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Slide 1
Investment risk management Traditional and alternative products
Luis A. Seco Sigma Analysis & Management
University of Toronto RiskLab
Slide 2
A hedge fund example
Slide 3
A hedge fund example
Slide 4
A hedge fund example
Slide 5
A hedge fund example
Slide 6
A hedge fund example
Slide 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.
Slide 8
A hedge fund example
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. Say the “insurance claim” is $1M. The fund would charge 20%
commission, but assume to take 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%
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.
Slide 11
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 Charges Fees: 1-20
Slide 12
The investment structure
The Management company “the hedge fund”
The Fund legal structure
The Bank Prime Broker The Administrator
Investor 1 Investor 2 Investor 3 Investor 4 Investor n
Slide 13
Risks per strategy
© Luis Seco. Not for dissemination without permission.
Slide 14
Slide 15
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 16
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 17
An convertible arbitrage strategy example
Consider a bond selling below par, at $80.00. 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 18
Scenario 1
Values of shares and bonds are unchanged:
Today 1 yr later Bonds 80 80 Stock -70 -70 T-Bill +70 +72.8 Coupon 4 Fee -3.5 Total $80 $83.3
Slide 19
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 80 73 70 Stock -70 -60 -60 T-Bill +70 +72.8 72.8 Coupon 4 4 Fee -3.5 -3.5 Total $80 $86.3 $83.3
Slide 20
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 80 91 88 85 Stock -70 -80 -80 -80 T-Bill +70 +72.8 +72.8 +72.8 Coupon 4 4 4 Fee -3.5 -3.5 -3.5 Total $80 $84.3 $81.3 $78.3
Slide 21
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): 10-5+4=9%
Standard deviation:
Slide 22
Long-short equity
William Holbrook Beard (1824-1900)
Slide 23
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
• -$900 + 9 A
• +$900 – 9 B
Assets at Prime Broker
(After one year)
• $1000
• 990
• -945
• -9
$ 1036
Slide 24
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
• -$900 + 9 A
• +$900 – 9 B
Assets at Prime Broker
(After one year)
• $500
• 990
• -945
• -9
$ 536
Slide 25
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):
Collateral required:
$450+$720=$1170
Cash from short sale: $900
Cash required: $270
Securities at Prime Broker
• 9 A ($990):
• – 9 B (-$945):
Profit: $36
Slide 26
Hedge Fund Correlation histogram
Slide 27
Risk and Performance Measurement
Slide 28
Measurement
Return: – from track records
Risks: – Volatility – Operational risk: due diligence – Business risk – Exposures to market factors
Slide 29
Sample Hedge Fund report
Slide 30
Data Issues (discussion)
www.hedgefundresearch.com www.hedgefund.net www.hedgefund-index.com www.barclaygrp.com www.eurekahedge.com sigma2.fields.utoronto.ca
Slide 31
The portfolio distribution function (CDF)
90% probability that annual returns are less than 3%
7% probability that annual losses exceed 5%
Slide 32
Probability density: histogram
Slide 33
Return
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 ri, where the subindex i denotes every consecutive month, the average monthly return is given by
Because returns are expressed in percentages, one has to be careful, as the following example shows.
Slide 34
Returns: careful.
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 35
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 36
Slide 37
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 38
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 39
Slide 40
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 41
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 42
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 43
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 Si is assumed a gaussian return profile, with mean µi, and standard deviation given by σi. 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 44
The efficient frontier
Risk
Return
Feasible
Region
EfficientPortfolios
Slide 45
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 46
Sharpe’s ratio: basic fact
Imagine one is looking for the portfolio that has the best chance of optimizing its performance against a benchmark given by LIBOR. That portfolio is the one with the highest Sharpe ratio, as defined in the previous paragraph.
Slide 47
Sharpe Ratio
The objective function to maximize is
Since φ is increasing, our optimization problem becomes that of maximizing
Probabilityofmeetingthebenchmark
Cummulativedistributionfunctionofthegaussian
Slide 48
Sharpe vs. Markowitz
Slide 49
Benchmarks
They are reference portfolios against which performance of other portfolios are measured:
Bonuses are paid on benchmark-based performance.
They can be constant or random
Slide 50
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 51
Alpha and beta
Consider a portfolio with returns given by
and a benchmark with returns given by.
Find the linear regression coefficients α, β, such that,
with ε with mean 0 and lowest standard deviation.
Slide 52
VaR and risk budgeting
Assume a portfolio represented by a vector θ which represents the percentage allocated to specific managers or investment instruments.
Each manager or security Si is assumed a gaussian return profile, with mean µi, and standard deviation given by σi. Correlations are given by a variance/covariance matrix V.
VaR and portfolio standard deviation are related to the fundamental expression
Slide 53
Risk budgeting
The previous expression allows us to do a risk allocation to each manager
in such a way that the overall risk of the portfolio is given by
This expression is useful when allocating risk or risk limits to each of the investments in a certain universe.
Slide 54
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.
© Luis Seco. Not to be reproduced without permission
Slide 55
Non-normal returns
Slide 56
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 57
Semi-standard deviation formula
Target return / benchmark
Gains give a value ot 0
Slide 58
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 59
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 60
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 61
Slide 62
Slide 63
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 64
Skewness is useless
Slide 65
Uselessness of skewness
Slide 66
L-moments
Slide 67
The Omega
Slide 68
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.
Omega: Shadwick, Keating (2002)
Slide 69
Slide 70
Wins vs. losses: the Omega
Omega tries to capture tail behavior avoiding moments, using the relative proportion of wins over losses:
Slide 71
Wins vs. losses: the Omega
Omega tries to capture tail behavior avoiding moments, using the relative proportion of wins over losses:
TruncatedFirstMoments
Slide 72
The Omega of a heavy tailed distribution
Correlation risk
Slide 73
Slide 74
Hedge fund diversification
Hedge funds are uncorrelated to traditional markets, and internally uncorrelated also.
CorrelationhistogramforDowstocks
Correlationhistogramforhedgefunds
Slide 75
Fact.
Hedge funds are uncorrelated to traditional markets, so they constitute excellent diversification strategies.
Yes, ... and no! Many hedge funds are indeed
uncorrelated to markets, but others are very correlated to simple portfolios of traditional markets, so they add little diversification.
Even those funds which exhibit low correlation to markets and macroeconomic factors, when combined into portfolios, they can be highly correlated to the market.
Slide 76
Normal correlations
Slide 77
Distressed correlations
Slide 78
Correlation switching
Slide 79
Distress analysis
Slide 80
Correlation switching
Slide 81
Correlation risk
We will deal with correlation sensitivity from a mixtures of multivariate gaussian approach
Its density is given by:
Slide 82
GM in pictures
Slide 83
Non-gaussian portfolio theory
Each portfolio is described by four performance numbers: mean and standard deviation, each under normal and distressed market assumptions. They are given by
and
Slide 84
Benchmark satisfaction
The objective function to maximize was
It is possible to have portfolios which are efficient from this point of view, which however are not efficient under either normal or distressed conditions.
Increasingfunctions
Slide 85
Other risks
Backfill bias Survivorship bias Liquidity risk Style risk Legal risk Non-linear effects: option writing.
Slide 86
Hedge Fund Products
Fund-of-funds: Indices Options on fund-of-funds Warrants Non-recourse loans with fund collaterals CPPI (Constant proportion portfolio
insurance) CFO’s
Slide 87
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 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.
Slide 88
Hedge Fund Indices
Investable Non-investable
Slide 89
Historical comparative analysis
Pro-Forma
Slide 90
Correlation analysis
Slide 91
Guaranteed notes
There are two main reasons for a guarantee: – Regulatory environments – Risk perceptions (not to confuse with risk appetite)
Some guarantees are provided by well-rated banks. Others are not (Portus).
Guarantees are obtainable by setting aside an interest-earning portion of the assets, and investing the remainder at higher levels of leverage, through a variety of different instruments.
Slide 92
Anatomy of a guarantee Guaranteesprincipalin
thefuture:Howmuchisneededisdeterminedby
• Interestrates
• Maturitydateofthenote
ObtainsexposuretotheHedgeFunds
Slide 93
The cost of the guarantee
About2%peryearcost
An underlying hedge fund portfolio that produces 6bps/month
Interest rates at 25bps per month A 5 year note that guarantees principal No management or performance fees
Leveraged structures
Loans Options
CPPI
Slide 94
Slide 95
Non-recourse loans
The bank lends to the investor and takes the investment in the hedge fund portfolio as collateral.
In a low interest rate environment, it allows investors to amplify good hedge fund performance. In high interest rate environments, if hedge fund performance is poor, they can lead to sustained losses.
It allows small investors to increase the asset base and diversify the portfolio better; it makes it easier to satisfy the minimum investment requirements of individual hedge funds.
The structurer may demand liquidation if performance drops below a certain floor.
Slide 96
Options
Options are delta-hedged; the liquidity of the underlying hedge fund portfolio contributes to a volatility spread.
They are hard to delta-hedge due to the low liquidity of the underlying portfolio. Implied volatilities will be much higher than historical volatilities.
They are path-independent. They are also insensitive to changes in interest rates.
Slide 97
CPPI
Investor provide equity to a fund; the structurer provides leverage Proceeds are invested in a reference portfolio If the performance of the reference portfolio is
below a reference curve, the strike price is increased.
If performance of the reference portfolio is above another reference curve, the strike price is decreased
Slide 98
CPPI options
Slide 99
Bank
Bond Investor (1)
Bond Investor (2)
Bond Investor (3)
Equity Investor
Fund Pool
Collateralized Fund Obligation (CFO)
Slide 100
A $500M CFO
Slide 101
CFO’s
Advantages Equity investors find a way
to obtain leverage. Debt holders find an
uncorrelated asset class to invest in.
Tranches can be packaged by volume and credit rating.
Disadvantages Hard to value Very dependent on
correlations amongst the funds constituents
Expensive structuring fees makes it difficult to find the equity investor sometimes.
Slide 102
S&P CTA CFO. A case study.
© Luis Seco. Not to be reproduced without permission
Slide 103
Blow-up risk
© Luis Seco. Not to be reproduced without permission
Slide 104
The Merton model of default
Slide 105
A double-layer rating system
A B C
A Infrequent, small losses
Frequent, small losses
B
C Infrequent, large losses
Large, probably losses
Slide 106
Rating and Due Diligence