Section 6 Short Sales, Yield Curves, Duration, Immunization, Etc.

Transcription

1 More Tuoral a age 1 of 9 Secon 6 Shor Sales, Yeld Curves, Duraon, Immunzaon, Ec. Shor Sales: Suppose you beleve ha Company X s sock s overprced. You would ceranly no buy any of Company X s sock, and n fac, you would probably sell any sock ha you dd own n Company X. However, wha f you don currenly own any sock n Company X? Is here a way for you o ake advanage of Company X s sock beng overprced n your opnon? The answer s yes. The erm sellng shor refers o an nvesor ha sells a sock a he curren prce of he sock and buys he sock back a a laer dae o cover he shor. An nvesor would shor sale a sock f he nvesor beleves he sock s overprced and hus he prce wll declne n he fuure. The followng s an example ha llusraes how he ransacon of sellng shor works. Invesor A beleves Company X s sock, currenly prced a $10 per share, s overprced. So A wans o go shor 100 shares of Company X s sock. Then 1. A borrows 100 shares of Company X s sock from a second pary and mmedaely sells he 100 shares n he marke for $ A wll have o make a depos n order o borrow he 100 shares above. Ths depos s called a margn and s usually a percenage of he amoun beng borrowed. For example, le s assume he margn s 50% of he amoun borrowed. Then he margn equals $500. Ths s he amoun ha A has o pu up fron. 3. A wll earn neres on he margn. However, A wll no have access o he $1000 proceeds from he shor sale unl A covers he shor sale by buyng 100 shares a a laer dae, hus compleng he ransacon. 4. If Company X s sock pays dvdends, hen A wll have o pay he dvdend amoun o he second pary above, snce he second pary len he sock o A and hus s no recevng he dvdend paymens drecly. 5. A he me he shor sale s covered, he ne prof earned by A wll be he sum of he gan on he shor sale and neres on he margn, offse by he amoun of dvdends pad by A o he second pary. The yeld rae on he ransacon wll equal he rao of he ne prof o he margn.

2 More Tuoral a age of 9 Yeld aes on Shor Sales: We wll be neresed n calculang he yeld rae on a shor sale. The formula for calculang he yeld rae on a shor sale s G I M D +, where M M margn G gan on he shor sale I M neres on he margn, and D dvdends pad The followng example, whch refers o he above scenaro, llusraes hs calculaon. Suppose afer 1 year Company X s sock has declned o $8 per share. Also suppose A has earned $40 n neres on he margn and pad $60 n dvdends durng he year. If A decdes o cover he shor sale a hs me, hen A wll buy 100 shares of Company X s sock a $8 per share. So A pays $800 bu hen ges he $1000 from he shor sale of 1 year earler for a gan of $00 from he shor sale. Addng he neres on he margn and backng off he dvdends pad by A resuls n a ne prof of on a margn of 500. So he yeld rae on hs ransacon s 180/ %. rcng Socks: The heorecal prce of a sock s calculaed usng he dvdend dscoun model. Ths jus means he prce of he sock s equal o he presen value of he dvdends, whch normally are assumed o connue forever.

3 More Tuoral a age 3 of 9 ecognon of Inflaon: The followng example llusraes how nflaon affecs he purchasng power of money. Suppose you have $100 now and a gallon of mlk currenly coss $4. Then he $100 wll buy currenly buy 5 gallons of mlk. Now suppose he $100 s nvesed for years a an 8% annual effecve rae of neres and he annual rae of nflaon over hs year perod s 5%. Then a he end of he wo years, you have bu mlk coss per gallon. So you can now buy / gallons of mlk. The real rae of reurn, ', s measured by solvng Thus we have: 51+ ' ' ' ' Le denoe he nomnal rae of neres and r denoe he rae of nflaon. The formula relang hese varables s r r 1+

4 age 4 of 9 Yeld Curves: No only are shor-erm and long-erm neres raes generally dfferen a any pon n me, bu hey also change over me. Ths phenomenon s called he erm srucure of neres raes. The followng able s a hypohecal able llusrang he erm srucure of neres raes. We can exend he values n hs able o a connuous graph, and he resulng graphcal llusraon s called he yeld curve correspondng o able. The neres raes n he able are called spo raes. Hypohecal Term Srucure of Ineres aes Lengh of Invesmen Ineres ae 1 year 7.00% years 8.00% 3 years 8.75% 4 years 9.5% 5 years 9.50% The followng examples llusrae how we use spo raes from a yeld curve. If person A nvess 100 for years and person B nvess 100 for 3 years, boh usng he correspondng spo raes n he able above, hen A would have a he end of years, and B would have & a he end of 3 years. The followng example llusraes he concep of a forward rae. A company needs o borrow money for wo years. The company can eher 1 borrow money for years a he spo rae n he able above, or borrow money for 1 year a he 1 year spo rae n he able above, and hen borrow money for he second year a he 1 year spo rae n effec 1 year from now, denoed by f [1, ]. We call f [1, ] a forward rae. Fnd f [1, ] such ha he company s ndfferen o he wo opons. We fnd he forward rae f [1, ] as follows: f [1,] f[1,] & %

5 age 5 of 9 Duraon: The duraon of a sequence of fuure paymens s a measure of he mng of he fuure paymens. We saw a smlar dea earler when we suded he mehod of equaed me. We use he followng example o help us recall he mehod of equaed me. Suppose paymens of 000, 4000, and are o be made a mes 1,, and 4, respecvely. The mehod of equaed me produces he value 000* * *4 / Snce he denomnaor equals 16000, we can rewre hs value as he sum [000 / 16000]*1 + [4000 / 16000]* + [10000 / 16000]* Wren hs way, we can hnk of he value produced by he mehod of equaed me as beng a weghed average of he mng of he paymens. Tha s, he average me of he paymens, akng no accoun he amouns of he paymens, s a me Noce ha he wegh gven o he paymen a me equals he rao of he amoun of he paymen a me o he oal amoun of all paymens. The concep of duraon s very smlar o he concep of equaed me. The calculaon of duraon s agan a weghed average of he mng of he paymens, excep wh duraon he wegh gven o he paymen a me equals he rao of he presen value of he paymen a me o he oal presen value of all paymens. If we dscoun he paymens above usng an annual effecve neres rae of 5%, hen he presen value of he paymen made a me 1 s 1600, he presen value of he paymen made a me s 560, and he presen value of he paymen made a me 4 s The sum of hese presen values s 856, and so he duraon of hs cash flow s d We usually don dscoun each ndvdual paymen lke n hs example, bu raher we use noaon and formulas developed earler.

6 More Tuoral a age 6 of 9 The formula for he Macaulay duraon of a cash flow conssng of paymens of a me s MacD Noe ha hs value wll depend upon he neres rae used for dscounng. If we use a 0% neres rae n he calculaon of duraon, we ge he same value as f we use he mehod of equaed me. The ne presen value funcon of a cash flow conssng of paymens of a me s 1 + Smlarly o how we defned he force of neres, we defne he volaly of a cash flow. v The negave sgn n fron ensures a posve value for he volaly.

7 age 7 of 9 An neresng resul appears f we carry ou he algebra n he defnng expresson for. v Takng a dervave wh respec o of, we ge Then pluggng hs back n he formula v, we ge v Bu he second facor n he las expresson s MacD, he Macaulay duraon of he cash flow. So we ge he followng relaonshp beween Macaulay duraon and volaly: For hs reason, volaly s also called modfed duraon. MacD MacD ModD + 1 More Tuoral a

8 More Tuoral a age 8 of 9 Invesng Asses versus Lables: Analogous o he concep of volaly modfed duraon s he concep of convexy. The formula for he convexy of a cash flow s c We sudy hree sraeges of nvesng asses versus lables. Ths means we wll nves money now and use our reurn on he nvesmen o pay fuure lables. Mehod 1: Immunzaon Immunzaon s a echnque o srucure asses and lables n a manner ha would reduce, or elmnae, he rsk of adverse effecs creaed by small changes n neres raes. Le A L he ne receps a me. Immunzaon s acheved a neres rae 0 f he ne presen value funcon 1 + has a local mnmum a 0 and 0 0. Thus, mmunzaon s acheved f > 0 Noe: An mmunzaon sraegy s o arrange asses so ha 1. The presen value of cash nflow from asses s equal o he presen value of cash ouflow from lables.. The volaly modfed duraon of he asses s equal o he volaly modfed duraon of he lables. 3. The convexy of he asses s greaer han he convexy of he lables.

9 More Tuoral a age 9 of 9 Mehod : Full Immunzaon Full mmunzaon s a echnque o srucure asses and lables n a manner ha would reduce, or elmnae, he rsk of adverse effecs creaed by all changes n neres raes. Suppose we have one lably ouflow L k a me k. The concep s o hold wo asses, one ha wll produce a cash nflow, A, pror o he lably ouflow say a me k a, and anoher ha wll produce a cash nflow subsequen o he lably ouflow say a me k + b. We use he force of neres δ ha s equvalen o and we use he lably paymen me as he comparson dae. We wan aδ bδ aδ bδ δ Ae + Be L 0 and δ Aae Bbe 0. k The full mmunzaon sraegy s obaned by solvng hs sysem of wo equaons. epea he above process for each lably ouflow. There wll be wo asse nflows for each lably ouflow. Mehod 3: Absolue Machng also called Dedcaon The dea here s o srucure asses such ha he resulng asse nflows wll exacly mach he lably ouflow.