VALUATION OF FIN ASSETS
VALUATION OF FIN ASSETS FI 8000
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Fi8000 Valuation of Financial Assets Spring Semester 2009 Dr Isabel Tkatch Assistant Professor of Finance Nation s Financial Industry Gripped by Fear NY Time September 15 2008 By BEN WHITE and JENNY ANDERSON Fear and greed are the stuff that Wall Street is made of But inside the great banking houses those high temples of capitalism fear came to the fore this weekend Nation s Financial Industry Gripped by Fear II As Lehman Brothers one of oldest names on Wall Street filed for bankruptcy protection anxiety over the bank s fate and over what might happen next gripped the nation s financial industry Part ofthe fear gripping Wall Street is the who s next game After the collapse of Bear Stearns it was Lehman After Lehman many worry about who might be next Nation s Financial Industry Gripped by Fear III Those who bet on a rebound in financials are getting clobbered In March O Shaughnessy Asset Management a 9 billion quantitative money management group started investing in financials and the group continues to add to its portfolio By contrast hedge funds that continue to short financials betting their prices will fall are still performing well The MeanVariance Criterion MV or 10 criterion ER STDR Fresh Bank Worries Batter Stocks By JACK HEALY and ZACHERY KOUWE NY Time January 212001 It s a growing lack of confidence and almost panic that s traveling around the world right now said Michael Holland chairman of Holland amp Company As volatility rises and markets tumble lower analysts said that Mr Obama and Congress could not move fast enough That fear that s creeping back into Wall Street is spreading to the broader market said Ryan Larson senior equity trader at Voyageur Asset Management Today 3 Portfolio Theory 9 The MeanVariance Criterion 9 Capital Allocation o The Mathematics of Portfolio Theory Capital Allocation Data There are three risky assets and one riskfree asset in the market The risk free rate is If 1 and the distribution of returns of risky assets is normal with the following parameters Asset A E 8 Expected Return 56 42 17 s d an 39t39 Effieni fn39 25 50 21 Capital Allocation n mutually exclusive assets State all the possible investments Assuming you can use the MeanVariance MV rule which investments are MV efficient ie which assets can not be thrown out of the set of desirable investments by a risk averse investorwho uses the MV rule Present your results on the uo mean standarddeviation plane The MeanVariance Criterion MV or 10 criterion Let A and B be two risky assets All risk averse investors prefer asset A to B if HAZ PB and 0A lt GB or if HAgt PB and OASOB Note that these rules apply only when we assume that the distribution of returns is normal The Expected Return and the STD of Return uo plane tmr eu A 4D a 2n c rf nut 2m 4D ranu ED Capital Allocation n mutually exclusive assets The investment opportunity set rt A B C The Mean Variance M v or p o ef cient investment set rt A C 39AL A dominates it one dominant investment is enough Capital Allocation One Risky Asset A and One Riskfree Asset State all the possible investments how many possible investments are there Assuming you can use the MeanVariance MV rule which investments are MV efficient Present your results on the uo mean standarddeviation plane The Expected Return and STD of Return of the Portfolio 1 the proportion invested in the risky asset A p the portfolio with a invested in the risky asset A and 1 a invested in the risk 39ee asset If Rp the return ofportfolio p up the expected return ofportfolio p op the standard deviation of return of portfolio p Rp aRA 1arf up E aRA 1arf auA 1arf U2PVa39RA1a39rfa39UA2 Or upaUA Capital Allocation One Risky Asset and One Riskfree Asset The investment opportunity set all portfolios with proportion 0L invested in A and 1u invested in the riskfree asset rf The MeanVariance MV or po efficient investment set all the portfolios in the opportunity set The Capital Allocation Line m ERP rf STMRA STDRP 0r GA The Expected Return and the STD of Return ll6 plane 1Kr The Capital Allocation Line CAL Four Basic Investment Strategies EmmaV Portfolios on the CAL Portfolio 1 ERp pp StdRp 5 rt 0 100 000 P1 025 215 0625 A 1 560 250 P2 15 790 375 Capital Allocation n Mutually Exclusive Risky Asset and One Riskfree Asset State all the possible investments how many possible investments are there Assuming you can use the MeanVariance MV rule which investments are MV efficient Present your results on the po mean standarddeviation plane The Expected Return and the STD of Return uo plane t w Eu smlm Capital Allocation One Risky Asset and One Riskfree Asset The investment opportunity set all the portfolios with proportion a invested in the risky assetj and 1a invested in the riskfree asset j A or B or C The MeanVariance M V or po efficient investment set all the portfolios with proportion a invested in the risky assetA and 1a invested in the riskfree asset Why A Capital Allocation Two Risky Assets State all the possible investments how many possible investments are there Assuming you can use the MeanVariance MV rule which investments are MV efficient Present your results on the po mean standarddeviation plane The Expected Retu rn and STD of Return of the Portfolio wA the proportion invested in the risky asset A wB 1wA the proportion invested in the risky asset B p the portfolio with wA invested in the risky asset A and 1wA invested in the risky asset B Rp the return ofportfolio p up the expected return ofportfolio p op the standard deviation of the return of portfolio p RP WA39RA 1WA39RB lip EWA39RA139WA39RB 52p V WA39RA 139WA39RB Two Risky Assets The Investment Opportunity Set ERp Two Risky Assets The MV Efficient Set Frontier STDRp ERp Two Mutually Exclusive Risky Assets The MV Efficient Set ER Ao STDR Two Risky Assets The MV Efficient Set Frontier ER Two Risky Assets The MV Efficient Set Frontier ER Two Risky Assets The MV Efficient Set Frontier STDR ER STDR Capital Allocation Two Risky Assets The investment opportunity set all the portfolios on the frontier with ro ortion wA Invested In the risky asset A and lwA Invested in the risky asset B The MeanVariance MV or po efficient investment set all the portfolios on the efficient frontier Two Risky Assets The MV Efficient Set Frontier ER Portfolios on the Efficient Frontier wA the proportion invested in the risky asset A wB 1wA the proportion invested in the risky asset B What is the value of wAfOI39 each one of the portfolios indicated on the graph Assume that pA10 pB5 oA12 o B6 pAB05 What is the investment strategy that each portfolio represents How can you find the minimum variance portfolio What is the expected return and the std of return of that portfolio Portfolios on the Frontier Portfolio WA ERp up stump 5 P1 13 1150 1657 A 1 1000 1200 P2 035 675 406 Pmm 7 7 7 B 0 500 600 P3 05 250 1308 The Minimum Variance Portfolio The Variance of a portfolio on the frontier 2 risky assets A and B is VRp 17 wi 0i w 1 ZWAWBGAGBpAB If you differentiate this expression With respect to WA and set the derivative equal to zero you Will get the minimum Variance portfolio 2 75 T GAG5PM WA z 2 2 GA T 75 T GAG5PM and w517wA The Minimum Variance Portfolio The minimum Variance portfolio in our case is of TGAGBPAB Gi0 20AGBpAB 7 62 7126705 7 122 032 7 2126705 WA 2 02857 Therefore ymm2643 and o 393 mm Practice Problems BKM 7th Ed Ch 6 1518 2021 25 32 3435 unwr ULII iu Jll u 1518 2627 21 CFA 6 89 Mathematics of Portfolio Theory Read and practice parts 610 Fi8000 Valuation of Financial Assets Spring Semester 2009 Dr Isabel Tkatch Assistant Professor of Finance Derivatives Overview Derivative securities are financial contracts that derive their value from other securities They are also called contingent claims because their payoffs are contingent on the prices of other securities Derivatives Overview Examples of underlying assets oCommon stock and stock index oForeign exchange rate and interest rate oAgricultural commodities and precious metals oFutures Examples of derivative securities 0 Options Call Put 0 Forward and Futures 0 Fixed income and foreign exchange instruments such Derivatives Overview Trading venues D Exchanges standardized contracts 0 Over the Counter OTC customtailored contracts Serve as investment vehicles for both 0 Hedgers decrease the risk level ofthe portfolio DSpeculators increase the risk Long Position in a Stock The payoff increases as the value price of the stock increases The increase is oneforone for each dollar increase in the price ofthe stock the value of the long position increases by one dollar Long Stock a Payoff Diagram Stock price Payoff ST ST 0 0 5 5 10 10 15 15 20 20 25 25 30 30 Short Position in a Stock The payoff decreases as the value price of the stock increases The decrease is oneforone for each dollar increase in the price ofthe stock the value of the short position decreases by one dollar Note that the short position is a liability with a value equal to the price ofthe stock mirror image ofthe long position Short Stock a Payoff Diagram Stock price Payoff m 5T ST a 0 0 I 5 5 j 10 10 15 15 w 20 20 2 25 25 j 30 30 Long vs Short Position in a Stock Payoff Diagrams Jmninamnn Jmnmnmun Long and Short Positions in the Riskfree Asset Bond The payoff is constant regardless of the changes in the stock price The payoff is positive for a lender long bond and negative forthe borrower short bond Lending a Payoff Diagram Borrowing a Payoff Diagram Stock price Payoff m ST X 0 20 u 5 20 3 10 20 J J m r m a m t 15 20 m 20 20 j 25 20 v 30 20 m Stock price Payoff m 5T X a 0 20 I 5 20 j 10 20 15 20 m 20 20 25 20 j 30 20 Lending vs Borrowing Payoff Diagrams A Call Option A European call option gives the buyer of the option a right to purchase the underlying asset at the contracted price the exercise or strike price on a contracted future date expiration An American call option gives the buyer ofthe option iong call a right to buythe underlying asset at the exercise price on orbefore the expiration date Call Option an Example A March European call option on Microsoft stock with a strike price of 20 entitles the owner with a right to purchase the stock for 20 on the expiration datequot What is the owner s payoff on the expiration date What is his profit if the call price is 7 Under what circumstances does he benefit from the position Note that exchange traded options expire on the third Friday or the expiration month The Payoff of a Call Option On the expiration date lf Microsoft stock had fallen below 20 the call would en le 0 expire wo ess lf Microsoft was selling above 20 the call owner would have found it optimal to exercise Exercise of the call is optimal if the stock price exceeds the exercise pr39ce Payoff at expiration is the maximum oftwo Max Stock price Exercise price 0 Max s X 0 Pro t at expiration Payoff at expiration Premium Notation S the price ofthe underlying asset Stock we will referto 80 S or ST C the price of a Call option premium we will referto Co C or CT X or K the eXercise or striKe price T the expiration date t a time index Buying a Call Payoff Diagram Stock price Payoff ST MaxS X 0 0 0 5 0 10 0 15 0 20 0 25 5 30 10 Buying a Call Profit Diagram Stock price Pro m ST MaxiSrXDH u o 7 i 10 7 3 20 7 j m u n 25 2 in 30 3 j 35 8 40 13 m Buying a Call Payoff and Profit Diagrams Writing a Call Option The seller of a call option is said to write a call and he receives the option price called a premium He is obligated to deliver the underlying asset on expiration date European for the exercise price The payoff of a short call position writing a call is the negative of long call buying a call Max Stock price Exercise price 0 Max 51 X 0 Writing a Call Payoff Diagram Buying a Call vs Writing a Call Payoff Diagrams Stock price Payoff m ST MaxSTX0 a 0 o I 10 0 j 15 0 j J m r m a m t n 20 o w 25 5 2 30 10 j 40 2o Moneyness We say that an option is inthemoney if the payoff from exercising is positive 9 A call options is intomoney if St X gt 0 ie if stock rice gt strike I rice We say that an option is outofthemoney ifthe payoff from exercising is zero 9 A call options is outofthemoney if St X lt 0 Le if the stock price lt the strike price Moneyness We say that an option is atthemoney if the price of the stock is equal to the strike price 8 Le the payoff is just about to turn positive We say that an option is Deepinthe Imoney if the payoff to exercise is extremely arge o A call options is deepin the money if S X gt gt 0 Le if the stock price gt gt the strike price A Put Option A European put option gives the buyer of the option a right to sellthe underlying asset at the contracted price the exercise or strike price on a contracted future date expiration An American put option gives tne buyer ofthe option lorig put a ngnt to sell tne underlying asset at tne exercise price on orbeforethe expiration date Put Option an Example A March European put option on Microsoft stock with a strike price 20 entitles the owner with a right to sell the stock for 20 on expiration date What is the owner s payoff on expiration date Under what circumstances does he benefit from the position The Payoff of a Put Option On the expiration date i If Microsoft stock was selling above 20 the put have een left to expire w ess i If Microsoft had fallen below 20 the put holder would ave found it optimal to exercise Exercise of the put is optimal if the stock price is below the exercise price i Payoff at expiration is the maximum oftwo Max Exercise price Stock price ii Max x 5 ii 0 Pro t at expiration Payoff at expiration Premium Buying a Put Payoff Diagram Stock price Payoff m ST MaxXST 0 a 0 20 I 5 15 j 10 10 15 5 w 20 0 2 25 0 j 30 0 Writing a Put Option The seller of a put option is said to write a put and he receives the option price called a premium He is obligated to buy the underlying asset on expiration date European for the exercise price The payoff of a short put position writing a put is the negative of long put buying a put Max Exercise price Stock price 0 Max X Sr 0 Writing a Put Payoff Diagram Buying a Put vs Writing a Put Payoff Diagrams Stock price Payoff ST MaxXST0 0 20 5 15 10 10 15 5 20 0 25 0 30 0 Buying a Call vs Payoff Diagrams Buying a Put Symmetry Writing a Call vs Writing a Put Payoff Diagrams Symmetry Jmnmnmun Juliannaquot Jmnmnmun Investment Strategies A Portfolio of Investment Vehicles 9 We can use more than one investment vehicle to from a portfolio with the desired payoff Q We can use any combination of the instruments stock bond put or call in any quantity or position long or short as our investment strategy 9 The payoff of the portfolio will be the sum of the payoffs ofthe instruments Investment Strategies Protective Put 9 Long one stock The payoff at time T is ST 9 Buy one European put option on the same stock with a strike price ofX 20 and expiration at T The payoff at time T is Max XST 0 Max 20ST 0 g The payoff of the portfolio at time T will be the sum of the payoffs of the two instruments 9 Intuition possible loses ofthe long stock position are bounded by the long put position Protective Put Individual Payoffs Stock Buy xmumum n gaseseeseuau Protective Put Portfolio Payoff Stock Buy All Purtfuliu xmumzxm n uaszsa Investment Strategies Covered Call 9 Long one stock The payoff at time T is ST 9 Write one European call option on the same stock with a strike price of X 20 and expiration at T The payoff at time T is MaXSTX 0MaXST20 0 g The payoff of the portfolio at time T will be the sum of the payoffs of the two instruments 9 Intuition the call is covered since in case of delivery the investor already owns the stock Covered Call Individual Payoffs Stock Long Write price Stock Call 0 0 0 5 5 10 10 0 15 15 0 20 20 0 25 25 5 30 30 10 Covered Call Portfolio Payoff Stock Long Write price Stock Call Pomona 0 0 0 0 5 5 0 5 10 10 0 10 15 15 0 15 20 20 0 20 25 25 5 20 30 30 1 0 20 1 u m u m x n xmumzxm n x m u m u 1 Other Investment Strategies 9 Long straddle a Buy a call option strike X expiration T a Buy a put option strike X expiration T 9 Write a straddle short straddle 0 Write a call option strike X expiration T 0 Write a put option strike X expiration T g Bullish spread a Buy a call option strike X expiration T 0 Write a Call option strke XZgtX expiration T The Put Call Parity Compare the payoffs of the following strategies 9 Strategy I a Buy one call option str ke X expiration T a Buy one riskfree bond face value X maturity T return n 9 Strategy ll 0 Buy one share of stock a Buy one put option strike X expiration T Strategy Portfolio Payoff Stock Buy All Strategy Portfolio Payoff Stock Buy Buy A price Stock Put Puma 0 0 20 20 5 5 15 20 10 10 10 20 Jmnmnmun 15 15 5 20 20 20 0 20 25 25 0 25 30 30 0 30 The Put Call Parity If two portfolios have the same payoffs in every possible state and time in the future their prices must be equal CLSP 1VfT Arbitrage the Law of One Price If two assets have the same payoffs in every possible state in the future but their prices are not equal there is an opportunity to make an arbitrage profit We say that there exists an arbitrage profit opportunity if we identify that There is no initial investment There is no risk of loss There is a positive probability of pro t Arbitrage a Technical Definition Let CF be the cash flow of an investment strategy at timet and state j Ifthe following conditions are met this strategy generates an arbitrage profit i all the possible cash flows in every possible state and time are positive or zero CF 2 0 for every t and j ii at least one cash flow is strictly positive there exists a pair t j forwhich CF gt 0 Example Is there an arbitrage profit opportunity ifthe following are the market prices of the assets The price of one share of stock is 39 The I rice of a call 0 tion on that stock which expires in one year and has an exercise price of 40 is 725 The price of a put option on that stock which expires in one year and has an exercise price of 40 is 650 The annual risk free rate is 6 Example In this case we should check whether the put ca parity holds Since we can see that this parity relation is violated we will show that there is an arbitrage profit opportunity CL 7254 0 44986 1rfT 10061 SP 39 650 455 The Construction of an Arbitrage Transaction Constructing the arbitrage strategy 1 Move all the terms to one side of the equation so their sum will be positive 2 For each asset use the sign as an indicator of the appropriate investment in the asset If the sign is negative then the cash flow at time t0 is negative which means that you buy the stock bond or option If the sign is positive reverse the position Example In this case we move all terms to the LHS Example In this case we should 1 Sell short one share of stock 2 Write one put option 3 Buy one call option 4 Buy a zero coupon riskfree bond lend SP7C j4557449860514gt0 119 SP7C7LT gt0 1H7 Example Example Example Practice Problems BKM 7th Ed Ch 20 1 12 14 23 BKM 8th Ed Ch 20 514 1622 26 CFA 12 Practice Set 116 Fi8000 Valuation of Financial Assets Spring Semester 2008 Dr Isabel Tkatch Assistant Professor of Finance Currency Exchange Rate Spot A spot currency transaction is an exchange of one currency for another The currency exchange rate is a simple conversion factor Q The direct exchange rate is the number of US to be paid for 1 unit of foreign currency usually forthe UK and the Euro 9 The indirect exchange rate is the number of foreign currency units paid for 1 US usually for the Swiss Franc and Japanese Yen Currency Exchange Rate Numeric Example The exchange rate between the US and UK is 16757 US UK ie one has to pay 16757 for 1 direct The same exchange rate can be presented as 116757 05968 UK US ie one has to pay 05968 for 1 indirect Currency Exchange Rate Example continued The exchange rate between the US and UK is 16757 US UK The exchange rate between the US and J is 0007331 US J What should be the exchange rate between the UK and the J Currency Arbitrage There are at least two ways to convert UK to J 3 Direct conversion of UKto J 3 Conversion using an intermediar currenc 3 Convert UK to 08 Convert US to J Q If there is no opportunity to make arbitrage profits both conversion methods must imply the same UK to J exchange rate Currency Exchange Rate Example data 16757 USI UK or 05968 UKUS 0007331 USJ or 13640J US We will use the noarbitrage argument to calculate the noarbitrage UKJ exchange rate Currency Exchange Rate Conversion using an intermediary currency Convert UK to US the price of 1 US is 05968 UK Convert US to J 1 the price of 1 J is 0007331 US The UK price of 1 J 1 05968 UKIUS 0007331 USJ 0004375 UKIJ Currency Exchange Rate The UKIJ noarbitrage exchange rate The UKJ exchange rate is 0004375 ie the price of1 J is 0004375 UK The J UK exchange rate is 10004375 2285641 ie the price of1 UK is 2285641 J Currency Exchange Arbitrage Example continued The USI UK exchange rate is 16757 The USJ exchange rate is 0007331 lsthere an arbitrage o ortunit ifthe market UKIJ exchange rate is 0004494 Yes The U KIJ exchange rate in the market is different from the noarbitrage rate twostage exchange rate Market 0004494 U KlJ gt 0004375 U KlJ Noarbitrage How can we make arbitrage pro ts Currency Exchange Arbitrage Cross currency triangle arbitrage strategy Sell the expensive J convert J to UK in one step 1 Sell J for UK convert J to U K Bu the chea J convert UK to J in two ste s using the US as an intermedia 2 Buy US with UK convert UK to US 3 Buy J with US convert US to J ote this is a round trip transaction You start with J before step 1and you end up with J a er step 3 Currency Exchange Arbitrage Cross currency triangle arbitrage strategy Sell the expensive J direct UK to J exchange rate 1 Convert 1 J to 0004494 UK Buy the cheap J two stages using the US as an intermediary 2 Convert 0004494 UK to US You will get 0004494 UK 16757 USEUK 000753 US 3 Convert 000753 US to J You will get 000753 US 13640J US 102717 J Arbitrage pro t you started With 1 J and ended up wrrm 02717 J Currency Exchange Arbitrage Cross currency arbitrage strategy end up with US 2 Convert 13640 J to UK You will get 13640 J 0004494 U KlJ 06130 UK 3 Convert 06130 UKto US You will et 06130 UK 16757 USI UK 102717 US 1 Convert 1 US to J You will get 1 US 13640J US 13640 J Arbitrage pro t you started with 1 US and ended up with 102717 US an arbitrage pro t of002717 US Currency Exchange Rate Forward Pricing Currency Forwards 0 Forward or Futures Contracts 0 There are at least two ways to invest money in a riskfree asset for one year agreement between a buyer and a seller to trade at a speci c date in the future a speci c 9 Domestic risk 39ee investment quantity of a speci c currency for an agreed QBW US Trea urv BIIIS exchange rate 9 Foreign riskfree investment Forward tailored OTC market contracts for ear crad39twonhy traders and large trades QConvert tne foreign currency back to 05 forward contract Futures formal markets of standardized contracts International Monetary Market in Q If there is OPPPOITUVIW to make arbitrage Chicago London International Financial Futures Pro quotSr bOth 39nVeStment trategle ShOUId haVe Exchange the same dollar denominated riskfree return Covered Interest Arbitrage Covered Interest Arbitrage Numeric Example Numeric Example Continued Suppose you would like to invest 100000 in a We need the SPOt and forward one Year riskfree security USI UK exchange rates to answer that question Note that if we do not use a forward contract to I the US the annual SK free rate IS 53900 Whlle lock in the exchange rate the foreign alternative m the UK the annual 5k free rate 395 520 Aquot be omes a risky rather than riskfree investment Is there an arbitrage opportunity Find a way to Strategy eXChange rate Sn compare the domestic and foreign investment Is there an opportunity to make arbitrage profits if strategies the spot exchange rate is 16750 USI UK and the one year forward rate is 16500 US UK Comparing the Two Strategies Comparing the Two Strategies 2 Foreign riskrfreeinvestment 1 DomeSt39C SKfree aneStmenti Za Convert US rortne foreign currency UK 1a Buy Us Treasury Bills 2o Buy foreign UK denominatedMskrfree bonds 2c Convert tne foreign currency UK back to US forward rate t0 t1 Arbitrage Strategy Buy Cheap Domestic riskfree 39 Buy US Treasury Bills get 5 dollar denominated risk free rate Sell F nensive39 Foreiqn risk free39 Convert UK to US Short sell UK riskfree bonds for 1 year Convert US back to UK forward contract pay 363 dollar denominated risk free rate A A Covered Interest Arbitrage t0 t1 NoArbitrage Forward Exchange Rate F0US UK 16718 NoArbitrage UK Risk Free Rate rW 65909 Interest Rate Parity Covered Interest Arbitrage Intuition lftwo investments are riskfree they must have the same rate of return Therefore any difference in the domestic and foreign riskfree rates must be ofSet by a difference in the spot and forward exchange rates Interest Rate Parity Covered Interest Arbitrage Notation ED spot exchange rate US U K or UKUS FD forward exchange rate US UK or U KIUS to reverse the ratio er interest rates Practice Problems Practice Problem 1 The annual riskfree rate in the US is 500 while in Japan it is 320 What should be the spot J US exchange rate ifthe one year forward J US exchange rate is 107875 Answer E0J US 1097565 Practice Problems Practice Problem 2 The annual riskfree rate in the US is 460 while in Japan it is 350 The spot J UK exchange rate is 20500 the spot USI UK exchange rate is 18825 the one year forward J UK exchange rate is 20400 and the forward USI UK exchange rate is 18900 Describe an arbitrage transaction write down the strategy in the table format presented in the lecture notes Practice Problems BKM Ch 23 7th Ed101214 8th Ed1112CFA23 Practice problems Forward and futures contracts 15 Currency exchange rates 69 Fi8000 Valuation of Financial Assets Spring Semester 2009 Dr Isabel Tkatch Assistant Professor of Finance Lending vs Borrowing Investment Strategies Lending vs Borrowing riskfree asset 9 Lending a positive proportion is invested in the riskfree asset cash outflow in the present CF0lt 0 and cash inflow in the future CF gt 0 o Borrowing a negative proportion is invested in the riskfree asset cash inflow in the present CF0gt 0 and cash outflow in the future CF lt 0 swim Investment Strategies 9A Long vs Short position in the risky asset 0 Long A positive proportion is invested in the risky asset cash outflow in the present CF0lt 0 and cash inflow in the future CF gt 0 0 Short A negative proportion is invested in the risky asset cash inflow in the present CF0 gt 0 and cash outflow in the future CF lt 0 Long vs Short ER LongA and Short a LongA and Long a Investment Strategies 3 Passive risk reduction The risk ofthe portfolio is reduced ifwe invest a larger proportion in the riskfree asset relative to the risky one eThe perfect hedge The risk ofasset A is offset can be reduced to zero by forming a portfolio with a risky asset B such that pAE1 Diversi cation The risk is reduced ifwe form a portfolio of at least two risky assets and B such that pAElt1 The risk is reduced ifwe add more risky assets to our portfolio such that pUlt1 One Risky Fund and one Riskfree Asset Passive Risk Reduction pomiio Risk a El A Reduction in portfolio ns Increase of eriRi Two Risky Assets with pAB1 The Perfect Hedge ER Minimum Variance is zero STDR The Perfect Hedge an Example What is the minimum variance portfolio if we assume tha uA10 uB5 6A12 6B6 and pAB 1 a 7 UAUBPAB 0i 0 7 ZUAO39BpAB 62 1261 l 122 62 7 212671 3 WA The Perfect Hedge Continued What is the expected return pm and the standard deviation of the return 5mm of that portfolio Hmquot Willi 1 WAWE l1o3 5 53 3 3 3 Gm Mi i 1WA2 2WA1 WAUAUEPAE Diversification the Correlation Coefficient and the Frontier STDR Diversification the Number of Risky assets and the Frontier STDR Diversification the Number of Diversification the Number of Risky assets and the Frontier Risky assets and the Frontier STDR STDR Diversification the Number of Capital Allocation Risky assets and the Frontier n Risky Assets State all the possible investments how many possible investments are there Assuming you can use the MeanVariance MV rule which investments are MV efficient Present your results in the po mean standarddeviation plane STDR The Expected Return and the Variance of The Set of Possible Portfolios the Return of the Portfolio in the 0 plane w the proportion invested in the risky asset I39r391n p the portfolio of n risky assets w invested in asset I Rp the return of portfolio p ER The Frontier up the expected return ofportfolio p 01p the variance of the return of portfolio p R wlRlw2Rzwnamp ZwtR 1919 Zwr39 r 11 VRp0 ZZZW1WJGU I J STDR The Set of Efficient Portfolios in the 10 Plane ER The Ef cient Frontier 00 STDR Capital Allocation n Risky Assets The investment opportunity set all the portfolios wk wn where Zwi1 The MeanVariance MV or po efficient investment set only portfolios on the efficient frontier The case of n Risky Assets Finding a Portfolio on the Frontier Optimization Find the minimum variance portfolio for a given ex ected return Constraints A given expected return The budget constraint The case of n Risky Assets Finding a Portfolio on the Frontier n Min Zimma 171 W W 1 7 n 51 Zwrwa 11 iw 1 11 Capital Allocation n Risky Assets and a Riskfree Asset State all the possible investments how many possible investments are there Assuming you can use the MeanVariance MV rule which investments are MV efficient Present your results in the uo mean standarddeviation plane The Expected Return and the Variance of the Return of the Possible Portfolios w the proportion invested in the risky asset I39I391n p the portfolio of n risky assets w invested in asset I Rp the return ofportfolio p up the expected return ofportfolio p 01p the variance ofthe return of portfolio p R w rerwlRl w2RZ wnamp w rfZw11 11 1319 ll War Ewe 11 2 n n VRpop0wtwjov I J The Set of Possible Portfolios in the 10 Plane only n risky assets ER The Frontier STDR The Set of Possible Portfolios in the 10 Plane risk free asset included ER STDR The Set of Efficient Portfolios in the 16 Plane The Capital Market e if rr umeruxdup The Separation Theorem The asset allocation process of the riskaverse investors can be separated into two stages 1Choose the optimal portfolio of risky assets m The allocation between risky securities is identical for all the investors 2Choose the optimal allocation offunds between the risky portfolio m and the riskfree asset rf choose a portfolio on the CML The allocation between the risky portfolio and the risk 39ee asset is personal and depends on the risk preferences of each investor Capital Allocation n Risky Assets and a Riskfree Asset The investment opportunity set all the portfolios w0 W1 wn where Zw The MeanVariance MV or 10 efficient investment set all the portfolios on the Capital Market Line CMu n Risky Assets and One Riskfree Asset Finding the Market Portfolio Optimization Find the minimum variance portfolio for a given ex ected return Constraints A given expected return The budget constraint n Risky Assets and One Riskfree Asset Finding the Market Portfolio n n Mm wiwjaij W W i1 j1 St in Lim rfyp i1 i1 n Risky Assets and One Riskfree Asset Finding the Market Portfolio Solve the following system of eqnasions and find the proportions m w invested in the risky assets Wi iiW2 i2 Wp ip i0 Wi 2iW2 22 Wp 2p r7f Wi piW2 p2 Wp p rrf Scale the proportions z lei and m 44 2 is the market portfolio Example Find the market portfolio ifthere are only two risky assets A and B an a nsk free asset rf pA10 lla5 oA12 oE6 pAEO5 and rf4 To nd the proportions WAWE invested in assets A and B use the system ofequations for two risky assets wn m WEUAE liar 7f WAUEA WHERE 5 r 7f Using our data we get two equations 122 wE12670 5 10 e 4 wA12670 5 WE62 5 e 4 Example Continued Ifwe solve the two equations wA12 2 WE 125eo5 10 e 4 wA125eo 5 WE 5 2 5 e 4 we get the proportions WA WE 006481 009259 Now we have to scale the proportions Lamp 041175 and 25 0 58824 wAwE 006481009259 and m zAzE 041176058824 is the market portfolio Example Continued The expected retum of the market portfolio is I m ZAHA Zg 5 041176100 58824 5 7 06 Th hint itin rthrtm t thmrktrr1t li i aquot zjtriz tr ZzAzEaAaEJAE lo 411762122 o 58824262 2041175o 58824126 0 5 441 Practice Problems BKM 7th Ed Ch 7 113 1722 2526 BKM 8th Ed Ch 7 419 CFA 46 1011 Mathematics of Portfolio Theory Read and practice parts 1113 Fi8000 Valuation of Financial Assets Fall Semester2006 Dr Isabel Tkatch Assistant Professor of Finance Debt instruments Types of bonds Ratings of bonds default risk Spot and forward interest rate The yield curve Duration Bond Characteristics 9 A bond is a security issued to the lender buyer by the borrower seller for some amount of cash 9 The bond obligates the issuerto make speci ed payments of interest and principal to the lender on speci ed dates 9 The typical coupon bond obligates the issuer to make coupon payments which are determined by the coupon rate as a percentage ofthe par value face value When the bond matures the issuer repays the parvalue Q Zerocou on bonds are issued at discount sold for a price below par value make no coupon payments and pay the parvalue at the maturity date Bond Pricing Examples The par value of a riskfree zero coupon bond is 100 If the continuously compounded riskfree rate is 4 per annum and the bond matures in three months what is the price of the bond today 99005 A risky bond with par value of 1000 has an annual coupon rate of 8 with semiannual installments If the bond matures 10 yearfrom now and the riskadjusted cost of capital is 10 per annum com ounded semiannually what is the price of the bond today 8753779 Yield to Maturity Examples What is the yield to maturity annual compounded semiannually of the risky coupon bond if it is selling at 1200 5387 What is the expected yield to maturity of the risky couponbond if we are certain that the issuer is able to make all coupon payments but we are uncertain about his ability to pay the par value We believe that he will pay it all with probability 06 pay only 800 with probability 035 and won t be able to pay at all with probability 005 4529 Default Risk and Bond Rating 9 Although bonds generally promise a xed ow of 39 me in most cases this cas ow stream is uncertain since the issuer may default on his obligation 9 US government bonds are usually treated as 39ee of default credit risk Corporate and municipal bonds are considered risky Q Providers ofbond quality rating Q Standard and Poor s Corporation Duffamp Phelps QFitch lnvestorSerVice Default Risk and Bond Rating QAAA Aaa is the top rating Bonds rated BBB Baa and above are considered investmentgrade bonds Bonds rated lowerthan BBB are considered speculativegrade orjunk bonds Risky bonds offer a riskpremium The greater the default risk the higher the default risk premium The yield spread is the difference between the yield to maturity of high and lower grade bond Estimation of Default Risk The determinants ofthe bond default risk the probability of bankruptcy and debt quality ratings are based on measures of financial stability 3 Ratios of earnings to fixed costs 3 Pro tability measures Cash ow to debt ratios A complimentary measure is the transition matrix estimates the probability of a change in the rating of The TermStructu re of Interest Rates The short interest rate is the interest rate for a specific time interval say one year which does not have to start today The yield to maturity spot rate is the internal rate of return say annual of a zero coupon bond that prevails today and corresponds to the maturity of the bond Example In our previous calculations we ve assumed that all the short interest rates are equal Let us assume the following Example What is the price ofthe 1 2 3 and 4 years zerocoupon bonds paying 1000 at maturity Maturity ZeroCoupon Bond Price 10 1o111 1011 Date Short Forthe time interval Interest rate 0 r8 t0tot1 1 r210 t1tot2 2 r311 t2tot3 3 r411 t3tot4 Example What is the yieldtomaturity of the 1 2 3 and 4 years zerocoupon bonds paying 1000 at maturity Maturity Price Yield to Maturity 1 92593 y1 8000 2 84175 y2 8995 3 75833 y3 9660 4 68318 y4 9993 The TermStructu re of Interest Rates The price of the zerocoupon bond is calculated using the short interest rates r t 12 T For a bond that matures in T years there may be up to T different short annual rates Price FV I 1r11r21r The yieldtomaturity yT of the zerocoupon bond that matures in T years is the internal rate of return of the bond cash flow stream Price FV I 1yquot39 The TermStructu re of Interest Rates The price of the zerocoupon bond paying 1000 in 3 years is calculated using the short term rates Price 1000 I 108110111 75833 The yieldtomaturity y3 of the zerocoupon bond that matures in 3 years solves the equation 75833 1000 I 1y33 y3 9660 The TermStructu re of Interest Rates Thus the yields are in fact geometric averages of the short interest rates in each period H39VTT 1r11r21TT 1 VT 1r11r21FT T The yield curve is a graph of bond yieldto maturity as a function of timetomaturity The Yield Curve Example The TermStructu re of Interest Rates If we assume that all the short interest rates r t 1 2 T are equal then all the yields yT of zerocoupon bonds with different maturities T 1 2 are also equal and the yield curve is fla A flat yield curve is associated with an expected constant interest rates in the future An upward sloping yield curve is associated with an expected increase in the future interest rates A downward sloping yield curve is associated with an expected decrease in the future interest The Forward Interest Rate 0 The yield to maturity is the internal rate of return of a zero coupon bond that prevails today and corresponds to the maturity of the bond 0 The forward interest rate is the rate of return a borrower will pay the lender for a specific loan loan is equivalent to a forward zero coupon bond The Forward Interest Rate Suppose the price of 1year maturity zerocoupon bond with face value 1000 is 92593 and the price of the 2year zerocoupon bond with 1000 face value is 84168 If there is no opportunity to make arbitrage profits what is the 1year forward interest rate for the second year How will you construct a synthetic 1year forward zerocoupon bond loan of 1 000 that commences att 1 and matures att 2 The Forward Interest Rate If there is no opportunity to make arbitrage profits the 1year forward interest rate for the second year must be the solution of the following equation 1W22 1y11f2 where yT yield to maturity of a Tyear zerocoupon bond f 1year forward rate for yeart The Forward Interest Rate In our example y1 8 and y2 9 Thus 10092 10081f2 f2 01001 1001 Constructing the loan borrowing 1 Time t 0 CF should be zero 2 Time t 1 CF should be 1000 3 Time t 2 CF should be 10001f2 11001 The Forward Interest Rate Constructing the loan we would like to borrow 1000 a year from now for a forward Interest rate of 1001 13 CFD 92593 but it should be zero We offset that a h owifwe u e 1 earzero coupon ondfor 92593 That is if we buy 9259392593 1 units of the 1yearzero coupon bond 21 CF should be equal to 1000 32 CF2 100011001 11 01 We generate that cas ow ifwe sell 11001 of the 2year zerocoupon bond for 1 1 001 84168 92593 Bond Price Sensitivity 0 Bond prices and yields are inversely related 0 Prices of longterm bonds tend to be more sensitive to changes in the inter required rate of return cost of capital than of shortterm bonds compare two zero coupon bonds with different maturities 0 Prices of high couponrate bonds are less sensitive to changes in interest rates than prices of low couponrate bonds compare a zerocoupon bond and a couponpaying bond of the same maturity w E m Duration The observed bond price properties suggest that the timing and magnitude of all cash flows affect bond prices not only timetomaturity Macaulay s duration is a measure that summarizes the timing and magnitude effects of all promised cash ows Cash ow Welght Clix1 y w B ondPrice r Macauley39s Duration D 1twr Example Textbook Page 524 Calculate the duration of the following bonds 8 coupon bond 1000 par value semiannual installments Two years to maturity The annual discount rate is 10 compounded semiannually M Zerocoupon bond 1000 par value Two year to maturity The annual discount rate is 10 compounded semi annually Example Textbook Page 524 PVCF CF Weight 855611 08871 17741 Example Textbook Page 524 Calculation of the durations Forthe coupon bond D Sum wt t 18852 years Forthe zero coupon bond D Time to maturity 2 years Properties of the Duration The duration of a zerocoupon bond equals its time to maturity Holding maturity and par value constant the bonds duration is lower when the coupon rate is higher Holding couponrate and par value constant the bonds duration generally increases with its time to maturity The Use of Duration Q It is a simple summary statistic of the effective average maturity of the bond or portfolio of fixed income instruments 0 Duration can be presented as a measure of bond portfolio price sensitivity to changes in the interest rate cost of capital 0 Duration is an essential tool in immunizing portfolios against interest rate risk Macaulay s Duration Bond price p changes as the bonds yield to maturity y changes We can show that the proportional price change is equal to the proportional change in the yield times the duration DM P 1y Modi ed Duration Practitioners commonly use the modified duration measure DD1y which can be presented as a measure ofthe bond price sensitivity to changes in the interest rate AP 9 D A P y Example Calculate the percentage price change for the followin bonds ifthe semiannual Interest rate increases 39om quotA to 5 01 8 coupon bond 1 000 par value semiannual installments Two years to maturi y The annual discount rate is 10 compounded semiannually 2 Zerocoupon bond 1000 par value Two yearto maturity The annual discount rate is 10 compoun ed semiannua y 3 A zerocoupon bond with the same duration as the 8 coupon bond 18852 years or 377 6 o h periods The modi d duration is 37704105 3591 period of 6months Example The percentage price change for he following bonds as a result ofan increase in he interest rate 39om 5 to 501 APP DA 37704105001 003591 APP DAy 4105001 003810 APP DAy 37704105001 003591 Note that When two bonds have the same duration not time to maturity they also have the same price sensitivity to changes in the interest rate 1 vs 3 hen the duration not timetomaturity of one bond is higher then the other it s price sensitivity is also high NT Fquot Practice Problems BKM Ch 14 12 1112 BKM Ch 15 Concept check 89 End of chapter 6 10 2324 BKM Ch 16 13