# Investment

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```					Investment

Chapter 14
Hong Kong Real Estate
   According to the planning department
rental yields on residential apartments
are more than 5%.
   Rental yields are the annual rent divided
by the price level.
   But interest rates on mortgage loans are
2.5%.
   Why don’t people want to borrow at 2.5%
to make 5.1% returns?
Objectives
   Consider the theory of investment in real
capital.
   Evaluate the components of the cost of capital.
   Calculate the optimal capital stock as a
function of the cost of capital.
   Apply Capital Theory to the real estate market
   Calculate Tobin’s q to estimate the
desirability of corporate investment.
    Evaluate relationship between leverage
and investment.
Terminology: Investment
   We use the term investment to refer to
real expenditure (public and/or private)
on tangible assets.
   We call the stock of tangible assets
capital or physical capital.
   The unit of measure of aggregate capital
is dollars.
   Gross Investment refers to purchases of
new investment.
   Net Investment is Gross Investment
minus depreciation.
Components of Investment

   Investment
   Fixed Investment
 Residential Investment
   Structures
   Machinery & Equipment
   Changes in Stocks – Inventory
Investment
Gross Fixed Capital Formation:
HK 2002

5%   3%
10%
14%
Transfer Costs of Land &
Building
Real Estate Developers'
Margin
Machinery & Equipment

14%
Public Construction

Private Residential

Private: Non-residential

54%
Investment Facts

   Investment expenditure is a
substantial share of GDP, but not as
large as consumption.
   Fixed and inventory investment are
cycle.
   Investment is an especially volatile
part of GDP.
Business Cycle Volatility of Real Investment
in Hong Kong
.20

.15

.10

.05

.00

-.05

-.10

-.15
1975   1980     1985      1990     1995      2000

GDP        INVESTMENT
Marginal Analysis
   Economists use marginal analysis to
determine an optimal level of an activity.
   Most activities have diminishing marginal
returns.
   Marginal returns are the extra benefit received
from doing a bit more of the activity.
   Do more of the activity until that point
when marginal returns from doing a bit
more of the activity start to become more
than the cost of the activity.
Optimal Capital
   Benefit of owning capital is that it allows
us to produce more goods.
   Marginal product of capital is the extra
revenue from the extra goods we could
produce if we had just a bit more capital.
   MPK can be measured in either nominal,
current price (PMPK) or real, constant
price (MPK) terms.
   Capital has diminishing returns. MPK is a
decreasing function of the capital stock.
Productivity of Capital
   The productivity or average productivity
of capital is the revenue generated per
dollar of capital.
   APK is value of output divided by the
capital stock.       Value of Output
APK 
Value of Capital
   Value can be measured in constant or
current price terms.
   Marginal productivity of capital is often
thought to be roughly proportional to
average productivity capital.
MPK

K
Cost of Capital
   Economists define the (time) cost of capital as the
cost of holding a unit of capital for a period of
time.
   A firm invests in capital equipment for a period.
   The firm borrows money upfront to finance the
purchase.
   The firm produces goods and generates revenues.
   The firm sells the capital at the end of the period,
typically at less than the purchase price due to wear
and tear.
   The firm repays loan.
   Cost of using capital includes interest payment
plus loss on the resale of capital.
Optimal Capital Example.
A firm borrows Pt                 Optimal Condition
K , NEW
                                
to buy 1 capital good              Pt 1,OLD  PMPKt  1  it  Pt K , NEW
K

at interest rate 1+i.
   The firm produces                Definition of Capital
PMPKt+1 worth of                  Cost
goods and sells the
capital good for P K ,OLD.       PMPK t 
t
   Optimal to buy capital           iPt K , NEW  [ Pt 1,OLD  Pt K , NEW ]
K

good as long as pay-
off is greater than the           Cost of Capital
cost.
Capital Cost
    We can divide the       ck  iPt K , NEW  [ Pt 1,OLD  Pt K , NEW ] 
K

capital cost into three
i   g    PK
Pt K , NEW
parts.
1.   Interest cost: Net
interest rate.                 Pt 1, NEW  Pt 1,OLD
K                  K

2.   Depreciation: Defined                   Pt K , NEW
as change in value
due to aging.
3.   Capital gain: Defined               Pt 1, NEW  Pt K , NEW
K
gP 
K

as change in value                          Pt K , NEW
due to change in price
of new goods.
Real Capital Cost
   We can convert the
PMPK  i    gtP 1 Pt K , NEW
K

optimal capital                             

equation into real     MPK  i    g tP 1 ptK , NEW
K


terms by dividing both
sides by the price     MPK  r    gtp 1 ptK , NEW  rckt
K


level.
   Define the real price
of capital good as                 Pt K , NEW
price of capital good       pt 
k

Pt
relative to the firm’s
output price.
PMPK
P K , NEW

K
Example
   A taxi agency can produce a certain
amount of revenue with larger numbers
of taxis.
   Assume earnings (revenues minus wages
minus costs) per year is given by the
schedule \$200, 000 4 N
3

   Assume that the purchase price of a new
taxi (with license) is \$1,000,000. The
borrowing interest cost is 4% and a taxi’s
value depreciates by 8% per year. We
assume that taxi’s prices increase by 2%
per year.
Optimum Number of Taxis
   The extra earnings
generated by                                                       Marginal Marginal

moving from 5
Taxis        Revenues Costs      Profits     Earnings Cost
1      200000 100000       100000      200000 100000

taxis to 6 taxis is            2
3
336358.6 200000
455901.4 300000
136358.6
155901.4
136358.6 100000
119542.8 100000

less than cost of              4
5
565685.4 400000
668740.3 500000
165685.4
168740.3
109784 100000
103054.9 100000

capital.                       6
7
766731.7 600000
860703.4 700000
166731.7
160703.4
97991.42 100000
93971.69 100000
8    951365.7 800000     151365.7    90662.28 100000
   Maximum profits                9     1039230 900000     139230.5    87864.79 100000

occurs where
10     1124683 1000000    124682.7    85452.17 100000

marginal cost
equals marginal
earnings.
Optimal Capital: Example

   Solve for Optimal Level of Capital
\$150, 000
PMPK      1            (.04  .08  .02) \$1, 000, 000  ck P K
4
N

\$150, 000 14 *
 N  N *  1.54  5.0625
\$100, 000
PMPK
P K , NEW

ck

K
K*
MPK & Optimal Capital
Q: Why does MPK         Q: What shifts the
slope down.             MPK curve.
A: Diminishing          A: Changes in
returns to capital.     productivity of
unit of capital         increase in
generates less          workforce or
at a given              make capital more
workforce and           productive and
technology level.       shift MPK curve
out.
PMPK '
PMPK
P K , NEW
P K , NEW

ck

K
K*         K**
PMPK
P K , NEW

ck’

ck

K
K**         K*
Investment Volatility
   The stock of capital may not be
cycle.
   Capital stock is much larger than the flow
of new investment in a given year,
perhaps 10-15 times as large.
   A 1% reduction in optimal capital stock
will require a 10% reduction in
investment.
Tax Rates
   Corporations frequently must pay taxes
on earnings. Define taxrate, .
   Corporations also receive deductions for
costs of capital Define deduction rates =
(s1, s2, s3, ….)
   Maximize after-tax profits implies that
after-tax marginal product of capital =
after-tax cost of capital.

(1   ) PMPKt  (1  s1 )i  (1  s2 )  (1  s3 ) g P
K
Pt K , NEW
Which cost of capital?

   Which interest rates should we use
to calculate the cost of capital.
   This depends on several things
including the risk of the investment
project & flexibility and duration.
   If capital project is risky, we might
apply a risk premium (i.e. use the
interest rate on a risky bond).
Duration & Flexibility

   If we must own capital for many
periods before resale, we might
want to equalize average marginal
product of capital during the period
to a long term interest rate plus
average depreciation less change in
the price of capital.
Real Estate
   Real Estate is an important type of
physical capital investment and a
special type of financial investment.
   Uniqueness- Each location of property
is unique, making it harder to value.
   Illiquid – Harder to sell than paper
financial assets, longer lead times for
building real property.
   Large share of the wealth of middle
income households.
Rental Yields & Cost of Capital
 Payoff to owning       Capital cost of real
property is rent, R.    estate includes
 Define rental yield,      Interest rate, i

y, as the ratio of        Depreciation/Maint

rent to property           enance Costs δ
 Property Taxes: τ
prices, PPE.
 Other Costs: c
PMPK RE  R                                   RE
P
 Capital Gains, g
R
y  RE
P
ck  (i    c    g ) P RE
RE                   P RE
Cost of Capital Theory & Real Estate

   Constructing New Buildings has long
lead times. For a fixed stock of
buildings we can use cost of capital
theory (y = ckRE) to derive prices as
a function of rents.
R         1               1
y  ck  RE  P 
RE
R                        R
P         ck    (i    c    g )
P RE
Ceteris Paribas
Curve PRE
R↑          →     ↑
i↑          ←     ↓
δ↑          ←     ↓
τ↑          ←     ↓
c↑          ←     ↓
g   P RE
↑ →    ↑
Real Estate Pricing

1
R
ck
45%

PRE
P*
Expectations

   A key determinant of the price of
real estate is the expected capital
gain.
   Expected deflation explains why
rental yields are so much higher in
HK than mortgage rates.
   Waves of optimism and pessimism
in property prices.
HK Property Prices
HK: Property Capital Value: Residential: Medium
HKD/Sq ft

7000

6500

6000

5500

5000

4500

4000

3500

3000

2500

2000

1500
Dec-1988   Dec-1990   Dec-1992      Dec-1994       Dec-1996        Dec-1998   Dec-2000   Dec-2002
q theory & Corporate Investment
   A benchmark theory of corporate investment is
that investment is a function of a quantity q.
   The measure of q for a firm is
Market Value of Firm
          q
Replacement Cost of Capital
   The market value of a publicly listed firm without
debt is market capitalization (stock price * shares
outstanding).
   The market value of a publicly listed firm with
debt is the market capitalization plus value of
debt (i.e. the cost of owning the firm lock, stock
and barrel).
q theory
   If value of firm is greater than the cost of
capital (q > 1) than the value of capital
inside the firm is greater than the value of
capital outside the firm.
   If q > 1, firm should have positive net
investment.
   If q = 1, firm should have zero net investment.
   If q < 1, firm should have negative net
investment.
q as Cost of Capital Theory
   We might think of q theory as similar to
cost of capital theory for firms that get
financing through the stock market.
   Owners of equity have a claim to the
profits of the firm. They might require a
certain amount of profits relative to what
Profits
they pay for the stock.   ck 
Market Capitalization

   A firm generates a certain amount of
Profits
profits per unit of capital     MPK 
Price of Capital

Profits
Price of Capital
Market Capitalization                                 MPK
q                                   Profits

Price of Capital                                      ck
Market Capitalization
Investment & the Stock Market
   Q theory suggests that a      Why?
rise in stock market              Many firms change their
theories could be                  capital stock
thought of as a decline            infrequently. Short-
in the cost of raising             term fluctuations in
funds through equity.              stock market may have
little effect.
   Empirically, q theory
   Stock market bubbles
seems to do a poor job             may keep stock prices
of explaining                      from reflecting a
connections between the            realistic assessment of
stock market and                   value of corporate
investment.                        capital.
   Firms may be limited in
ability to raise funds in
stock market.
Corporate Finance
   Two kinds of Finance
   External Finance – Funds for investment raised
through loans or issuing securities.
   Internal Finance – Funds for investment raised
through retaining profits instead of paying
dividends.
   Benchmark M-M Theory says investment
decisions and firm value should not
depend on sources of financing.
Requirements:
   No distortionary taxation
   Perfect financial markets with perfect
information.
Reality

   Internal Funds are cheaper form of
financing than external funds.
   Much of corporate financing is through
internal finance.
   Investment is more strongly affected
by cash flow than q.
   Cost of capital depends on collateral
value that firms can pay if they
default on loans or bonds.
Credit Cycles: Real Estate
   Much of corporate collateral is real
estate.
   Real estate is good collateral
because it cannot be easily moved
and its value is relatively easy for
outsiders to derive.
   Fluctuations in property price have
effects on cost of capital and
investment.

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