代写代考 MATH3090/7039: Financial mathematics Lecture 10

MATH3090/7039: Financial mathematics Lecture 10

(Single-period) MV portfolio optimization
All risky assets: formulation

All risky-assets: solution
Risk-free asset and market portfolio

(Single-period) MV portfolio optimization
All risky assets: formulation
All risky-assets: solution
Risk-free asset and market portfolio

Review of risky and risk-free assets
Risky asset: an asset whose future return is uncertain.
Example: stock – prices are stochastic, i.e. unpredictable (risky). Risk-free asset: an asset which has certain future return. Example: Australian government bond, ANZ bank account

Portfolio optimization: a single-period example
Suppose that an investor has AUD$100K, and would like to invest in a portfolio consisting of two stocks for T = 1 year.
The investor can choose how to invest at time t = 0, and keep the holdings of the portfolio constant until time t = 1.
Possible trading strategies: 0. t = 0: (80%,20%) in
0. t = 0: (50%,50%) in (S1, S2)
There are infinitely many of different investment trading strategies. Which one is the “best”?
This is an example of the single-period portfolio optimization problem.

Portfolio optimization: a multi-period example
Suppose that an investor has AUD$100K, and would like to invest in a portfolio consisting of two stocks for, say T = 20 years.
The investor can change the holdings of the portfolio yearly.
Possible trading strategies:
0. t = 0: (80%,20%) in (S1,S2) 1. t = 1: (30%,70%) in (S1,S2)
20. t = 20: (0%, 100%) in (S1, S2)
0. t = 0: (50%,50%) in (S1,S2) 1. t = 1: (36%,64%) in (S1,S2)
20. t = 20: (90%, 10%) in (S1, S2)

Portfolio optimization: a multi-period example
Suppose that an investor has AUD$100K, and would like to invest in a portfolio consisting of two stocks for, say T = 20 years.
The investor can change the holdings of the portfolio yearly.
Possible trading strategies:
0. t = 0: (80%,20%) in (S1,S2) 1. t = 1: (30%,70%) in (S1,S2)
20. t = 20: (0%, 100%) in (S1, S2)
There are infinitely many of different investment portfolios.
Which one is the “best”?
This is an example of the multi-period portfolio optimization problem.
0. t = 0: (50%,50%) in (S1,S2) 1. t = 1: (36%,64%) in (S1,S2)
20. t = 20: (90%, 10%) in (S1, S2)

Mean-variance (MV) portfolio optimization
The wealth of the portfolio at time T is a random variable (due to the randomness in the stocks’ prices).
The mean of this random variable is essentially the reward.
The variance of this random variable is essentially the risk.
Criteria for “best”:
• given a level of risk (variance) that the investor is willing to
accept, s/he wants to maximize the reward (mean), or
equivalently,
• given a level of reward (mean) the investor wishes to obtain, s/he
wants to minimize the risk (variance).
This is the mean-variance (MV) portfolio optimization problem. We can replace mean by expected return.

MV portfolio theory
MV portfolio theory is a mathematical technique to determine the asset weights in a portfolio which
• maximize the portfolio expected return, given a target portfolio variance
• or minimize the portfolio variance, given a target portfolio return.
We assume that the investor only care about expected return and variance, and not other risk-measures (such as VaR, or CVaR). We will focus on single-period MV portfolio optimization.

(Single-period) MV portfolio optimization
All risky assets: formulation
All risky-assets: solution
Risk-free asset and market portfolio

Risky assets
• Times 0 and T.
• There are N risky assets in the market. We will introduce the
risk-free asset later.
• Rn : Ω → R: random return on n-th asset.
• R = (R1, . . . , RN ): vector of all assets’ random returns.
• μn = E[Rn]: expected return of the n-th asset (non-random).
• μ = (μ1, . . . , μN ): vector of all assets’ expected returns (non-random).

Risky assets (cont.)
• σnm = E[(Rn − μn)(Rm − μm)]: the covariance between the n-th and m-th assets, n,m = 1,…,N.
Note that the variance of the n-th asset is
σn2 = σnn = E[(Rn − μn)2].
• Σ: the N × N variance-covariance matrix, where
􏰌 σ11 … σ1N 􏰍 Σ= . . .
σN1 … σNN We assume that Σ is invertible.

• A (static) portfolio is a vector
w = (w1,…,wN) ∈ RN,
where wn, n = 1,…,N, is the percentage of weight of the total
wealth invested in the n-th asset (at time 0). N
• RP =􏰏wnRn =w·R:Ω→Ristheportfolio’srandom
n=1 return.
• μP = 􏰏 wnμn = w · μ is the portfolio’s expected return.
• σP2 = 􏰏􏰏wiwjσij = wTΣw is the portfolio’s variance. i=1 j=1

MV optimization problem
The MV portfolio optimization problem is either:
• given the portfolio’s target expected return, find the weights which minimize its variance, or
• given the portfolio’s target variance, find the weights which maximize the its expected return.
These problem statements are equivalent. It is more common to work with the first problem statement.

MV portfolio optimization: statement
• μ􏰒P : portfolio’s target expected return. • w􏰒: optimal weights.
We want to
􏰊1T 􏰋 Findw􏰒=argmin wΣw
subjectto  w·μ =μ􏰒P
w·1 =􏰎Nn=1wn=1
This is a quadratic programming problem:
• The objective function is quadratic in terms of w • The constraints are linear in w

(Single-period) MV portfolio optimization
All risky assets: formulation
All risky-assets: solution
Risk-free asset and market portfolio

Method of Lagrange multipliers: review
Find (x∗,y∗)=argmaxf(x,y) (x,y)
subject to g(x, y) = 0, (1) for sufficiently smooth f and g. Here g(x, y) = 0 is a set of constraints.
Introduce the Lagrange multiplier λ and study the Lagrange function
L(x, y, λ) = f (x, y) − λ · g(x, y) .
If f(x∗, y∗) is a maximum of f(x, y) for (1), then ∃λ∗ such that (x∗, y∗, λ∗)
is a stationary point of L(x, y, λ). (The converse is not necessarily true.) Necessary solutions: ⇒ set partial derivatives of L(x, y, λ) to zero to find all stationary points.

Method of Lagrange multipliers (cont.)

Method of Lagrange multipliers: example
Wewanttomaximizef(x,y)=x+ysubjecttox2+y2 =1. In this case
L(x,y,λ)=x+y+λ(x2 +y2 −1), and hence need to solve
􏰆∂L ∂L ∂L􏰇 􏰂 2 2 􏰃 ∂x,∂y,∂λ =0⇔ 1+2λx,1+2λy,x +y −1 =0.
These yield λ = ±√1 which implies stationary points 2
􏰆11􏰇 􏰆11􏰇 (x,y)= √ ,√ and (x,y)= −√ ,−√
Thusthemaxisf􏰄√1 ,√1 􏰅=√2 22

MV optimization: the Lagrange multipliers
Recall we want to
􏰊1T 􏰋 Findw􏰒=argmin wΣw
subjectto w·μ =μ􏰒P
w·1 =􏰎Nn=1wn=1.
The Lagrange function is
L(w,λ,γ)= 12wTΣw+λ(μ􏰒P −w·μ)+γ(1−w·1).
Setting the partial derivatives of L to 0 yields
∂L =Σw−λμ−γ1=0, (i)
∂L = μ􏰒P − w · μ = 0, (ii)
∂L =1−w·1=0. (iii) ∂γ

MV optimization (cont.)
Σw−λμ−γ1=0 ⇒
We need to find λ and γ. • From (ii):
w􏰒 =λΣ−1μ+γΣ−11.
• From (iii):
μ􏰒P =μ·w􏰒=λ(μTΣ−1μ)+γ(μTΣ−11).
1 = 1 · w􏰒 = λ(1TΣ−1μ) + γ(1TΣ−11).

Find λ and γ
From the previous slide
λ(μTΣ−1μ) + γ(μTΣ−11) = μ􏰒P λ(1TΣ−1μ) + γ(1TΣ−11) = 1.
 T −1 T −1     μ Σ μ μ Σ 1 λ = μ􏰒P ,
1TΣ−1μ 1TΣ−11γ
which yields
􏰚λγ􏰛=A−1􏰚μ􏰒P 􏰛,
􏰚 a22 −a12 􏰛 −a21 a11 .
a11a22 − a12a21

Find λ and γ (cont.) Substitutions give
μT Σ−1 μ det 1TΣ−1μ
1TΣ−11 = (μTΣ−1μ)(1TΣ−11) − (μTΣ−11)2,
1TΣ−11μ􏰒P − μTΣ−11 (μTΣ−1μ)(1TΣ−11) − (μTΣ−11)2
μTΣ−1μ − μTΣ−11μ􏰒P . (μTΣ−1μ)(1TΣ−11) − (μTΣ−11)2

Optimal weights w􏰒 Recall from L10.24
w􏰒 = λΣ−1μ + γΣ−11. This, together with L10.26, give
w􏰒 = a + b μ􏰒 P
(μTΣ−1μ)Σ−11 − (μTΣ−11)Σ−1μ
where a = (μTΣ−1μ)(1TΣ−11) − (μTΣ−11)2 , (1TΣ−11)Σ−1μ − (μTΣ−11)Σ−11
b = (μTΣ−1μ)(1TΣ−11) − (μTΣ−11)2 . The optimal portfolio variance is given by
1TΣ−11􏰂μ􏰒 − μTΣ−11􏰃2 2T1 P1TΣ−11
σ􏰒P = w􏰒 Σw􏰒 = 1TΣ−11 + (μTΣ−1μ)(1TΣ−11) − (μTΣ−11)2 .

Efficient frontier
Efficient Frontier

(Single-period) MV portfolio optimization
All risky assets: formulation
All risky-assets: solution
Risk-free asset and market portfolio

Introducing a risk-free asset
Let there be a risk-free asset with certain return Rf . • Invest 1 − w · 1 in the risk-free asset.
◦ 1−w·1>0: lending.
◦ 1−w·1<0: borrowing. • The portfolio’s expected return μP =(1−w·1)Rf +w·μ =Rf +w·(μ−Rf1). • Since Rf is risk free, the portfolio’s variance is still σP2 = wTΣw. MV portfolio optimization The MV portfolio optimization problem with a risk-free asset is 􏰊1T 􏰋 Findw􏰒=argmin wΣw subjectto Rf +w·(μ−Rf1) =μ􏰒P Using the method of Lagrange multipliers, we have the optimal weights w􏰒 = λ Σ − 1 ( μ − R f 1 ) , where the Lagrange multiplier is λ = μ􏰒 P − R f . (μ−Rf1)TΣ−1(μ−Rf1) Capital market line The portfolio’s variance is 2 T ( μ􏰒 P − R f ) 2 σ􏰒P =w􏰒 Σw􏰒 =g(μ􏰒P)= (μ−Rf1)TΣ−1(μ−Rf1). Expressing μ􏰒P in terms of σ􏰒P give μ􏰒P =f(σ􏰒P)=Rf +􏰡(μ−Rf1)TΣ−1(μ−Rf1)σ􏰒P. This is a linear function in σ􏰒P , called the capital market line (CML). Capital market line Wehaveμ􏰒P =Rf +􏰟(μ−Rf1)TΣ−1(μ−Rf1)σ􏰒P Capital Market Line Efficient Frontier Geometrically: μ􏰒P = Rf + μM − Rf σ􏰒P μM is the expected return and σM is the standard deviation of the market portfolio wM . ButhowdowefindμM andσM? Market portfolio (cont.) The market portfolio wM is when there is no borrowing or lending. Recall that the optimal weights are w􏰒 = λΣ−1(μ − Rf 1). We invest 1 − 1 · w􏰒 in Rf . The market portfolio is derived by setting: 1=1·w􏰒 =1Tw􏰒 =λ1TΣ−1(μ−Rf1) λ=1 1TΣ−1(μ − Rf 1) wM =λΣ−1(μ−Rf1) = Σ−1(μ−Rf1) . 1TΣ−1(μ − Rf 1) Market portfolio (cont.) We also find that • The expected return of the market portfolio is μM =wM ·μ sincewehave0=1−1·wM investmentintherisk-freerateRf. • The variance of the market portfolio is σM2 =wTMΣwM = (μ−Rf1)TΣ−1(μ−Rf1). [1TΣ−1(μ − Rf 1)]2 程序代写 CS代考加微信: powcoder QQ: 1823890830 Email: powcoder@163.com

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