Investment risk management
Traditional and alternative products
Luis A. Seco
Sigma Analysis & Management
University of Toronto
RiskLab
Slide 1
A hedge fund example
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
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 7
A hedge fund example
Slide 8
The snow swap
Snow
City
$10M
No snow
© Luis Seco. Not to be distributed
without permission.
Resort
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.
So we create do the snow investment…
… 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.
Why you need to be smart
© Luis Seco. Not to be distributed
without permission.
Hedge Fund: definition
An investment partnership that exploits
investment opportunities
Unregulated
Bound to an Offering Memorandum
Seeks returns independent of market
movements
Reports NAV monthly
Charges Fees: 1-20
Slide 14
The investment structure
Investor 1
Investor 2
Investor 3
Investor 4
Investor n
The Fund
legal structure
The Bank
Prime Broker
The Administrator
The Management company
“the hedge fund”
Slide 15
Risks per strategy
Slide 16
Slide 17
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without permission.
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. ZachariasLanghans).
The Galmer Arbitrage GT
Slide 18
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 19
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 20
Scenario 1
Values of shares and bonds are unchanged:
Bonds
Stock
T-Bill
Coupon
Fee
Total
Today
80
-70
+70
$80
1 yr later
80
-70
+72.8
4
-3.5
$83.3
Slide 21
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
$86.3
$83.3
Total
$80
Slide 22
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
$84.3
$81.3
$78.3
Total
$80
Slide 23
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 24
Long-short equity
William Holbrook Beard (1824-1900)
Slide 25
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
Assets at Prime Broker
Assets at Prime Broker
(Before trade)
(After trade)
(After one year)
• $1000
•  $1000
•  $1000
•  -$900 + 9 A
•  990
•  +$900 – 9 B
•  -945
•  -9
$ 1036
Slide 26
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
Assets at Prime Broker
Assets at Prime Broker
(Before trade)
(After trade)
(After one year)
• $500
•  $500
•  $500
•  -$900 + 9 A
•  990
•  +$900 – 9 B
•  -945
•  -9
$ 536
Slide 27
A long-short pair trade (v3)
Assumptions: 50% collateral for long trades, 80%
collateral for short trades.
Securities at Prime Broker
Securities at Prime Broker
•  9 A ($900):
•  9 A ($990):
•  – 9 B (-$900):
•  – 9 B (-$945):
Collateral required:
Profit: $36
$450+$720=$1170
Cash from short sale: $900
Cash required: $270
Slide 28
Risk and Performance Measurement
Slide 29
Measurement
Return:
–  from track records
Risks:
–  Volatility
–  Operational risk: due diligence
–  Business risk
–  Exposures to market factors
Slide 30
Return
Starting from share value observations Si on a
monthly basis, we define the return as
Simple Returns: Ri = (Si - Si-1)/Si-1
Log Returns Ri = ln(Si/Si-1)
Slide 31
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without permission
TWR and IRR
Over a period of time, the time-weighted-rate of
return is defined by
1+TWR = (1+R1)(1+R2)… (1+Rk)
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 Ni denote the cashflows at month i.
Slide 32
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Return statistics
Heaven (Probability)
Earth (Statistics)
Assumes a probability distribution
Derives distributions from history
Assumes total knowledge
Only knows the past
Expressed with mathematical formulas
Implementable on a computer
Slide 33
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The portfolio distribution function (CDF)
90% probability that annual
returns are less than 3%
7% probability that annual
losses exceed 5%
Slide 34
Probability density: histogram
Slide 35
Mean 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
Slide 36
following example shows.
Mean return estimators
Heaven (Probability)
Earth (Statistics)
Usually measured monthly, and reported annually
Slide 37
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without permission
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 38
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 39
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 40
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 41
Volatility
Heaven (Probability)
Earth (Statistics)
1 n
µˆ = r = ∑ ri
n i=1
σ=
1 n
(ri − µ) 2
∑
n i=1
s=
1 n
(ri − r ) 2
∑
n −1 i=1
€
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without permission
Population s.d.
Sample s.d.
Slide 42
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 43
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 44
Fund-of-Fund Risk: volatility
Volatility of a portfolio with weights w
Slide 45
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without permission
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 46
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 47
The efficient frontier
Efficient
Portfolios
Risk
Feasible
Region
Return
Slide 48
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 49
Optimal Sharpe Ratio Portfolios
The objective function to maximize is
Probability of
meeting the
benchmark
Cummulative
distribution function
of the gaussian
Since φ is increasing, our optimization problem
becomes that of maximizing
Slide 50
Sharpe vs. Markowitz
Slide 51
Benchmark-based investments
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 52
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 53
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 54
Non-normal returns
Slide 55
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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 56
Semi-standard deviation formula
Target return / benchmark
Gains give a value ot 0
Slide 57
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 58
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 59
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 60
Slide 61
Slide 62
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 63
Skewness is useless
Slide 64
Uselessness of skewness
Slide 65
The Omega
Slide 66
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 67
Omega: Shadwick, Keating (2002)
Slide 68
Wins vs. losses: the Omega
Omega tries to capture
tail behavior
avoiding moments,
using the relative
proportion of wins
over losses:
Slide 69
Wins vs. losses: the Omega
Omega tries to capture
tail behavior
avoiding moments,
using the relative
proportion of wins
over losses:
Truncated First Moments
Slide 70
The Omega of a heavy tailed distribution
Slide 71
Correlation risk
Slide 72
Hedge fund diversification
Hedge funds are
uncorrelated to
traditional markets,
and internally
uncorrelated also.
Correlation
histogram for
Dow stocks
Correlation
histogram for
hedge funds
Slide 73
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 74
Hedge Fund Correlation histogram
Slide 75
Normal correlations
Slide 76
Distressed correlations
Slide 77
Correlation switching
Slide 78
Distress analysis
Slide 79
Correlation switching
Slide 80
Correlation risk
We will deal with correlation sensitivity from a
mixtures of multivariate gaussian approach
Its density is given by:
Slide 81
GM in pictures
Slide 82
Other risks
Backfill bias
Survivorship bias
Liquidity risk
Style risk
Legal risk
Non-linear effects: option writing.
Slide 83
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 84
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 85
Hedge Fund Indices
Investable
Non-investable
Slide 86
Historical comparative analysis
Pro-Forma
Slide 87
Correlation analysis
Slide 88
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 89
Anatomy of a guarantee
Guarantees principal in
the future:
How much is needed is determined
by
Obtains exposure to
the Hedge Funds
• Interest rates
• Maturity date of the note
Slide 90
Leveraged structures
Loans
Options
CPPI
Slide 91
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 92
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 93
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 94
Collateralized Fund Obligation (CFO)
Equity
Investor
Bank
Bond
Investor (1)
Bond
Investor (2)
Fund Pool
Bond
Investor (3)
Slide 95
A $500M CFO
Slide 96
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 97
S&P CTA CFO. A case study.
Slide 98
Blow-up risk
Slide 99
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without permission
The Merton model of default
Slide 100
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without permission
Rating and Due Diligence
Slide 101