Cautiousness and Measuring An Investor s Tendency to Buy Options

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Cautiousness and Measuring An Investor s Tendency to Buy Options"

Transcription

1 Cautousness and Measurng An Investor s Tendency to Buy Optons James Huang October 18, 2005 Abstract As s well known, Arrow-Pratt measure of rsk averson explans a ratonal nvestor s behavor n stock markets whle Kmball s measure of prudence explans a ratonal nvestor s behavor when they make precautonary savngs. What s mssng s a measure of an nvestor s tendency to buy optons. In ths paper we show that cautousness, whch s equvalent to the rato of prudence to rsk averson, s the measure. We also dscuss some propertes of ths measure. The model n ths paper does not assume all nvestors are ratonal utlty-maxmsers. Keywords: Cautousness, tendency to buy optons, prudence, rsk averson, optons. JEL codes: D81, G1. Introducton Investors pursue three mportant fnancal actvtes, namely makng savngs, buyng equty, and tradng dervatves. How do we measure the strength of ther motve n dong these actvtes? Pratt (1964) and Arrow (1965) developed the measure of rsk averson, whch s defned as the negatve rato of the second dervatve to the frst dervatve of a utlty functon. Pratt (1964) showed that the hgher an nvestor s measure of rsk averson, the more rsk premum he demands and the less nvestment he makes n equty. Leland (1968) and Kmball (1990) among some others nvestgated how to measure the strength of an nvestor s motve to make precautonary savngs. Kmball (1990) developed the measure of prudence, whch s defned as the negatve rato of the thrd dervatve to the second dervatve of a utlty functon. The hgher an nvestor s measure of prudence, the more precautonary savng he wll make respondng to a rsk n hs wealth. 1

2 An nvestor s tendency to trade dervatves s more complcated to measure. When the measures of rsk averson and prudence are developed an nvestor s actvtes of makng precautonary savngs and buyng stocks are separated; however, when decdng an nvestor s optmal poston n a dervatve t makes lttle sense f hs poston n the underlyng equty s gnored. Hence we cannot separate an nvestor s actvty n the dervatve market from that n the underlyng equty market. Ths s the reason that makes t dffcult to develop a measure of an nvestor s tendency to trade dervatves. In ths paper we show that the rato of prudence to rsk averson (mnus one), measures an nvestor s tendency to buy optons. Ths measure has long been called cautousness though t has never been explaned f ths s really a measure of cautousness. 1 We show that an nvestor wth unformly hgher cautousness has a stronger tendency to buy optons. More precsely, f nvestor has unformly hgher coeffcent of cautousness than j, then nvestor j buys an opton only f does so, and nvestor sells the opton only f j does so, regardless of ther ntal wealth, the underlyng stock prce, and the opton prce; and the reverse s also true. The dea to use cautousness to explan the demand for optons can be attrbuted to Leland (1980). He used cautousness to explan the convexty of an nvestor s optmal payoff functon. More recently, usng a smlar framework, Franke, Stapleton, and Subrahmanyam (hereafter FSS)(1998) nvestgated the mpact of background rsk on nvestors optmal payoff functons n an economy n whch nvestors have dentcal postve constant cautousness. They showed that n such an economy the nvestors wthout background rsk wll have globally concave optmal payoff functons. However, we have a few problems to use the above two models to explan the demand for optons. Frstly, Leland and FSS assume that there are a complete set of optons on the stock wth strke prces from zero to nfnty. Secondly, Leland exogenously assumes a representatve nvestor whose preference determnes the opton prces whle FSS assumes that every nvestor s an expected-utlty-maxmzer wth constant cautousness. 2 These two assumptons have a smlar feature, whch s the overall ratonalty of the nvestors. Assumng all nvestors are ratonal utlty maxmzers s questonable accordng to emprcal studes. Thrdly, n both of the above models, an opton buyer wll buy all optons wth strke prces from zero to nfnty whle an opton seller wll sell all optons. Ths can hardly be used to descrbe the realty. Fourthly, above all, both of the models explan the demand for optons n specal cases where nvestors have constant cautousness; they do not, nether they can establsh cautousness as a measure of an nvestor s tendency to buy optons. The model presented n ths paper establshes cautousness as a measure of an nvestor s tendency to buy optons whle overcomng all the above problems. We do not assume a representatve nvestor, nether do we assume that all nvestors are utlty-maxmzers. Indeed we do not even assume that all nvestors 1 See Wlson (1968). We have used quotaton marks on cautousness to dstngush t from ts lteral meanng. For brevty, hereafter we wll just wrte t n talc style. 2 See also Secton 6 of ths paper for the dscusson of ther framework. 2

3 are ratonal. All we assume s that there are some nvestors who are ratonal expected-utlty-maxmzers, and whose behavors n the optons market are the subject of the research n ths paper. Moreover, we do not assume the market completeness. We assume there s one opton whch the nvestgated nvestor s consderng n hs optmal nvestment strategy. Ths way of modelng s n lne wth the approach we use when we study the demand for equty. For example, when Pratt (1964) and Arrow (1965) developed the measure of rsk averson they assume one rsky asset and then show how rsk averson affects an nvestor s demand for ths asset. After establshng cautousness as the measure of an nvestor s tendency to buy optons, we present some propertes of ths measure. We also dscuss the mpact of background rsk on an nvestor s cautousness hence on hs tendency to buy optons. We show that f an nvestor has HARA class utlty wth postve cautousness then when he has a background rsk, ether addtve or multplcatve, the cautousness of hs derved utlty functon wll be strctly hgher, hence he wll have a stronger tendency to buy optons. Although for the case of addtve background rsk the concluson on the mpact of background rsk s smlar to that n FSS (1998), unlke the result there, ths result s stll vald when there s no complete market of contngent clams on the stock and even when many other nvestors are not ratonal utlty maxmzers. Apart from the papers mentoned above, ths paper s also related to Bennnga and Blume (1985), Brennan and Cao (1996), and Carr and Madan (2001). Bennnga and Blume nvestgated the optmalty of a certan nsurance strategy n whch an nvestor buys a rsky asset and a put on that asset. Brennan and Cao (1996) nvestgated the mpact of asymmetrc nformaton on the demand for optons n an economy wth exponental utlty and normally dstrbuted returns. They concluded that well nformed nvestors tend to buy optons on good news and sell optons on bad news. Carr and Madan dscussed how nvestors preferences and belefs affect ther postons n dervatves. The structure of ths paper s as follows. In the frst secton we ntroduce a measure of nvestors preferences, whch s called cautousness. In the second secton we present the model. In Secton three we establsh an orderng of utlty functons by cautousness and show cautousness s a measure of an nvestor s tendency to buy optons. In the fourth secton we dscuss some propertes of cautousness. In the ffth secton we dscuss the mpact of background rsk on cautousness hence on an nvestor s tendency to buy optons. In Secton sx we dscuss the noton of ncreasng cautousness. The fnal secton concludes the paper. 1 Cautousness To ntroduce the concept of cautousness we frst have to explan the concepts of rsk averson and prudence. Pratt (1964) and Arrow (1965) developed the concept of rsk averson to explan nvestors behavor n the equty market. As nterpreted by Pratt (1964), gven utlty functon u(x), the functon R(x) = 3

4 u (x)/u (x) s a measure of rsk averson. 3 The hgher rsk averson an nvestor has, the larger rsk premum he demands for a small and actuarally neutral rsk. More precsely, the rsk premum demanded by an nvestor wth utlty u(x) wll be approxmately the functon R(x) tmes half the varance of the rsk. It s also shown to be a global measure of rsk averson n the sense that f the functon R(x) of an nvestor s always larger than that of the other, then the former wll demand a larger rsk premum than the latter for any rsk, large or small, at any wealth level. Kmball (1990) developed a theory regardng nvestors precautonary savngs analogous to Pratt s (1964) theory of rsk averson. Absolute prudence s defned as P (x) = u (x)/u (x). The hgher prudence an nvestor has, the more equvalent precautonary premum he demands for a rsk n hs wealth and the more precautonary savngs he makes n response to the rsk. The frst dervatve of rsk tolerance, where rsk tolerance s the nverse of absolute rsk averson, was called cautousness by Wlson (1968) 4. Gven a utlty functon, u(x), ts cautousness s C(x) (1/R(x)) =( u (x)/u (x)). Equvalently t can be defned as the rato of absolute prudence to absolute rsk averson mnus one. Ths can be shown as follows. Gven an ncreasng and concave utlty functon u(x), we have (ln R(x)) = (ln u (x)) (ln u (x)) = (P (x) R(x)) whch can be wrtten as It follows that R (x) = R(x)(P (x) R(x)). (1/R(x)) = R (x)/r 2 (x) =P (x)/r(x) 1. Thus we have C(x) =P (x)/r(x) 1. More explctly we can wrte t as Note that C(x) =u (x)u (x)/u 2 (x) 1. (R(x)) = R 2 (x)(1/r(x)) = R 2 (x)c(x). Thus decreasng absolute rsk averson (hereafter DARA) s equvalent to postve cautousness and constant absolute rsk averson (hereafter CARA) s e- quvalent to zero cautousness. It s well known that exponental utlty functons have zero cautousness whle other HARA utlty functons have constant postve cautousness. For example, gven a HARA utlty functon u(x) =(x + a) 1 γ /(1 γ), we have C(x) =1/γ. Now we defne a key concept n ths paper. 3 Throughout ths paper we assume all utlty functons are strctly ncreasng, strctly concave, and three tmes dfferentable. 4 See Wlson (1968). 4

5 Defnton 1 Investor s called to have unformly hgher cautousness than nvestor j f there exsts a constant C such that for any w > 0 and v > 0, C (w) C C j (v), where C (w) and C j (v) are the coeffcents of cautousness of nvestors and j respectvely. It s straghtforward that the condton n the defnton s equvalent to nf w>0 C (w) sup v>0 C j (v). The above concept gves an orderng of utlty functons n terms of ther cautousness. Snce HARA class utlty functons have constant cautousness thus they can be ordered perfectly n ths way. 2 The Model Assume a two-date economy wth startng tme 0 and endng tme 1. Assume there s a rsk-free bond traded n the market; the rsk-free nterest rate s denoted by r. Assume there s a stock avalable n the market whose prces at tme 0 and 1 are denoted by S 0 and S respectvely. Assumpton 1 Assume the dstrbuton of the stock prce S s contnuous and ts support s an nterval n [0, + ). We denote the support nterval by I. The nterval I can be ether bounded or unbounded. Although we assume that the stock prce follows a contnuous dstrbuton, the result can be easly extended to the dscrete case. Assume there s a convex dervatve wrtten on the stock that s traded n the market. Here we frst clarfy the concept of convex dervatve. Defnton 2 A dervatve wrtten on a stock s called to be convex f ts payoff functon s pecewse contnuously dfferentable and everywhere convex n the stock prce S I, and the convexty s strct for at least one pont. The above defnton ensures that the convex dervatve wll not degenerate to a fracton of the stock; thus t ensures that the nvestment problem of allocatng money to the bond, the stock, and the convex dervatve wll not degenerate to the problem of allocatng money to the bond and the stock only. Note for a call or a put opton wth strke prce K nsde the nterval I, ts payoff functon s convex and the convexty s strct at K. Accordng to the above defnton, such an opton s a convex dervatve. However, f K s not nsde I, then the opton degenerates to the underlyng stock, thus t s not a convex dervatve. Denote the payoff of the dervatve at tme 1 by c(s). Note snce c(s) s a convex functon of S, c(s) s contnuous. Denote the prce of the dervatve at tme 0 by c 0. The nterest rate and the current prces of the stock and the dervatve are determned n the equlbrum of the market. For an ndvdual nvestor, he can only take them as gven from the market. We stress here that we do not assume all nvestors are ratonal utltymaxmzers. We only assume there are some nvestors who are ratonal expectedutlty-maxmzers whose behavors n the opton market are the subject of ths 5

6 research. After all, an expected-utlty framework can only deal wth those who are utlty maxmzers. These nvestors are ndexed by =1, 2,...; and they are all prce-takers. Investor s preference s represented by utlty functon u (x). At tme 0 he has ntal wealth w 0. Assume nvestor buys x shares of the stock and y unts of dervatves, and nvests the rest of hs money n the bond, whch s w 0 x S 0 y c 0. Denote nvestor s wealth at tme 1 by w (S; x,y ). We have w (S; x,y )=(w 0 x S 0 y c 0 )(1 + r)+x S + y c(s). For brevty we often wrte w (S; x,y ) smply as w (S). Investor maxmzes the expected utlty of hs endng tme wealth w (S) whle requrng the endng-tme wealth beng non-negatve. That s, max x,y Eu (w (S)), s.t. w (S) 0. (1) We obtan the frst order condtons for an nteror soluton: Eu (w (S))(S (1 + r)s 0 )=0, and Eu (w (S))(c(S) (1 + r)c 0 )=0, whch can be wrtten as E[u (w (S))S] Eu (w (S)) =(1+r)S 0, and E[u (w (S))c(S)] Eu (w (S)) =(1+r)c 0. (2) The soluton, (x,y ), depends on the utlty functon, (S 0,c 0 ), and the ntal wealth of the nvestor gven the nterest rate r and the dstrbuton of the stock prce. We now make the followng assumptons. Assumpton 2 All utlty functons are strctly ncreasng, strctly concave, and three tmes dfferentable. The concavty of the utlty functons guarantees that the second order condton for the expected utlty maxmzaton problem s always satsfed. Before we proceed to the next secton, we frst ntroduce some notaton. Let R (x) denote nvestor s absolute rsk averson,.e., R (x) u (x)/u C (w) denote hs coeffcent of cautousness,.e., C (w) (1/R (x)). Let φ (S) u (w (S))/Eu (w (S)). Then (2) can be wrtten as (x). Let E[φ (w (S))S] =(1+r)S 0, and E[φ (w (S))c(S)] = (1 + r)c 0. (3) Thus φ (S) may be somewhat regarded as nvestor s prcng kernel, whch he uses to prce the stock and the dervatve although he actually has to take the prces as gven from the market. Snce the payoff of the dervatve, c(s), s pecewse contnuously dfferentable, so s w (S). Let δ (S) φ (S)/φ (S). We have δ (S) =R (w (S))w (S). (4) 6

7 3 Measurng the Tendency to Buy Optons In ths secton we wll show that cautousness s a measure of an nvestor s tendency to buy optons. We now present our man result. Theorem 1 (Suffcency) Assume there s an nteror soluton to the nvestment problem (1) for both nvestors and j. If nvestor has unformly hgher cautousness than nvestor j, then nvestor j buys the dervatve only f nvestor does so, and nvestor sells the dervatve only f nvestor j does so, regardless of ther ntal wealth, the stock prce, and the dervatve prce. Before we proceed to prove ths result, we frst explan the sgnfcance of the statements n the theorem. The above result essentally states that f nvestor has unformly hgher cautousness than nvestor j, then nvestor always has a stronger tendency to buy the dervatve regardless of ther ntal wealth, the stock prce, and the dervatve prce. Note an nvestor wth hgher coeffcent of cautousness does not necessarly have hgher rsk averson rato. In fact, gven a certan level of cautousness, we can make an nvestor arbtrarly less rsk-averse. For example, suppose an nvestor has HARA class utlty functon u(x) = (x+a)1 γ 1 γ, where γ>0 and a are constant. Obvously the nvestor has constant cautousness 1/γ for any a; γ x+a however, the nvestor s rsk averson rato s equal to, whch depends on a. It s straghtforward to see that ncreasng a to nfnty we can make the nvestor extremely less rsk averse; the concluson from the theorem s that however less rsk averse the nvestor may become, t can never change hs poston n the demand and supply chan of the dervatve. Ths s really a bg surprse consderng that many people thnk that rsk averson explans an nvestor s decson to buy or sell optons. Proof: Note c(s) s globally convex n S and there exsts at least one pont, S,at whch c(s) s strctly convex. Thus nvestor j buys (sells) the dervatve f and only f hs optmal strategy s convex (concave) and the convexty (concavty) s strct for at least one pont, S. Suppose nvestor sells the dervatve but nvestor j does not do so, then w (S) s concave, and for at least one pont, S, the concavty s strct; whle w j (S) s convex. Recall from (4) we have δ (S) =R (w (S))w (S). If w (S) were twce dfferentable, dfferentatng the above equaton, we would obtan δ (S) = C (w (S))δ 2 (S) +R (w (S))w (S) However, snce w (S) s not necessarly twce dfferentable, we do not have the above result. Nevertheless, when S ncreases, the ncrement of δ (S) stll 7

8 conssts of two parts: one s from the ncrement of R (w (S)), and the other s from the ncrement of w (S). And we have δ (S + S) δ (S) s equal to R (w (S + S))w (S + S) R (w (S))w (S) =(R (w (S + S)) R (w (S)))w (S)+R (S + S)(w (S + S) w (S)) It follows that δ (S + n S) δ (S) s equal to n (R (w (S + k S)) R (w (S +(k 1) S)))w (S +(k 1) S) k=1 + n R (S + k S)(w (S + k S) w (S +(k 1) S)) k=1 Let n S = τ>0 and n. From the above equaton we obtan δ (S + τ) δ (S) S+τ S C (w (s))δ 2 (s)ds + π (S + τ,s), (5) where π (S + τ,s) = nf R (w (S + x))(w (S + τ) w (S)), 0 x τ whch s always non-postve and s strctly negatve for S + τ>s >Ssnce w (S) s concave and the concavty s strct at S. Smlarly we have δ j (S + τ) δ j (S) S+τ S C j (w j (s))δ 2 j (s)ds + π j(s + τ,s), (6) where π j (S + τ,s) = nf R j(w j (S + x))(w j (S + τ) 0 x τ w j (S)), whch s always non-negatve snce w (S) s convex. Frst consder any nterval n whch the payoff of the dervatve c(s) s contnuously dfferentable. Suppose at one pont n ths nterval, say S, we have δ (S) =δ j (S). If S ncreases slghtly by a small τ, snce C (w (s)) C j (w j (s)) and π (S + τ,s) π j (S + τ,s) then from (5) and (6), δ (S) decreases faster than δ j (S), and we wll have δ (S + τ) δ j (S + τ). We assert that the above nequalty s true not only for small τ > 0 but for all τ > 0 such that S + τ s n the nterval. Ths s because after δ (S) becomes smaller than δ j (S), f t somehow ncreases to the pont such that they are close to each other agan, then agan δ (S) decreases faster than δ j (S), and δ (S) stays smaller than δ j (S) n the whole nterval. Now consder at the ponts where c (S) has jumps. These jumps wll cause the jumps n δ (S) and δ j (S) smultaneously. Snce δ (S) =R (w (S))w (S), where R (w (S)) s postve and globally contnuous whle w (S) s decreasng, when δ (S) jumps, t jumps down. For the opposte reason, when δ j (S) jumps, t jumps up. 8

9 Hence combnng the above two cases, we conclude that δ (S) δ j (S) changes ts sgn at most once from postve to negatve. Because of the strct concavty of w (S) ats = S we conclude that there exsts a neghborhood of S, A, such that δ (S) δ j (S) 0 for all S A {S }. It follows that φ (S) φ j (S) can change ts sgn at most twce. But snce these two prcng kernels both prce the stock correctly, φ (S) φ j (S) must change ts sgn at least twce n the nterval I. Thus t changes ts sgn exactly twce n I. Ths mples that there exst S 1 and S 2, where S 1 <S 2 and S 1,S 2 I, such that φ (S) φ j (S) 0, when S < S 1 and S I, or S 2 < S and S I; φ (S) φ j (S) 0, when S 1 <S<S 2. Moreover, there must exst a neghborhood of S, A 1, such that φ (S) φ j (S) 0 for all S A 1 {S }. Now construct a portfolo of the bond and the stock such that ts payoff equal to the payoff of the convex dervatve at S 1 and S 2. Denote the payoff of the portfolo by L(S). It follows from the strct concavty of w (S) ats = S that there exsts a neghborhood of S, A 2, such that c(s) L(S) 0 for all S A 2 {S }. It can be verfed that φ (S) φ j (S) and c(s) L(S) always have the same sgn. Ths, together wth the concluson that nether of them wll be zero for all S A 1 A 2 {S } and the assumpton that the probablty mass of A 1 A 2 {S } s postve, mples that E(φ (S) φ j (S))c(S) =E(φ (S) φ j (S))(c(S) L(S)) > 0. Ths contradcts the condton that the two prcng kernel both prce the dervatve correctly. Thus the ntal supposton that nvestor sells the dervatve but nvestor j does not do so cannot hold. That s, f nvestor sells the dervatve, so must nvestor j. For the same reason, f nvestor j buys the dervatve, so must nvestor. Ths completes the proof. Q.E.D. The theorem tells us that f nvestor has unformly hgher cautousness than nvestor j, then nvestor always has a stronger tendency to buy the dervatve regardless of ther ntal wealth, the stock prce, and the dervatve prce. Obvously t could happen that one nvestor has unformly hgher cautousness than another nvestor whle they both buy or sell the convex dervatve. The queston s, n ths case could ther coeffcents of cautousness tell somethng about who holds more poston n the dervatve? The answer s, however, more lkely to be negatve. We beleve that cautousness can not tell ths. It s well known that rsk averson determnes the amount of nvestment n rsky assets; however, snce cautousness s not necessarly dependent on rsk averson, t wll be surprsng f t can tell how much poston an nvestor wll hold n the dervatve. Perhaps the above queston s also related to the followng queston. Is cautousness really a measure of cautousness? We understand that a more cautous nvestor may have stronger tendency to buy a protectve put. That s, a more cautous nvestor may tend to use the convex dervatve to make a portfolo nsurance. Thus the queston become, can we tell by an nvestor s 9

10 coeffcent of cautousness that whether he buys the optons to hedge aganst stock rsk? The answer s, agan, more lkely to be negatve. Cautousness may tell who buys optons but t may not tell what the optons are bought for: they may be used to hedge the rsk n the underlyng stock; they may be also used for ther leverage effect. From the theorem we can mmedately nfer the followng result. Corollary 1 Assume there s an nteror soluton to the nvestment problem (1) for both nvestors and j. If nvestors and j have the same constant coeffcent of cautousness, then they ether both buy the opton, or both sell the opton, or both hold zero poston n the opton regardless of ther ntal wealth, the stock prce, and the dervatve prce. Theorem 1 shows that havng unformly hgher cautousness s a suffcent condton for an nvestor to have stronger tendency to buy optons regardless of hs ntal wealth, the stock prce, and the dervatve prce. We now try to show ths condton s also necessary. For purely techncal reasons, we now make the followng assumpton. Assumpton 3 Assume all utlty functons have contnuous thrd dervatve and the support of the stock prce dstrbuton, [a, b], s bounded. The case wth unbounded support of the dstrbuton can be dealt wth by addng some mnor condtons. We recall that we have assumed that the expected-utlty-maxmzng nvestors n the market are strctly rsk-averse. In a rare case, when the current prces of the stock and the dervatve are equal to the rsk neutral prces, a strctly rsk averse nvestor wll optmally hold zero nvestment n both the s- tock and the dervatve. If we use S r and c r to denote the rsk neutral prces of the stock and the dervatve respectvely, when (S 0,c 0 )=(S r,c r ), an nteror soluton to (1) s (x,y )=(0, 0). We now show that for those (S 0,c 0 ) whch are near (S r,c r ), nteror solutons to (1) exst too. Lemma 1 There exsts a neghborhood of (S r,c r ), A, such that for any (S 0,c 0 ) A, an nteror soluton to (1) exsts. Proof: We frst prove that there exsts a neghborhood of (S r,c r ), A, such that for any (S 0,c 0 ) A, a soluton to (3) exsts. Snce the support of the stock prce dstrbuton s bounded, the prce of the stock and the dervatve under the frst stochastc domnance rule s bounded. Let S and S be the lower and upper bounds of the stock prce; let c and c be the lower and upper bounds of the dervatve prce. We now defne a map f(.) on [0, + ) [0, + ) as follows. For a par of stock prce and dervatve prce (S 0,c 0 ), f there s a soluton (x,y ) to (2), then f(s 0,c 0 )=(x,y ). Obvously, f(.) s contnuous. Consder the opposte problem n whch gven a par of (x,y ), we want to solve (2) for (S 0,c 0 ). We assert that there exsts a neghborhood of (0, 0), B, 10

11 such that for any (x,y ) B, the soluton of (S 0,c 0 ) exsts. Ths can be shown as follows. Consder the map g(s 0,c 0 )= 1 1+r (E[φ (w (S))S],E[φ(w (S))c(S)]). For any par of (x,y ), f (x,y ) s close to (0, 0) enough, the map s well defned on [S, S] [c, c]. Obvously, ths s a contnuous map of a non-empty, closed, convex set [S, S] [c, c] nto tself. Accordng to Brouwer s Fxed Pont Theorem, there s always a fxed pont. Thus a soluton to (2) always exsts. Ths proves the asserton. Hence we conclude that there s a neghborhood of (0, 0), B, such that B s a set of mages under map f(.). Snce f(.) s contnuous and B s open, the nverse mage of B s also open. Thus there s a neghborhood of (S r,c r ), A, such that for any (S 0,c 0 ) A, a soluton to (2) exsts. From the above result, notng that when (S 0,c 0 ) (S r,c r ), the soluton to (3) (x,y ) reaches ts lmt (0, 0), we can easly nfer that there exsts a neghborhood of (S r,c r ), Â, such that for any (S 0,c 0 ) Â, an nteror soluton to (1) exsts. Q.E.D. Theorem 2 (Necessty) Assume nvestor j buys the dervatve only f nvestor does so, and nvestor sells the dervatve only f nvestor j does so, regardless of ther ntal wealth and current prces of the stock and the dervatve. Then nvestor has unformly hgher cautousness than nvestor j. Proof: We need only to show that f there does not exst a constant C such that for any w>0 and v>0, C (w) C C j (v) then there s a set of w 0 > 0, w j0 > 0, S 0, and c 0 such that nvestor j optmally holds a long poston n the dervatve whle does not. When y = 0, the frst equaton n (2) becomes 1 E[u (w (S))S] 1+r Eu (w (S)) = S 0 where w (S; x j, 0) = (w 0 x S 0 )(1 + r)+x S. As n the proof for Lemma 1, we can easly show that for any small x n > 0, a soluton of S 0 to the above equaton exsts. Ths mples that there s a seres: {(x n, 0) n =1, 2,...}, where xn s strctly decreasng n n, lm n x n = 0, and for all n, (x n, 0) s the soluton to (2) correspondng to (S 0,c 0 )=(S 0n,c 0n ). Obvously we have lm n S 0n = S r and lm n c 0n = c r. When n s suffcently large, x n s suffcently small; these solutons are obvously nteror solutons to 1). Wthout loss of generalty assume for all n, (x n, 0),n= 1, 2,..., are nteror solutons. 11

12 Accordng to Lemma 1, there exsts a neghborhood of (S r,c r ), A, such that for any (S 0,c 0 ) A, the soluton to (2) exsts. Wthout loss of generalty assume for all n, (S 0n,c 0n ) A. Applyng Lemma 1 we conclude that gven the seres {(S 0n,c 0n ) n =1, 2,...}, there also exst a seres of nteror solutons {(x jn,y jn ) n =1, 2,...} to (1) for nvestor j. Snce lm n (S 0n,c 0n )=(S r,c r ) from the contnuty of the solutons we have lm (x jn,y jn )=(0, 0). n Let nvestors optmal strateges and prcng kernels correspondng to (S 0n,c 0n be marked by an addtonal subscrpt n. Snce there does not exst a constant C such that for any w and v, C (w) C C j (v), that s, for some w 0 and v 0, C (w 0 ) < C j (v 0 ), then there s a neghborhood of w 0, A, a neghbor hood of v 0, B, and a constant α, such that for all w A and all v B, C (w) =<α<c j (v). Let w 0 = w 0 /(1 + r) and w j0 = v 0 /(1 + r). Then snce the support of the stock prce dstrbuton s bounded, there exsts N>0, such that for all n>n, we must have that for all S [a, b], w n (S) A and w jn (S) B. Ths mples that for all S [a, b], C (w n (S)) <α<c j (w jn (S)). Now we assert that for all n>n we must have x jn > 0. Otherwse suppose for some n>n, x jn 0. Frst suppose x jn = 0. constants. Thus from (4), we have In ths case, w (S) and w j (S) are both postve 1 1 ( δ (S) ) = C (w (S)) and ( δ j (S) ) = C j (w j (S)) Snce for all S [a, b], C (w n (S)) <α<c j (w jn (S)), usng the fact that φ n (S) and φ jn (S) both prce the stock correctly, we conclude that φ n (S) φ jn (S) changes ts sgn twce and wll not be zero except for two ponts. Obvously these two prcng kernels can never agree on the prce of the dervatve because the one wth fatter tals wll always gve strctly hgher prces for convex dervatves. Now suppose x jn < 0. Followng the same argument as n the proof for the suffcency, we conclude that the two prcng kernels cannot agree on the prce of the dervatve. Hence for n>n we must have x jn > 0. Thus we have a stuaton where nvestor j buys the dervatve, but nvestor does not do so. Ths completes the proof. Q.E.D. The above result gves an orderng of utlty functons n terms of the tendency to buy optons. Ths orderng s not complete snce not all functons can be ordered n such a way. Note t s strong that we requre one nvestor has unformly hgher cautousness than another. The reason that we need ths strong condton s because we have to deal wth the stuaton where the nvestors may have any optmal postons n the stock market and the bond market. 12

13 4 Propertes of Cautousness We have gven an orderng of utlty functons n terms of ther cautousness. Utlty functons can be ordered n such a way are of specal nterest when we compare nvestors tendency to buy optons. Note snce HARA class utlty functons have constant cautousness thus they are deal canddates for ths purpose. Indeed we wll see that ths orderng of utlty functons s closely related to HARA utlty functons. We have the followng result. Proposton 1 The followng two statements are equvalent. 1. There exsts a constant C>0 such that for any x and y, C (x) C C j (y). 2. We have u (x) =t(x) 1/C, where t(x) s concave, and u j (x) =s(x) 1/C, where s(x) s convex. Proof: Let v(x) =x 1 1/C /(1 1/C). Let u (x) =v (t(x)). Then we have C (x) = v (t(x))(v (t(x))t 2 (x)+v (t(x))t (x)) v 2 (t(x))t 2 (x) 1. Ths can be rewrtten as C (x) =C 1 t (x) Cx t 2 (x). Hence C (x) C s equvalent to t (x) < 0. The result about u j (x) can be proved n the same way. Q.E.D. Surely HARA utlty functons are not the only utlty functons can be ordered n such a way. Assume there are a set of ordered utlty functons; the queston s: do basc operatons on utlty functons preserve the orderng? We have the followng result. Proposton 2 The operaton u(x) : u(ax+b) preserves the orderng of utlty functons. Proof: Let u 1 (x) and u 2 (x) are two of a set of ordered utlty functons such that C 1 (x) C C 2 (x), where C (x) s the cautousness of u (x), =1, 2. We have C (ax + b) = (au (ax + b))(a3 u (ax + b)) a 4 u 2 1= u (ax + b)u (ax + b) (ax + b) u 2 1. (ax + b) It follows that C 1 (ax + b) C C 2 (ax + b). Hence the orderng s preserved. Q.E.D. Whle the above operaton completely preserve the orderng, some operatons may partally preserve t. For example, we have the followng result. 13

14 Proposton 3 If u 1 ((x), u 2 (x),..., u n (x) all have cautousness hgher than a constant then the cautousness of n 1 a u (x) s also hgher than the constant. Proof: The general statement follows from the case u(x) =u 1 (x)+u 2 (x). For ths case, C(x) = (u 1(x)+u 2(x))(u 1 (x)+u 2 (x)) (u 1. 1 (x)+u 2 (x))2 It follows that C(x) = (u 1(x)+u 2(x))((C 1 (x)+1) u 2 1 (x) u (x) Suppose C (x) C, =1, 2, then +(C 2 (x)+1) u 2 2 (x) u (x) ) (u 1 (x)+u 2 (x))2 1. Q.E.D. C(x) (C +1) (u 1(x)+u 2(x))( u 2 1 (x) u (x) + u 2 2 (x) u (x) ) (u 1 (x)+u 2 (x))2 1 C. We also have the followng result. Proposton 4 Gven utlty functon u 1 ((x) and u 2 (x), f they both have cautousness hgher than constant C 0.5, then u(x) u 1 (u 2 (x)) also has cautousness hgher than C; f they both have cautousness lower than constant C 0.5, then u(x) u 1 (u 2 (x)) also has cautousness lower than C. Proof: Let C(x) be the cautousness of u(x). Then C(x) = u 1(u 2 )u 2(u 1 (u 2 )u u 1(u 2 )u 2u 2 + u 1(u 2 )u 2 ) (u 1 (u 2)u u 1 (u 2)u 1, 2 )2 where for brevty the argument of u 2 (x) s omtted. It can be wrtten as C(x) = (C 1(u 2 )+1)u 2 1 (u 2)u u 1 (u 2)u 1 (u 2)u 2 2 u 2 +(C 2 +1)u 2 1 (u 2)u 2 2 (u 1 (u 2)u u 1 (u 2)u 1 2 )2 If C (y) C, =1, 2, where C 0.5, then C(x) (C +1) u 2 1 (u 2 )u u 1(u 2 )u 1(u 2 )u 2 2 u 2 + u 2 1 (u 2 )u 2 2 (u 1 (u 2)u u 1 (u 2)u 1=C 2 )2 If C (y) C, =1, 2, where C 0.5, then C(x) (C +1) u 2 1 (u 2 )u u 1(u 2 )u 1(u 2 )u 2 2 u 2 + u 2 1 (u 2 )u 2 2 (u 1 (u 2)u u 1 (u 2)u 1=C 2 )2 Q.E.D. 14

15 5 Impact of Background Rsk We have shown that cautousness measures an nvestor s tendency to buy optons. In ths secton we nvestgate the mpact of background rsk on an nvestor s cautousness hence on hs tendency to buy optons. We consder both addtve and multplcatve background rsks. Gven a utlty functon, u(x), when there s an addtve (multplcatve) background rsk ɛ, as usual, we call û(x) =Eu(x + ɛ) (û(x) =Eu(xɛ)) the derved utlty functon. We have the followng result. Proposton 5 Assume u(x) has postve thrd dervatve and ts cautousness s hgher than a constant. Then gven a background rsk, ether addtve or multplcatve, the cautousness of the derved utlty functon wll also be hgher than the constant. Proof: We denote the background rsk as ɛ, a random varable. Assume the cautousness of u(x) s hgher than constant C. Let R(x) and P (x) be the rsk averson and prudence of the utlty functon. Then we have P (x)/r(x) C +1. We frst assume the background rsk s addtve. Note for postve a and b we have a + b 2 ab. Thus for any e 1 and e 2, P (x + e 1 ) R(x + e 2 ) + P (x + e 2) R(x + e 1 ) 2 Rearrangng the terms n (7), we have, for any e 1 and e 2, P (x + e 1 )P (x + e 2 ) 2(C + 1) (7) R(x + e 1 )R(x + e 2 ) u (x+e 1 )u (x+e 2 )+u (x+e 2 )u (x+e 1 ) 2(C +1)u (x+e 1 )u (x+e 2 ) (8) Assumng e 1 and e 2 are ndependent and have dentcal dstrbutons as ɛ and takng the expectaton of (8) wth respect to e 1 and e 2, we obtan 2E(u (x + ɛ))e(u (x + ɛ)) 2(C + 1)(E(u (x + ɛ))) 2 (9) Rearrangng the terms n (9), we have ˆP (x)/ ˆR(x) C +1, where ˆR(x) and ˆP(x) are the rsk averson and prudence of the derved utlty. Hence the cautousness of the derved utlty, Ĉ(x) = ˆP (x)/ ˆR(x) 1 C. The proof for the case where the background rsk s multplcatve s vrtually the same. Q.E.D. Corollary 2 Assume a utlty functon s HARA class wth postve cautousness. Gven a background rsk, ether addtve or multplcatve, the cautousness of the derved utlty functon wll be strctly hgher. Proof: Note that a HARA utlty functon has constant cautousness, say C. It follows from Proposton 5 that the cautousness of the derved utlty functon s hgher than C. Note n the proof of Proposton 5 the nequaltes are strct unless the utlty functon has constant R(x) and P (x), that s, t s power utlty whch has zero cautousness. Hence f a utlty functon s HARA class 15

16 wth postve cautousness then gven a background rsk, the cautousness of the derved utlty functon wll be strctly hgher. Q.E.D. The above result shows that f an nvestor has HARA class utlty wth postve cautousness then when he has a background rsk, the cautousness of hs derved utlty functon wll be strctly hgher, hence he wll have a stronger tendency to buy optons. FSS (1998) also studed the mpact of an addtve background rsk on an nvestor s demand for optons. They showed that n an economy n whch nvestors have dentcal constant postve cautousness the nvestors wthout background rsk wll have globally concave optmal payoff functons, whch they nterpreted that an addtve background rsk makes an nvestor more lkely to buy optons. The dfference between Corollary 2 and FSS s man result s worth notng although they gve the smlar concluson. FSS s model reles on the assumpton that there s a complete market of contngent clams on the stock and the assumpton that all nvestors have dentcal cautousness whle Corollary 2 does not need these two assumptons at all. Note Corollary 2 s even vald when many other nvestors are not ratonal utlty maxmzers. Corollary 2 can also be used to gve a smple proof of FSS (1998) man result that n an economy n whch nvestors have dentcal constant postve cautousness the nvestors wthout background rsk wll have globally concave optmal payoff functons. The proof s shown n the appendx. 6 Increasng Cautousness Huang (2000) showed that f the margnal utlty of zero wealth s nfnty then ncreasng (decreasng) cautousness mples decreasng (ncreasng) relatve rsk averson. Snce decreasng relatve rsk averson s slghtly more popular, t may be reasonable to favor ncreasng cautousness. In the followng we gve another argument for ncreasng cautousness. We show that when all nvestors have constant cautousness then the prcng representatve nvestor wll have ncreasng cautousness. The framework used s from Leland (1980) and FSS (1998). Assume n a one-perod economy there are N nvestors and every nvestor s wealth conssts of a portfolo of state-contngent clams on the market portfolo. Let X be the payoff of the market portfolo at the end of the perod. Assumpton 1 Assume that there s a complete market for state-contngent clams on X. Ths assumpton ensures that all nvestors can buy and sell state-contngent clams on X so that, as dscussed n Leland (1980), any nvestor can choose a payoff functon x (X). Assumpton 2 Assume all nvestors are ratonal expected-utlty-maxmzers. 16

17 Ths assumpton wll ensure that we can derve a prcng representatve nvestor whose preference determnes the prce of contngent clams. As s well known, the prcng representatve nvestor s preference wll be reflected n the unque prcng kernel, φ(x), whch s determned n the equlbrum of the economy. Note the dfference between ths model and the model used n Secton 2 to develop the measure of an nvestor s tendency to buy optons. There we do not assume the above two assumptons. Let u (x) denote nvestor s utlty functon. Let w 0 be nvestor s ntal endowment, expressed as the fracton of the spot value of the total wealth n the economy. Let x be hs optmal payoff functon respectvely. Then the nvestor has the followng utlty maxmzaton problem: max x Eu (x ). (10) Subject to E(φx )=w 0 E(φX). (11) where E(.) denotes the expectaton operator. In equlbrum, the market s cleared, thus we have x (X) =X. (12) We have the frst order condton u (x )=λ φ(x). (13) Dfferentatng both sdes of (13) wll lead to the followng result: x (X) =R e (X)/R (x ), (14) where R (x) = u /u (x) s nvestor s absolute rsk averson and R e(x) = φ (X)/φ(X) s the prcng representatve nvestor s absolute rsk averson. Dfferentatng both sdes of (14), we obtan x (X) =R 2 e(x)[c (x ) C e (X)]/R (x ), (15) where C (x) s nvestor s cautousness and C e (X) =(1/R e (X)) s the prcng representatve nvestor s cautousness. From (12) and (14) we obtan R e (X) =( R 1 (x )) 1 (16) From (12) and (15), we obtan C e (X) = s C (x ). (17) where s = R 1 (x )/ R 1 (x ). We have the followng result. 17

18 Proposton 6 Assume all nvestors have ncreasng cautousness. Then the prcng representatve nvestor also has ncreasng cautousness. Moreover, the cautousness of the prcng representatve nvestor s strctly ncreasng unless all nvestors have dentcal constant cautousness. Proof: Dfferentatng both sdes of (15), we have x x (X) (X) = e (X) 2R R e (X) R (x ) R (x ) x (X)+C (x )x (X) C e (X). C (x ) C e (X) It can be rewrtten as: x /x =2(P e R e )+(P R )x +(C x C e)/(c C e ), where we have omtted the arguments of the functons. Applyng (14) and (15) to the above equaton and rearrangng the terms, we obtan x (X) = R2 e (C C e )(2(P e R e )+(P R ) R e ) R2 e C e + C R R R x Snce x R e =0wehave R 1 (C C e )(2(P e R e )+(P R ) R e )+ R R 2 e R. C x 2 C e =0. Snce C 0, we have C e 2R e(p e R e ) (C C e )/R + R 2 e (C C e )C /R. From (17) we obtan (C C e )/R = 0. Thus we have C e R 2 e C (C C e )/R. Applyng (16) and (17) we can rewrte t as C e R 2 e( C 2 /R R e ( C /R ) 2 ). Rearrangng the terms, we obtan C e R 3 e( R 1 C 2 /R ( C /R ) 2 ). Applyng the Cauchy nequalty, we obtan C e 0. C e = 0 f and only f C (x) =C j (x) =C s a constant for any and j. Q.E.D. Ths shows f every nvestor has ncreasng cautousness then so does the prcng representatve nvestor. As a specal case, when every nvestor has constant cautousness we have the followng corollary. 18

19 Corollary 3 Assume all nvestors have constant cautousness. Then the prcng representatve nvestor has strctly ncreasng cautousness unless all nvestors have dentcal constant cautousness. Proof: It drectly follows from Proposton 6. The above result shows clearly that f nvestors have dfferent postve constant cautousness, then the prcng representatve nvestor wll not have constant cautousness. Instead he wll have strctly ncreasng cautousness. 7 Conclusons In ths paper we have shown that cautousness, whch s equvalent to the rato of prudence to rsk averson, s a measure of an nvestor s tendency to buy optons. We have shown that f nvestor has unformly hgher coeffcent of cautousness than j, then nvestor j buys an opton only f does so, and nvestor sells the opton only f j does so, regardless of ther ntal wealth, the underlyng stock prce, and the opton prce; and the reverse s also true. It s nterestng to see that the measure of an nvestor s tendency to buy optons s closely related to both measures of rsk averson and prudence whch explan nvestors actvtes n the bond market and stock market. Regardng the latter two actvtes, t s now wdely accepted that nvestors should have decreasng absolute rsk averson whle Kmball (1993) proposed decreasng absolute prudence. It s also sad that nvestors are more lkely to have decreasng relatve rsk averson. As Huang (2000) showed that ncreasng (decreasng) cautousness mples decreasng (ncreasng) relatve rsk averson gven that margnal utlty of zero wealth s nfnty, then ncreasng cautousness may be more lkely. Accordng to the model n ths paper, an nvestor s cautousness can tell f he has a stronger tendency to buy optons than others. However, f two nvestors both buy optons, ther coeffcents of cautousness cannot tell f one buys more optons than the other. Moreover, although the model n ths paper shows cautousness tells who buys optons but t does not tell what the optons are bought for: they may be used to hedge the rsk n the underlyng stock; they may be also used for ther leverage effect. Thus cautousness s not necessarly a measure of cautousness. We have also showed that background rsk wll ncrease the cautousness of HARA class utlty. Further research wll be nterestng to show the mpact of background rsk on the cautousness of a general utlty functon. 19

20 Appendx A Proof of Theorem 3 n FSS (1998) In the economy s descrbed n Secton 6, we have the followng result. [Theorem 3, FSS (1998)] Assume all nvestors have dentcal postve constant cautousness and some nvestors have unnsurable background rsk. Then the nvestors wthout background rsk have concave optmal payoff functons. Proof: Assume all nvestors have dentcal postve constant cautousness C. When nvestor s exposed to background rsk ɛ, the utlty functon u (x )n the utlty maxmzaton problem (10) s replaced by the ndrect utlty functon û (x )=E ɛ (u (x + ɛ )). Thus on the rght hand sde of (17) C (x ) s replaced by the cautousness of nvestor s derved utlty functon, Ĉ (x ), f nvestor has background rsk. For the nvestors wthout background rsk, C (x )=C s a postve constant. From Corollary 2, we know that for every nvestor who has background rsk, Ĉ (x ) > C. Thus from Equaton (17), we easly verfy that the prcng representatve nvestor s cautousness s strctly hgher than those of the nvestors wthout background rsk. From (15) the optmal payoff functons of those wthout background rsk are strctly concave. Q.E.D. 20

21 REFERENCES 1. Amersh, A. H. and J. H. W. Stoeckenus, The Theory of Syndcates and Lnear Sharng Rules, Econometrca 51 (1983), Arrow, Kenneth J. (1965), Aspects of a Theory of Rsk Bearng, Yrjo Jahnsson Lectures, Helsnk. Reprnted n Essays n the theory of Rsk Bearng (1971). Chcago: Markham Publshng Co. 3. Bennnga S. and M. Blume (1985), On the Optmalty of Portfolo Insurance, Journal of Fnance 40, No. 5, Bennnga, S. and J. Mayshar, (2000), Heterogenety and Opton Prcng, Revew of Dervatves Research 4, Borch, K. (1962), Equlbrum n a Rensurance Market, Econometrca 30, Brennan, M. J. and H. H. Cao (1996), Informaton, trade, and dervatve securtes, Revew of Fnancal Studes 9, No.1, Brennan, M. J. and R. Solank (1981), Optmal Portfolo Insurance, Journal of Fnancal and Quanttatve Analyss 16, Carr P. and D. Madan, 2001, Optmal Postonng n Dervatve Securtes, Quanttatve Fnance 1, Eeckhoudt L. and H. Schlesnger (1994), A Precautonary Tale of Rsk Averson and Prudence. B. Muner and M. J. Machna (eds.), Models and Experments n Rsk and Ratonalty, 75-90, Kluwer Academc Publshers. Prnted n the Netherlands. 10. Franke, G., R. C. Stapleton, and M. G. Subrahmanyam (1999), When are Optons Overprced: The Black-Scholes Model and Alternatve Characterzatons of the Prcng Kernel, European Fnance Revew 3, Franke, G., R. C. Stapleton and M. G. Subrahmanyam (1998), Who Buys and Who Sells Optons, Journal of Economc Theory 82, Goller, C. and C. J. W. Pratt (1996), Rsk Vulnerablty and the temperng Effect of Background Rsk, Econometrca 49, Huang C. and R. Ltzenberger (1988), Foundatons for Fnancal Economcs. Prentce-Hall Canada, Incorporated. 14. Huang, James (2000). Relatonshps between Rsk Averson, Prudence, and cautousness. Lancaster Unversty Management School workng paper. 15. Huang, James (2004). Rsk Neutral Probabltes and Opton Bounds: A Geometrc Approach. Lancaster Unversty Management School workng paper. 21

22 16. Huang, James (2004a). Opton Bounds and Second Order Arbtrage Opportuntes. Lancaster Unversty Management School workng paper. 17. Khlstrome, Rchard E., Davd Rome, and Steve Wllams (1981): Rsk Averson wth Random Intal Wealth, Econometrca 49, Kmball, Mles S. (1990), Precautonary Savng n the Small and n the Large, Econometrca 58, Kmball, Mles S. (1993), Standard Rsk Averson, Econometrca 61, Leland, H. E. (1980), Who Should Buy Portfolo Insurance? Journal of Fnance 35, Mas-Colell, A. (1985), The Theory of General Economc Equlbrum: A Dfferentable Approach. Econometrc Socety Monograph, Cambrdge, Cambrdge Unversty Press. 22. Pratt, J. W. (1964), Rsk Averson n the Small and n the Large, Econometrca 32, Pratt, J. W., and R. Zeckhauser (1987), Proper Rsk Averson, Econometrca 55, Rubnsten, M. E. (1976), An Aggregaton Theorem for Securtes Markets, Journal of Fnancal Economcs 1, Wlson, R. (1968), The Theory of Syndcates, Econometrca 36,

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ). REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

An Alternative Way to Measure Private Equity Performance

An Alternative Way to Measure Private Equity Performance An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

More information

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo

More information

OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL. Thomas S. Ferguson and C. Zachary Gilstein UCLA and Bell Communications May 1985, revised 2004

OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL. Thomas S. Ferguson and C. Zachary Gilstein UCLA and Bell Communications May 1985, revised 2004 OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL Thomas S. Ferguson and C. Zachary Glsten UCLA and Bell Communcatons May 985, revsed 2004 Abstract. Optmal nvestment polces for maxmzng the expected

More information

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 3: Force of Interest, Real Interest Rate, Annuity Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and

More information

The Cox-Ross-Rubinstein Option Pricing Model

The Cox-Ross-Rubinstein Option Pricing Model Fnance 400 A. Penat - G. Pennacc Te Cox-Ross-Rubnsten Opton Prcng Model Te prevous notes sowed tat te absence o arbtrage restrcts te prce o an opton n terms o ts underlyng asset. However, te no-arbtrage

More information

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy 4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Luby s Alg. for Maximal Independent Sets using Pairwise Independence Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

More information

Joe Pimbley, unpublished, 2005. Yield Curve Calculations

Joe Pimbley, unpublished, 2005. Yield Curve Calculations Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward

More information

BERNSTEIN POLYNOMIALS

BERNSTEIN POLYNOMIALS On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

More information

The Short-term and Long-term Market

The Short-term and Long-term Market A Presentaton on Market Effcences to Northfeld Informaton Servces Annual Conference he Short-term and Long-term Market Effcences en Post Offce Square Boston, MA 0209 www.acadan-asset.com Charles H. Wang,

More information

The Application of Fractional Brownian Motion in Option Pricing

The Application of Fractional Brownian Motion in Option Pricing Vol. 0, No. (05), pp. 73-8 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qng-xn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com

More information

Can Auto Liability Insurance Purchases Signal Risk Attitude?

Can Auto Liability Insurance Purchases Signal Risk Attitude? Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang

More information

Communication Networks II Contents

Communication Networks II Contents 8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

More information

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by 6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

More information

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt. Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces

More information

Support Vector Machines

Support Vector Machines Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

More information

FINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals

FINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals FINANCIAL MATHEMATICS A Practcal Gude for Actuares and other Busness Professonals Second Edton CHRIS RUCKMAN, FSA, MAAA JOE FRANCIS, FSA, MAAA, CFA Study Notes Prepared by Kevn Shand, FSA, FCIA Assstant

More information

Financial Mathemetics

Financial Mathemetics Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,

More information

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange

More information

Implied (risk neutral) probabilities, betting odds and prediction markets

Implied (risk neutral) probabilities, betting odds and prediction markets Impled (rsk neutral) probabltes, bettng odds and predcton markets Fabrzo Caccafesta (Unversty of Rome "Tor Vergata") ABSTRACT - We show that the well known euvalence between the "fundamental theorem of

More information

Kiel Institute for World Economics Duesternbrooker Weg 120 24105 Kiel (Germany) Kiel Working Paper No. 1120

Kiel Institute for World Economics Duesternbrooker Weg 120 24105 Kiel (Germany) Kiel Working Paper No. 1120 Kel Insttute for World Economcs Duesternbrooker Weg 45 Kel (Germany) Kel Workng Paper No. Path Dependences n enture Captal Markets by Andrea Schertler July The responsblty for the contents of the workng

More information

Buy-side Analysts, Sell-side Analysts and Private Information Production Activities

Buy-side Analysts, Sell-side Analysts and Private Information Production Activities Buy-sde Analysts, Sell-sde Analysts and Prvate Informaton Producton Actvtes Glad Lvne London Busness School Regent s Park London NW1 4SA Unted Kngdom Telephone: +44 (0)0 76 5050 Fax: +44 (0)0 774 7875

More information

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence 1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh

More information

7.5. Present Value of an Annuity. Investigate

7.5. Present Value of an Annuity. Investigate 7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on

More information

Pricing Overage and Underage Penalties for Inventory with Continuous Replenishment and Compound Renewal Demand via Martingale Methods

Pricing Overage and Underage Penalties for Inventory with Continuous Replenishment and Compound Renewal Demand via Martingale Methods Prcng Overage and Underage Penaltes for Inventory wth Contnuous Replenshment and Compound Renewal emand va Martngale Methods RAF -Jun-3 - comments welcome, do not cte or dstrbute wthout permsson Junmn

More information

An Overview of Financial Mathematics

An Overview of Financial Mathematics An Overvew of Fnancal Mathematcs Wllam Benedct McCartney July 2012 Abstract Ths document s meant to be a quck ntroducton to nterest theory. It s wrtten specfcally for actuaral students preparng to take

More information

Finite Math Chapter 10: Study Guide and Solution to Problems

Finite Math Chapter 10: Study Guide and Solution to Problems Fnte Math Chapter 10: Study Gude and Soluton to Problems Basc Formulas and Concepts 10.1 Interest Basc Concepts Interest A fee a bank pays you for money you depost nto a savngs account. Prncpal P The amount

More information

Differences of Opinion of Public Information and Speculative Trading in Stocks and Options

Differences of Opinion of Public Information and Speculative Trading in Stocks and Options Dfferences of Opnon of Publc Informaton and Speculatve Tradng n Stocks and Optons H. Henry Cao Cheung Kong Graduate School of Busness CKGSB Hu Ou-Yang Lehman Brothers and CKGSB We analyze the effects of

More information

A Probabilistic Theory of Coherence

A Probabilistic Theory of Coherence A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want

More information

Using Series to Analyze Financial Situations: Present Value

Using Series to Analyze Financial Situations: Present Value 2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

More information

8 Algorithm for Binary Searching in Trees

8 Algorithm for Binary Searching in Trees 8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the

More information

Hedging Interest-Rate Risk with Duration

Hedging Interest-Rate Risk with Duration FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton

More information

The Stock Market Game and the Kelly-Nash Equilibrium

The Stock Market Game and the Kelly-Nash Equilibrium The Stock Market Game and the Kelly-Nash Equlbrum Carlos Alós-Ferrer, Ana B. Ana Department of Economcs, Unversty of Venna. Hohenstaufengasse 9, A-1010 Venna, Austra. July 2003 Abstract We formulate the

More information

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression. Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook

More information

ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING

ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 610-519-4390,

More information

Adverse selection in the annuity market when payoffs vary over the time of retirement

Adverse selection in the annuity market when payoffs vary over the time of retirement Adverse selecton n the annuty market when payoffs vary over the tme of retrement by JOANN K. BRUNNER AND SUSANNE PEC * July 004 Revsed Verson of Workng Paper 0030, Department of Economcs, Unversty of nz.

More information

A Two Stage Stochastic Equilibrium Model for Electricity Markets with Two Way Contracts

A Two Stage Stochastic Equilibrium Model for Electricity Markets with Two Way Contracts A Two Stage Stochastc Equlbrum Model for Electrcty Markets wth Two Way Contracts Dal Zhang and Hufu Xu School of Mathematcs Unversty of Southampton Southampton SO17 1BJ, UK Yue Wu School of Management

More information

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

More information

Production. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem.

Production. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem. Producer Theory Producton ASSUMPTION 2.1 Propertes of the Producton Set The producton set Y satsfes the followng propertes 1. Y s non-empty If Y s empty, we have nothng to talk about 2. Y s closed A set

More information

Simon Acomb NAG Financial Mathematics Day

Simon Acomb NAG Financial Mathematics Day 1 Why People Who Prce Dervatves Are Interested In Correlaton mon Acomb NAG Fnancal Mathematcs Day Correlaton Rsk What Is Correlaton No lnear relatonshp between ponts Co-movement between the ponts Postve

More information

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001. Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.

More information

On Competitive Nonlinear Pricing

On Competitive Nonlinear Pricing On Compettve Nonlnear Prcng Andrea Attar Thomas Marott Franços Salané February 27, 2013 Abstract A buyer of a dvsble good faces several dentcal sellers. The buyer s preferences are her prvate nformaton,

More information

Stock Profit Patterns

Stock Profit Patterns Stock Proft Patterns Suppose a share of Farsta Shppng stock n January 004 s prce n the market to 56. Assume that a September call opton at exercse prce 50 costs 8. A September put opton at exercse prce

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 5.4 Annuities, Present Value, and Amortization Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

More information

What should (public) health insurance cover?

What should (public) health insurance cover? Journal of Health Economcs 26 (27) 251 262 What should (publc) health nsurance cover? Mchael Hoel Department of Economcs, Unversty of Oslo, P.O. Box 195 Blndern, N-317 Oslo, Norway Receved 29 Aprl 25;

More information

A Model of Private Equity Fund Compensation

A Model of Private Equity Fund Compensation A Model of Prvate Equty Fund Compensaton Wonho Wlson Cho Andrew Metrck Ayako Yasuda KAIST Yale School of Management Unversty of Calforna at Davs June 26, 2011 Abstract: Ths paper analyzes the economcs

More information

Chapter 15: Debt and Taxes

Chapter 15: Debt and Taxes Chapter 15: Debt and Taxes-1 Chapter 15: Debt and Taxes I. Basc Ideas 1. Corporate Taxes => nterest expense s tax deductble => as debt ncreases, corporate taxes fall => ncentve to fund the frm wth debt

More information

Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc.

Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc. Underwrtng Rsk By Glenn Meyers Insurance Servces Offce, Inc. Abstract In a compettve nsurance market, nsurers have lmted nfluence on the premum charged for an nsurance contract. hey must decde whether

More information

Interest Rate Fundamentals

Interest Rate Fundamentals Lecture Part II Interest Rate Fundamentals Topcs n Quanttatve Fnance: Inflaton Dervatves Instructor: Iraj Kan Fundamentals of Interest Rates In part II of ths lecture we wll consder fundamental concepts

More information

EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR

EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR EXAMPLE PROBLEMS SOLVED USING THE SHARP EL-733A CALCULATOR 8S CHAPTER 8 EXAMPLES EXAMPLE 8.4A THE INVESTMENT NEEDED TO REACH A PARTICULAR FUTURE VALUE What amount must you nvest now at 4% compoune monthly

More information

1 Example 1: Axis-aligned rectangles

1 Example 1: Axis-aligned rectangles COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

More information

When Network Effect Meets Congestion Effect: Leveraging Social Services for Wireless Services

When Network Effect Meets Congestion Effect: Leveraging Social Services for Wireless Services When Network Effect Meets Congeston Effect: Leveragng Socal Servces for Wreless Servces aowen Gong School of Electrcal, Computer and Energy Engeerng Arzona State Unversty Tempe, AZ 8587, USA xgong9@asuedu

More information

2.4 Bivariate distributions

2.4 Bivariate distributions page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

More information

Addendum to: Importing Skill-Biased Technology

Addendum to: Importing Skill-Biased Technology Addendum to: Importng Skll-Based Technology Arel Bursten UCLA and NBER Javer Cravno UCLA August 202 Jonathan Vogel Columba and NBER Abstract Ths Addendum derves the results dscussed n secton 3.3 of our

More information

Extending Probabilistic Dynamic Epistemic Logic

Extending Probabilistic Dynamic Epistemic Logic Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set

More information

Chapter 11 Practice Problems Answers

Chapter 11 Practice Problems Answers Chapter 11 Practce Problems Answers 1. Would you be more wllng to lend to a frend f she put all of her lfe savngs nto her busness than you would f she had not done so? Why? Ths problem s ntended to make

More information

BANCO DE PORTUGAL Economics Research Department

BANCO DE PORTUGAL Economics Research Department BANCO DE PORUGAL Economcs Research Department he Estmaton of Rsk Premum Implct n Ol Prces Jorge Barros Luís WP -00 February 000 he analyses, opnons and fndngs of ths paper represent the vews of the author,

More information

Kiel Institute for World Economics Duesternbrooker Weg 120 24105 Kiel (Germany) Kiel Working Paper No. 1119

Kiel Institute for World Economics Duesternbrooker Weg 120 24105 Kiel (Germany) Kiel Working Paper No. 1119 Kel Insttute for World Economcs Duesternbrooker Weg 120 24105 Kel (Germany) Kel Workng Paper No. 1119 Under What Condtons Do Venture Captal Markets Emerge? by Andrea Schertler July 2002 The responsblty

More information

Captive insurance companies and the management of non-conventional corporate risks

Captive insurance companies and the management of non-conventional corporate risks Captve nsurance companes and the management of non-conventonal corporate rsks Jean-Baptste Lesourd a and Steven Schlzz b* a EJCM, Unversté de la Médterranée, Marselle, France b School of Agrcultural and

More information

ErrorPropagation.nb 1. Error Propagation

ErrorPropagation.nb 1. Error Propagation ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then

More information

Pricing index options in a multivariate Black & Scholes model

Pricing index options in a multivariate Black & Scholes model Prcng ndex optons n a multvarate Black & Scholes model Danël Lnders AFI_1383 Prcng ndex optons n a multvarate Black & Scholes model Danël Lnders Verson: October 2, 2013 1 Introducton In ths paper, we consder

More information

greatest common divisor

greatest common divisor 4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no

More information

1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)

1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP) 6.3 / -- Communcaton Networks II (Görg) SS20 -- www.comnets.un-bremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes

More information

Thursday, December 10, 2009 Noon - 1:50 pm Faraday 143

Thursday, December 10, 2009 Noon - 1:50 pm Faraday 143 1. ath 210 Fnte athematcs Chapter 5.2 and 4.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

More information

Adverse selection in the annuity market with sequential and simultaneous insurance demand. Johann K. Brunner and Susanne Pech *) January 2005

Adverse selection in the annuity market with sequential and simultaneous insurance demand. Johann K. Brunner and Susanne Pech *) January 2005 Adverse selecton n the annuty market wth sequental and smultaneous nsurance demand Johann K. Brunner and Susanne Pech *) January 005 Revsed Verson of Workng Paper 004, Department of Economcs, Unversty

More information

Quasi-Hyperbolic Discounting and Social Security Systems

Quasi-Hyperbolic Discounting and Social Security Systems Quas-Hyperbolc Dscountng and Socal Securty Systems Mordecha E. Schwarz a and Eytan Sheshnsk b May 22, 26 Abstract Hyperbolc countng has become a common assumpton for modelng bounded ratonalty wth respect

More information

1. Math 210 Finite Mathematics

1. Math 210 Finite Mathematics 1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

Optimal Consumption and Investment with Transaction Costs and Multiple Risky Assets

Optimal Consumption and Investment with Transaction Costs and Multiple Risky Assets THE JOURNAL OF FINANCE VOL. LIX, NO. 1 FEBRUARY 2004 Optmal Consumpton and Investment wth Transacton Costs and Multple Rsky Assets HONG LIU ABSTRACT We consder the optmal ntertemporal consumpton and nvestment

More information

Multi-Product Price Optimization and Competition under the Nested Logit Model with Product-Differentiated Price Sensitivities

Multi-Product Price Optimization and Competition under the Nested Logit Model with Product-Differentiated Price Sensitivities Mult-Product Prce Optmzaton and Competton under the Nested Logt Model wth Product-Dfferentated Prce Senstvtes Gullermo Gallego Department of Industral Engneerng and Operatons Research, Columba Unversty,

More information

Loss analysis of a life insurance company applying discrete-time risk-minimizing hedging strategies

Loss analysis of a life insurance company applying discrete-time risk-minimizing hedging strategies Insurance: Mathematcs and Economcs 42 2008 1035 1049 www.elsever.com/locate/me Loss analyss of a lfe nsurance company applyng dscrete-tme rsk-mnmzng hedgng strateges An Chen Netspar, he Netherlands Department

More information

The Proper Use of Risk Measures in Portfolio Theory

The Proper Use of Risk Measures in Portfolio Theory The Proper Use of Rsk Measures n Portfolo Theory Sergo Ortobell a, Svetlozar T. Rachev b, Stoyan Stoyanov c, Frank J. Fabozz d,* and Almra Bglova e a Unversty of Bergamo, Italy b Unversty of Calforna,

More information

IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS

IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,

More information

1 Approximation Algorithms

1 Approximation Algorithms CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons

More information

The Market Organism: Long Run Survival in Markets with Heterogeneous Traders

The Market Organism: Long Run Survival in Markets with Heterogeneous Traders The Market Organsm: Long Run Survval n Markets wth Heterogeneous Traders Lawrence E. Blume Davd Easley SFI WORKING PAPER: 2008-04-018 SFI Workng Papers contan accounts of scentfc work of the authors) and

More information

Formula of Total Probability, Bayes Rule, and Applications

Formula of Total Probability, Bayes Rule, and Applications 1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.

More information

Copulas. Modeling dependencies in Financial Risk Management. BMI Master Thesis

Copulas. Modeling dependencies in Financial Risk Management. BMI Master Thesis Copulas Modelng dependences n Fnancal Rsk Management BMI Master Thess Modelng dependences n fnancal rsk management Modelng dependences n fnancal rsk management 3 Preface Ths paper has been wrtten as part

More information

We are now ready to answer the question: What are the possible cardinalities for finite fields?

We are now ready to answer the question: What are the possible cardinalities for finite fields? Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the

More information

New bounds in Balog-Szemerédi-Gowers theorem

New bounds in Balog-Szemerédi-Gowers theorem New bounds n Balog-Szemeréd-Gowers theorem By Tomasz Schoen Abstract We prove, n partcular, that every fnte subset A of an abelan group wth the addtve energy κ A 3 contans a set A such that A κ A and A

More information

Joint Resource Allocation and Base-Station. Assignment for the Downlink in CDMA Networks

Joint Resource Allocation and Base-Station. Assignment for the Downlink in CDMA Networks Jont Resource Allocaton and Base-Staton 1 Assgnment for the Downlnk n CDMA Networks Jang Won Lee, Rav R. Mazumdar, and Ness B. Shroff School of Electrcal and Computer Engneerng Purdue Unversty West Lafayette,

More information

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60 BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true

More information

Calculation of Sampling Weights

Calculation of Sampling Weights Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

More information

Logistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification

Logistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson

More information

Power-of-Two Policies for Single- Warehouse Multi-Retailer Inventory Systems with Order Frequency Discounts

Power-of-Two Policies for Single- Warehouse Multi-Retailer Inventory Systems with Order Frequency Discounts Power-of-wo Polces for Sngle- Warehouse Mult-Retaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)

More information

Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio

Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio Vascek s Model of Dstrbuton of Losses n a Large, Homogeneous Portfolo Stephen M Schaefer London Busness School Credt Rsk Electve Summer 2012 Vascek s Model Important method for calculatng dstrbuton of

More information

Efficient Project Portfolio as a tool for Enterprise Risk Management

Efficient Project Portfolio as a tool for Enterprise Risk Management Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse

More information

Interest Rate Futures

Interest Rate Futures Interest Rate Futures Chapter 6 6.1 Day Count Conventons n the U.S. (Page 129) Treasury Bonds: Corporate Bonds: Money Market Instruments: Actual/Actual (n perod) 30/360 Actual/360 The day count conventon

More information

FINANCIAL MATHEMATICS

FINANCIAL MATHEMATICS 3 LESSON FINANCIAL MATHEMATICS Annutes What s an annuty? The term annuty s used n fnancal mathematcs to refer to any termnatng sequence of regular fxed payments over a specfed perod of tme. Loans are usually

More information

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

More information

Multiple-Period Attribution: Residuals and Compounding

Multiple-Period Attribution: Residuals and Compounding Multple-Perod Attrbuton: Resduals and Compoundng Our revewer gave these authors full marks for dealng wth an ssue that performance measurers and vendors often regard as propretary nformaton. In 1994, Dens

More information

Probability and Optimization Models for Racing

Probability and Optimization Models for Racing 1 Probablty and Optmzaton Models for Racng Vctor S. Y. Lo Unversty of Brtsh Columba Fdelty Investments Dsclamer: Ths presentaton does not reflect the opnons of Fdelty Investments. The work here was completed

More information

Traffic-light a stress test for life insurance provisions

Traffic-light a stress test for life insurance provisions MEMORANDUM Date 006-09-7 Authors Bengt von Bahr, Göran Ronge Traffc-lght a stress test for lfe nsurance provsons Fnansnspetonen P.O. Box 6750 SE-113 85 Stocholm [Sveavägen 167] Tel +46 8 787 80 00 Fax

More information

Problem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative.

Problem Set 3. a) We are asked how people will react, if the interest rate i on bonds is negative. Queston roblem Set 3 a) We are asked how people wll react, f the nterest rate on bonds s negatve. When

More information

NON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia

NON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia To appear n Journal o Appled Probablty June 2007 O-COSTAT SUM RED-AD-BLACK GAMES WITH BET-DEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate

More information

Number of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000

Number of Levels Cumulative Annual operating Income per year construction costs costs ($) ($) ($) 1 600,000 35,000 100,000 2 2,200,000 60,000 350,000 Problem Set 5 Solutons 1 MIT s consderng buldng a new car park near Kendall Square. o unversty funds are avalable (overhead rates are under pressure and the new faclty would have to pay for tself from

More information

Most investors focus on the management

Most investors focus on the management Long-Short Portfolo Management: An Integrated Approach The real benefts of long-short are released only by an ntegrated portfolo optmzaton. Bruce I. Jacobs, Kenneth. Levy, and Davd Starer BRUCE I. JACOBS

More information

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6)

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6) Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called

More information