16 Investment
Based on Sørensen and Whitta-Jacobsen: "Investment and asset prices"
More about asset pricing, the stock market, and why stock prices vary so much.
The q theory of business investment (Tobins q ): Investment is a positive function of the ratio of the market valuation of existing assets to the replacement
cost of those assets.
Stock prices
The arbitrage condition for capital market equilibrium:
e
(r + ") Vt = Dte + Vt+1
Vt
(1)
where Dte is dividends (at the end of a period), Vt is the actual market value
of shares in the …rm (at the start of a period), r is the real interest rate and "
is the risk premium. This is the same as
E (r) pt = yt+1 + pt+1
pt
using the notation from the previous lecture. The price of stocks/value of
shares is equal to the discounted value of the future payo¤.
We re-write the pricing equation (1) so that
Vt =
e
Dte + Vt+1
(2)
1+r+"
Since this condition must hold for all subsequent periods, future stock prices
will be determined the same way. So, inserting for Vt+1 =
gives
e +V e
Dt+1
t+2
1+r+"
e
e
Dt+1
Vt+2
Dte
Vt =
+
+
2
1 + r + " (1 + r + ")
(1 + r + ")2
and
e
e
e
Dt+2
Vt+3
Dt+1
Dte
+
+
+
Vt =
1 + r + " (1 + r + ")2 (1 + r + ")3 (1 + r + ")3
in (2)
so on until we end up with
e
e
Dt+1
Dt+2
Dte
Vt =
+
+
+ :::
2
3
1 + r + " (1 + r + ")
(1 + r + ")
1
X
Dse
Vt =
s t+1
s=t (1 + r + ")
where we have assumed that
lim
e
Vt+n
n!1 (1 + r
+ ")
n
=0
Equation (3) is similar to
2
pt = Et 4
1
X
s=t+1
in the previous lecture.
3
msys5
(3)
The stock price index in Norway 1914-2001
19.05.2010
26.01.2009
04.10.2007
21.06.2006
01.03.2005
11.11.2003
25.07.2002
28.03.2001
09.12.1999
25.08.1998
06.05.1997
11.01.1996
26.09.1994
14.06.1993
19.02.1992
25.10.1990
11.07.1989
21.03.1988
27.11.1986
12.08.1985
13.04.1984
03.01.1983
The stock price index in Norway 1983-2011
600
500
400
300
200
100
0
19.10.2010
16.04.2010
08.10.2009
02.04.2009
26.09.2008
27.03.2008
18.09.2007
13.03.2007
08.09.2006
02.03.2006
01.09.2005
24.02.2005
25.08.2004
18.02.2004
14.08.2003
06.02.2003
02.08.2002
24.01.2002
20.07.2001
11.01.2001
11.07.2000
03.01.2000
The stock price index in Norway 2000-2011
600
500
400
300
200
100
0
Why are stock prices so volatile?
Fluctuations in expected future dividends (De) - optimism/pessimism
Fluctuations in the real interest rate (r) - before 2001: linked to currency
speculation
Fluctuations in the risk premium (") - changing attitudes to risk
Also: overreacting to news, herd behavior, short-termism, bubbles (deviation
from fundamentals)
Adjustment costs
In a simple investment model, a change in any of the exogenous variables should
lead to an instantaneous adjustment of capital. But because capital is a state
variable (and thus adjusts sluggish), this is not quite what we observe in the
data. We need to introduce costs to the adjustment of capital.
For example, the costs of installing new capital or training new workers, socalled adjustment costs or installation costs:
c (I ) ; c (0) = 0; c0 > 0
where I = investment spending/retained pro…ts. This slows down the adjustment speed of capital. The investment theory described in the following is
known as Tobin’s (1969) q theory.
How do …rms …nance investment?
Retained pro…ts (internal funds)
Debt (short term or long term)
Issuing new equity
The pecking order of …nance: …rms use internal funds …rst, then debt, and
…nally equity as a last resort.
Tobin’s q theory
q is the ratio between the market value of the …rm and the replacement value
of the …rm’s capital stock:
Vt
or qtKt = Vt
qt =
Kt
where Kt is the real capital stock. Note that the price of capital is set equal
to 1 so the replacement value of capital is simply Kt.
For example, a low q (between 0 and 1) means that the cost to replace a …rm’s
assets is greater than the value of its stock. This implies that the stock is
undervalued. Conversely, a high q (greater than 1) implies that a …rm’s stock
is more expensive than the replacement cost of its assets, which implies that
the stock is overvalued.
Expected payo¤
Next period’s capital stock is equal to current capital stock plus investment
Kt+1 = Kt + It
(4)
So, if next period investment in known and next period’s q is assumed to be
e
the same, qt+1
= qt, then the expected value of the …rm is
e =q K
Vt+1
t t+1
(5)
Expected dividends are equal to expected pro…ts ( et) minus retained pro…ts
used for investment (It) with adjustment costs
Dte =
e
t
[It + c (It)]
(6)
The cost function
It seems reasonable that adjustment costs, or installation costs, rise more than
proportionately with investment spending. This can be represented by the
following cost function
a
c (It) = It2
(7)
2
where a is a positive constant. This implies that
@c
= aIt
@It
i.e. the marginal installation cost increases proportionately with the level of
investment.
Optimal investment
Inserting (4)-(7) into (2) we get
e
t
[It + c (It)] + qt (Kt + It)
1+r+"
The …rm will maximize Vt with repect to It, taking qt as given. The …rst order
condition yields
Vt =
qt
=
|{z}
expected capital gain tomorrow
@c
1+
| {z@It}
foregone dividend today
The …rst-order condition on investment says that the marginal cost of investment should equal the capital gain from holding one unit of capital. A rise in
q will therefore stimulate investment.
1
It = (qt
a
1)
Tobins q around the world 1999-2004
Investment and the interest rate
Conventional assumption: investment depends negatively on the real interest
rate. The q theory: investment depends positively on q . Use equation (3),
assume constant De and multiply both sides with (1 + r + ") :
#
"
1
1
1
+
+
+ ::: (1 + r + ")
2
3
1 + r + " (1 + r + ")
(1 + r + ")
#
"
1
1
+
+ :::
(1 + r + ") Vt = Dte + Dte
2
(1 + r + ") (1 + r + ")
(1 + r + ") Vt = Dte
Subtracting (3) from this result yields
(1 + r + ") Vt
Vt = Dte
Dte
Vt =
(r + ")
i.e. q depends negatively on r
Dte=Kt
qt =
(r + ")
(8)
Sørensen and Whitta-Jacobsen show in …gures that stock returns E (r) and
bond returns (r) seem to move together, and that investment tend to move
in line with stock prices, thus giving indirect evidence of a negative impact of
interest rates on investment.
According to equation (8), q will also vary inversely with the risk premium.
Investment and pro…t
Assume that shareholders expect the …rm to pay out a fraction
as dividends
Dte =
of its pro…ts
t
Equation (8) can then be rewritten as
qt =
( t=Kt)
(r + ")
where ( t=Kt) is the …rm’s current rate of pro…t (pro…t to capital ratio).
We would expect a positive relationship between the rate of pro…t and investment.
If output is given by a Cobb-Douglas production function
Y = BK L1
where L is labor input, and markets are competitive, then total pro…ts will
be equal to Y . In that case the rate of pro…t is Y =K which is directly
proportional to the output-capital ratio Y =K .
The more …rms can produce and sell on the basis of a given capital stock,
the higher their pro…t rate will be. However, expected pro…t will also depend
on expectations about future market conditions (Et), so that a more general
formulation will be
!
e
Yt
Dt
=g
; Et
Kt
Kt
where g 0 (Yt=Kt) > 0 and g 0 (E ) > 0.
The investment function
The …rst order condition for investment using the speci…c cost function (7) is
qt = 1 + aIt
Combined with the elaborations above this gives us the following investment
function
1
It =
a
2
Yt
g K
; Et
t
4
(r + ")
3
15
An increase in Yt or Et will stimulate investment by raising qt, through an
Dte
Yt
increase in the expected dividend ratio, K = g K
; Et . An increase in
t
t
the current capital stock reduces investment by driving down Dte=Kt, and an
increase in the interest rate r likewise discourages investment via a negative
impact on qt.