Description
Present Value and Opportunity Cost of Capital
Contents
? Present Value
? Net Present Value
? NPV Rule
? ROR Rule
? Opportunity Cost of Capital
? Managers and the Interests of Shareholders
2
Time Value of Money
? Time Value of Money analysis involves:
? What is Rs.100 worth 10 years from today (Future
Value)?
? What is Rs.100 to be received in 10 years worth today
(Present Value)?
? Applications
? Loan amortization
? stated vs. effective interest charged
? rebate vs. low financing
? pricing of bonds
? pricing of stocks/firms
? What is the value of a particular division within a
firm?
3
Time Value of Money
? Time Value of Money analysis involves:
4
Time Value of Money
? Time Value of Money analysis involves:
? What is Rs.100 worth 10 years from today (Future
Value)?
? What is Rs.100 to be received in 10 years worth today
(Present Value)?
? Applications
5
Time Value of Money
? Time Value of Money analysis involves:
? What is Rs.100 worth 10 years from today (Future
Value)?
? What is Rs.100 to be received in 10 years worth today
(Present Value)?
? Applications
? Loan amortization
? stated vs. effective interest charged
? rebate vs. low financing
? pricing of bonds
? pricing of stocks/firms
? What is the value of a particular division within a
firm?
6
Present Value
7
Present Value
Value today of a
future cash
flow.
Discount Rate
Interest rate used to
compute present
values of future cash
flows.
Discount Factor
Present value of a
Rs.100 future
payment.
Present Value
8
1
factor discount = PV
PV = Value Present
C ×
Present Value
9
Discount Factor = DF = PV of Rs.100
Discount Factors can be used to compute the present value
of any cash flow.
DF
r
t
=
+
1
1 ( )
Net Present Value
10
r
C
+
+
1
C = NPV
investment required - PV = NPV
1
0
Valuing an Office Building
An office building costs Rs. 350 lacs now. The building
would be worth Rs. 400 lacs after a year. The equally
risky investment in the stock market offers a return of
7% per annum. What is the value of the Office
Building?
11
Valuing an Office Building
Step 1: Forecast cash flows
Cost of building = C
0
= 350
Sale price in Year 1 = C
1
= 400
Step 2: Estimate opportunity cost of capital
If equally risky investments in the capital market
offer a return of 7%, then
Cost of capital = r = 7%
12
Valuing an Office Building
Step 3: Discount future cash flows
Step 4: Go ahead if PV of payoff exceeds
investment
13
374
) 07 . 1 (
400
) 1 (
1
= = =
+ +r
C
PV
24 374 350 = + ÷ = NPV
Risk and Present Value
? Higher risk projects require a higher rate of return
? Higher required rates of return cause lower PVs
14
Risk and Present Value
?Suppose the investment becomes
riskier and the returns of a similar risk
profile investment in the stock market
is 12%, what is its PV?
15
Risk and Present Value
16
374
.07 1
400
PV
7% at 400 Rs. C of PV
1
=
+
=
=
357
.12 1
400
PV
12% at 400 Rs. C of PV
1
=
+
=
=
Rate of Return Rule
? Accept investments that offer rates of return in
excess of their opportunity cost of capital
? What is Opportunity cost of capital?
17
Opportunity Cost of Capital
? It is that rate of return offered by an equivalent
investment alternative in the capital market
? It is also known as discount rate, hurdle rate
? It is the rate at which we discount the future cash
flows of a given project.
18
Rate of Return Rule
? Accept investments that offer rates of return in excess of
their opportunity cost of capital
19
Example
In the project listed below, the foregone investment
opportunity is 12%. Should we do the project?
14.3% or .143
350,000
350,000 400,000
investment
profit
Return =
÷
= =
Net Present Value Rule
? Accept investments that have positive net
present value
20
Example
Suppose we can invest Rs. 50 today and receive
Rs. 60 in one year. Should we accept the project
given a 10% expected return?
55 . 4
1.10
60
+ -50 = NPV Rs =
Opportunity Cost of Capital
Example
You may invest Rs. 100,000 today. Depending on
the state of the economy, you may get one of three
possible cash payoffs:
21
140,000 110,000 80,000 Payoff
Boom Normal Slump Economy
000 , 110 .
3
000 , 140 000 , 110 000 , 80
C payoff Expected
1
Rs =
+ +
= =
Opportunity Cost of Capital
Example - continued
The stock is trading for Rs.95.65. Next year’s price,
given a normal economy, is forecast at Rs.110. What
is the expected pay-off?
22
Opportunity Cost of Capital
Example - continued
The stocks expected payoff leads to an expected
return.
23
15% or 15 .
65 . 95
65 . 95 110 profit expected
return Expected =
÷
= =
investment
Opportunity Cost of Capital
Example - continued
Discounting the expected payoff at the expected
return leads to the PV of the project
24
650 , 95 .
1.15
110,000
PV Rs = =
Investment vs. Consumption
? Some people prefer to consume now. Some
prefer to invest now and consume later.
? Calculating the PV of borrowing and lending
allows us to reconcile these opposing desires
which may exist within the firm’s shareholders.
25
Managers and Shareholder Interests
26
? Tools to Ensure Management Pays
Attention to the Value of the Firm
? Manger’s actions are subject to the
scrutiny of the board of directors.
? Shirkers are likely to find they are ousted
by more energetic managers.
? Financial incentives such as stock options
Question
? If C
0
is Initial investment, C
1
is the cash flow at the
end year 1, and r is the discounting rate, then if the
investment is risk-free, what is the appropriate
measure of r?
27
Question
? If present value of Rs. 15,000 paid after a year is Rs.
13,000, what is the discount factor?
28
Question
? A trader pays Rs. 1 lac for a truck-load of rice and is
certain of selling the same after a year for Rs. 1.32 lacs.
? What is the return on investment?
? If the return is lower than the rate of interest, then is the
NPV +ve or –ve?
29
Question
? A parcel of land costs Rs. 5 lacs. For an additional Rs. 8
lacs you can build a motel on the property. The value
of the land and the motel after is expected to be Rs. 15
lacs. Would it make business sense to invest in the
motel, if another project yielding 10% pa is available
for investment?
30
Question
? Shantanu, a leading analyst with Meghraj
TramellCrow, a leading real estate firm,
recommends an investment in a plot of land
worth Rs. 8.5 lacs. He is certain that the land
would be worth Rs. 9.1 lacs after a year. Given the
guaranteed returns available on FDs at 7.5%,
should the investors buy the plot of land? To get
the same return as that on FDs, at what rate
should the investor purchase the plot of land?
31
Question
? Gaurav has received Rs. 2 cr in inheritance. How
should he invest it? 4 alternatives:
1. Invest in 1 yr T-bill yielding 8%
2. Loan to his nephew, Nitin, who has aspired to start his
own business. Nitin has arranged a 1 yr loan of Rs. 90
lacs at 12%, but has approached Gaurav for a loan at 10%
3. Invest in the stock market, where the expected rate of
return is 18%.
4. Investment in local real estate, which Gaurav judges is
as risky as the stock market. The opportunity at hand
would cost Rs. 100 lacs and is forecasted to be worth Rs.
1.15 cr after 1 yr.
? Which of these has positive NPV? Which one
would you advice Gaurav to take?
32
Question
? Gaurav has received Rs. 2 cr in inheritance. How
should he invest it? In alternative 4
“Investment in local real estate, which Gaurav judges is as
risky as the stock market. The opportunity at hand
would cost Rs. 100 lacs and is forecasted to be worth Rs.
1.15 cr after 1 yr.” –
If Gaurav takes a bank loan of Rs. 60 lacs at 12% (Gaurav has
a long term relationship with the bank) and invest Rs. 1
cr in the real estate and the balance in the stock market.
? Is this a smart move? Explain.
33
Question
? Calculate the NPV and rate of return for the
following investments. The opportunity cost of
capital is 20%.
? Figures in Rs. lacs
34
Investment Initial CF CF in Year 1
1 10,000 18,000
2 5,000 9,000
3 5,000 5,700
4 2,000 4,000
Individual Assignment
? Madhura has retired and depends on her
investments for retirement income. Akhil is a young
executive who wants to save for the future. They are
both stockholders of Tata Motors, which is
investing Rs. 1200 crores to develop the new Nano.
This investment’s pay-off is many years in future.
Assume that the investment is +ve NPV for Akhil.
Should the NPV be +ve also for Madhura?
35
Individual Assignment
? Define Opportunity Cost of Capital. How in principle
would you find the opportunity cost of capital for a risk
free asset? For a risk asset?
? Respond to-
? My company’s cost of capital is the rate of the bank when
we borrow money.
? NPV is just theory. We must maximize profit. That’s what
shareholders want.
? Why is a reputation for honesty and fair business
practice important to financial value of the corporation?
36
Group Assignment
? Chapter 2 Brearley Myers 7
th
edition
? Challenge Questions: 1, 2, 4, 5
37
38
Topics Covered
? Valuing Long-Lived Assets
? PV Calculation Short Cuts
? Compound Interest
? Nominal and Real Rates of Interest (inflation)
? Present Values and Bonds
39
Present Values
40
Discount Factor = DF = PV of Re.1
Discount Factors can be used to compute
the present value of any cash flow.
DF
r
t
=
+
1
1 ( )
Present Values
41
Discount Factors can be used to compute
the present value of any cash flow.
DF
r
t
=
+
1
1 ( )
1
1
1
1 r
C
C DF PV
+
= × =
Present Values
42
? Replacing “1” with “t” allows the formula to
be used for cash flows that exist at any point
in time
t
t
t
r
C
C DF PV
) 1 ( +
= × =
Present Value
? In general for a single CF,
? where, i = discount rate
? PVIF
i, n
= PV of Interest Factor which depends on i, n.
= discount factor
43
Present Value
44
Notes:
(a) If i = 0, then PV of a CF, say CF = Rs. 1, is constant at Rs.1, irrespective of
how far in the future it is received.
(b) For a given "n", the higher the "i", the lower is PV.
(c) For a given "i", the larger the "n", the smaller the PV.
PV and "i" are inversely related ] They move in opposite direction.
PV and "n" are inversely related ]They move in the opposite direction
Value depends on
45
Value depends on
? Amount of Cash Flow
? Timing of Cash Flow
? Risk of Cash Flow
46
How is risk reflected?
? Higher risk implies higher risk premium implies
higher “i.”
? Given the CFs, the higher the i, the lower the value
(PV).
47
Present Values
48
Example
You just bought a new computer for Rs. 30,000.
The payment terms are 2 years same as cash.
If you can earn 8% on your money, how much
money should you set aside today in order to
make the payment when due in two years?
Present Values
49
Example
You just bought a new computer for Rs. 30,000. The
payment terms are 2 years same as cash. If you can
earn 8% on your money, how much money should
you set aside today in order to make the payment
when due in two years?
16 . 720 , 25 .
2
) 08 . 1 (
30000
Rs PV = =
Present Values
50
? PVs can be added together to evaluate
multiple cash flows.
PV
C
r
C
r
= + +
+ +
1
1
2
2
1 1 ( ) ( )
....
Present Values
? Given Rs. 2, Re. 1 received a year from now and
the other Re. 1 two years from now, the value of
each is commonly called the Discount Factor.
Assume r1 = 20%
r2 = 7%.
51
Present Values
? Given Rs. 2, Re. 1 received a year from now and
the other Re. 1 two years from now, the value of
each is commonly called the Discount Factor.
Assume r1 = 20%
r2 = 7%.
52
87 .
83 .
2
1
) 07 . 1 (
00 . 1
2
) 20 . 1 (
00 . 1
1
= =
= =
+
+
DF
DF
Present Values
53
From the example, there is a flaw. What is the
flaw? What is the result of the finding?
Present Values
54
Example
Assume that the cash flows from the construction
and sale of an office building is as follows. Given a
7% required rate of return, create a present value
worksheet and show the net present value.
000 , 300 000 , 100 000 , 150
2 Year 1 Year 0 Year
+ ÷ ÷
Present Values
55
Example - continued
Assume that the cash flows from the construction and sale of an
office building is as follows. Given a 7% required rate of return,
create a present value worksheet and show the net present
value.
( )
400 , 18
900 , 261 000 , 300 873 . 2
500 , 93 000 , 100 935 . 1
000 , 150 000 , 150 0 . 1 0
Value
Present
Flow
Cash
Factor
Discount
Period
2
07 . 1
1
07 . 1
1
= =
+ + =
÷ ÷ =
÷ ÷
Total NPV
Perpetuity
Perpetuity - Financial concept in which a cash flow
is theoretically received forever.
56
PV
C
r =
=
lue present va
flow cash
Return
Perpetuity
Perpetuity - Financial concept in which a cash flow
is theoretically received forever.
57
r
C
PV
1
rate discount
flow cash
Flow Cash of PV
=
=
Annuity
? Definition: Equal CF over equal length periods,
paid at end of period.
58
Annuity
Annuity - An asset that pays a fixed sum each year
for a specified number of years.
59
Future Value of an Annuity
60
Annuity Example
? Given a loan with:
Amount of loan equal to Rs. 35,000;
Payment = Rs. 4,998.1 per year
n =30 years
What is the interest rate on the loan?
61
Annuity Example
? PV = CF [ PVIFA i,30 ]
35,000 = 4,998.1 [ PVIFA i,30 ]
PVIFAi,30 = 35000/4,998.1 = 7.0027
From Table: looking for row for 30 periods,
PVIFAi,30;
i = 14%.
62
Annuity
Annuity - An asset that pays a fixed sum each year
for a specified number of years.
63
( )
(
¸
(
¸
+
÷ × =
t
r r
r
C
1
1 1
annuity of PV
Annuity
64
Example
You agree to lease a car for 4 years at Rs.
3000 per month. You are not required to pay
any money up front or at the end of your
agreement. If your opportunity cost of capital
is 12% per annum, what is the cost of the
lease?
Annuity
65
Example - continued
You agree to lease a car for 4 years at Rs.3000 per
month. You are not required to pay any money up
front or at the end of your agreement. If your
opportunity cost of capital is 12% per year, what is
the cost of the lease?
( )
922 , 113 .
01 . 1 01 .
1
01 .
1
3000 Cost Lease
48
Rs Cost =
(
¸
(
¸
+
÷ × =
Growing Perpetuities
? Suppose the pay-offs are expected to increase
by g% every year, then present value of the
perpetuity is:
g r
C
PV
÷
=
66
Growing Annuities
(
(
¸
(
¸
|
|
.
|
\
|
+
+
÷
÷
=
T
r
g
g r
C
PV
) 1 (
1
1
67
? Suppose the pay-offs are expected to increase by g%
every year, then present value of the perpetuity is:
Compound Interest
68
0
2
4
6
8
10
12
14
16
18
0 3 6 9
1
2
1
5
1
8
2
1
2
4
2
7
3
0
Number of Years
F
V
o
f
R
e
.
1
10% Simple
10% Compound
Future Value
69
Future Value
? In general for a single CF,
? where, i / re-investment rate, return on investment,
cost of borrowing, opportunity cost, compounding
rate, interest rate
? n = number of periods in the future
? (1+i)^
n
= FVIF
i, n
= FV of interest Factor
= compounding factor
70
Future Value
71
Notes:
(a) If interest rate "i" = 0, then FV of a CF is constant irrespective of how far
in the future you would be receiving it. The horizontal line above represents
this.
(b) Given "i", the greater the "n", # of periods in the future, the greater the
FV. Thus, FV and "n" are positively related.
(c) Given "n", the higher the "i" the higher the FV. Thus, FV and "i" are
positively related. i.e., they move in the same direction.
Impact of Frequency
? If X has Rs. 1000 to invest @12%pa for 2 years,
his returns would be as follows:
? Annual
? FV
2
= ?
? Semi Annual
? FV
2
= ?
? Quarterly
? FV
2
= ?
? Monthly
? FV
2
= ?
? Daily
? FV
2
= ?
72
Impact of Frequency
? If X has Rs. 1000 to invest @12%pa for 2 years, his returns
would be as follows:
? Annual
? FV
2
= 1000x[1+(.12/1)]
(1)(2)
= 1254.40
? Semi Annual
? FV
2
= 1000x[1+(.12/2)]
(2)(2)
= 1262.48
? Quarterly
? FV
2
= 1000x[1+(.12/4)]
(4)(2)
= 1266.77
? Monthly
? FV
2
= 1000x[1+(.12/12)]
(12)(2)
= 1269.73
? Daily
? FV
2
= 1000x[1+(.12/365)]
(365)(2)
= 1271.20
73
Effective Annual Rate
? The actual rate of interest earned after adjusting the
nominal rate for factors such as the number of
compounding periods per year.
? EAR = (1 + i/m)
m
– 1
? Where m is the number of compounding periods per
year.
74
Compound Interest
75
Example
Suppose you are offered an automobile
loan at an APR of 12% per year. What
does that mean, and what is the true
rate of interest, given monthly
payments?
Compound Interest
76
Example - continued
Suppose you are offered an automobile loan at an
APR of 12% per year. What does that mean, and what
is the true rate of interest, given monthly payments?
Assume Rs.10,000 loan amount.
% 6825 . 12
25 . 268 , 11
) 01 . 1 ( 000 , 10 Pmt Loan
12
=
=
× =
APR
Rule of 72
? The Rule of 72 is a handy rule of thumb that states the
following:
? If you earn r % per year, your money will double in about
72/r % years.
So, for example, if you invest at 6%, your money will double in 12
years.
? Why do we say .about?. Because at higher-than-normal
rates, the rule breaks down.
What if r = 72%? ? FVIF(72,1) = 1.72, not 2.00
And if r = 36%? ? FVIF(36,2) = 1.8496
? The lesson? The Rule of 72 is a useful rule of thumb, but it
is only a rule of thumb!
77
Inflation
Inflation - Rate at which prices as a whole are
increasing.
Nominal Interest Rate - Rate at which money invested
grows.
Real Interest Rate - Rate at which the purchasing power
of an investment increases.
78
Interest Rates and Inflation
? One of the key functions of the financial markets is to allocate
capital to different borrowers.
? The rates charged in the markets are a function of the risk of the
particular borrowers & also on the rates of inflation present in
the borrower's market.
? The base rate for a particular country is the rate at which the
central banks can borrow.
? From that base rate, interest rates are adjusted for risk of
individual firms.
? The premium or spread above the base rate depends on
economic conditions.
? The base rate is affected by the expected inflation in a country.
79
Inflation
80
1 + real interest rate =
1+nominal interest rate
1+inflation rate
approximation formula
Real int. rate nominal int. rate-inflation rate ~
Inflation
Example
If the interest rate on one year govt. bonds is 5.9%
and the inflation rate is 3.3%, what is the real
interest rate?
81
Inflation
Example
If the interest rate on one year govt. bonds is 5.9%
and the inflation rate is 3.3%, what is the real
interest rate?
82
2.6% or .026 = .033 - .059 = ion Approximat
2.5% or .025 = rate interest real
1.025 = rate interest real 1
= rate interest real 1
+.033 1
+.059 1
+
+
Bond
? Bond: A promised stream of CFs. The promissory
agreement is traded on an exchange.
? Borrower/issuer: promises to pay future CFs for
current CF
? Lender/buyer: Receives the promised CFs for
current CF.
? Coupon: Equal periodic CFs over life of bond
(annual terms).
? Par Value: CF to be received at maturity, typically
Rs.1,000
? YTM: Yield To Maturity = required return on debt
83
YTM Example
? The following is information on a bond that is trading
in the secondary market with the following
characteristics:
coupon rate = 6%;
years to maturity = 30;
Current market bond price = 1,153.72
YTM = ?
84
YTM Example
? coupon = coupon rate × par value =
6% × 1,000 = 60
? PV ’ SUM of discounted CFs
1,153.75 = 60(PVIFA
YTM,30
) + 1,000(PVIF
YTM,30
)
? Solution is to use the Trial & Error Method: choose
YTM such that
Left Hand Side = Right Hand Side of equation
? If you try YTM = 5%, you will get it right.
YTM = 5%
85
Valuing a Bond
86
Example
If today is July 2008, what is the value of the
following bond?
? An ABC Bond pays Rs.115 every June for 5 years.
In June 2013 it pays an additional Rs. 1000 and
retires the bond.
? The bond is rated AAA (AAA YTM is 7.5%)
? How would the cash flows be?
Valuing a Bond
87
Example
If today is July 2008, what is the value of the following bond?
? An ABC Bond pays Rs.115 every June for 5 years. In June 2013 it
pays an additional Rs. 1000 and retires the bond.
? The bond is rated AAA (AAA YTM is 7.5%)
Cash Flows
June 09 10 11 12 13
115 115 115 115 1115
Valuing a Bond
88
Example continued
If today is July 2005, what is the value of the following bond?
? An ABC Bond pays Rs. 115 every June for 5 years. In June 2010 it pays
an additional Rs. 1000 and retires the bond.
? The bond is rated AAA (AAA YTM is 7.5%)
( ) ( ) ( ) ( )
84 . 161 , 1 .
075 . 1
115 , 1
075 . 1
115
075 . 1
115
075 . 1
115
075 . 1
115
5 4 3 2
Rs
PV
=
+ + + + =
Question
? If the cost of capital is 9%, what is the PV of Rs. 3.74
lacs paid after 9 years?
89
doc_851088723.pptx
Present Value and Opportunity Cost of Capital
Contents
? Present Value
? Net Present Value
? NPV Rule
? ROR Rule
? Opportunity Cost of Capital
? Managers and the Interests of Shareholders
2
Time Value of Money
? Time Value of Money analysis involves:
? What is Rs.100 worth 10 years from today (Future
Value)?
? What is Rs.100 to be received in 10 years worth today
(Present Value)?
? Applications
? Loan amortization
? stated vs. effective interest charged
? rebate vs. low financing
? pricing of bonds
? pricing of stocks/firms
? What is the value of a particular division within a
firm?
3
Time Value of Money
? Time Value of Money analysis involves:
4
Time Value of Money
? Time Value of Money analysis involves:
? What is Rs.100 worth 10 years from today (Future
Value)?
? What is Rs.100 to be received in 10 years worth today
(Present Value)?
? Applications
5
Time Value of Money
? Time Value of Money analysis involves:
? What is Rs.100 worth 10 years from today (Future
Value)?
? What is Rs.100 to be received in 10 years worth today
(Present Value)?
? Applications
? Loan amortization
? stated vs. effective interest charged
? rebate vs. low financing
? pricing of bonds
? pricing of stocks/firms
? What is the value of a particular division within a
firm?
6
Present Value
7
Present Value
Value today of a
future cash
flow.
Discount Rate
Interest rate used to
compute present
values of future cash
flows.
Discount Factor
Present value of a
Rs.100 future
payment.
Present Value
8
1
factor discount = PV
PV = Value Present
C ×
Present Value
9
Discount Factor = DF = PV of Rs.100
Discount Factors can be used to compute the present value
of any cash flow.
DF
r
t
=
+
1
1 ( )
Net Present Value
10
r
C
+
+
1
C = NPV
investment required - PV = NPV
1
0
Valuing an Office Building
An office building costs Rs. 350 lacs now. The building
would be worth Rs. 400 lacs after a year. The equally
risky investment in the stock market offers a return of
7% per annum. What is the value of the Office
Building?
11
Valuing an Office Building
Step 1: Forecast cash flows
Cost of building = C
0
= 350
Sale price in Year 1 = C
1
= 400
Step 2: Estimate opportunity cost of capital
If equally risky investments in the capital market
offer a return of 7%, then
Cost of capital = r = 7%
12
Valuing an Office Building
Step 3: Discount future cash flows
Step 4: Go ahead if PV of payoff exceeds
investment
13
374
) 07 . 1 (
400
) 1 (
1
= = =
+ +r
C
PV
24 374 350 = + ÷ = NPV
Risk and Present Value
? Higher risk projects require a higher rate of return
? Higher required rates of return cause lower PVs
14
Risk and Present Value
?Suppose the investment becomes
riskier and the returns of a similar risk
profile investment in the stock market
is 12%, what is its PV?
15
Risk and Present Value
16
374
.07 1
400
PV
7% at 400 Rs. C of PV
1
=
+
=
=
357
.12 1
400
PV
12% at 400 Rs. C of PV
1
=
+
=
=
Rate of Return Rule
? Accept investments that offer rates of return in
excess of their opportunity cost of capital
? What is Opportunity cost of capital?
17
Opportunity Cost of Capital
? It is that rate of return offered by an equivalent
investment alternative in the capital market
? It is also known as discount rate, hurdle rate
? It is the rate at which we discount the future cash
flows of a given project.
18
Rate of Return Rule
? Accept investments that offer rates of return in excess of
their opportunity cost of capital
19
Example
In the project listed below, the foregone investment
opportunity is 12%. Should we do the project?
14.3% or .143
350,000
350,000 400,000
investment
profit
Return =
÷
= =
Net Present Value Rule
? Accept investments that have positive net
present value
20
Example
Suppose we can invest Rs. 50 today and receive
Rs. 60 in one year. Should we accept the project
given a 10% expected return?
55 . 4
1.10
60
+ -50 = NPV Rs =
Opportunity Cost of Capital
Example
You may invest Rs. 100,000 today. Depending on
the state of the economy, you may get one of three
possible cash payoffs:
21
140,000 110,000 80,000 Payoff
Boom Normal Slump Economy
000 , 110 .
3
000 , 140 000 , 110 000 , 80
C payoff Expected
1
Rs =
+ +
= =
Opportunity Cost of Capital
Example - continued
The stock is trading for Rs.95.65. Next year’s price,
given a normal economy, is forecast at Rs.110. What
is the expected pay-off?
22
Opportunity Cost of Capital
Example - continued
The stocks expected payoff leads to an expected
return.
23
15% or 15 .
65 . 95
65 . 95 110 profit expected
return Expected =
÷
= =
investment
Opportunity Cost of Capital
Example - continued
Discounting the expected payoff at the expected
return leads to the PV of the project
24
650 , 95 .
1.15
110,000
PV Rs = =
Investment vs. Consumption
? Some people prefer to consume now. Some
prefer to invest now and consume later.
? Calculating the PV of borrowing and lending
allows us to reconcile these opposing desires
which may exist within the firm’s shareholders.
25
Managers and Shareholder Interests
26
? Tools to Ensure Management Pays
Attention to the Value of the Firm
? Manger’s actions are subject to the
scrutiny of the board of directors.
? Shirkers are likely to find they are ousted
by more energetic managers.
? Financial incentives such as stock options
Question
? If C
0
is Initial investment, C
1
is the cash flow at the
end year 1, and r is the discounting rate, then if the
investment is risk-free, what is the appropriate
measure of r?
27
Question
? If present value of Rs. 15,000 paid after a year is Rs.
13,000, what is the discount factor?
28
Question
? A trader pays Rs. 1 lac for a truck-load of rice and is
certain of selling the same after a year for Rs. 1.32 lacs.
? What is the return on investment?
? If the return is lower than the rate of interest, then is the
NPV +ve or –ve?
29
Question
? A parcel of land costs Rs. 5 lacs. For an additional Rs. 8
lacs you can build a motel on the property. The value
of the land and the motel after is expected to be Rs. 15
lacs. Would it make business sense to invest in the
motel, if another project yielding 10% pa is available
for investment?
30
Question
? Shantanu, a leading analyst with Meghraj
TramellCrow, a leading real estate firm,
recommends an investment in a plot of land
worth Rs. 8.5 lacs. He is certain that the land
would be worth Rs. 9.1 lacs after a year. Given the
guaranteed returns available on FDs at 7.5%,
should the investors buy the plot of land? To get
the same return as that on FDs, at what rate
should the investor purchase the plot of land?
31
Question
? Gaurav has received Rs. 2 cr in inheritance. How
should he invest it? 4 alternatives:
1. Invest in 1 yr T-bill yielding 8%
2. Loan to his nephew, Nitin, who has aspired to start his
own business. Nitin has arranged a 1 yr loan of Rs. 90
lacs at 12%, but has approached Gaurav for a loan at 10%
3. Invest in the stock market, where the expected rate of
return is 18%.
4. Investment in local real estate, which Gaurav judges is
as risky as the stock market. The opportunity at hand
would cost Rs. 100 lacs and is forecasted to be worth Rs.
1.15 cr after 1 yr.
? Which of these has positive NPV? Which one
would you advice Gaurav to take?
32
Question
? Gaurav has received Rs. 2 cr in inheritance. How
should he invest it? In alternative 4
“Investment in local real estate, which Gaurav judges is as
risky as the stock market. The opportunity at hand
would cost Rs. 100 lacs and is forecasted to be worth Rs.
1.15 cr after 1 yr.” –
If Gaurav takes a bank loan of Rs. 60 lacs at 12% (Gaurav has
a long term relationship with the bank) and invest Rs. 1
cr in the real estate and the balance in the stock market.
? Is this a smart move? Explain.
33
Question
? Calculate the NPV and rate of return for the
following investments. The opportunity cost of
capital is 20%.
? Figures in Rs. lacs
34
Investment Initial CF CF in Year 1
1 10,000 18,000
2 5,000 9,000
3 5,000 5,700
4 2,000 4,000
Individual Assignment
? Madhura has retired and depends on her
investments for retirement income. Akhil is a young
executive who wants to save for the future. They are
both stockholders of Tata Motors, which is
investing Rs. 1200 crores to develop the new Nano.
This investment’s pay-off is many years in future.
Assume that the investment is +ve NPV for Akhil.
Should the NPV be +ve also for Madhura?
35
Individual Assignment
? Define Opportunity Cost of Capital. How in principle
would you find the opportunity cost of capital for a risk
free asset? For a risk asset?
? Respond to-
? My company’s cost of capital is the rate of the bank when
we borrow money.
? NPV is just theory. We must maximize profit. That’s what
shareholders want.
? Why is a reputation for honesty and fair business
practice important to financial value of the corporation?
36
Group Assignment
? Chapter 2 Brearley Myers 7
th
edition
? Challenge Questions: 1, 2, 4, 5
37
38
Topics Covered
? Valuing Long-Lived Assets
? PV Calculation Short Cuts
? Compound Interest
? Nominal and Real Rates of Interest (inflation)
? Present Values and Bonds
39
Present Values
40
Discount Factor = DF = PV of Re.1
Discount Factors can be used to compute
the present value of any cash flow.
DF
r
t
=
+
1
1 ( )
Present Values
41
Discount Factors can be used to compute
the present value of any cash flow.
DF
r
t
=
+
1
1 ( )
1
1
1
1 r
C
C DF PV
+
= × =
Present Values
42
? Replacing “1” with “t” allows the formula to
be used for cash flows that exist at any point
in time
t
t
t
r
C
C DF PV
) 1 ( +
= × =
Present Value
? In general for a single CF,
? where, i = discount rate
? PVIF
i, n
= PV of Interest Factor which depends on i, n.
= discount factor
43
Present Value
44
Notes:
(a) If i = 0, then PV of a CF, say CF = Rs. 1, is constant at Rs.1, irrespective of
how far in the future it is received.
(b) For a given "n", the higher the "i", the lower is PV.
(c) For a given "i", the larger the "n", the smaller the PV.
PV and "i" are inversely related ] They move in opposite direction.
PV and "n" are inversely related ]They move in the opposite direction
Value depends on
45
Value depends on
? Amount of Cash Flow
? Timing of Cash Flow
? Risk of Cash Flow
46
How is risk reflected?
? Higher risk implies higher risk premium implies
higher “i.”
? Given the CFs, the higher the i, the lower the value
(PV).
47
Present Values
48
Example
You just bought a new computer for Rs. 30,000.
The payment terms are 2 years same as cash.
If you can earn 8% on your money, how much
money should you set aside today in order to
make the payment when due in two years?
Present Values
49
Example
You just bought a new computer for Rs. 30,000. The
payment terms are 2 years same as cash. If you can
earn 8% on your money, how much money should
you set aside today in order to make the payment
when due in two years?
16 . 720 , 25 .
2
) 08 . 1 (
30000
Rs PV = =
Present Values
50
? PVs can be added together to evaluate
multiple cash flows.
PV
C
r
C
r
= + +
+ +
1
1
2
2
1 1 ( ) ( )
....
Present Values
? Given Rs. 2, Re. 1 received a year from now and
the other Re. 1 two years from now, the value of
each is commonly called the Discount Factor.
Assume r1 = 20%
r2 = 7%.
51
Present Values
? Given Rs. 2, Re. 1 received a year from now and
the other Re. 1 two years from now, the value of
each is commonly called the Discount Factor.
Assume r1 = 20%
r2 = 7%.
52
87 .
83 .
2
1
) 07 . 1 (
00 . 1
2
) 20 . 1 (
00 . 1
1
= =
= =
+
+
DF
DF
Present Values
53
From the example, there is a flaw. What is the
flaw? What is the result of the finding?
Present Values
54
Example
Assume that the cash flows from the construction
and sale of an office building is as follows. Given a
7% required rate of return, create a present value
worksheet and show the net present value.
000 , 300 000 , 100 000 , 150
2 Year 1 Year 0 Year
+ ÷ ÷
Present Values
55
Example - continued
Assume that the cash flows from the construction and sale of an
office building is as follows. Given a 7% required rate of return,
create a present value worksheet and show the net present
value.
( )
400 , 18
900 , 261 000 , 300 873 . 2
500 , 93 000 , 100 935 . 1
000 , 150 000 , 150 0 . 1 0
Value
Present
Flow
Cash
Factor
Discount
Period
2
07 . 1
1
07 . 1
1
= =
+ + =
÷ ÷ =
÷ ÷
Total NPV
Perpetuity
Perpetuity - Financial concept in which a cash flow
is theoretically received forever.
56
PV
C
r =
=
lue present va
flow cash
Return
Perpetuity
Perpetuity - Financial concept in which a cash flow
is theoretically received forever.
57
r
C
PV
1
rate discount
flow cash
Flow Cash of PV
=
=
Annuity
? Definition: Equal CF over equal length periods,
paid at end of period.
58
Annuity
Annuity - An asset that pays a fixed sum each year
for a specified number of years.
59
Future Value of an Annuity
60
Annuity Example
? Given a loan with:
Amount of loan equal to Rs. 35,000;
Payment = Rs. 4,998.1 per year
n =30 years
What is the interest rate on the loan?
61
Annuity Example
? PV = CF [ PVIFA i,30 ]
35,000 = 4,998.1 [ PVIFA i,30 ]
PVIFAi,30 = 35000/4,998.1 = 7.0027
From Table: looking for row for 30 periods,
PVIFAi,30;
i = 14%.
62
Annuity
Annuity - An asset that pays a fixed sum each year
for a specified number of years.
63
( )
(
¸
(
¸
+
÷ × =
t
r r
r
C
1
1 1
annuity of PV
Annuity
64
Example
You agree to lease a car for 4 years at Rs.
3000 per month. You are not required to pay
any money up front or at the end of your
agreement. If your opportunity cost of capital
is 12% per annum, what is the cost of the
lease?
Annuity
65
Example - continued
You agree to lease a car for 4 years at Rs.3000 per
month. You are not required to pay any money up
front or at the end of your agreement. If your
opportunity cost of capital is 12% per year, what is
the cost of the lease?
( )
922 , 113 .
01 . 1 01 .
1
01 .
1
3000 Cost Lease
48
Rs Cost =
(
¸
(
¸
+
÷ × =
Growing Perpetuities
? Suppose the pay-offs are expected to increase
by g% every year, then present value of the
perpetuity is:
g r
C
PV
÷
=
66
Growing Annuities
(
(
¸
(
¸
|
|
.
|
\
|
+
+
÷
÷
=
T
r
g
g r
C
PV
) 1 (
1
1
67
? Suppose the pay-offs are expected to increase by g%
every year, then present value of the perpetuity is:
Compound Interest
68
0
2
4
6
8
10
12
14
16
18
0 3 6 9
1
2
1
5
1
8
2
1
2
4
2
7
3
0
Number of Years
F
V
o
f
R
e
.
1
10% Simple
10% Compound
Future Value
69
Future Value
? In general for a single CF,
? where, i / re-investment rate, return on investment,
cost of borrowing, opportunity cost, compounding
rate, interest rate
? n = number of periods in the future
? (1+i)^
n
= FVIF
i, n
= FV of interest Factor
= compounding factor
70
Future Value
71
Notes:
(a) If interest rate "i" = 0, then FV of a CF is constant irrespective of how far
in the future you would be receiving it. The horizontal line above represents
this.
(b) Given "i", the greater the "n", # of periods in the future, the greater the
FV. Thus, FV and "n" are positively related.
(c) Given "n", the higher the "i" the higher the FV. Thus, FV and "i" are
positively related. i.e., they move in the same direction.
Impact of Frequency
? If X has Rs. 1000 to invest @12%pa for 2 years,
his returns would be as follows:
? Annual
? FV
2
= ?
? Semi Annual
? FV
2
= ?
? Quarterly
? FV
2
= ?
? Monthly
? FV
2
= ?
? Daily
? FV
2
= ?
72
Impact of Frequency
? If X has Rs. 1000 to invest @12%pa for 2 years, his returns
would be as follows:
? Annual
? FV
2
= 1000x[1+(.12/1)]
(1)(2)
= 1254.40
? Semi Annual
? FV
2
= 1000x[1+(.12/2)]
(2)(2)
= 1262.48
? Quarterly
? FV
2
= 1000x[1+(.12/4)]
(4)(2)
= 1266.77
? Monthly
? FV
2
= 1000x[1+(.12/12)]
(12)(2)
= 1269.73
? Daily
? FV
2
= 1000x[1+(.12/365)]
(365)(2)
= 1271.20
73
Effective Annual Rate
? The actual rate of interest earned after adjusting the
nominal rate for factors such as the number of
compounding periods per year.
? EAR = (1 + i/m)
m
– 1
? Where m is the number of compounding periods per
year.
74
Compound Interest
75
Example
Suppose you are offered an automobile
loan at an APR of 12% per year. What
does that mean, and what is the true
rate of interest, given monthly
payments?
Compound Interest
76
Example - continued
Suppose you are offered an automobile loan at an
APR of 12% per year. What does that mean, and what
is the true rate of interest, given monthly payments?
Assume Rs.10,000 loan amount.
% 6825 . 12
25 . 268 , 11
) 01 . 1 ( 000 , 10 Pmt Loan
12
=
=
× =
APR
Rule of 72
? The Rule of 72 is a handy rule of thumb that states the
following:
? If you earn r % per year, your money will double in about
72/r % years.
So, for example, if you invest at 6%, your money will double in 12
years.
? Why do we say .about?. Because at higher-than-normal
rates, the rule breaks down.
What if r = 72%? ? FVIF(72,1) = 1.72, not 2.00
And if r = 36%? ? FVIF(36,2) = 1.8496
? The lesson? The Rule of 72 is a useful rule of thumb, but it
is only a rule of thumb!
77
Inflation
Inflation - Rate at which prices as a whole are
increasing.
Nominal Interest Rate - Rate at which money invested
grows.
Real Interest Rate - Rate at which the purchasing power
of an investment increases.
78
Interest Rates and Inflation
? One of the key functions of the financial markets is to allocate
capital to different borrowers.
? The rates charged in the markets are a function of the risk of the
particular borrowers & also on the rates of inflation present in
the borrower's market.
? The base rate for a particular country is the rate at which the
central banks can borrow.
? From that base rate, interest rates are adjusted for risk of
individual firms.
? The premium or spread above the base rate depends on
economic conditions.
? The base rate is affected by the expected inflation in a country.
79
Inflation
80
1 + real interest rate =
1+nominal interest rate
1+inflation rate
approximation formula
Real int. rate nominal int. rate-inflation rate ~
Inflation
Example
If the interest rate on one year govt. bonds is 5.9%
and the inflation rate is 3.3%, what is the real
interest rate?
81
Inflation
Example
If the interest rate on one year govt. bonds is 5.9%
and the inflation rate is 3.3%, what is the real
interest rate?
82
2.6% or .026 = .033 - .059 = ion Approximat
2.5% or .025 = rate interest real
1.025 = rate interest real 1
= rate interest real 1
+.033 1
+.059 1
+
+
Bond
? Bond: A promised stream of CFs. The promissory
agreement is traded on an exchange.
? Borrower/issuer: promises to pay future CFs for
current CF
? Lender/buyer: Receives the promised CFs for
current CF.
? Coupon: Equal periodic CFs over life of bond
(annual terms).
? Par Value: CF to be received at maturity, typically
Rs.1,000
? YTM: Yield To Maturity = required return on debt
83
YTM Example
? The following is information on a bond that is trading
in the secondary market with the following
characteristics:
coupon rate = 6%;
years to maturity = 30;
Current market bond price = 1,153.72
YTM = ?
84
YTM Example
? coupon = coupon rate × par value =
6% × 1,000 = 60
? PV ’ SUM of discounted CFs
1,153.75 = 60(PVIFA
YTM,30
) + 1,000(PVIF
YTM,30
)
? Solution is to use the Trial & Error Method: choose
YTM such that
Left Hand Side = Right Hand Side of equation
? If you try YTM = 5%, you will get it right.
YTM = 5%
85
Valuing a Bond
86
Example
If today is July 2008, what is the value of the
following bond?
? An ABC Bond pays Rs.115 every June for 5 years.
In June 2013 it pays an additional Rs. 1000 and
retires the bond.
? The bond is rated AAA (AAA YTM is 7.5%)
? How would the cash flows be?
Valuing a Bond
87
Example
If today is July 2008, what is the value of the following bond?
? An ABC Bond pays Rs.115 every June for 5 years. In June 2013 it
pays an additional Rs. 1000 and retires the bond.
? The bond is rated AAA (AAA YTM is 7.5%)
Cash Flows
June 09 10 11 12 13
115 115 115 115 1115
Valuing a Bond
88
Example continued
If today is July 2005, what is the value of the following bond?
? An ABC Bond pays Rs. 115 every June for 5 years. In June 2010 it pays
an additional Rs. 1000 and retires the bond.
? The bond is rated AAA (AAA YTM is 7.5%)
( ) ( ) ( ) ( )
84 . 161 , 1 .
075 . 1
115 , 1
075 . 1
115
075 . 1
115
075 . 1
115
075 . 1
115
5 4 3 2
Rs
PV
=
+ + + + =
Question
? If the cost of capital is 9%, what is the PV of Rs. 3.74
lacs paid after 9 years?
89
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