Transcript Lecture 17: Taxation & Risky Assets and Mean

Last week saw consumer with wealth W,
chose to invest an amount x* when returns
were
rg in the good state with probability p
and rb in the bad state with probability 1-p.
• EU=pU(W+x* rg ) + (1-p)U(W+x* rb)
What is the effect of taxation on the amount
we choose to invest in the risky
investment?
Now only get (1-t)rg in the good state with
probability p
and (1-t)rb in the bad state with probability
1-p.
• EU=pU(W+x(1-t)rg) + (1-p)U(W+x (1-t)rb)
• EU=pU(W+x(1-t)rg) + (1-p)U(W+x (1-t)rb)
What are the FOC’s?
U g
U b
(1  t )rb
(1  t) rg  (1  p )
p
x
x
• EU=pU(W+x(1-t)rg) + (1-p)U(W+x (1-t)rb)
What are the FOC’s?
U g
U b
p
(1  t )rg  (1  p )
(1  t )rb  0
x
x
U g
U b
p
rg  (1  p )
rb  0
x
x
• EU=pU(W+x(1-t)rg) + (1-p)U(W+x (1-t)rb)
What are the FOC’s?
U g
U b
p
rg  (1  p )
rb  0
x
x
Like last case except that there is no guarantee
that x with taxes is the same as x*
Why? Payoff in good and bad states different.
• EU=pU(W+x(1-t)rg) + (1-p)U(W+x (1-t)rb)
What are the FOC’s?
U g
^
U b
p
[(W  x(1  t )rg ] rg  (1  p )
[(W  x(1  t )rb ] rb  0
x
x
^
Like last case except that there is no guarantee
that x with taxes is the same as x*
Why? Payoff in good and bad states different.
W+x[1-t]rs
^
• What is the relationship between x* and x
Usually think, if return on asset goes
down, want less of it. That is, might
^
think x*> x
To see why this is wrong set
U g
Then MRS between
the two states is:
x*
x
1 t
^
^
[W  x(1  t )rg ]
(1  p )rb
x

^
U b
prg
[W  x(1  t )rb ]
x
x*
x
1 t
^
To see this suppose set
U g
MRS between the two
states is:
^
[W  x(1  t )rg ]
x
^
U b
[W  x(1  t )rb ]
x
x*
x
1 t
^
To see this suppose set
MRS between the two
states is:
but since 0<t<1,
(1-t)<1
x*
x
 x*
1 t
^
So
U g
x*
[W 
(1  t )rg ]
x
1 t
U b
x*
[W 
(1  t )rb ]
x
1 t
U g
[W  x * rg ]
(1  p )rb
x

U b
p
r
g
[W  x * rb ]
x
Why does investment rise?
• When the good state occurs t is a tax.
• However, if losses can be offset against tax
then t is a subsidy when rb occurs
rb
(1-t)rb
(1-t)rg
rg
•So t reduces spread of returns and therefore
risk
•Only way can recreate original spread is to
invest more (i.e. x up)
Mean-Variance Analysis
Suppose W = £100 and bet £50
on flip of a coin
Probability
0.5
Outcomes
£50
£150
• EU=0.5U(50) + 0.5 U(100)
More outcomes => More complexity
• E.g W = £30, Bet £30 on throw of dice
• Prizes 1=£10, 2= £20, 3=£30, 4=£40,
5=£50,6=£60,
• each with probability 1/6
Probability
1/6
Outcomes
£10
£20 £30 £40 £50 £60
More outcomes => More complexity
• EU= 1/6 U(10)+ 1/6 U(20)+ 1/6 U(30)+
1/6 U(40)+ 1/6 U(50)+ 1/6 U(60)
Probability
1/6
Outcomes
£10
£20 £30 £40 £50 £60
More complex still: Probability of
returns on investment follows a
normal distribution
• With EU need to consider every possible
outcome and probability
• Too complex, need something simpler
Probability
Returns
Probability of returns on investment
follows a normal distribution
• If we can use some representative information
that would be simpler.
Probability
Returns
Probability of returns on investment
follows a normal distribution
Called Mean-Variance Analysis
U=U(m,s2)
Probability
Returns
Like returns to be high
Like Risk to be low
U=U(m,s2)
Return
U=U(,)
U2
U0
Risk
(measured by Variance s2
or Standard Deviation, s)
The Mean-Variance approach
says that consumers preferences
can be captured by using two
summary statistics of a
distribution :
• Mean
• Variance
Mean
m=p1w1+ p2w2+ p3w3+ p4w4+ …
…….+psws+ ...
Or in other words
S
m   ps ws
s1
Mean
S
m   ps ws
s1
Similarly Variance is
 s2=p1(w1-m)2 p2 (w2-m)2 +
………...p3 (w3-m)2 + p4 (w4-m)2 +
…
…….+ps (ws-m)2 + ...
Or in other words
S
s 2   p s (ws  m)2
s1
Variance
S
2
2
s   p s (ws  m)
s1
We like return- measured by the
Mean
We dislike risk - measured by the
Variance
2
U  U (m ,s )
U  U(,  )
Suppose we have two assets, one
risk-free and one risky, e.g. Stock
The return on the risk-free asset is rf and its
variance is sf2=0
The return on the risky asset is rs with
probability s , but on average it is rm and its
variance is sm2
If we had a portfolio composed of
x of the risky asset and (1-x) of
the risk-free asset, what would its
properties be?
Return
S
rx   p s( xrs  (1 x)r f )
s1
S
rx  x  p s( rs)  (1 x)r f
s1
rx  xrm  (1 x)r f
Variance
S
s 2x   p s ( xrs  (1 x)rf  rx )2
s1
S
s 2x   p s ( xrs  (1 x)r f  xrm  (1 x)rf )2
s1
S
s 2x   p s ( xrs  (1 x)r f  xrm  (1 x)r f )2
s1
S
2   p s ( xrs  xrm)2
sx
s1
S
2  x2  p s (rs  rm)2
sx
s1
2s 2
s2

x
x
m
Results
Return
rx  xrm  (1 x)r f
Variance:
2
s 2x  x2sm