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OPTIMAL PORTFOLIO SELECTION WITH TRANSACTION COSTS Phelim P. Boyle* and Xiaodong Lin†
ABSTRACT This paper examines the lifetime portfolio-selection problem in the presence of transaction costs. Using a discrete time approach, we develop analytical expressions for the investor’s indirect utility function and also for the boundaries of the no-transactions region. The economy consists of a single risky asset and a riskless asset. Transactions in the risky asset incur proportional transaction costs. The investor has a power utility function and is assumed to maximize expected utility of end-of-period wealth. We illustrate the solution procedure in the case in which the returns on the risky asset follow a multiplicative binomial process. Our paper both complements and extends the recent work by Gennotte and Jung (1994), which used numerical approximations to tackle this problem.
1. INTRODUCTION
in his book Continuous Time Finance (Merton 1990). By making certain simplifying assumptions, he was able to obtain closed-form solutions in a number of cases. The two-asset problem is of particular interest. Merton assumed that one asset has a lognormal distribution and that the other asset is a riskless money market account earning a constant return. The optimal investment decision also depends on the preferences of the investor and the problem is simplified if it is assumed that the investor has a so-called power utility function. In this case the utility function1 is of the form
The asset allocation problem is an important one in investment finance, and it also has applications in actuarial science. This is the problem faced by an investor who has to decide how to allocate his or her wealth across different assets or asset classes. Asset allocation decisions are important for pension plans. One strategic choice in the investment of pension plan assets concerns the split between stocks and bonds. In the case of final salary plans, a rule of thumb that is sometimes quoted is 60% in stocks and 40% in bonds, but the scientific support for this rule is not obvious. We can appeal to the theoretical models in financial economics to develop a framework for tackling this problem. Although these models are based on simplifying assumptions, they can provide useful insights. This paper is written in this spirit. We show how the traditional asset allocation problem can be solved when there are transaction costs. In particular, we use a multiperiod discrete time model and show how explicit solutions can be obtained in some cases. The asset allocation problem is a variant of the classic consumption investment problem in modern finance. This problem was solved in a series of brilliant papers by Robert Merton; these papers are reproduced
U(W) 5
Wa , a , 1. a
Under these assumptions Merton obtained an expression for the optimal investment in the risky asset. This fraction, also known as the Merton ratio, is µ2r , s (1 2 a) 2
where µ is the expected return on the stock, s is the volatility of the stock, and r is the riskless return. This solution indicates that it is optimal for the investor to
Utility functions are discussed in Chapter 1 of Bowers et al. (1986). The power utility function is often used in financial economics because it has a particularly simple form and has reasonable properties concerning both absolute and relative risk aversion. It has increasing risk aversion and constant relative risk aversion. These concepts are discussed by Arrow (1971), and he also provides an intuitive discussion of their implications for investment decisions.
1
*Phelim P. Boyle, F.C.I.A., Ph.D., is at the Centre for Advanced Studies in Finance, University of Waterloo, Ontario, N2L 3G1, Canada. †Xiaodong Lin, A.S.A., is Assistant Professor in the Department of Statistics, University of Toronto, Ontario, M5S 3G3, Canada.
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NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 2
keep a constant fraction in the risky asset. As time passes, the portfolio is assumed to be adjusted so that this fraction is maintained. This can be accomplished without cost because we are assuming no transaction costs. The Merton ratio has a number of intuitive properties. First, note that the optimal fraction is proportional to (µ2r), the risk premium on the stock. Other things being equal, the lower the risk premium, the lower the optimal fraction invested in the stock. In the limit, when µ5r, the fraction invested in the stock is zero. This makes intuitive sense. Second, the optimal fraction invested in the stock is inversely proportional to the variance of the rate of return on the stock. Other things being equal, the optimal fraction invested decreases as the variance increases. Third, the optimal fraction invested in stock is inversely proportional to (12a). Now since 2
WU" (W) 5 1 2 a, U'(W)
(12a) is a measure of the investor’s relative risk aversion. The intuition here is that as the investor becomes more risk averse, he or she will invest less in the stock. We can use the Merton ratio to estimate numerical values of the optimal fraction in stock for representative values of the parameters. In the case of a diversified common stock portfolio, historical estimates might be µ50.1, s50.2. For the short-term riskless rate, let us take r50.05. There are different estimates of a in the empirical literature, but based on the paper2 by Constantinides (1990), let us assume a521, so that (12a)52. With these estimates, the Merton ratio works out to be 62.5%, which is remarkably close to the 60% referred to earlier. The introduction of transaction costs adds considerable complexity to the problem. It is no longer optimal to maintain a fixed constant proportion in the risky asset at all times. The investor has to trade off the benefits of moving closer to the desired position through costly transactions against the costs associated with making these transactions. The problem is simplified if we assume that the transaction costs are proportional to the amount of the risky asset traded and that there are no transaction costs on trades in the riskless asset. We will maintain these assumptions. It is also convenient to assume that the investor has a power utility function. When we include transaction costs, the optimal strategy is defined in terms of
Constantinides uses a value of (12a) equal to 2.2 in his paper.
2
bounds on the fraction to be held in the stock. As long as the proportion invested in the stock lies within these bounds, the portfolio is not adjusted. If the fraction strays outside these bounds, a transaction is made to restore the fraction to the nearest boundary. As the transaction rate goes to zero, both bounds converge to the Merton ratio. Conversely, as the transaction cost rate increases, the bounds widen. Several authors have made important contributions to the effect of transaction costs in this setting, including Kamin (1975), Magill and Constantinides (1976), and Constantinides (1979). Constantinides (1979) showed that in the case of proportional transaction costs and power utility, the no-transaction regions are convex cones. Consequently, the derivation of the boundaries of the no-transaction region is of central importance for all practical applications. It is generally believed that these boundaries cannot be obtained analytically and hence approximations are needed. Constantinides (1986) has developed approximate solutions for the case of an investor with a power utility. Often these solutions deal with the case in which the investor has an infinite horizon. For practical applications it may be more realistic to assume a finite terminal date. For this case, Gennotte and Jung (1994) have developed a numerical approach that can be used to obtain approximate values of the boundaries. This paper provides an explicit closed-form solution to the finite horizon problem when there are proportional transaction costs and the investor has a power utility function. Our procedure enables us to derive the boundaries of the no-transaction region in a systematic fashion. We derive an explicit representation of the optimal trading strategy. As previous authors have shown, the optimal strategy involves refraining from trading if the fraction of the risky asset lies within the no-transaction region. The layout of the rest of the paper is as follows. We set up the model in Section 2 and convert it into a dynamic programming problem. Also in this section we introduce the concept of indirect utility function. We develop the optimal trading strategy in Section 3 and obtain explicit expressions for the indirect utility functions. This enables us to develop analytical solutions for the boundaries of the no-transaction region. In Section 4 we indicate how the method can be applied to the multiperiod binomial model. We provide numerical examples in Section 5 to illustrate the approach and indicate how transaction costs affect the optimal asset allocation. Section 6 summarizes the paper.
OPTIMAL PORTFOLIO SELECTION
WITH
TRANSACTION COSTS
2. THE MODEL Consider an economy with one risky asset and one riskless asset available for trading. The risky asset pays no dividends. There are T11 trading dates indexed as t50, 1, z z z, T over a certain period. We denote St and Bt as the prices of the risky asset and the riskless asset, respectively, at time t. Hence, for t50, 1, z z z, T21, zt 5
St11 B , r 5 t11 St Bt
(2.1)
are the total rates of return on the risky asset and the riskless asset, respectively. Thus, zt is a random variable and r is a constant. It is assumed that z0, z1, z z z, zT21 are independent discrete random variables with a finite number of states. In this case, the price process of the risky asset has independent increments. For instance, if we assume that at each trading period, the return of the risky asset has only two possible outcomes: moving up by u units or moving down by d units, that is, Pr(zt 5 u) 5 p, (2.2) Pr(zt 5 d) 5 1 2 p, d , r , u, 0 , p , 1, then by properly choosing u and d, we will obtain the binomial process as in Cox, Ross and Rubinstein (1979). Suppose that the agent holds a portfolio with dollar amounts x00 in the riskless asset and x0 in the risky asset at time 0. He may adjust the portfolio at each trading date t50, 1, z z z, T21 to maximize his expected utility of terminal wealth. We assume that the utility function U has the following form U(W )5 (1/a)W a for a≤1.3 Thus, U is concave differentiable and homogeneous of degree a. Hence, the agent has constant relative risk aversion. Let x0t and xt be the dollar amounts of the riskless asset and the risky asset in the portfolio at time t before trading. Let (y0t , yt) be the corresponding portfolio holdings after trading at time t. We assume that there is a transaction cost every time we trade the risky asset. The cost is proportional to the dollar amount of the risky asset traded. Let u be the dollar amount paid whenever we buy or sell one dollar of the risky asset. Then the following relations hold: yt 5 xt 1 vt , yt0 5 xt0 2 vt 2 u|vt|,
a50 corresponds to logarithmic utility.
3
(2.3) (2.4)
29
where vt is the dollar amount of the risky asset traded at time t. A positive vt indicates that we buy the risky asset and a negative vt indicates that we sell the risky asset. We further assume that y0t and yt are non-negative.4 The portfolio amounts prior to trading at time t11 are then xt11 5 yt zt 5 (xt 1 vt) zt 0 xt11 5 y0t r 5 (x0t 2 vt 2 u|vt|) r.
(2.5) (2.6)
Thus, v0, v1, z z z, vT21 represent a trading strategy, and 0 (x00, x0), (x01, x1), z z z, (xT21 , xT21), (x0T, xT), which satisfy (2.5) and (2.6), are the portfolios held over the entire trading period corresponding to the strategy. One objective is to find an optimal trading strategy v0, v1, z z z, vT21 that maximizes the expected utility of terminal wealth, namely, max vt,t50,1,z z z,T21
E[U(x0T 1 xT)],
(2.7)
for the given initial portfolio (x00, x0). This problem can be solved by a dynamic programming technique.5 To apply dynamic programming, we define the indirect utility functions Jt, t50, 1, 2, z z z, T, as follows: JT(x0T, xT) 5 U(x0T 1 xT),
(2.8)
and for t50, 1, 2, z z z, T21, Jt(x0t , xt) 5 max Et Jt11(x0t11, xt11),
(2.9)
vt
where x0t11 and xt11 are given by (2.5) and (2.6) and Et denotes the expectation over zt conditional on x0t and xt. The method here is similar to those used in intertemporal portfolio selection without transaction costs (see Ingersoll 1987, Chap. 11). By the Bellman principle of optimality (Bellman and Dreyfus 1962 or Bertsekas 1987), we have J0(x00, x0) 5
max vt,t50,1,z z z,T21
E[U(xT0 1 xT)]
(2.10)
for any given initial portfolio (x00, x0). The variables vt, which maximize Et Jt11 (x0t11, xt11), t50, 1, z z z, T21, form the optimal trading strategy. In the next section, we will see that introducing indirect utility functions not only enables us to find the optimal strategy for a particular initial portfolio but also gives us more insights about the optimal trading during the trading period.
This restriction can be relaxed. See the discussion at the end of this section. 5 For the theory of dynamic programming, refer to Hillier and Lieberman (1990, Chap. 11) for an introductory level and Bertsekas (1987) for a more advanced level. 4
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NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 2
3. OPTIMAL TRADING STRATEGY In this section, we develop an analytical method for solving the indirect utility functions. The optimal trading strategy is subsequently given for any portfolio at any time. We show that at time t, the portfolio space is divided into three disjoint positive cones,6 which can be specified as the buying region, the selling region and the no-transaction region. If a portfolio lies in the buying region, the optimal strategy is to buy the risky asset until the portfolio reaches the boundary of the buying region, while if a portfolio lies in the selling region, the optimal strategy is to sell the risky asset until the portfolio reaches the boundary of the selling region. If a portfolio lies in the no-transaction region, it is not adjusted at that time. Furthermore, the procedure used in the proof enables us to solve for the boundaries. Let Gt be the set of portfolios defined by Gt 5 {(xt0, xt); (3.1) 0 for all vt, Et Jt11 (xt11 , xt11) ≤ Et Jt11 (x0t r, xt zt)}. For any portfolio (x0t , xt) in Gt, vt50 is a maximum point of Et Jt11 (x0t11, xt11). Thus, the expected value will not be increased by buying or selling the risky asset. In other words, it is optimal not to transact a portfolio if it lies in Gt. For this reason, Gt is called the no-transaction region at time t. The following function plays a central role in our analysis: ft (vt, x0t , xt) 5 Et Jt11 (x0t11, xt11),
In this case, the portfolio space is divided into a finite number of cones and on each cone the utility function J is a convex combination of the utility function U. We now present our main result. Theorem Suppose that the terminal utility function U (W) is a concave, homogeneous differentiable function with degree a. Then for each, t, t50, 1, 2, z z z, T, (i) Jt(x0t , xt) is a concave homogeneous differentiable function (ii) Jt(x0t , xt) is a piece-wise linear utility function with respect to U (iii) There is at≤bt such that the no-transaction region at time t is Gt 5
$
(x0t , xt); at ≤
%
xt ≤ bt x0t
If xt /xt0,at, we buy the risky asset with amount vt1, which is given in (3.17). If xt /x0t .bt, we sell the risky asset with amount v2t , which is given in (3.19) (Figure 1).
PORTFOLIO SPACE
Figure 1 TRADING STRATEGY
AND
AT
TIME t
(3.2)
where x0t11 and xt11 are given by (2.5) and (2.6). We have already seen that Gt is the set of portfolios (xt0, xt) with ft(vt, x0t , xt)≤ft (0, x0t , xt). We next introduce the definition that will be used to characterize the indirect utility functions. Definition We say a utility function J is a piece-wise linear utility function with respect to a utility function U if there is a sequence of increasing numbers qj, j50, 1, z z z, s, and non-negative constants aij and bij with respect to the underlying probability space {wi; i51, z z z, I} such that J(x , x) 5
Σ U(a x
i51
ij
0
1 bij x) Pr(wi),
x for qj ≤ 0 , qj11. x
(3.3) (3.4)
A cone G is a set such that for any g∈G and any non-negative a, ag∈G. A positive cone is a cone contained in the positive orthant.
6
]+ ft (0, 1, at) 5 0, ]vt
(3.5)
and bt is the largest solution of
I
0
Furthermore, if 0,at≤bt,`, then at is the smallest solution of
]2 ft (0, 1, bt) 5 0, ]vt
(3.6)
where ]+ft /]vt and ]2ft /]vt denote the right and left derivatives of ft, respectively.
OPTIMAL PORTFOLIO SELECTION
WITH
TRANSACTION COSTS
We now make the following observations on these results. The result (i) and the first part of (iii) have been derived earlier by Constantinides (1979) and by Gennotte and Jung (1991). However, their methods do not provide an analytical solution for at and bt. An advantage of our approach is that the indirect utility functions can explicitly be solved and characterized. The boundary at and bt can also be solved analytically from the equalities in the theorem. Hence, they are exact. We illustrate the procedure in the next section. The proof of the theorem is based on the following lemmas. The proof of all these lemmas is constructive. These constructive proofs provide us with a procedure for calculating the indirect utility functions and the boundaries of the no-transaction regions. Throughout the rest of this section, we always assume that the utility function U (W) is concave, differentiable and homogeneous of degree a.
31
Hence ]+ ft (0, x0t , xt) ]2 ft (0, x0t , xt) ≤ . ]vt ]vt Therefore, ft is concave everywhere.
]2 ft (0, x0t , xt) ]+ ft (0, x0t , xt) ≥ 0 and ≤ 0. ]vt ]vt
(3.10)
Now we define at 5 min
$
xt ≥ 0;
]2 ft (0, 1, xt) ≥ 0, ]vt
%
]+ ft (0, 1, xt) ≤0 , ]vt
If Jt11 is a concave homogeneous differentiable function of degree a, ft(vt, xt0, xt) is a concave function with respect to vt and is a homogeneous function of degree a with respect to vt, x0t , and xt.
Homogeneity of ft follows from its definition. Obviously, for vt≠0, ft is a concave function. At vt50
n
Since ft is concave with respect to vt, any local maximum of ft will be a global maximum. Thus a portfolio (x0t , xt)∈Gt if and only if vt50 is a maximum point for ft (vt, x0t , xt). Equivalently, we have that (x0t , xt)∈Gt if and only if
Lemma 1
Proof
(3.9)
bt 5 max
$
xt ≥ 0;
(3.11)
]2 ft (0, 1, xt) ≥ 0, ]vt
%
]+ ft (0, 1, xt) ≤0 . ]vt
(3.12)
]2ft(0,1,0) ]+ft(0,1,`) 5` and 52`. ]vt ]vt The next lemma characterizes the no-transaction region. It shows the no-transaction region is a positive cone. with
]+ ft (0, x0t , xt) 5 lim 2 r (1 1 u) ] vt vt→01 Et
Lemma 2
] Jt11[(xt0 2 (1 1 u)vt) r, (xt 1 vt) zt] ]x0t11
A portfolio (x0t , xt) belongs to Gt if and only if the ratio xt /xt0 satisfies at ≤xt /xt0≤bt. If we consider the proportion of the value of the risky asset to the value of total asset, xt /(xt1x0t ), we have
] Jt11[(x0t 2 (1 1 u) vt) r, (xt 1 vt) zt] 1 Et zt z ]xt11 5 2 r (1 1 u) Et 1 Et zt z
]Jt11(xt0r, xt zt) 0 ]xt11
] Jt11(x0t r, xt zt) ]xt11
where z denotes the inner product in the two-dimensional Euclidean space. Similarly, ] ft (0, x , xt) ] J (x r, x z ) 5 2 r (1 2 u) Et t11 0 t t ] vt ]xt11 2
0 t
0 t
1 Et zt z
at xt bt ≤ ≤ . 0 1 1 at xt 1 xt 1 1 bt
(3.7)
] Jt11(x0t r, xt zt) . ]xt11
(3.8)
Remark Since Gt denotes the no-transaction region at time t [see (3.1)], Lemma 2 implies that the no-transaction region at time t is a positive cone.
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NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 2
Proof
Lemma 3
] ft(0,x ,xt) ≥0. Since ft is homo]vt ]2ft(0,x0t ,xt) geneous with degree a, is homogeneous ]vt with degree a21. Hence, 2
0 t
Given (x0t , xt)∈Gt,
]2 ft(0, 1, xt /x0t ) ]2 ft(0, x0t , xt) 5 (x0t )12a ≥ 0. ]vt ]vt
Suppose Jt11 is a differentiable function. Then, if 0,at,`,
m 2 bt , 0, 1 1 (1 2 u) m (1 2 u) mbt 1 m g5 . 0. (1 2 u) mxt 1 xt
]2 ft(0, 1, bt) 5 0. ]vt
(3.15)
Proof
Thus, at ≤xt /x0t ≤bt. Conversely, suppose that m5xt /x0t and at,m,bt. Let v*t 5
(3.14)
If 0,bt,`,
Similarly, we can show ]+ ft(0, 1, xt /x0t ) ≤ 0. ]vt
]+ ft(0, 1, at) 5 0. ]vt
(3.13)
Suppose that
]+ft(0,1,at) ,0. There is xt,at such that ]vt
]+ft(0,1,xt) ≤0 by continuity. From the proof of ]vt Lemma 2, there is v*t,0 and g.0 replacing m and bt by xt and at, such that ]2 ft(0, 1, xt) ]f (v*, 1, at) 5 g 12a t t ]vt ]vt
Then g a21
≥ g 12a
]2 ft(0, xt0, xt) ] vt
5 g a21
$
%
5
Lemma 3 provides a method to calculate the boundaries of the no-transaction regions. In fact, they can be obtained by solving two sets of algebraic equations. In Lemma 4, the amount of the risky asset to buy or to sell at time t is calculated when a portfolio is outside the no-transaction region.
] Jt11(gx0t r, gxt zt) ] x0t11
] Jt11(gx0t r, gxt zt) ] xt11
5 2 r (1 2 u) Et 1 Et zt z
] Jt11(xt0r, xt zt) ] x0t11
] Jt11(x0t r, xt zt) ] xt11
5 2 r (1 2 u) Et 1 Et zt z
]+ft(0,1,xt) ]2ft(0,1,xt) ≤0 and ≥0, xt∈Gt. Thus, ]vt ]vt at is not a minimum, which is a contradiction. By a similar argument, the second conclusion can be shown. n Since
2r (1 2 u) Et
1 Et zt z
]2 ft(0, 1, at) ≥ 0. ]vt
] Jt11((1 2 (1 2 u)v*) t r, (bt 1 v*)z t t) 0 ] xt11
] Jt11((1 2 (1 2 u)v*)r, (bt 1 v*)z t t t) ] xt11
Lemma 4 If xt /x0t ,at, then
]ft(v*,t 1, bt) ]2 ft(0, 1, bt) ≥ ≥0 ] vt ] vt
]2ft(0,xt0,xt) Thus, ≥0. By properly choosing g and v*t ]vt again, we can show that ]+ ft(0, xt0, xt) ≤0 ]vt Hence, (x0t , xt)∈Gt.
max ft(vt, x0t , xt) 5 ft(v1t , x0t , xt) v t
1 5 ft(0, y01 t , yt ),
(3.16)
where x0t at 2 xt , 1 1 (1 1 u) at yt1 5 xt 1 vt1, y01 5 x0t 2 (1 1 u)v1t , t vt1 5
n
1 1 01 with (y01 t , yt )∈Gt and yt /yt 5at.
(3.17)
OPTIMAL PORTFOLIO SELECTION
WITH
TRANSACTION COSTS
If xt /xt0.bt, then
Lemma 5
max ft(vt, x , xt) 5 ft(v , x , xt) v 2 t
0 t
t
If Jt11 is differentiable, so is Jt.
0 t
2 5 ft(0, y02 t , yt )
(3.18) Proof
where x0t bt 2 xt v 5 , 1 1 (1 2 u)bt y2t 5 xt 1 v2t , y02 5 x0t 2 (1 2 u)v2t . t 2 t
(3.19)
2 2 02 with (y02 t , yt )∈Gt and yt /yt 5bt.
Proof We shall prove the first half of the lemma, and the proof of the second half logically follows. 01 1 It is easy to see v1t .0, y1t /y01 t 5at, (yt , yt )∈Gt, and 1 ft(v1t , x0t , xt)5ft(0, y01 t , yt ). First, we assume at,`. To show v1t is a maximum it suffices to show ]ft(v1t , x0t , xt) 5 0, ]vt
(3.20)
since v .0. Now, 1 t
]2 Jt(x0t , xt) ]+ ft(0, xt0, xt) ]vt1 5 0 ]xt ]vt ]xt0 ]f (0, xt0, xt) 1 t ]x0t ]+ ft(0, 1, at) ]v1t 5 (x0t )a21 ]vt ]x0t ]f (0, xt0, xt) 1 t ]x0t ]f (0, xt0, xt) ]+ Jt(xt0, xt) 5 t 5 , ]x0t ]x0t [by (3.22)]
(3.23)
]+ft(0,1,at) 50 by Lemma 3. ]vt Now we differentiate Jt with respect to the second argument xt.
5 2 r(1 1 u) Et
] Jt11((x0t 2 (1 1 u)v1t ) r, (xt 1 v1t ) zt) ]x0t11
] Jt11((x0t 2 (1 1 u)v1t ), r, (xt 1 v1t ) zt) ]xt11
5 (x0t 2 (1 1 u)v1t )a21
~2 r (1 1 u) E ] J
(r, at zt) 0 ]xt11
t11
t
1 Et zt z
Obviously, Jt is differentiable in the interior of three disjoint sets: xt /x0t ,at, at,xt /xt0,bt, and xt /xt0.bt. We now show that Jt is differentable on the lines xt /x0t 5 at and xt /xt05bt. For any given (x0t , xt) with xt /x0t 5at,
since
]ft(v1t , x0t , xt) ] vt
1 Et zt z
33
!
] Jt11(r, at zt) ]xt11
5 (xt0 2 (1 1 u)vt1)a21
]+ ft(0, 1, at) . ] vt
]ft(v , x , xt) 5 0. ]vt 0 t
(3.21)
If at5`, ft is nondecreasing, v1t is the right endpoint of its domain and so v1t is the maximum point. n From Lemmas 2 and 4, we have Jt(x0t , xt) 5
$
ft(v1t , x0t , xt), xt /x0t , at, ft(0, x0t , xt), at ≤ x0 /xt0 ≤ bt, 2 0 ft(vt , xt , xt), xt /x0t . bt.
(3.24)
The differentiability of Jt on the line xt /x0t 5bt can be shown similarly. n
By Lemma 3, 1 t
]2 Jt(x0t , xt) ]f (0, x0t , xt) 5 t ]xt ]xt ]+ ft(0, 1, at) ]v1t 5 (x0t )a21 ]vt ]xt ]ft(0, x0t , xt) 1 ]xt ]+ Jt(xt0, xt) 5 . ]xt
(3.22)
The next lemma characterizes the indirect utility functions Jt, t50, 1, z z z, T. It shows that the piecewise linearity property is preserved as we move back in time. This property plays a key role in the derivation of the boundaries and the indirect utilities. Lemma 6 If Jt11(x0t11, xt11) is a piece-wise linear utility function with respect to U, so is Jt(xt0, xt).
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NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 2
where (r/z1t ) qj ≤ q˜ h, q˜ h11≤(r/z1t ) qj11; for g5I11, z z z, 2I,
Proof Let {wi} be all possible outcomes from (zt11, zt12, z z z, zT), where {wi} represents the set of all future paths of the underlying tree structure starting at a node at time t11. Hence if {zkt ; k51, 2, z z z} are all possible outcomes for zt, then all the paths of the underlying tree structure starting at time t can be written as {(zkt , wi)}. We first show that ft(0, x0t , xt) is a piece-wise linear utility function with respect to U. Suppose that7
Σ U(a x I
Jt11(x0t11, xt11) 5
ij
i51
0 t11
1 bij xt11) Pr(wi),
xt11 , qj11, j 5 0, 1, z z z, s. 0 xt11 q0 5 0, qs11 5 `. qj ≤
a˜ gh 5 ag2I, jr, b˜ gh 5 bg2I, j z2t , where (r/z2t )qj ≤ q˜ h,q˜ h11≤(r/z2t ) qj11.8 We next show that Jt (x0t , xt) is a piece-wise linear utility function with respect to U. From Lemma 4,
$
ft(0, yt01, yt1), xt /xt0 , at, ft(0, xt0, xt), at ≤ xt /xt0 ≤ bt, 02 2 ft(0, yt , yt ), xt /xt0 . bt.
Jt(x , xt) 5 0 t
(3.29)
It follows from the fact that ft(0, x0t , xt) is a piece-wise linear utility function with respect to U, that Jt (x0t , xt) is a piece-wise linear utility function on the cone at ≤xt /xt0≤bt. We now analyze the region xt /x0t ,at. Suppose that q˜ j1,at ≤ q˜ j111. From (3.29), 1 Jt(x0t , xt) 5 ft(0, y01 t , yt )
Let ck(xt0, xt) 5 Jt11(xt0r, xt ztk) Pr(zt 5 ztk).
5
(3.25)
Then
Σ U(a˜
i j1
i
y01 1 b˜ i j1 y1t ) Pr(w ˜ i), t
(3.30)
w ˜ i is one of {(z kt , wi)}. Since
Σ U(a I
ck(x0t , xt) 5
i51
ij
rx0t 1 bij zkt xt) Pr{(zkt , wi)}
r x r q ≤ t , k qj11. ztk j xt0 zt
(3.26)
yt1 5 xt 1
(3.27)
Hence ck(x0t , xt) is a piece-wise linear utility function with respect to U. We now show ft(0, xt0, xt) 5
Σ c (x , x ) k
k
0 t
t
(3.28)
is also a piece-wise linear utility function with respect to U. It suffices to show the sum of two piece-wise linear functions is a piece-wise linear function. Then, by induction the sum of any finite number of piecewise linear functions is a piece-wise linear function. Relabel all (r/zkt )qj, k51, 2, j50, 1, z z z, as q˜h in order of magnitude. Then for
a˜ gh 5 ag jr, b˜ gh 5 bg j zt1,
We begin with t115T. Thus, in this case s50, q15` and JT (x0T, xT)5U(x0T1xT).
7
5
at (1 1 u) at x0t 1 xt, 1 1 (1 1 u)at 1 1 (1 1 u)at
(3.31)
y01 5 t
1 11u x0t 1 xt, 1 1 (1 1 u)at 1 1 (1 1 u)at
(3.32)
Jt (x0t , xt)
Σ
(3.33)
@
a˜ i j1 1 at b˜ i j1 0 5 U x i 1 1 (1 1 u)at t (1 1 u) (a˜ i j1 1 at b˜ i j1) 1 xt 1 1 (1 1 u)at
#
Pr(w ˜ i).
(3.34)
for xt /xt0,at. This is obviously in the form of (3.3). Similarly, for xt /x0t .bt, let q˜ j2≤bt,q˜ j 211. Then, Jt(x0t , xt) 5
x q˜ h ≤ 0t , q˜ h11, xt and g51, z z z, 2I, we define a˜ gh and b˜ gh as follows: for g51, z z z, I,
x0t at 2 xt 1 1 (1 1 u)at
Σ U@ 1 a˜1 (11 2b u)b b i j2
i
1
t
˜ij
2
x0t
t
#
(1 2 u) (a˜ i j2 1 bt b˜ i j2) xt 1 1 (1 2 u) bt
Pr (w ˜ i).
(3.35)
Therefore Jt (x0t , xt) is a piece-wise linear utility function with respect to U. n We are now ready to prove the main theorem.
This procedure will become somewhat clearer when it is illustrated in the next section.
8
OPTIMAL PORTFOLIO SELECTION
WITH
TRANSACTION COSTS
35
Proof
where zH and zL are the highest and lowest returns of z, respectively. Then all the arguments in this section apply.
Results (i) and (ii) are a direct corollary of Lemmas 5 and 6. The first part of (iii) is proved in Lemmas 2 and 4, and the second part is proved in Lemma 3. n We now make some additional comments about the results. 1. Suppose that J0 (x00, x0) 5
Σ U(a
ij
i
x00 1 bi j x0) Pr(wi)
(3.36)
qj ≤ x0 /x , qj11 0 0
It is easy to see that ai j x001bi j x0 is the terminal wealth if the path wi is realized. Therefore, ai j and bi j are marginal rates of the riskless and risky assets, respectively, for those portfolios whose ratio is between qj and qj11. 2. If the equations in the theorem have no solutions, then at and bt can be easily identified; at takes one of two values. Either at 5 0, if
]1 ft(0, 1, 0) , 0, ]vt
(3.37)
at 5 `, if
]1 ft(0, 1, `) . 0. ]vt
(3.38)
or
In the second case, we must have bt5`. In the same way, bt can only take one of two values. Either bt 5 0, if
]2 ft(0, 1, 0) , 0, ]vt
(3.39)
4. AN APPLICATION In this section, we illustrate how to use the method developed in the last section to calculate the indirect utility function Jt, and the boundary points at and bt of the no-transaction region Gt, for t50, 1, z z z, T21. We consider the case in which the rate of return for the risky asset in each period is independent of t and has only two states. In other words, the price of the risky asset either goes up by u or down by d, that is, Pr(zt 5 u) 5 p, Pr(zt 5 d) 5 1 2 p, d , r , u, 0 , p , 1,
We further assume that the rate of return r for the riskless asset satisfies d,r,u (no arbitrage condition). If we choose u5es=1/n, d5e2s=1/n, r5ed/n and p5 1/2 [11(µ/s) =1/n], we obtain the binomial model proposed by Cox, Ross, and Rubinstein (1979) with T5n. An alternative choice is u511µ/n1s/=n, d511 µ/n2s/=n, r5ed/n, and p51/2, which was proposed by Hua He (1990). The price movement in both models will lead to the Black-Scholes economy, when n goes to infinity. In the Black-Scholes economy the prices are described by the stochastic differential equation9 dBt 5 dBt dt dSt 5 µSt dt 1 sSt dWt,
or bt 5 `, if
]2 ft(0, 1, `) . 0. ]vt
(4.1)
(3.40)
In the first case (when bt50), we must have at50. 3. If the terminal utility function is strictly concave, then the solution of the two equations is unique. Furthermore, if there is no transaction cost, that is, u50, then at5bt. 4. In this paper, we require that y0t ≥0 and yt ≥0; that is, buying and selling assets short is not allowed. However, our results and methodology can also be extended to the general case, which allows negative investment, but the agent is required to be solvent during the trading period (see Constantinides 1979, p. 1129). In that case, the domain of Jt is no longer the positive orthant but the following: {(x0t , xt); u1 , arg(x0t 1 ixt) , u2, u1 5 2 arctan (r/zH), u2 5 p 2 arctan (r/zL)},
where Wt is an one-dimensional standard Brownian motion (Black and Scholes 1973). In this section, we use Hua He’s approach since it is the simplest for our purpose. However, our method can be applied to any other discrete model. The terminal utility function used in this example is the power function U(W) 5
Wa , a ≤ 1, a ≠ 0. a
We now calculate Jt, at and bt recursively starting at t5T. At t5T,
For the theory of stochastic differential equations, see Oksendal (1985).
9
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NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 2
JT(x0T, xT) 5 U(x0T 1 xT) 5
(x0T 1 xT)a . a
(4.2)
Now, suppose that Jt11(x0t11, xt11) is solved, that is,
for vt ≤ 0. Hence from (3.5) and (3.6) in the main theorem, at is a solution of one of the equations
Σ (a˜ i
ij
1 b˜ ij at)a21 [b˜ ij 2 (1 1 u)a˜ ij] 5 0, q˜ j ≤ at , q˜ j11, j 5 0, 1, 2, z z z
2T2t21
Σ
Jt11(x0t11, xt11) 5
U(ai j x0t11 1 bi j xt11)Pr (wi)
i51
Σ
1
5
and bt is a solution of one of the equations (ai j x0t11 1 bi j xt11)a,
(4.3)
2T2t21a i xt11 qj ≤ 0 , qj11. xt11
Σ (a˜ i
ij
1 b˜ ij bt)a21 [b˜ ij 2 (1 2 u)a˜ ij] 5 0, q˜ j , bt ≤ q˜ j11, j 5 0, 1, 2, z z z.
We then define ft(0, xt0, xt) 5 Et Jt11(xt0r, xt zt) 1 5 [Jt11(x0t r, xtu) 1 Jt11(x0t r, xtd)]. 2
(4.4)
Rearrange all (r/u)qj and (r/d)qj, j50, 1, z z z, from smallest to largest and relabel them as
q˜ j1 , at ≤ z z z , q˜ j2 ≤ bt , z z z. Define
Then for
q¯ 0 5 0, q¯ 1 5 at, q¯ 2 5 q˜ j111, z z z, q¯ j22j112 5 bt, q¯ j22ji113 5 `,
x q˜ h ≤ 0t , q˜ h11, xt and g51, z z z, 2T2t, we define a˜ gh and b˜ gh as follows: for g51, z z z, 2T2t21,
where (r/u)qj ≤ q˜ h,˜qh11≤(r/u)qj11; for g52 2T2t,
a¯ i0 5
11, z z z,
j 5 1, 2, z z z, j2 2 j1 1 1; a¯ i, j22j112
ft(0, x , xt) 5 0 t
2T2t
1 2T2ta q˜ j ≤
Σ
i51
(a˜ ij x 1 b˜ ij xt)a. 0 t
(4.5)
xt , q˜ j11, x0t
ft(vt, x0t , xt)
Σ {a˜ [x ij
i
0 t
2 (1 1 u)vt] 1 b˜ ij(xt 1 vt)}a,
(4.6)
for vt ≥ 0, and ft(vt, x0t , xt) 5
1 2T21a
Σ {a˜ [x i
ij
0 t
2 (1 2 u)vt] 1 b˜ ij(xt 1 vt)}a,
Jt(x0t , xt) 5 q¯ j ≤
here we change g and h back to i and j to avoid too much notation. Thus,
a
(4.12)
(4.13)
Then,
Then,
2
(4.11)
a˜ i j2 1 bt b˜ i j2 5 , 1 1 (1 2 u) bt b¯ i, j22j112 5 (1 2 u)a¯ i, j22j112.
where (r/d)qj ≤ q˜ h,q˜ h11≤(r/d)qj11.
1
a˜ ij1 1 at b˜ ij1 ¯ , b 5 (1 1 u)a¯ i0; 1 1 (1 1 u)at i0
a¯ ij 5 a˜ i,j11j21, b¯ ij 5 b˜ i, j11j21,
T2t21
a˜ gh 5 ag22T2t21, j r, b˜ gh 5 bg22T2t21, j d,
(4.10)
and
a˜ gh 5 ag jr, b˜ gh 5 bg ju,
T21
(4.9)
Since W a/a is strictly concave, at most one equation in each set has a solution and the solution is unique if it exists. If there is no solution for all the equations, then at or bt will be either 0 or `, depending on the sign of the first or last equation in each set (see comment 2 in the previous section). Assume that
0 5 q˜ 0 , q˜ 1 , z z z.
5
(4.8)
(4.7)
1 2T2ta
2T2t
Σ
i51
(a¯ i j x0t 1 b¯ i j xt)a,
(4.14)
xt , q¯ j11, j 5 0, 1, z z z. x0t
5. NUMERICAL EXAMPLES In this section we illustrate the approach by using a simple numerical example. First, we solve the notransaction costs case because this provides a useful benchmark. Then we see how things change when we assume that there are transaction costs on the risky asset. We just consider a three-period problem for simplicity. Preliminary investigations of high-dimensional problems suggest that it may be challenging to
OPTIMAL PORTFOLIO SELECTION
WITH
TRANSACTION COSTS
write a simple program to implement our algorithm. We plan to explore this issue in subsequent research. We use the notation of the last section. Suppose we have n periods each of length h. We use the Cox Ross Rubinstein parametrization for u, d, r and p, viz, u 5 es=h, d 5 e2s=h, r 5 edh and p 5
eµh 2 d . u2d
We assume the investor maximizes the expected value of utility of terminal wealth with the following utility function U(W ) 5
Wa , a ≤ 1, a ≠ 0. a
In the no-transaction case, the investor’s problem can be solved as a series of one-period problems. The optimal fraction to invest in the risky asset can be obtained from maximizing the expected utility of end of period wealth. (W0(r 1 x(u 2 r))) a (W0(r 1 x(d 2 r)))a 1 (1 2 p) a a
E {U (W )} 5 p
We can treat this as a function of x and solve for the value of x that maximizes this function. The optimal value is r (1 2 u) , (r 2 d) 1 u (u 2 r) where
~(1 2p (up) 2(r r)2 d)!
1/(12a)
u5
If we assume that µ50.1, s50.25, d50.05, h50.25, and a521, then the optimal fraction to be invested in the risky asset is 0.41302 and the ratio of the optimal investment in the risky asset to the optimal investment in the riskless asset is 0.703645. Each period the same allocation is made. When we analyze the corresponding problem in which there are transaction costs, the boundaries of the no-transaction region will contain the point 0.703645. In this case in which there are no transaction costs, the initial allocation of the investor’s wealth between the two assets does not affect the investor’s welfare as long as the total wealth is held constant. This is because with no transaction costs, the risky asset can be converted without cost into the riskless asset. When there are transaction costs, this will no longer be the case. For future reference we record the
37
expected utility of the investor computed as at time zero when there are no transaction costs; it is 20.955693. The expected utility is negative because a is equal to 21 in our example. We now contrast this case with the situation in which there are transaction costs. To start with, we assume that the transaction costs are 0.1% of the risky asset. We can use the procedure developed earlier in the paper to compute the no-transaction region boundaries. We use Equations (3.5) and (3.6) to find at and bt at each step, working recursively backwards from the last period. The values we obtain are given in Table 1. TABLE 1 NO TRANSACTION BOUNDS WHEN u 5 0.001 a0 5 0.639132
b0 5 0.770427
a1 5 0.643571
b1 5 0.765155
a2 5 0.614274
b2 5 0.803352
To interpret these results, note that if the investor’s initial position consists of 0.05 in the risky asset and 0.95 in the riskless asset, then the investor should buy enough of the risky asset to reach the boundary, so that the proportion of the risky asset to the riskless asset is a050.639132. Allowing for transaction costs, the amount of the risky asset to be purchased at time zero is 0.339788. Note that the no-transaction region tends to widen as we approach maturity. We also note that the Merton ratio of 0.703645 lies within the no-transaction region. These results agree with the findings of Gennotte and Jung (1994). If we increase the transaction cost rate to 1%, the bounds widen considerably. In this case the values are given in Table 2. TABLE 2 NO TRANSACTION BOUNDS WHEN u 5 0.01 a0 5 0.433005
b0 5 1.097272
a1 5 0.331171
b1 5 1.363041
a2 5 0.093558
b2 5 2.789939
The increase in transaction costs has a dramatic impact on the width of the bounds. We would expect that the investor would be better off as the level of transaction costs declines. We would
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NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 2
also expect that the investor’s expected utility will depend on the initial holdings in the risky asset. It should be better to start with a position in the risky asset that corresponds to the interior of the no-transaction region. The next set of results shows how the investor’s expected utility changes with the initial position in the risky asset and also with the level of transaction costs. We assume three initial situations corresponding to 5%, 40%, and 95% in the risky asset. The corresponding ratios of the risky to the riskless asset are 0.0526, 0.6667, and 19, respectively. The three transaction cost rates are 0, 0.1% and 1%. The investor’s time zero value of expected utility for these cases are shown in Table 3. TABLE 3 INVESTOR’S EXPECTED UTILITY FOR DIFFERENT INITIAL ASSET ALLOCATIONS AND FOR DIFFERENT LEVELS OF TRANSACTION COSTS
Transaction Cost Rate
Initial Percentage in Risky Asset
0
0.1%
1%
5 40 95
20.955693 20.955693 20.955693
20.956059 20.955727 20.956229
20.958668 20.955740 20.960386
The procedures developed in our paper may be useful in considering more general versions of the portfolio problem as well as other situations, such as more general utility functions and the inclusion of intermediate consumption. Another interesting extension would be the case of two or more risky assets. The two-asset case presents formidable problems because in general we do not know what the no-transaction regions look like. On the computational front we would like to develop an efficient algorithm to implement the method developed in the current paper. These issues are left for further research.
ACKNOWLEDGMENTS Xiaodong Lin was partly supported by the Natural Sciences and Engineering Research Council of Canada. The authors are grateful to Inmoo Lee for research assistance. They are grateful to Elias Shiu and anonymous referees for helpful suggestions. They also appreciate the helpful input of the following master’s students at the University of Waterloo: Sean Finucane, Peter Fitton, Lau Sok Hoon, and Vincent Lee and especially Claire Bilodeau.
REFERENCES We observe that these results confirm our intuition. The highest levels of expected utility occur when there are no transaction costs and decline as we move across the table from left to right in the direction of increasing transaction costs. When there are transaction costs, the expected utility levels are highest for initial positions that lie within the no-transaction region.
6. SUMMARY In this paper, we have analyzed the portfolio selection problem of an investor when there are proportional transaction costs on the risky asset. Since the work of Constantinides, it has been known that under certain assumptions the no-transaction region consists of a cone. Furthermore, Gennotte and Jung (1994) have provided a numerical procedure to obtain numerical solutions for the values of the boundaries of the notransaction region. This paper shows explicitly the functional form of the indirect utility function. The proof of our main theorem provides a constructive analytical procedure for determining the no-transaction region. Once we know this region, the investor’s problem is solved.
ARROW, K.J. 1971. Essays in the Theory of Risk Bearing. Amsterdam: North Holland. BELLMAN, R., AND DREYFUS, S. 1962. Applied Dynamic Programming. Princeton, N.J.: Princeton University Press. BENSAID, B., LESNE, J.P., PAGES, H., AND SCHEINKMAN, J. 1992. ‘‘Derivative Asset Pricing with Transaction Costs,’’ Mathematical Finance 2:63–87. BERTSEKAS, D.P. 1987. Dynamic Programming: Deterministic and Stochastic Models. Englewood Cliffs, N.J.: Prentice-Hall. BLACK, F., AND SCHOLES, M.J. 1973. ‘‘The Pricing of Options and Corporate Liabilities,’’ Journal of Political Economy 81: 637–54. BOWERS, N., GERBER, H., HICKMAN, J., JONES, D., AND NESBITT, C. 1986. Actuarial Mathematics, Itasca, Ill.: Society of Actuaries. BOYLE, P.P., AND VORST, T. 1992. ‘‘Option Replication in Discrete Time with Transaction Costs,’’ Journal of Finance 47:271–93. CONSTANTINIDES, G.M. 1979. ‘‘Multiperiod Consumption and Investment Behavior with Convex Transaction Costs,’’ Management Science 25:1127–37. CONSTANTINIDES, G.M. 1986. ‘‘Capital Market Equilibrium with Transaction Costs,’’ Journal of Political Economy 94:842–62. CONSTANTINIDES, G.M. 1990. ‘‘Habit Formation: A Resolution of the Equity Premium Puzzle,’’ Journal of Political Economy 98:519–43.
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TRANSACTION COSTS
DAVIS, M., AND NORMAN, A. 1990. ‘‘Portfolio Selection with Transaction Costs,’’ Mathematics of Operations Research 15:676–713. DUMAS, B., AND LUCIANO, E. 1991. ‘‘An Exact Solution to a Dynamic Portfolio Choice Problem under Transaction Costs,’’ Journal of Finance 46:577–95. EDIRINGSHE, C., NAIK, V., AND UPPAL, R. 1993. ‘‘Optimal Replication of Options with Transaction Costs and Trading Restrictions,’’ Journal of Financial and Quantitative Analysis 28:117–38. GARMAN, M.B. 1976. ‘‘An Algebra for Evaluating Hedge Portfolios,’’ Journal of Financial Economics 3:403–27. GENNOTTE, G., AND JUNG, A. 1994. ‘‘Investment Strategies under Transaction Costs: The Finite Horizon Case,’’ Management Science 3:385–404. GILSTER, J.E., JR., AND LEE, W. 1984. ‘‘The Effect of Transaction Costs and Different Borrowing and Lending Rates on the Option Model: A Note,’’ Journal of Finance 39:1215–21. HE, H. 1990. ‘‘Convergence from Discrete to Continuous Time Contingent Claim Prices,’’ Review of Financial Studies 3: 523–46. HILLIER, H., AND LIEBERMAN, G. 1990. Introduction to Operations Research. New York: McGraw Hill. HODGES, S.D., AND NEUBERGER, A. 1989. ‘‘Optimal Replication of Contingent Claims under Transaction Costs.’’ Working paper, Financial Options Research Center, University of Warwick, Coventry, U.K. HUANG, C., AND LITZENBERGER, R.H. 1988. Foundations for Financial Economics. Englewood Cliffs, N.J.: Prentice-Hall. INGERSOLL, J.E., JR. 1987. Theory of Financial Decision Making. Savage, Md.: Rowman & Littlefield. KAMIN, J. 1975. ‘‘Optimal Portfolio Revision with a Proportional Transaction Cost,’’ Management Science 21:1263–71.
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LELAND, H.E. 1985. ‘‘Options Pricing and Replication with Transaction Costs,’’ Journal of Finance 40:1238–1301. MAGILL, M., AND CONSTANTINIDES, G. 1976. ‘‘Portfolio Selection with Transaction Costs,’’ Journal of Economic Theory 13: 245–63. MERTON, R.C. 1969. ‘‘Lifetime Portfolio Selection under Uncertainty: The Continuous Time Case,’’ Review of Economics and Statistics 51:247–57. MERTON, R.C. 1971. ‘‘Optimum Consumption and Portfolio Rules in a Continuous Time Model,’’ Journal of Economic Theory 3:373–413. MERTON, R.C. 1973. ‘‘Theory of Rational Option Pricing,’’ Bell Journal of Economics and Management Science 4:141– 83. MERTON, R.C. 1976. ‘‘Option Pricing When Underlying Stock Returns Are Discontinuous,’’ Journal of Financial Economics 3:144–52. MERTON, R.C. 1990. Continuous Time Finance. Cambridge, Mass.: Basil Blackwell. OKSENDAL, B. 1985. Stochastic Differential Equations. New York: Springer-Verlag. SHEN, Q. 1990. ‘‘Bid Ask Prices for Call Options with Transaction Costs.’’ Working paper, the Wharton School, University of Pennsylvania, Philadelphia. STULZ, R.M. 1982. ‘‘Options on the Minimum or Maximum of Two Risky Assets: Analysis and Applications,’’ Journal of Financial Economics 10:161–85.
Discussions on this paper will be accepted until October 1, 1997. The authors reserve the right to reply to any discussion. See the Table of Contents page for detailed instructions on the preparation of discussions.
ABSTRACT This paper examines the lifetime portfolio-selection problem in the presence of transaction costs. Using a discrete time approach, we develop analytical expressions for the investor’s indirect utility function and also for the boundaries of the no-transactions region. The economy consists of a single risky asset and a riskless asset. Transactions in the risky asset incur proportional transaction costs. The investor has a power utility function and is assumed to maximize expected utility of end-of-period wealth. We illustrate the solution procedure in the case in which the returns on the risky asset follow a multiplicative binomial process. Our paper both complements and extends the recent work by Gennotte and Jung (1994), which used numerical approximations to tackle this problem.
1. INTRODUCTION
in his book Continuous Time Finance (Merton 1990). By making certain simplifying assumptions, he was able to obtain closed-form solutions in a number of cases. The two-asset problem is of particular interest. Merton assumed that one asset has a lognormal distribution and that the other asset is a riskless money market account earning a constant return. The optimal investment decision also depends on the preferences of the investor and the problem is simplified if it is assumed that the investor has a so-called power utility function. In this case the utility function1 is of the form
The asset allocation problem is an important one in investment finance, and it also has applications in actuarial science. This is the problem faced by an investor who has to decide how to allocate his or her wealth across different assets or asset classes. Asset allocation decisions are important for pension plans. One strategic choice in the investment of pension plan assets concerns the split between stocks and bonds. In the case of final salary plans, a rule of thumb that is sometimes quoted is 60% in stocks and 40% in bonds, but the scientific support for this rule is not obvious. We can appeal to the theoretical models in financial economics to develop a framework for tackling this problem. Although these models are based on simplifying assumptions, they can provide useful insights. This paper is written in this spirit. We show how the traditional asset allocation problem can be solved when there are transaction costs. In particular, we use a multiperiod discrete time model and show how explicit solutions can be obtained in some cases. The asset allocation problem is a variant of the classic consumption investment problem in modern finance. This problem was solved in a series of brilliant papers by Robert Merton; these papers are reproduced
U(W) 5
Wa , a , 1. a
Under these assumptions Merton obtained an expression for the optimal investment in the risky asset. This fraction, also known as the Merton ratio, is µ2r , s (1 2 a) 2
where µ is the expected return on the stock, s is the volatility of the stock, and r is the riskless return. This solution indicates that it is optimal for the investor to
Utility functions are discussed in Chapter 1 of Bowers et al. (1986). The power utility function is often used in financial economics because it has a particularly simple form and has reasonable properties concerning both absolute and relative risk aversion. It has increasing risk aversion and constant relative risk aversion. These concepts are discussed by Arrow (1971), and he also provides an intuitive discussion of their implications for investment decisions.
1
*Phelim P. Boyle, F.C.I.A., Ph.D., is at the Centre for Advanced Studies in Finance, University of Waterloo, Ontario, N2L 3G1, Canada. †Xiaodong Lin, A.S.A., is Assistant Professor in the Department of Statistics, University of Toronto, Ontario, M5S 3G3, Canada.
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NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 2
keep a constant fraction in the risky asset. As time passes, the portfolio is assumed to be adjusted so that this fraction is maintained. This can be accomplished without cost because we are assuming no transaction costs. The Merton ratio has a number of intuitive properties. First, note that the optimal fraction is proportional to (µ2r), the risk premium on the stock. Other things being equal, the lower the risk premium, the lower the optimal fraction invested in the stock. In the limit, when µ5r, the fraction invested in the stock is zero. This makes intuitive sense. Second, the optimal fraction invested in the stock is inversely proportional to the variance of the rate of return on the stock. Other things being equal, the optimal fraction invested decreases as the variance increases. Third, the optimal fraction invested in stock is inversely proportional to (12a). Now since 2
WU" (W) 5 1 2 a, U'(W)
(12a) is a measure of the investor’s relative risk aversion. The intuition here is that as the investor becomes more risk averse, he or she will invest less in the stock. We can use the Merton ratio to estimate numerical values of the optimal fraction in stock for representative values of the parameters. In the case of a diversified common stock portfolio, historical estimates might be µ50.1, s50.2. For the short-term riskless rate, let us take r50.05. There are different estimates of a in the empirical literature, but based on the paper2 by Constantinides (1990), let us assume a521, so that (12a)52. With these estimates, the Merton ratio works out to be 62.5%, which is remarkably close to the 60% referred to earlier. The introduction of transaction costs adds considerable complexity to the problem. It is no longer optimal to maintain a fixed constant proportion in the risky asset at all times. The investor has to trade off the benefits of moving closer to the desired position through costly transactions against the costs associated with making these transactions. The problem is simplified if we assume that the transaction costs are proportional to the amount of the risky asset traded and that there are no transaction costs on trades in the riskless asset. We will maintain these assumptions. It is also convenient to assume that the investor has a power utility function. When we include transaction costs, the optimal strategy is defined in terms of
Constantinides uses a value of (12a) equal to 2.2 in his paper.
2
bounds on the fraction to be held in the stock. As long as the proportion invested in the stock lies within these bounds, the portfolio is not adjusted. If the fraction strays outside these bounds, a transaction is made to restore the fraction to the nearest boundary. As the transaction rate goes to zero, both bounds converge to the Merton ratio. Conversely, as the transaction cost rate increases, the bounds widen. Several authors have made important contributions to the effect of transaction costs in this setting, including Kamin (1975), Magill and Constantinides (1976), and Constantinides (1979). Constantinides (1979) showed that in the case of proportional transaction costs and power utility, the no-transaction regions are convex cones. Consequently, the derivation of the boundaries of the no-transaction region is of central importance for all practical applications. It is generally believed that these boundaries cannot be obtained analytically and hence approximations are needed. Constantinides (1986) has developed approximate solutions for the case of an investor with a power utility. Often these solutions deal with the case in which the investor has an infinite horizon. For practical applications it may be more realistic to assume a finite terminal date. For this case, Gennotte and Jung (1994) have developed a numerical approach that can be used to obtain approximate values of the boundaries. This paper provides an explicit closed-form solution to the finite horizon problem when there are proportional transaction costs and the investor has a power utility function. Our procedure enables us to derive the boundaries of the no-transaction region in a systematic fashion. We derive an explicit representation of the optimal trading strategy. As previous authors have shown, the optimal strategy involves refraining from trading if the fraction of the risky asset lies within the no-transaction region. The layout of the rest of the paper is as follows. We set up the model in Section 2 and convert it into a dynamic programming problem. Also in this section we introduce the concept of indirect utility function. We develop the optimal trading strategy in Section 3 and obtain explicit expressions for the indirect utility functions. This enables us to develop analytical solutions for the boundaries of the no-transaction region. In Section 4 we indicate how the method can be applied to the multiperiod binomial model. We provide numerical examples in Section 5 to illustrate the approach and indicate how transaction costs affect the optimal asset allocation. Section 6 summarizes the paper.
OPTIMAL PORTFOLIO SELECTION
WITH
TRANSACTION COSTS
2. THE MODEL Consider an economy with one risky asset and one riskless asset available for trading. The risky asset pays no dividends. There are T11 trading dates indexed as t50, 1, z z z, T over a certain period. We denote St and Bt as the prices of the risky asset and the riskless asset, respectively, at time t. Hence, for t50, 1, z z z, T21, zt 5
St11 B , r 5 t11 St Bt
(2.1)
are the total rates of return on the risky asset and the riskless asset, respectively. Thus, zt is a random variable and r is a constant. It is assumed that z0, z1, z z z, zT21 are independent discrete random variables with a finite number of states. In this case, the price process of the risky asset has independent increments. For instance, if we assume that at each trading period, the return of the risky asset has only two possible outcomes: moving up by u units or moving down by d units, that is, Pr(zt 5 u) 5 p, (2.2) Pr(zt 5 d) 5 1 2 p, d , r , u, 0 , p , 1, then by properly choosing u and d, we will obtain the binomial process as in Cox, Ross and Rubinstein (1979). Suppose that the agent holds a portfolio with dollar amounts x00 in the riskless asset and x0 in the risky asset at time 0. He may adjust the portfolio at each trading date t50, 1, z z z, T21 to maximize his expected utility of terminal wealth. We assume that the utility function U has the following form U(W )5 (1/a)W a for a≤1.3 Thus, U is concave differentiable and homogeneous of degree a. Hence, the agent has constant relative risk aversion. Let x0t and xt be the dollar amounts of the riskless asset and the risky asset in the portfolio at time t before trading. Let (y0t , yt) be the corresponding portfolio holdings after trading at time t. We assume that there is a transaction cost every time we trade the risky asset. The cost is proportional to the dollar amount of the risky asset traded. Let u be the dollar amount paid whenever we buy or sell one dollar of the risky asset. Then the following relations hold: yt 5 xt 1 vt , yt0 5 xt0 2 vt 2 u|vt|,
a50 corresponds to logarithmic utility.
3
(2.3) (2.4)
29
where vt is the dollar amount of the risky asset traded at time t. A positive vt indicates that we buy the risky asset and a negative vt indicates that we sell the risky asset. We further assume that y0t and yt are non-negative.4 The portfolio amounts prior to trading at time t11 are then xt11 5 yt zt 5 (xt 1 vt) zt 0 xt11 5 y0t r 5 (x0t 2 vt 2 u|vt|) r.
(2.5) (2.6)
Thus, v0, v1, z z z, vT21 represent a trading strategy, and 0 (x00, x0), (x01, x1), z z z, (xT21 , xT21), (x0T, xT), which satisfy (2.5) and (2.6), are the portfolios held over the entire trading period corresponding to the strategy. One objective is to find an optimal trading strategy v0, v1, z z z, vT21 that maximizes the expected utility of terminal wealth, namely, max vt,t50,1,z z z,T21
E[U(x0T 1 xT)],
(2.7)
for the given initial portfolio (x00, x0). This problem can be solved by a dynamic programming technique.5 To apply dynamic programming, we define the indirect utility functions Jt, t50, 1, 2, z z z, T, as follows: JT(x0T, xT) 5 U(x0T 1 xT),
(2.8)
and for t50, 1, 2, z z z, T21, Jt(x0t , xt) 5 max Et Jt11(x0t11, xt11),
(2.9)
vt
where x0t11 and xt11 are given by (2.5) and (2.6) and Et denotes the expectation over zt conditional on x0t and xt. The method here is similar to those used in intertemporal portfolio selection without transaction costs (see Ingersoll 1987, Chap. 11). By the Bellman principle of optimality (Bellman and Dreyfus 1962 or Bertsekas 1987), we have J0(x00, x0) 5
max vt,t50,1,z z z,T21
E[U(xT0 1 xT)]
(2.10)
for any given initial portfolio (x00, x0). The variables vt, which maximize Et Jt11 (x0t11, xt11), t50, 1, z z z, T21, form the optimal trading strategy. In the next section, we will see that introducing indirect utility functions not only enables us to find the optimal strategy for a particular initial portfolio but also gives us more insights about the optimal trading during the trading period.
This restriction can be relaxed. See the discussion at the end of this section. 5 For the theory of dynamic programming, refer to Hillier and Lieberman (1990, Chap. 11) for an introductory level and Bertsekas (1987) for a more advanced level. 4
30
NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 2
3. OPTIMAL TRADING STRATEGY In this section, we develop an analytical method for solving the indirect utility functions. The optimal trading strategy is subsequently given for any portfolio at any time. We show that at time t, the portfolio space is divided into three disjoint positive cones,6 which can be specified as the buying region, the selling region and the no-transaction region. If a portfolio lies in the buying region, the optimal strategy is to buy the risky asset until the portfolio reaches the boundary of the buying region, while if a portfolio lies in the selling region, the optimal strategy is to sell the risky asset until the portfolio reaches the boundary of the selling region. If a portfolio lies in the no-transaction region, it is not adjusted at that time. Furthermore, the procedure used in the proof enables us to solve for the boundaries. Let Gt be the set of portfolios defined by Gt 5 {(xt0, xt); (3.1) 0 for all vt, Et Jt11 (xt11 , xt11) ≤ Et Jt11 (x0t r, xt zt)}. For any portfolio (x0t , xt) in Gt, vt50 is a maximum point of Et Jt11 (x0t11, xt11). Thus, the expected value will not be increased by buying or selling the risky asset. In other words, it is optimal not to transact a portfolio if it lies in Gt. For this reason, Gt is called the no-transaction region at time t. The following function plays a central role in our analysis: ft (vt, x0t , xt) 5 Et Jt11 (x0t11, xt11),
In this case, the portfolio space is divided into a finite number of cones and on each cone the utility function J is a convex combination of the utility function U. We now present our main result. Theorem Suppose that the terminal utility function U (W) is a concave, homogeneous differentiable function with degree a. Then for each, t, t50, 1, 2, z z z, T, (i) Jt(x0t , xt) is a concave homogeneous differentiable function (ii) Jt(x0t , xt) is a piece-wise linear utility function with respect to U (iii) There is at≤bt such that the no-transaction region at time t is Gt 5
$
(x0t , xt); at ≤
%
xt ≤ bt x0t
If xt /xt0,at, we buy the risky asset with amount vt1, which is given in (3.17). If xt /x0t .bt, we sell the risky asset with amount v2t , which is given in (3.19) (Figure 1).
PORTFOLIO SPACE
Figure 1 TRADING STRATEGY
AND
AT
TIME t
(3.2)
where x0t11 and xt11 are given by (2.5) and (2.6). We have already seen that Gt is the set of portfolios (xt0, xt) with ft(vt, x0t , xt)≤ft (0, x0t , xt). We next introduce the definition that will be used to characterize the indirect utility functions. Definition We say a utility function J is a piece-wise linear utility function with respect to a utility function U if there is a sequence of increasing numbers qj, j50, 1, z z z, s, and non-negative constants aij and bij with respect to the underlying probability space {wi; i51, z z z, I} such that J(x , x) 5
Σ U(a x
i51
ij
0
1 bij x) Pr(wi),
x for qj ≤ 0 , qj11. x
(3.3) (3.4)
A cone G is a set such that for any g∈G and any non-negative a, ag∈G. A positive cone is a cone contained in the positive orthant.
6
]+ ft (0, 1, at) 5 0, ]vt
(3.5)
and bt is the largest solution of
I
0
Furthermore, if 0,at≤bt,`, then at is the smallest solution of
]2 ft (0, 1, bt) 5 0, ]vt
(3.6)
where ]+ft /]vt and ]2ft /]vt denote the right and left derivatives of ft, respectively.
OPTIMAL PORTFOLIO SELECTION
WITH
TRANSACTION COSTS
We now make the following observations on these results. The result (i) and the first part of (iii) have been derived earlier by Constantinides (1979) and by Gennotte and Jung (1991). However, their methods do not provide an analytical solution for at and bt. An advantage of our approach is that the indirect utility functions can explicitly be solved and characterized. The boundary at and bt can also be solved analytically from the equalities in the theorem. Hence, they are exact. We illustrate the procedure in the next section. The proof of the theorem is based on the following lemmas. The proof of all these lemmas is constructive. These constructive proofs provide us with a procedure for calculating the indirect utility functions and the boundaries of the no-transaction regions. Throughout the rest of this section, we always assume that the utility function U (W) is concave, differentiable and homogeneous of degree a.
31
Hence ]+ ft (0, x0t , xt) ]2 ft (0, x0t , xt) ≤ . ]vt ]vt Therefore, ft is concave everywhere.
]2 ft (0, x0t , xt) ]+ ft (0, x0t , xt) ≥ 0 and ≤ 0. ]vt ]vt
(3.10)
Now we define at 5 min
$
xt ≥ 0;
]2 ft (0, 1, xt) ≥ 0, ]vt
%
]+ ft (0, 1, xt) ≤0 , ]vt
If Jt11 is a concave homogeneous differentiable function of degree a, ft(vt, xt0, xt) is a concave function with respect to vt and is a homogeneous function of degree a with respect to vt, x0t , and xt.
Homogeneity of ft follows from its definition. Obviously, for vt≠0, ft is a concave function. At vt50
n
Since ft is concave with respect to vt, any local maximum of ft will be a global maximum. Thus a portfolio (x0t , xt)∈Gt if and only if vt50 is a maximum point for ft (vt, x0t , xt). Equivalently, we have that (x0t , xt)∈Gt if and only if
Lemma 1
Proof
(3.9)
bt 5 max
$
xt ≥ 0;
(3.11)
]2 ft (0, 1, xt) ≥ 0, ]vt
%
]+ ft (0, 1, xt) ≤0 . ]vt
(3.12)
]2ft(0,1,0) ]+ft(0,1,`) 5` and 52`. ]vt ]vt The next lemma characterizes the no-transaction region. It shows the no-transaction region is a positive cone. with
]+ ft (0, x0t , xt) 5 lim 2 r (1 1 u) ] vt vt→01 Et
Lemma 2
] Jt11[(xt0 2 (1 1 u)vt) r, (xt 1 vt) zt] ]x0t11
A portfolio (x0t , xt) belongs to Gt if and only if the ratio xt /xt0 satisfies at ≤xt /xt0≤bt. If we consider the proportion of the value of the risky asset to the value of total asset, xt /(xt1x0t ), we have
] Jt11[(x0t 2 (1 1 u) vt) r, (xt 1 vt) zt] 1 Et zt z ]xt11 5 2 r (1 1 u) Et 1 Et zt z
]Jt11(xt0r, xt zt) 0 ]xt11
] Jt11(x0t r, xt zt) ]xt11
where z denotes the inner product in the two-dimensional Euclidean space. Similarly, ] ft (0, x , xt) ] J (x r, x z ) 5 2 r (1 2 u) Et t11 0 t t ] vt ]xt11 2
0 t
0 t
1 Et zt z
at xt bt ≤ ≤ . 0 1 1 at xt 1 xt 1 1 bt
(3.7)
] Jt11(x0t r, xt zt) . ]xt11
(3.8)
Remark Since Gt denotes the no-transaction region at time t [see (3.1)], Lemma 2 implies that the no-transaction region at time t is a positive cone.
32
NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 2
Proof
Lemma 3
] ft(0,x ,xt) ≥0. Since ft is homo]vt ]2ft(0,x0t ,xt) geneous with degree a, is homogeneous ]vt with degree a21. Hence, 2
0 t
Given (x0t , xt)∈Gt,
]2 ft(0, 1, xt /x0t ) ]2 ft(0, x0t , xt) 5 (x0t )12a ≥ 0. ]vt ]vt
Suppose Jt11 is a differentiable function. Then, if 0,at,`,
m 2 bt , 0, 1 1 (1 2 u) m (1 2 u) mbt 1 m g5 . 0. (1 2 u) mxt 1 xt
]2 ft(0, 1, bt) 5 0. ]vt
(3.15)
Proof
Thus, at ≤xt /x0t ≤bt. Conversely, suppose that m5xt /x0t and at,m,bt. Let v*t 5
(3.14)
If 0,bt,`,
Similarly, we can show ]+ ft(0, 1, xt /x0t ) ≤ 0. ]vt
]+ ft(0, 1, at) 5 0. ]vt
(3.13)
Suppose that
]+ft(0,1,at) ,0. There is xt,at such that ]vt
]+ft(0,1,xt) ≤0 by continuity. From the proof of ]vt Lemma 2, there is v*t,0 and g.0 replacing m and bt by xt and at, such that ]2 ft(0, 1, xt) ]f (v*, 1, at) 5 g 12a t t ]vt ]vt
Then g a21
≥ g 12a
]2 ft(0, xt0, xt) ] vt
5 g a21
$
%
5
Lemma 3 provides a method to calculate the boundaries of the no-transaction regions. In fact, they can be obtained by solving two sets of algebraic equations. In Lemma 4, the amount of the risky asset to buy or to sell at time t is calculated when a portfolio is outside the no-transaction region.
] Jt11(gx0t r, gxt zt) ] x0t11
] Jt11(gx0t r, gxt zt) ] xt11
5 2 r (1 2 u) Et 1 Et zt z
] Jt11(xt0r, xt zt) ] x0t11
] Jt11(x0t r, xt zt) ] xt11
5 2 r (1 2 u) Et 1 Et zt z
]+ft(0,1,xt) ]2ft(0,1,xt) ≤0 and ≥0, xt∈Gt. Thus, ]vt ]vt at is not a minimum, which is a contradiction. By a similar argument, the second conclusion can be shown. n Since
2r (1 2 u) Et
1 Et zt z
]2 ft(0, 1, at) ≥ 0. ]vt
] Jt11((1 2 (1 2 u)v*) t r, (bt 1 v*)z t t) 0 ] xt11
] Jt11((1 2 (1 2 u)v*)r, (bt 1 v*)z t t t) ] xt11
Lemma 4 If xt /x0t ,at, then
]ft(v*,t 1, bt) ]2 ft(0, 1, bt) ≥ ≥0 ] vt ] vt
]2ft(0,xt0,xt) Thus, ≥0. By properly choosing g and v*t ]vt again, we can show that ]+ ft(0, xt0, xt) ≤0 ]vt Hence, (x0t , xt)∈Gt.
max ft(vt, x0t , xt) 5 ft(v1t , x0t , xt) v t
1 5 ft(0, y01 t , yt ),
(3.16)
where x0t at 2 xt , 1 1 (1 1 u) at yt1 5 xt 1 vt1, y01 5 x0t 2 (1 1 u)v1t , t vt1 5
n
1 1 01 with (y01 t , yt )∈Gt and yt /yt 5at.
(3.17)
OPTIMAL PORTFOLIO SELECTION
WITH
TRANSACTION COSTS
If xt /xt0.bt, then
Lemma 5
max ft(vt, x , xt) 5 ft(v , x , xt) v 2 t
0 t
t
If Jt11 is differentiable, so is Jt.
0 t
2 5 ft(0, y02 t , yt )
(3.18) Proof
where x0t bt 2 xt v 5 , 1 1 (1 2 u)bt y2t 5 xt 1 v2t , y02 5 x0t 2 (1 2 u)v2t . t 2 t
(3.19)
2 2 02 with (y02 t , yt )∈Gt and yt /yt 5bt.
Proof We shall prove the first half of the lemma, and the proof of the second half logically follows. 01 1 It is easy to see v1t .0, y1t /y01 t 5at, (yt , yt )∈Gt, and 1 ft(v1t , x0t , xt)5ft(0, y01 t , yt ). First, we assume at,`. To show v1t is a maximum it suffices to show ]ft(v1t , x0t , xt) 5 0, ]vt
(3.20)
since v .0. Now, 1 t
]2 Jt(x0t , xt) ]+ ft(0, xt0, xt) ]vt1 5 0 ]xt ]vt ]xt0 ]f (0, xt0, xt) 1 t ]x0t ]+ ft(0, 1, at) ]v1t 5 (x0t )a21 ]vt ]x0t ]f (0, xt0, xt) 1 t ]x0t ]f (0, xt0, xt) ]+ Jt(xt0, xt) 5 t 5 , ]x0t ]x0t [by (3.22)]
(3.23)
]+ft(0,1,at) 50 by Lemma 3. ]vt Now we differentiate Jt with respect to the second argument xt.
5 2 r(1 1 u) Et
] Jt11((x0t 2 (1 1 u)v1t ) r, (xt 1 v1t ) zt) ]x0t11
] Jt11((x0t 2 (1 1 u)v1t ), r, (xt 1 v1t ) zt) ]xt11
5 (x0t 2 (1 1 u)v1t )a21
~2 r (1 1 u) E ] J
(r, at zt) 0 ]xt11
t11
t
1 Et zt z
Obviously, Jt is differentiable in the interior of three disjoint sets: xt /x0t ,at, at,xt /xt0,bt, and xt /xt0.bt. We now show that Jt is differentable on the lines xt /x0t 5 at and xt /xt05bt. For any given (x0t , xt) with xt /x0t 5at,
since
]ft(v1t , x0t , xt) ] vt
1 Et zt z
33
!
] Jt11(r, at zt) ]xt11
5 (xt0 2 (1 1 u)vt1)a21
]+ ft(0, 1, at) . ] vt
]ft(v , x , xt) 5 0. ]vt 0 t
(3.21)
If at5`, ft is nondecreasing, v1t is the right endpoint of its domain and so v1t is the maximum point. n From Lemmas 2 and 4, we have Jt(x0t , xt) 5
$
ft(v1t , x0t , xt), xt /x0t , at, ft(0, x0t , xt), at ≤ x0 /xt0 ≤ bt, 2 0 ft(vt , xt , xt), xt /x0t . bt.
(3.24)
The differentiability of Jt on the line xt /x0t 5bt can be shown similarly. n
By Lemma 3, 1 t
]2 Jt(x0t , xt) ]f (0, x0t , xt) 5 t ]xt ]xt ]+ ft(0, 1, at) ]v1t 5 (x0t )a21 ]vt ]xt ]ft(0, x0t , xt) 1 ]xt ]+ Jt(xt0, xt) 5 . ]xt
(3.22)
The next lemma characterizes the indirect utility functions Jt, t50, 1, z z z, T. It shows that the piecewise linearity property is preserved as we move back in time. This property plays a key role in the derivation of the boundaries and the indirect utilities. Lemma 6 If Jt11(x0t11, xt11) is a piece-wise linear utility function with respect to U, so is Jt(xt0, xt).
34
NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 2
where (r/z1t ) qj ≤ q˜ h, q˜ h11≤(r/z1t ) qj11; for g5I11, z z z, 2I,
Proof Let {wi} be all possible outcomes from (zt11, zt12, z z z, zT), where {wi} represents the set of all future paths of the underlying tree structure starting at a node at time t11. Hence if {zkt ; k51, 2, z z z} are all possible outcomes for zt, then all the paths of the underlying tree structure starting at time t can be written as {(zkt , wi)}. We first show that ft(0, x0t , xt) is a piece-wise linear utility function with respect to U. Suppose that7
Σ U(a x I
Jt11(x0t11, xt11) 5
ij
i51
0 t11
1 bij xt11) Pr(wi),
xt11 , qj11, j 5 0, 1, z z z, s. 0 xt11 q0 5 0, qs11 5 `. qj ≤
a˜ gh 5 ag2I, jr, b˜ gh 5 bg2I, j z2t , where (r/z2t )qj ≤ q˜ h,q˜ h11≤(r/z2t ) qj11.8 We next show that Jt (x0t , xt) is a piece-wise linear utility function with respect to U. From Lemma 4,
$
ft(0, yt01, yt1), xt /xt0 , at, ft(0, xt0, xt), at ≤ xt /xt0 ≤ bt, 02 2 ft(0, yt , yt ), xt /xt0 . bt.
Jt(x , xt) 5 0 t
(3.29)
It follows from the fact that ft(0, x0t , xt) is a piece-wise linear utility function with respect to U, that Jt (x0t , xt) is a piece-wise linear utility function on the cone at ≤xt /xt0≤bt. We now analyze the region xt /x0t ,at. Suppose that q˜ j1,at ≤ q˜ j111. From (3.29), 1 Jt(x0t , xt) 5 ft(0, y01 t , yt )
Let ck(xt0, xt) 5 Jt11(xt0r, xt ztk) Pr(zt 5 ztk).
5
(3.25)
Then
Σ U(a˜
i j1
i
y01 1 b˜ i j1 y1t ) Pr(w ˜ i), t
(3.30)
w ˜ i is one of {(z kt , wi)}. Since
Σ U(a I
ck(x0t , xt) 5
i51
ij
rx0t 1 bij zkt xt) Pr{(zkt , wi)}
r x r q ≤ t , k qj11. ztk j xt0 zt
(3.26)
yt1 5 xt 1
(3.27)
Hence ck(x0t , xt) is a piece-wise linear utility function with respect to U. We now show ft(0, xt0, xt) 5
Σ c (x , x ) k
k
0 t
t
(3.28)
is also a piece-wise linear utility function with respect to U. It suffices to show the sum of two piece-wise linear functions is a piece-wise linear function. Then, by induction the sum of any finite number of piecewise linear functions is a piece-wise linear function. Relabel all (r/zkt )qj, k51, 2, j50, 1, z z z, as q˜h in order of magnitude. Then for
a˜ gh 5 ag jr, b˜ gh 5 bg j zt1,
We begin with t115T. Thus, in this case s50, q15` and JT (x0T, xT)5U(x0T1xT).
7
5
at (1 1 u) at x0t 1 xt, 1 1 (1 1 u)at 1 1 (1 1 u)at
(3.31)
y01 5 t
1 11u x0t 1 xt, 1 1 (1 1 u)at 1 1 (1 1 u)at
(3.32)
Jt (x0t , xt)
Σ
(3.33)
@
a˜ i j1 1 at b˜ i j1 0 5 U x i 1 1 (1 1 u)at t (1 1 u) (a˜ i j1 1 at b˜ i j1) 1 xt 1 1 (1 1 u)at
#
Pr(w ˜ i).
(3.34)
for xt /xt0,at. This is obviously in the form of (3.3). Similarly, for xt /x0t .bt, let q˜ j2≤bt,q˜ j 211. Then, Jt(x0t , xt) 5
x q˜ h ≤ 0t , q˜ h11, xt and g51, z z z, 2I, we define a˜ gh and b˜ gh as follows: for g51, z z z, I,
x0t at 2 xt 1 1 (1 1 u)at
Σ U@ 1 a˜1 (11 2b u)b b i j2
i
1
t
˜ij
2
x0t
t
#
(1 2 u) (a˜ i j2 1 bt b˜ i j2) xt 1 1 (1 2 u) bt
Pr (w ˜ i).
(3.35)
Therefore Jt (x0t , xt) is a piece-wise linear utility function with respect to U. n We are now ready to prove the main theorem.
This procedure will become somewhat clearer when it is illustrated in the next section.
8
OPTIMAL PORTFOLIO SELECTION
WITH
TRANSACTION COSTS
35
Proof
where zH and zL are the highest and lowest returns of z, respectively. Then all the arguments in this section apply.
Results (i) and (ii) are a direct corollary of Lemmas 5 and 6. The first part of (iii) is proved in Lemmas 2 and 4, and the second part is proved in Lemma 3. n We now make some additional comments about the results. 1. Suppose that J0 (x00, x0) 5
Σ U(a
ij
i
x00 1 bi j x0) Pr(wi)
(3.36)
qj ≤ x0 /x , qj11 0 0
It is easy to see that ai j x001bi j x0 is the terminal wealth if the path wi is realized. Therefore, ai j and bi j are marginal rates of the riskless and risky assets, respectively, for those portfolios whose ratio is between qj and qj11. 2. If the equations in the theorem have no solutions, then at and bt can be easily identified; at takes one of two values. Either at 5 0, if
]1 ft(0, 1, 0) , 0, ]vt
(3.37)
at 5 `, if
]1 ft(0, 1, `) . 0. ]vt
(3.38)
or
In the second case, we must have bt5`. In the same way, bt can only take one of two values. Either bt 5 0, if
]2 ft(0, 1, 0) , 0, ]vt
(3.39)
4. AN APPLICATION In this section, we illustrate how to use the method developed in the last section to calculate the indirect utility function Jt, and the boundary points at and bt of the no-transaction region Gt, for t50, 1, z z z, T21. We consider the case in which the rate of return for the risky asset in each period is independent of t and has only two states. In other words, the price of the risky asset either goes up by u or down by d, that is, Pr(zt 5 u) 5 p, Pr(zt 5 d) 5 1 2 p, d , r , u, 0 , p , 1,
We further assume that the rate of return r for the riskless asset satisfies d,r,u (no arbitrage condition). If we choose u5es=1/n, d5e2s=1/n, r5ed/n and p5 1/2 [11(µ/s) =1/n], we obtain the binomial model proposed by Cox, Ross, and Rubinstein (1979) with T5n. An alternative choice is u511µ/n1s/=n, d511 µ/n2s/=n, r5ed/n, and p51/2, which was proposed by Hua He (1990). The price movement in both models will lead to the Black-Scholes economy, when n goes to infinity. In the Black-Scholes economy the prices are described by the stochastic differential equation9 dBt 5 dBt dt dSt 5 µSt dt 1 sSt dWt,
or bt 5 `, if
]2 ft(0, 1, `) . 0. ]vt
(4.1)
(3.40)
In the first case (when bt50), we must have at50. 3. If the terminal utility function is strictly concave, then the solution of the two equations is unique. Furthermore, if there is no transaction cost, that is, u50, then at5bt. 4. In this paper, we require that y0t ≥0 and yt ≥0; that is, buying and selling assets short is not allowed. However, our results and methodology can also be extended to the general case, which allows negative investment, but the agent is required to be solvent during the trading period (see Constantinides 1979, p. 1129). In that case, the domain of Jt is no longer the positive orthant but the following: {(x0t , xt); u1 , arg(x0t 1 ixt) , u2, u1 5 2 arctan (r/zH), u2 5 p 2 arctan (r/zL)},
where Wt is an one-dimensional standard Brownian motion (Black and Scholes 1973). In this section, we use Hua He’s approach since it is the simplest for our purpose. However, our method can be applied to any other discrete model. The terminal utility function used in this example is the power function U(W) 5
Wa , a ≤ 1, a ≠ 0. a
We now calculate Jt, at and bt recursively starting at t5T. At t5T,
For the theory of stochastic differential equations, see Oksendal (1985).
9
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NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 2
JT(x0T, xT) 5 U(x0T 1 xT) 5
(x0T 1 xT)a . a
(4.2)
Now, suppose that Jt11(x0t11, xt11) is solved, that is,
for vt ≤ 0. Hence from (3.5) and (3.6) in the main theorem, at is a solution of one of the equations
Σ (a˜ i
ij
1 b˜ ij at)a21 [b˜ ij 2 (1 1 u)a˜ ij] 5 0, q˜ j ≤ at , q˜ j11, j 5 0, 1, 2, z z z
2T2t21
Σ
Jt11(x0t11, xt11) 5
U(ai j x0t11 1 bi j xt11)Pr (wi)
i51
Σ
1
5
and bt is a solution of one of the equations (ai j x0t11 1 bi j xt11)a,
(4.3)
2T2t21a i xt11 qj ≤ 0 , qj11. xt11
Σ (a˜ i
ij
1 b˜ ij bt)a21 [b˜ ij 2 (1 2 u)a˜ ij] 5 0, q˜ j , bt ≤ q˜ j11, j 5 0, 1, 2, z z z.
We then define ft(0, xt0, xt) 5 Et Jt11(xt0r, xt zt) 1 5 [Jt11(x0t r, xtu) 1 Jt11(x0t r, xtd)]. 2
(4.4)
Rearrange all (r/u)qj and (r/d)qj, j50, 1, z z z, from smallest to largest and relabel them as
q˜ j1 , at ≤ z z z , q˜ j2 ≤ bt , z z z. Define
Then for
q¯ 0 5 0, q¯ 1 5 at, q¯ 2 5 q˜ j111, z z z, q¯ j22j112 5 bt, q¯ j22ji113 5 `,
x q˜ h ≤ 0t , q˜ h11, xt and g51, z z z, 2T2t, we define a˜ gh and b˜ gh as follows: for g51, z z z, 2T2t21,
where (r/u)qj ≤ q˜ h,˜qh11≤(r/u)qj11; for g52 2T2t,
a¯ i0 5
11, z z z,
j 5 1, 2, z z z, j2 2 j1 1 1; a¯ i, j22j112
ft(0, x , xt) 5 0 t
2T2t
1 2T2ta q˜ j ≤
Σ
i51
(a˜ ij x 1 b˜ ij xt)a. 0 t
(4.5)
xt , q˜ j11, x0t
ft(vt, x0t , xt)
Σ {a˜ [x ij
i
0 t
2 (1 1 u)vt] 1 b˜ ij(xt 1 vt)}a,
(4.6)
for vt ≥ 0, and ft(vt, x0t , xt) 5
1 2T21a
Σ {a˜ [x i
ij
0 t
2 (1 2 u)vt] 1 b˜ ij(xt 1 vt)}a,
Jt(x0t , xt) 5 q¯ j ≤
here we change g and h back to i and j to avoid too much notation. Thus,
a
(4.12)
(4.13)
Then,
Then,
2
(4.11)
a˜ i j2 1 bt b˜ i j2 5 , 1 1 (1 2 u) bt b¯ i, j22j112 5 (1 2 u)a¯ i, j22j112.
where (r/d)qj ≤ q˜ h,q˜ h11≤(r/d)qj11.
1
a˜ ij1 1 at b˜ ij1 ¯ , b 5 (1 1 u)a¯ i0; 1 1 (1 1 u)at i0
a¯ ij 5 a˜ i,j11j21, b¯ ij 5 b˜ i, j11j21,
T2t21
a˜ gh 5 ag22T2t21, j r, b˜ gh 5 bg22T2t21, j d,
(4.10)
and
a˜ gh 5 ag jr, b˜ gh 5 bg ju,
T21
(4.9)
Since W a/a is strictly concave, at most one equation in each set has a solution and the solution is unique if it exists. If there is no solution for all the equations, then at or bt will be either 0 or `, depending on the sign of the first or last equation in each set (see comment 2 in the previous section). Assume that
0 5 q˜ 0 , q˜ 1 , z z z.
5
(4.8)
(4.7)
1 2T2ta
2T2t
Σ
i51
(a¯ i j x0t 1 b¯ i j xt)a,
(4.14)
xt , q¯ j11, j 5 0, 1, z z z. x0t
5. NUMERICAL EXAMPLES In this section we illustrate the approach by using a simple numerical example. First, we solve the notransaction costs case because this provides a useful benchmark. Then we see how things change when we assume that there are transaction costs on the risky asset. We just consider a three-period problem for simplicity. Preliminary investigations of high-dimensional problems suggest that it may be challenging to
OPTIMAL PORTFOLIO SELECTION
WITH
TRANSACTION COSTS
write a simple program to implement our algorithm. We plan to explore this issue in subsequent research. We use the notation of the last section. Suppose we have n periods each of length h. We use the Cox Ross Rubinstein parametrization for u, d, r and p, viz, u 5 es=h, d 5 e2s=h, r 5 edh and p 5
eµh 2 d . u2d
We assume the investor maximizes the expected value of utility of terminal wealth with the following utility function U(W ) 5
Wa , a ≤ 1, a ≠ 0. a
In the no-transaction case, the investor’s problem can be solved as a series of one-period problems. The optimal fraction to invest in the risky asset can be obtained from maximizing the expected utility of end of period wealth. (W0(r 1 x(u 2 r))) a (W0(r 1 x(d 2 r)))a 1 (1 2 p) a a
E {U (W )} 5 p
We can treat this as a function of x and solve for the value of x that maximizes this function. The optimal value is r (1 2 u) , (r 2 d) 1 u (u 2 r) where
~(1 2p (up) 2(r r)2 d)!
1/(12a)
u5
If we assume that µ50.1, s50.25, d50.05, h50.25, and a521, then the optimal fraction to be invested in the risky asset is 0.41302 and the ratio of the optimal investment in the risky asset to the optimal investment in the riskless asset is 0.703645. Each period the same allocation is made. When we analyze the corresponding problem in which there are transaction costs, the boundaries of the no-transaction region will contain the point 0.703645. In this case in which there are no transaction costs, the initial allocation of the investor’s wealth between the two assets does not affect the investor’s welfare as long as the total wealth is held constant. This is because with no transaction costs, the risky asset can be converted without cost into the riskless asset. When there are transaction costs, this will no longer be the case. For future reference we record the
37
expected utility of the investor computed as at time zero when there are no transaction costs; it is 20.955693. The expected utility is negative because a is equal to 21 in our example. We now contrast this case with the situation in which there are transaction costs. To start with, we assume that the transaction costs are 0.1% of the risky asset. We can use the procedure developed earlier in the paper to compute the no-transaction region boundaries. We use Equations (3.5) and (3.6) to find at and bt at each step, working recursively backwards from the last period. The values we obtain are given in Table 1. TABLE 1 NO TRANSACTION BOUNDS WHEN u 5 0.001 a0 5 0.639132
b0 5 0.770427
a1 5 0.643571
b1 5 0.765155
a2 5 0.614274
b2 5 0.803352
To interpret these results, note that if the investor’s initial position consists of 0.05 in the risky asset and 0.95 in the riskless asset, then the investor should buy enough of the risky asset to reach the boundary, so that the proportion of the risky asset to the riskless asset is a050.639132. Allowing for transaction costs, the amount of the risky asset to be purchased at time zero is 0.339788. Note that the no-transaction region tends to widen as we approach maturity. We also note that the Merton ratio of 0.703645 lies within the no-transaction region. These results agree with the findings of Gennotte and Jung (1994). If we increase the transaction cost rate to 1%, the bounds widen considerably. In this case the values are given in Table 2. TABLE 2 NO TRANSACTION BOUNDS WHEN u 5 0.01 a0 5 0.433005
b0 5 1.097272
a1 5 0.331171
b1 5 1.363041
a2 5 0.093558
b2 5 2.789939
The increase in transaction costs has a dramatic impact on the width of the bounds. We would expect that the investor would be better off as the level of transaction costs declines. We would
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NORTH AMERICAN ACTUARIAL JOURNAL, VOLUME 1, NUMBER 2
also expect that the investor’s expected utility will depend on the initial holdings in the risky asset. It should be better to start with a position in the risky asset that corresponds to the interior of the no-transaction region. The next set of results shows how the investor’s expected utility changes with the initial position in the risky asset and also with the level of transaction costs. We assume three initial situations corresponding to 5%, 40%, and 95% in the risky asset. The corresponding ratios of the risky to the riskless asset are 0.0526, 0.6667, and 19, respectively. The three transaction cost rates are 0, 0.1% and 1%. The investor’s time zero value of expected utility for these cases are shown in Table 3. TABLE 3 INVESTOR’S EXPECTED UTILITY FOR DIFFERENT INITIAL ASSET ALLOCATIONS AND FOR DIFFERENT LEVELS OF TRANSACTION COSTS
Transaction Cost Rate
Initial Percentage in Risky Asset
0
0.1%
1%
5 40 95
20.955693 20.955693 20.955693
20.956059 20.955727 20.956229
20.958668 20.955740 20.960386
The procedures developed in our paper may be useful in considering more general versions of the portfolio problem as well as other situations, such as more general utility functions and the inclusion of intermediate consumption. Another interesting extension would be the case of two or more risky assets. The two-asset case presents formidable problems because in general we do not know what the no-transaction regions look like. On the computational front we would like to develop an efficient algorithm to implement the method developed in the current paper. These issues are left for further research.
ACKNOWLEDGMENTS Xiaodong Lin was partly supported by the Natural Sciences and Engineering Research Council of Canada. The authors are grateful to Inmoo Lee for research assistance. They are grateful to Elias Shiu and anonymous referees for helpful suggestions. They also appreciate the helpful input of the following master’s students at the University of Waterloo: Sean Finucane, Peter Fitton, Lau Sok Hoon, and Vincent Lee and especially Claire Bilodeau.
REFERENCES We observe that these results confirm our intuition. The highest levels of expected utility occur when there are no transaction costs and decline as we move across the table from left to right in the direction of increasing transaction costs. When there are transaction costs, the expected utility levels are highest for initial positions that lie within the no-transaction region.
6. SUMMARY In this paper, we have analyzed the portfolio selection problem of an investor when there are proportional transaction costs on the risky asset. Since the work of Constantinides, it has been known that under certain assumptions the no-transaction region consists of a cone. Furthermore, Gennotte and Jung (1994) have provided a numerical procedure to obtain numerical solutions for the values of the boundaries of the notransaction region. This paper shows explicitly the functional form of the indirect utility function. The proof of our main theorem provides a constructive analytical procedure for determining the no-transaction region. Once we know this region, the investor’s problem is solved.
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Discussions on this paper will be accepted until October 1, 1997. The authors reserve the right to reply to any discussion. See the Table of Contents page for detailed instructions on the preparation of discussions.
