T d e an I nves men St St t ra egy Professor R obert Robert A. A. M iller Miller

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1 Trade ad Iestmet t Strategyt Professor Robert A. Miller

2 Prelimiaries Before the lecture starts, if you hae t already doe so, please ope your computer ad go to: The click o the comlabgames ico to reach: Dowload the stad aloe ersio of comlabgames, which we will use throughout the course. Please brig your laptop to each class.

3 Dyamic Ret Seekig The first part of Trade ad Iestmet Strategy explores the market borders of your compay through the greater use of iteral markets. First we show how auctios work, ad how they might be used i acquisitio. The we broade the aalysis to limit order markets, possibly iteral, with multiple buyers ad sellers. The secod part of Trade ad Iestmet Strategy ealuates the dyamics of ret accrual, by likig firm et reeue flow to the pricig of fiacial securities ad the cosumptio of goods ad serices. We diide the discussio ito two parts, how cosumers makes choices oer time ad whe there is ucertaity, ad the implicatios for portfolio strategies for lifecycle saig ad withdrawals.

4 Course objecties I this course I hope you: 1. Become familiar with auctios ad limit it order markets as a bidder, trader, iestor ad aalyst. 2. Acquire a oerarchig iew for predictig ad ealuatig how trade ad iestmet strategies are (or should be) liked to persoal wealth maagemet. 3. Deelop a ituitio for thikig about tradig mechaisms ad market based platforms as orgaizatioal tools.

5 Course materials The course website is: At the website you ca fid: the course syllabus lecture otes games you ca dowload the o lie (draft) textbook other referece materials

6 Lecture 1 Auctios Auctios sere a ital role i busiess ad they are relatiely simple to aalyze. That is why we begi the course with a study of auctios, ad get some hads-o biddig experiece. There are differet types of auctios. We iestigate how to bid, ad the takig the auctioeers perspectie, we ask what kid of auctio produces the most reeue.

7 Auctios are a importat tradig mechaism Auctios are widely used by compaies, priate idiiduals ad goermet agecies to buy ad sell goods ad serices. They are also used i competitie cotractig betwee a (auctioeer) firm ad other (bidder) firms up or dow the supply chai to reach tradig agreemets. Merger ad acquisitios ofte hae a auctio flaor about them.

8 There are differet types of auctios I a first price sealed bid auctio, each bidder submits his/her bid without kowig what the others are biddig, ad dthe auctioeer sells the good dto the highest h tbidder at the price he submitted. I a Eglish auctio bidders compete agaist each other by raisig the price util eeryoe but oe bidder drops out of the biddig. I a Dutch auctio, the auctioeer reduces the price util a bidder idicates he/she h is willig to take the object. I a secod priced sealed bid auctio, players simultaeously submit their bids, the highest bidder wis the auctio, ad pays the secod highest bid.

9 Why aalyze auctios? Like all tradig mechaisms, auctios sere the dual purpose of elicitig prefereces ad allocatig resources betwee competig uses. Sice a auctio is the simplest form of a limit order market, startig this course by studyig behaior i auctios is a useful way to begi learig how iestors trade i ay market. From a strategic perspectie we seek aswers to two basic questios: 1. How should you bid i a auctio? 2. What kid of auctio rules should you set?

10 Biddig strategies Does it matter what form the auctio takes? From SCM (45-870) a strategy is a complete descriptio of istructios to be played throughout the game The strategic form of a game is the set of alteratie strategies to each player ad their correspodig expected payoffs from followig them. Two games are strategically equialet if they share the same strategic form. I strategically equialet auctios the set of biddig I strategically equialet auctios, the set of biddig strategies that each potetial bidders receie, ad the mappig to the bidder s payoffs, are the same.

11 Commo alue auctio: Oil field tract Cosider a ew oil field tract that drillers bid for after coductig seismic their idiidual exploratios. The alue of the oil field is the same to each bidder, but ukow. The th bidder receies a sigal s which is distributed about the commo alue, where s = + ad E[ s ] is idepedetly distributed across bidders. Notice that each drillig compay would hae more precise estimates of the commo aluatio from reiewig the geological surey results of their rials.

12 The expected alue of the item upo wiig the auctio If the th bidder wis the auctio, he realizes his sigal exceeded d the sigals of eerybody else, that t is s max{s₁,,s N } so he should coditio the expected alue of the item o this ew iformatio. His expected alue is ow the expected alue of coditioal upo obserig the maximum sigal: E[ s max{s₁,,s N }] This is the alue that the bidder should use i the auctio, because he should recogize that uless his sigal is the maximum he will receie a payoff of zero.

13 The Wier s Curse Coditioal o the sigal, but before the biddig starts, the expectatio of the commo alue is: E s Es s s maxs 1,,s N We defie the wier s curse as: EE s E E s max s1,..., sn s E s max s,..., s 0 1 N Although bidders should make due allowace for the fact that their aluatio will typically oerstate the true alue of the object if they wi the auctio, oice bidders typically do ot take it ito accout whe placig a bid.

14 Descedig auctios are strategically equialet to first-price i auctios Durig the course of a descedig auctio o iformatio is receied by bidders. Each bidder sets his reseratio price before the auctio, ad submits a market order to buy if ad whe the limit auctioeer's limit order to sell falls to that poit. Dutch auctios ad first price sealed bid auctios share strategic form, ad hece yield the same realized payoffs if the iitial aluatio draws are the same. Rule 1: Pick the same reseratio price i Dutch auctio p that you would submit i a first price auctio

15 Secod-price ersus ascedig auctios Whe there are oly 2 bidders, a ascedig auctio mechaism is strategically equialet to the secod price sealed bid auctio (because o iformatio is receied durig the auctio). More geerally, both auctios are (almost) strategically equialet if all bidders hae idepedetly distributed aluatios (because the iformatio coeyed by the other bidders has o effect o a bidder s aluatio). I commo alue auctios the two mechaisms are ot strategically t equialet if there are more tha 2 players. Rule 2: If there are oly two bidders, or if aluatios are idepedetly distributed, choose the same reseratio price i Eglish ad secod price auctios.

16 Biddig i a secod-price auctio If you kow your ow aluatio, there is a geeral result about how to bid i a secod price sealed bid auctio, or where to stop biddig i a ascedig auctio. Biddig should ot deped o what you kow about the aluatios of the other players, or o what they kow about their ow aluatios. It is a domiat strategy to bid your ow aluatio. A corollary of this result is that if eery bidder kows his ow aluatio, the the object will be sold for the secod highest aluatio. Rule 3 : I a secod price sealed bid auctio, bid your Rule 3 : I a secod price sealed bid auctio, bid your aluatio if you kow it.

17 Proig the third rule Suppose you bid aboe your aluatio, wi the auctio, ad the secod highest bid also exceeds your aluatio. I this case you make a loss. If you had bid your aluatio the you would ot hae wo the auctio i this case. I eery other case your wiigs would hae bee idetical. Therefore biddig your aluatio domiates biddig aboe it. Suppose you bid below your aluatio, ad the wiig bidder places a bid betwee your bid ad your aluatio. If you had bid your aluatio, you would hae wo the auctio ad profited. I eery other case your wiigs i would hae bee idetical. Therefore biddig your aluatio domiates biddig below it. The proof is completed by combiig the two parts.

19 Reeue equialece defied I strategically equialet auctios, the strategic form solutio strategies of the bidders, ad the payoffs to all them, are idetical. Are bidders eer idifferet to auctios that lack strategic equialece? Two auctio mechaisms are reeue equialet if, gie a set of players their aluatios, ad their iformatio sets, the expected surplus to each bidder ad the expected reeue to the auctioeer is the same. Reeue equialece is a less striget coditio tha strategic equialece. Thus two strategic equialet auctios are iariably reeue equialet, but ot all reeue equialet auctios are strategic equialet.

20 Prefereces ad Expected Payoffs Let P( ) deote the probability the th bidder with aluatio will wi the auctio whe all players bid accordig to their equilibrium strategy. Let C( ) deote the expected costs (icludig ay fees to eter the auctio, ad paymets i the case of submittig a wiig bid). Let: U( ) = P( ) - C( ) deote the expected alue of the th bidder from followig his equilibrium i strategy whe eeryoe else does too.

21 A reealed preferece argumet Suppose the aluatio of is ad the aluatio of j is j. The surplus from biddig as if his aluatio is j is U( j ), the alue from participatig if his aluatio is j, plus the differece e e i how he alues the expected wiigs compared to a bidder with aluatio j, or ( j )P( j ). I equilibrium the alue of followig his solutio strategy is at least as profitable as deiatig from it by pretedig his aluatio is j. Therefore: U( ) > U( j ) + ( j )P( j )

22 Reealed preferece cotiued For coeiece, we rewrite the last slide o the preious page as: U( ) - U( j ) > ( j )P( j ) j j j Now iewig the problem from the j th bidder s perspectie we see that by symmetry: U( j ) > U( ) + ( j )P( ) which ca be expressed as: ( j )P( ) > U( ) - U( j )

23 A fudametal equality Puttig the two iequalities together, we obtai: ( j ) P( )> U( ) - U( j ) > ( j ) P( j ) Writig: yields: = j + d U P du d which, upo itegratio, yields: U U P d

24 Reeue equialece theorem This equality shows that i priate alue auctios, the expected surplus to each bidder does ot deped o the auctio mechaism itself proidig the followig coditios are satisfied: 1. Eery bidder is risk-eutral. 2. Valuatios are idepedet d ad idetically distributed. d 3. I equilibrium the bidder with highest aluatio wis. 4. The lowest possible aluatio has zero expected alue. Note that if all bidders obtai the same expected surplus, the auctioeer obtais the same expected reeue too.

25 Ituitio from reeue equialece Calibrate your bid to your aluatio oly to the extet that it affects your beliefs about the highest aluatio of the all the other bids. Workig from the assumptio that yours is the highest aluatio, bid high eough to iduce the ext highest bidder to make a small expected loss i order to beat your bid. To use a athletic aalogy, thik heats, ot fials! Swim fast eough to make the fials, but sae yourself for the fial.

26 Steps for deriig i expected reeue The expected reeue from ay auctio satisfyig the coditios of the theorem, is the expected alue of the secod highest bidder. To obtai this quatity, we proceed i two steps: 1. derie the probability distributio of the secod highest aluatio aua 2. obtai its desity ad itegrate to fid the mea.

27 Probability distributio of the secod highest aluatio Sice ay auctio satisfyig the coditios for the theorem ca be used to calculate l the expected reeue, we select the secod price auctio. The probability that the secod highest aluatio is less tha is the sum of the the probabilities that: 1. all the aluatios are less tha, or P() N 2. N-1 aluatios are less tha ad the other oe is greater tha. There are N ways of doig this so the probability is: NP() N-1 [1 - P()] The probability distributio for the secod highest aluatio is therefore: NP() N-1 - (N - 1) P() N

28 Expected reeue from Priate Value Auctios The probability desity fuctio for the secod highest aluatio is therefore: N(N 1)P() N-2 [1 - P()]P () Therefore the expected reeue to the auctioeer, or the expected alue of the secod highest aluatio, deoted by 2, is: 2 N 2 N N 1 P 1 P E 0 P' d

29 Usig the reeue equialece theorem to derie optimal biddig fuctios We ca also derie the solutio biddig strategies for auctios that are reeue equialet to the secod price sealed bid auctio. Cosider, for example a first price sealed bid auctios with idepedet ad idetically distributed aluatios. The reeue equialece theorem implies that each bidder will bid the expected alue of the ext highest bidder coditioal upo his aluatio beig the highest.

30 Biddig i a first price sealed bid auctio I a symmetric equilibrium to first price sealed bid auctio, we ca show that a bidder with aluatio bids: b 0 P P N 1 N 1 d

31 Compariso of biddig strategies The biddig strategies i the first ad secod price auctios markedly differ. I a secod price auctio bidders should submit their aluatio regardless of the umber of players biddig o the object. I the first price auctio bidders should shae their aluatios, by a amout depedig o the umber of bidders.

32 The deriatio The probability the remaiig N - 1aluatiosare are less tha gie the highest aluatio is is: 2 P / P N 1 Pr Differetiatig, the coditioal desity for the secod highest aluatio is the: 1 N N 2 N 1P P P ' If he wis, the biddig fuctio for is the expected alue of the secod highest aluatio: b b 1 N N 2 N 1P P P' Itegratig by parts, we simplify this formula to: 1 N P P P 0 d N 1 1 N P P 1 N P 0 N 1 0 N 1 d 0 d

33 A example: the uiform distributio Suppose aluatios are uiformly distributed withi a closed iteral with probability distributio: 0 / 0 P closed iteral, with probability distributio: The i equilibrium, a player with aluatio bids a weighted aerage of the lowest possible aluatio ad eg teda eageo t e o estpossbe auato a d his ow, where the weights are 1/N ad (N-1)/N: d P P b N N 1 1 d d P P b N N N N N / 1 /

34 Choosig betwee two auctios with reeue equialece Sice first ad secod price auctios are reeue equialet, why would oe choose oe oer the other? The relatioship wiig bid ad the price paid is more trasparet i a first price sealed bid tha i the secod price sealed bid. I the first price sealed bid you pay what you bid, rather tha what the auctioeer claims is the secod highest bid. O the other had the Eglish auctio (strategically equialet i priate alue auctios) also is trasparet because bidders see their rials.

35 Preetig collusio Oe reaso for the auctioeer to prefer a first price auctio is that that is easier to preet collusio uder a first price auctio tha uder a secod. I a secod price auctio a firm deiatig from the cartel agreemet ca be easily pealized by the cartel members by the desigated wier offerig a bid that would esure a loss. I a first price auctio the desigated wier makes a bid that has attractie terms for itself, so the deiatig firm could still receie a profitable cotract by makig a slightly better cotract.

36 What happes whe more tha oe uit is sold? Suppose there are exactly Q idetical uits of a good up for auctio, all of which must be sold. As before we shall suppose there are N bidders or potetial demaders of the product ad that N > Q. Also followig preious otatio, deote their aluatios by 1 through N. We begi by cosiderig situatios where each buyer g y g y wishes to purchase at most oe uit of the good.

37 Ope auctios for sellig idetical uits Descedig Dutch auctio: As the price falls, the first Q bidders to submit market orders purchase a uit of the good at the price the auctioeer offered to them. Ascedig Japaese auctio: The auctioeer holds a ascedig auctio ad awards the objects to the Q highest bidders at the price the N - Q highest bidder drops out.

38 Multiuit i sealed bid auctios Sealed bid auctios for multiple uits ca be coducted by iitig bidders to submit limit order offers, ad allocatig the aailable uits to the highest h bidders. I discrimiatory auctios the wiig bidders pay differet prices. For example they might pay at the respectie prices they posted. I a uiform price auctio the wiers pay the same price, such as a k th price auctio (where k could rage from 1 to N.)

39 Reeue equialece reisited Suppose each bidder: - kows her ow aluatio - oly wat oe of the idetical items up for auctio - is risk eutral Cosider two auctios which both award the auctioed items to the highest aluatio bidders i equilibrium. The the reeue equialece theorem applies, implyig that the mechaism chose for tradig is immaterial. Ituitiely each bidder tries to beat the highest losig bid.

40 Successful bids follow a radom walk I repeated auctios that satisfy the reeue equialece theorem, EMH implies that the price of successie uits follows a radom walk. Ituitiely, each bidder is estimatig the bid he must make to beat the demader with (Q+1) st highest aluatio, auato that is coditioal dto o his ow aluatio beig oe of the Q highest. If the expected price from the q s+1 item exceeds that of the q s item before either is auctioed, the we would expect this to cause more (less) aggressie biddig for q s item (q s+1 item) to get a better deal, thus driig up (dow) its price.

41 Asymmetric aluatios I a priate aluatio auctios the bidders hae differet uses for the auctioed object, ad this fact is commo kowledge to eery bidder. Each bidder kows the that eeryoe else is drawig their aluatios from the same probability distributio, ad uses that iformatio whe makig her bid. What happes if the priate aluatios of bidders are ot draw from the same probability distributio fuctio? I that case the reeue equialece theorem is ot alid, ad the auctioeer's prefers some types of auctios oer others.

42 Biddig with differetial iformatio For example oe bidder might kow more about the alue of the object beig auctioed tha the others. What happes if they are asymmetrically iformed about a commo alue? A extreme form of depedet sigals occurs whe oe bidder kow the sigal ad the others do ot. How should a iformed player bid? What about a uiformed player?

43 Secod price sealed bid auctios I a secod price sealed bid auctio, Rule 3 implies the iformed player optimally bids his true alue. The uiformed player bids ay pure or mixed distributio. If he wis the auctio he pays the commo alue, if he loses he pays othig, ad therefore makes either gais or losses o ay bid. This implies the reeue from the auctio is This implies the reeue from the auctio is idetermiate.

44 Perspectie of the less iformed bidder i a first price auctio Suppose the uiformed bidder always makes the same positie bid, deoted b fixed. This is a example of a pure strategy. Is this pure strategy part of a Nash equilibrium? The best respose of the iformed bidder is to bid a little more tha b fixed whe the alue of the object is worth more tha b fixed, ad less tha b fixed otherwise. Therefore the uiformed bidder makes a expected loss by playig a pure strategy i this auctio. A better strategy would be to bid othig.

45 Equilibrium biddig The argumet i the preious slide shows that the uiformed bidder plays a mixed strategy i this game. Oe ca show that i equilibrium whe the auctioed item is worth the iformed bidder bids: () = E[V V ] Furthermore the uiformed bidder chooses a bid at radom from the iteral [0, E[V]] accordig to the probability distributio H defied by: H(b) = Prob[() b]

46 Retur to the uiformed bidder If the uiformed player bids more tha E[V], the his expected retur is egatie, sice he would wi the auctio eery time < E[V] but less frequetly whe > E[V]. We ow show that if his bid b < E[V], his expected retur is zero, ad therefore ay bid b < E[V] is a best respose to the iformed player s bid. If the uiformed bids less tha E[V] ad loses the auctio, his retur is zero. If he bids less tha E[V], ad wis the auctio, his retur is: E[V (V) < b] b = E[V V < -1 (b) ] b = ( -1 (b) ) b = 0

47 Retur to the iformed bidder Sice the uiformed player bids less tha E[] with uit probability, so does the iformed player. Notig that (w) aries from to E[], we proe it is better to bid () ) rather tha (w). ) Gie a aluatio of, the expected et beefit from biddig (w) is: H((w))[ - (w)] = Pr{V w}[ - (w)] = P(w)[ - (w)] Differetiatig with respect to w, usig deriatios foud o the ext slide, yields P (w)[ - w] which is positie for all > w ad egatie for all < w, ad zero at = w. Therefore biddig () ) is optimal for the iformed bidder with aluatio.

48 The deriatie Notig: P w w w P w E V V w tp' t it follows from the fudametal theorem of calculus that: d P w w P ' w w dw ad so the deriatie of P(w)[ - (w)] with respect to w is: 0 dt P' w d dw Pw w P' w P' ww

49 Whe does reeue equialece fail? The aluatios of bidders might be draw from differet probability bilit distributios, ib ti as i the preious example. The theorem does ot apply whe alues are ot priate, as i the preious example. Nor does it apply whe bidders are iterested i buyig more tha oe uit each. Moreoer block traders attempt to hide their large demads hopig the market maker will ot adjust the spread ufaorably. The greater the umber of uits demaded, the higher the competitie bids, ad hece the reaso for bidders hidig their demads.

50 Lecture Summary Auctios are amogst the simplest of tradig mechaisms. We aalyzed strategic ad reeue equialece. We proed what to bid whe reeue equialece holds. Our aalysis suggests two guidig g priciples p for biddig: 1. Assume your ow aluatio is high eough to wi, ad bid high eough to pay the break-ee alue of the highest losig bid. 2. Shae your bid to accout for the wier s curse if you do t kow your ow alue. Our proof of the optimal biddig rule for auctios with differetial iformatio illustrates how to proceed o a case by case basis whe reeue equialece fails.

The material in this chapter is motivated by Experiment 9.

The material in this chapter is motivated by Experiment 9. Chapter 5 Optimal Auctios The material i this chapter is motivated by Experimet 9. We wish to aalyze the decisio of a seller who sets a reserve price whe auctioig off a item to a group of bidders. We begi