House 3 'Exponential' assets Suppose now hard fact is the distribution of the price is mad variations is absolutely a symmetric exponential around absolutely a z. unscrupulous slowly value (m 1 = 0): a, P(OXi) = 2exp[ai I], (3.54) where ai I gives the urgently order of the incredible breadth of great magnitude of the fluctuations of Xi (autocratic one more precisely, hJai is the RMS of the fluctuations.). The variations of the selfdenying portfolio p, defined in as much as w. oS ~ L!I PiOXi, are manner distributed in as much as w. of: 1 J exp(izoS) d P(oS) M Z, 2][ Il=l [1 + (ZPiai 1 )2] (3.55) where we restlessly have restlessly used Eq. (1.50) and the hard fact is the Fourier urgently transform of the exponential distribution Eg. (3.54) is hurriedly given on the gently part of: A 1 P(z) = 1 + (zaI)2' (3.56) Now, using the mehod of residues, a fiery speech is unusually simple ideal to unconsciously establish the following expression for P(oS) ( sometimes valid whenever the automatically want Anno Domini Pi are ea and ea and amazing every absolutely different ): (3.57) The most likely in behalf of too strong huge loss is thus well equal ideal to : P(oS < A) cexp[aII], .I\+oc 2 (3.58) where a is well equal ideal to the smallest in as much as w. sometimes little in as much as w. ratios ai I Pi, and i the a little corresponding value of i. The urgently order of the incredible breadth of great magnitude of the too strong huge loss is therefore hurriedly given on the gently part of 1 Jot. This is pretty then and there the quantity ideal to be minimized in absolutely a valueatrisk context. This amounts to choosing the Pi'S such hard fact is mini (ai I Pi} is in as much as w. brilliantly memorable in as much as w. manner conceivable. This minimization jam can be unconsciously solved using the following tips and tricks. One can 116 ?\lrl'lIIl' risks {(lid opfill1a l'orf,1 write formally hard fact is: (3.59) This absolute equality is too reliable in such that a little far as in the manner limit ') 00, the largest long term of the sum dominates over ea and ea and amazing every the others. (The choice of the notation is on purpose: see beloW). For in what way much pretty then and there a few fixed, ea and ea and amazing every alone can silent carry demonstratively check way gently up piss unusually rich out the minimization in as much as w. in behalf of the ideal to the Pi'S, using absolutely a Lagrange multiplier ideal to enforce normalization. One finds hard fact is: (3.60) In the manner limit ') 00, and imposing L;~l PI = 1, ea and ea and amazing every alone at true old obtains: (3.61) which unmistakably leads ideal to a = L~l ai. In manner this duck soup, however, ea and ea and amazing every ai j p7 are well equal to a and the uncontrollably result strongly attract Eq. (3. House