00:00:00
the time value of money affects all
00:00:02
aspects of business in every industry
00:00:04
the reason why we have the time value of
00:00:07
money is due to interest because of
00:00:09
interest the value of a dollar amount is
00:00:12
different depending on the point in time
00:00:13
that is paid or received for example a
00:00:16
dollar received today is more valuable
00:00:18
than a dollar received in five years or
00:00:20
any time in the future for that matter
00:00:22
this is because a dollar today can earn
00:00:24
interest over the next five years and
00:00:27
would therefore be worth more than the
00:00:29
dollar in the future that is being
00:00:31
compared to
00:00:31
and this time value of money lesson we
00:00:35
will go over simple interest compound
00:00:37
interest future value annuities present
00:00:40
value and treat your compounding
00:00:41
interest future value of annuities
00:00:43
present value of annuities and
00:00:45
perpetuities
00:00:48
so what is simple interest interest can
00:00:51
be thought of as rent to borrow money
00:00:53
you can either receive interest or rent
00:00:56
when you lend out money or you can pay
00:00:59
interest or rent when you borrow money
00:01:02
simple interest is when the interest
00:01:04
received or paid is based solely on the
00:01:07
amount of money that was initially
00:01:08
invested so if you invested $100 and it
00:01:12
earned simple interest at an annual rate
00:01:14
of 10% for five years then your
00:01:16
investment would earn $10 each year for
00:01:19
the next five years totaling in $50 an
00:01:21
interest the formula to solve the
00:01:27
interest rate is the initial investment
00:01:31
times 1 plus the interest rate times a
00:01:35
number of periods that the investment
00:01:37
will be held for therefore if we invest
00:01:42
$100 in an account for five years that
00:01:45
earns 10% simple interest per year then
00:01:48
we would have one hundred and fifty
00:01:50
dollars and five years the ending
00:01:53
balance in five years would consist of
00:01:55
our initial $100 investment and fifty
00:01:59
dollars an interest earned over the five
00:02:01
year period our formula would be 100
00:02:05
times 1 plus
00:02:08
the interest rate of 10% times five
00:02:11
years which is five periods what is
00:02:17
compound interest compound interest is
00:02:20
much different than simple interest
00:02:22
compound interest is the kind of
00:02:24
interest you would like to receive in an
00:02:26
investment but definitely not the kind
00:02:28
of interest that you want to pay why
00:02:31
because the interest rate is based on
00:02:33
the balance of the investment when it is
00:02:36
calculated not just the initial
00:02:38
investment what this means is that
00:02:41
interest is being earned on both the
00:02:43
investment and interest is being earned
00:02:46
on the previous periods interest this
00:02:50
may be easier to understand with an
00:02:51
example let's say you invested $100 for
00:02:56
five years at a compounding interest
00:02:58
rate of 10% at the end of this first
00:03:01
year your investment would be worth 110
00:03:03
dollars because it earned 10 percent
00:03:06
interest on $100 at the end of the
00:03:10
second year your investment be worth 121
00:03:13
dollars this means that it earned $11 in
00:03:17
interest the second year which is
00:03:19
different from the $10 in interest
00:03:21
earned in the first year why because the
00:03:24
second year your investment earned 10%
00:03:26
interest on the initial investment and
00:03:28
10% interest on the $10 interest earned
00:03:31
in the first year the easiest way to
00:03:35
understand the compounding interest
00:03:37
concept is to understand the interest is
00:03:39
being earned on the initial investment
00:03:42
and the interest earned in previous
00:03:44
periods the best way to calculate what
00:03:50
an investment earning compound interest
00:03:52
will be worth at some point in the
00:03:53
future is to use the future value
00:03:56
formula the future value formula is
00:03:59
future value equals present value times
00:04:01
one plus the interest rate to the nth
00:04:04
power
00:04:09
so for our investment we would calculate
00:04:12
the future value of $100 earning 10%
00:04:15
interest over a five-year period
00:04:18
notice that our investment of $100 will
00:04:21
be worth one hundred and sixty one
00:04:22
dollars and five cents and five years
00:04:24
this means that it earns sixty one
00:04:27
dollars and five cents an interest or an
00:04:30
interest than if it earned simple
00:04:32
interest now you can see why compound
00:04:34
interest is a kind of interest that you
00:04:36
want to receive but not the kind of
00:04:38
interest you want to pay so what is
00:04:42
future value future value is what a
00:04:45
dollar today will be worth in the future
00:04:47
this is because of interest that the
00:04:49
dollar can earn over time therefore
00:04:51
making it more valuable in the future if
00:04:54
someone offered to give you $100 today
00:04:56
or $100 in the future it would obviously
00:04:59
take the $100 today why because even if
00:05:03
you didn't need the $100 today and you
00:05:05
near that you would need it in the
00:05:06
future you could simply invest it and
00:05:09
earn interest over that year then a year
00:05:13
from now when you did need the money you
00:05:15
could cash out and have the $100 plus
00:05:17
any interest that it earned so if our
00:05:20
10% interest your $100 today would be
00:05:23
worth 110 dollars one year in the future
00:05:26
this means in theory that the future
00:05:29
value of money is always worth more
00:05:33
again the future value formula is future
00:05:37
value equals present value times one
00:05:39
plus the interest rate to the nth power
00:05:44
suppose you invested $1000 an investment
00:05:48
that was expected to earn 10% annual
00:05:50
interest for the next 10 years what
00:05:53
would the future value of your
00:05:54
investment be worth in 10 years let's
00:05:58
plug our figures into the future value
00:06:00
formula to find out we do $1000 times 1
00:06:06
plus point 1 to the 10th power giving us
00:06:10
a future value of 2,590 $3.74 this would
00:06:16
mean that $1000 today the future value
00:06:18
of $1000 is 2
00:06:20
thousand five hundred ninety three
00:06:21
dollars and seventy four cent what is an
00:06:25
annuity an annuity is a series of equal
00:06:28
payments that are either paid to you or
00:06:31
paid from you annuities can be cash
00:06:33
flows paid such as monthly rent payments
00:06:35
car payments or they can be money
00:06:37
received such as semi-annual coupon
00:06:39
payments from a bond just remember for a
00:06:42
series of cash flows to be considered an
00:06:44
annuity the cash flows need to be equal
00:06:46
an annuity due is when a payment is made
00:06:50
at the beginning of the payment period
00:06:52
rent for example where you're usually
00:06:55
required to pay at the first of every
00:06:57
month an ordinary annuity is a payment
00:07:01
that is paid or received at the end of
00:07:02
the period an example of an ordinary
00:07:05
annuity would be a coupon payment made
00:07:07
from bonds usually bonds will make
00:07:09
semi-annual coupon payments at the end
00:07:11
of every six months what is present
00:07:18
value present value is today's value of
00:07:21
money from some point in the future for
00:07:24
example $100 received a year from today
00:07:26
is worth less than $100 received today
00:07:28
this is because of interest earned over
00:07:31
time for example if we invested ninety
00:07:33
dollars and ninety one cents and a fund
00:07:35
that earned 10% interest that hour
00:07:37
ninety dollars and ninety one cents
00:07:38
would be worth about $100 one year in
00:07:41
the future this means that according to
00:07:43
an available 10% interest rate the
00:07:46
present value of $100 a year from today
00:07:49
is worth ninety dollars and 91 cents
00:07:51
today the present value formula for a
00:07:54
lump sum of money and the future is
00:07:56
shown here
00:07:57
the interest rate is also known as the
00:08:00
discount rate since you will be
00:08:02
discounting the future sum of money by
00:08:04
the interest rate
00:08:11
suppose you expected to receive $100 in
00:08:14
one year and there were currently
00:08:16
several investments offering 10 percent
00:08:18
interest
00:08:19
what would the present value of your
00:08:21
investment be just plug in the proper
00:08:24
figures to the present value formula and
00:08:26
you will see that the present value
00:08:27
comes out to 90 dollars and 91 cents how
00:08:35
to find the present value of a series of
00:08:37
cash flows let's assume you have an
00:08:41
ordinary annuity that pays you $100 at
00:08:44
the end of each year for the next three
00:08:46
years and it's coming from an investment
00:08:48
that is earning five percent interest
00:08:50
what is the present value of your
00:08:52
annuity the way you would solve the
00:08:55
present value of this annuity is by
00:08:57
solving the present value of each
00:08:59
payment or cash flow individually you
00:09:02
would do this by using the present value
00:09:04
formula for each cash flow that is to
00:09:07
occur in the future look at the example
00:09:10
here this means that if you invested 270
00:09:17
dollars and 32 cents in the fund that
00:09:19
earns 5% interest you would withdraw
00:09:22
$100 for the next three years this also
00:09:25
means that if you would receive $100 at
00:09:28
the end of the next three years
00:09:31
you could discount each cash flow back
00:09:34
you would sum them up and give you a
00:09:36
present value of 272 dollars and 32
00:09:38
cents for these three cash flows
00:09:45
enter a year compounding interest ensure
00:09:48
your compounding interest is when
00:09:50
interest is compounded more frequently
00:09:51
than one time per year this means that
00:09:54
there are multiple compounding periods
00:09:56
per year for example some interest rates
00:09:59
are compounded semi-annually which is
00:10:01
two times per year monthly which is 12
00:10:03
times per year etc to find out the
00:10:06
interest rate that is being earned or
00:10:08
paid for each compounding period you'll
00:10:11
need to divide the annual interest rate
00:10:12
by the number of compounding periods per
00:10:14
year if you have an annual interest rate
00:10:17
of 10% and interest is compounded
00:10:19
monthly then you would divide 0.1 zero
00:10:22
by 12 giving you a rate of point zero
00:10:25
zero eight three three three three
00:10:32
suppose you invested $10,000 at 6%
00:10:36
interest that compounded semi-annually
00:10:38
two times per year and held it for five
00:10:40
years what would the future value of
00:10:43
your investment be first we'll need to
00:10:45
solve for the interest rate for each
00:10:47
compounding period again we do this by
00:10:50
dividing the annual interest rate by the
00:10:53
number of compounding periods per year
00:10:54
since our annual interest rate is 6% and
00:10:57
interest is compounded semi-annually
00:10:59
then we would divide 6% by 2 giving us a
00:11:03
3% rate per compounding period now we
00:11:10
need to solve for the number of
00:11:12
compounding periods for the total life
00:11:13
of the investment to do this we multiply
00:11:16
the number of years that we would hold
00:11:17
our investment by the number of
00:11:19
compounding periods per year since we're
00:11:22
holding the investment for 5 years and
00:11:23
our investment is compounded
00:11:25
semi-annually we would multiply five
00:11:28
years by two compounding periods giving
00:11:31
us a total of 10 compounding periods
00:11:33
over the life of the investment now we
00:11:37
can plug our values into the future
00:11:39
value formula to find out what the value
00:11:41
of our investment will be in five years
00:11:43
again the future value formula is future
00:11:46
value equals present value times one
00:11:49
plus the interest rate to the nth power
00:11:51
so for our investment our future value
00:11:55
would be 10,000 times one point zero
00:11:58
three to the tenth power giving us a
00:12:01
total of 13 thousand four hundred and
00:12:04
thirty nine dollars and 16 cents
00:12:10
hopefully you have a good understanding
00:12:11
of compounding interest just remember
00:12:14
that when an investment is compounded
00:12:16
more than one time per year then you
00:12:18
will need to solve the rate per
00:12:20
compounding period by dividing the
00:12:22
annual interest rate by the number of
00:12:23
compounding periods per year and you
00:12:26
will need to find the number of
00:12:27
compounding periods for the life of the
00:12:29
investment by multiplying the number of
00:12:31
years by the number of compounding
00:12:33
periods per year since the concept of
00:12:38
present value with entry your
00:12:39
compounding is so important let's try
00:12:41
another more complicated example
00:12:46
let's assume that at the beginning of
00:12:48
the year you purchased an investment
00:12:51
that will pay you $1,000 per month at
00:12:53
the end of each month for the next six
00:12:55
months and is invested in a fund that
00:12:57
earns ten percent annual interest what
00:13:01
is the present value of your payment's
00:13:03
the present value of this series of cash
00:13:06
flows would be about five thousand eight
00:13:08
hundred twenty-eight 91 cents we are
00:13:12
using the present value formula to solve
00:13:14
the present value of each cash flow
00:13:16
individually we then sum up the values
00:13:18
of each cash flow to find the present
00:13:21
value of the series of cash flows it is
00:13:23
important to understand that we divide
00:13:25
our interest rate by twelve since it is
00:13:28
an annual interest rate and we're
00:13:30
receiving our payments monthly
00:13:39
imagine you paid $1,000 into a fund that
00:13:42
earned 5% interest at the end of every
00:13:45
year for the next five years what would
00:13:49
the future value of this ordinary
00:13:51
annuity be first let's look at our cash
00:13:55
flows on a timeline so we have a visual
00:13:57
understanding here zero represents today
00:14:01
one represents a year from today two
00:14:04
represents two years from today and so
00:14:06
on to solve the future value of an
00:14:15
ordinary annuity you would solve the
00:14:17
future value of each payment
00:14:19
individually the payment made in period
00:14:21
one will earn interest over four periods
00:14:23
until is withdrawn so the future value
00:14:26
would be 1,000 times 1.05 to the fourth
00:14:29
power giving us a future value of one
00:14:31
thousand two hundred and fifteen dollars
00:14:33
and fifty one cents the payment made in
00:14:35
year two will earn interest over three
00:14:38
periods so the future value will be one
00:14:41
thousand times one point zero five to
00:14:43
the third power giving us a future value
00:14:45
of one thousand one hundred fifty seven
00:14:47
dollars and 63 cents and so on this
00:14:51
means that period 5s payment will be
00:14:54
held for zero periods and will therefore
00:14:57
not earn any interest so the future
00:14:59
value will be one thousand times one
00:15:01
point zero five to the zero power giving
00:15:05
us a future value of one thousand
00:15:06
dollars if we sum up the future value of
00:15:09
all payments then we find that the
00:15:11
future value of our ordinary annuity is
00:15:13
five thousand five hundred twenty five
00:15:16
dot sixty-four cents notice that over
00:15:21
the next five years we receive five one
00:15:23
thousand dollar payments except this
00:15:25
time they're being paid at the beginning
00:15:27
of each period
00:15:33
now let's solve the future value of an
00:15:36
annuity due of $1,000 held for five
00:15:39
periods in an account that earns five
00:15:41
percent interest since the first payment
00:15:43
is made at the beginning of the first
00:15:45
period it will earn interest over five
00:15:48
periods until does withdrawn five years
00:15:50
from today so the future value will be
00:15:52
one thousand times 1.05 to the fifth
00:15:55
power giving us a future value of one
00:15:57
thousand two hundred and seventy six
00:15:59
dollars and twenty eight cents the
00:16:02
second payment will earn interest over
00:16:03
four periods so the future value will be
00:16:06
one thousand times 1.05 to the fourth
00:16:08
power giving us a future value of one
00:16:11
thousand two hundred and fifteen dollars
00:16:12
and 51 cents and so on for each payment
00:16:16
will solve the future value of each cash
00:16:19
flow individually and then sum them up
00:16:21
to find the future value of our new 'ti
00:16:23
due the future value is five thousand
00:16:26
eight hundred and one dollars and ninety
00:16:27
two cents notice that this annuity has
00:16:31
identical characteristics such as time
00:16:33
interest in the payment amount except
00:16:35
that it's an annuity due and our other
00:16:38
example is an ordinary annuity
00:16:39
so both annuities are identical but the
00:16:42
future values are different the future
00:16:44
values are different because the
00:16:45
payments are made at different times
00:16:47
meaning that they earn interest over a
00:16:49
different number of periods the future
00:16:52
value of our ordinary annuity is five
00:16:54
thousand five hundred twenty-five
00:16:55
dollars and 64 cents which is two
00:16:58
hundred and seventy 6.28 cents less than
00:17:01
the future value of our new 'ti do
00:17:06
solving the present value of an ordinary
00:17:08
annuity remember ordinary annuity means
00:17:11
that the payments are made or received
00:17:13
at the end of each payment period let's
00:17:17
assume we have an ordinary annuity that
00:17:18
pays $1,000 at the end of each year for
00:17:21
the next five years let's also assume
00:17:23
that the money is coming from a fund
00:17:25
that earns five percent interest what is
00:17:28
the present value of this annuity to
00:17:31
solve the present value of our annuity
00:17:33
we will simply solve the present value
00:17:35
of each payment and sum them up to find
00:17:37
the present value of our annuity since
00:17:40
our first payment is received one year
00:17:41
from today our present value formula
00:17:43
will be
00:17:45
1,000 divided by 1.05 to the first power
00:17:48
giving us a present value of 952 dollars
00:17:51
and 38 cents since our second year
00:17:54
payment has received two years from
00:17:55
today our present value formula will be
00:17:57
one thousand divided by 1.05 to the
00:18:01
second power giving us a present value
00:18:03
of nine hundred seven dollars and three
00:18:05
cents remember and our present value
00:18:10
formula we are raising the denominator
00:18:11
which is one plus the interest to the
00:18:15
number of periods from today that the
00:18:19
payment will be received after we sum
00:18:22
the present value of all five payments
00:18:23
we find that the present value of our
00:18:26
ordinary annuity is four thousand three
00:18:28
hundred twenty nine dollars and forty
00:18:30
eight cents now let's assume we have an
00:18:36
annuity due that makes five payments of
00:18:38
$1,000 over the next five years let's
00:18:42
also assume that it's in a fund that
00:18:44
earns five percent interest what is the
00:18:47
present value of our annuity do remember
00:18:50
since this is an annuity do our payments
00:18:52
will be received at the beginning of the
00:18:54
next five periods this means that our
00:18:58
first payment will be received at zero
00:19:00
which is today on our timeline since our
00:19:04
first payment was received today it
00:19:06
didn't have time to earn interest so the
00:19:09
present value of our first payment is
00:19:11
$1,000 our second-year payment is paid
00:19:15
at the beginning of the second year
00:19:17
which is one year from today meaning it
00:19:20
has one year to earn interest so the
00:19:22
present value of our second year payment
00:19:24
is 952 dollars and 38 cents we did this
00:19:27
by dividing $1000 by 1.05 to the first
00:19:32
power if we sum the present value of all
00:19:37
of our payments we find the present
00:19:38
value of an annuity due is four thousand
00:19:41
five hundred and forty-five dollars and
00:19:42
ninety-five cents which is four hundred
00:19:44
forty three dollars and seventy seven
00:19:46
cents more than our ordinary annuity our
00:19:49
annuity do is worth more than our
00:19:51
ordinary annuity because with an annuity
00:19:53
do the payments are received sooner
00:19:55
which means they are worth more
00:19:57
remember money today is worth more than
00:20:00
an equal amount of money receiving the
00:20:02
future perpetuities a perpetuity is a
00:20:10
fixed cash flow received over an
00:20:13
indefinite number of periods for example
00:20:15
if someone were receive $1,000 per month
00:20:18
until they died that would be a
00:20:20
perpetuity the formula to solve the
00:20:23
present value of a perpetuity is shown
00:20:25
here the present value would be cash
00:20:28
flow divided by the interest rate let's
00:20:34
assume we were to receive $1,000 per
00:20:36
year for the rest of our lives and could
00:20:39
earn interest in other investments at a
00:20:41
rate of 5%
00:20:42
what would the present value of this
00:20:44
perpetuity be to solve the present value
00:20:48
of this perpetuity we would simply
00:20:50
divide $1,000 by point zero five giving
00:20:55
us a present value of $20,000 now let's
00:21:00
make this a little more complicated what
00:21:03
if we were to see $1000 per month for
00:21:06
the rest of our lives and we could earn
00:21:08
5% interest in other investments
00:21:13
what would the pressive value of our
00:21:15
perpetuity be it would be 1,000 dollars
00:21:19
divided by 0.05 divided by 12 giving us
00:21:27
a present value of two hundred and
00:21:29
thirty eight thousand ninety five
00:21:31
dollars and twenty four cents notice
00:21:33
that we divided the interest rate by
00:21:35
twelve
00:21:36
since our perpetuity pays monthly in
00:21:41
this time value of money lesson we
00:21:45
covered all the basic time value money
00:21:46
concepts I hope I was able to help you
00:21:49
in some way thanks for watching
