FM I Chapter 3: Time Value of Money .ppt

Discussion Points:
Introduction
Compound Interest and Future Value
Future Value of An Annuity
Discounting Techniques and Present Value
Annuities – A Level Stream

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Chapter ThreeChapter Three
Time value of moneyTime value of money..

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What is Time Value?What is Time Value?
We say that money has a time value because that
money can be invested with the expectation of
earning a positive rate of return
In other words, “a dollar received today is worth
more than a dollar to be received tomorrow”
That is because today’s dollar can be invested so that
we have more than one dollar tomorrow
This concept is referred to as the TIME VALUE OF TIME VALUE OF
MONEYMONEY

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The Terminology of Time ValueThe Terminology of Time Value
Present Value - An amount of money today, or the
current value of a future cash flow
Future Value - An amount of money at some future
time period
Period - A length of time
Interest Rate - The compensation paid to a lender (or
saver) for the use of funds expressed as a percentage
for a period (normally expressed as an annual rate)

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Simple versus compound interestSimple versus compound interest
There are two types of interest: simple vs. compound
a. Simple interest:
In order to borrow money from a bank, we have to pay interest on
the money, which is usually a percentage of the amount borrowed.
Simple interest is a type of interest that is paid only on the amount
borrowed.
If we deposit P dollars at an annual interest rate of r%, for a time
period t, then the future value or maturity value of the principal P is
given by
FV = P(1+r*t).
Note that the interest is given by
I = P*r*t
Eg; Assume that you deposit $1,000 in an account earning 7% simple
interest for 2 years. What is the accumulated interest at the end of the
2nd year?

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Calculating the Future ValueCalculating the Future Value

Future ValueFuture Value is the value at some future time of a present amount
of money, or a series of payments, evaluated at a given interest
rate.
e.g. 1. Suppose that you have an extra $100 today that you wish to
invest for one year. If you can earn 10% per year on your
investment, how much will you have in one year?

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Present ValuePresent Value

Present ValuePresent Value is the current value of a future amount of
money, or a series of payments, evaluated at a given interest
rate.
Suppose we wish to have a certain amount of money at a
future date, based on money deposited today.
The amount needed today is called the present value of the
future amount.
If future amount A is obtained by investing amount P today at
simple interest rate r% for t years, then present value P can be
found from the future value formula above by solving for
principal P:
P = FV/(1+r*t)

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Examples Examples
What is the Present Value Present Value (PVPV) of the previous
problem(eg2) ?
The Present Value is simply the $1,000 you originally deposited.
That is the value today!
Example 2: Tuition of $6,000 will be due when the spring semester
starts in 6 months.
What amount should be deposited today at 8% interest to have
enough to cover the tuition?

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Compound InterestCompound Interest
When interest is paid on not only the principal amount
invested, but also on any previous interest earned, this is
called compound interest.
Note from the example that the future value is increasing at
an increasing rate
In other words, the amount of interest earned each year is
increasing
The reason for the increase is that each year you are earning
interest on the interest that was earned in previous years in
addition to the interest on the original principle amount

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Compound Interest (cont.)Compound Interest (cont.)
Suppose you put $10,000 into a bank account earning 10% annual
compound interest.
After 1 year, the account will have:10,000 +10,000*(0.1)
= 10,000*(1+0.1) dollars
After 2 years, the account will have: 10,000(1+0.1) + 10,000(1+0.1)*(0.1)
= 10,000*(1+0.1)
2
dollars…
After n years, the account will have: 10,000(1+0.1)
n
dollars.
In general, if P dollars are deposited for n consecutive compounding
periods at an interest rate i per period, the compound amount A is given
by
A = P(1+i)
n
. where;
A= Future value, p= present value, i= interest rate per period stated
annually

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Examples Examples
1. If you invested $2,000 today in an account that pays 16$2,000 today in an account that pays 16% interest, with
interest compounded annually, how much will be in the account at the
end of two years if there are no withdrawals?
2. John wants to know how large his $5,000$5,000 deposit will become at an
annual compound interest rate of 18% at the end of 5 years5 years.

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Calculating the Present Value under compounding interest Calculating the Present Value under compounding interest
Based on the previous relation ship, present value can be
calculated as:
Examples
1.Assume that you need $10,000$10,000 in 2 years.2 years. Let’s examine
the process to determine how much you need to deposit
today at a discount rate of 15% compounded annually

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Cont’dCont’d
2. Assume that you need to have exactly $40,000$40,000 saved 10 years from now. years from now. How much
must you deposit today in an account that have 16% interest, compounded
semiannually, so that you reach your goal of $4,000?.
3.Suppose that your six-years old sister has just announced her desire to attend college.
After some research, you determine that you will need about $100,000 on her 18th
birthday to make advance payment for four years of college. If you can earn 12 %
interest per year on your investments, how much do you need to invest today to
achieve your goal?

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AnnuitiesAnnuities
An annuity is an amount of money that occurs (received or paid) in equal
amounts at equally spaced time intervals.
Annuities are very common:
•Rent
•Mortgage payments
•Pension income
Annuities require :-
The period payments or receipts (called rents) always be the same
amount.
The interval b/n such rents always be the same.
The interest compounded once each interval.
Interest rate per time period remains constant
on the basis of timing occurrence of the rents, annuities may be
classified
into three:
a.Ordinary annuity
b.Annuity due
c.Deferred annuity

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Ordinary annuityOrdinary annuity
An ordinary annuity assumes the first payment occurs at the
end
of the first year and the total amount on deposit is determined
at
the time the final rent is made.

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Future value of ordinary annuityFuture value of ordinary annuity
An annuity is a stream of payments made through time.
 A stream of equal payments at equal time intervals is a
fixed annuity. If those payments are made at the end of each
time period (month, quarter, year, etc...) it is an Ordinary
Annuity. If the payments are due at the beginning of each
period, it is an Annuity Due.
The future value of an annuity is the sum of all payments
made plus all interest earned.
The payment amount, interest rate, and number of payments
all contribute to the future value of the annuity.  

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Cont’dCont’d
Future Value (FV) - Ordinary Annuity
Where:
FV = Future Value of an Ordinary Annuity
PMT = Amount of each payment
i = Interest Rate Per Period
 n =Number of Periods
Example 1; Equal annual deposits of $10,000 are made into an
account that pays 10% interest compound annually. How much
will be the future value of ordinary annuity after 4 years ?
Example2: What amount will accumulate  if we deposit
$5,000 at the end of each year for the next 5 years? 
Assume an interest of 16% compounded annually.

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Present Value of an Ordinary AnnuityPresent Value of an Ordinary Annuity
The Present Value of an Ordinary Annuity (PVoa) is the value
of a stream of expected or promised future payments that have
been discounted to a single equivalent value today.   
PVoa can also be thought of as the amount you must invest
today at a specific interest rate so that when you withdraw an
equal amount each period, the original principal and all
accumulated interest will be completely exhausted at the end of
the annuity.
The Present Value of an Ordinary Annuity could be solved by
calculating the present value of each payment in the series
using the present value formula and then summing the results.
A more direct formula is:
PVoa = PMT [(1 - (1 / (1 + i)
n
)) / i]
Where: PVoa = Present Value of an Ordinary Annuity, PMT =
Amount of each payment, i = Discount Rate Per Period, n =
Number of Periods

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Cont’dCont’d
Example 1: What amount must you invest today at 10% compounded annually
so that you can withdraw $5,000 at the end of each year for the next 5 years?
PMT = 5,000 , i = .06, n = 5
Example 2:a computer dealer offers to lease a system to you for $50 per
month for two years.  At the end of two years, you have the option to buy the
system for $500.  You will pay at the end of each month.  He will sell the same
system to you for $1,200 cash.  If the going interest rate is 12%, which is the
better offer?
You can treat this as the sum of two separate calculations: 
The present value of an ordinary annuity of 24 payments at $50 per monthly
period Plus
the present value of $500 paid as a single amount in two years.
PMT = 50 per period, i = .12 /12 =  .01   Interest per period (12% annual rate / 12
payments per year), n = 24 number of periods
PVoa = 50 [ (1 - ( 1/(1.01)
24
)) / .01] = 50 [(1- ( 1 / 1.26973)) /.01] = 1,062.17 +
FV = 500 Future value (the lease buy out), i = .01 Interest per period, n = 24
PV = FV [ 1 / (1 + i)
n
]   = 500 ( 1 / 1.26973 ) =  393.78
The present value (cost) of the lease is $1,455.95 (1,062.17 + 393.78).

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Annuities DueAnnuities Due
Thus far, the annuities that we have looked at begin their
payments at the end of period ; these are referred to as regular
annuities
Annuity due is the same as a regular annuity, except that its
cash flows occur at the beginning of the period rather than at
the end
0 1 2 3 4 5
100100100100100
1001001001001005-period Annuity Due
5-period Regular Annuity

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Future Value of an Annuity Due (FVad)Future Value of an Annuity Due (FVad)
The Future Value of an Annuity Due is identical to an ordinary
annuity except that  each payment occurs at the beginning of a
period rather than at the end. Since each payment occurs one
period earlier, we can calculate the future value of an ordinary
annuity and then multiply the result by (1 + i).
FVad = FVoa(1+i)
Where:
FVad = Future Value of an Annuity Due , FVoa = Future Value of an
Ordinary Annuity i = Interest Rate Per Period
Example: What amount will accumulate  if we deposit $5,000 at
the beginning of each year for the next 5 years?  Assume an
interest of 6% compounded annually.
PV = 5,000, i = .06, n = 5
FVoa = FVoa(1+i) , FV= 5,000 [ (1.3382255776 - 1) /.06 ](1.06) =
5,000 (5.637092)(1.06) =28,185.46 (1.06) = 29,876.59

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Present Value of an Annuity DuePresent Value of an Annuity Due
The Present Value of an Annuity Due is identical to an ordinary
annuity except that  each payment occurs at the beginning of a period
rather than at the end. Since each payment occurs one period earlier,
we can calculate the present value of an ordinary annuity and then
multiply the result by (1 + i).
PVad = PVoa (1+i)
Where:
PV-ad = Present Value of an Annuity Due
PV-oa = Present Value of an Ordinary Annuity
i = Discount Rate Per Period
Example: What amount must you invest today a 6% interest rate
compounded annually so that you can withdraw $5,000 at the
beginning of each year for the next 5 years?
PMT = 5,000, i = .06, n = 5
PVoa = PVoa (1+i) , PVoa = 5,000 [(1 - (1/(1 + .06)
5
)) / .06] (1+.06) = 5,000
(4.212364) (1.06)= 21,061.82 (1.06) = 22,325.53

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Thank you for
your time !!!