FM210 Lecture Notes
Lecture 1 – Time Value of Money I
Time Value of Money Concepts:
Discount rate: The rate used to calculate the present value of future cash flows; also
known as the interest rate or opportunity cost of capital.
Present value (PV): The value today of future cash flows.
Future value (FV): The value that an investment grows to after earning interest.
Discount factor: Present value of a $1 future payment. D1 = ( )
1
1+r
(one-year
discount factor)
Discount Rates:
A discount rate is the reward that investors demand for accepting delayed rather than
immediate gratification.
Aka interest rate/required rate of return/opportunity cost of capital.
If you lend someone money for a year, you demand interest as you cannot instantly
spend the money you have lent on consuming goods.
If you lend money to a less trustworthy person/company, you require a greater interest
rate as you’re less confident that you’ll get your money back.
It is called the opportunity cost of capital because it is the result foregone by investing
in a capital project rather than investing in freely-available securities.
The higher the risk of an investment the higher the return required by a risk-averse
investor.
Present Value – One Time Period:
Discounting
The present value (PV) is the value today of future cash flows.
With the relevant interest rate for the cash flow at t = 1 of 10%, $100 received at t = 1
is equivalent to $90.91 today.
, In other words, we can generate $100 at t = 1 by investing $90.91 today at an interest
rate of 10% i.e. $90.91 (1.10) = $100.
Future Value – One Time Period:
Accumulating
The future value (FV) is the amount an investment will grow to after earning interest.
An investment of $90.91 today, is worth more than $90.91 at t = 1 i.e. $90.91 (1.10) =
$100
In other words, we can generate $100 at t = 1 by investing $90.91 today at an interest
rate of 10% i.e. $90.91 (1.10) = $100.
Due to the time value of money, i.e. positive discount rates, before we can add cash
flows together to make decisions we need to accumulate/discount cash flows to the
same point in time.
Simple Interest vs. Compounds Interest
Simple interest
Simple interest only pays interest on the original principal (principal is the term used for the
original amount of money invested).
Example
Original amount invested = $100 at t = 0, annual simple interest rate = 10%, total amount up
to year 3 are:
Year 1: 100 +100(0.1) = $110
Year 2: 100 +100(0.1) +100(0.1) = $120
Year 3: 100 +100(0.1) +100(0.1) +100(0.1) = $130
Compound interest
Compound interest pays interest not only the original principal but also on accumulated
interest.
Example
Original amount invested = $100 at t = 0, annual simple interest rate = 10%, total amount up
to year 3 are:
,Year 1: 100(1.1) = $100
Year 2: 100 (1.1)2 = 121
Year 3: 100 (1.1)3 = 133.1
With compound interest since we also earn interest on interest the terminal value wealth at
the end of three years is greater than with simple interest ($133.1>130).
This may not seem like a significant difference, but can become large over longer periods of
time.
Over 20 years the terminal wealth using compound interest of 10% is more than double that
when using simple interest of 10% ($672.75>$300) due to earning interest on interest.
How can we state this as an annual rate?
1. Stated annual interest rate aka annual percentage rate (APR)
2. Effective annual rate (EAR)
What is the stated annual interest rate?
The simplest way to convert an effectively monthly rate to an annual figure is to multiply the
effective monthly rate by 12 (12 monthly periods a year).
- e.g. Effective compounded monthly rate = 0.5%, 12 (0.005) = 0.06 or stated annual
interest rate = 6%.
The stated annual interest rate indicates the amount of simple interest earned in a year and
does not take into consideration interest earned on interest through compounding. Therefore,
we do not discount with SAIRs. When accumulating or discounting cash flows to calculate
, present values or future values, we always use compound interest. Quoting the annual rate
with simple interest only is simply a convention.
If a stated annual interest rate is quoted then the number of compounding periods per year
also has to be stated.
What is the EAR?
The effective annual rate (EAR) indicates the actual amount of interest that will be earned at
the end of the year after taking into consideration compounding.
Converting a SAIR to an EAR
Define k as the number of compounding periods in a year:
( )
k
Stated annualinterest rate
1+ EAR= 1+
k
The greater the number of compounding periods in a year, k, the larger the EAR will be,
holding the SAIR constant as your interest on interest will be more.
The SAIR in the formula has to be the SAIR for k compounding periods per year.
Example
SAIR = 5%
Annual compounding = 1+ ( 0.05 1
1 )
−1 = 0.05
Quarterly compounding = 1+ ( 0.05 4
4 )
−1 = 0.0509
( ) −1 = 0.05116
12
0.05
Monthly compounding = 1+
12
( )
365
0.05
Daily compounding = 1+ −1 = 0.051268
365
Effective to Effective rate
Equivalent n time period effective discount rate = ( 1+r )n −1
r is the effective rate for the time period
If the effective monthly rate is 1% then the effective rate for 2 months is:
(1+0.01)2 -1 = 0.0201 = 2.01%
Lecture 1 – Time Value of Money I
Time Value of Money Concepts:
Discount rate: The rate used to calculate the present value of future cash flows; also
known as the interest rate or opportunity cost of capital.
Present value (PV): The value today of future cash flows.
Future value (FV): The value that an investment grows to after earning interest.
Discount factor: Present value of a $1 future payment. D1 = ( )
1
1+r
(one-year
discount factor)
Discount Rates:
A discount rate is the reward that investors demand for accepting delayed rather than
immediate gratification.
Aka interest rate/required rate of return/opportunity cost of capital.
If you lend someone money for a year, you demand interest as you cannot instantly
spend the money you have lent on consuming goods.
If you lend money to a less trustworthy person/company, you require a greater interest
rate as you’re less confident that you’ll get your money back.
It is called the opportunity cost of capital because it is the result foregone by investing
in a capital project rather than investing in freely-available securities.
The higher the risk of an investment the higher the return required by a risk-averse
investor.
Present Value – One Time Period:
Discounting
The present value (PV) is the value today of future cash flows.
With the relevant interest rate for the cash flow at t = 1 of 10%, $100 received at t = 1
is equivalent to $90.91 today.
, In other words, we can generate $100 at t = 1 by investing $90.91 today at an interest
rate of 10% i.e. $90.91 (1.10) = $100.
Future Value – One Time Period:
Accumulating
The future value (FV) is the amount an investment will grow to after earning interest.
An investment of $90.91 today, is worth more than $90.91 at t = 1 i.e. $90.91 (1.10) =
$100
In other words, we can generate $100 at t = 1 by investing $90.91 today at an interest
rate of 10% i.e. $90.91 (1.10) = $100.
Due to the time value of money, i.e. positive discount rates, before we can add cash
flows together to make decisions we need to accumulate/discount cash flows to the
same point in time.
Simple Interest vs. Compounds Interest
Simple interest
Simple interest only pays interest on the original principal (principal is the term used for the
original amount of money invested).
Example
Original amount invested = $100 at t = 0, annual simple interest rate = 10%, total amount up
to year 3 are:
Year 1: 100 +100(0.1) = $110
Year 2: 100 +100(0.1) +100(0.1) = $120
Year 3: 100 +100(0.1) +100(0.1) +100(0.1) = $130
Compound interest
Compound interest pays interest not only the original principal but also on accumulated
interest.
Example
Original amount invested = $100 at t = 0, annual simple interest rate = 10%, total amount up
to year 3 are:
,Year 1: 100(1.1) = $100
Year 2: 100 (1.1)2 = 121
Year 3: 100 (1.1)3 = 133.1
With compound interest since we also earn interest on interest the terminal value wealth at
the end of three years is greater than with simple interest ($133.1>130).
This may not seem like a significant difference, but can become large over longer periods of
time.
Over 20 years the terminal wealth using compound interest of 10% is more than double that
when using simple interest of 10% ($672.75>$300) due to earning interest on interest.
How can we state this as an annual rate?
1. Stated annual interest rate aka annual percentage rate (APR)
2. Effective annual rate (EAR)
What is the stated annual interest rate?
The simplest way to convert an effectively monthly rate to an annual figure is to multiply the
effective monthly rate by 12 (12 monthly periods a year).
- e.g. Effective compounded monthly rate = 0.5%, 12 (0.005) = 0.06 or stated annual
interest rate = 6%.
The stated annual interest rate indicates the amount of simple interest earned in a year and
does not take into consideration interest earned on interest through compounding. Therefore,
we do not discount with SAIRs. When accumulating or discounting cash flows to calculate
, present values or future values, we always use compound interest. Quoting the annual rate
with simple interest only is simply a convention.
If a stated annual interest rate is quoted then the number of compounding periods per year
also has to be stated.
What is the EAR?
The effective annual rate (EAR) indicates the actual amount of interest that will be earned at
the end of the year after taking into consideration compounding.
Converting a SAIR to an EAR
Define k as the number of compounding periods in a year:
( )
k
Stated annualinterest rate
1+ EAR= 1+
k
The greater the number of compounding periods in a year, k, the larger the EAR will be,
holding the SAIR constant as your interest on interest will be more.
The SAIR in the formula has to be the SAIR for k compounding periods per year.
Example
SAIR = 5%
Annual compounding = 1+ ( 0.05 1
1 )
−1 = 0.05
Quarterly compounding = 1+ ( 0.05 4
4 )
−1 = 0.0509
( ) −1 = 0.05116
12
0.05
Monthly compounding = 1+
12
( )
365
0.05
Daily compounding = 1+ −1 = 0.051268
365
Effective to Effective rate
Equivalent n time period effective discount rate = ( 1+r )n −1
r is the effective rate for the time period
If the effective monthly rate is 1% then the effective rate for 2 months is:
(1+0.01)2 -1 = 0.0201 = 2.01%
