annuity
Future value of an ordinary annuity
If you deposit $100 annually, at the end of each of the next 3 years,
into a savings account paying 5% annual interest. How much money
will you have at the end of 3 years?

1+𝑟 𝑛 −1 (1 +0.05)3 − 1
𝐹𝑉𝑛 = 𝐶𝐹 × = 100 × = $315.25
𝑟 0.05

=FV(I,N,PMT,PV,Type)
 FV, PV of an annuity
Application
Suppose that you want to buy an apartment. You have intention to deposit $2500
at the end of each of the next 5 years, into a savings account paying 4% annual
interest. How much money will you have at the end of 5 years?

April Peel deposits $12,000 at the end of each year for 9 years in an account paying
8% interest compounded annually.
(a) Find the final amount she will have on deposit.
(b) April’s brother-in-law works in a bank that pays 6% compounded annually. If
she deposits money in this bank instead of the one above, how much will she
have in her account?
(c) How much would April lose over 9 years by using her brother-in-law’s bank?
 FV, PV of an annuity
Present value of an ordinary annuity
General equation:

𝐶𝐹 1
𝑃𝑉 = × 1− 𝑛
𝑟 1+𝑟
 FV, PV of an annuity
Present value of an ordinary annuity
What is the present value of the ordinary annuity of $100 in 3 years,
suppose that the interest rate is 5% per year?

𝐶𝐹 1 100 1
𝑃𝑉 = × 1− = × 1− = $272.32
𝑟 1+𝑟 𝑛 0.05 1+0.05 3

=PV(I,N,PMT,FV,Type)
 FV, PV of an annuity
Application
A production line will generate $700 cash flow at the end of each year
for 5 years. The rate of return is 8%. Calculate the present value of this
annuity.

Find the present value of a $1 million lottery jackpot distributed in

equal annual payments over 20 years, using an interest rate of 5%
 FV, PV of an annuity
Present value & Future value of an annuity due
Future value:
1+𝑟 𝑛 −1
𝐹𝑉𝑛 = 𝐶𝐹 × × 1+𝑟
𝑟
=FV(I,N,PMT,PV,Type)
Present value:
𝐶𝐹 1
𝑃𝑉 = × 1− × 1+𝑟
𝑟 1+𝑟 𝑛
=PV(I,N,PMT,FV,Type)
 FV, PV of an annuity
Application
Marian Kirk wishes to select the better of two 10-year annuities, C and D.
• Annuity C is an ordinary annuity of $2,500 per year for 10 years.
• Annuity D is an annuity due of $2,200 per year for 10 years.
a. Find the future value of both annuities at the end of year 10 assuming
that Marian can earn (1) 10% annual interest and (2) 20% annual
interest.
b. Use your findings in part a to indicate which annuity has the greater
future value at the end of year 10 for both the (1) 10% and (2) 20%
interest rates.
c. Find the present value of both annuities, assuming that Marian can earn
(1) 10% annual interest and (2) 20% annual interest.
d. Use your findings in part c to indicate which annuity has the greater
present value for both (1) 10% and (2) 20% interest rates.
e. Briefly compare, contrast, and explain any differences between your
findings using the 10% and 20% interest rates in parts b and d.
 FV, PV of an annuity
Special Applications of Time Value
• Find CF:
=PMT(I,N,PV,FV,Type)

• Find N:
=NPER(I,PMT,PV,FV,Type)

• Find r:
=RATE(N,PMT,PV,FV)
 FV, PV of an annuity
Application
Let's say we need to accumulate money to have $10,000 in 5 years with a
6%/year interest rate from the bank savings account today. How much do we
need to deposit if we start doing it right away? What if we send at the end of
each year?

Let's say you save $1,500 at the end of each year at the interest rate of
8%/year. How many years do you need to save to get $13000? What if you
start saving at the beginning of the year?

Let's say you save 150 million at the beginning of each year. What interest
rate will help you reach 3 billion in 5 years?
 FV, PV of an annuity
A perpetuity is an annuity with an infinite life. In other words, it is an
annuity that never stops providing its holder with a cash flow at the
end of each year
Present value of a perpetuity
𝐶𝐹
𝑃𝑉 =
𝑟
Present value of a growing perpetuity
𝐶𝐹
𝑃𝑉 =
𝑟 −𝑔
g: growth rate
 FV, PV of an annuity
Application

Calculate the present value of a perpetuity of $30/year and the interest

rate of 6%/year.

Calculate the present value of the above perpetuity, but with a growth
rate of 2% per year.
 FV, PV of Mixed Streams

A mixed stream is a stream of unequal periodic cash flows that reflect

no particular pattern

Attention:
• To simplify the problem, I use end-of-years cash flow in the examples
• The NPV function assumes that the first payment in the stream arrives 1 year
in the future and that all subsequent payments arrive at 1-year intervals
 FV, PV of Mixed Streams
Present value of Mixed Streams
Calculate the present value of the below mixed streams assuming that
the interest rate is 12%/year.

=NPV(r, value1, value 2,…)

 FV, PV of Mixed Streams
Future value of Mixed Streams
Calculate the future value of the below mixed streams assuming that the
interest rate is 12%/year.

Step 1: =NPV(r, value1, value 2,…)

Step 2: =FV(rate, n, 0, PV)
 FV, PV of Mixed Streams
Application
For each of the mixed streams of cash flows shown in the following
table, determine the future value at the end of the final year if deposits
are made into an account paying annual interest of 12%, assuming that
no withdrawals are made during the period and that the deposits are
made Cash flow stream

a. At the end of each year. Year A B C

1 $900 $30,000 $1,200

b. At the beginning of each year. 2 $1,000 $25,000 $1,200

3 $1,200 $20,000 $1,000

4 $10,000 $1,900

5 $5,000
 FV, PV of Mixed Streams
Application

In July 2012, Beijing had the heaviest rains in over six decades. More than 2 million
people were affected by the rainfall, roads were flooded, and the whole transport
system had to be suspended for days. The government now is offering a flood
recovery project that requires the tender to draw the flood waters out within a
week. CCTech is a large manufacturer of high-pressure industrial water pumps, and
the firm decided to bid for the flood recovery project. The government will pay $5
million this year and $2 million for the following four years.
a. Draw the timeline for the stream of cash flows.
b. If the discount rate is 8% per year, what is the present value of the project?
c. Suppose the project is expected to cost $10 million today. Should CCTech take
the project if it is offered? Why or why not?
Compounding Interest More Frequently Than
Annually

Interest is often compounded more frequently than once a year.

Savings institutions compound interest semiannually, quarterly,
monthly, weekly, daily, or even continuously
Compounding Interest More Frequently Than
Annually
Example
Compounding Interest More Frequently Than
Annually
The compounding frequency
The compounding frequency m (360 basis):

360
𝑚=
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑦𝑠 𝑜𝑓 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 𝑝𝑒𝑟𝑖𝑜𝑑

- Annual Compounding : m= 1
- Semiannual compounding: m= 2
- Quarterly compounding : m= 4
- Monthly compounding : m= 12
- Daily compounding : m= 360
Compounding Interest More Frequently Than
Annually
The compounding frequency
Ms. C wished to find the future value of $100 invested at 3% interest
compounded quarterly for 2 years.

𝑟 𝑛×𝑚
𝐹𝑉𝑛 = 𝑃𝑉 × 1 +
𝑚
Compounding Interest More Frequently Than
Annually
Application
Suppose $10,000 is invested at an annual rate of 5% for 10 years. Find
the future value if interest is compounded as follows.
(a) Annually
(b) Quarterly
(c) Monthly
(d) Daily (365 days)
Compounding Interest More Frequently Than
Annually
Application
David buys a car costing $14,000. He agrees to make payments at the
end of each monthly period for 4 years. He pays 7% interest,
compounded monthly.
(a) What is the amount of each payment?
(b) Find the total amount of interest David will pay
Compounding Interest More Frequently Than
Annually
Application
TV Town sells a big screen smart HDTV for $600 down and monthly
payments of $30 for the next 3 years. If the interest rate is 1.25% per
month on the unpaid balance, find
(a) the cost of the TV.
(b) the total amount of interest paid
Compounding Interest More Frequently Than
Annually
Application
You want to borrow $600,000 to buy an apartment, and you can only
afford $4,000 a month to repay the loan. Suppose the bank charges you
a fixed interest rate of 4% with monthly compounding. How long will it
take you to pay off the loan?
Compounding Interest More Frequently Than
Annually
Application
To save for retirement, Karla put $300 each month into an ordinary
annuity for 20 years. Interest was compounded semiannually. At the
end of the 20 years, the annuity was worth $147,126. What annual
interest rate did she receive?
Compounding Interest More Frequently Than
Annually
Effective Annual Rates Of Interest/Effective Annualized Rate (EAR)

𝐴𝑃𝑅 𝑚
𝐸𝐴𝑅 = (1 + ) −1
𝑚

EAR: is the annual rate of interest actually paid or earned

APR: The nominal, or stated, annual rate is the contractual annual rate
of interest charged by a lender or promised by a borrower
m: Compounding frequency
Compounding Interest More Frequently Than
Annually
Effective Annual Rates Of Interest/Effective Annualized Rate (EAR)

Compounding frequency’s effect on $100

Compounding Interest More Frequently Than
Annually
Application
For each of the cases in the following table:
a. Calculate the future value at the end of the specified deposit
period.
b. Determine the effective annual rate, EAR.
Case Amount of initial deposit Nominal annual rate Compounding frequency Deposit period

A $ 2,500 6% 2 5

B $ 50,000 12% 6 3

C $ 1,000 5% 1 10

D $ 20,000 16% 4 6
 Compounding Interest More Frequently Than
Annually
Application
The New York Times posed a scenario with two individuals, Sue and Joe, who each have $1200 a month to
spend on housing and investing. Each takes out a mortgage for $140,000. Sue gets a 30-year mortgage at a rate
of 6.625%. Joe gets a 15-year mortgage at a rate of 6.25%. Whatever money is left after the mortgage payment
is invested in a mutual fund with a return of 10% annually. Source: The New York Times.
(a) What annual interest rate, when compounded monthly, gives an effective annual rate of 10%?
(b) What is Sue’s monthly payment?
(c) If Sue invests the remainder of her $1200 each month, after the payment in part (b), in a mutual fund
with the interest rate in part (a), how much money will she have in the fund at the end of 30 years?
(d) What is Joe’s monthly payment?
(e) You found in part (d) that Joe has nothing left to invest until his mortgage is paid off. If he then invests the
entire $1200 monthly in a mutual fund with the interest rate in part (a), how much money will he have at
the end of 30 years (that is, after 15 years of paying the mortgage and 15 years of investing)?
(f) Who is ahead at the end of the 30 years and by how much?
(g) Discuss to what extent the difference found in part (f) is due to the different interest rates or to the
different amounts of time.