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Stock market and the
Economy: introduction
The purpose of this introduction is two-fold:
i.
Introduce the concept of Expected Present
Discounted Value, which is the foundation for
IS/LM extensions;
ii.
Distinguish between nominal & real interest
rates and consider the impact that money growth
has on these variables by extending the IS/LM
model.
Expected Present Discounted
Values

The expected present value of a sequence of future
payments is the value today of this expected sequence of
payments.

In order to decide whether I make an investment, I
compute the value today of the expected returns
(benefits) of the investment and compare with the cost of
investing.

If this value exceeds costs then I make the investment


If the one-year nominal interest rate is it lending $1
today yields $(1+ it) next year
Hence $1 next year is worth
today
1
$
1
1  it
1  it
is the discount factor which is the present
discounted value of $1 next year.
 The discount factor lies between zero and one since
the discount rate (the nominal interest rate) it > 0
 The higher the discount rate the lower the discount
factor


Therefore $1 in two years time is worth
$(1+it)(1+it+1)
and thus $1 two years from today must be worth
1
$
(1  it )(1  it 1 )
•
The present discounted value of a sequence of payments,
or value in today’s dollars equals:
1
1
$Vt  $ zt 
$ zt 1 
$ zt  2    
(1  it )
(1  it )(1  it 1 )
where $zt is today’s payment; $zt+1 is the payment next year
etc…

If $z t= $zt+1 =…..= $zt+n then


1
1
Geometric series
$Vt  $ zt 1 
  
n 1 
(1  i ) 
 (1  i )
thus
1  [1 / (1  i ) ]
$Vt  $zt
1  [1 / (1  i )]
n

We are interested in computing the expected
present discounted value when future payments and
interest rates are uncertain
1
1
e
e
$Vt  $ zt 
$ z t 1 
$
z
t2    
e
(1  it )
(1  it )(1  i t 1 )

This equation implies:
1. Present value depends positively on today’s actual
payment and expected future payments
2. Present value depends negatively on current and
expected future interest rates
La formazione del prezzo di
un’attività finanziaria:



il problema della allocazione delle risorse tra
consumo e risparmio (investimento) per un
individuo
obiettivo dell’individuo è distribuire il proprio
consumo nel tempo in modo ottimale
(massimizzando la propria utilità)
a tal fine l’individuo effettua transizioni in
attività reali e finanziarie
Problema del consumatore:
beneficio marginale di consumare un euro in
più oggi
=
beneficio marginale di consumare domani un
euro investito oggi in qualche attività
(confrontare in ogni periodo l’utilità marginale che
deriva da una unità di consumo aggiuntiva oggi con
quella derivante dalla rinuncia a consumare quella
unità aggiuntiva oggi per consumarla domani)
Soluzione del problema di
massimizzazione dell’utilità del
consumatore:
pt u' Ct  = 1+  E t pt 1  dt 1u' Ct 1
1
Perdita di utilità presente (sottraendo pt al
consumo odierno) = guadagno atteso di
utilità (consumando nel periodo successivo
i proventi attesi dell’attività finanziaria)
Tasso di preferenza
intertemporale ():
ct+1
ct+1
ct
ct
Curva di indifferenza intertemporale. Il valore assoluto della
pendenza di una curva di indifferenza intertemporale in un punto è
detto saggio marginale di preferenza intertemporale, cioè |ct+1/ct|.
Se la funzione di utilità del
consumatore è lineare
l’utilità marginale sarà costante, cioè
u'(Ct )  u' Ct 1   u' C
Se l’attività finanziaria oggetto
dell’investimento del consumatore è
un’azione, avremo:
pt = v t
dove vt indica il valore dell’azione, quindi
E t v t 1  dt 1 
vt =
1 +  
Da cosa dipende il valore di
un’azione?

Tasso di preferenza intertemporale
()

Rendimento futuro atteso dell’attività
Et (vt+1+ dt+1)
La determinazione del coefficiente
di attualizzazione
Ipotesi: le attività più rischiose dovranno
scontare i flussi di reddito attesi a tassi
maggiori rispetto a quelli relativi ad
attività meno volatili
Coefficiente di attualizzazione
():
  irf  
dove
irf = tasso di interesse risk-free
r = premio per il rischio dell’attività
The components of an interest
rate: the risk-free rate

The risk-free rate (denoted as irf) is approximately the yield
on short-term Treasury bills
– Includes the pure rate and an allowance for inflation


Viewed as a conceptual floor for the structure of interest
rates
The Inflation Adjustment
– Inflation refers to a general increase in prices
– Refers to the fact that, if prices rise, $100 at the beginning of the
year will not buy as much at the end of the year
– If you loaned someone $100 at the beginning of the year, you need
to be compensated for what you expect inflation to be during the
year

Interest rates include estimates of average annual inflation over loan
periods
The components of an interest
rate: risk premiums

Default risk: refers to the chance that the lender
will not receive the full amount of principal and
interest payments agreed upon;

Liquidity risk: refers to the extra interest
demanded by lenders as compensation for
bearing liquidity risk (associated with being unable to
sell the bond of an little known issuer);

Maturity risk: long-term bond prices change
more with interest rate swings than short-term
bond price. Gives rise to maturity risk (investors
demand a maturity risk premium ranging from 0% to 2%
or more for long-term issues).
Incorporating Expectations


Basic IS/LM model (& AD/AS extension) no
expectations are made in the goods and money
markets.
Core of modern macroeconomics is based on the
foundation that different agents in the economy
form expectations about future events.
e.g.
i.
ii.
Consumers have expectations about future
income;
Firms have expectations about future sales &
profits.
Stock market and the economy:
outline



Distinguish between nominal and real interest rates (IS
curve);
Consider the role of expectations in financial markets in
the determination of bond and stock prices (LM curve);
Consider the role of expectations in consumption and
investment decisions (IS curve)
Therefore our aim is to modify the basic IS/LM model (&
AD/AS extension) to examine how fluctuations in
economic conditions and macroeconomic policy may
account for movements in the stock market.
Nominal vs Real Interest
Rates





Nominal Interest Rate is the interest rate
expressed in units of money (it)
It tells us how much money we have to pay in the
future in exchange for having one more unit of
money today (1+it )
Real Interest Rate is the interest rate expressed in
terms of a basket of goods (rt)
It tells us how many goods we have to give up in
the future in exchange for having one more basket
of goods today (1+it )
The Real Interest Rate is important since agents
consume goods and not money!
Deriving the Real Interest
Rate
Borrow money today to buy a good of price Pt then
you have to repay (1+it ) Pt next year
 In terms of the good you have to repay
(1 it )Pt
Pe
t

1
where Pet+1 denotes the price expectation of the good
next year.


Therefore it follows that the one-year real interest
rate is
(1 it )Pt
1 rt 
Pe
t 1
 Expected inflation is defined as
 et
P e t 1  Pt

Pt
 Therefore (1  r )  1  it
t
1  et
 Note that: (1 it )  (1 i  e ) provided πet & it are small
t
t
(1  te )
 Thus the real interest rate is (approximately) equal to the
nominal interest rate minus the expected rate of inflation.
rt  it   e t
Fisher equation
rt  it  

e
t
Implications of the Fisher Equation:
 te  0; it  rt
 te  0; it  rt
 te  it ; rt  0
i ; te, rt
Nominal and Real Interest
Rates in the United States
Since 1978
Nominal and Real OneYear T-bill Rates in the
United States, 19782001
While the nominal
interest rate has
declined considerably
since the early 1980s,
the real interest rate is
actually higher in 2001
than it was then.
Nominal & Real Interest Rates & the
IS/LM Model

So far you have encountered the following version of the
IS curve:
Y=C(Y-T)+I(Y,i)+G
•
Here investment depends on the nominal interest rate.
•
However since firms produce goods, their investment
decision should depend on how many goods they have to
repay (and not how much money).
•
Therefore investment spending depends on the real interest
rate and the modified IS curve is
Y=C(Y-T)+I(Y,r)+G
Modified IS/LM model
IS: Y = C(Y-T)+I(Y,i-πe)+G
LM: M/P = YL(i)
 Money market equilibrium (LM curve) still
depends on the nominal interest rate, since the
opportunity cost of holding money is the rate of
return from holding bonds i
 Goods market equilibrium now also depends on πe
Equilibrium Output and
Interest Rates
The equilibrium level of
output and the
equilibrium nominal
interest rate are given
by the intersection of the
IS curve and the LM
curve. The real interest
rate equals the nominal
interest rate minus
expected inflation.
If r = i - e, then ∆r = ∆ i - ∆ e, and if e is
constant then ∆ e = 0 and ∆r = ∆ i.
Assumptions: short-run





πe is fixed in the short-run.
Sticky-prices such that ∆M > ∆ P. Therefore an increase
in money supply leads to an increase in the real money
stock (M/P).
Phillips Curve: π – πe = -a(U – Un).
Increase in money supply results in a higher rate of
growth of nominal money and an increase in the real
money stock.
To maintain money market equilibrium nominal interest
rate must fall (LM curve shifts to the right).
The Short-run Effects of
an Increase in Money
Growth
An increase in money
growth increases the
real money stock in the
short run. This increase
in real money leads to
an increase in output
and a decrease in both
the nominal and the real
interest rate.




Output increases as the nominal interest rate falls.
IS curve does not shift since πe is fixed.
The increase in output results in a fall in
unemployment below its natural rate. From the
Phillips Curve this implies that inflation increases.
The Fisher equation isr  i   e t
t
t

Thus the real interest rate must fall with the
nominal interest rate.

Thus higher money growth leads to lower nominal
interest rates and lower real interest rates in the
short-run.
Medium-Run






πe = π (expected inflation = actual inflation)
Phillips Curve: π – πe = 0 = - a(U – Un). Thus,
unemployment remains at is natural rate.
Yn  C(Yn  T )  I (Yn , r )  G
Output remains at its natural rate
and thus the real interest rate is unaffected.
The Quantity Theory of Money holds such that π
= gm (inflation = money growth rate) assuming for
simplicity that the rate of growth of output equals
zero.
Fisher equation: i = r + πe = r + π = r + gm
In the medium-run, an increase in money growth
leads to an equal increase in the nominal interest
rate. The real interest rate is unchanged.
Short-run to medium-run
adjustment
Short-run: gm, i, r
r < rn  Y > Yn  u < un  π 

Over-time:
π > gm  gm - π < 0  i 

Medium-run:
r = rn; Y=Yn; U=Un; π = gm; i = rn + gm

Fisher Hypothesis

To summarize, in the medium run money
growth does not affect the real interest rate, but
the nominal interest rate increases one for one
with inflation. This result is known as the
Fisher effect, or the Fisher Hypothesis.

This hypothesis is important for two reasons.
1. In offers a testable theory in explaining changes in
interest rates.
2. The Fisher Hypothesis supports the Quantity Theory
of Money, i.e. money is neutral in the medium run.
Evidence on the Fisher
Hypothesis

To see if increases in inflation lead to one-forone increases in nominal interest rates,
economists look at:
1. Nominal interest rates and inflation across countries.
The evidence of the early 1990s finds substantial
support for the Fisher hypothesis
2. Swings in inflation, which should eventually be
reflected in similar swings in the nominal interest
rate. Again, the data appears to fit the hypothesis
quite well

This offers further support that monetary policy
should primarily be used to maintain low and
stable inflation
Evidence on the Fisher
Hypothesis
Nominal Interest Rates
and Inflation Across
Latin America in the
Early 1990s
Roughly half of the
points are above the
line, the other half
below. This is
evidence that a 1%
increase in inflation
should be reflected in a
1% increase in the
nominal interest rate.
Evidence on the Fisher
Hypothesis
The Three-Month
Treasury Bill Rate and
Inflation, 1927-2000
The increase in
inflation from the early
1960s to the early
1980s was associated
with an increase in the
nominal interest rate.
The decrease in
inflation since the mid1980s has been
associated with a
decrease in the
nominal interest rate.
Nominal vs Real Interest Rates,
and Present Values
1
1
e
e
$Vt  $ zt 
$ z t 1 
$
z
t2    
e
(1  it )
(1  it )(1  i t 1 )

If we want to compute the present value of a
sequence of real payments, can either use the
nominal sequence above and divide by Pt
$Vt
 Vt
Pt
or as before we can construct the sequence using rt
1
1
e
e
$Vt  $ zt 
$z t 
$
z
t 2    
e
(1  rt )
(1  rt )(1  r t 1 )

1.
2.
We have two formula’s for computing the
present value of sequence of payments:
Based on nominal interest rate which we will
use when considering the pricing of financial
assets. Why? Since bonds offer nominal returns.
Based on real interest rate which we will use in
considering consumption and investment
decisions. Why? Base expectations on future
real income.
Where do we go from here?




We will consider the role of expectations in
determining bond and stock prices using are
present value formula.
Derive the yield curve used to predict the future
short-term interest rate
Consider the relationship between economic
activity and stock prices
Speculative bubbles in the stock market