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The Time Value of Money
Which would you prefer -- $10,000 today
or $10,000 in 5 years?
Obviously, $10,000 today.
Time value of money -- Money to be paid out or received in the
future is not equivalent to money paid out or received today
You already recognize that there is TIME VALUE TO
MONEY!!
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
Intuition Behind Present Value
• There are three reasons why a dollar tomorrow is
worth less than a dollar today
– Individuals prefer present consumption to future consumption. To induce people
to give up present consumption you have to offer them more in the future.
– When there is monetary inflation, the value of currency decreases over time. The
greater the inflation, the greater the difference in value between a dollar today and
a dollar tomorrow.
– If there is any uncertainty (risk) associated with the cash flow in the future, the
less that cash flow will be valued.
• Other things remaining equal, the value of cash
flows in future time periods will decrease as
– the preference for current consumption increases.
– expected inflation increases.
– the uncertainty in the cash flow increases.
The 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 (often a year, but can be a month,
week, day, hour, etc.)
• 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)
Abbreviations
• PV - Present value
• FV - Future value
• Pmt - Per period payment amount
• N - Either the total number of cash flows or
the number of a specific period
• i - The interest rate per period
Importance
• Shareholders’ wealth maximization requires to address timing
of cash flow associated with any investment alternatives. So, it
is the basic foundation for the goal.
• To compare the initial cash out flow (investment) and the
future cash flow generated from it in long run. Such cash flow
occurs at several point of time in future so, to make it
comparable it must be discounted to present value.
• This concept also facilitates investors of securities to evaluate
the price of securities before investing on them.
• Also used to evaluate the cost benefit of credit.
• Helps to compare cost of different sources of financing and
help in financing decisions.
• To calculate effective rate on different cash flow compounded
for different period.
Cash flow time line
Graphical presentation of cash flow at different
point of time.
Each tick represents one time period
0 period means today
i=10%
PV 0 1 2 3 4 5 FV 6
-100 20 25 30 40 25 30
Today
Calculating the Future Value
• 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?
-100 ?
0 1 2 3 4 5
FV1 1001 0.10 110
Calculating the Future Value (cont.)
• Suppose that at the end of year 1 you decide to extend
the investment for a second year. How much will you have
accumulated at the end of year 2?
-110 ?
0 1 2 3 4 5
FV2 1001 0.101 010 121
.
or
FV2 1001 0.10 121
2
Generalizing the Future Value
• Recognizing the pattern that is developing, we can
generalize the future value calculations as follows:
• FV=PV(FVIF,i%,n)
FVN PV1 i
N
If you extended the investment for a third
year, you would have:
FV3 1001 010 13310
3
. .
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
– Year 1: $10
– Year 2: $11
– Year 3: $12.10
• 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
Calculating the Present Value
• So far, we have seen how to calculate the
future value of an investment
• But we can turn this around to find the
amount that needs to be invested to achieve
some desired future value:
PV=FV(PVIF,i%,n)
FVN
PV
1 i N
Present Value: An Example
• Suppose that your five-year old daughter 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 pay for four years of
college. If you can earn 8% per year on your investments,
how much do you need to invest today to achieve your
goal?
100,000
PV $36,769.79
108
.
13
• Find the future value of an initial Rs. 500 compounded for
1 year at 10%.
• Find the future value of an initial Rs. 500 compounded for
5 year at 10%.
• Find the present value of Rs. 500 due in 1 year at 10%.
• Find the present value of Rs. 500 due in 5 years at 10%.
• Repeat all above if compounding is for each six months.
• If you deposit Rs. 20000 at XYZ bank’s saving account
which provide 10%, how long will it take to be the deposit
double?
• If you borrow Rs. 700 today and promise to pay back Rs.
749 at the end of year what is the interest rate you are
paying?
Annuities
• An annuity is a series of nominally equal payments occurred at equal
interval of time for a given number of periods.
• There are two types of annuities:
a) ordinary annuity : Series of equal payment at the end of each period.
b) Annuity due: Series of equal payment at the beginning of each period.
• Annuities are very common:
– Rent
– Mortgage payments
– Car payment
– Pension income
• The timeline shows an example of a 5-year, $100 annuity
100 100 100 100 100
0 1 2 3 4 5
The Principle of Value Additivity
• How do we find the value (PV or FV) of an
annuity?
• First, you must understand the principle of
value additivity:
– The value of any stream of cash flows is equal to
the sum of the values of the components
• In other words, if we can move the cash flows
to the same time period we can simply add
them all together to get the total value
Present Value of an Ordinary Annuity
• We can use the principle of value additivity to find the
present value of an annuity, by simply summing the
present values of each of the components:
N
1 i
Pmt t Pmt 1 Pmt 2 Pmt N
PVA
t 1
t
1 i 1
1 i 2
1 i N
Present Value of an Ordinary Annuity (cont.)
• Using the example, and assuming a discount rate of 10%
per year, we find that the present value is:
100 100 100 100 100
PVA 379.08
110
.
1
110
.
2
110
.
3
110
.
4
110
.
5
62.09
68.30
75.13
82.64
90.91
379.08 100 100 100 100 100
0 1 2 3 4 5
Present Value of an Ordinary Annuity (cont.)
• Actually, there is no need to take the present
value of each cash flow separately
• We can use a closed-form of the PVA equation
instead:
• PVA=Pmt(PVIFA,i%,n)
1 1
N
1 i
N
1 i
Pmt t
PVA t
Pmt
t 1 i
Present Value of an Ordinary Annuity (cont.)
• We can use this equation to find the present
value of our example annuity as follows:
1 1
PVA Pmt
110 379.08
.
5
.
010
This equation works for all regular annuities,
regardless of the number of payments
Future Value of an Ordinary Annuity
• We can also use the principle of value additivity to find the
future value of an annuity, by simply summing the future
values of each of the components:
N
Pmt t 1 i Pmt 1 1 i Pmt 2 1 i
Nt N 1 N 2
FVA Pmt N
t 1
The Future Value of an Ordinary Annuity (cont.)
• Using the example, and assuming a discount rate of 10%
per year, we find that the future value is:
FVA 100110 100110 100110 100110 100 610.51
4 3 2 1
. . . .
}
146.41
133.10
121.00 = 610.51
110.00
at year 5
100 100 100 100 100
0 1 2 3 4 5
The Future Value of an Annuity (cont.)
• Just as we did for the PVA equation, we could
instead use a closed-form of the FVA equation:
• FVA=Pmt(FVIFA,i%,n)
N
1 i N 1
Pmt 1 i
Nt
FVA t Pmt
t 1
i
This equation works for all regular annuities,
regardless of the number of payments
The Future Value of an Annuity (cont.)
• We can use this equation to find the future
value of the example annuity:
1105 1
.
FVA 100 610.51
010
.
Annuities Due
• Thus far, the annuities that we have looked at begin their
payments at the end of period 1; these are referred to as
regular annuities
• A 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
5-period Annuity Due 100 100 100 100 100
5-period Regular Annuity 100 100 100 100 100
0 1 2 3 4 5
Present Value of an Annuity Due
• We can find the present value of an annuity due in the
same way as we did for a regular annuity, with one
exception
• Note from the timeline that, if we ignore the first cash
flow, the annuity due looks just like a four-period regular
annuity
• Therefore, we can value an annuity due with:
• PVAD=Pmt(PVIFA,i%,n-1)+Pmt or
• PVAD=Pmt(PVIFA,i%,n)(1+i)
1 1 N 1
Pmt
1 i Pmt
PVAD
i
Present Value of an Annuity Due (cont.)
• Therefore, the present value of our
example annuity due is:
1 1 51
100
110 100 416.98
.
PVAD
010.
Note that this is higher than the PV of the,
otherwise equivalent, regular annuity
Future Value of an Annuity Due
• To calculate the FV of an annuity due, we
can treat it as regular annuity, and then
take it one more period forward:
• FVAD=Pmt(FVIFA,i%,n)(1+i)
1 i N 1
FVAD Pmt 1 i
i
Pmt Pmt Pmt Pmt Pmt
0 1 2 3 4 5
Future Value of an Annuity Due (cont.)
• The future value of our example annuity is:
1105 1
.
FVAD 100 110 67156
. .
010
.
Note that this is higher than the future value
of the, otherwise equivalent, regular annuity
Deferred Annuities
• A deferred annuity is the same as any other
annuity, except that its payments do not
begin until some later period
• The timeline shows a five-period deferred
annuity
100 100 100 100 100
0 1 2 3 4 5 6 7
PV of a Deferred Annuity
• We can find the present value of a deferred annuity in the
same way as any other annuity, with an extra step
required
• Before we can do this however, there is an important rule
to understand:
When using the PVA equation, the resulting PV
is always one period before the first payment
occurs
PV of a Deferred Annuity (cont.)
• To find the PV of a deferred annuity, we
first find use the PVA equation, and then
discount that result back to period 0
• Here we are using a 10% discount rate
PV2 = 379.08
PV0 = 313.29
0 0 100 100 100 100 100
0 1 2 3 4 5 6 7
PV of a Deferred Annuity (cont.)
1 1
PV2 100
110 379.08
.
5
Step 1:
010
.
379.08
Step 2: PV0 313.29
110
.
2
FV of a Deferred Annuity
• The future value of a deferred annuity is
calculated in exactly the same way as any
other annuity
• There are no extra steps at all
Uneven Cash Flows
• Very often an investment offers a stream of
cash flows which are not either a lump sum or
an annuity
• We can find the present or future value of
such a stream by using the principle of value
additivity
Uneven Cash Flows: An Example (1)
• Assume that an investment offers the following cash
flows. If your required return is 7%, what is the maximum
price that you would pay for this investment?
100 200 300
0 1 2 3 4 5
100 200 300
PV 513.04
107
.
1
107
.
2
107
.
3
Uneven Cash Flows: An Example (2)
• Suppose that you were to deposit the following amounts
in an account paying 5% per year. What would the
balance of the account be at the end of the third year?
300 500 700
0 1 2 3 4 5
FV 300105 500105 700 1,555.75
2 1
. .
Class Discussion
• Here are the three contracts offered by different
companies for an Engineer who has to select one. The
cash flow (Rs. In million) from each contract is as follows:
Year CF Contract 1 CF Contract 2 CF Contract 3
1 3 2 7
2 3 3 1
3 3 4 1
4 3 5 1
• If discounting interest rate is 10%, which contract should
the engineer accept?
Class Discussion
• Assume that it is now Jan-1, 2010 and you need Rs. 1000
on Jan-1, 2014. Your bank account interest rate is 8%
annual. If you want to make equal payment on each Jan-1,
2011 through 2014 to accumulate the Rs. 1000, how large
must each deposit be? If your father offer either to make
the payments calculated above or to give you a lump sum
of Rs. 750 on Jan-1, 2011, which would you choose?
• If you won a lottery which will pay you Rs. 1.75 million per
year over next 20 years and the first installment is
received immediately, find the following:
– If interest rate is 8% what is the present value of the lottery?
– If interest rate is 8% what is the future value of the lottery?
– How would your answer change if the payments were received at
the end of each year?
Non-annual Compounding
• So far we have assumed that the time period is equal to a
year
• However, there is no reason that a time period can’t be any
other length of time
• We could assume that interest is earned semi-annually,
quarterly, monthly, daily, or any other length of time
• The only change that must be made is to make sure that the
rate of interest is adjusted to the period length
Non-annual Compounding (cont.)
• Suppose that you have $1,000 available for investment.
After investigating the local banks, you have compiled the
following table for comparison. In which bank should you
deposit your funds?
Bank Interest Rate Compounding
First National 10% Annual
Second National 10% Monthly
Third National 10% Daily
Non-annual Compounding (cont.)
• To solve this problem, you need to determine which bank
will pay you the most interest
• In other words, at which bank will you have the highest
future value?
• To find out, let’s change our basic FV equation slightly:
Nm
i
FV PV 1
m
In this version of the equation ‘m’ is the number of
compounding periods per year
Non-annual Compounding (cont.)
• We can find the FV for each bank as follows:
FV 1,000110 1100
1
First National Bank: . ,
12
010
.
Second National Bank: FV 1,000 1 1104.71
,
12
365
010
.
Third National Bank: FV 1,000 1 110516
, .
365
Obviously, you should choose the Third National Bank
Effective Annual Rate
• If a rate is quoted at 16%, compounded semiannually, then the actual
rate is 8% per six months. Is 8% per six months the same as 16% per
year?
• Normally there are two kinds of interest rate: simple interest rate and
effective annual rate.
• Simple interest rate also called annual percentage rate (APR) is the
quoted interest rate which is used to compute the interest paid per
period.
• Effective annual rate (EAR) is the annual interest rate actually being
earned, as opposed to the quoted rate, considering the compounding
of interest.
• EAR is calculated to compare between two simple interest rate when
the compounding period is different.
m
i
EAR 1 1
m
Class Discussion
• Suppose Bank of Kathmandu pays 7 percent interest,
compounded annually on saving deposit. Everest Bank Ltd
pays 6.5 percent interest compounded quarterly. Based on
the effective annual rate in which bank would you prefer
to deposit your money?
• You have an option to invest at a development bond
issued by Nepal Rastra Bank paying interest 8% but the
interest is paid semiannually. Similarly, a bank is offering
you a fixed deposit scheme with 7.5% interest rate
compounded monthly. If you have Rs. 100000 investable
fund where will you invest and what will be the total
amount with you after one year?
• A bank charges 1% per month on car loans. What is the
APR? What is the EAR?
Loan Amortization
• One of the most important applications of compound
interest involves loans that are paid off in installments
over time.
• Example: automobile loans, student loans, home
mortgage loan etc.
• If a loan is to be repaid in equal periodic amounts
(monthly, quarterly or annually) then it is said to be
amortized loan.
Loan Amortization (Cont.)
• Suppose a loan of Rs. 10000 is to be repaid in 4 equal
installments including principal and 10% interest per
annum.
• The time line for the loan will be:
Inst1 Inst 2 Inst 3 Inst 4
10000
0 1 2 3 4
Loan Amortization (Cont.)
• All four installment will be equal and they contain
principal repayment as well as interest amount.
• Installment amount is calculated by using the following
formula:
Amount of Loan 10000
Installment 3154 .67
(PVIFA, i%, n) 3.1699
• The installment of Rs. 3154 includes both principal and
interest.
• But the interest and principal amount differs each year.
• Loan amortization schedule provides the amount of
interest and principal repayment for each period.
Loan Amortization (Cont.)
Year (A) Loan (B) Installment Interest (D) Principal Loan
(C) =Bx10% Repayment Balance
(E)=C-D
1 10000 3154.67 1000 2154.67 7845.30
2 7845.30 3154.67 784.53 2370.14 5475.16
3 5475.16 3154.67 547.52 2607.15 2868.01
4 2868.01 3154.67 286.66 2868.01 0
Class Discussion
• ABC inc. just borrowed Rs. 25000. Loan is to be repaid in
equal installment at the end of each of the next 5 years
and the interest rate is 10%.
Set up an amortization schedule for the loan.
How large must each annual payment be if the loan is for Rs.
50000?
How large be the installment be if the loan is Rs. 50000 and equal
installment would be paid at the end of next 10 years?
Although the loan is for the same amount but the payment is
spread over twice as many periods. Why are these payments not
half as large as the payment of the loan in the earlier case?
Class Discussion
• You want to buy a Mazda. It costs $25,000. With a 10%
down payment, the bank will loan you the rest at 12% per
year (1% per month) for 60 months. What will your monthly
payment be?
• Consider Bill’s retirement plan
Assume he just turned 40, but, recognizing that he has a lot
of time to make up for, he decides to invest in some high-risk
ventures that may yield 20% annually. (Or he may lose his
money completely!) Anyway, assuming that Bill still wishes
to accumulate $1 million by age 65, and will begin making
equal annual deposits in one year and make the last one at
age 65, now how much must each deposit be?
Class Discussion
• To complete your last year in Business school and then go
through Law school, you need Rs. 10000 per year for next
4 years starting from next year i.e. you need to withdraw
first 10000 one year from now. Your rich uncle offers to
put you through the school and he will deposit in a bank
paying 7% interest a sum of money that will be sufficient
to provide the four payments of Rs. 10000 each. His
deposit will be made today.
How large must the deposit be?
How much will be in the account immediately after you make the first withdrawal?
After the last withdrawal?
