Unformatted text preview: The Time Value of Money –
Present and Future Values
Ding Ding
University of Toronto
September 21, 2011 D.Ding (UofT) Time Value of Money Sep/2011 1/3 Problem
$1 today =? = $1 in the future Example
You just hit the $20 million jackpot in the Happy Million Lottery. The
terms are that you get a payment of $1 million every year for the next 20
years. You are ecstatic (who wouldn’t be?) and begin to plan how you will
spend the money. But, have you really won $20 million? D.Ding (UofT) Time Value of Money Sep/2011 2/5 Future Values I
Suppose you have C0 = $10, 000 (C0 is cash at date 0) to invest
at an interest rate of r = 5% for one year.
How much will you have in a year from now?
You have the original $10, 000.
You will be paid I interest:
I = C0 r =? Your investment would grow to ? The total amount due at the end of the investment is called the
Future Value (FV). L. Sun (UofT) The Time Value of Money May/11 3 / 41 Future Values II So your FV is
Principal
z}{
FV = C0 + C0 r = C0 (1 + r )
 {z }
Interest We use C0 to indicate the cash‡ow at date zero; in this case, it
is also called the principal investment amount. L. Sun (UofT) The Time Value of Money May/11 4 / 41 Future Values III Suppose now you leave the money in the bank for T years, and
suppose that the interest rate stays constant at r = 5%.
Moreover, suppose you can also reinvest the interest that you
were paid at the same rate.
Question: How much would you have after T years? L. Sun (UofT) The Time Value of Money May/11 5 / 41 Future Values IV
Let’ do it successively:
s
1 After 1 year you have C0 plus interest I1 so that
C1 = C0 + I1 = C0 + C0 2 r = C0 (1 + r ) After 2 years you have the principal, the interest on the
principal on year 1, the interest on the principal on year 2, and
interest on the interest from year 1:
C2 = C0 + C0 r + C0 = C0 1 + 2r + r r + (C0 r) r 2 = C0 (1 + r )2
Put di¤erently
C2 = C1 (1 + r ) = [C0 (1 + r )]
L. Sun (UofT) (1 + r ) = C0 (1 + r )2 . The Time Value of Money May/11 6 / 41 Future Values V 1 After 1 year
C1 = C0 (1 + r ) 2 After 2 years C2 = C0 (1 + r )2 3 Iterating the argument, after T years, the Future Value of the
investment is
CT = C0 (1 + r )T 4 The e¤ect of interest being paid on interest is called
compounding L. Sun (UofT) The Time Value of Money May/11 7 / 41 Present Value I
Supposed you are promised C1 = $10, 000 due in one year. The
current interest rate that the bank o¤ers is r = 5%.
How much are you willing to invest for this promise of
repayment? Call this amount C0 .
Answer: You should be at least as well o¤ as when investing X
into a bank account, i.e.
C0 (1 + r ) = C1 ! C0 = C1
=?
1+r In economic terms, C0 is the Present Value (PV) of amount C1 . L. Sun (UofT) The Time Value of Money May/11 8 / 41 Present Value II
In general, the formula for PV of a oneperiod case can be
written as:
C1
PV =
1+r
Now suppose we are looking at an investment which pays out
CT in T periods. What is its present value?
Answer:
CT
PV =
(1 + r )T
The process of calculating the present value of a future cash
‡ow is called discounting . It is the opposite of compounding. L. Sun (UofT) The Time Value of Money May/11 9 / 41 Present Value: Jackpot ReEvaluated Example
You just hit the $20 million jackpot in the Happy Million Lottery. The
terms are that you get a payment of $1 million every year for the next 20
years. You are ecstatic (who wouldn’t be?) and begin to plan how you will
spend the money. But, have you really won $20 million? D.Ding (UofT) Time Value of Money Sep/2011 3/5 Net Present Value I
The Net Present Value (NPV) of an investment is the present
value of the expected cash ‡ows, less the present value of the
cost of the investment. Example
Suppose an investment that promises to pay $10, 000 in one year is
o¤ered for sale for $9, 500. The interest rate is r = 5%. Should you
buy?
NPV = =
= C1
1+r
$10, 000
$9, 500 +
1.05
$9, 500 + $9, 523.81 C0 + $23.81 So yes, you should buy.
L. Sun (UofT) The Time Value of Money
May/11 10 / 41 Net Present Value I
The Net Present Value (NPV) of an investment is the present
value of the expected cash ‡ows, less the present value of the
cost of the investment. Example
Suppose an investment that promises to pay $10, 000 in one year is
o¤ered for sale for $9, 500. The interest rate is r = 5%. Should you
buy?
NPV = =
= C1
1+r
$10, 000
$9, 500 +
1.05
$9, 500 + $9, 523.81 C0 + $23.81 So yes, you should buy.
L. Sun (UofT) The Time Value of Money May/11 10 / 41 Net Present Value II
More generally:
Suppose a project pays you cash ‡ows C1 , ..., CT for the next T
years.
The interest rate is constant at r .
The initial cost/investment is C0 .
Then the NPV of this project is
NPV =
= L. Sun (UofT) C1
CT
+ ... +
1+r
(1 + r )T
T
Ct
C0 + ∑
t
t =1 ( 1 + r )
C0 + The Time Value of Money May/11 11 / 41 A few words about NPV NPV can be used to assess which of mutually exclusive projects
should be undertaken — simply pick the one with the highest
NPV.
This is also called the Net Present Value criterion.
In principle, accepting positive NPV projects bene…ts
shareholders, and in an ideal world, all positive NPV projects will
be undertaken. L. Sun (UofT) The Time Value of Money May/11 13 / 41 Internal Rate of Return I
The idea is simple: the IRR is the discount rate that sets a
project’ NPV to zero:
s
T 0= Ct
t
t =1 (1 + rIRR ) C0 + ∑ Intuitively, the higher the rIRR , the smaller the summation term.
A high IRR means that even for a very large discount rate, the
project still yields a nonnegative NPV.
Minimum Acceptance Criteria: Accept if the IRR exceeds the
required return.
Ranking Criteria: Select alternative with the highest IRR L. Sun (UofT) The Time Value of Money May/11 14 / 41 Internal Rate of Return II
Disadvantages:
Does not distinguish between investing and borrowing (i.e.
yields the same IRR if all signs are reversed).
There may be multiple IRR (this ‘
problem’ however, is often
,
overemphasized).
Problems with mutually exclusive investments that have
di¤erent timing of CFs.
The scale of investments/CFs does not matter. Advantages:
Easy to understand and communicate Note: Often used by consultancy …rms L. Sun (UofT) The Time Value of Money May/11 15 / 41 Compounding periods I Interest rates and time are usually measured in years, i.e. one
unit of time is a year, and the interest rate counts for a oneyear
investment.
Most credit payments however are monthly or biweekly.
Often compounding occurs more frequently than just once a
year.
How does that a¤ect your investment and credit payment? L. Sun (UofT) The Time Value of Money May/11 16 / 41 Compounding periods II
Example
Case 1: Mr. Smith has deposited $1000 in TD bank. The bank pays
a 10% annual interest rate. How much will Mr. Smith have at the
end of one year?
WSmith = $1000 (1 + 0.10) = $1100 Example
Case 2: Mr. Black has put $1000 in a savings account in ING bank.
The bank pays a 10% annual interest rate, but compounding
semiannually. How much will Mr. Black have at the end of one year?
WBlack = $1000
L. Sun (UofT) 0.10
1+
2 The Time Value of Money 2 = $1102.50
May/11 17 / 41 E¤ective Annual Interest Rate I Question: Which rate of interest r would yield the same wealth
but with only one payment?
Answer: Solve for e¤ective annual interest rate re¤ective :
1 + re¤ective =
re¤ective = rstated
m
rstated
1+
m 1+ m
m 1 for instance, if rstated = 10%, re¤ective = 10.25% (if m = 2) L. Sun (UofT) The Time Value of Money May/11 18 / 41 Useful for Economic Applications: Continuous
Compounding I
Suppose an investment project pays you C1,1 , ..., C1,12 in the
next 12 months; let the interest rate be r .
Suppose also that a comparable bank account would pay interest
to you monthly.
Then the PV of this cash‡ow stream is
PV = L. Sun (UofT) C1,1
C1,12
r 1 + ... +
r
(1 + 12 )
(1 + 12 )12 The Time Value of Money May/11 19 / 41 Useful for Economic Applications: Continuous
Compounding II
More generally, suppose interest in a year is paid for n
subperiods. Then the PV of the nth payment C is
PVn = C
r
1+ n n PV = lim C
r
1+ n n in the limit,
n !∞ =? Letting n go to in…nity is the same thing as saying that time is
continuous. Consequently, when using the exponential function,
we say that we are using continuous compounding/discounting.
L. Sun (UofT) The Time Value of Money May/11 20 / 41 Useful for Economic Applications: Continuous
Compounding III In general, with continuous discounting, a payment Ct that
occurs at time t (measured continuously, t = 1 indicating 1
year) is worth at time zero
PV = Ct L. Sun (UofT) e The Time Value of Money rt May/11 21 / 41 Returns and Rates of Return I You will hear about Returns and Rates of Return (RoR) over
and over.
However, people are often not very precise when using these
terms.
Make sure you understand which of these people are referring to
when mentioning ‘
returns’
. L. Sun (UofT) The Time Value of Money May/11 22 / 41 Returns and Rates of Return II
The return is merely
C1
C0
The raw return is the cash amount earned on an investment
return = raw return = C1 C0 The rate of return describes which percentage gain you
generated in one year on an investment. Formally
rate of return = C1 C0
C0 Moreover, rates of return are, by convention “annualized”
L. Sun (UofT) The Time Value of Money May/11 23 / 41 Returns and Rates of Return III
The rates of return earned on investments that are not for an
entire year are referred to as holding period returns ( hpr).
Holding period return= Ending PricingBeginning Price
Beginning Price
As a convention, you should always convert holding period
returns into annualized returns.
Thus if there are n holding periods per year and your holding
period return is hpr , then
rannual = (1 + hpr )n L. Sun (UofT) The Time Value of Money 1 May/11 24 / 41 Example
You bought a stock at $50/share. 3 months later, you sold the stock
at $55/share.
$50 = $5 Raw Return =$55
Return= $55 = 110%
$50 $
holding period return= $55 50 50 = 10%
$ Annual rate of return
rannual = (1 + hpr )n = (1 + 10%)
L. Sun (UofT) 1 12/3 The Time Value of Money 1 = 46.41%
May/11 25 / 41 Some useful simpli…cations I Perpetuity: A constant stream of cash ‡ows that lasts forever.
Growing perpetuity: A stream of cash ‡ows that grows at a
constant rate forever.
Annuity: A stream of constant cash ‡ows that lasts for a …xed
number of periods.
Growing annuity: A stream of cash ‡ows that grows at a
constant rate for a …xed number of periods. L. Sun (UofT) The Time Value of Money May/11 26 / 41 Perpetuity I A constant stream of cash ‡ows that lasts forever.
Doesn’ this sound silly? — One side of such a bargain has an
t
obligation to pay a …xed amount for all eternity! E¤ectively the
stream of payments is in…nite!
But this it is not so much out of this world as it seems: British
government debt was –for centuries– issued in the form of
perpetuities.
Question: Is the value of this stream of payments in…nite?
Answer: No! You have to take discounting into account! L. Sun (UofT) The Time Value of Money May/11 27 / 41 Perpetuity II
Example: What is the value of a British Consol that promises to
pay C = $ 5 each year, every year until the end of time?
Suppose the prevailing market interest rate is 10%.
Then the present value of this stream is
PV = C
C
+ ...
+
1+r
(1 + r )2 C
=
1+r 1
1
+ ...
1+
+
(1 + r ) (1 + r )2 ! =?
So what is the market price for the British Consol in the
example?
L. Sun (UofT) The Time Value of Money May/11 28 / 41 Side notes I
Summation formula of geometric progression
Sn = a + aq + aq2 + ... + aqn
qSn = aq + aq2 + aq3 ... + aqn +1 (1)
(2) (1)(2) (1 q ) Sn = a 1
Sn q n +1 a 1 q n +1
=
(1 q ) when n ! ∞, if 0 < q < 1
a 1 q n +1
a
=
n !∞
1q
(1 q ) lim sn = lim n !∞
L. Sun (UofT) The Time Value of Money May/11 29 / 41 Side notes II PV = C
C
C
+
+
+ ...
2
1+r
(1 + r )
(1 + r )3 =
= C
1+r = L. Sun (UofT) C
1+r 1
1
+
+ ...
(1 + r ) (1 + r )2
1 ! C
r 1+ 1 1
(1 +r ) The Time Value of Money May/11 30 / 41 Side notes III
Alternatively:
C
C
C
+
+ ...
+
2
1+r
(1 + r )
(1 + r )3
PV
C
C
=
+
+ ...
2
1+r
(1 + r )
(1 + r )3
C
PV
PV =
+
1+r
1+r
C
PV =
r
PV = L. Sun (UofT) The Time Value of Money May/11 31 / 41 Growing Perpetuity I
Suppose now that the stream of cash ‡ows grows over time and
lasts forever. Surely now the value must be in…nite, right?
Answer: Wrong again! Discounting still has bite.
Assume that the growth rate g is below the interest rate r
The …rst payment to come in a year is C , and it will grow
thereafter at rate g annually.
PV = C
C (1 + g ) C (1 + g )2
+
+ ...
+
1+r
(1 + r )2
(1 + r )3 Again
PV
L. Sun (UofT) C (1 + g ) C (1 + g )2
(1 + g )
=
+
+ ...
(1 + r )
(1 + r )2
(1 + r )3
The Time Value of Money May/11 32 / 41 Growing Perpetuity II Thus
PV = C
(1 + g )
+ PV
1+r
(1 + r ) solve for PV we get
PV =? L. Sun (UofT) The Time Value of Money May/11 33 / 41 Example
General Electric pays an annual dividend of roughly $0.9. Dividends
grow at about 9% per annum. Suppose the discount rate for this
kind of stock is 12%. What is the fundamental value of this promised
dividend stream?
Answer? In class. L. Sun (UofT) The Time Value of Money May/11 34 / 41 Annuity I Instead of a perpetual cash‡ow stream, suppose that there is a
…xed payment A for T years.
This is an annuity.
Some insurance payments and pensions (de…ned bene…ts) come
in the form of annuities.
Also, payments that you make to pay of a credit are annuities
(for the creditor). L. Sun (UofT) The Time Value of Money May/11 35 / 41 Annuity II
Let’ …rst develop the general formula
s
We want to …nd the PV for the Tperiod payment stream
PV = A
A
A
+
+ ... +
2
1+r
(1 + r )
(1 + r )T (3) A
A
A
PV
=
+
... +
2
3
1+r
(1 + r )
(1 + r )
(1 + r )T +1 (4) (3) (4)
PV 1 1
1+r = PV =
L. Sun (UofT) A
1+r
A
r The Time Value of Money 1 A (1 + r )T +1
!
1 (1 + r )T May/11 36 / 41 Annuity III And this gives us the annuity formula
A
PV =
r 1 1 (1 + r )T ! Note: This formula applies when the …rst payment occurs
precisely after one year has passed.
As for the example: L. Sun (UofT) The Time Value of Money May/11 37 / 41 Annuity VI
Example
You want to buy a car and you calculate that you can a¤ord a $400
monthly car payment. Question: how much of a car can you a¤ord if
the interest rate is 7% for 36month loans?
!
A
1
PV =
1
r
(1 + r )T
!
$400
1
= 0.07
1
.07 36
1 + 012
12
= $12, 955.
A hint: whenever you derive or develop a theoretical formula,
make sure that the units (dollars etc.) are correct.
L. Sun (UofT) The Time Value of Money May/11 38 / 41 Annuity V Example
Your friend is planning to buy a house. He will borrow $200, 000
mortgage at an annual interest of 5% and he plans to pay o¤ the
mortgage in 20 years. What is his monthly mortgage payment?
!
A
1
PV =
1
r
(1 + r )T
A = $1319.9 L. Sun (UofT) The Time Value of Money May/11 39 / 41 Growing Annuity An annuity with …nite number of growing cash ‡ows
Formula for present value of growing Annuity is
"
1
1+g
1
PV = C
rg
rg
1+r L. Sun (UofT) The Time Value of Money T # May/11 40 / 41 Summary Future Values (one period & multiperiod case)
Present Values (one period & multiperiod case)
NPV and IRR
Compounding Periods (E¤ective annual interest rate)
Simpli…cations
Perpetuity
Growing Perpetuity
Annuity
Growing Annuity L. Sun (UofT) The Time Value of Money May/11 41 / 41 ...
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 Fall '11
 DingDing
 Economics, Interest Rates, Time Value Of Money, Net Present Value, time value, L. Sun

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