Signal Processing - May 2017 - 17
Expected Utility and RP
U (w 1)
Take a Bet: Source of the Excess Return
U (w )
U ( [W ])
[U (W )]
U (w 0)
RP
w0
CE ( [W ])
w1
w
FIGURE 2. The relationship among expected wealth E [W ] , expected utility
E [U (W )] , CE, and RP.
The RP is therefore the amount of money that a rational person
would demand for taking on a risk asset, or the amount of money
that a rational person is willing to pay to eliminate risk. Thus,
RP = E [W ] - CE.
(4)
According to the expected utility theory, the source of the
excess return of a risk asset, such as stock, is the RP. The RP is
the reason people pay insurance premiums to insurance companies to remove risk, and it is also reflected by the interest
rate of a loan. Figure 2 shows the relationship among expected
wealth E [W ], expected utility E [U (W )], CE, and RP. See
"Take a Bet: Source of the Excess Return" for a gambling
example for the risk-averse behavior and the RP.
Asset correlations and portfolio optimization
p i, t - p i, t -1 + d i, t
,
p i, t - 1
return) but not the noise (risk). If we have a pool of securities, we
can take advantage of the correlations among individual securities and improve the signal-to-noise ratio (SNR) by investing in
a portfolio (i.e., a basket of n securities). The portfolio return is
n
R p = / i = 1 x ip R i , where x ip is the weight of the ith security,
/ i x ip = 1. The portfolio expected return is
n
E [R p] = / x ip E [R i],
i =1
and the portfolio variance is
2
vRp =
n
n
/ / x ip x jp v ij = x Tp R x p ,
i =1 j =1
where v ij = Cov (R i, R j) is the covariance of R i and R j and
R is the covariance matrix. We can rewrite
For a security/asset i, the net return R i, t at time t is
R i, t =
Assume you win the lottery and have the following two
choices:
1) the bet: flip a coin (equal probability on both sides);
heads, you get US$2,000, tails, you get US$0
2) cash US$1,000.
Would you take the bet or the cash? What if the cash
amount were changed to US$900? US$800? Or more?
The cash amount for which you are indifferent between
choices 1) and 2) is the CE for the risky bet 1). The RP is then
1,000-CE, which is the extra money you need to be compensated for the risk you are taking in 1) or the price you
are willing to pay to eliminate the risk in 1) in exchange for
the certainty in 2). Are you risk averse? Do you believe the
saying that a bird in the hand is worth two in the bush?
2
(5)
where p i, t is the price at time t and d i, t is the dividend during the period t - 1 to t. [The logarithm of the total return
log (1 + R i, t) is often used because of asymmetry of the net
return. It is easy to show that the log return and the net return
are essentially the same when the net return is small. The log
return is more commonly used in empirical research.]
Assume that R i, t is stationary and its return is represented by
a random variable R i. The expected return is E [R i] = E [R i, t].
If the utility function can be approximated by quadratic form
[15], [16], the expected utility maximization becomes a meanvariance investment criterion: maximize the expected return
for a given variance, or minimize the variance for a given
expected return. The risk of an asset is represented by the
variance v 2i = Var (R i) or the square root of variance v i , also
called volatility in finance.
Using SP concepts, we can rephrase mean-variance criterion: because people are risk averse, they like the signal (expected
vRp =
/ x ip e / x jp v ij o = / x ip Cov (R i, R p).
n
n
n
i =1
j =1
i =1
We see that the contribution of security i to the risk or variance of the return on portfolio p is x ip Cov (R i, R p), i.e., the
risk of security i in portfolio p or the weighted average of
covariances. From the SP perspective, this risk is a projection
of R i on R p . We can therefore formulate a portfolio optimization problem (see "Mean-Variance Portfolio") to find the best
portfolio (weights) [17].
To find the optimal mean-variance portfolios (MVPs),
we use the Lagrangian expression:
n
J = v 2R p + 2m e c re - / x ip E [R i] m + 2z e c 1 - / x ip m,
i =1
(6)
i
where 2m e and 2z e are the Lagrange multipliers. We then
take derivative and set it to zero:
IEEE Signal Processing Magazine
2v 2R e
- 2m e E [R i] - 2z e = 0, 6i = 1, f, n,
2x ie
|
May 2017
|
17
Table of Contents for the Digital Edition of Signal Processing - May 2017
Signal Processing - May 2017 - Cover1
Signal Processing - May 2017 - Cover2
Signal Processing - May 2017 - 1
Signal Processing - May 2017 - 2
Signal Processing - May 2017 - 3
Signal Processing - May 2017 - 4
Signal Processing - May 2017 - 5
Signal Processing - May 2017 - 6
Signal Processing - May 2017 - 7
Signal Processing - May 2017 - 8
Signal Processing - May 2017 - 9
Signal Processing - May 2017 - 10
Signal Processing - May 2017 - 11
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Signal Processing - May 2017 - 14
Signal Processing - May 2017 - 15
Signal Processing - May 2017 - 16
Signal Processing - May 2017 - 17
Signal Processing - May 2017 - 18
Signal Processing - May 2017 - 19
Signal Processing - May 2017 - 20
Signal Processing - May 2017 - 21
Signal Processing - May 2017 - 22
Signal Processing - May 2017 - 23
Signal Processing - May 2017 - 24
Signal Processing - May 2017 - 25
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Signal Processing - May 2017 - 28
Signal Processing - May 2017 - 29
Signal Processing - May 2017 - 30
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Signal Processing - May 2017 - 33
Signal Processing - May 2017 - 34
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Signal Processing - May 2017 - 101
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Signal Processing - May 2017 - 105
Signal Processing - May 2017 - 106
Signal Processing - May 2017 - 107
Signal Processing - May 2017 - 108
Signal Processing - May 2017 - 109
Signal Processing - May 2017 - 110
Signal Processing - May 2017 - 111
Signal Processing - May 2017 - 112
Signal Processing - May 2017 - Cover3
Signal Processing - May 2017 - Cover4
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