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FINM8006 Advanced Investments

创建日期：2024-05-15 17:43:03

所属分类：金融Finance

标签： FINM8006

Advanced Investments

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FINM8006 Advanced Investments

Exercise Questions
Short Answer Questions
Suppose the CAPM holds and that there are no borrowing
restrictions, so the mean- variance ecient frontier consists of
combinations of Treasury bills and the market portfolio.
Nonetheless, some households foolishly hold stock portfolios that
are less than perfectly diversified. Show that the Sharpe ratio of an
imperfectly diversified portfolio, divided by the Sharpe ratio of the
market portfolio, equals the correlation of the imperfectly
diversified portfolio with the market portfolio.
solution
SRi
SRM
=
ERei
i
M
EReM
=
M
i
SRi
SRM
=
cov(ERei ,ER
e
M)
2M
M
i
=
⇢iM
2M
M
i
= ⇢
Show graphically that small risk around reference point in prospect
theory value function v(x) indicates risk aversion in the first order.
Be specific about what features of v(x) results in this loss
avoidance. Is the theory of first order stochastic dominance, i.e.,
w1
FOSD
w2 i↵ F1(w)  F2(w), still valid under prospect theory?
Suppose the factor structure of two diversified portfolios a and b
are given by
Ra = 0.16 + 1.2f˜1 + 0.4f˜2
Rb = 0.26 + 0.8f˜1 + 1.6f˜2
0.16 = 0.04 + 1.21 + 0.42
0.26 = 0.04 + 0.81 + 1.62
We get 1 = 0.065, 2 = 0.105
If you try to run the OLS regression
Rit Rf = i ◆+ ✏it (1)
where ◆ is a vector of 1 with dimension T , for each of the portfolio,
that is, if you run time-series excess return on a single vector of 1s,
what is the interpretation of your estimated i? Why?
Recall the general OLS estimate
ˆ = (X 0X )1X 0Y
in exercise 1. Here we have
ˆ = (◆0◆)1◆0(Ri Rf ) =
PT
1 (Rit Rf )
T
so the estimate ˆ is the mean value of excess return for each
portfolio.
Now if you plot your estimated ˆi (on Y axis) against your
estimated E (Ri Rf ) from CAPM model (on X axis), which
portfolios (in terms of size and book-to-market) in the
Fama-French portfolios do you expect to lie above the 45 degree
line?
Value portfolios, especially small-value portfolio will lie above the
line, that’s the so-called value premium.
Consider N risky assets with mean vector R¯ and variance
covariance matrix ⌃. An arbitrary portfolio p with portfolio
weights wp (N ⇥ 1 vector ) on the mean-variance frontier has
mean µp and variance 2p. find the zero-beta portfolio z for any
mean-variance ecient portfolio
min
1
2
w 0z⌃wz s.t. w
0
z ◆ = 1, w
0
z⌃wp = 0
where ◆ is a N ⇥ 1 vector of 1s. For simplicity in expression, let
define ◆0⌃1◆ ⌘ C and R¯ 0⌃1◆ ⌘ B as in the lecture. Solve for the
expected return of the zero-beta portfolio µz .
min
1
2
w 0z⌃wz s.t. w
0
z ◆ = 1w
0
z⌃wp = 0
FOC
⌃wz ◆ ⌃wp = 0
wz = ⌃
1◆+ wp
plug into the first constraint:
◆0⌃1◆+ w 0p◆ = 1
Notice w 0p◆ = ◆0wp = 1, so
C + = 1
plug into second constraint:
◆0wp + w 0p⌃wp = 0
+ 2p = 0
solving the two equations we have
=
1
1 C2p
, =
2p
1 C2p
µz = R¯
0wz =
µp 2pB
1 C2p
There is one risky asset with payo↵ v that is normally distributed
with zero mean and variance 2v . An informed investor who has
zero endowment of the risky asset, observes v and places a market
order of amount x at price p. The informed investor has constant
absolute risk aversion a, so she maximizes expected CARA utility
E [eaW ]
, where W is terminal wealth and a is the absolute risk aversion.
Risk neutral market makers observe the total order flow x + u,
where u is the demand of noise traders and is normally distributed
with mean zero and variance 2u. The informed trader is assumed
to behave strategically; that is, in deciding on her optimal trade x ,
she takes the dependence of the price on the order flow into
account. Assume risk free return is zero.
The informed investor’s terminal wealth is given by W = (v p)x .
Assume that the market makers uses a linear pricing rule
p = (x + u). What is the conditional distribution of wealth,
W (x , v), conditional on v and x?
W (x , v) = (v (x + u))x
so W (x , v) is normally distributed with conditional expectation
and variance:
E [W (x , v)|x , v ] = (v x)x
VAR[W (x , v)|x , v ] = 2x22u

p = E [v |x + u] = E [v |v + u]
By the projection theorem for jointly normal variables and the fact
that E [v ] = E [u] = 0, we have
p =
cov(v ,v + u)
Var(v + u)
(v + u) =
2v
22v +
2
u
(x + u)
with a = 0 we also have = 12 and x =
1
2v .
p =
1
2
2
v
1
42
2
v +
2
u
(x + u)
compare to pricing rule p = (x + u), we must have
1
2
2
v
1
42
2
v +
2
u
=
therefore, (0) = v2u and (0) =
u
v
measures the sensitivity of
price to order flows. It is increasing in standard deviation of stock
value and decreasing in standard deviation of noise trading.

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