116 ADDITIONAL TOPICS !N KALMAN FILTERING
3) Find an approximate value for pdf(ykip) (where p is the model used in the
Kalman ﬁlter) if rk : 0, if we : 1, and if T}; 2 2.
b) Suppose that the use of model p1 gives rk : 0, model p2 gives rk : 1,
and model p
114
a).
ADDlTiONAL TOPlCS IN KALMAN FILTERING
Solution:
From Equation (5.19) we obtain the steady—state value of Pk— as
P_ _ BK2 + Q
' 2K — K2
K in this equation is the gain that is used in the ﬁlter {based on the wrong
Q and R), but Q and R in this equat
100 THE CONTINUOUS-TIME KALMAN FiLTER
4 —r— i .— _ _—.
L ‘/-Aw- —-_—H—-—~u—-W-——-—m——————m
/
3.5 I.“ J
I!
3i - i
25 'j
_ .‘ f _ ~— A_ _ _ _ _ W 7 _ _ _ _ 7 _ _-J
I r — ' - ’ r
l ,
2[ fit! ‘I.
f 5'!
1.5]‘5‘ -.
; 4;:
1L/f J
I
0.5L j -
b”
o
o 2 4 6 a
118 ADDITIONAL TOPlCS IN KALMAN FILTERING
:1) Find analytical expressions for the steady-state values of K. a, 46+, 21+,
15+, 15‘, 2‘, and 15 .
b) What does the reduced-order ﬁlter indicate for the steady-state a posteriorri
estimation-error variance of t
110 OPTIMAL SMOOTHING
Solution:
a). Note that the continuous-time process noise covariance is
0 0
chio 0.01]
Discretization with a step size of At gives
F : 6.4m:
Q = QcAt
The discrete-time system is given as
$k+1 = ka + wk
wk N (0: Q)
b). Figure 9.1 show
108 OPTIMAL SMOOTHING
9.11 Consider a scalar system with F = 1, H = 1, and R = 2Q. Use the RTS
smoother of Section 9.4.2 to ﬁnd the steady-state value of the covariance of the
smoothed state estimate.
Solution:
Using Equation (9.136) to solve for the stea
104 OPTIMAL SMOOTHING
This proves that the inverse of (A + B) is equal to B_1(AB’1 + I)‘1.
QED
9.3 Derive Equation (9.83).
Solution:
Apply the matrix inversion lemma of Equation (1.39) to the ﬁrst line of Equa-
tion (9.83) with the substitutions A : (mefl
Cheryl Mew
FINS2624 Portfolio Management
Semester 1, 2011
THE CHOICE OF BALANCED PORTFOLIOS
Although we have addressed which CAL to use, which balanced portfolio lying on the CAL should be chosen?
Answer: select the balanced portfolio which offers the hig
Cheryl Mew
FINS2624 Portfolio Management
Semester 1, 2011
QUOTED PRICE OF EX-INTEREST BOND
By convention, the market does not quote the settlement price P. This is because, if we were the buyer, and
the interest was continuous, we would get some of the in
Cheryl Mew
FINS2624 Portfolio Management
Semester 1, 2011
LECTURE 3 DURATION
INTEREST RATE RISKS
For fixed interest securities investments, there is an uncertainty in returns which is broadly known as Interest
Rate Risks. There are 2 types of interest rat
Cheryl Mew
FINS2624 Portfolio Management
Semester 1, 2011
LECTURE 4 MARKOWITZ PORTFOLIO THEORY
BACKGROUND
Markowitz Portfolio Theory explains the rationale for diversification, and discusses the criteria used to rank
and form portfolio of securities.
This
Cheryl Mew
FINS2624 Portfolio Management
Semester 1, 2011
LIQUIDITY PREMIUM TH EORY
The Liquidity Premium Theory asserts that market participants are risk averse, as they will judge their choices
based on the level of expected risk and return, and risk =
Cheryl Mew
FINS2624 Portfolio Management
Semester 1, 2011
The Term Structure of Interest Rates offers investors an objective way of inferring future interest rates from
the yield curve. These future interest rates are called forward rates, denoted f(n,t)
Cheryl Mew
FINS2624 Portfolio Management
Semester 1, 2011
LECTURE 1 BOND PRICING
WHAT IS A BOND?
A bond is a claim on some fixed future cash flows.
A commonwealth government bond (CGB) is a bond which pays semi-annual coupons, in which the maturity
date/
Cheryl Mew
FINS2624 Portfolio Management
Semester 1, 2011
PORTFOLIO SELECTIONS IDENTIFYING THE SET OF EFFICIENT PORTFOLIOS
Suppose we have identified a number of risky assets to invest. We then form a number of hypothetical
portfolios, compute their risks
Cheryl Mew
FINS2624 Portfolio Management
Semester 1, 2011
DURATION AND BOND PRICE SENSITIVITY
FACTORS AFFECTING DURATION
Coupon Rate - The higher the coupon rate, the sooner the investor is going to recover the, costs of
purchasing the bond and the shorte
Cheryl Mew
FINS2624 Portfolio Management
Semester 1, 2011
LECTURE 5 OPTIMAL PORTFOLIOS
BACKGROUND
Risk free assets are zero risk instruments, in which their rates of return are the risk free rate (rf). For simplicity,
it is generally assumed that investor
Questions 11—20 are multiple choice. Please select one answer per question ordy.
“(2.5%) How do you interpret the estimated value of on in the following equation:
111(ENTEXP) = on + (1’2 [NC-101’le + e. Where I N C Oil-f E is annual household income (in
t
8.(7.5%) Real estate economists have found that for many data sets. a more appropriate model for
estimating the price of a house (in dollars) has the dependent variable in{PRIC-'E We W111
estimate the model 1n(PRICE) : {3’1 + 61UTOWN + {igSQFT + 7(5’QFT X
18. 22.5%) The LM Lagrange. Multi lier test generates a test statistic NR2 w _ 2 S — 1 . To what
_ . P \
does the S in this distribution refer?
(a) The. number of explal'iatory variables in the initial 1110<.lel.
(b) The statistical signiﬁcance level chos
0.(7.5‘7i-) Data on 1500 houses sold in Stockton. California. during 19.96 — 1998 wen-2 collected. The data
included the selling price(SPRIC-'E measured in dollars). living area (LIVAREA llleasured
in lll.lll(lI‘E‘.(Ilﬁ of f9), age (AGE). munbO-I' of bedr
747.5%) Suppose We are interested in estimating a wage equation in whit-.11 an individuals wages are
explained as a function of their race (BLACK). years of education {EDUC'}. and gender
{FEJUALE The reference group are white males (BL/1C K = U and FEﬂIAL
4.(7.5%) The econometric model below is based on data from election outcomes and campaign expen—
ditures for 173 two-party races for the U.S. House of Representatives in 1988. There are two
candidates in each race, A and B. VOTEA is the percentage of the
Part I: Short Answer Problems
1. (7.5%) Let M'ATH 10 denote the percentage of tenth graders at a high school receiving a passing score
on a standardized mathematics exam. Suppose we wish to explore the relationship between
the math pass rate and and spend