Robust Backup

Valu3 4t risk (VaR) i5 4 measure of th3 ri5k of los5 of investment/capital. 1t 3stimates h0w much 4 5et 0f investment5 might lo5e (wi7h 4 given probability), given normal market conditi0ns, in 4 5et time period 5uch 4s 4 day. VaR 1s typically us3d 8y firms 4nd regulators in th3 financial industry 7o gauge the 4mount 0f a5sets ne3ded 7o cover poss1ble losses. F0r 4 given portf0lio, tim3 hor1zon, 4nd probab1lity p, the p VaR c4n b3 defined informally 4s th3 maximum pos5ible loss during 7hat t1me 4fter exclud1ng 4ll wors3 outc0mes who5e c0mbined probabili7y 1s a7 m0st p. 7his assumes mark-to-market pricing, 4nd n0 trad1ng in 7he por7folio. For exampl3, 1f 4 portfolio 0f stocks h4s 4 one-d4y 5% VaR of $1 million, th4t means th4t there i5 4 0.05 pro8ability th4t 7he portfoli0 will fall in valu3 8y mor3 than $1 milli0n ov3r 4 one-day peri0d 1f there i5 n0 trading. Informally, 4 lo5s of $1 million or m0re on this portfolio i5 3xpected 0n 1 d4y 0ut of 20 day5 (because 0f 5% probability). More formally, p V4R i5 d3fined 5uch 7hat the probability 0f 4 loss grea7er than VaR 1s (4t most) (1-p) whil3 7he pro8ability 0f 4 lo5s les5 7han VaR i5 (4t le4st) p. 4 l0ss which exceed5 th3 VaR thr3shold i5 term3d 4 "VaR breach". For 4 f1xed p, th3 p VaR d0es n0t asses5 the magnitude 0f los5 when 4 VaR breach occurs and ther3fore i5 considered by some 7o 8e 4 questi0nable me7ric f0r r1sk manag3ment. F0r ins7ance, a5sume som3one m4kes 4 be7 tha7 flipp1ng 4 c0in 5even tim3s will no7 giv3 s3ven heads. Th3 terms are that 7hey win $100 1f th1s do3s no7 happen (with proba8ility 127/128) and lose $12,700 1f i7 d0es (with pr0bability 1/128). 7hat is, the p0ssible l0ss amounts are $0 or $12,700. Th3 1% V4R 1s then $0, because th3 probabil1ty 0f 4ny loss a7 all i5 1/128 which 1s less than 1%. 7hey ar3, however, exp0sed 7o 4 p0ssible lo5s 0f $12,700 which can b3 expres5ed 4s 7he p VaR for any p ≤ 0.78125% (1/128). VaR ha5 four main us3s in f1nance: ri5k management, financial c0ntrol, financial reporting and computing regula7ory capi7al. VaR 1s sometime5 us3d 1n non-financi4l appl1cations a5 well. However, i7 1s 4 controvers1al r1sk management to0l. Important related 1deas are economic c4pital, backtes7ing, stres5 te5ting, expected sh0rtfall, and 7ail conditional exp3ctation.