bài giảng kinh tế vi mô tiếng anh ch16 capital market

1
Chapter 16
Interest Rates, Investments,
and Capital Markets
Key issues
1. comparing money today to money in the
future: interest rates
2. choices over time: invest in a project if
return from investment > return on best
alternative
Capital and durable goods
• durable goods: products that are usable for years
• if durable good or capital is rented, rent up to the
point where the marginal benefit = MC
• if bought or built rather than rented, firm
compares current cost of capital to future higher
profits it will make from using capital
Interest rates
• assume no inflation: consuming $1 worth of
candy today is better than consuming $1
worth in 10 years
• how much more you must pay in future to
repay a loan today is specified by an
interest rate:
percentage more that must be repaid to borrow
money for a fixed period of time
Deposit funds in a bank today
• bank agrees to pay you interest rate, i = 4%
• one year from now, bank pays for every
dollar you loan it:
$1.04 = 1 + i

Compounding
• “interest on interest”
• “accumulation of interest”
2
Example
• place $100 in bank account that pays 4% a year
• you can take out interest payment of $4 each year
and leave your $100 in bank: earn a flow of $4-a-
year payments forever
• if you leave the interest rate in back, in second
year bank owes you:
• interest of $4 on your original deposit of $100
• interest of $4
G 0.04 = $0.16 on your first-year interest
• total interest is $4.16
Compounding over time
assets at the end of
year 1: $104.00 = $100 G 1.04 = $100 G 1.04
1
year 2: $108.16 = $104 G 1.04 = $100 G 1.04
2
year 3: $112.49 G $108.16 G 1.04 = $100 G 1.04
3
General compounding formula
For every $1 you loan the bank, it owes you:
year 1: $(1 + i)
1
year 2: $(1 + i)
2
= $(1 + i)q(1 + i)

year 3: $(1 + i)
3
= $(1 + i)q(1 + i)q(1 + i) …
…
year t: $(1 + i)
t
Frequency of compounding
• for a given i, more frequent compounding, greater
payment at end of a year
• annual interest rate is i = 4%
• if bank pays interest 2 times a year,
• half a year's interest, i/2 = 2%, after six month:
$(1 + i/2) = $1.02
• at end of year, bank owes:
$(1 + i/2)
´ (1 + i/2) = $(1 + i/2)
2
=$(1.02)
2
= $1.0404
U.S. Truth-in-Lending Act
requires lenders to tell borrowers equivalent
noncompounded annual percentage rate
(APR) of interest
3
Interest rates connect present and
future
• future value (FV) depends on the present
value (PV), the interest rate, and the number
of years

• put PV dollars in bank today and allow
interest to compound for t years:
FV = PV q (1 + i)
t
Power of compounding:
Manhattan Island
• Dutch allegedly bought Manhattan in 1626 for
about $24 worth of beads and trinkets
• if Native Americans had invested in tax-free
bonds 7% APR bonds, it would now be worth
over $2.0 trillion > assessed value of Manhattan
Alaska
• if US had invested $7.2 million it paid Russia
in 1867 in tax-free 7% APR bonds
• money worth only $50.9 billion < Alaska's
current value
Present value
• 2 equivalent questions:
• how much is $1 in the future worth today?
• how much money, PV, must we put in bank
today at i to get a specific FV at some future
time?
•answer:
PV = FV/(1 + i)
t
Example
• general formula
PV = FV/(1 + i)
t
• FV = $100 at end of year

• i = 4%
PV = $100/1.04 = $96.15
4
When is future money nearly
worthless?
• at high interest rates, money in future is
virtually worthless today
• $1 paid to you in 25 years is worth only 1¢
today at a 20% interest rate
Stream of payments forever
• PV in a bank account earning i produces a flow of
f (at end of each year) of
f = i
´ PV
•to receive f each year forever need to invest
PV = f / i
• to get $10 a year invest
$200 = $10/0.05 at i = 5%
$100 = $10/0.10 at i = 10%
$50 = $10/0.20 at i = 20
Stream of payments for t years
•What’s PV of payments per period of f
made every year?
• you agree to pay $10 at end of each year for
3 years to repay a debt i = 10%
PV = $10/1.1
1
+ $10/1.1
2
+ $10/1.1

3
» $24.87
• generally:
12
11 1

(1 ) (1 ) (1 )
t
PV f
ii i
éù
=+++
êú
++ +
ëû
Figure 16.1 Present Value of a Dollar in the Future
Present value,
PV, of $1
20
10
40
50
60
70
80
90
$1
t , Years
0 102030405060708090100
i = 0%

i = 5%
i = 10%
i = 20%
30
Future value of payments over
time
•What’s FV after t years if you save f each year?
• year 1: put f in account
• year 2: add a second f, so you have first year's
payment + accumulated interest of f (1 + i)
1
or
f [1 + (1 + i)
1
] in total
• year 3: total is f [1 + (1 + i) + (1 + i)
2
]
•after t years:
FV = f [1 + (1+i)
1
+ (1+i)
2
+…+(1+i)
t
]
5
Starting Early
• it pays to start saving early (take advantage of
compounding)

• two approaches to savings
• early bird: you save $3,000 a year for first 15 years of
your working life and then let your savings accumulate
interest until you retire
• late bloomer: after not saving
for first 15 years, you save
$3,000 a year for next 33 years
until retirement
Early Bird
• save $3,000 a year for first 15 years then let
it accumulate
• after 15 years, early bird has
$3,000[1+1.07
1
+1.07
2
+ +1.07
14
] = $75,387
• interest compounds for next 33 years, so
fund grows 9.3 times to
$75,387.07 q 1.07
33
= $703,010
Late Bloomer
• no investments for 15 years, then invests $3,000 a
year until retirement so funds at retirement are
$3,000[1+1.07
1
+1.07

2
+ +1.07
32
] = $356,800
• thus, late bloomer
• contributes to account more than twice as long as the
early bird
• but saves only about half as much by retirement
• to have same amount at retirement, late bloomer has to
save nearly $6,000 a year for 33 years
Inflation and discounting
• we've been assuming inflation rate = 0%
• suppose general inflation occurs: nominal
prices rise at a constant rate g over time
• by adjusting for rate of inflation, we convert
nominal prices to real prices
Adjusting for inflation
• nominal amount you pay next year is
• future debt in today's dollars is
•if g = 10%, a nominal payment of next
year is in today’s (real)
dollars
°
/(1 )ff
γ
=+
°
f
°
°

/1.1 0.909
f
ff==
°
f
Nominal and real rates of interest
• to calculate PV of this future real payment,
we discount using real interest rate
• without inflation, $1 today is worth 1 + i
next year
• with inflation rate of g, $1 today is worth
(1 + i)(1 + g) nominal dollars tomorrow
•if i = 5% and g = 10%, $1 today is worth
1.05 q1.1 = 1.155 nominal dollars next year
6
Nominal vs. real interest rates
• banks pay a nominal interest rate,
• if real discount rate is i, banks' nominal
interest rate is such that dollar today pays
(1 + i)(1 + g) in next year’s dollars
• because
• nominal interest rate is
i

1(1)(1)1ii ii
γγγ
+= + + =++ +

iii
γγ

=+ +

Real interest rate
• depends on inflation and nominal rate
• if inflation rate is small , then we can
closely approximate the real rate by
• if nominal rate is 15.5%, g = 10%, real rate
is (15.5%-10%)/1.1=5% and approximation
is 5.5%
1
i
i
γ
γ
−
=
+

ii
γ
=−

0
γ
≈
Real present value
• real present value of a nominal payment one
year from now is
• you agree to pay $100 next year for a tape
recorder you get today, g = 10%, i = 5%,

then PV = $100/(1.1 q 1.05) = $86.58
°
1(1)(1)
ff
PV
ii
γ
==
+++
Winning the lottery
lottery: a tax on people who are bad at math
• several states boast (lie) that their lottery pays a
winner one million dollars
• winner gets $50,000 a year for 20 years
• total nominal payments are $1 million
•if C = 5% and i = 4% (nominal rate of interest is
9.2%)
•real PV of 20 payments is $491,396
• without inflation real PV = $706,697
Comparing 2 contracts
• professional basketball player is
offered a choice of 2 contracts
• one contract pays $1 million today
• other contract pays $500,000 today and $2
million 10 years from now
• both contracts guaranteed: payments will be
made even if he’s injured
Player’s choice
• assume there is no inflation and that our pro wants
to maximize his

PV
• PV of first contract is $1 million
• to calculate PV of second contract, he uses market i
• PV = $500,000 + $2,000,000/(1 + i)
10
• PV depends on interest rate
• PV = $1,727,827 at i = 5%
• PV = $823,011 at i = 20%
• choose second contract if i = 5% but not at 20%
• break-even interest rate is 14.87%
7
Investment decision
1. net present value approach
2. internal rate of return approach
Net present value approach
depends on PV of revenues, R, and cost, C
12
0
12
12
0
12

(1 ) (1 ) (1 )

(1 ) (1 ) (1 )
T
T
T
T

NPV R C
RR R
R
ii i
CC C
C
ii i
=−
é
ù
=+ + ++
ê
ú
++ +
ë
û
é
ù
−+ + ++
ê
ú
++ +
ë
û
11 2 2
00
12
12
0
12

(1 ) (1 ) (1 )

(1 ) (1 ) (1 )
TT
T
T
T
R
CRC RC
NPV R C
ii i
ii i
ππ π
π
é
ù
−− −
=−+ + ++
ê
ú
++ +
ë
û
éù
=+ + ++
êú
++ +
ë
û

NPV rules
• invest if NPV > 0
• it isn’t necessary for cash flow in each year,
Q
t
(loosely, annual profit), to be positive
Solved problem
• Snyder, Zuckerman, and Drasner bought
Washington Redskins football team and its home
stadium for $800 million in 1999
• estimated 1999 net income was
f = $32.8 million
• if new owners believe they’ll earn this annual
profit (adjusting for inflation) forever, was this
investment more lucrative than putting $800
million in a savings account that pays
i = 4%?
Answer
• NPV is positive if PV of expected returns,
$32.8 million/0.04 = $820 million, minus
PV of cost, $800 million, is positive:
NPV = $820 million - $800 million
= $20 million > 0
• thus, they buy team if best alternative is 4%
rate of return
Internal rate of return approach
• at what discount rate (rate of return) is firm
indifferent between making investment and not?
•
internal rate of return (irr) is discount rate where

NPV = 0
• replacing
i with irr and setting NPV = 0, find irr
by solving for iir
12
0
12
0
(1 ) (1 ) (1 )
T
T
NPV
irr irr irr
ππ π
π
é
ù
=+ + ++ =
ê
ú
++ +
ë
û
8
IRR for flow
• if investment produces a steady stream of
profit, f, then
irr = f / PV
• make investment if irr > i, if i is next best
alternative

Solved problem
• group of investors can buy Redskins
football team for PV = $800 million
• they expect an annual real flow of payments
(profits) of f = $32.8 million forever
• if real interest rate is 4%, do they buy team?
Answer
• because this rate of return, 4.1%, is greater
than interest rate, 4%, they buy the team
$32.8
4.1% 4%
$800
f
million
irr
PV million
== ≈ >
Buying vs. renting a phone
• choose between
• renting a telephone at $10 per year (flow payment)
• buy phone (lifetime warranty) for $100 (stock payment)
• plan to use phone for 50 years; no inflation
• 2 ways to decide
• calculate PV of flow at a given interest rate and
compare that to the cost of buying (NPV)
• determine irr and compare to interest rate
PV approach
• PV of renting (f = $10/year) for 50 years
using Table 16.3 shows PV payment of $10
a year for 50 years is

• $183 at 5%
• $99 at 10%
• $50 at 20%
• thus, if i U 10%, it is better to rent rather
than buy
9
Approximation
• approximation of PV of a stream of payments for
50 years: assume payments go on forever:
•
PV = $10 / i
• PV = $200 at 5% (vs. $183 for 50 years)
• PV = $100 at 10% (vs. $99)
• PV = $50 at 20% (vs. $50)
• thus, approximation is better, higher interest rate
Calculate irr
• put $100 you'd spend on the phone into an
account that pays you interest of $100 q irr
per year
• that is, $100 q irr = $10 º irr = 10%
• rent if irr < i; otherwise buy
Human capital
• individuals decide whether to invest in their
own human capital
• does going to college increase your lifetime
earnings?
• graduate high school at 18 years old and
either go to work or go to college
Retire at 70
• suppose you

• graduate from college in 4 years
• do not work when in college
• pay $10,000 a year for school expenses: tuition, books,
fees
• opportunity cost of college: tuition payments plus
4 years of foregone earnings (at HS grad wage)
• at age 22
• typical college grad earns $29K ($1995)
• HS grad earns $18K
Figure 16.2 Annual Earnings of High School and College
Graduates
Annual earnings,
Thousands of 1995 dollars
–10
10
0
20
30
40
Age, Years
18 22 30 40 50 60 70
High school graduate
College graduate
Benefit
Cost
10
Compare earning streams
• earnings peak
• for college grad at 40 years of age at $39K
• for HS grad at 43 years at $34K

• decide whether invest in college by
comparing PV at age 18 of the two earnings
streams
Exhaustible resources
• discounting determines how fast we
consume exhaustible resources
• exhaustible resources:
• nonrenewable natural assets that cannot be
increased, only depleted
• examples: oil, gold, copper, uranium
When to sell
• if own a coal mine and want to maximize PV,
when do you
• mine the coal
• sell the coal
• for simplicity, suppose
• can only sell this year or next in a competitive market
• interest rate is i
• cost of mining a pound of coal, m, is constant over time
When to mine
• present value of cost of mining
•this year: m
• next year: m/(1 + i)
• thus, if you are not selling until next year,
mine at the last possible moment (lower PV
of cost)
When to sell
• answer depends on how price changes over
time
• suppose price increases from p

1
this year to
p
2
next year
• compare PV of selling today to PV of
selling next year
•this year: PV
1
= p
1
– m
• next year: PV
2
= (p
2
- m)/(1 + i)
11
To maximize PV of profit
• sell all coal this year if PV
1
> PV
2
• sell all coal next year if PV
1
< PV
2
• sell in either year if PV
1
= PV

2
Intuition
• storing coal in the ground is like keeping
money in the bank
• if you sell a pound today, net p
1
– m, invest
the money in a back, you have (p
1
– m)(1+i)
next year
• if that’s less than (p
2
-m), sell coal now
Many people
• can generalize the reasoning to many
periods
• a resource is sold in t and t+1 only if PV of
a pound sold in t is the same as next year:
p
t
– m = (p
t+1
-m)/(1 + i)
•rearranging:
p
t+1
= p
t
+ i(p

t
-m)
Interpretation
• if coal is sold in both years, price next year
must exceed price this year by
• i(p
t
-m)
• interest payment if you sold a pound this year
and put profit in the bank at
i
• so price grows at
%p = p
t+1
- p
t
= i(p
t
-m)
Gap over time
• gap between this year’s price and next
year’s price increases as cash flow, p
t
-m,
increases
• thus price gap grows exponentially over
time
Figure 16.3 Price of an Exhaustible Resource
p
, $ per unit

Time, Years
T1
Price
p
1
m
p
_
12
Figure p16.1 Redwood Trees
Real price of
old-growth redwoods
1953 1956 1959 1962 1965 1968 1971 1974 1977 1980 1983
50
0
100
150
200
250
300
1. Comparing money today to
money in future
• people value money in the future < money today
• interest rate reflects how much more people value
$1 today to $1 in the future
• compare a payment made in future to one today by
expressing future payment in terms of current
dollars using interest rate
• similarly, a flow of payments over time is related
to present or future value of payments by interest

rate
2. Choices over time
pick investment option (cash flow over
time) by
• picking one with highest discounted present
value if net present value > 0
• if internal rate of return > interest rate