Part 1: Review of Hedge Funds

Transcription

1 Part 1: Review of Hedge Funds and structured products Luis A. Seco Sigma Analysis & Management University of Toronto RiskLab Luis Seco. Not to be distributed without permission.

2 A hedge fund example Luis Seco. Not to be distributed without permission.

3 A hedge fund example Luis Seco. Not to be distributed without permission.

4 A hedge fund example Luis Seco. Not to be distributed without permission.

5 A hedge fund example Luis Seco. Not to be distributed without permission.

6 A hedge fund example Luis Seco. Not to be distributed without permission.

7 The snow swap! Track the snow precipitation in late fall and early spring;! If the precipitation is high, the ski resort pays to the City of Montreal a prescribed amount.! If the precipitation is low, the City pays the resort another pre-determined amount.! The dealer keeps a percentage of the cash flows. Luis Seco. Not to be distributed without permission.

8 A hedge fund example Luis Seco. Not to be distributed without permission.

9 The snow swap Snow City $10M Resort No snow Luis Seco. Not to be distributed without permission.

10 The snow fund! Modify the snow swap so the City pays when precipitation is low in the city, and the resort pays when precipitation is high in the resort.! The fund takes the spread risk, and earns a fee for the risk.! Consider an atomic insurance claim of $1M. The fund would charge 20% commission, but assume the spread risk.! Setting aside $2M, and charging $200K, the fund could Lose nothing: 75% Make $2M: 12.5% Lose $2M: 12.5%! Expected return=10%. Std=50% Luis Seco. Not to be distributed without permission.

11 A diversified fund: a hedge fund.! If we do the swap across 100 Canadian cities:! Expected return:10%! Std: 5%.! Better than investing in the stock market. Luis Seco. Not to be distributed without permission.

12 So we create the snow fund with some of the best known ski resorts:! Blue Mountain (Toronto)! Mountain Creek (New Jersey)! Panorama Mountain Village (Calgary)! Snowshoe Mountain (West Virginia)! Steamboat Ski Resort (Hayden, Denver)! Stratton Mountain Resort (Vermont)! Tremblant (Montreal)! Whistler Blackcomb (Vancouver) and then: Luis Seco. Not to be distributed without permission.

13 Intrawest goes Bankrupt Luis Seco. Not to be distributed without permission.

14 Hedge Fund: definition! An investment partnership; seeks return niches by taking risks, which they may hedge or diversify away (or not).! Unregulated! Bound to an Offering Memorandum! Seeks returns independent of market movements! Reports NAV monthly Can give rise to valuation issues.! They can be illiquid. Lock-ups. Redemption restrictions.! Capacity restrictions.! Charges Fees: 1-20 Luis Seco. Not to be distributed without permission.

15 The investment structure Investor 1 Investor 2 Investor 3 Investor 4 Investor n The Fund legal structure The Administrator The Bank Prime Broker The Management company the hedge fund Luis Seco. Not to be distributed without permission.

16 Hedge Fund Fees! The management company charges about 1-2% of the NAV. The fund issues payments to the management company usually monthly or quarterly.! The management company usually charges about 20% of the net gains to the fund for a given period, usually a year. The payment occurs at year-end, usually.! Some hedge funds only receive a performance fee if their return exceeds a certain objective (hurdle rate). This can be fixed (say 7%) or variable (LIBOR, for example) Luis Seco. Not to be distributed without permission.

17 Properties of hedge funds! They can be illiquid. Lock-ups. Redemption restrictions.! They give rise to valuation issues.! Capacity restrictions.! Vulnerable to attacks.! Legal risk.! etc Luis Seco. Not to be distributed without permission.

18 Share value! Starting from the Net Asset Value observations (NAV) of the Fund N i on a monthly basis, and the number of outstanding shares n i, we define the share value S i as S i = N i /n i Luis Seco. Not to be distributed without permission.

19 Investments w/o hedge funds +10% -6% Luis Seco. Not to be distributed without permission.

20 Hedge fund diversification! Hedge funds are uncorrelated to traditional markets, and internally uncorrelated also. Luis Seco. Not to be distributed without permission.

21 Databases Luis Seco. Not to be distributed without permission.

22 Hedge Fund Information! Hedge funds are private partnerships, and hence have no obligation to report except to their own investors.! Moreover, publication can be considered illegal marketing.! But databases exist: Luis Seco. Not to be distributed without permission.

23 Database issues! Include funds with certain characteristics! Many hedge funds do not want to report into them Good funds with ample assets do not want to be subject to database requirements. Closed funds have no incentive to report.! Backfill bias: Hedge funds select when they enter the database! Survivorship bias: defunct funds do not appear in databases. Luis Seco. Not to be distributed without permission.

24 Sample Hedge Fund report Luis Seco. Not to be distributed without permission.

25 Sample HF report part 2 Luis Seco. Not to be distributed without permission.

26 Hedge Fund Indices Luis Seco. Not to be distributed without permission.

27 Hedge Fund indices! They offer fund-of-fund investments that try to track the performance of the hedge fund sector (global and style specific) investing in liquid funds with high capacity.! The result is often a fund that tracks nothing and lags performance.! In contrast with equity indices, investors in a fund don t like it when their fund is included in an index. Luis Seco. Not to be distributed without permission.

28 Hedge Fund Indices! Investable! Non-investable Luis Seco. Not to be distributed without permission.

29 Style correlations Luis Seco. Not to be distributed without permission.

30 X/I correlations Luis Seco. Not to be distributed without permission.

31 Convertible arbitrage Fig. 1: A graphical analysis of a convertible bond. The different colors indicate different exercise strategies of call and put options. Risk management for financial institutions (S. Jaschke, O. Reiß, J. Schoenmakers, V. Spokoiny, J.-H. Zacharias- Langhans). The Galmer Arbitrage GT Slide 31

32 Convertible arbitrage! The convertible arbitrage strategy uses convertible bonds.! Hedge: shorting the underlying common stock.! Quantitative valuations are overlaid with credit and fundamental analysis to further reduce risk and increase potential returns.! Growth companies with volatile stocks, paying little or no dividend, with stable to improving credits and below investment grade bond ratings. Slide 32

33 An convertible arbitrage strategy example! Consider a bond selling below par, at $ It has a coupon of $4.00, a maturity date in ten years, and a conversion feature of 10 common shares prior to maturity. The current market price per share is $7.00.! The client supplies the $80.00 to the investment manager, who purchases the bond, and immediately borrows ten common shares from a financial institution (at a yearly cost of 1% of the current market value of the shares), sells these shares for $70.00, and invests the $70.00 in T-bills, which yield 4% per year. The cost of selling these common shares and buying them back again after one year is also 1% of the current market value. Slide 33

34 Scenario 1 Values of shares and bonds are unchanged: Today 1 yr later Bonds Stock T-Bill Coupon 4 Fee -3.5 Total $80 $83.3 Slide 34

35 Scenario set 2 In the next two examples, the share price has dropped to $6.00, and the bond price has dropped to either $73.00 or $70.00, depending on the reason for the drop in share market values. The net gain to the client is 7.87% and 4.12% respectively, again after deducting costs and fees. Today 1 yr later (a) 1 yr later (b) Bonds Stock T-Bill Coupon 4 4 Fee Total $80 $86.3 $83.3 Slide 35

36 Scenario set 3 In the following three examples, the share price increased to $8.00, and the bond price increased either to $91.00, $88.00 or $85.00, depending on the expectations of investors, keeping in mind that we have one less year to maturity. The net gain to the client is 5.37% and 1% in the first two examples, with an unlikely net loss of 2.12% in the last example. Today 1 yr later(a) 1 yr later(b) 1 yr later(c) Bonds Stock T-Bill Coupon Fee Total $80 $84.3 $81.3 $78.3 Slide 36

37 A Risk Calculation: normal returns If returns are normal, assume the following: Bond mean return: 10% Equity mean return: 5% Libor: 4% Bond/equity covariance matrix (50% correlation):! Mean return (gross): =9%! Standard deviation: Slide 37

38 Long-short equity William Holbrook Beard ( ) Slide 38

39 A long-short pair trade! The fund has $1000. The manager is going to purchase stock 9 units of stock A, and sell-short 9 units of stock B. Both are valued at $100 each. After a year, A is worth $110, B is $105. Assets at Prime Broker (Before trade) $1000 Assets at Prime Broker (After trade) $1000 -$ A +$900 9 B Assets at Prime Broker (After one year) $ $ 1036 Slide 39

40 A long-short pair trade (v2)! The fund has $500. The manager is going to purchase stock 9 units of stock A, and sell-short 9 units of stock B. Both are valued at $100 each. After a year, A is worth $110, B is $105. Assets at Prime Broker (Before trade) $500 Assets at Prime Broker (After trade) $500 -$ A +$900 9 B Assets at Prime Broker (After one year) $ $ 536 Slide 40

41 A long-short pair trade (v3)! Assumptions: 50% collateral for long trades, 80% collateral for short trades. Securities at Prime Broker 9 A ($900): 9 B (-$900): Securities at Prime Broker 9 A ($990): 9 B (-$945): Collateral required: $450+$720=$1170 Cash from short sale: $900 Cash required: $270 Profit: $36 Slide 41

42 Hedge Fund Correlation histogram Slide 42

43 Performance and risk measures Luis Seco. Not to be distributed without permission.

44 Return! Starting from share value observations S i on a monthly basis, we define the return as! Simple Returns: R i = (S i - S i-1 )/S i-1! Log Returns R i = ln(s i /S i-1 ) Luis Seco. Not to be reproduced without permission Slide 44

45 TWR and IRR! Over a period of time, the time-weighted-rate of return is defined by 1+TWR = (1+R 1 )(1+R 2 ) (1+R k )! Over the same period of time, the Internal Rate of Return is defined as IRR=(1+R) n where the number R is defined as and N i denote the cashflows at month i. Luis Seco. Not to be reproduced without permission Slide 45

46 Return statistics! Average return is usually measured on a monthly basis, and quoted on an annualized basis.! If the series of monthly returns (in percentages) is given by numbers r i, where the sub-index i denotes every consecutive month, the average monthly return is given by Slide 46

47 Expected Return: Heavenly version! For a discrete random variable with P[R=x i ]=p i! For a continuous distribution with probability density p(x): Luis Seco. Not to be reproduced without permission Slide 47

48 Statistics vs. Accounting Imagine a hedge fund with a monthly NAV given by $1, $2, $1, $2, $1, $2, etc. The monthly return series is given by 100%, -50%, 100%, -50%, 100%, -50%, etc. Its average return (say, after one year) is 25% monthly, or an annualized return in excess of 300%. Slide 48

49 Returns: from monthly to annual There is no standard method of quoting annualized returns: One possibility is multiplying returns by 12 (annual return with monthly compounding) Another, is to annualize using the formula Slide 49

50 Portfolio returns The big advantage of return, is that the return of a portfolio is the average of the returns of its constituents. More precisely, if a portfolio has investments with returns given by with percentage allocations given by then, the return of the portfolio is given by Slide 50

51 Volatility! Like returns, volatility is usually measured on a monthly basis, and quoted on an annual basis.! If the series of monthly returns (in percentages) is given by numbers ri, where the subindex i denotes every consecutive month, the monthly volatility is given by Slide 51

52 Earthly versions: Sample Mean Population s.d. Sample s.d. Standard deviation has the same units as the data. Luis Seco. Not to be reproduced without permission Slide 52

53 Variance: Heavenly Version Luis Seco. Not to be reproduced without permission Slide 53

54 Slide 54

55 Slide 55

56 Covariances and correlations! They measure the joint dependence of uncertain returns. They are applied to pairs of investments.! If two investments have monthly return series given by numbers ri and si respectively, where the subindex i denotes every consecutive month, and their average returns are given by r and s, their covariance is given by! If they have volatilities given respectively by! Then, their correlation is given by Slide 56

57 Covariance: Heavenly Version Luis Seco. Not to be reproduced without permission Slide 57

58 Covariance and correlation matrices Because correlations and covariances are expressed in terms of pairs of investments, they are usually arranged in matrix form. If we are given a collection of investments, indexed by i, then the matrix will have the form Slide 58

59 Portfolio Optimization: Markowitz Markowitz optimization allows investors to construct portfolios with optimal risk/return characteristics. Risk is represented by the portfolio expected return Risk is represented by the standard deviation of returns. The optimization problem thus created is LQ, it is solved using standard techniques. Slide 59

60 Risk/return space A portfolio is represented by a vector θ which represents the number of units it holds in a vector of securities given by S. Each security S i is assumed a gaussian return profile, with mean µ i, and standard deviation given by σ i. Correlations are given by a variance/covariance matrix V. The portfolio return is represented by its return mean and its risk is given by its standard deviation Slide 60

61 The efficient frontier Slide 61

62 Sharpe s ratio A way to bring return and risk into one number is by the information ratio, and by the Sharpe s ratio. If a certain investment has a return given by r, and a volatility given by σ, then the information ratio is given by r/ σ. If interest rates are given by i, then Sharpe s ratio is given by (r-i)/ σ. It measures the average excess return per unit of risk. Portfolios with higher Sharpe s ratios are usually better. Slide 62

63 Sharpe Ratio The objective function to maximize is Since φ is increasing, our optimization problem becomes that of maximizing Slide 63

64 Sharpe vs. Markowitz Slide 64

65 Tracking error! It is the standard deviation of the difference between the portfolio returns and the benchmark returns.! A performance indicator often times used in traditional investments is Slide 65

66 The normality assumption Under the normal assumption, a portfolio with a 1% standard deviation will have annual returns which will vary no more than 1%, up or down, from its expected return, with a 65% probability. If a higher degree of certainty about portfolio performance is desired, then one can say that the portfolio return will vary more than 2% from its expected return only 1% of the times. These probabilities are linear in the standard deviation; in other words, if the portfolio volatility is 3% (instead of 1% as in the example above), one will expect the returns to oscillate within a 6% band of its average return 99% of the time. Slide 66

67 Non-normal returns Luis Seco. Not to be reproduced without permission Slide 67

68 Gain/loss deviation It measures the deviation of portfolio returns from its expected return, taking into account only gains. In other words, portfolio losses are not taken into account with calculating the deviation. Loss deviation is the corresponding thing when losses only are taken into account in calculating portfolio deviations. Both of these are used when one is trying to get a feeling as to the asymmetry of the gain/loss distribution. They are not statistically conclusive amounts per se, like standard deviation is. Slide 68

69 Semi-standard deviation formula Target return / benchmark Gains give a value ot 0 Slide 69

70 Sortino ratio It is the substitute of the Sharpe ratio when one looks only at the loss deviation, instead of looking at the combined standard deviation. Many people believe that by not punishing unusual gains, like the Sharpe ratio does indirectly, one maximizes the upside while maintaining the downside. There is however no evidence that the Sortino ratio, as such, actually achieves this but it still remains to be a curious quantity to look at. Slide 70

71 Moments One of the criticisms of the use of volatilities and correlations as risk measures is the presence of extreme events in portfolio returns, which will go un-noticed in those calculations. From a certain viewpoint, volatilities and correlations are magnitudes inherited from normal distributions, according to which events such as the ones in 1987, 1995, 1998, etc. should have never occurred. One attempt to capture tail events is by introducing higher moments to measure large deviations: higher moments are defined as follows: Slide 71

72 Skew and kurtosis! Skew is a measure of asymmetry. It is the normalized third moment.! Kurtosis is a measure of spread. It is the fourth moment, minus 3. Platykurtotic: k<0 Leptokurtotic: k>0 Mesokurtotic: k=0. Slide 72

73 Slide 73

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75 Biased estimators! The estimator for the skewness and kurtosis introduced earlier is biased: Its expected value can even have the opposite sign from the true skewness (or kurtosis).! Intuitively speaking, the third and fourth powers are so large, that one or two events will dominate the value of the formula, making all other observations irrelevant.! Skew and kurtosis should not be used in critical situations Slide 75

76 Skewness is useless Slide 76

77 Uselessness of skewness Slide 77

78 L-moments Slide 78

79 The Omega Slide 79

80 Omega! Shadwick introduced the concept of Omega a few years ago, as the replacement of the Sharpe ratio when returns are not normally distributed.! His aim was to capture the fat tail behavior of fund returns.! Once the fat tail behavior has been captured, one then needs to optimize investment portfolios to maximize the upside, while controlling the downside. Slide 80

81 Omega: Shadwick, Keating (2002) Slide 81

82 Wins vs. losses: the Omega Omega tries to capture tail behavior avoiding moments, using the relative proportion of wins over losses: Slide 82

83 Wins vs. losses: the Omega Omega tries to capture tail behavior avoiding moments, using the relative proportion of wins over losses: Slide 83

84 The Omega of a heavy tailed distribution Slide 84