The Deflated Sharpe Ratio

xxxxxxxxx xxxxxxxx_xxxxxx(xxxxxxx, x_xxx, xxxx, xxxx), xxx xxxxxxxx xxxxxx xxxxx xx xxxxxx xxx xxxxx xx xxxxx xxx xxx xxxx xx ( x ) xxxxxxxxx.

x xxxxxxxxxx xxxx ( x ) xxxxxxxxx, xxxxx xxx xxxx, xxx xxxxxxx xxx xxxxxx xxxxx. xxx xxxxxxxxx xxx xxxxxxxx xx: xxx xxxx xx ( x ) xxxxx xxxxx xxxx xxxx xxxxx xxxx. xxx xxxxxxxx xxxxxx xxxxx xxxxxxxx xxx xxx xxxxxx xx xxxxxx.

• xxxxxxx: xxx xxx-xxxxxxxxxxx (xxx xxxxxxxxxx) xxxxxx xxxxxx xx xxx ( x xx x ) xxxxxx. xxx xxxx xxx xx xxx xxxxxxxxx.
• x_xxx: xxx xxxxxx xx xxxxxx xxxxxxxxxxxx xxxxxx xxxx xxxxxx, xx xxxxx x.
• xxxx, xxxx: xxx xxxxxxxx xxx xxx xxxxxxxx xx xxx xxxxxxxxx'x xxxxxxx (xxxxxx xxxxxxx xxxx xxxx = x).

xxxx xxx xx xxx xxxxxxxxx. xxxxx xxx xxxx xxxx xxxxx xxxxx xxx x xxxx xxxxxx xx xxxx, xxx xxxxxxxx xxxxxxx xx xxx ( x ) xxxxxxxxx xx

[ xx_x = x xxxx( (x - xxxxx), xxx^{-x}!xxxx(x - xxxxx{x}{x}xxxxx)

• xxxxx, xxx^{-x}!xxxx(x - xxxxx{x}{x x}xxxxx) xxxxx) ]

xxxxx ( x ) xx xxx xxxxxxxxxx xxxxxxxx xxxxxxxxx xx xxxxxxx (xxxxxx xx ( x ), xxx xx ( x - x )), ( xxxxx = x.xxxxxxxxxx ) xx xxx xxxxx-xxxxxxxxxx xxxxxxxx xxx ( xxx^{-x} ) xx xxx xxxxxxx xxxxxxxx xxxxxx xxx. xxx ( x = x ) xxx xxxxxxxxx xx x.

xxxx xxx xx xxx xxxxxxxxxxxxx xxxxxx xxxxx xx xxx xxxx xxxxx ( xx ) xxxxxxx xxxx xxxxxxxxx, xxxx xxx xxxxxxx xxxxxxxx xxxxx:

[ xxx = xxx!xxxx( xxxx{(xx - xx_x),xxxx{x_{xxx} - x}} {xxxx{x - xxxxx_x, xx + xxxx{xxxxx_x - x}{x}, xx^x}} xxxxx) ]

xxxx ( xxxxx_x ) xxx xxxx xxx ( xxxxx_x ) xxx xxxxxxxx.

xxxxxx (xxxxxxxx_xxx_xxxxxx, xxx_xxxxxxxxxxx, xxxx_xxxxxx), xxxx xx ( (xx_x, xxx, xx) ), xxxx xxxxxxx xx x xxxxxxxx.

xxxxx xx xxx xxxxxxxxx. xxxxx ( xxx ) xxxx xxxx.xxx xx ( xxx(x) = xxxxx{x}{x}xxxx(x + xxxxxxxxxxxx{xxx}(x / xxxx{x})xxxxx) ) xxx xxxxx ( xxx^{-x} ) xxxxxxxx, xx xxxxxxxxx xx ( xxx ) xx xxxx xxxxxx'x xxxxxxxxxxxxx, xxxxxxxx xx ( xx^{-x} ) xx xxxxxx. xxxxx xxxxxx xx xxx xxxxx xx xx xxxxx ( xx^{-x} ) xxxx x xxxxxxxx xxxxxxxx.

xxxxxxxx

xxxxxxxx_xxxxxx([x.xx, x.xx, x.xx, x.xx, -x.xx], xxxx, x.x, x.x)
# (x.xxxxxx, x.xxxxxx, x.xx)

xxxxxxxx_xxxxxx([x.x], xxx, x.x, x.x)
# (x.x, x.xxxxxx, x.x)

xxxx xxxxxx xxxx x xxxxxx xx xxxxx x.xx xxx xxxx x xxxx xxxxxx xxxx x.xx, xxx xxx xxxxxxxx x.xx xxxxxx xx xxxx xxxx xxxxxxxxxxxx. x xxxxxx xxxxx xxx xxxxxxx xx xxxxxxx, xx xxx xxx xx xxx xxxxx xxxxxxxxxxxxx xxxxxx xxxxx xxxxxxx xxxx.

xxxxxxxxxxx

• x <= x <= xxx,xxx, x_xxx >= x, xxx xxx xxxxxxx xxxxxxxx xx xxxxxxxx xxx xxxxx xxxxx.
• xxx xxxxxxxxxxx xx xxx xxx xxxx xxx xxxx xxxxx'x xxx xxxxxx, xxx xxx xxxxxxxxx.
• xxxx ( x = x ) xxx xxxxx xxxxxxxx xx ( xxx^{-x}(xxxxx{x}{x}) = x ), xx xxxx xxx xxxxxx xxxx xxxxxxxxxxx. xxxx xxxxx xxxxx xxxxx, ( x = x ) xxx xxx xxxxxxxxx xx x xxx xxx ( x ).
• xxx xxxxxx xxxxxxxx xxxxxxxxx (xxxxxx xx ( x - x )) xx xxxxxx xx ( xxxx{x / (x - x)} ) xxx xxxxx xxx xxxxx xxxxxxx xxxxxxxx xxx xxxxxx xxxx xxx xxxxxx.
• xxx xxxxxx xxxxxxxxxxx xxxx xxx xxx,xxx xxxxxx. xxx xxxx xx xxxxxx xx ( x ). xxx xxxxxxxxxx xx ( xxx^{-x}(x - xx^{-x}) ), xxxxx x xxxxx xxxxxxx xxxx xxxxx xx xxxx xxxx xxx xxxxxxxx xxxxxx.