数理解析研究所講究録 1477 巻 2006 年 83-91

83

Optimal Portfolio Selection by CVaR-Based Sharpe Ratio
Sequential Linear Programming Approach
Shan LIN ( 林 杉 ) and Masamitsu OHNISHI ( 大西 匡光 )Graduate School of Economics, Osaka University(大阪大学大学院・経済学研究科)

1

Introduction

We address an optimal portfolio selection problem of maximizing so we call CVaR (Conditional-based Sharpe ratio of portfolio’s return rate, which is defined as the ratioof the expected excess return to CVaR. The Sharpe ratio defined as the ratio of expectedexcess return to standard deviation, the most common traditional performance measure, takes
standard deviation as a risk measure, however its has been received a lot of criticisms. In our
CVaR-based Sharpe ratio, the standard deviation is replaced with CVaR, which is a remarkablecoherent risk measure which overcomes essential defects of standard deviation. Although our
new performance measure is expected to enlarge the applicable area of practical investment
problems for which the original Sharpe ratio is not suitable, however, we should device effective
computational methods to solve optimal portfolio selection problems with very large numberof investment opportunities.
In order to deal with rather complicated non-concave objective function, which comes fromthe introduction of CVaR, we propose the following SLP (Sequential Linear Programming)
approach: By introducing a real parameter, we make the non-concave maximization problem
to a parametric family of concave maximization problems. Then, for each of these problems,
utilizing the results of Rockafellar and Uryasev (2000), we introduce an auxiliary decisionvariable to obtain a tractable concave maximization proble 1n. Furthermore, if we estimate or
approximate required expected values by sampling methods or historical data, we can reduce theparametric concave maximization problems to LP (Linear Programming) problems. Therefore,
our problem could be finally reduced to a sequence of LP problems. Numerical experimentsfrom real Japanese financial data are conducted to test our SLP approach.
The paper is organized as follows. In Section 2, we introduce downside risk measures:and CVaR, in Section 3, we make a brief review of parametric approach to fractional
programming. Sample approach will be presented in Section 4. Further, in Section 5 anempirical study is given.
$\mathrm{V}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}-\mathrm{a}\mathrm{t}-\mathrm{R}\mathrm{i}\mathrm{s}\mathrm{k})$$\mathrm{V}\mathrm{a}\mathrm{R}$VaR and CVaR

2

assets.
Let denote a random variable denoting a rate of return on an asset or a portfolio of, is defined as theof with confidence level $\beta\in[0,1]$ , denoted
Value at Risk$\tilde{r}$$\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\neg r$$\overline{r}$$(\mathrm{V}\mathrm{a}\mathrm{R})$84

negative of $(1-\beta)$ -quantile of$\overline{r}.\cdot$$\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\neg r$$:=$$- \inf\{r\in \mathbb{R} :\mathrm{P}(\overline{r}\leq r)\geq 1-\beta\}$$=$$\sup\{u\in \mathbb{R} :\mathrm{P}(-\mathrm{r}\geq u)\geq 1-\beta\}$

Conditional Value at Risk (CVaR) of
is then defined as follows:$\overline{r}$(1).

with confidence level $\beta\in[0,1]$ , denoted$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\urcorner r$,(2)$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\neg r:=\frac{1}{1-\beta}\int_{0}^{1-\beta}.\mathrm{V}\mathrm{a}\mathrm{R}_{\alpha}[\hat{r}]\mathrm{d}\alpha.$

It can be shown that$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\neg r=\frac{1}{1-\beta}\mathrm{E}$where$\mathrm{E}[Y;\mathrm{A}]$[$-\overline{r}\cdot-)r-\geq \mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[r\urcorner]-\mathrm{V}\mathrm{a}\mathrm{R}\mathrm{O}[\mathrm{r}]${(-r$\mathrm{P}$$\geq \mathrm{V}\mathrm{a}\mathrm{R}_{\beta}$

denotes the partial expectation of a random variable. Although this expression is somewhat complex, if[$r\neg)-(1-\beta)$ } ,$Y$on an event

$A$

(3)
; that is$\mathrm{E}[Y\cdot a]\};=\mathrm{E}[Y1_{A}]$$\mathrm{P}$,(4)$-\neg r-\overline{r}\geq \mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[r\neg]$(5)$(-\overline{r}\geq \mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\neg r )=1-\beta$

then the second term vanishes and it becomes$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\neg r=\frac{1}{1-\beta}\mathrm{E}[-\overline{r},\cdot-\overline{r}\geq \mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\neg r ]=\mathrm{E}$[Average Value at Risk (AVaR), Expected Shortfall (ES), Tail Conditional Expectation (TCE),
and others are similar concepts, not few researchers prefer one of these terms to CVaR, but these
become identical when the above condition holds (whose sufficient condition is the continuityof cumulative distribution function (cdf) of .
A very useful characterization is obtained by Pflug (2000), Uryasev (2000), and Rockafellar
and Uryasev $(2000, 2001)$ . Let us introduce a function:$\gamma r$$F_{\beta}(a,\cdot\gamma r$$:=a+ \frac{1}{1-\beta}\mathrm{E}[(-\tilde{r}-a)^{+}]$

then the following theorem holds (for a real number
part of ).

,$a\in \mathbb{R}$$c\in \mathbb{R}$(6),, ( $c \rangle^{+}:=\max\{c, 0\}$ is the positive$c$

Theorem 1.
(1)$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}\beta[\neg r$

coincides with the minimum of function$F_{\beta}(\cdot;\overline{r})$$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\neg r’=\min\{F_{\beta}(a,\cdot r\gamma :(2) The minimum of function$F_{\beta}(\cdot;\tilde{r})$$\min\{F_{\beta}(a;\overline{r}) :(3)$F_{\beta}(a;\tilde{r})$is convex both in$a\in \mathbb{R}$:a\in \mathbb{R}\}$(7).

is attained at when the variable is equal toa\in \mathbb{R}\}=F_{\beta}(\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\tilde{r}(x)];r\gamma$

and .

.

$\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\neg r$:(8)$\overline{r}$$\square$Now let us consider a portfolio optimization of investments in financial assets numbered,, . We use the following notation$\mathrm{i}=1$$\cdots$$n$$\mathrm{s}$85,$\mathrm{i}=1$,$\bullet$$r-i$$\bullet$$\overline{r}_{i}:=\mathrm{E}[\overline{r_{i}}]$$\bullet$$x_{i}(\in \mathbb{R})$, : the random rate of return on financial asset ;$n$$\cdots$,,$\mathrm{i}$$\mathrm{i}=1$$\mathrm{i}=1$,,$\cdots$$\cdots$, : the mean (or expected) rate of return on financial asset ;$n$$\mathrm{i}$, : a portfolio weight, that is, a proportion of investment in financial$n$asset ;$\mathrm{i}$$\bullet$$\bullet$$\bullet$$r-.–$$(\tilde{r}_{1}, \cdots,\overline{r}_{n})^{\mathrm{T}}$$\overline{r}:=(\overline{r}_{1}, \cdots)$

$x:=$

1,

$\cdots$$\overline{r}_{n})^{\mathrm{T}}$: the random vector of return rates on financial assets: the vector of mean return rates on financial assets$(x_{1}, \cdots, x_{n})^{\mathrm{T}}$$\mathrm{i}=1$$\mathrm{i}=1$,,$\cdots$$\cdots$,$n,\cdot$, ;$n$: the portfolio of investment proportions in financial assets$\mathrm{i}=$

, .
$n$

Further we let.

(9)$\mathrm{r}(\mathrm{x})$$:=$$\mathrm{r}(\mathrm{x})$$:= \mathrm{E}[\overline{r}(x)]=\mathrm{E}[\tilde{r}^{\mathrm{T}}x]=\sum_{i=1}^{n}\mathrm{E}[\tilde{r}_{i}]x_{i}=\overline{r}^{\mathrm{T}}x=\sum_{i=1}^{n}\overline{r}:x_{0}$$\tilde{r}^{\mathrm{T}}x=\sum_{l=1}^{n}\overline{r_{i}}x_{i)}$.

(1)

The above theorem is particularly useful when we must consider the minimization of, for every evalon portfolio . According to the definition ofof return rate
$x\in
X$
in
the
order:
valuesthe, we must evaluate
uation of the objective function at$\mathrm{C}_{J}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\overline{r}(x)]$$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\overline{r}(x)]$$x$$\mathrm{r}(\mathrm{x})$(1)$\supset$$\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\overline{r}(x)]$(2)$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\tilde{r}(x)]$

(11)

,

but these are tremendous tasks. The following theorem implies that the evaluation and minican be done by the simultaneous minimization of function
mization of.
with respect to the original decision variable $x\in X$ and an auxiliary variable$\Gamma\prec(a;\overline{r}(x))$$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\tilde{r}(x)]$$a\in \mathbb{R}$Theorem 2.
(1)$\min\{\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{E}_{\beta}[\overline{r}(x)] :x\in X\}$$= \min\{F\beta(a;\overline{r}(x)) :a\in \mathbb{R},\cdot x\in X\}$

.

(12)

(2) For $x^{*}\in X$ ,$\min\{\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\overline{r}(x)] :x\in X\}$(13)$=\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}\beta[\overline{r}(x^{*})]$

if and only if$\min\{F_{\beta}(a;\tilde{r}(x)) :(3)$F_{\beta}(a;\tilde{r}(x))$

a\in \mathbb{R}_{\mathrm{j}}x\in X\}=F_{\beta}(\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\tilde{r}(x^{*})];\overline{r}(x^{*}))$

is convex both in$a\in \mathbb{R}$.

and $x\in X$ .

(14)$\square$Accordingly, the original convex program ming problem with $n+1$ decision variables;$x\in X\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta},[\tilde{r}(x)]$$|\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{z}\mathrm{e}\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}$(15)could be reduced to the following convex programming problem with $n+1$ decision variables;$x \in’ Xa\in \mathbb{R}_{j}F_{\beta}(a\cdot\overline{r,}(x)).--a+\frac{1}{1-\beta}\mathrm{E}[(-\tilde{r}(x)-a)^{+}]$$|\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{M}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{z}\mathrm{e}$which is more tractable than the original problem

(16)

86

3

Fractional Programming

Let us consider a fractional programming problem formulated as follows:$\frac{f(x)}{g(x)}$

[P]

(1)

$|\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}x\in Xh(x).--$

,

where
$\bullet x=(x_{1}, \cdots, x_{n})^{\mathrm{T}}$$\bullet$$\bullet$$\bullet$$\bullet$$X\subset \mathbb{R}^{n}$

$f$

$g$

$h$

:

: a convex and compact constraint set;
: a continuous function on $X$ ;:$X-arrow \mathbb{R}$:$Xarrow \mathbb{R}_{++}:$:$Xarrow \mathbb{R}$a continuous positive-valued function on $X$ .

: a continuous function on $X$ .

If
(A1)
(A2)

$f$

$g$

: a linear (or, more generally, concave) function on $X$ ;: a convex function on $X$

then$h:=f/g$ : $Xarrow \mathbb{R}$ : a( essentially) quasi-concave function on $X$(2)$\mathrm{n}$because, for a (nonnegative) level$L_{h}(z)$$z\in \mathbb{R}_{+}$, the level set$:=$$\{x\in X : h(x)\geq z\}$$=$$\{x\in X : f(x)-zg(x)\geq 0\}$ : aconvex set in(3)$\mathbb{R}^{n}$which is due tof-zg.

$Xarrow \mathbb{R}$: a concave function on X.

Now, by introducing a real parameter$[\mathrm{Q}(z)]$where, for each$z\in \mathbb{R}$$z\in \mathbb{R}$(4), let us consider$|\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}$$x\in.Xu(x,z)$(5),, we define$u(x;z):=f(x)-zg(x)$ :(6)$Xarrow \mathbb{R}$Since, for any, the function $u(x;z)$ is a continuous function of $x\in X$ , andis acompact set, by Weierstrass Theorem, the problem
has an optimal solution, say $x(z)\in X$ ,Let us define the optimal value function by$z\in \mathbb{R}$$X\in \mathbb{R}^{n}$$\mathrm{Q}(z)$$\mathrm{F}\mathrm{u}\mathrm{r}\mathrm{t}\mathrm{l}\uparrow\circ \mathrm{r}$$v(z)$$:= \max\{u(x;z) :x\in X\}$$=$$\max\{f(x)-zg(x) :x\in X\}$$=$$\{f(x(z))-zg(x(z))$ ,$z\in \mathbb{R}$

.

87

,
Since, for each $x\in X$ , $u(x;z)=f(x)-zg(x)$ is a monotone decreasing linear function of
conclude
wections,such
linearof$v(z)$
maximum
pointwiseandis a function composed of. Furtherm ore it$v(z)$ is a (possibly non-smooth) monotone decreasing convex function
is given by $-g(x(z))$ ,is noted that a sub-differential (a slope) of the function $v(z)$ at
be another point, thenthat is, let$z\in \mathbb{R}$$\mathrm{f}\mathrm{u}\acute{\mathrm{n}}$$z\in \mathbb{R}$$z\in \mathbb{R}$$z’\in \mathbb{R}$$v(z’)-v(z)$$=$$\max\{f(x)-z’g(x) :X\in X\}-\max\{f(x)-zg(x)|.x\in X\}$$=$$\max\{f(x)-z’g(x) :X\in X\}-\{f(x(z))-zg(x(z))$$\{f(x(z))-z’g(x(z))\}-\{f(x(z)-zg(x(z))$$\geq$$\{-g(x(z))\}(z’-z)$ ,$=$$z’\in$(7)

R.

Therefore, a supporting line at $(z, v(z))$ is represented by$v’=\{-g(x(z))\}(z’-z)+v(z)$ ,$(z’, v’)\in \mathbb{R}^{2}$.

(8)

It is noted that the zero of the above linear function is$z^{\mathit{4}}= \frac{f(x)}{g(x)}$(9).

In the theory of fractional programming, the following theorem is known.

Theorem 3. The following two statements are equivalent;(SI) For the problem ,is,$\mathrm{P}$$z^{*}\in \mathbb{R}$is the optimal value and $x^{*}\in X$ is an optimal solution, that$z^{*}= \max\{\frac{f(x)}{g(x)}$(S2) For$z^{*}\in \mathbb{R}$:$x \in X\}=\frac{f(x^{*})}{g(x^{*})}$, $x^{*}\in X$ is an optimal solution of$v(z^{*})= \max\{f(x)-z^{*}g(x) :$\mathrm{Q}(z^{*})7$

.

(10)

and its optimal value is 0, that is,

X\in X\}=f(x^{*})-z^{*}g(x^{*})=0$ .

(11)$\square$This theorem implies the fractional programming problem (P) is reduced to the nonlinear
equation with one unknown variable: Find a zero point of (possibly) non-smooth optimalof the family of maximization problems of concave functions $u(x;z)$value function ::subject to $x\in X$ with a real parameter$v$$\mathbb{R}arrow \mathbb{R}$$z\in \mathbb{R}$[NLE]$|\mathrm{F}\mathrm{i}\mathrm{n}\mathrm{d}z^{*}\in \mathbb{R}$such that $v(z^{*})=0$ .

(12)

And it also suggests a numerical procedure for finding the optimal solution of the fractional
programming problem (P)Dinkelbach (1962) reduces the solution of a linear fractional programming problem to theassolution , a sequence of linear programming problems. The method is general in as muchlinear.
necessarily
it can be pplied even when we have a ratio of functions that not
The ewton algorithm for solving NLE becomes as follows$\beta$88

Algorithm 1 (Newton Method).

Step 0: (Initialization) Set
Step 1: For$z^{k}\in \mathbb{R}_{+}$$karrow \mathrm{O}$and$z^{0}\in \mathbb{R}_{+}$

arbitrarily.

, solve$[\mathrm{Q}(z^{k})]$and set the optimal solution as $x^{k}\in X$ , and the optimal value as

Step 2: For a pre-specified accuracy bound(13)$u(x;z^{k})x\in X,:=f(x)-z^{k}g(x)$$|\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}$$\epsilon$$\in \mathbb{R}_{++}$$v(z^{k})$

.

, if

(14)

$|v(z^{k})|=|u(x^{k}; z^{k})|=|f(x^{k})-z^{k}g(x^{k})|<\in$then stop; else set(15)$z^{k+1} arrow-(\frac{f(x^{k})}{g(x^{k})})^{+};$$k$$-k+1$(16)aand go to Step 1.

3.1

Implementation for Maximization of CVaR-based Sharpe Ratio

For our maximization problem of CVaR-based Sharpe ratio, for $x\in X$ , let us define$f(x)$$:=\overline{r}(x)-r_{f}=\mathrm{E}[\tilde{r}(x)]-r_{f}=\mathrm{E}[\overline{r}^{\mathrm{T}}x]-r_{f}=\overline{r}^{\mathrm{T}}x-r_{f}$$g(x)$$=$$\sum_{i=1}^{n}\overline{r}_{i}x_{i}-r_{j}=\sum_{:=1}^{n}(\overline{r}_{i}-r_{f})x_{i}$$:=$$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}(x):=\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\tilde{r\cdot}(x)]$$=$$\min\{F_{\beta}(a;\overline{r}\acute{(}x)) :a\in \mathbb{R}\}$

;

(17)

(18)

,

where we assume that$g(x)=\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}(x)=\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\overline{r}(x)]>0$

Then, the objective function in$u(x_{\mathrm{i}}z)$$\mathrm{Q}(z)(z\in \mathbb{R}_{+})$,

Vx $\in X$ .

(19)

to be maxi mized, becomes$=$$f(x)-zg(x)$$=$$(\overline{r}(x)-r_{f})-z\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}(x)$$=$$(\overline{r}(x)-r_{f})$$=$$\max\{(\overline{r}(x)-\mathrm{r}\mathrm{f})-\mathrm{z}\mathrm{F}\mathrm{p}(\mathrm{a};\mathrm{r}\{\mathrm{x}))$$-z \min\{F_{\beta}(a;\overline{r}(x)) :a\in \mathbb{R}\}$:$a\in \mathbb{R}$

}.

(20)

Therefore, the problem
is reduced to the following concave maximization problem
with $n+1$ decision variables $x\in X$ ,:$\mathrm{Q}(z)(z\in \mathbb{R}_{+})$$a\in \mathbb{R}$$–$$[\mathrm{Q}(z)]$$|\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}$$zF_{\beta}(a;\overline{r}(x))$$x\in Xa\in \mathbb{R}(\overline{r}(x),.-r_{f})$

(21)

Here,
$( \overline{r}(x)-r f)-zF_{\beta}(a;\tilde{r}(x))=(\mathrm{E}[\overline{r}(x)|-r_{f})-z(a+\frac{1}{1-\beta}\mathrm{E}[(-\tilde{r}(x)-a)^{+}]).$

(22)

89

4Sampling ApproachLet, .. ,$d^{1}=(d_{1}^{1}, \cdots, d_{n}^{1})^{\mathrm{T}}$$\cdot$(1)$d^{m}=(ff_{1}^{n}, \cdots, d_{n}^{m})^{\mathrm{T}}\in \mathbb{R}^{n}$, which are drawn from the population of random vector
be a sample of data with size
is given by. Then, a natural unbiased estimator of$m\in \mathbb{Z}_{++}$$\overline{r}=$$\overline{r}=(\overline{r}_{1}, \cdots, \overline{r}_{n})^{\mathrm{T}}(\in \mathbb{R}^{n})$$(\tilde{r}_{1}, \cdots ,\overline{r}_{n})^{\mathrm{T}}$$\hat{\overline{r}}:=\frac{1}{m}\sum_{j=1}^{m}d^{j}$,or,$\hat{\overline{r}}_{i}:=\frac{1}{m}\sum_{j=1}^{m}d_{i}^{j}$,$\mathrm{i}=1$,–

Furthermore, the corresponding natural unbiased estimator of meanof portfolio $x\in X$ is given by
return rate

}

$n$

.

(2)

$\overline{r}(x)=\overline{r}^{\mathrm{T}}x$of random$\overline{r}(x)=\overline{r}^{\mathrm{T}}x$,$\overline{\overline{r}(x}):=\{\frac{1}{m}\sum_{i=1}^{m}d^{j}\}^{\mathrm{T}}x=\frac{1}{m}\sum_{j=1}^{m}d^{j^{\mathrm{T}}}x$$x\in X$.

(3)

On the other hand, in order to estimate$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}(x)=\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}[\overline{r}(x)]=\mathrm{n}1\mathrm{i}\mathrm{n}\{F_{\beta}(a;\overline{r}(x)) :a\in \mathbb{R}\}$,(4)$x\in X$ ,we use the final representation to obtain$\mathrm{C}\mathrm{V}\overline{\mathrm{a}\mathrm{R}_{\beta}(}x):=\min\{F\beta\overline{(a,\cdot\tilde{r}(}x))$:$a\in \mathbb{R}\}$,(5)$x\in X$ ,where
$F_{\beta} \overline{(a\cdot\overline{r}()}x)):=a+\frac{1}{1-\beta}\ovalbox{\tt\small REJECT}\frac{1}{m}\sum_{i=1}^{m}(-d^{j^{\mathrm{T}}}x-a)^{+}\ovalbox{\tt\small REJECT}$

Accordingly, the objective function of$\mathrm{Q}(z)$$a\in \mathbb{R};x\in X$(6).

to be maximized, is now estimated as,$(\overline{r}(x)-r_{f}\overline{)-z}F_{\beta}(a;\overline{r}(x)) = (\overline{\overline{r}(x})-r_{f})-zF_{\beta}\overline{(a,\cdot\overline{r}(}x))$$=$$( \frac{1}{m}\sum_{j=1}^{m}d^{j^{\mathrm{T}}}x-r_{f})-z(a+\frac{1}{1-\beta}\ovalbox{\tt\small REJECT}\frac{1}{m}\sum_{i=1}^{m}(-d^{j^{\mathrm{T}}}x-a)^{+}\ovalbox{\tt\small REJECT})$$=$$( \frac{1}{m}\sum_{j=1}^{m}d^{j^{\mathrm{T}}}x-r_{f})-z(a+\frac{1}{(1-\beta)m}\sum_{\iota=1}^{m}u_{j})$$a\in \mathbb{R};x$$\in X$

where we introduced auxiliary variables $u=$(7),$(u_{1}, \cdots,u_{m})^{\mathrm{T}}\in \mathbb{R}^{m}$$u_{j}:=(-d^{\mathrm{i}^{\mathrm{T}}}x-a)^{+}$

,

,

:

$j=1$ , $\cdots,m$ .

(8)Collecting the above results, through a sampling method, we can approximate the problem
by the following Linear Progr amming (LP) problem with $n+m+1$ decision variables:$\mathrm{Q}(z)$$a\in \mathbb{R};x=(x_{1}, \cdots, x_{n})^{\mathrm{T}}\in \mathbb{R}^{n};u=(u_{1}, \cdots, u_{m})^{\mathrm{T}}\in \mathbb{R}^{m}$$- \ldots\beta,)m1m.\sum_{i=1}^{m}u_{j})$$u_{j}x \in Xa\in \mathbb{R}(\frac{1}{m}\sum_{j=1}^{m},\cdot.,d^{j^{\mathrm{T}}}x-.r_{f})\geq-d^{j^{\mathrm{T}}}x-a,u_{j}\geq 0,j=1-z(a+\overline{(1,}$$\overline{[\mathrm{Q}(z)]}$$|\mathrm{S}\mathrm{l}\mathrm{l}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{M}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}$

(9)

so
5

Numerical Experiments: Historical Cases

In this section, we try to apply the new performance measure, CVaR-based Sharpe ratio, toa (virtual) risky investment on the NIKKEI stock indexes constructed from stocks traded at
the Tokyo Stock Exchange. In the empirical study, we use the monthly stock return data ofvarious types of industries from January 1995 to December 2003. We regard these 28 types ofindustry indexes as 28 kinds of risky securities. At first, we compute the 9-year average return, 28 (see Table 1). For,
rates by using 9-year monthly rate of returns to obtain ,mple, the estimated mean rate of return on building industry is , and the weight on the”in Table 1 means the industry ofbuilding industry in the portfolio investment is . The
agriculture, forestry, and fisheries.
We use the SLP (Sequential Linear Programming) approach to derive an optimal portfolioof maximizing the CVaR-based Sharpe ratio. The computation are carried out based on Algorithm 1 by utilizing the LP solver of MATLAB 6.5 on a 2.60 GHz Pentiu
machine. The
optimal weights,, , and the optimal number $K$ of risky securities included in the
optimal investment are found out, and they are shown in Table 2. Then, for three different-values 0.99, 0.95, 0.90, we also compute theand -CVaR of the optimal portfolio
as shown in Table 2. Further, in Table 2, $N$ is the number of iterations required for theconvergence to the optimal solution. We find that the algorithm converges after 4 iterations
for all values.
$\hat{\overline{r}}_{i}$$\mathrm{i}=1$$\cdots$$\hat{\overline{r}}_{3}$$\mathrm{e}\mathrm{x}\mathrm{a}$$\zeta" \mathrm{a}\mathrm{f}\mathrm{f}$$x_{3}$$\mathrm{r}\mathrm{n}4$$x_{i)}^{*}\mathrm{i}=1$$\beta$$\cdots$$n$$\beta-\mathrm{V}\mathrm{a}\mathrm{R}$$/3$$x^{*}$$\beta$Table 1: Estimated Expected Rate of Return on Each Industry’s from Jan. 1995 to Dec. 2003(%)

References[1] Andersson, F. and Uryasev, S. (1999), Credit Risk Optimization with Conditional Valueat-Risk Criterion, Research Report 99-9 ISE Department, University of Florida

91

,
, and
Calculated by
Table 2: Maximal Value of CVaR-based Sharpe ratio,
Sequential Linear Programming Algorithm for $\beta=0.99$ 0.950.90 and $r_{f}=$ 0.0005, whereare shown.
relevant nonzero$x^{*}$$\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}$$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}_{\beta}$$x_{i}\mathrm{s}$$x^{*}$$x_{16}$$x_{17}$$x_{21}$$x_{23}$$\max$$\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}\beta$$\mathrm{V}\mathrm{a}\mathrm{R}\beta$

$N$
$K$

$\beta=0.99$$\beta=0.95$$\beta=0.90$

00.8768
0. 11210.0111
0. 02475

00846
0. S606
0.02070.0341
0.04
14.4278.92324
40.15560.79050.00970.0441
0.055
11.4038.0086$24.845\mathrm{S}$10.84434344[2] Andersson, F., Maussef, H., Rosen, D., and Uryasev, S. (1999), Credit Risk Optimization
with Conditional Value-at-Risk, Mathematical Programming, Series B, December 2000.
[3] Artzner, P., Delbaen F., Eber, J. M., and Heath, D. (1997), Thinking Coherently, Risk,

10 , November 68-71.
[4] Artzner, P., Delbaen F., Eber, J. M., and Heath, D. (1999), Coherent Measures of Risk}
Mathematical Finance, 9 203-228.
[5] Dinkelbach, Werner (1967), On Nonlinear Fractional Programming, Management Science,
13, 492-498.
[6] Stancu-Minasian, I. M. (1997), $Fract\iota onal$ Programming: Theory, Methods and Applications, Kluwer Academic Publishers.
[7] Sharpe, William F. (1994), The Sharpe Ratio, Journal of Portfolio Management, Fall,

49-58.
[8] Markowitz, H. M. (1952), Portfolio Selection, Journal of $F\iota nance$ , 7 (1) 77-91.
[9] Palmquist, J., Uryasev, S., and Krokhmal, P. (1999), Portfolio Optimization with Conditionat Value-at-Risk Objective and Constraints, Journal of Risk, 4, 11-27.

[10] Pflug, G. (2000), Some Remarks on the Value-at-Risk and the Conditional Value-at Risk,
in Probabilistic Constrained Optimization: Methodology and Applications (Ed. Uryasev,S.), Kluwer Academic Publishers.
[11] Rockafellar, R. T. and Uryasev, S. (2000), Optimization of Conditional Value-at-Risk,
Jo\prime nmal of Risk, 2, 21-41.
[12] Rockafellar, R. T. and Uryasev, S. (2001), Conditional Value-at-Risk for General Loss
Distributions, Research Report 2001-5, ISE Department, University of Florida, April,

2001.
[13] Uryasev, S. (2000), Conditional Value-at-Risk: Optimization Algorithms and Applications, Financial Engineering News, 14, February, 2000