2.1 Mean-variance Analysis: Single-period Model

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1 Chapter Portfolio Selection The theory of option pricing is a theory of deterministic returns: we hedge our option with the underlying to eliminate risk, and our resulting risk-free portfolio then earns the risk free rate of interest Banks make money from this hedging process; they sell something for a bit more than its worth and hedge away the risk to make a guaranteed profit But not everyone is hedging Fund managers buy and sell assets including derivatives with the aim of beating the bank s rate of return In so doing they take risk 1 Mean-variance Analysis: Single-period Model Let us take into account a single period model 11 Without a Riskfree Investment Suppose there are n risky assets The value today of the ith asset is S i and its random return rate is R i over the period considered Here R i is Normally distributed with mean μ i and standard deviation σ i The correlation coefficient between these returns on the ith and jth assets is ρ ij with ρ ii = 1, that is In other words, the covariance matrix is ρ ij = E [R i μ i R j μ j σ i σ j ρ ij σ i σ j n n Δ = V =v ij n n Without loss of generality, we assume V is positive definite A portfolio: w 1,w,, w n =w T with w i =1, 1

2 CHAPTER PORTFOLIO SELECTION where w i denotes the proportion invested in the ith asset Then the reward and risk of the portfolio can be described by the mean and the variance σ w = w i μ i w i w j v ij = w T Vw, respectively Markowitz s problem mean-variance analysis: we want subject to min σ w T w w i μ i = μ, w i = 1, where μ> is a given constant Clearly it is required that there exist some i and j such that μ i μ j In the matrix form, the problem can be written as subject to min w T Vw, w T w T μ = μ w T e = 1, where μ =μ 1,μ,, μ n T, and e =1, 1,, 1 T To solve this problem, we introduce Lagrangian multipliers λ 1 and λ It can be shown that the problem is equivalent to Let w, λ 1,λ be the true solution Then min w T Vw+ λ 1 μ w T μ + λ 1 w T e w T,λ 1,λ V w = λ 1 μ + λ e 1 w T μ = μ w T e = 1 3

3 1 MEAN-VARIANCE ANALYSIS: SINGLE-PERIOD MODEL 3 By 1, we have w = 1 V 1 λ 1 μ + λ e Note that and 3 can be rewritten as Combination of 4 and 5 yields or where In terms of 4 and 6, we get The corresponding variance = 1 V 1 μ, eλ 1,λ T 4 μ, e T w =μ, 1 T 5 1 μ, et V 1 μ, eλ 1,λ T =μ, 1 T 1 λ 1,λ T = A 1 μ, 1 T, 6 A =μ, e T V 1 μ, e 7 w = V 1 μ, e A 1 μ, 1 T σ = w T V w = μ, 1 A 1 μ, e T V 1 VV 1 μ, e A 1 μ, 1 T = μ, 1 A 1 μ, e T V 1 μ, e A 1 μ, 1 T = μ, 1 A 1 μ, 1 T 8 Here the last inequality is due to 7 Let us denote a b μ A = = T V 1 μ e T V 1 μ b c e T V 1 μ e T V 1 e It follows, A 1 1 c b = ac b b a As a result, 8 gives σ a bμ + μ = ac b This is the relation between risk and reward of the optimal portfolio In the σ μ plane, it is one branch of a hyperbola, which is called the frontier Question: What is an efficient frontier?

4 4 CHAPTER PORTFOLIO SELECTION 1 With a Riskfree Investment Now let us allow a riskfree asset in our portfolio Let S be the value today of riskfree asset and r be the riskfree rate We keep the previous notation and denote the portfolio by w,w T, where w + w i =1 Consequently, the reward becomes rw + w i μ i = w i μ i r+r Δ = w T μ + r, where The risk variance is still subject to μ =μ 1 r, μ r,, μ n r T σ w = w i w j v ij = w T Vw Mean-variance analysis with a riskfree investment: min w T Vw, w T w T μ = μ r = μ Note that there is no constraint on w Likewise, we can introduce a Lagrangian function Let w, λ be the true solution Then min w T Vw+ λ μ w T μ w T,λ V w = λμ 9 w T μ = μ 1 It follows or By 9, we have w = λ V 1 μ 11 λ μt V 1 μ = μ λ = μ μ T V 1 μ,

5 CONTINUOUS-TIME MODEL WITH UTILITY FRAMEWORK 5 Consequently, and w = μ μ T V 1 μ V 1 μ σ = w T V w = μ μ T V 1 μ μ T V 1 μ V μ T V 1 μ V 1 μ = μ μ T V 1 μ That is, μ σ = ± μ T V 1 μ = ± μ r μ T V 1 μ In this case the frontier becomes two straight lines The upward straight line σ = is the efficient frontier also called the Capital Market Line And μ r μ T V 1 μ μ r σ is called Sharp ratio, which is a measure for the portfolio s performance Continuous-time Model with Utility Framework 1 Optimal Investment Suppose that there are only two assets available for investment: a riskless asset bank account and a risky asset stock Their prices, denoted by R t and S t, respectively, evolve according to the following equations: dr t = rr t dt, ds t = S t [μdt + σdw t where r> is the constant riskless rate, μ> and σ> are constants called the expected rate of return and the volatility, respectively, of the stock Now we consider an investment problem associated with the market Assume that an investor has an initial wealth Z Let Z t be a self-financing process At time t, the investor holds and Z t in stock and bank respectively Then, with the initial wealth Z dz t =[rz t +μ r dt + σ dw t

6 6 CHAPTER PORTFOLIO SELECTION Assume the investor s risk preference is described by a strictly increasing and concave utility function U : R 1 R 1 The investor s problem is to choose an admissible strategy so as to maximize the expected utility of terminal wealth, that is, sup E z [UZ T In the following we would like to choose the logarithm utility function UZ =logz In addition, we are confined within the solvency region We define the value function by S = {Z : Z>} ϕz,t =supet Zt=z [UZ T,Z t >, t [,T It can be shown see Oksendal 3 that ϕ satisfies the Hamilton-Jacobi-Bellman HJB equation [ ϕ sup y t + 1 σ y ϕ ϕ +rz +μ r y = 1 z z Let y be the maximum Then σ y ϕ ϕ +μ r z z =, namely Substituting into 1, we get or ϕ t 1 ϕ y μ r z = σ ϕ z ϕ t 1 σ y ϕ z + rz ϕ z = μ r ϕ z + rz ϕ σ ϕ z z =,z>, t,t The terminal condition is ϕz,t = log z It can be verified that the true solution to the above problem is μ r ϕz,t = r + T t + log z σ Correspondingly, y = μ r z σ This means that the optimal investment policy is to keep a constant fraction of the total wealth in the risky asset Remark 11 a The above result can be extended to the power utility function Uz = zγ γ, γ<1, γ 1 b We can similarly take into consideration the multi-assets case

7 3 CONTINUOUS-TIME MEAN-VARIANCE ANALYSIS 7 Optimal Investment and Consumption If the consumption is allowed, then dz t =[rz t +μ r C t dt + σ dw t, where C t is the consumption rate The investor s problem is [ T sup E z e βs U C s ds + e βt UZ T,C t Denote the value function [ T ϕz t,t=supet Zt,C t The resulting HJB equation becomes [ ϕ sup y,c t e βs t U C s ds + e βt t UZ T,Z t >, t [,T t + 1 σ y ϕ ϕ +rz +μ r y C βϕ + U C z z = 13 The analytic solution to the above equation with boundary condition and the corresponding optimal investment/consumption strategy can be obtained I leave it as an exercise 3 Continuous-time Mean-variance Analysis Assume the wealth process dz t =[rz t +μ r dt + σ dw t The continuous-time mean-variance problem is min Var [Z T subject to E [Z T =a This is equivalent to min E [ ZT subject to E [Z T =a By introducing a Lagrangian multiplier, we instead consider min E [ ZT λ E[ZT a [ = min E ZT λ λ +λa, 14 where λ is a constant Now we define two value functions V z,t = inf Y s,s [t,t EZt=z t [ ZT λ,

8 8 CHAPTER PORTFOLIO SELECTION and [ u z,t = inf Et Zt=z Z T Y s,s [t,t It is easy to see V z,t =u z λe rt t,t Let us find analytical representations of u and V Note that u satisfies u t μ r u z σ u zz + rzu z =, for z>, t [,T subject to u z,t =z Assume the solution takes the form u z,t =Atz It follows or A tz μ r A t 4z 4σ A t A t = μ r [ So A t =exp μ r r T t Then σ σ +rz A t = r A t V z,t =At z λe rt t Next, we go back to 14 and find the optimal strategy I leave this as an assignment 4 Continuous-time Model: Probabilistic Approaches Let us confine to the utility framework Wealth process: dz t =[rz t +μ r dt + σ dw t We want to max E [U Z T, give Z = z Without loss of generality, we only consider power utility, ie, U Z T = 1 γ Zγ T for γ<1, γ 41 An Approach Based on Girsanov Transformation For Z t >, we can rewrite = π t Z t, where π t is the fraction in the risky asset Then dz t Z t =[r +μ r π t dt + σπ t dw t

9 4 CONTINUOUS-TIME MODEL: PROBABILISTIC APPROACHES 9 So, the portfolio selection problem becomes max E π t [ 1 γ Zγ T, where [ T Z T = z exp r +μ r π s σ πs T ds + σπ s dw s Note that [ 1 E = 1 γ zγ E γ Zγ T { [ T exp = 1 { [ T γ zγ E exp T γ σ πs + = 1 { [ T γ zγ Ê exp ds γr + γ μ r π s γσ πs γr + γ μ r π s γσ πs γ T T ds + ds γ σ πs T } ds + γσπ s dw s r +μ r π s 1 γ σ π s where we have used the Girsanov transformation by taking [ t γ σ π t s Gt =exp ds + γσπ s dw s } ds, γσπ s dw s } and Now, it is enough to consider max γ π s P A = A G T dp r +μ r π s 1 γ σ π s Clearly, the optimal π s, denoted by π s, π s = μ r 1 γ σ The corresponding optimal value function is [ { [ } 1 E γ Zγ T = 1 T μ r γ zγ Ê exp γ r + ds 1 γ σ [ = 1 μ r γ zγ exp γ r + T 1 γ σ

10 3 CHAPTER PORTFOLIO SELECTION 4 Martingale Approach For any strategy, we can think of the associated Z T as the payoff of a contingent claim Then the initial wealth z is its fair value, and z = Ê [ e rt Z T = E [e rt Ĝ T Z T 15 Here Ê is taken under the risk neutral world ie martingale measure P, P A = Ĝ T dp μ r W σ s The idea is the following: we first find an optimal Z T, ie, with Ĝt =exp [ μ r t σ max E Z T A [ 1 γ Zγ T ; 16 then we can replicate the optimal Z T to get the optimal strategy Bear in mind that the problem 16 is subject to the constraint 15 We then introduce a Lagrangian multiplier λ and instead take into consideration [ 1 max E λ E [e rt Ĝ T Z T z Z T It suffices to consider Clearly, the optimal Z T is = maxe Z T γ Zγ T [ 1 γ Zγ T λe rt Ĝ T Z T 1 max Z T γ Zγ T λe rt Ĝ T Z T Z T,λ = + λz λe rt Ĝ T 1/γ 1 17 Notice that ZT,λ is the solution to the problem 16 if [ γ/γ 1 z = E e rt Ĝ T ZT,λ = E [λ e 1/γ 1 rt Ĝ T So Ĝγ/γ 1 T γ 1 λ = z γ 1 e [E γrt [ = z γ 1 e [E γrt γ exp γ 1 [ = z γ 1 e exp γrt γ γ 1 [ = z γ 1 exp γ r 1 1 γ μ r T σ μ r T σ + γ μ r T σ γ 1 γ 1 μ r W T σ μ r T σ γ 1

11 5 ADVANCED TOPICS 31 Substitute to 17, we then obtain [ ZT = z exp r 1 γ γ μ r 1 γ σ [ μ r = z exp r + 1 γ T + 1 σ 1 γ T μ r T σ μ r W T σ 1/γ 1 μ r W T σ A question: what is the optimal strategy implied by the above expression? 5 Advanced topics Portfolio selection with portfolio constraints, transaction costs, and/or non-concave utility