Morningstar - Q1 2020 - 50

~
~t ~
Xt+1
Pt Xt+1 = Et Qt+1

Strategies

A Kernel of Truth
The stochastic discount factor helps us
understand how assets are priced.

QUANT U

Paul D. Kaplan

A central fact about financial assets is the
uncertainty of their future values. In principle, that
uncertainty should be fully reflected in the
prices that they trade at in the financial markets.
How those prices are determined is one of the
main questions that financial economics
seeks to answer. It answers it using a construct
called the stochastic discount factor, or
SDF, also called the pricing kernel. In this issue
of Quant U, I explain the concept of the
stochastic discount factor and what it means
for asset prices, expected returns, and
portfolio selection.
Probability Theory
Financial economics models uncertainty
using probability theory, so it's not surprising
that concepts from probability theory are
at the heart of the SDF construct. A key concept
of probability theory is that of a random
variable. A random variable takes on a
potentially different value under each possible
scenario. The likelihood of each scenario
occurring is known before the actual scenario
occurs. For each scenario, the SDF assigns
a price today to a $1 payoff should the given
scenario occur. Both the payoff from
holding an asset and the SDF can be modeled
as random variables. The price of the
asset today is the probability-weighted average
of the SDF times the asset's future payoff.
This probability-weighted average is called the
expected value.

Let's express the asset pricing formula
mathematically with two points in time, today
(time t) and one period from today (time t+1). Let:
~t
Q t+1
~
X t+1	

= the SDF (a random variable)
= payoff of the asset in question1
(a random variable)
~
Pt Xt+1 	 = the price of the asset at time t
(a value)
We have:
~
~t ~
Pt Xt+1 = Et Qt+1
Xt+1
~t E [.] means
1 the expected value, taking all
where
t=
Et Qt+1
+ Rft
information 1available
at time t into account.
~
Xt+1 applies to all securities. Also, there
~ that SDF
Note
-1
Rt+1 =
~
are
no arbitrage
Pt Xt+1 opportunities because any
two assets with identical payoffs across scenarios
have the same price. This is sometimes called
~t
~
the
of
1 +one
Rft price.
=1
Et Qlaw
t+1
A risk-free
~t ~ asset Risft one that pays the same in all
Rft = ~ ~
Et Q~t+1
scenarios.
The price
t R of a risk-free asset that pays
ftX
Pt Xt+1 = Et 1Q+t+1
t+1
$1 in all scenarios is:
~~
~~
~
~
E X~Y = Cov X,Y + E X E Y
1
t
Et Qt+1 =
1 + Rft
~
~
~t ~
, Rt+1
Et Rt+1 - Rft~ = - 1 + Rft Covt Qt+1
X
~
where
the
t+1 one-period risk-free rate of
-1
Rt+1 = Rft is ~
return atPtime
Xt+1t.
t
~
~
Et Rt+1 - Rft = t Et RBt+1- Rft
We can restate the asset pricing formula in
~t
~
+ Rft The
= 1 return on the asset over the
Et Qt+1
terms
of 1returns.
~t is: ~
period t to t+1
Cov Q , R
~t ~ t t+1 Rft t+1
=
Ett Qt+1 Rft =~t ~
Cov Q 1,+RRft

t
Bt+1
t+1
~
1 In a multiperiod setting, X t+1 could include the price of an asset at time t+ 1, which is also a random variable.

~~
~~
~
~
t+1 X,Y v -1
E~t X=YQ~t = Q~Cov
Q
... Qv + E X E Y
v
t+1 t+2
50

Morningstar Q1 2020

~t ~ ~
~t ~~ ~
~ +
+
Qt+2
t
- c1t+1+ R EtCov
,...
R
Qct+2
EWt R= c-t R Et =Qt+1

~t ~
~
Xt+1
Xt+1 = Et1 Qt+1
Pt ~
t
Et Qt+1 =
1 + Rft
~
~ ~
= E1 Qt X
Xt+1
P ~
t
= X~ t t+1 t+1
Ett Qt+1
~
1t+1+ Rft - 1
Rt+1 =
~
P
X
~t t ~ t+11
E~t Qt+1 = Xt+1
Rt+1 =
1~+ Rft - 1
~~t Pt X~t+1 ~t ~
Therefore,
= 1Xt+1
EPtt QXt+1 1=+~REftt Qt+1
Xt+1
~
-1
Rt+1 =
~
~
~tt ~P1t + XR~t+1 R=ft 1
EEt Q
ft
Q~t+1
t Rft = 1
t+1
=
Ett Qt+1
1 + Rft
+
~ ~ 1~ R~ftRt ~
~
t
ft X
= E Qt+1
EPtt Q~Xcan
t+1 Rrewrite
t+1
We
t+1
~ ft1 +~=Rftt 1this
~=R~1as:
~
~
+X,
E~ X Y = XCov
Yft + E X E Y
t+1
-
1
Rt+1 =
~tt ~P X~ 1 Rft
~ R t =t+1 ~ ~
~
~
Q~~t+1
EEEtt XQ
~t+1
Y ft== Cov1 +X,RY +~E X E ~
Yt ~
, Rt+1
Et Rt+1 - Rft 1 += R-ft 1ft + Rft Covt Qt+1
~
~t ~
~
E
=
Xt+1
Q
X
Pt ~
t
t+1
=~1oft+1the
E~t Q~expected
~ 1 +~Rfttvalue
~
~
The
~ product
~~ of two
~ random
t+1 X,Y ++ E X E Y t
ER XR~t+1
Y - =R XCov
~~-t -1~1~ Rft Covt Qt+1 , Rt+1
=
=
ft
t
t+1
t+1
~
variables,
say
X
and
Y
,
is
related
to
their
~
Xt+1-P= XEt Qt+1 Xt+1
P ~
Rt t+1
= Rt Et RBt+1- Rft
Ett R~t+1
covariance
measure
of how much they move
t ~ ft (a 1
ft
Rft= =
Et Q~t+1
~
~t ~
+
+
1
R
1
R
-
+
together)
as
follows:
, Rt+1
Et R~t+1 Rft = -ft 1ft ~Rft Covt Qt+1
~
E R tt - =Rft ~~=1 t ~Et RBt+1- Rft
+
1
R
=
1
Ettt Qt+1
~Qft+t , R
t+1
t+1
Rft~ ~t+1
~
~
t+1
~ ~ ~ Covt X1t+1
+
ERtt+1=X =
Y = Cov
~~t X,~Y- 1~ E X E Y
~
Rft~EBt+1R - R
-~PRt tftX~XQ~t+1
=
E R~t Cov
t , tR
t+1
Rft t =t+1
E~tt Qt+1
Cov
,
R t 1 Bt+1 ft
Q
t+1
t+1
Rt+1~=
1 + R-t+1
~
~
~t ~
ft
=
+ R Cov
~ 1earlier
Applying
our
equation
,
X~t+1
=to
,on
Rt+1
Qt+1
E~tt Rt+1~-PRtthis
t -
t
t ~t Cov
t ft~Q
t+1 , R ~v -1 ft
~
t
Bt+1
t+1
Qt+1
Qrearranging
~t ...=~Q1v of terms, we have an equa+
with
some
1
R
EQtv Q~=t+1
t+2
~ Covt Qftt+1 ,~R~t+1
~
~
+ E X return
= Cov
E Xfor
Y the
X,Y excess
E Y on the asset:
tion
expected
=
~
~
~t
~t ~t ~t 1 +~R~
~~~1v -1~+
t+1t ~t =
EQ
+ Et+2
c-~t+1
Wtt Q
ct+2 ...
Q...
Qt+1v R -Et R Qt+2 ~
Cov
,RRcBt+1
t+1
t Q
t Rt QQ
t ft=
t+1 t+1ftE
R~=t+1
R
EEttvt Q
ft =
ft
Bt+1
t t
~t+1 ft
~
~
1 -+ R1ft ~+ R Cov ~Q t , R~
Et Rt+1 - Rft =~
~tt ~)t+1 t+1
)ft u(c
tRftu(c
~
~
~
t+1+ E Q t+2
~tt Q==t u(c
~cRtt +)E~+
W
c~v -1
=tt+1EQt...
EQ
+t t+2 ct+22 +......
U
t+1 Qt+1
ft t Qt+2
tv =t+1Qt+1
v
+
+
(1 + )
~t 1 ~~R1~ft
~~
,X,RYt+1~+ E X~ E~ Y~
Qt+1
= t Cov
E X Y Cov
u(ccan
u(c
The
=~above ~equation
~t t+2~) to+ any asset or
~ ) be applied
t+1
+~)=tEQ~t t ~~E~
==~u(c
...
U
+R)ftEt+1
c-u'(c
c~1~t+1
Qt+2
R +it-E+tot~R(1
EWttt t1R~
t tWe
t can
t+1
ft
t+1Cov
t t+ Bt+1
tRapply
+ Yc~t+2
,
Q
portfolio.
a
) 2 ... portfolio,
Bt+1
t+1=X,Q
E X Y =t Cov
Yt+1+ E X Ebenchmark
~
~
~
~
1 + as-au'(c
t combine the
) market
such
t = -
~Q)t+1
Rbroad
1~+ Rindex,
Covtthen
, Rt+1
Et Rt+1
~ft ) u(ct+1 )ft u(c
t+2
u'(c
~t t )~t+1
~~tv -1equation
+t+1
result
the
to get:
+
+...
u(c
Et tabove
U~tt1= with
~
~
+ ~ (1 + ~)t 2 ~
= Qt+1 Qt+2
Q ~
...
= Q1Qt+1
- u'(c
Rftt tQ) =t+1-, Rt+1
1v+ Rft Covt Qt+1
, Rt+1
E1tv+Rt+1Cov
=
t
~
~ ~)=t ~,t R~E~ R~ - R ~t ~
EWt 1R=t+1Cov
~t t Bt+1
+ Et ftQt+2 ct+2 ...
+Rt EQ
t+1
c-u'(c
Qt+1t cBt+1
t
t ftt+1
t
= Qt+1
t+1
1 +~ u'(c
1
)
~
~ t-O
~ )
~tv~-1
= t Eu(c
R ) - Ru(c
E~tt1Rt+1~-~t cRt+1
ft~t+1
ft
t+1Bt+1
= ~return
QQ~tt+1
Q1v+== Qu(c
...
~Etthe
where
R
on
benchmark
+ the t+2
+ ... and:
)tQ+is
U
t+1
t+2
v
c
Bt+1		
t
t
t
Covt Qt+1
, R1t+1+
(1 + ) 2
~
-1
=
ct+1 ~ O~ ~~t
t1
~t ~
~
~ttQ=,t R~Q
+- OEt Qt+2
+c~E1Q
= Cov
cCov
W
ct+2 ...
c~
c1~t+1
t+1
t+
tu'(c
t+1
t+1
,
R
t
ttt tQ)t+1
Bt+1
~
t+1
t+1
1 = t+1
Qt
ct +t = (1 + ~) 1O= ~Qt+1
t+1
1
~ t)-t O u(c~ )
u'(c
u(c~t+2)
~~tv -1
Cov
Qt+1 , RBt+1
t+1- O
~c~t1= u(c
~t ct+1
t ~+t+1
~
+
+...
=
)
E
Q
U
1
t
= Q Q ...Q1Q
Q1tt+1
t+1
v+ = t+1tct t+2~t
v
(1 + ) 2
- t+1
O+
Ot
c
+
(1
)
~ t ~ ~ Qt+1 ~
The
equations
t above
t two
t+1 ~
v -1 -present
1 ~t ~a generalized
R~~Ot+1
-O
= =Qt+1
Q
tt ~
v = cu'(c
t+2
v +
+ ~QE1Capital
)~Q...
- OE Q
W
ct+2 ...
c~1Q
ct+1
t1 of the
t Et t+1
t Q
t
t+1
t+1
t+2 Model
form
Asset
Pricing
t
-
~t+1
t= Q O
=
Q
Ot+1 t+1
+
c
~
1
+
(1
)
)
u'(c
derived
solely from
t
~t ~the
~t~ ~ asset pricing
~ )SDF
-E 1-based
ROt+1
= =ct +c~ Ett -~~QO1t tt+1
W
c) ...
c1~-t+1
Qt+2
-OO +
u(c
1u(c
t1
t
t+2 t+2
formula.
t t+1 is+a generalized
Ett+1
+QQEt+1
EIn
+...
= u(c
)t t particular,
Uc~tt+1
t
~
=
Q
tt+1
+ 1 + (1
+ )2
~t 1- Ot+1
ct systematic
1 + = of
ROt+1
measure
risk.
~
~
O
Q
~ct
u(c )
(1 +t+1) u(c )
R~Ot+1
= =u(cEt )~+ ~E~ttt 11--OOt+1 -+1 t+2 2 + ...
U
Q
+
cc~tt+11 u'(c
(1
1
)
~ -O + ~+ )
Et t t+1
t+1 ~tt
1 Qt+1
t+1 =
1 ROt+1
=
Q
=
Q
t+1
O
(1
1cc+t
)-)+1O ) t+1
u'(c
~
+
(1
t
~
t
ct+1 ~
) Ot 1~~- tOt
11 u'(c
Ect t+1Qt+1
== QQt+1
c1~t+1
t+1
+
+
O = 11 + R~
1 = u'(c
t ln(c),
)
t ~t O- O
Ot+1
c
+
(1
)
1
Q
t
~
1-
u(c)
-1 1, -OO ≠ 1- 1
Rc~Ot+1 == c O~t+1
-O
t1 ~t
1ln(c),
t+1
O=1
Qt+1
= Et 1-
O Qt+1
O
c
+ 1

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Morningstar - Q1 2020

Table of Contents for the Digital Edition of Morningstar - Q1 2020
