Download Chapter 14. Investment and asset prices

Chapter 14. Investment and
asset prices
ECON320
Prof Mike Kennedy
What is the relationship between asset
prices and investment
• The idea is simple:
– If the market price of an investment opportunity is greater than its
replacement costs, then there exists an incentive to invest – it becomes
profitable to invest
– We will see that prices embody information about the future
– This applies to non-residential and residential investment whether we use
the stock market or the price of a house as a measure of value
– The replacement cost of a firm’s assets equals the price at which the firm
can purchase new machinery etc.
– If the stock market price of the assets of a firm is greater than these costs
then there is an incentive to invest
– Note that the market price can be greater (or less) than replacement value
for a long time since it takes time for production to adjust
– Note as well that additions to the stocks of housing or capital will not
occur instantly as there are adjustment or installation costs, which likely
rise more than proportionally with the scale of the project
• We will examine the relationship between asset prices and investment
Changes in the real S&P tend to lead
the business cycle
5
Cycle GDP
15
4
∆Real S&P (right scale)
10
3
2
1
5
0
0
-1
-2
-5
-10
-3
-15
-4
-5
-20
The change in real S&P and the
cycle in total investment
20
15
10
5
0
-5
-10
-15
-20
∆Real S&P (right scale)
Cycle INV
The importance of asset prices:
Canadian non-residential investment and the real value of the S&P
12.5
5
Non-residential investment
S&P in real terms (right scale)
Correlation = 0.95
4.6
Elasticity = 1.01 SE = 0.14
4.4
11.5
4.2
4
11
3.8
3.6
10.5
3.4
3.2
10
3
Log of real A&P
Log of real non-residenntial investment
12
4.8
The importance of asset prices:
Canadian M&E investment and the real value of the S&P
11.5
5
Log of real investment in M&E
11
10.5
S&P in real terms (right scale)
4.8
Correlation = 0.95
4.6
Elasticity = 1.30 SE = 0.17
4.4
4.2
4
10
3.8
3.6
9.5
3.4
3.2
9
3
Log of real S&P
M&E
Changes in real house prices and the
business cycle
5
4
3
2
1
0
-1
-2
-3
-4
-5
Cycle GDP
∆Real house prices
The importance of asset prices:
Canadian residential investment and the real value of house prices
11.8
4.9
Residential structures
Real house prices (right scale)
4.7
Correlation = 0.92
Elasticity = 0.88 SE = 0.11
4.5
11.4
4.3
11.2
4.1
11
3.9
10.8
10.6
3.7
3.5
Log of real house prices
Log of real housing investment
11.6
US stock market over time
2000
50
1800
45
1600
Real price
40
1400
P/E10 or CAPE (right scale)
35
1200
30
1000
25
800
20
600
15
400
10
200
5
0
0
Source Shiller website CAPE = cyclically adjusted price-earnings ratio
The Canadian stock market since the
mid 1950s
140
120
100
Real stock…
80
60
40
20
0
Facts about the stock market
•
•
•
•
Volatility
Stock market capitalisation has risen
Ownership of stocks has become more widespread
Likely more extensive than the data suggest due to
pensions
• Its importance is also due to its influence on
investment and consumption and through these on
employment
• So what drives stock prices and how do these
prices affect investment?
The value of a firm and the fundamental stock price
• We will assume that firms want to maximize the
wealth of shareholders – which is equivalent to
maximizing the share price
• In doing so the firm will maximize the
consumption possibilities of shareholders
• This is the same as assuming that they maximize
profits
• Our theory of investment will be based on a
theory of the value of the firm – that is the price
of its stocks
The arbitrage condition
•
•
•
•
•
In equilibrium investors will be indifferent between investing in shares
and bonds
Investors know that share prices are much more volatile than those of
bonds and they will want to be compensated for this
The opportunity costs times the interest rate on safe bonds
The required return (compensating for risk, ε) is (r + ε)Vt
The arbitrage condition is then
(1)
•
The stock market will only be in equilibrium when this condition is met
The arbitrage condition, con’t
• We can re-arrange (1) to get
(2)
• This condition, important for what follows, must hold in all
subsequent periods
(3)
• By successive forward substitution we get:
The arbitrage condition, con’t
(6)
This is the fundamental share price: The market value of a firm
equals the present value of expected future dividends. Note that
we are assuming that r and ε are constant over time
How well do the facts fit the theory?
There should be a close correlation between profits (earnings) and dividends
5.00
ln(Real dividends)
4.50
ln(Real earnings)
4.00
3.50
3.00
2.50
2.00
1.50
1.00
Data from Shiller's website
How well do the facts fit the theory?
• The behaviour of the stock market differs form
the Efficient Market Theory (EMT)
• It turns out that stock are much more volatile
than suggested by the EMT possibly due to
fluctuations in:
– The expected change in dividends (∆Det)
– The real rate of interest (rt)
– The required risk premium (εt)
The risk premium (the appetite for risk as measured
by implied volatilities) seems to vary a lot
70
Peak of financial crisis
60
VIX
50
40
30
20
10
0
Sep 11 2002
terrorist attack
NASDAQ
peaks
Euro area
debt crisis
Some reasons why equation (6) is not
that bad of a representation
• It says nothing about rationality
• The theory in (6) is compatible with the notion
that investors frequently revise their forecasts
• The possibility of investors becoming overly
optimistic is not excluded
• It only assumes that the expected return on
shares is systematically related to the return
on bonds, which seems to be the case over
longer periods of time
Our strategy for deriving an investment equation
• Look again at the equation for the current market value of the firm
• We will assume that the managers of the firm want to maximize
the current value of the firm (Vt)
• We will proceed by finding relationships for both expected
dividends and the expected value of the firm in the next period
(the numerator in the above relationship) that are in some way
related to investment spending
• The managers will then choose a level of investment spending that
maximizes Vt – this will give us a simple investment function
• We will then use this simple investment function to examine how
investment responds to changes in real interest rates, risk, sales
and business confidence
• We will see that everything will work through equity prices


Stock prices, investment and Tobin’s “q”
• As noted above, the wealth of owners today (Vt) depends on
e
Dte  Vt1
as well as the discount factor
• The question is: What level of It will maximise Vt?
• We now introduce the following variable
qt  Vt / Kt
where we have assumed that the price of capital is unity so
that Kt = the replacement value of the capital stock
• The firm is assumed to inform markets about its investment
plans, however shareholders’ expected q is assumed to
e
remain unchanged q t1
 q t which implies that
• This is one part of Vt above, the other part being expected
dividends 
Stock prices, investment and Tobin’s “q” con’t
• When thinking about Det, the other determinant of Vt, note that a
firm’s profits consist of dividends that it will pay out plus its
retained earnings, which it can invest
– This is called the uses of profits, the sources being sales minus costs
• Assume that the firm finances its investment spending from
retained earnings and that there are adjustments cost associated
with installing new capital
• The total cost of investment is then It + c(It), where c(It) is the
installation cost function is to be defined
• Expected dividends in this period are (by definition) expected
profits (Πet) less these expenditures on investment
Stock prices, investment and Tobin’s “q” con’t
• Installation cost are assumed to rise more than one-for-one with
investment and we capture this by
a 2
c(I t )  I t
2
• The marginal installation cost (dc/dIt = aIt) rises with investment
• Going back to Vet +1 we can use the capital accumulation identity
Kt+1 = Kt + It to eliminate Kt+1 and which gets current
investment into the relationship
e
Vt1
 qt K t1  q t (I t  K t )

Stock prices, investment and Tobin’s “q” con’t
• Return to the question of maximizing the market value of stocks and use
the information we have on both expected dividends and the expected
value of the firm in the next period as they are related to investment
• The firm now chooses It so as to maximize Vt which is given by ∂Vt/∂It = 0.
• Recalling that dc/dIt = aIt we get:
•
qt 1
It 
a
The higher the value of qt the further the firm can raise investment, while
the higher are installation costs the lower is investment
Stock prices, investment and Tobin’s “q” con’t
Stock prices, investment and Tobin’s “q” con’t
• Note that the previous figure is consistent with the simple graphs shown
earlier in that there is a positive relationship between investment
spending and stock prices
• In the case where the firm issues new shares we simply subtract these
costs from the value of shares held by existing shareholders
• It is optimal for investors to let the firm expand the capital stock (invest)
until the expected marginal gain on their shares is driven down to zero,
that is qt – (1 + dc/dIt) = 0, which is the investment function
– The profit maximizing firm will invest up to the point where the rise in the
stock market value of the firm equals acquisition and installation costs
– The level of investment depends on the value that markets place on the firm
– Investment will be sensitive to the variables the changes qt
• Going forward we will examine how qt responds to various economic
variables and how this affects investment spending
The role of interest rates and risk and their
effect on q
• Interest rates will be negatively correlated with investment and in
the q theory it works through share prices
• To show this simply, suppose that dividends are expected to stay
constant, then
• Which simplifies to
• Since Vt = qtKt it follows that qt will be negatively related to both
interest rates and risk
The role of profits and sales and how
they affect q
• To see the effect of profits and sales we start with the definition of
qt given on the previous slide
• Dividends are assumed to be a fraction (θ) of profits which implies
that the numerator in the above is Dte / K t   ( t / K t )
• The term in brackets is the profit rate (profits to capital) of the firm
• Based on a Cobb-Douglas production function the profit rate would

be
θ(Πt/ Kt) = αY/K
• Higher sales generates higher profits and a rise in qt and in
investment
The role of confidence and how it affects q
• Firms would also look at other variables than
the profit rate – θ(Πt/Kt) – in deciding how
much to invest
• These could be anything that might have a
bearing on expected future sales
• One measure that captures these hard-tomeasure effects would be business confidence
(E) which is shown for a few countries on the
following graph
Business confidence can also affect investment
spending as we saw in the great recession
103
102
101
100
99
98
Recession
United Kingdom
97
United States
Euro area (18 countries)
96
OECD - Total
95
Source OECD
Summing up: The investment function
• Recall again the investment equation
q t 1 1 
It 
  (q t 1) where
a 
a

• In a general sense we can think of the Dte/Kt as a function of
sales and confidence g(Yt/Kt, Et) where both the first and
second derivatives are positive
• A rise in Y will increase expected dividends while an increase
in Kt will lower them
• An increase in business confidence will also raise expected
dividends to the extent that such measures contain additional
information regarding the outlook for activity
• We can substitute this into our definition of qt to get
The investment function, con’t
• Writing this in a general format we get
• The general investment function has the following properties:
– An increase in sales (Y) or confidence (E) will raise investment spending
by raising q through its effect on dividends [Dte/Kt = g(Yt/Kt, Et)] and
possibly through a fall in risk (ε)
– An increase in the current level of capital reduces investment since it
reduces Dte/Kt
– A rise in the interest rate and/or the risk premium discourages
investment by lowering qt – the rate at which future dividends are
discounted will have risen
A q-theory of housing investment
• The model will be a special case of the q-theory
adapted to fit the housing market
• A theory of house price determination will also
be developed
• We will start with the production of housing (the
supply) and then go on to develop a demand for
housing equation based on consumers
maximizing behaviour
• Of note is that the stock of housing wealth is an
important determinate of household wealth and
therefore consumption

The production and pricing of new housing
• We will follow a similar qt-strategy for housing investment –
that is, we will derive a measure of qt for this sector
• We start with the production side of the construction sector
which will give us the supply of housing
I H  A X , 0   1
• New housing is constructed with a composite input (Xt) and
the production function exhibits diminishing returns to scale
(β < 1)
• The composite input uses labour (L) and building materials
(Q) in fixed proportions
L  aX , Q  bX



The production and pricing of new housing con’t
• Given a wage rate (W) and a price for materials (pQ) the price of the
composite index, X, is equal to
P  aW  bP Q
• Before setting up the definition of profits, note that, based on the
production function, we can write X as*:
I H  1 H 1/  1 (1  )/ 
X    I   
A 
 A  A
• Profits in the construction industry (Π = pHIH – PX) can be written as
6 4 4Production
4 7 4 cost
4 48
Sales
(1  )/  
67 8
1


P
1/

1
  p H I H  PX  p H I H   I H   

A 
A A

* See final slide for a way to do this easily
The production and pricing of new housing con’t
• The first order condition to maximize construction profits
(∂Π/∂IH = 0) yields
H  /(1  )

p
I H  k   
,
 P 

k    /(1  ) A1/(1  )
where k is a positive constant
• For housing, Tobin’s q is given by (pH/P) and housing
investment will rise proportionally with it
Looking at the demand of housing and the role
of interest rates and income
• We need to determine the demand for housing, which will
allow us to introduce interest rates and income
• After buying a house, a consumer must maintain its value by
doing maintenance and repairs (δ).
• If the house was completely financed by debt then the total
cost of housing consumption will be (r + δ)pHH where H is the
stock of housing
• The cost of maintaining the original value of the house will be
smaller (larger) if the prices are expected to rise (fall), which
ˆ  ge
gives us a repair function   

The role of interest rates and income con’t
• Ignoring saving, the consumer’s budget constraint is
where C is non-durable consumption and Y is income
• Utility is Cobb-Douglas and is given by
• Using the definition for C from the budget constraint we have
• The first order condition for a maximum is given by the usual
condition dU/dH = 0



The steps taken to maximize utility
• To do this simply, start with the definition of consumption
C  Y  (r   ) p H H
• And note that
dC
 (r   )p H
dH
• Next look at the utility function U  H C1 and taking the total
derivative wrt to H we get
6 4U7/H48
6 44U7/C4 48
dU
1 1
  dC
 H C  (1  )H C
 H 1C1  (r   ) p H (1  )H C 
dH
dH
• Which is equivalent to equation (25) in the text
• The utility maximization condition then yields the MRS
between housing and non-durable consumption as
U /H
 (r  ) pH
U /C


Deriving the demand curve for housing
• We can write out the maximization condition as
(r   ) p H U /C  U /H
• Which then becomes
(r   ) p H (1  )H C   H 1C1
• We can simplify the above by substituting in the definition of
utility U = HηC(1-η)
U
U
(r   ) p (1 )   
C
H
H
(1  )(r   ) p H H  C   (Y  (r   ) p H H )
• Now bring over to the left-hand side all the terms associated
with H which yields
(1  )[(r   ) p H H ]  [(r   ) p H H ]  Y

The demand curve for housing
• The last equation on the previous slide simplifies to equation
(28) in the text
Y
H is the user cost of housing
d
where
(r
+
δ)p
H 
(r   )p H
• Housing demand varies positively with income and negatively
with the user cost and house prices
• Note that even if the purchase of a house was out of saving
(not borrowing) the user cost still matters
– In that case the r represents money that could have been earned on
bonds and the homeowner is still responsible for maintenance
• The relationship between the housing stock and its price in
the short-run (when the stock of housing is fixed) is
Short-run equilibrium in the housing market
Housing stocks are slow to respond to demand
changes implying price have to adjust
2.5
Real house prices (average growth rate)
2
Housing stock (average growth rate)
1.5
1
0.5
0
-0.5
1986-91
Growth rates are 5-year averages
1991-96
1996-01
2001-06
2006-11
The price of housing
• From the figure on slide 42 we see that the larger is the stock
of housing, the lower is the price
• From slides 6 and 7 we know that house prices fluctuate a lot
• The figure, as well as the equation for the short-run level of
house prices (when the stock of housing is fixed), suggest that
these fluctuations are due to changes in income and interest
rates
• There is also an expectations channel, which works through
ˆ  ge
the depreciation term   
• Expected capital gains and losses will impact current house
prices

House prices seem to respond weakly to real
interest rates
40
35
30
25
Change in hosue prices
20
15
10
5
0
-6.00
-4.00
-2.00
0.00
2.00
-5
-10
-15
Change in real interest rates
4.00
6.00
8.00

The housing investment function
• In slide 36 we derived the investment function
H  /(1  )

p
I H  k   
 P 
• Substituting into this equation the price relationship we
derived from our demand equation gives
• In a general format we have
Developing a dynamic model of housing
• We start with the price equation
• Next we need the equation for housing investment (IH)
H  /(1  )

p
H
I  k   
 P 
• We can complete the model with the relationship between
the stock of housing the next period and current investment


Developing a dynamic model of housing, con’t
• In the short run, the stock of housing is fixed and assuming no change
in income or the user cost of housing, the price of housing (pH) is
determined
• Given pH and a value for P (the price of the composite input) the
second equation determines the level of housing investment (IH)
• Now that we have IH, the third equation determines the new stock of
housing for the next period (Ht +1)
• The new higher (lower) housing stock will give will give a new pH (from
the first equation) which will feed into the investment equation and so
on…
H
H
ptH  I tH  H t1  pt1
 I t1
 H t2  ...
• The process comes to an end when Ht +n = Ht +n+1 = H which, from the
third equation, implies that IH = δH – investment is just sufficient to
cover depreciation and house prices stop changing


Simplifying the housing production function
• In slide 35, it was found helpful to simplify the production
function for housing investment
I H 1/ 
1
1/ 
X     I H 
A
 A 
1 ( 1) / 
 
A 
• We can use logs on just the 1/A part of the above to see how
this is done
1 1/   1 1  1
 1  1   1  1 
ln    ln    11ln   
ln   ln  
 A     A  A 
A    A  
• Taking the exponent of this expression we get
1 1/  1 (1 )/  1 
    
 
A 
A 
A 