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1、Long-Term Capital Management,L.P.,Jiao Yunfan Peng XiaoliJin Long Yu Shuoning,Outline,Background Information about LTCMTrading StrategiesSelling VolatilityRisk ArbitrageSwap Spread-Convergence StrategyYield-Curve relative-value tradesA uniform analysis frameA recent empirical casePortfolio Risk Mana
2、gementThe FailWhere did LTCM Go WrongRescuing LTCM,Background Information,Background Information,Established in early 1994 with a initial capital size of$1 billion.Led by Mr.John Meriwether,once the Vice Chairman of Salomon Brothers in charge of the fixed income business.Engaged in trading strategie
3、s that would exploit market pricing discrepancies.Relied heavily on sophisticated analytical models.Capital doubled after three and a half years in 1997.Then,what happened?,Background Information,On 23 Aug.,1998,LTCM lost$553 million,15%of its capital,Correlations of all trades=1,“Dont ever call me
4、at home again!”,What Happened to LTCM in Aug.,1998?!,They must sell something-in a market WITHOUT A BUTER,Trading Strategies,Selling VolatilityRisk ArbitrageSwap Spread-Convergence StrategyYield-Curve relative-value tradesA uniform analysis frameA recent empirical case,SAMPLE TRADE 1:Selling Volatil
5、ity,Trade IntroductionRisks,Selling Volatility,The market expected that the implied volatility was higher than that of LTCMs perception.So LTCM thought that the option price was overvalued according to B/S model.They bet that option prices would fall in the future.So LTCM sold options to earn the pr
6、ice discrepancy.LTCM simultaneously sold long-term puts and calls on an index.,Selling Volatility:profit and loss,If the stock volatility was small,that is to say,the price was near the strike price K,LTCM would make a large profit.However,if the stock price moved largely in either direction-the vol
7、atility was large,LTCM would suffer a loss.The smaller the volatility was,the more profit LTCM would gain.But if the index level moved outside the lower limit or the upper limit,a loss would occur.,P,S,K,K was equal to the forward price of the index.,Selling Volatility:Risks,Daily settlement needed
8、a lot of working capital.This was difficult to maintain over five years.It was difficult to predict the market volatility.Using historical data was risky.And the model did not include extreme event.Incorrect comparison of volatility(Heston and Nandi,2000)Someone comparing the implied volatility with
9、 the simple measure of historical volatility might often be tempted to sell straddles.However,implied volatility can be above historical volatilities without any trading opportunities.(Implied volatility reflects the drift or the expected value of the variance under the risk-neutral distribution,whi
10、ch tends to be higher than the drift of the variance under the data-generating distribution,under negative correlation between returns and volatility.),Selling Volatility:Risks(continue),skew(smirk)implied volatilities In-the-money call has higher implied volatility than a near-the-money callcorrela
11、tion between return and volatilityNegative correlation between equity market returns and implied volatilities could make the straddle values highly sensitive to the direction of the market.rebalancing the straddle to maintain minimal exposure to the direction of the market is theoretically feasible,
12、the rebalancing process exposes a trader to model risk and may not always help.,SAMPLE TRADE 2:Risk Arbitrage,Risk arbitrage,Risk arbitrage/merger arbitrageRisk arbitrageurs would attempt to capture the spread by purchasing the shares of the target company and selling short of the shares of the acqu
13、irer.Tend to provide stable returns(about twice the T-bills index)in most market environments.In 1997,the funds risk arbitrage portfolio:contained about$5 billion of long positioninvolved over 30 different merger situations,most of which US companiesA 5%net spread on these positions represent$250 mi
14、llion of expected annual incremental profits,Risk arbitrage:rationale,Following the announcement of a acquisition,the targets shares tended to trade at a discount to the consideration offered usually the shares of the acquirer or cash.Only some of the discount could explained by the time value of th
15、e money or the risk of the acquisition not being consummated.LTCM believed the spread exist becausethere were fewer natural holders of the targets shares while the merger was being completed,many of targets shareholders preferred to take profits and not bear the risk of a deal break.,Risk arbitrage:
16、strategy,Participate in situations where the risk of a break was manifestly very smallavoided hostile takeovers avoided mergers that faced significant regulatory hurdlesPreferred stock deals to cash dealsCash deals were more likely to break or be renegotiatedStock deals tended to have higher expecte
17、d profits because selling short was difficult and/or costly for many investors,Risk arbitrage:could it profit?,Less risky deals had lower spread.Could the fund still profit?The fund could finance the trades very efficiently.(zero haircuts)For any particular merger,LTCM considered the risk of a break
18、 to be nearly uncorrelated with its other strategies.The spreads therefore did not have to be large in order for risk arbitrage to enhance the Sharpe Ratio of the fund.,Risk arbitrage:risks and drawbacks,Negative factor:delay of the announced deal as it reduces the annualized returnBreakup of the an
19、nounced deal:regulatory reasons,lack of agreement,shareholder rejection,unexpected event,counterbidA decrease in the deals flow:all managers chasing the same deals,SAMPLE TRADE 3:Swap Spread,Convergence Trade,Swap Spread,Net Flow X to the Fund:X=(FIX T-BOND INTEREST REPO)(FIXED INTEREST LIBOR)=(LIBO
20、R REPO)(FIXED INTEREST-FIX T-BOND INTEREST)=(LIBOR REPO)SWAP SPREAD=20bps-SWAP SPREAD,Trade Introduction,REPO,U.S.Treasury,BOND COLLATERAL,Investors,FUND,FUND,BOND,LTCM,FIX T-BOND INTEREST,LIBOR,FIXED INTEREST,Counterparties,Swap Spread,Fix T-bond Interest Falls,Swap Spread Widen,Bond Value Increase
21、s,Unwind the Position at a PROFIT before the Maturity,Betting on the Widening,REPO,U.S.Treasury,BOND COLLATERAL,Investors,FUND,FUND,BOND,LTCM,FIX T-BOND INTEREST,LIBOR,FIXED INTEREST,Counterparties,Profit and loss,Swap Spread,Fix T-bond Interest Rises,Swap Spread Narrow,Bond Value Decreases,Stay wit
22、h the position or Suffer a mark-to-market LOSS,Betting on the Widening,REPO,U.S.Treasury,BOND COLLATERAL,Investors,FUND,FUND,BOND,LTCM,FIX T-BOND INTEREST,LIBOR,FIXED INTEREST,Counterparties,Profit and loss,Swap Spread:Example,DATA:,Swap Spread:Example,Here we consider the swap-spread risk,so we wil
23、l consider the to fixed cash flows duration,we just consider two factors influence:The swap spread The interest of Treasury bond.,Cash flow of repo:,Cash flow of swap:,6.77%,6.94%,Libor-20bp,Libor,Swap Spread:Example,Valuation of repo,Swap Spread:Example,Valuation of swap,The term structure curve,Sw
24、ap Spread:Example,If we assume:1,The T-bonds yield curve is flat r.2,ss is also flat ss.,The definition of swap spread:,We use discrete calculation to calculate the result,so we have,Swap Spread:Example,Value of repo:,Value of swap:,So the value of the portfolio:,Swap Spread:Example,Only from mathem
25、atic analysis,we can get the following conclusion:,Swap Spread:Example,The parameters are:,We assume that T=20 years,then we get that:,From the notes of the case file,we assume T=20 is only a kind of approximation,and it works well.,Swap Spread:Example,At the beginning we assume that we the principa
26、l is$100,so we can get the following conclusions:,If ss(swap spread)changes 1bp,that will make the value of the portfolio change about 0.144%of the nominal principalIf the interest of the Treasure bond changes 1bp,the value of the portfolio will only change about 0.0014%,nearly zero.,From the case w
27、e know LTCM built the position to an exposure of$5 million per basis spread,the nominal principal size was about$5 billion,that was about 0.1%=5 million/5 billion.LTCM mainly considered the risk of SWAP SPREAD,but not the interest risk.,If LTCM holds a position whose nominal size is$1 billion,that m
28、eans if the swap spread increase 1bp,the portfolio will increase about$1.44 million.While if the T-bonds interest increases 1bp,the portfolio will decrease only about$14 thousand.,SAMPLE TRADE 4:Relative Value Strategy,Yield Curve Relative Value Strategy,Relative Value Strategy,The yield curve is co
29、ncave in the middle terms.As a normal yield curve the curve should slope up,when you find such a curve shape,you can do arbitrage following the below strategy-Butterfly Strategy:,Preferences of specific maturities.,Pay fixed-rate in 3 year swapsReceiving fixed-rate in 7 year swapsPay fixed-rate in 1
30、0 year swaps.,Libor,Libor,Libor,Fixed,Fixed,Fixed,Relative Value Strategy,If we assume that the principal of the 3 securities are all$1,and the coupon rate is 10%,then we can calculate the sensitive factor of different security.We get the following result:,If the yield curve moves parallel,we can go
31、t the above 2 equations.We need one more equation,The portfolios duration should be zero.,Relative Value Strategy,If the yield curve rotates around the point(0,spot),If we assume as before when the YC rotates,we should calculate the duration from a totally new perspective:When the curve turns a very
32、 small angle,we can conclude that the interest-change should be positive proportional to the Time.The cash follow will change now,we should first decide the cash flow.,Relative Value Strategy,Assume that the portfolio size is one dollar,and we spend dollars on each security.,Then we know that we can
33、 buy,Cash Flow Table:(for each shares investment),How can we get this table?The numbers are got from the following formula,In each row end,we should consider the principal.,Relative Value Strategy,Then we can get:,That is:,This should be zero,so we get the third equation.,Relative Value Strategy,Sub
34、stitute all the data,we get:,Also we know:,Relative Value Strategy,The result is,Until now we have analyze the portfolios sensitive,we build the positions that will not change when the Yield Curve moves,parallel or rotate.Because the yield curve in a 2-dimension space,it can only move parallel or ro
35、tate.So we know that if the curves shape do not change,the portfolio will not change,only the curve moves relatively-that is the curves shape changes,the portfolios value will change,so we call it Relative-Value Strategy!,Relative Value Strategy,Uniform Analysis Frame,Uniform Analysis Frame,In fact
36、the idea has been used in the management of the asset and liability,in Reitano s thesis Non-parallel Yield Curve Shifts and Immunization(Journal of Portfolio Management 1992),he discussed the immunization strategy in the portfolio management so that the portfolios value will not change when the yiel
37、d curve shifts nonparallel.Here we also used this kind of idea to do arbitrage,when the yield curve shifts,we make sure that the value do not change,but and only when the yield curve changes its shape,then we gain or loss.,Uniform Analysis Frame,If you find that there are n factors will influence yo
38、ur portfolios value,that means:,Now we choose n securities:,Each security,we hold omega i shares,If we want to use the immunization strategy,we should promise that:,Immunization Matrix,Uniform Analysis Frame,So we can solve the above equation,to get the immunization strategy.But our purpose is to do
39、 arbitrage,so we will find that we do not want to use this immunization strategy to hedge all the risk.So we can have the following relax-equation.,Uniform Analysis Frame:Look back!,How to use our model to analyze the relative strategy?,The relative strategy should be a three factors model:1.The par
40、allel movement.Delta r2.The rotating angle.-Delta Theta3.The curvature of the curve.CurvatureThis is the relax-variable!,Uniform Analysis Frame:Look back!,Now we only have n-1 immunization equations,and n variables,so there is one variable is free!And we have another equation:So we can solve it!1.Th
41、e portfolio is also immune to the n-1 factors.2.The factor j will influence the portfolios value.3.If the factor j goes as our judgment,then we will lose,This model can also be used in the Swap Spread strategy,and other multi-factors arbitrage model.,Uniform Analysis Frame,How to extend your model t
42、o a dynamic model-consider the Time.How to add the influence of trading cost-it might be very practicalHow to consider a continual model-very interesting in theoryWhether the equations have solution?Under what condition the solutions do not exit?Whether can the extend-immunization matrix be diagonal
43、?Why?,A Recent Empirical Case,From American Markets Prove,A Recent Empirical Case,We prepare some data got from American market to prove the strategies illustrated are efficient.Data date:5/19/2008Data resource:,US Treasury Bonds Rates,Municipal Bonds,Corporate Bonds,US Treasury Bonds Rates,Black:On
44、e month agoGreen:One week agoRed:One day agoBlue:Today,Municipal Bonds,Municipal Bonds:Notes,First row:aaaaa;Column:Different time point;Second row:aaaa;Column:Different time point;Third row:aaa;Column:Different time point;Time Point:2y,5y,10y,20y;Red line has a higher credit quality;Green one is lo
45、wer.,Corporate Bonds(No AAA),AAA;Red is AA,Green is A,Portfolio Risk Management,How did the Fund manage its overall portfolio risk?What is liquidity?How did LTCM manage its liquidity risk?,Portfolio Risk Management,100%financing and long-short structure implications for how the firm should think abo
46、ut risk management:No explicit equity investment and it is impossible to work directly with measures such as return on equity;Risk could not be measured by the notional sizes of the positions.In particular,the risk of a long-short position depended entirely on the degree to which the profits on the
47、long position could deviate from the profits on the short position.,Portfolio Risk Management,Value-at-risk MeasuresPricing Discrepancies BeliefEconomic Stress TestingPositions CorrelationLong term vs Short term RiskRisk Estimation,Portfolio Risk Management,Value-at-risk(VaR)MeasuresVaR can be defin
48、ed as the worst loss that can happen under normal market conditions over a specified horizon at a specified confidence level.More formally,VaR measures the shortfall from the quantile of the distribution of trading revenues.As before VaR is interpreted as the largest acceptable loss the bank is will
49、ing to suffer over a specified period.To cover this loss,the bank must maintain adequate equity capital.In other words,VaR is the amount of capital a firm allocated to self-insurance.,Portfolio Risk Management,Equity Capital as a VaR measureVaR Example1:swap-spread tradeNotional position:$5 billionG
50、ain or loss 5 million for 1 b.p.spread changeExpected:23 b.p.widen in swap spread with a standard deviation of 7 b.p.Results:VaR=$25 million,standard deviation=$35 million.The amount of equity capital that needs to be set aside to cover most of the potential losses is$25 million.,Portfolio Risk Mana
