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bombolles especulatives. Jaume Ventura CREI, UPF & Barcelona GSE Bojos per l’economia ! Març 2014. An Economic Model of Asset Prices. Assumptions Time is a sequence of dates t=0,1,2,…,∞. - PowerPoint PPT Presentation
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AN ECONOMIC MODEL OF ASSET PRICES
Assumptions1. Time is a sequence of dates t=0,1,2,…,∞.2. There is a market with traders willing to borrow
and lend at the expected return of 1+r per period.
Question1. What is the price of an asset that delivers pay-
offs dt in date t?
2. Assume this asset is traded only in date 0.
...,, 321 ddd
What future payments does the asset promise?
What is the expected value of these future payments?
How much are traders willing to pay today for these future payments?
,...,...,, 21 nddd
,...,...,, 02010 ndEdEdE
,...)1(
,...,)1(
,1
022010
nn
r
dE
r
dE
r
dE
MARKET EQUILIBRIUM!!!
Let xn be today’s value of payment at time n. Then,
.
.
.
101 )1( dErx r
dEx
1
101
202 )1)(1( dErrx 2)1(20
2r
dEx
nn dErrrx 0)1)...(1)(1( nr
dEx nn
)1(0
How much are traders willing to pay today for these future payments?
How much are traders willing to pay today for the asset?
Thus, the price of the asset is
,...)1(
,...,)1(
,1
022010
nn
r
dE
r
dE
r
dE
...)1(
...)1(1
022010
nn
r
dE
r
dE
r
dE
1
00 )1(t
tt
r
dEp
o Asset prices are high when expected payments are high and interest rates are low.
1
00 )1(t
tt
r
dEp
MODIFYING OUR ECONOMIC MODEL OF ASSET PRICES
We have assumed so far that the asset is traded at time 0 only.
Assume from now on that the asset is traded in all periods.
Can the ability to resell the asset modify its value?
NEW MARKET EQUILIBRIUM!!!
Let pn be the price of the asset in date n. Then:
.
.
.
.
.Iterating forever…
1010)1( pEdErpo r
pE
r
dEp
111010
0
202010 )1( pEdErpE 220
22010
0 )1()1(1 r
pE
r
dE
r
dEp
10100 )1( nnn pEdErpE nn
nn
r
pE
r
dE
r
dE
r
dEp
)1()1(...
)1(100
22010
0
1
000 )1(
lim)1(t
nn
ntt
r
pE
r
dEp
Asset prices have a fundamental and a bubble component.
The bubble component is a pyramid scheme.
Self-fulfilling expectations play a crucial role in asset price fluctuations.
1
000 )1(
lim)1(t
nn
ntt
r
pE
r
dEp
Fundamental Bubble
CALCULATING FUNDAMENTAL AND BUBBLE COMPONENTS
1. Measure the cash-flows that US productive assets generate as capital income, net of taxes and investment.
2. Compute the expected present discounted value of these cash-flows by assuming –
a. The interest rate is constant for all time horizons (equal to the 1950-2010 period average); and
b. Out-of-sample cash-flows grow at a constant rate (equal to the 1950-2010 period average), and resort to perfect foresight for within-sample cash-flows.
BACK TO THEORY (WITHOUT EQUATIONS!!)
Two effects of bubbles on investments1. An increase in the size of bubbles today absorbs credit
and lowers investment – CROWDING-OUT EFFECT.2. An increase in the size of bubbles tomorrow provides
collateral and raises investment – CROWDING-IN EFFECT.
What effects dominates?1. If bubbles are not too large, the crowding-in effect
dominates and bubbles raise investment and growth.2. If bubbles become too large, the crowding-out effect
dominates and bubbles lower investment and growth.
POLICY IMPLICATIONS FOR CENTRAL BANKS
The behavior of the economy depends on self-fulfilling expectations and the bubble is sometimes too small and sometimes too large.
Central Banks can manage bubbles by taxing credit when the bubble is too large and subsidizing credit when the bubble is too low.
This policy raises welfare and has no fiscal cost.