Exam 4 is Th. April 24. You are allowed 13 shees of noes and a calculaor. ch. 7: 137) Unless old oherwise, duraion refers o Macaulay duraion. The duraion of a single cashflow is he ime remaining unil mauriy, eg n. The greaer he duraion, he more sensiive he PV is o changes in he ineres rae. The unis of a duraion are he lengh of he ime period, usually years. 138) Le P = = PV = price of he invesmen as a funcion of he ineres rae i. The Macaulay duraion = MacD = D Mac = D = P d (i) (1 + i) = di (1 + i). d di 139) The modified duraion = ModD = D Mod = DM = P (i) = = price sensiiviy wih respec o ineres. The change in price = he change in PV = P(i + h) h D Mod for a small change h in ineres. 140) Suppose an invesmen is bough a ime 0 and has cashflows C a imes = 1,..., n. Then PV = = n =1 C vi = n =1 C (1 + i) = n =1 C e δ, and he C v i C (1 + i) C (1 + i) = C e δ. Noe C e δ Macaulay duraion D = D Mac = = C vi ha he numeraor summand = ()(denominaor summand). Given he presen value is a sum, pu he sum in he denominaor and pu a sum wih summand = () (denominaor summand) in he numeraor o find he duraion. Noe ha is he exponen on v i and is he exponen on (1 + i) in he denominaor summand. 141) D Mac = D = d dδ = price sensiiviy wih respec o force of ineres. 142) D = D Mac = D Mod (1 + i) = D Mod vi 1, and D Mod = D Mac 1 + i = D Mac v i. 143) Noe ha D Mac = n =1 w C where w = C (1 + i) and n =1 w = 1. Hence he Macaulay duraion is a weighed average of he cashflow imes wih weighs w. 144) Suppose C K for = 1,..., n. Ofen he duraion D = D Mac can be compued by recognizing ha erm in he sums are PVs of annuiies. For example, he PV of an n annuiy immediae wih K = 1 is a n = vi. See 35). The PV of an increasing annuiy immediae wih K = 1 is (Ia) n = can be useful. n =1 =1 vi. See 71). Using a ime diagram o ge he PV a) The duraion of an annuiy immediae wih PV = Ka n is D = D Mac = K m =1 vi K m = =1 vi (Ia) n i. a n i b) Consider a bond ha maures in n years wih redempion value C ha pays annual coupons of Fr a imes 1,..., n wih yield rae j = i. Then PV = = Fr n =1 v +Cv n. Hence he duraion D = D Mac = Fr n =1 v + C n v n Fr n = Fr(Ia) n i + C n vi n. =1 v + Cv n Fra n i + Cv n c) An n year bond wih annual coupons K = Fr ha sells a is par value has P = = F = C and r = g = j = i. Then he duraion is D = ä n i. 1
145) Suppose here is a porfolio of m invesmens wih PV = P k (i) = X k for k = 1,..., m. Denoe he cashflows or he kh invesmen by C 1k,..., C nk. Le D k be he Macaulay duraion of he kh invesmen, and le he PV of he porfolio be P = = X = m k=1 X k = m k=1 P k (i). Le D = D Mac be he Macaulay duraion of he porfolio. Then D = mk=1 D k P k (i) mk=1 P k (i) = m k=1 w k D k where w k = P k(i) and m k=1 w k = 1. Hence he duraion of he porfolio is he weighed average of he duraions D k of he invesmens wih weighs w k. 146) Suppose an invesmen is bough a ime 0 and has cashflows C a imes = 1,..., n. Then PV = = n =1 C vi = n =1 C (1 + i) = n =1 C e δ, and he convexiy of he asse is d 2 di 2. Hence he convexiy = ( + 1)C vi +2. Noe ha C vi he numeraor summand = [( + 1)v 2 ] (denominaor summand). 147) A financial enerprise will have asses A 0 and liabiliies L 0. Le he PV of he asses = price of he asses be P A (i) = n =0 A vi, and le he PV of he liabiliies be P L (i) = n =0 L vi. Wan asses o be enough o pay off liabiliies, and immunizaion is he process of proecing a financial enerprise from small changes in he ineres rae in ha P A (i) P L (i) for i (i 0 ɛ, i 0 + ɛ) where i 0 is he ineres rae a ime = 0. 148) The ne presen value of he asses and liabiliies is NPV (i) = g(i) = P A (i) P L (i) = n =0 C vi where C = A L. Redingon immunizaion has hree sufficien condiions, and here are hree equivalens ses of he sufficien condiions. condiion se I) se II) se III) i) P A (i 0 ) = P L (i 0 ) P A (i 0 ) = P L (i 0 ) NPV (i 0 ) = 0 ii) duraion of asses = duraion of liabiliies P A (i 0) = P L (i 0) NPV (i 0 ) = 0 iii) convexiy asses > convexiy liabiliies P A (i 0) > P L (i 0) NPV (i 0 ) > 0 For he firs se of sufficien condiions, he convexiy and duraions are evaluaed a i = i 0. The Macaulay duraion or he modified duraion can be used. For condiion i), equaliy can be replaced by, a leas for he ses of condiions II) and III), bu mos exs use equaliy. Condiion i) makes g(i 0 ) = NPV (i 0 ) = 0. Condiions ii) and iii) insure ha g(i) = NPV (i) has a relaive min a i = i 0. Hence g(i) g(i o ) = 0 for i in a small neighborhood of i 0, say for i (i 0 ɛ, i 0 + ɛ). Hence P A (i) P L (i) for small changes in i from i 0, and Redingon immunizaion proecs he financial enerprise from such small changes in ineres. n n 149) A porfolio is fully immunized if A vi L vi for any i > 0. Then he =0 =0 enerprise is proeced agains any change in he ineres rae (provided i > 0). There are also 3 sufficien condiions: i) P A (i 0 ) = P L (i 0 ) ii) D Mod (i 0 ) of asses = D Mod (i 0 ) of liabiliies or, equivalenly, P A(i 0 ) = P L(i 0 ) iii) There is one asse cash flow before and afer any liabiliy cash ouflow. 150) Full immunizaion implies Redingon immunizaion. 151) Exac maching (also known as dedicaion, asse-liabiliy maching, absolue maching and cash flow maching) is a form of immunizaion. Exac maching has A = L for all (eg = 0, 1,..., n). Ofen can no predic when he cash inflows and ouflows will occur, so exac maching is no possible. 2
152) The dividend discoun model for he heoreical price of a sock is he PV of a perpeuiy of dividends. This price is for a sock ha will be held many years. The heoreical price from he model is no a very accurae esimae of he acual price. If he sock is bough a ime 0 and he dividends are D for = 1, 2, 3,..., hen he price P = PV = =1 D vi. Ofen dividends are given quarerly, biannually or annually. The ineres rae i should be for he ime period. If an ineres rae is given for a period oher han he ime period, conver ha ineres rae ino i. 153) Some special cases of 152) are a) D D for = 1, 2,... Then P = PV = Da = D/i. b) The firs dividend a ime = 1 is D and he dividend increases by a consan raio (1 + k) for each subsequen period where k < i. Then P = PV = ( ) Dv + Dv 2 (1 + k) + Dv 3 (1 + k) 2 + Dv 4 (1 + k) 3 D 1 + k + = = D =1 1 + k 1 + i i k. 154) If he dividend D has spo raes s and year forward raes f, hen he price of he sock from he dividend discoun model is P = PV = D vs = D (1 + s ) 1 = D [ (1 + f j )] 1. =1 =1 =1 155) Be familiar wih muual funds (including index funds), money marke funds, and cerificaes of deposi. Financial Derivaives including Ch. 9 and 6.4: 156) A derivaive or financial derivaive is a financial insrumen (conrac or agreemen) whose value is based on an underlying asse such as a sock. A derivaive derives is value from he underlying asse. Derivaives can be hough of as bes on he price of somehing. Forwards, opions, fuures and swaps are derivaives. Consrucing derivaives from oher financial producs is known as financial engineering. 157) Uses of derivaives include i) risk managemen and hedging: reduce risk, for example due o price changes, by enering ino a derivaives conrac; ii) pure speculaion: wan o make a be, no reduce risk; iii) reduce ransacion coss: ofen he underlying asse is no bough and sold, so he ransacion coss for buying and selling he asse are avoided; iv) risk sharing: insurance, for example, is a financial derivaive where he vas lucky majoriy lose he premium and he unlucky few ge money o cover heir losses and damages; v) can someimes defer axes, e ceera. 158) I is assumed ha here is a risk free rae of ineres (i or r or δ) such as he yield rae on US reasury bills. This risk free rae is ypically quoed as a coninuously compounded rae = force of ineres = δ. I will be assumed ha any fracion of an asse, such as a half share of sock, can be bough. 159) Usually need o pay a commission for a purchase or sale: eg $10 for boh buying and selling shares of sock, or a percenage fee such as 0.1% of he value of he purchase or sale. You sell shares o he broker a he bid price and buy shares from he broker a he ask price (> bid price). The bid and ask price are from he perspecive of he marke makers (brokers) who make i possible o buy and sell he asse (eg shares of sock), almos insananeously. The bid ask spread = ask price bid price. 160) Suppose he financial derivaive is a be on he fuure price of an underlying asse. The buyer of he asse a marke value (ime = 0) has he long posiion and is j=0 3
being ha he price of he asse will increase. The shor seller has he shor posiion and is being ha he price of he asse will decrease. If he price of he asse goes down, he holder of he long posiion will have a loss while he shor seller will have a profi (if ransacion fees, ineres, e ceera are no oo high). If he price of he asse goes up, he holder of he long posiion will have a profi while he holder of he shor posiion will have a loss. 161) A shor posiion is aken by borrowing he asse from he holder of he long posiion. The asse is immediaely sold, bu he holder of he shor posiion purchases he asse and reurns i o he holder of he long posiion a he ime he shor sale is erminaed. The ac of buying he asse a he ime ha he shor sale is erminaed is known as closing or covering he shor posiion. 162) i) The borrowed asses are sold immediaely by he shor seller. The proceeds belong o he shor seller bu are held by he lender (holder of he long posiion) or by a designaed financial insiuion. Ofen ineres paid on he proceeds is a cos o he shor seller if he ineres rae is less han ha of he risk free ineres rae. ii) Addiional collaeral, known as a haircu, may also be required. Ineres a a rae known as he shor rebae for socks and he repo rae for bonds is paid o he shor seller. If his rae is less han he risk free ineres rae, hen he loss of ineres income is a cos for he shor seller and a source of profi for he holder of he long posiion. iii) The shor seller mus pay he lender any dividends paid during he lending period. 163) Ignoring ineres gains or losses from a shor sale of sock, he cash inflow from selling K shares of borrowed sock = K(bid price) commission, while he cash ouflow from buying K shares of sock o close he shor posiion = K(ask price) + commission. If he commission is based on a commission rae ha is a small decimal like 0.00c, hen he cash inflow is K(bid price)(1 0.00c), while he cash ouflow is K(ask price)(1 + 0.00c). Then he cash inflow uses he smaller numbers (bid price and (1 0.00c)), while he cash ouflow uses he larger numbers (ask price and (1 + 0.00c)). See HW11 2. 164) A forward conrac is an agreemen o buy or sell a cerain underlying asse a a specific fuure dae called he expiraion dae or delivery dae T for a specific price called he forward price or delivery price F = F 0,T. Le ime = 0 be he ime when he conrac is made and le he spo price S 0 be he value of he asse a ime 0 and le S T be he value of he asse a ime T. The buyer a ime T has he long posiion and he seller a ime T has he shor posiion. A ime T he long payoff = S T F 0,T and he shor payoff = F 0,T S T. Noe ha he 2 payoffs always sum o 0. If hese payoffs are for one uni and here are K unis, hen he long payoff = K(S T F 0,T ) and shor payoff = K(F 0,T S T ). Noe ha he holder of he shor posiion again makes a profi if he price goes down and a loss if he price goes up. The holder of he long posiion makes a profi if he price goes up (S T > F 0,T ) and a loss if he price goes down (S T < F 0,T ). The holder of he shor posiion sells he asse for KF 0,T o he holder of he long posiion a ime T when he marke price of he asse is KS T. The forward conrac is called a long forward for he long posiion and a shor forward for he shor posiion. 165) The payoff is he value of he conrac o one of he paries a a paricular dae. 166) In general, he long posiion makes a profi if he price of he underlying asse increases, while he shor posiion makes a profi if he price of he underlying asse decreases. 4
167) For a call opion or call, he purchaser (buyer or holder) of he call opion has he righ (opion) o buy or no buy, from he wrier = seller of he opion, he underlying asse a a prespecified ime (expiraion dae) a a specified price K called he srike price or exercise price. The wrier of he opion mus sell he asse a he specified price K if he purchaser decides o buy he asse, called exercising he call opion. If he purchaser decides no o exercise he opion, hen he opion expires. A mnemonic for a call opion is COB: a call is an opion o buy he underlying asse for he purchaser of he call. 168) There are wo main syles of (call or pu) opions. The names of he syles have nohing o do where he opions are wrien or raded. Eiher syle can be raded in he US or Europe. i) The European (call or pu) opion can only be exercised a he expiraion dae. ii) The American (call or pu) opion can be exercised anyime during he life of he opion. 169) A payoff a ime does no depend on cashflows a imes oher han. The ime = T is especially imporan. 170) The purchased E. call opion payoff = long call payoff = max(0, S T K) while he wrien E. call opion payoff = shor call payoff = max(0, S T K) where S T is he marke price of he asse a ime T. Noe ha he sum of he wo payoffs is 0. I is assumed ha he call opion will be exercised (he asse will be bough) if he spo price of he asse a expiraion dae S T > K = exercise price, and no exercised oherwise. 171) Le C = C(K, T) be he call opion premium. Le he AV of C a ime T be F V (C). If he ineres rae is i and he force of ineres = rae of ineres compounded coninuously is r, hen FV (C) = C(1 + i) T = Ce rt. Someimes T is he end of one ime period (like a year or 6 monhs), and he ineres rae i and force of ineres r are given for ha ime period. Then FV (C) = C(1 + i) = Ce r. 172) The purchased E. call opion profi = long call profi = max(0, S T K) FV (C) while he wrien E. call opion profi = shor call profi = FV (C) max(0, S T K). Noe ha he sum of he wo profis is 0. 173) If he premium, payoffs and profis in 170) and 172) are for one uni, and here are J unis, hen he purchased E. call opion payoff = long call payoff = J max(0, S T K), wrien E. call opion payoff = shor call payoff= J max(0, S T K), he purchased E. call opion profi = long call profi = J[max(0, S T K) FV (C)], and he wrien E. call opion profi = shor call profi = J[FV (C) max(0, S T K)]. 174) For he call opion, he purchaser has he long posiion (wans he price o increase from K), and he wrier has he shor posiion (wans he price o decrease from K). 175) For a pu opion or pu, he purchaser (buyer or holder) of he pu opion has he righ (opion) o sell or no sell, o he wrier = seller of he opion, he underlying asse a a prespecified ime (expiraion dae) a a specified price K called he srike price or exercise price. The wrier of he opion mus buy he asse a he specified price K if he purchaser decides o sell he asse, called exercising he pu opion. If he purchaser decides no o exercise he opion, hen he opion expires. A mnemonic for a pu opion is POS: a pu is an opion o sell he underlying asse for 5
he purchaser of he pu. 176) Le P = P(K, T) be he pu opion premium. Le he AV of P a ime T be FV (P). If he ineres rae is i and he force of ineres = rae of ineres compounded coninuously is r, hen FV (P) = P(1 + i) T = Pe rt. Someimes T is he end of one ime period (like a year or 6 monhs), and he ineres rae i and force of ineres r are given for ha ime period. Then FV (P) = P(1 + i) = Pe r. 177) The purchased E. pu opion payoff = long pu payoff = max(0, K S T ) while he wrien E. pu opion payoff = shor pu payoff = max(0, K S T ) where S T is he marke price of he asse a ime T. Noe ha he sum of he wo payoffs is 0. I is assumed ha he pu opion will be exercised (he asse will be sold) if he spo price of he asse a expiraion dae S T < K = exercise price, and no exercised oherwise. 178) The purchased E. pu opion profi = long pu profi = max(0, K S T ) FV (P) while he wrien E. pu opion profi = shor pu profi = FV (P) max(0, K S T ). Noe ha he sum of he wo profis is 0. 179) If he premium, payoffs and profis in 177) and 178) are for one uni, and here are J unis, hen he purchased E. pu opion payoff = long pu payoff = J max(0, K S T ), wrien E. pu opion payoff = shor pu payoff= J max(0, K S T ) he purchased E. pu opion profi = long pu profi = J[max(0, K S T ) FV (P)], and he wrien E. pu opion profi = shor pu profi = J[FV (P) max(0, K S T )]. 180) For he pu opion, he purchaser has he shor posiion (wans he price o decrease from K), and he wrier has he long posiion (wans he price o increase from K). erm based on posiion in opion alernaive erm posiion in underlying asse long call purchased call long shor call wrien call shor long pu purchased pu shor shor pu wrien pu long 181) The owner of he opion has he long posiion in he opion since he owner benefis if he price of he opion goes up. The wrier has he shor posiion in he opion since he seller benefis if he price of he opion decreases. For a call, he long and shor posiions in he opion are he same as hose in he underlying asse. For a pu, a long posiion in he pu has a shor posiion in he underlying asse, and a shor posiion in he pu has a long posiion in he underlying asse. The erminology in he lefmos column of he above able refers o he long or shor posiion in he opion, no in he underlying asse. Purchased opions have he long posiion in he opion while wrien opions have he long posiion in he opion. 182) Given premiums, srike prices, ineres raes i, force of ineres r = rae of ineres rae compounded coninuously, be able o find he spo price S T given he profi or payoff. Be able o find he profi and payoff, given he spo price S T, using 170), 172), 173), 177), 178) and 179). The premium is he price of he opion. 183) The call opion is similar o a long forward, bu proecs he purchaser of he call opion from a large price decrease in he underlying asse. The pu opion is similar o a shor forward, bu proecs he purchaser of he pu opion from a large price increase in he underlying asse. The cos for his insurance is he premium. 6