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FMAT3888-latex代写

时间：2023-10-03

The University of Sydney

School of Mathematics and Statistics

FMAT3888 Projects in Financial Mathematics Semester 2, 2023

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.

School of Mathematics and Statistics

FMAT3888 Projects in Financial Mathematics Semester 2, 2023

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.