Discrete time mean variance portfolio selection with transaction costs

This thesis incor-porates proportional transaction costs into the mean-variance formulation,and studies the optimal asset allocation policy in two kinds of single-periodmarkets under the

PORTFOLIO SELECTION

WITH TRANSACTION COSTS

XIONG DAN B.Sci (Hons), NUS

A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE

DEPARTMENT OF MATHEMATICS

NATIONAL UNIVERSITY OF SINGAPORE

2008

Transaction cost is a realistic feature in financial markets, which however isoften ignored for the convenience of modeling and analysis This thesis incor-porates proportional transaction costs into the mean-variance formulation,and studies the optimal asset allocation policy in two kinds of single-periodmarkets under the influence of transaction costs The optimal asset alloca-tion strategy is completely characterized in a market consisting of one risklessasset and one risky asset Analytical expression for the optimal portfolio isderived, and the so-called “burn-money” phenomenon is observed by exam-ining the stability of the optimal portfolio In the market consisting of oneriskless asset and two risky assets, we provide a detailed scheme for obtain-ing the optimal portfolio, whose analytical solution can be very complicated.

We also study the no-transaction region and some special asset allocationstrategies by the scheme

Key Words: asset allocation, portfolio section, mean-variance formulation,transaction costs, no-transaction region, Sharpe Ratio

iii

Notations and Assumptions

b: the coefficient of buying transaction cost

s: the coefficient of selling transaction cost

For every $1 worth of stock you buy, you pay $(1 + b); for every $1 stock yousell, you receive $(1 − s)

e0: the single-period deterministic return of the bank account

ei: the single-period random return of a stock

σi: volatility of a stock

ρ: the correlation between the return of stock 1 and stock 2

x0: holdings in the bank account

Assume E[Ai] > 0, for i = 1, 2

If you own $(1+b) in bank account, E[Ai] means the expected excess monetaryprofit if you were to invest the money in stock i

iv

3.1 Characterization of optimal strategies 333.2 Sharpe Ratio with transaction costs 503.3 No-transaction region 56

v

Chapter 1

Introduction

One prominent problem in mathematical finance is portfolio selection folio selection is to seek the best allocation of wealth among a basket of secu-rities The mean-variance model by Markowitz (1959,1989) [7] [8] provided afundamental framework for the study of portfolio selection in a single-periodmarket The most important contribution of this model is that it quantifiesthe risk by using the variance, which enables investors to seek the highestreturn after specifying their acceptable risk level (Zhou and Li (2000) [15])

Port-As a tribute to the importance of his contribution, Markowitz was rewardedthe Nobel Prize for Economics in 1990 An analytical solution of the mean-variance efficient frontier in the single period was obtained in Markowitz(1956) [6] and in Merton (1972) [10]

After Markowitz’s pioneering work, single-period portfolio selection wassoon extended to multi-period settings See for example, Mossin (1968) [11],Samuelson (1969) [12] and Hakansson (1971) [2] Researches on multi-periodportfolio selections have been dominated by those of maximizing expectedutility functions of the terminal wealth, namely maximizing E[U(X(T ))]

1

where U is a utility function of a power, log, exponential or quadratic form.The term (E[x(T )])2 in Markowitz’s original mean-variance formulation how-ever, is of the form U(E[x(T )]) where U is nonlinear This posed a ma-jor difficulty to multi-period mean-variance formulations due to their non-separability in the sense of dynamic programming This difficulty was solved

by Li and Ng (2000) [3] by embedding the original problem into a tractableauxiliary problem In a separate paper, similar embedding technique wasused again to study the continuous-time mean-variance portfolio selection byZhou and Li (2000) [15]

Another development in portfolio selection is the extension of a less market to one with transaction costs Historically, Merton (1971) [9] pio-neered in applying continuous-time stochastic models to the study of portfolioselection In the absence of transaction costs, he showed that the optimalinvestment policy of a CRRA investor is to keep a constant fraction of totalwealth in the risky asset In 1976, Magil and Constantinides [5] incorporatedproportional transaction costs into Merton’s model and proposed that theshape of the no-transaction region is a wedge Almost all the subsequentwork along this direction has concentrated on the infinite horizon problem.See for example, Shreve and Soner (1994) [13] Theoretical analysis on thefinite horizon problem has been possible only very recently See Liu andLoewenstein (2002) [4], Dai and Yi (2006) [1] The continuous-time mean-variance model with transaction costs have recently been studied in Xu [14]

friction-To the best of our knowledge, no results have been reported in the ture with regard to the discrete-time mean-variance model with transactioncosts The work presented in this thesis is an effort to extend Markowitz’s

litera-mean-variance formulation to incorporate transaction costs in a discrete-timemarket setting Li and Ng (2000) [3] solved the multi-period discrete-timemean-variance problem without transaction costs In their paper, the originalnon-separable problem is embedded into a tractable auxiliary problem, andthe method of dynamic programming is then applied to the auxiliary problem

to obtain the solution In this thesis, we consider proportional transactioncosts, where transaction fees are charged as a fixed percentage of the amounttransacted We will follow the embedding technique in Li and Ng (2000) [3],and provide solution to the last investment stage of the multi-period prob-lem with transaction costs The solution we obtained will be needed whenapplying dynamic programming going backward in time-steps to solve themulti-period problem We leave this to future research work

We first look at the market consisting of one risky asset and one risklessasset, and then we move on to examine the market consisting of two riskyassets and one riskless asset In the market consisting one risky and oneriskless asset, we present a complete analytical solution We also derive theanalytical expressions of the boundaries of the “no-transaction region” Weshow that if the initial holdings fall out of this no-transaction region, thenthe optimal asset allocation strategy is to bring the allocation to the nearestboundary of the no-transaction region

It is to be noted that a feature results from transaction costs is that wealthcan be disposed of by the investor of his own free will This is achieved bycontinuingly buying and selling a stock and paying for the transaction fees

In the market consisting of one risky and one riskless asset, such phenomenon

is indeed observed It happens when the target investment return is too low

In this case, the one-step solution is found to be unstable As a result, asequence of continuing buying and selling of the stock is required until thesolution reaches stable state As money is deliberately disposed of in thisprocess, we call this phenomenon the “burn-money phenomenon”.

To rule out the burn-money phenomenon, we assume the target ment return is of a sufficient high level in the market consisting of 2 riskyassets and 1 riskless asset In this market, we work out a complete scheme

invest-to find the optimal asset allocation strategy We also derive a necessary andsufficient condition for a certain asset allocation strategy to be within theno-transaction region One particular strategy is discussed in this market:when the Sharpe Ratio (with transaction costs) of the first stock is muchhigher then the Sharpe Ratio of the second stock, we find out that the op-timal strategy implies we should not invest in the second stock at all Thisconfirms our intuition that stocks with higher Sharpe Ratio is preferable overstocks with lower Sharpe Ratio Before we move on to examine the first mar-ket, we introduce the general problem settings in the rest of this introductorychapter

1.1 Multi-period mean-variance formulation

Mathematically, a general mean-variance formulation for multi-period folio selection without transaction costs can be posed as one of the following

port-two forms:

(P 1(σ)) max

u t

E[xT]s.t V ar[xT] ≤ σ

ut is our control An equivalent formulation to either (P 1(σ)) or (P 2(ǫ)) is

(E(ω)) max

u t

E[xT] − ωV ar[xT]s.t xt+1 =

(E(ω)) This auxiliary problem takes the following form.

(A(λ)) max

u t

E{−x2T + λxT}s.t xT =

1.2 The last stage with transaction costs

When transaction cost is considered, total wealth xt will not be enough todescribe the state of the current investment Instead, we have to specify theholdings xiin each individual asset at each time period The terminal wealthwill be calculated as the monetary value of the final portfolio, which is equal

to the total cash amount when long stocks are sold and short stocks arebought back In addition, the constraints in the optimization problem willbecome non-smooth Despite these differences, it is still possible to applythe method of dynamic programming to the problem setting with transac-tion costs, if we adapt the objective function maxu tE{−x2

T + λxT} from theseparable auxiliary problem constructed above In order to obtain solutions

to the multi-period problem by the method of dynamic programming, weshould start from the last investment stage of the problem After we obtain

the solution to the last stage, we can then go backwards stage by stage andobtain the sequence of optimal investment strategies The solution to thelast investment stage of the multi-period problem with transaction costs iswhat we deal with in this thesis.

In a market consisting of one riskless asset and n risky assets, the problemsetting for the last stage of the multi-period mean-variance formulation withtransaction costs can be written as

max

u i

E{−x2T + λxT}s.t xT = e0u0+ (1 − s)e1u+1 − (1 + b)e1u−1

+ (1 − s)e2u+2 − (1 + b)e2u−2+ (1 − s)e3u+3 − (1 + b)e3u−3

· · · ·+ (1 − s)enu+n − (1 + b)enu−n

Here xi denotes the initial amount invested in the i-th asset ui’s are ourcontrols, namely, we would like to adjust each xi to the amount ui xT isthe final total monetary wealth λ is the same as in the multi-period settingwithout transaction costs It is to be noted that the value of λ is chosen

at the very beginning of the investment horizon and will remain constantthroughout all investment stages In particular, if we assume the investor’sposition is known at the beginning of the last investment stage, then weshould have no information about how big λ is, relative to the investor’sposition As it turns out, in our subsequent discussions, this relation between

λ and the investor’s current position is critical in determining the investor’sstrategies

in stock from x1 to an optimal amount u1 (In case u1 = x1, no adjustment

is needed.) In the process of buying or selling stocks, transaction fees arecharged We treat transaction costs in the following manner: when we buy

$1 worth of stock, we pay $(1 + b); when we sell $1 worth of stock, we receive

$(1 − s) The optimization problem in this market can be written as

P2 = B2+ x1E[(A

′

1)2]E[A′

′2, B2 ≤ λ′ < P2

λ′2, λ′ < B2

(2.1.3)(2) When x1 <0, the optimal u∗

λ′2, λ′ < B1

(2.1.5)

The rest of this chapter is mostly to establish this theorem Part (1) inTheorem 2.1.1 corresponds to the case when the investor starts with a longposition in the stock; part (2) corresponds to the case when the investorstarts with a short position in the stock In order to obtain the results inTheorem 2.1.1, we look at the following 6 cases.

We examine the two parts separately in subsequent discussion

In this part, we assume

x1 ≥ 0

We distinguish the following 3 kinds of strategies, each of which corresponds

to a different form of objective function

sell), and therefore assumes a short position in stock.

Under different parameter settings (parameters include b, s, e0, e1 and λ),

we wish to identify the strategy that dominates all other strategies, namelygives a better objective value than the rest to E{−x2

T + λxT} For a givenparameter setting, the best strategy among the 3 is the optimal strategy

Case 1 x1 ≥ 0, u1 > x1 The strategy of buying more stocks

$(1 + b) cash amount in his hands He has two investment options If heputs the money in the bank, he will get a sure return of $(1 + b)e0 at theend of the single-period investment horizon; If he invests the money in thestock, with the money he can purchase $1-worth of stock due to buying

transaction costs At the end of the investment horizon, the $1-worth ofstock will become $e1 After he cashes in the holdings in stock, $(1 − s)e1

is what he will get in monetary terms due to selling transaction costs So

A1 means the excess return of investment in the risky asset over the risklessasset It is thus reasonable to assume

E[A1] > 0,for otherwise, investing in stock will yield a lower expected return yet theinvestor has to bear a higher level of risk, making investment in stocks muchlike a lottery game or a unfair gambling game

To solve the maximization problem, we have

(λ′− B1)E[A1]E[(A1)2] > x1,

u1 = x1, when (λ

′− B1)E[A1]E[(A1)2] ≤ x1.

′ > B1+ x1E[(A1)

2]E[A1] ,

u1 = x1, when λ′ ≤ B1+x1E[(A1)

2]E[A1] .

(2.1.11)

In the following, for simplicity reason let us denote

P1 = B. 1+x1E[(A1)

2]E[A1] .With the values of u1 obtained in (2.1.1), we can now calculate the optimalobjective value of V1 = E{−x2

T + λxT} under the strategy of buying morestocks The optimal objective values are summarized below followed by adetailed calculation

V1{λ′ ≤P 1 } corresponds to the case when u1 = x1

V1{λ′ >P 1 } In this case, u1 = (λ′−B1 )E[A 1 ]

E[(A 1 ) 2 ]

V1{λ′ >P 1 } = E[−x2T + λxT]

= E[−(A1u1+ B1)2 + λ(A1u1+ B1)]

= E[−A21u21− 2A1B1u1− B12+ λ(A1u1+ B1)]

(both u1 and B1 are deterministic numbers)

= −E[A21]u21− 2E[A1]B1u1− B12+ λ(E[A1]u1+ B1)

= −(λ

′− B1)2E2[A1]E[A2

′2.Case 2 x1 ≥ 0, 0 ≤ u1≤ x1 The strategy of selling some stocks

So now xT can be written as

xT = A′1u1+ B2 (2.1.15)With similar calculations as in case 1, we can derive that if we adopt thisstrategy, the best values of u1 are given by

1)2] <0,

u1 = (λ

′− B2)E[A′

1]E[(A′

1)2] , when 0 ≤

(λ′− B2)E[A′

1]E[(A′

1)2] ≤ x1,

u1 = x1, when (λ

′− B2)E[A′

1]E[(A′

1)2] > x1.

(2.1.16)

Since E[A1] > 0 ⇒ E[A′

1] > 0, the above results are equivalent to

1)2] , when B2 ≤ λ

′

1)2]E[A′

u1 = x1, when λ′ > B2+ x1E[(A

′

1)2]E[A′

The optimal objective value of V2 = E{−x2

T + λxT} under the strategy ofselling some stocks can be calculated in the same way as in case 1 Theseoptimal objective values are summarized below

′2,

V2{λ′ >P 2 } = −E[(e0x0+ (1 − s)e1x1− λ′)2] + λ′2

(2.1.18)

Case 3 x1 ≥ 0, u1 <0 The strategy of short selling.

A′′1 = (1 + b)e1− (1 − s)e0 (2.1.20)

So now xT can be written as

xT = A′′1u1+ B2 (2.1.21)With similar calculations as before, we can derive that if we adopt this strat-egy, the best values of u1 are given by

(λ′− B2)E[A′′1]E[(A′′

1)2] <0,

u1 = 0, when (λ

′− B2)E[A′′

1]E[(A′′

(2.1.22)

Again E[A1] > 0 ⇒ E[A′′

1] > 0, the above results are equivalent to

1)2] , when λ

′ < B2,

u1 = 0, when λ′ ≥ B2

(2.1.23)

The optimal objective value of V3 = E{−x2

T + λxT} under the strategy ofshort selling stocks can be calculated in the same way as before Theseoptimal objective values are summarized below

′2,

V3{λ ′ ≥B 2 } = −(λ′− B2)2+ λ′2

(2.1.24)

The division of regions

P2 = B2+ x1E[(A

′

1)2]E[A′

x1E[(A′

1)2]E[A′

1]

= (b + s)x1e0 +x1E[(A1)

2]E[A1] −

x1E[(A′

1)2]E[A′

1]

= x1(E[A′1] − E[A1]) + x1E[(A1)

2]E[A1] −

x1E[(A′

1)2]E[A′

1]

= x1

(E[(A1)

2]E[A1] − E[A1]) − (

E[(A′

1)2]E[A′

1]



= x1(1 − s)

2V ar[e1]E[A1]E[A′

′

1] − E[A1]) ≥ 0

Because

E[A1] > 0, E[A′1] > 0 and E[A′1] > E[A1]

In the first 3 cases, we have assumed that x1 ≥ 0, so it is clear that

P2 > B2 

With Lemma 2.1.2, the first 3 cases are summarized graphically here.Case 1.

′2,

V3{λ ′ ≥B 2 } = −(λ′− B2)2+ λ′2

The dominate strategy

2 , the two strategies u1 = 0 and

u1 = x1 will yield the same objective value

Proof The objective value can be written as

Since both E[A′

1] and x1 are greater than 0, the results follow immediately Lemma 2.1.4 Among the 3 strategies, (i) When λ′ > P1, u1 > x1 domi-nates; (ii) When P1 ≤ λ′ ≤ P2, u1 = x1 dominates; (iii) When P2 ≥ λ′ ≥ B2,

0 ≤ u1 ≤ x1 dominates; (iv) When λ′ < B2, u1<0 dominates;

Proof The result for the case when λ′ ≥ B2 is self-evident The case when

λ′ < B2 can be deduced from Lemma (2.1.3) 

Making use of lemma (2.1.2) (2.1.3) and (2.1.4), case 1, 2 and 3 can now

′2 (unstable), λ′ < B2

(2.1.27)

The no-transaction region

Remark 2.1.5 P1 and P2 can be seen as a sort of buying and selling aries respectively The interval λ′ > P1 is the buying region; the interval

bound-P2 ≤ λ′ ≤ P1 corresponds to no transaction region; the interval λ′ < P2 isthe selling region When transactions costs are zero, b = s = 0, we have

P1 = P2, hence the no transaction region vanishes without transaction costs.Proof We have seen in Lemma(2.1.2) that

P1− P2 = x1(1 − s)

2V ar[e1]E[A1]E[A′

′

1] − E[A1])

When b = s = 0, we have E[A1] = E[A′

1], the result follows 

Remark 2.1.6 Both B1 and B2 are combinations of our positions in bankand in stock

The value of B1 remains unchanged when we buy stocks; the value of B2

remains unchanged when we sell stock

Theorem 2.1.7 The optimal strategy in λ′ > P1 and B2 ≤ λ′ < P2 bringsthe current position in bank and stock to the buying and selling boundaries

λ′ = P1 and λ′ = P2 respectively

Proof In λ′ > P1, our original position x0 and x1 gives

λ′ > P1 = B1 +x1E[(A1)

2]E[A1]

= e0[x0+ (1 + b)x1] + x1E[(A1)

2]E[A1] .The optimal strategy in this (buying) region is to increase x1 to

u1 = (λ

′− B1)E[A1]E[(A1)2] .

Let us denote the new position in bank by u0, we have

= e0[x0− (1 + b)(u1− x1) + (1 + b)u1] + u1E[(A1)

2]E[A1]

= e0[x0+ (1 + b)x1] + u1E[(A1)

2]E[A1]

= B1+ u1E[(A1)

2]E[A1]

= B1+ (λ′− B1)

= λ′ The above calculation shows that when λ′ > P1, our optimal strategy bringsour positions in bank and in stock to the buying boundary λ′ = P1 In thecase when B2 ≤ λ′ < P2, the optimal strategy brings current position tothe selling boundary λ′ = P2 The calculation is similar to above In thecase when λ′ < B2, the optimal strategy is to short stocks This falls intothe case when our new position in stock is negative It will be seen in thefollowing discussion that this strategy is unstable It will result in a sequence

of continuing buying and selling of the stock until the holdings in the stockbecome 0, in which case, we again have λ′ = P2

In this case, we assume

x1 <0

We distinguish the following 3 kinds of strategies, each of which corresponds

to a different form of objective function

Case 4 represents the strategy to purchase more stocks and eventually avoid

a short position in stock; Case 5 represents the strategy to buy some morestocks but still maintain a short position in stock; Case 6 represents thestrategy to sell even more stocks All the calculations in this part are similar

to the previous part, and hence are omitted We provide the summary of theresults of these 3 cases here

In the following, for simplicity reason let us denote

1 ] Case 4

1)2] , when P3 ≤ λ

′ ≤ B1,

u1 = x1, when λ′ < P3

(2.1.30)

1)2]E[A′′

1]

= x1(E[A′′1] − E[A′′′1]) + x1E[(A

′′′

1 )2]E[A′′′

x1E[(A′′

1)2]E[A′′

1]

= x1(E[(A

′′′

1)2]E[A′′′

V ar[A′′

1]E[A′′

= (b + s)x1e0T−1− x1E[(A

′′

1)2]E[A′′

1]

= x1(E[A′′1] − E[A′′′1]) −x1E[(A

′′

1)2]E[A′′

1]



>0.Because x1 <0, E[A′′

′2,

V4{λ ′ <B 1 } = −(λ′− B1)2+ λ′2

2 , the two strategies u1 = 0 and

u1 = x1 will yield the same objective value

Proof Same as lemma 2.1.3 

Lemma 2.1.10 Let

P5 = B1pE[(A′′

1)2] − B2pE[(A′′′

1)2]pE[(A′′

1)2] −pE[(A′′′

1)2] ,then we have P3 < P5 < P4

Proof Since we have B2 = B1− (b + s)e0x1, and E[A′′1] − E[A′′′1] = (b + s)e0,

P5 can be rewritten as

P5 = B1+ (b + s)e0pE[(A′′′

1)2]pE[(A′′

1] x1

= B1+ KE[(A

′′′

1)2]E[A′′′

1] x1,where

K =

q

E[(A ′′

1 ) 2 ] E[(A ′′′

1] > E[A′′′

1] > 0 and V AR[A′′

1] = V AR[A′′′

1] = (1 + b)2E[e1], wehave

E[(A′′

1)2]E[(A′′′

1] x1,

we conclude P5 > P3 The proof for P5 < P4 is the same 

Lemma 2.1.11 When x1 <0, (i) if λ′ > B1, the strategy u1 >0 dominates;(ii) if P5 ≤ λ′ ≤ B1, the strategy x1 < u1 < 0 dominates; (iii) if λ′ < P5,the strategy u1 < x1 dominates In particular, when λ′ = P5, the investor

is indifferent between the strategy of buying more stocks and the strategy ofselling some stocks

Proof When x1 <0, we look at case 4, 5 and 6 In the region λ′ > B1, byLemma 2.1.9, case 5 dominates case 6 The strategy of case 5 in this region

is u1= 0 Case 4 clearly shows that the strategy of u1 >0 is better than thestrategy of u1 = 0 in this region Hence u1 >0 dominates all if λ′ > B1 Bycomparing the objective value of V5 and V6, it can be seen that on the right

of P5 case 5 dominates case 6; on the left of P5 case 6 dominates case 5 Theargument for the rest of the result is thus similar 

2.2 The burn-money phenomenon

In case 4, 5 and 6 in the previous section, it is observed that the strategy of

u1 = x1 never dominates This means no-transaction region does not existwhen the initial holding in stock is negative In other words, we should con-tinue trading for as long as the holding in stock is negative, until it eventuallybecomes 0 In fact, we have the following theorem

Theorem 2.2.1 When x1 < 0, if λ′ = P3, case 6 dominates and the beststrategy is to sell some stocks so that λ′ = P4 On the other hand, when

λ′ = P4, case 5 dominates the best strategy is to buy some stocks so that

λ′ = P3

Proof Same procedure as in Theorem 2.1.6 

Remark 2.2.2 Equation 2.1.27 revisited In equation 2.1.27 we rized the optimal strategy when the investor starts off with a long position