Embed Size (px)
Citation preview
Main line: 1 A partial-equilibrium one-period m
odel 2 A general intertemporal equilibrium
model of the asset market, includes three models(model 1 is based on a constant interest rate assumption, model 2 is a no-riskless-asset case, model 3 is the general model).
Ⅰ A partial-equilibrium one-period model We follows the warrant pricing approa
ch used in Chapter 7, that is, investors choose among three assets: the warrant, the stock of the firm and a riskless asset to form optimal portfolios which maximize their expected utility.
Consider an economy made up of only one firm with current value , and there exists a “representative man” acts so as to maximize the expected utility of wealth at the end of a period of length, that is, …… ①
Define a random variable Z by and assume its probability distribution
is known at present, more importantly,
max ( )tE U V t ( )
( )
V tZ
V t
( , )P Z
V t
is independent of the particular structure of the firm, this is consistent with the MM(Modigliani-Miller) theorem.
Define as the current value of the i th type of security issued by the firm. The different types of securities are distinguishable by their terminal value
. For a debt issue(i=1), …②
( , )P Z
,iF V
,0iF VZ
1 ,0 min( , )F VZ B VZ
Because each of the securities appears separately in the market, so:
and Define , so we can rewrite ①
as a maximization under constraint: …③
1
,n
iV F V 1
,n
iVZ F VZ ,i iw F V V
1 1
,0max 1
,i
n ni
t i iw
i
F VZEU V w w
F V
The corresponding first-order conditions are:
This can be rewritten in terms of util-prob distributions Q as:
…④
1
,0 ,0
, ,
ni i
t ii i
F VZ F VZE U V w
F V F V
0 0
,0,0exp( )
, ,ji
i j
F VZF VZdQ dQ
F V F V
Where and is a new multiplier related to .
dQ is independent of the functions by the assumption that the value of the firm is independent of its capital structure, so ④ is a set of integral equations linear in the , and we can rewrite ④ as
0
,
,
U ZV dP ZdQ
U ZV dP Z
exp
iF
iF
…* Suppose the firm issues just one
type of security--equity, then
Substituting in ④, we have
1 1, , , ,0F V V and F VZ VZ
0
exp ,ZdQ Z
0
, exp ,0 ,i iF V F VZ dQ Z
From ④, we can see that the expected return on all securities in util-prob space must be equated. If U was linear, then dQ=dP and ④ would imply the result for the risk-neutral case. Hence, the util-prob distribution is the distribution of returns adjusted for risk.
Some examples Example 1: Firm issues two types of securities,
debt and equity with current value and
respectively. From ② and ④, we have ⑤:
1 ,F V
2 ,F V
/
1 0 /
/
0
, exp , ,
exp exp ,
B V
B V
B V
F V ZVdQ Z BdQ Z
B B ZV dQ Z
Suppose or for then as .
In the limit, the debt becomes riskless, so will be replaced by r. Another useful form of ⑤ is
Since in equilibrium
0 B V , 0dQ Z 0 Z B V 1 , expF V B 0B V
1 0 /
/
, exp , ,
exp ,
B V
B V
F V r ZVdQ Z ZV B dQ Z
V r ZV B dQ Z
1 2, ,V F V F V
So, . This is identical to the warrant pricing equation derived in Chapter 7.
This equation can also be derived directly from the terminal value of equity
in the same way as debt.
2 /, exp ,
B VF V r ZV B dQ Z
2 ,0 max 0,F VZ VZ B
Example 2: Firm’s capital structure made up from
three types of securities: debt, equity(N shares outstanding with current price per share of S, i.e. ), warrants (exercise price is ). Assume there are n warrants outstanding with current market value per warrant of W,
2 ,F V NS
S
i.e. . Because the warrant is a junior security to the debt, the current value of the debt will be the same as in the first example. The current value of the equity will be
…⑥
3 ,F V nW
/
2 /
/
, exp [ ,
, ]
V
B V
V
F V r ZV B dQ Z
NZV nS B dQ Z
n N
Where . Rewrite ⑥ as
…⑦
NS B
2 /
/
1 /
, exp [ ,
1 , ]
exp, ,
B V
V
V
F V r ZV B dQ Z
N n NnS ZV B dQ Z
n N N
N r nV F V nS ZV B dQ Z
n N N
In equilibrium, . So from ⑦ we have
⑧ If we define normalized price of the
firm as
3 1 2, , ,F V V F V F V
3 /, exp ,
V
nF V r ZV dQ Z
n N
V n NVy
NS B n N
And define the normalized price of a warrant as , then ⑧ can be rewritten as
which is of the same form as equation (7.24).
3 ,F V
wn n N
/
, exp 1 ,V
w y r Zy dQ Z
Example 3: Firm’s capital structure contains two
securities:convertible bonds with a total terminal claim on the firm of either B dollars or alternatively the bonds can be exchanged for a total of n shares of equity; and N shares of equity with current price per share of S dollars.
So, , and
Where is determined to be .
2 ,0 max 0,min ,F VZ VZ B NVZ n N
/
2 /
/
, exp ,
,
V
B V
V
F V r VZ B dQ Z
NVZdQ Z
n N
n N B n
/
1 2 0
/ /
, , exp [ ,
, , ]
B V
B V V
F V V F V r VZdQ Z
nBdQ Z VZ dQ Z
n N
By inspection of this equation, we have the well-known result that the value of a convertible bond is equal to its value as a straight bond plus a warrant with exercise price .
S B n
Example 4: A “dual” fund: it issues two types of
securities to finance its assets: namely, capital shares(equity) which are entitled to all the accumulated capital gains(in excess of the fixed terminal payment); and income-shares(a type of bond) which are entitled to all the ordinary
income in addition to a fixed terminal payment.
Let be the instantaneous fixed proportion of total asset value earned as ordinary income, V be the current asset value of the fund and Z the total return on the fund.
Let be the current value of the income shares with terminal claim on the fund of B dollars plus all interest and dividends earned, be the current value of the capital shares.
So, from definition, we have
1 ,F V
2 ,F V
2 ,0 max 0,expF VZ VZ B
And Where .
The current value of the capital shares can be less than the current net asset value of the capital shares, defined to
2 /, exp ,
VF V r VZ dQ Z
exp( )B
/
1 2 0
/ /
, , exp { ,
, 1 exp ,
V
V V
F V V F V r VZdQ Z
BdQ Z VZ dQ Z
be V-B, because
If , that is, then, .
2 /
/
, exp ,
exp , exp
V
V
F V r VZ dQ Z
r VZdQ Z V
exp V V B 1 expV B
2 ,F V V B
Ⅱ A general intertemporal equilibrium model Consider an economy with K
consumers –investors and n firms with current value .Each consumer acts so as to
Define , where is the number of shares and is the price per share at time t.
, 1, ,iV i n
0 0max , ,
kk k k k k kE U C s s ds B W T T
i i iN t P t V t iN t
iP t
Assume that expectations about the dynamics of the prices per share in the futures are the same for all investors and can be described by the stochastic differential equation:, 1,...,i
i i ii
dPdt dZ i n
P
Further assume that one of the n assets (the nth one) is an instantaneously riskless asset with instantaneous return :
For simplicity, we assume that and are functions only of .
r t
, ,dr f r t dt g r t dq
i i r t
From , divide both side by and substitute for ,then
The accumulation equation for the kth investor can be written as
…⑨
i i iN t P t V t i i i i i idV N dP dN P dP
i idP P
1i ii i i i i i
i i
dV dNdt dZ dt dZ
V N
iV
1
nk k k k ki
ii
dPdW w W y C dt
P
Where is his wage income and is the fraction of his wealth invested in the ith security.
So, his demand for the ith security can be written as
Where is the number of shares of the ith security demanded by investor k.
k k k ki i i id w W N P
kid
kiw
ky
kiN
Substituting for , we can
rewrite ⑨ as
From the budget constraint, and from ⑨, we have
i.e. the net value of shares purchased must equal the value of savings from wage income.
i idP P
1 1
n nk k k k k k k
i i i i idW w r r W dt w W dZ y C dt
1
nk k
i iW N P
1
nk k k
i i iy C dt dN P dP
According to Chapter 4 and 5, the necessary optimality conditions for an individual who acts to maximize his expected utility are
(10)
subject to .
3 2( , )
21 22
1
2
11 121 1 1
0 max ( , , ,
1
2
1)
2
k k
k k k k k
C w
mk k k k k k
i i
m m mk k k k k k k
i j ij ir i
U C t J W r t J f
J w r r W y C J g
J w w W J w W
, , ,k k k k k kJ W r T B W T
From (10), we can derive m+1 first-order conditions
(11) (12) Equation (12) can be solved explicitly
for the demand functions for each risky security as
1 10 , , ,k k k kU C t J W r t
1 11 121
0 , 1,...,m
k k k k ki j ij irJ r J w W J i m
1 1
, 1,...,m m
k k ki ij j ij jrd A v r H v i m
where the are the elements of the inverse of the instantaneous variance-covariance matrix of returns ,
and . Applying the Implicit Function
Theorem to (11), we have
ijv
ij
1 11k k kA J J 12 11
k k kH J J
1 11 0
0
kk
k
k kk
k
CA U U
W
C CH
r W
The aggregate demands are (13) If it is assumed that the asset market
is always in equilibrium, then ,where M is the total
value of all assets. So, from ⑨
(14)
iD
1 1 1
, 1,...,K m m
ki i ij j ij jrD d A v r H v i m
1 1
n n
i i iN P D M
1 1 1
1 1
n n Kk
i i i i i
n Kk ki
ii
dM N dP dN P dP dW
dPD y C dtP
Let be the price per “share” of the market portfolio and N be the number of “shares”, i.e. . Then,
N and are defined by
MP
MM NP
M M MdM NdP dN P dP
MP
1
1
( )
n
M i i
n
M M i i i
NdP N dP
dN P dP dN P dP
From and , then
And from (13), we get (15) Define .Divide equation
(14) by M and substitute for
1
nk k k
i i iy C dt dN P dP 1
K ki idN dN
1
Kk k
M MdN P dP y C dt
1
ni
M ii
dPNdP D
P
i i i iw N P M D M
i idP P
We can rewrite (15) as (16) And from this we can determine
1 1
m mM
j j j j jM
dPw r r dt w dZ
P
1
m
M j jw r r
1
, 1,...,m
iMiM j ij
M i
dPdPdt w dt i m
P P
2 M MM
M M
dP dPdt
P P
1
mM
Mr j jrM
dPdt dr w dt
P
From (13), we can solve for the yields on individual risky assets in matrix-vector form:
(17) Since in equilibrium, , it can
be rewritten in scalar form as
(18)
1r
Hr D
A A
i iD wM
1
, 1,...,m
i j ij ir iM ir
M H M Hr w i m
A A A A
Multiplying both side by and summing from 1 to m, we have
(19) Hence, from (18) and (19), if we
know the equilibrium prices, then the equilibrium expected yields of the risky assets and the market portfolio can be determined. The equilibrium interest rate can be determined from (11).
iw
2M M Mr
M Hr
A A
Model 1: A constant interest rate assumption
With a constant interest rate, the ratios of an investor’s demands for risk assets are the same for all investors. Hence, the “mutual-fund” or separation theorem holds, and all optimal portfolios can be represented as a linear combination of any two distinct efficient portfolios(we can choose them to be the market portfolio and the riskless asset).
By combining (18) and (19) we have (20) With a slightly different interpretation
of the variables, (20) is the equation for the Security Market Line of CAPM.
If it is assumed that the are constant over time, then from (16), is log-normally distributed.
2iM
i MM
r r
iw
MP
We can integrate the stochastic process for to get conditional on
where is a normal variate with zero mean and variance .
Similarly, we can integrate (16) to get
idP
21exp ;
2i i i i i iP t P Z t
i iP t P
;t
i itZ t dZ
21exp ;
2M M M M MP t P X t
where is a normal variate with zero mean and variance .
Define the variables
1
;t m
j j i MtX t w dZ
21
log ;2
ii i i i i
i
P tp t Z t
P t
21
log ;2
MM M M M
M
P tp t X t
P t
Then consider the ordinary least-squares regression , if Model 1 is the true specification, then the following must hold:
; ; is a normal variate with zero mean a
nd a covariance with the market return of zero.
ii i M i tp t r p t r
2iM
iM
2 21
2i i M i 1 221 ;it iM i iY t
;iY t
Reconsider the first example where firm’s capital structure consists of two securities: equity and debt, and it is assumed that the firm is enjoined from the issue or purchase of securities prior to the redemption date of the debt namely,
So, (21)
0idN dV dP
dt dZV P
Let be the current value of the debt with years until maturity and with redemption value at that time of B.
Let be the current value of equity and the dynamics of the return on equity can be written as
(22)
;D t
,F V
e e
dFdt dZ
F
(20)
Like every security in the economy, the equity of the firm must satisfy (20) in equilibrium, hence,
(23) By Ito’s lemma, subs
titute dV from (21), we get: (24)
2iM
i MM
r r
2e M
e MM
r r
21
2V VVdF F dV F d F dV
2 21
2 VV V VdF V F VF F dt VF dZ
Comparing (24) with (22), it must be that
(25) (26) And also, in equilibrium, (27) Substituting for and from (23) and
(27), we have the Fundamental Partial
e e
dFdt dZ
F
2 21
2 VV V VdF V F VF F dt VF dZ
2 21
2e VV V
e V
F V F VF F
F VF
2M
MM
r r
e
Differential Equation of Security Pricing
(28) subject to the boundary condition
The solution is (29) Z is a log-normally distributed
random variable with mean and variance of , and is the log-normal density function.
2 210
2 VV VV F rVF F rF
,0 max 0,F V V B
, exp ,B V
F V r VZ B d Z
exp r
2log Z d
(28) is the Fundamental Partial Differential Equation of Asset Pricing because all the securities in the firm’s capital structure must satisfy it. And each securities are distinguished by their terminal claims.
Equation (29) can be rewritten in general form as
(30) 0
, exp ,0 ,F V r F VZ d Z
Although (30) is a kind of discounted expected value formula, one should not infer that the expected return on F is r. From (23),(26) and (27), the expected return on F can be written as V
e
F Vr r
F
(28) was derived by Black and Scholes (1973) under the assumption of market equilibrium when pricing option contracts, but it actually holds without this assumption.
Consider a two-asset portfolio which contains the firm as one security and any one of the security in the firm’s capital structure as the other.
Let P be the price per unit of this portfolio, the fraction of the total portfolio’s value invested in the firm and the fraction in the particular security chosen from the firm’s capital structure.
So,
1
(1 ) e e e e
dP dV dFdt dZ
P V F
Suppose is chosen such that Then the portfolio will be perfectly hedged and the instantaneous return on the portfolio will be certain(equal to r),that is,
. So . Then, as was done previously, we can
arrived at (28). Nowhere was the market equilibrium assumption needed!
0e e
e e r ee r r
Remarks: Although the value of the firm follows a
simple dynamic process with constant parameters, the individual component securities follow more complex processes with changing expected returns and variances. Thus, in empirical examinations using a regression, if one were to use equity instead of firm values, systematic biases would be introduced.
ii i M i tp t r p t r
Model 2: The “no riskless asset” case
If there exists uncertain inflation and there are no future markets in consumption goods or other guaranteed “purchasing power” securities available, there will be no perfect hedge against future price changes, i.e. no riskless asset exists.
Follow the same procedure as in section 11.4, we can derive analogous equilibrium conditions, namely,
(31)
(32) The nth security must satisfy (31) in
equilibrium (33)
, 1,...,i iM
MG i n
A
2M M
MG
A
n Mn
MG
A
Solve and G in (32),(33) and substitute them into (31), we have
(34) In a similar fashion to the analysis in
Model 1, we get the Fundamental Partial Differential Equation for Security Pricing ,where
M A
2
2 2, 1,...,Mi Mn M Mi
i M nM Mn M Mn
i m
2 210
2 VV VV F VF F F
2 2M n Mn M M Mn
2iM
i MM
r r
If security n is a zero-beta security, i.e.
,then ,and (34) can be rewritten as
where .
0Mn n
1 , 1,...,i i M i n i m 2
i Mi M
Model 3: The general model In this model, the interest rate
varies stochastically over time. In section 11.4, we have (35)
(36)
, 1,...,i iM ir
M Hr i m
A A
2M M Mr
M Hr
A A
So, (37) Solve (36) and (37) for and , and
substitute them into (35), we have
(38) where and
m mM mr
M Hr
A A
M A H A
2
mr Mk Mm kr M kr Mr Mkk M mr r r
Q Q
2M mr Mr MmQ 1,..., 1k m
Theorem 11.1(Three “Fund” Theorem) Given n assets satisfying the conditions of t
he model in section 11.4, there exist three portfolios (“mutual funds”) constructed from these n assets such that all risk-averse individuals, who behave to maximize their expected utilities, will be indifferent from these three funds.Further, a possible choice for the three funds is the market portfolio, the riskless asset, and a portfolio which is (instantaneously)perfectly correlated with changes in the interest rate.
Since in this model, the interest rate varies stochastically, we can determine the term structure from this model, and nowhere in the model is it necessary to introduce concepts such as liquidity, transactions costs,time horizon or habit to explain the existence of a term structure.
Suppose there exists a security(mth security) is perfectly correlated with changes in the interest rate, and its dynamics are described by
(39) and from , we have
mm m
m
dPdt dq
P
, ,dr f r t dt g r t dq Mm Mm M m Mr M m
m Mr m Mr mMr M r
r r g
mr mg
Let be the price of a discounted loan which pays a dollar at time in the future when the current interest rate is r. Then the dynamics of P can be written as . And also we have
,P r
dPdt dq
P
M M M Mr M Mr M rr
Mr Mr
r g
r g
(38)
Set in (38) and substitute
, we have (40) By Ito’s lemma, and must satisfy
(41)
, , ,Mm mr M r
2
mr Mk Mm kr M kr Mr Mkk M mr r r
Q Q
k
mm
r r
21
02 rr rg P fP P P
rP g
P
So, given , (41) can be solved to
determine and hence the term structure of interest rate.
Suppose the Expectations Hypothesis holds, then for all and the term structure is completely determined by
(42) subject to .
,P r
r
210
2 rr rg P fP P rP
,0 1P r
(38)
From (40), it must be that in equilibrium ,and the equilibrium condition (38) simplifies to
m r
2
mr Mk Mm kr M kr Mr Mkk M mr r r
Q Q
2
2 2 2
2 2
2 2 2 2 2
21
r Mk Mr krk M
M r Mr
r Mk M k Mr kr M k rM
M r Mr M r
k Mk Mr krM
M Mr
r r
r
r
And further if we assume f and g are constants, we have the explicit solution for (42):
Note that as ,which is not at all reasonable.
2
2 3, exp2 6
f gP r r
P
Just in the similar way as Model 1 and Model 2,we can derive the Fundamental Equation of Security Pricing as
subject to an appropriate boundary condition .
2 2 21 10
2 2VV rr rV V rV F g F g F rVF fF F rf
, ,0F V r
Remark 1: the model does not allow for nonhomogeneous expectations, non-serially independent preferences, or transactions costs.
Remark 2:the fundamental assumption in this model is continuous-time assumption.If the model were formulated in discrete time, then the results derived in this chapter no longer hold.
The end Thanks!
