MA 592: Mathematical Finance
Lecture 30
Prof. Siddhartha Pratim Chakrabarty
Department of Mathematics
Indian Institute of Technology Guwahati
Best Linear Predictor (Contd ...)
1
The best fit is generally considered as the one that minimizes the mean
square error:
h
i
MSE = E (ϵ2 ) = E (y − βx − α)2
=
2
E (y 2 ) − 2βE (xy ) − 2αE (y ) + β 2 E (x 2 ) + 2αβE (x) + α2 .
The minimum value is obtained by setting the partial derivatives of MSE
w.r.t α and β to be equal to zero, which gives:
βE (x 2 ) + αE (x) = E (xy ) and βE (x) + α = E (y ).
3
Solving we get:
β=
σxy
and α = E (y ) − βE (x).
σx2
4
The best linear predictor (BLP) of y w.r.t x is the linear function, βx + α
that minimizes the mean square error E (ϵ2 ).
5
β is called the beta of y w.r.t. x.
6
The line y = βx + α is called the regression line.
The Risk-Return of an Asset Compared with the Market Portfolio
1
Let us consider any particular asset ak in the market portfolio.
2
We want to use the best linear predictor to approximate the return Rk of
asset ak , by a linear function of the return RM of the entire market
portfolio.
3
We can write: Rk = βk RM + αk + ϵk , where βk =
αk = E (Rk ) − βk E (RM ).
4
Cov (Rk , RM )
and
2
σM
Here ϵk is the residual random variable, and βk is the beta of the assets
return w.r.t. the market portfolios return and is the slope of the linear
regression line. Therefore:
σk2 = Var (βk RM + αk + ϵk ) =
2
βk2 σM
| {z }
Systematic Risk
+
σϵ2k
|{z}
Unsystematic Risk
.
The Risk-Return of an Asset Compared with the Market Portfolio (Contd ...)
1
2
2
Note that the systematic risk βk2 σM
of asset ak is proportional to the
market risk, with a proportionality factor of βk2 .
According to economic theory, when adding an asset to a diversified
portfolio, the unique risk of that asset gets canceled out by other assets in
that portfolio.
Security Market Line
1
The expected return and risk of an asset ak , in the market portfolio is
related to the assets beta w.r.t. the market portfolio as follows:
2
µk = βk (µM − µrf ) + µrf and σk2 = βk2 σM
+ σϵ2k .
2
The graph of the line: µk = βk (µM − µrf ) + µrf is called the Security
Market Line.
3
This equation shows that the expected return of an asset is equal to the
return of the risk free asset + the risk premium βk (µM − µrf ) of the asset.
Derivation of Security Market Line
1
Consider a portfolio where weight s is invested in asset ak , and the weight
(1 − s) is invested in the market portfolio.
2
Then, the expected return and risk on the portfolio are:
µP
=
σP
=
sµk + (1 − s)µM ,
1/2
2
s 2 σk2 + 2s(1 − s)σkM + (1 − s)2 σM
.
3
In particular, s = 0 corresponds to the market portfolio.
4
The curve cannot cross the CML, since it would violate the definition of
CML, as the efficient boundary of the feasible set.
5
The tangent condition can be translated into the condition that the slope
of the curve is equal to the slope of the CML:
dµP /ds
(µk − µM )σP
dµP
=
=
.
2
2
dσP
dσP /ds
s(σk2 + σM
− 2σkM ) + (σkM − σM
)
Derivation of Security Market Line (Contd ...)
1
But the equilibrium market portfolio contains asset k since it contains all
assets.
2
Therefore s = 0 and σP = σM , which gives:
(µk − µM )σM
dµP
=
.
2
dσP
(σkM − σM
)
3
This must be equal to the slope of the market line.
4
Thus:
(µk − µM )σM
µM − µrf
=
,
2
σM
(σkM − σM
)
⇒
⇒
⇒
2
2
(µk − µM )σM
= (µM − µrf )(σkM − σM
),
σkM (µM − µrf )
µk = µrf +
,
2
σM
µk = µrf + βk (µM − µrf ).
Futures on Stock Index
1
A stock exchange index is a weighted average of a selection of stock
prices, with weights proportional to the market capitalization of stocks.
2
An index is approximately proportional to the value of the market
portfolio. For the purpose of futures markets, the index can be treated as
a security.
3
The futures prices f (n, T ), expressed in index points, are assumed to
satisfy the conditions outlined earlier.
4
Marking to market is given by the difference f (n, T ) − f (n − 1, T )
multiplied by a fixed amount.
5
Our purpose is to study applications of index futures for hedging, based
on CAPM.
6
Recall that the expected return on a portfolio over a time step of length τ
is given by:
µV = rF + (µM − rF )βV ,
where rF is the risk-free rate for a single period, µM is the expected return
on the market portfolio and βV is the beta coefficient of the portfolio.
Futures on Stock Index (Contd ...)
1
Let V (n) denote the value of the portfolio at the n-th time step.
2
We assume for simplicity that the index is equal to value of the market
portfolio, so that the futures prices are given by:
f (n, T ) = M(n)(1 + rF )T −n ,
where M(n) is the value of the market portfolio at the n-th time step a .
3
4
We now form a new portfolio (with value Ve (n), at the n-th time step), by
adding to the original portfolio, N short futures contracts on the index,
with delivery time T .
The value Ve (n) of the new portfolio is the same as the value V (n) of the
original portfolio, since there is no investment made to initiate N futures
contracts.
a Note that here discrete compounding has been used to be consistent with portfolio
theory
Futures on Stock Index (Contd ...)
1
At the (n + 1)-th step, the value of the new portfolio is given by:
Ve (n + 1) = V (n + 1) − N(f (n + 1, T ) − f (n, T )),
includes the cash flow resulting from marking-to-market.
2
The return on the new portfolio over the first step is given by:
KVe
=
=
=
3
Ve (n + 1) − Ve (n)
,
Ve (n)
V (n + 1) − N(f (n + 1, T ) − f (n, T )) − V (n)
,
V (n)
N(f (n + 1, T ) − f (n, T ))
KV −
.
V (n)
It can be shown that the beta of the new portfolio, βVe can be modified by
an appropriate choice of N. Accordingly, we have the result:
Result
If:
N = (βV − a)
(1 + rF )V (n)
,
f (n, T )
then βVe = a for any given number a.
Proof of the Result
1
We calculate the beta from the definition:
βVe
=
=
=
Cov (KVe , KM )
,
2
σM
Cov (KV , KM )
1 Cov (N(f (n + 1, T ) − f (n, T )), KM )
−
,
2
2
V (n)
σM
σM
1 Cov (N(f (n + 1, T ) − f (n, T )), KM )
βV −
,
2
V (n)
σM
where KM is the return on the market portfolio, and KV is the return on
the portfolio without futures.
Proof of the Result (Contd ...)
1
Since Cov (f (n, T ), KM ) = 0, and covariance is linear with respect to each
argument, therefore:
Cov (N(f (n + 1, T ) − f (n, T )), KM ) = NCov (f (n + 1, T ), KM ).
2
But f (n + 1, T ) = M(n + 1)(1 + rF )T −(n+1) .
3
Thus, we have:
Cov (f (n + 1, T ), KM ) = (1 + rF )T −(n+1) Cov (M(n + 1), KM ).
4
Again, by linearity of covariance in each argument:
M(n + 1) − M(n)
2
Cov (M(n + 1), KM ) = M(n)Cov
, KM = M(n)σM
.
M(n)
Proof of the Result (Contd ...)
1
Substituting, we get:
βVe = βV −
2
(1 + rF )T −(n+1) NM(n)
f (n, T )
= βV − N
.
V (n)
V (n)(1 + rF )
This implies:
N = βV − βVe
(1 + rF )V (n)
.
f (n, T )