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FMAT3888 Projects in Financial Mathematics

创建日期：2024-06-04 17:12:04

所属分类：金融Finance

标签： FMAT3888

Projects in Financial Mathematics

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FMAT3888 Projects in Financial Mathematics

Interdisciplinary Project: Portfolio Optimisation with Market Data
This is an open-ended project and there are a number of questions here you can investigate. You are welcome
to go beyond the questions asked here (e.g., also considering portfolios where short selling is not allowed, or
alternative risk measures such as Value-at-Risk). You may adjust the numbers in red in these questions, in order
to obtain more interesting results.
Data and Setup. The spreadsheet (available on canvas) contains monthly returns data for different assets1. In
this project you need to consider the following 5 assets:
1. Emerging Market Equities (EMEQ)
2. Australian Listed Property (ALP)
3. Hedge Funds (HF)
4. Global Fixed Interest (GOV)
5. Cash (CASH)
Let Si = (Sit)t∈N be the price process for Asset i for i = 1, . . . , 5. Here time t is in months and S
i
t is the price of
Asset i at the end of month t. For i = 1, . . . , 5, assume the dynamics of the price of Asset i satisfies
Sit = S
i
t−1 · eX
i
t , t = 1, 2, 3, . . . . (1)
DenoteX t = (X
1
t , . . . , X
5
t )
T . AssumeX 0,X 1,X 2, . . . are i.i.d., and each admits multivariate normal distribution
with mean a = (a1, . . . , a5) ∈ R5 and covariance matrix B = (bij)i,j=1,... ,5. For i = 1, . . . , 5, denote αit the
monthly return of Asset i in month t. By (1),
αit =
Sit
Sit−1
− 1 = eXit − 1 =⇒ Xit = ln(1 + αit).
Note that the realised monthly returns αit since January 2001 are provided in the spreadsheet.
Remark: Here we use lognormal distribution (instead of normal distribution) to model the return αit, because
we would like αit > −1 to always hold (why?).
1 Parameter Estimation
Q1. Estimate the parameters a and B using market data for the two time intervals:
(A) from 1/1/2007 to 31/12/2010, (B) from 1/1/2011 to 31/12/2014.
Q2. For n ∈ N, by (1) the return of Asset i from the end of month t (equiv. beginning of month t + 1) to the
end of month t+ n is given by
Rit,n :=
Sit+n
Sit
− 1 = exp
(
t+n∑
k=t+1
Xik
)
− 1.
1In reality each of them is an asset class which contains multiple assets. For simplicity we will consider each asset class as a single
asset.
Show that
Rit,n ∼ eY
i − 1, i = 1, . . . , 5, (2)
where (Y 1, . . . , Y 5) admits multivariate normal distribution with mean na and covariance matrix nB .
Q3. Let the random vector R(1) := (R
(1)
1 , . . . , R
(1)
5 )
T (resp. R(2) := (R
(2)
1 , . . . , R
(2)
5 )
T ) model the joint annual
(resp. two-year) returns for the five assets. For k = 1, 2 denote
µ
(k)
i := E
[
R
(k)
i
]
, c
(k)
ij := Cov
(
R
(k)
i , R
(k)
j
)
, ρ
(k)
ij :=
c
(k)
ij√
c
(k)
ii
√
c
(k)
jj
, i, j = 1, . . . , 5.
Use the results in Q1 and Q2 to compute/estimate µ
(k)
i , c
(k)
ij , ρ
(k)
ij for i, j = 1, . . . , 5 and k = 1, 2 for the two time
intervals (A) and (B) from Q1.
Approximating lognormal by normal: From (2) we know that R(1) and R(2) follow a lognormal distribution.
This could make some of the computations below (Q4 and Q6) difficult. If so, you can approximate R(1) and
R(2) as normally distributed using the following reasoning.
Recall that ex − 1 ≈ x when x ≈ 0. Hence for Y ∼ N(µ, σ2), if Y ≈ 0 with large probability, i.e., when
µ, σ2 ≈ 0 (why?), then with large probability eY − 1 ≈ Y . In this case it is reasonable to approximate eY − 1
using Y . (However, since E[eY −1] ̸= E[Y ] and Var[eY −1] ̸= Var[Y ], a better approach involves moment matching
and approximates eY − 1 by a normal random variable with mean E[eY − 1] and variance Var[eY − 1].) The same
applies to the multivariate case.
2 Static Portfolio Optimisation
Q4. Consider an investor who statically invests all her wealth in these five assets for two years. Answer the
following questions using both sets of parameters from Q1, namely for periods (A) and (B).
(a) Solve the utility maximisation problem:
max E[U(wTR(2))]
subject to
5∑
i=1
wi = 1,
where w = (w1, . . . , w5)
T is the vector of weights, and U(x) = −e−γx with γ = 1.
(b) Comment on the differences of your results corresponding to the two datasets (A) and (B).
(c) Compare your result from (a) (with dataset (B)) with the realised return on her portfolio using the market
data for the period from 1/1/2021 to 31/12/2022.
Q5. Under the setup of Q4, answer the following questions for both datasets (A) and (B) from Q1.
(a) Find the efficient frontier in the (σ, µ)-plane for the market consisting of these five assets, using the estimated
parameters µi := µ
(2)
i , cij := c
(2)
ij , ρij := ρ
(2)
ij for i, j = 1, . . . , 5.
(b) Find the portfolio with minimum variance which yields at least 12% for the expected return. That is, solve
min wTCw
subject to
5∑
i=1
wiµi ≥ 0.12,
5∑
i=1
wi = 1,
2
where w = (w1, . . . , w5)
T is the vector of weights and C = [cij ] is the covariance matrix for R
(2).
(c) Comment on the differences between your results for the two datasets.
(d) Compare your result from (b) (with dataset (B)) with the realised return on her portfolio using the market
data for the period from 1/1/2021 to 31/12/2022.
(e) Compare your results to those from Q4.
3 Dynamic Portfolio Optimisation
Q6. Consider an investor who invests all her wealth in these five assets for two years, during which she will adjust
her portfolio weights at the beginning of the second year. For k = 1, 2, denote ξk := (ξk1 , . . . , ξ
k
5 )
T the returns
of the five assets for the k-th year. Note that ξ1 and ξ2 are i.i.d. copies of R(1). Let w = (w1, . . . , w5)
T (resp
u = (u1, . . . , u5)
T ) be the portfolio weights at the beginning of the first year (resp. second year). Then the return
of the profolio over the two-year investment period is given by
G(w,u) = (1 +wTξ1)(1 + uTξ2)− 1. (why?)
Suppose the investor believes that parameters estimated using the dataset (B) are valid. Answer the following
questions.
(a) Solve the utility maximisation problem:
max E[U(G(w,u))]
subject to
5∑
i=1
wi =
5∑
i=1
ui = 1,
where U(x) = −e−γx with γ = 1. Note u = u(ξ1) may depend on the realisation of ξ1.
(b) Compare your result with that for Q4(a).
Q7. Under the setup of Q6, answer the following questions.
(a) Solve the portfolio optimisation problem:
min Var[G(w,u)]
subject to E[G(w,u)] ≥ 0.12,
5∑
i=1
wi =
5∑
i=1
ui = 1.
Note here the control u = u(ξ1) may depend on the realisation of ξ1.
(b) Compare your result with that for Q5(b)
(b) Compare your results to those from Q6.

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MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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