Luis A. Seco Sigma Analysis & Management University of Toronto RiskLab
Transcript of Luis A. Seco Sigma Analysis & Management University of Toronto RiskLab
Investments
Luis Seco University of Toronto
Sigma Analysis & Management Ltd.
Market inefficiencies• The city of Montreal spends over
one $100M each year cleaning snow
• The cleaning is done via public tender for a flat rate for selected months
• The monthly charge is costly, but snow cleaning outside the contract season is extremely expensive
2
Ski Resorts
Ski resorts have the opposite problem:
• Snow outside the normal season provides a good gains
• Late arrival of snow, or an early spring leads to substantial losses.
3
The snow swap
4
If there is no snow, the city pays the ski resort a fee:
$10M
If there is no snow, the ski resort pays the city a fee:
$10M
The city and the resort -in the end-
did not agree on where to measure snow precipitation
The deal died 😩😩
The business of Risk transfer
5
$10M payment
ski resort d
ependent
The Snow Fund
$10M payment
city dependent
For $2M fee (10% of the cashflows)
Fund cash flowsAssume correlation of 50% between city and resort precipitation. Then
• with 75% probability, both swaps yield opposite flows, and we just collect our fee: $2M.
• With 12.5% probability, we receive payments from both: $22M.
• With 12.5% probability, we have to pay both: -$18M
Investment parameters:
Invested amount: $20M Average return: 10%
Volatility: 50%
Sharpe ratio = 0.2
A bad deal
6
Consider a portfolio ⇧ that allocates wi to assets with returns given by randomvariables Xi, i = 1, . . . , n.If the covariance matrix of Xi is given by V = (�i,j , the portfolio volatility is
�2⇧ =
*nX
i=1
wi ·Xi ,nX
j=1
wi ·Xj
+
=X
i,j
wi wj hXi , Xji
=X
i,j
wi wj �i,j
= w · V · wT .
Therefore,�⇧ =
pw · V · wT .
We seek a portfolio ⇧ that maximizes the probability of exceeding a fixed bench-mark r.If returns are normally distributed:
Prob {⇧ � r} = Prob
⇢⇧� µ
�� r � µ
�
�
= 1� �
✓r � µ
�
◆
µ� r
�
�2 = 0.75 ⇤ 02 + 0.125 ⇤ 1.102 + 0.125 ⇤ 0.92
⇡ 0.25
2
A diversified fund
7
We do the same deal in 100 cities and ski resorts
in North America
Cashflows are independent of each other (independent of global warming)
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 several others
Investment parameters: Invested amount: $2Bn Average return: 10%
Volatility: 5%
Sharpe ratio = 2
Fees• Management fees: proportional to NAV
• Often paid monthly, quarterly (1%, 2%, 0%, …): accrued monthly
• Performance fees: proportional to the P&L • Often paid annually (10%, 20%, 30%,…): accrued monthly
• Hurdles • Stops performance fees when P&L is below a “hurdle” rate, or
• modifies performance fees to P&L over a hurdle rate.
• M or P • Performance fees will be discounted by management fees paid
• Makes the management fee a loan agains the performance fee
• First Loss fees • Managers give investors protection agains loses in exchange to higher performance fees
8
1-20 Fees
9
Investor Manager
Investedamount 2000 0
Gross investment
gains200 0
Management fee -20 +20
Performance fee -36 +36
Total gains 144 56
Profitability 7.2% Infinity
The fund structure
10
FUND
Investor 1 Investor 2 Investor 3 Investor 4
Management Company
Administrator
Auditor
Bank
Broker (custodian)
Owners
Service Providers
Conflict free
Directors
The fund structure
11
FUND
Investor 1 Investor 2 Investor 3 Investor 4
Management Company
Administrator
Auditor
Bank
Broker (custodian)
Owners
Service Providers
Conflict free
Directors
The Master/Feeder structure
12
Master FUND
Investor 1Investor 2 Investor 3 Investor 4
Management Company
Administrator
AuditorBankBroker
(custodian)
Feeder 1 FUND Feeder 2 FUND
Administrator
Auditor
Risk
13
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 several others
Investor Manager
Investedamount 2000 0
Gross investment
gains-60 0
Management fee -20 +20
Performance fee 0 0
Total gains -80 20
Profitability -4% Infinity
Understanding Performance
14
Turning data into information• There is very little information coming from hedge funds
• Some hedge funds are even closed and do not report anything to anyone except their own investors
• However, some data can be obtained from certain sources, some times the manager themselves
• The data is usually reduced to: • Monthly return data per fund
• AUM, firm-wide and per strategy
• A rough description of their strategy, one or two successful past trades
• Qualitative information
15
Portfolio Return
16
This is a free offprint provided to the author by the publisher. Copyright restrictions may apply.
HEDGE FUNDS 19
concepts. The first is the return. Intuitively, the return is a mathematical termthat embodies the growth characteristics of the fund’s share price over time. In itssimplest characterization, the monthly return is defined as follows:
Definition 2.1. For a given month k, the fund’s return rk in that month is calcu-lated as
rk =Sk − Sk−1
Sk−1,
where Sj denotes the fund’s share price for month j.
Definition 2.2. For a fund with monthly returns given by rk, from k = 0, . . . ,m,the arithmetic mean return is given by
Ramr =
∑mk=1(rk)
mIn Example 2.3 and Example 2.4, it shows that when simple return measure
might not be in line with the real investment performance. As intuitive as thisdefinition may be, it has serious limitations.
Example 2.3. Imagine I set up my own fund with my own $1 as the only initialinvestment, with one share, in January 1, and imagine also that every month untilDecember I manage to double the value of my investment, without the inflow ofany other assets; in other words, the share value at the end of November is $210 =$1, 024. The monthly return is equal to 100%. A wealthy friend of mine, impressedby my rate of return, decides to invest $1,000,000=$1M on December 1st. Thetotal asset value at that time is $1,001,024. At that time, I lose half of the assetsof the fund, and my return for December is -50%. My average monthly return is
11
12100% +
−50%
12= 87 .5%
This number shows a significant positive return, when in fact the investment endedup with a net investment loss of $499,489.00 during the year.
Example 2.4. Consider a fund that has $1 in assets invested in securities, withthe following results; the first month, the value of the securities double, to a total of$2; the next month, the value of the securities is cut in half, back to $1; the monthafter they double again, to be cut in half again the month after that; and so onand so forth. The monthly returns of the fund will then be +100%, -50%, +100%,-50%, etc. Nothing wrong with these numbers, but if one decides to calculate theiraverage, one will find that the average return of this fund is +25% per month; alittle strange for a fund that makes no money.
Later, when we tackle the problem of doing statistics properly on fund returns,we will see that this is an issue that we have to live with. However, there is analternative definition of returns that is sometimes used, and referred to as log-returns.
Definition 2.5. For a given month k, the fund’s log-return rlogk in that month iscalculated as
rlogk = logSk
Sk−1(2.1)
= log(1 + rk).(2.2)
where Sj denotes the fund’s share price for month j.
65
This is a free offprint provided to the author by the publisher. Copyright restrictions may apply.
HEDGE FUNDS 19
concepts. The first is the return. Intuitively, the return is a mathematical termthat embodies the growth characteristics of the fund’s share price over time. In itssimplest characterization, the monthly return is defined as follows:
Definition 2.1. For a given month k, the fund’s return rk in that month is calcu-lated as
rk =Sk − Sk−1
Sk−1,
where Sj denotes the fund’s share price for month j.
Definition 2.2. For a fund with monthly returns given by rk, from k = 0, . . . ,m,the arithmetic mean return is given by
Ramr =
∑mk=1(rk)
mIn Example 2.3 and Example 2.4, it shows that when simple return measure
might not be in line with the real investment performance. As intuitive as thisdefinition may be, it has serious limitations.
Example 2.3. Imagine I set up my own fund with my own $1 as the only initialinvestment, with one share, in January 1, and imagine also that every month untilDecember I manage to double the value of my investment, without the inflow ofany other assets; in other words, the share value at the end of November is $210 =$1, 024. The monthly return is equal to 100%. A wealthy friend of mine, impressedby my rate of return, decides to invest $1,000,000=$1M on December 1st. Thetotal asset value at that time is $1,001,024. At that time, I lose half of the assetsof the fund, and my return for December is -50%. My average monthly return is
11
12100% +
−50%
12= 87 .5%
This number shows a significant positive return, when in fact the investment endedup with a net investment loss of $499,489.00 during the year.
Example 2.4. Consider a fund that has $1 in assets invested in securities, withthe following results; the first month, the value of the securities double, to a total of$2; the next month, the value of the securities is cut in half, back to $1; the monthafter they double again, to be cut in half again the month after that; and so onand so forth. The monthly returns of the fund will then be +100%, -50%, +100%,-50%, etc. Nothing wrong with these numbers, but if one decides to calculate theiraverage, one will find that the average return of this fund is +25% per month; alittle strange for a fund that makes no money.
Later, when we tackle the problem of doing statistics properly on fund returns,we will see that this is an issue that we have to live with. However, there is analternative definition of returns that is sometimes used, and referred to as log-returns.
Definition 2.5. For a given month k, the fund’s log-return rlogk in that month iscalculated as
rlogk = logSk
Sk−1(2.1)
= log(1 + rk).(2.2)
where Sj denotes the fund’s share price for month j.
65
return log-return
Share value
In either case, we can collect a time series of portfolio returns Daily Monthly
Portfolio Stats
17
0
0.25
0.5
0.75
1
Consider a portfolio ⇧ that allocates wi to assets with returns given by randomvariables Xi, i = 1, . . . , n.If the covariance matrix of Xi is given by V = (�i,j , the portfolio volatility is
�2⇧ =
*nX
i=1
wi ·Xi ,nX
j=1
wi ·Xj
+
=X
i,j
wi wj hXi , Xji
=X
i,j
wi wj �i,j
= w · V · wT .
Therefore,�⇧ =
pw · V · wT .
We seek a portfolio ⇧ that maximizes the probability of exceeding a fixed bench-mark r.If returns are normally distributed:
Prob {⇧ � r} = Prob
⇢⇧� µ
�� r � µ
�
�
= 1� �
✓r � µ
�
◆
µ� r
�
�2 = 0.75 ⇤ 02 + 0.125 ⇤ 1.102 + 0.125 ⇤ 0.92
⇡ 0.25
St =Net Asset Value at time t
Total number of shares at time t
⇢(x) =dF
dx
2
Cumulative return distribution function (CDF)
Consider a portfolio ⇧ that allocates wi to assets with returns given by randomvariables Xi, i = 1, . . . , n.If the covariance matrix of Xi is given by V = (�i,j , the portfolio volatility is
�2⇧ =
*nX
i=1
wi ·Xi ,nX
j=1
wi ·Xj
+
=X
i,j
wi wj hXi , Xji
=X
i,j
wi wj �i,j
= w · V · wT .
Therefore,�⇧ =
pw · V · wT .
We seek a portfolio ⇧ that maximizes the probability of exceeding a fixed bench-mark r.If returns are normally distributed:
Prob {⇧ � r} = Prob
⇢⇧� µ
�� r � µ
�
�
= 1� �
✓r � µ
�
◆
µ� r
�
�2 = 0.75 ⇤ 02 + 0.125 ⇤ 1.102 + 0.125 ⇤ 0.92
⇡ 0.25
St =Net Asset Value at time t
Total number of shares at time t
⇢(x) =dF
dx
F
⇢
2
Return density
Consider a portfolio ⇧ that allocates wi to assets with returns given by randomvariables Xi, i = 1, . . . , n.If the covariance matrix of Xi is given by V = (�i,j , the portfolio volatility is
�2⇧ =
*nX
i=1
wi ·Xi ,nX
j=1
wi ·Xj
+
=X
i,j
wi wj hXi , Xji
=X
i,j
wi wj �i,j
= w · V · wT .
Therefore,�⇧ =
pw · V · wT .
We seek a portfolio ⇧ that maximizes the probability of exceeding a fixed bench-mark r.If returns are normally distributed:
Prob {⇧ � r} = Prob
⇢⇧� µ
�� r � µ
�
�
= 1� �
✓r � µ
�
◆
µ� r
�
�2 = 0.75 ⇤ 02 + 0.125 ⇤ 1.102 + 0.125 ⇤ 0.92
⇡ 0.25
St =Net Asset Value at time t
Total number of shares at time t
⇢(x) =dF
dx
F
⇢
2
Consider a portfolio ⇧ that allocates wi to assets with returns given by randomvariables Xi, i = 1, . . . , n.If the covariance matrix of Xi is given by V = (�i,j , the portfolio volatility is
�2⇧ =
*nX
i=1
wi ·Xi ,nX
j=1
wi ·Xj
+
=X
i,j
wi wj hXi , Xji
=X
i,j
wi wj �i,j
= w · V · wT .
Therefore,�⇧ =
pw · V · wT .
We seek a portfolio ⇧ that maximizes the probability of exceeding a fixed bench-mark r.If returns are normally distributed:
Prob {⇧ � r} = Prob
⇢⇧� µ
�� r � µ
�
�
= 1� �
✓r � µ
�
◆
µ� r
�
�2 = 0.75 ⇤ 02 + 0.125 ⇤ 1.102 + 0.125 ⇤ 0.92
⇡ 0.25
St =Net Asset Value at time t
Total number of shares at time t
⇢(x) =dF
dx
F
⇢
F (x) = Prob {return x}
2
x
Qua
ntile
s
Returns
Statistics - ProbabilityEarth - Heaven
18
0
0.25
0.5
0.75
1
Consider a portfolio ⇧ that allocates wi to assets with returns given by randomvariables Xi, i = 1, . . . , n.If the covariance matrix of Xi is given by V = (�i,j , the portfolio volatility is
�2⇧ =
*nX
i=1
wi ·Xi ,nX
j=1
wi ·Xj
+
=X
i,j
wi wj hXi , Xji
=X
i,j
wi wj �i,j
= w · V · wT .
Therefore,�⇧ =
pw · V · wT .
We seek a portfolio ⇧ that maximizes the probability of exceeding a fixed bench-mark r.If returns are normally distributed:
Prob {⇧ � r} = Prob
⇢⇧� µ
�� r � µ
�
�
= 1� �
✓r � µ
�
◆
µ� r
�
�2 = 0.75 ⇤ 02 + 0.125 ⇤ 1.102 + 0.125 ⇤ 0.92
⇡ 0.25
St =Net Asset Value at time t
Total number of shares at time t
⇢(x) =dF
dx
2
Excel: histogramMath: formulas
“On earth as it is in heaven”
Earth and Heaven
19
Earth
Heaven
Portfolio statistics
r =1
n
nX
i=1
ri (1)
� =
vuut 1
n� 1
nX
i=1
(ri � r)2
1
r =1
n
nX
i=1
ri (1)
� =
vuut 1
n� 1
nX
i=1
(ri � r)2
1
Average return, mean return
Volatility, standard deviation
months (or days)
return
20
µ =
Z 1
�1x⇢(x) dx
= E(X)
�2 =
Z 1
�1(x� µ)2⇢(x) dx
= E(X � µ)2
3
Heaven
Earth
Careful with StatisticsFund balance Simple
ReturnLogarithmic
returnJanuary $1 0.00%
February $2 100% 69.31%
March $1 -50% -69.31%
April $2 100% 69.31%
May $1 -50% -69.31%
June $2 100% 69.31%
July $1 -50% -69.31%
August $2 100% 69.31%
September $1 -50% -69.31%
October $2 100% 69.31%
November $1 -50% -69.31%
December $2 100% 69.31%
Average return 25% 0
Standard Deviation 75% 70%
21
Running mean
22
Figure 3: Two year-rolling window Sharpe Ratio for hedge fund indices
Besides the evolution of the Sharpe ratio we look at the first four moments of returns:
Mean, standard deviation, skewness and kurtosis. To compare the moments of hedge fund
returns to the moments of stock market’s returns, Figure 4 - Figure 7 include the appropriate
measures of the S&P500, running a two year-rolling window approach again. Note that the
mean return and standard deviation have been calculated again for yearly returns, whereas
skewness and kurtosis refer to daily returns, since standardizing is much more complicated
in this case and has no e↵ect on the interpretation since one only mutiplies everything with
a constant.
Figure 4: Two year-rolling window mean return (yearly) for hedge fund indices and the S&P500
10
Mean of the prior 2 year time window
2 year window runs ahead daily
Conflict between heaven and earth
Running standard deviation
23
Figure 5: Two year-rolling window standard deviation (yearly) for hedge fund indices and the S&P500
Figure 6: Two year-rolling window skewness (daily) for hedge fund indices and the S&P500
11
Conflict between heaven and earth
Financial Crisis 2 years after Financial Crisis
Averages over time
24
Figure 8: Two year-rolling window Sharpe ratio (yearly) for the Global Hedge Fund Index, the S&P500 and
the AGG Index
Figure 9: µ/� pairs for the Global Hedge Fund Index, the S&P500 and the AGG Index
13
Lower risk
Medium risk
Higher risk
Returns over timeTime-Weighted-Rate-of-Return
• Monthly returns can be compounded over time
• Time-weighted rate of return, over n periods, is defined as
r =1
n
nX
i=1
ri (1)
� =
vuut 1
n� 1
nX
i=1
(ri � r)2
For two funds, with monthly returns given by random variables X and Y , thecovariance is obtained as
Cov(X,Y ) =1
n� 1
nX
i=1
�Xi �X
� �Yi � Y
�
and the correlation
⇢(X,Y ) =Cov(X,Y )
�X · �Y
By the Cauchy-Schwartz inequality,
�1 ⇢ 1.
1 +R = (1 + r1)(1 + r2) · · · (1 + rn)
1
Internal-Rate-of-Return
• Takes into account the amounts invested over time
r =1
n
nX
i=1
ri (1)
� =
vuut 1
n� 1
nX
i=1
(ri � r)2
For two funds, with monthly returns given by random variables X and Y , thecovariance is obtained as
Cov(X,Y ) =1
n� 1
nX
i=1
�Xi �X
� �Yi � Y
�
and the correlation
⇢(X,Y ) =Cov(X,Y )
�X · �Y
By the Cauchy-Schwartz inequality,
�1 ⇢ 1.
1 +R = (1 + r1)(1 + r2) · · · (1 + rn)
If we make
• investments worth pk
• at time tk ago
• and the final value of the fund is V ,
• the Internal Rate of Return is definedimplicitly by the expression
V =X
k
pk(1 +R)tk
1
25
r =1
n
nX
i=1
ri (1)
� =
vuut 1
n� 1
nX
i=1
(ri � r)2
For two funds, with monthly returns given by random variables X and Y , thecovariance is obtained as
Cov(X,Y ) = E(X �X)(Y � Y )
=1
n� 1
nX
i=1
�Xi �X
� �Yi � Y
�
and the correlation
⇢(X,Y ) =Cov(X,Y )
�X · �Y
By the Cauchy-Schwartz inequality,
�1 ⇢ 1.
1 +R = (1 + r1)(1 + r2) · · · (1 + rn)
If we make
• investments worth pk
• at time tk ago
• and the final value of the fund is V ,
• the Internal Rate of Return is definedimplicitly by the expression
V =X
k
pk(1 +R)tk
500, 001.024 = 1 · (1 +R)11 + 1, 000, 000, 000 · (1 +R)1/12
R ⇡ �100%
1
Time weighted returnsFund balance Inflows P&L Monthly Return Cummulative
returnJanuary $1 $1 $1 100% 100%
February $2 $0 $2 100% 300%
March $4 $0 $4 100% 700%
April $8 $0 $8 100% 1500%
May $16 $0 $16 100% 3100%
June $32 $0 $32 100% 6300%
July $64 $0 $64 100% 12700%
August $128 $0 $128 100% 25500%
September $256 $0 $256 100% 51100%
October $512 $0 $512 100% 102300%
November $1,024 $0 $1,024 100% 204700%
December $1,000,002,048 $1,000,000,000 -$500,001,024 -50% 102300%
January $500,001,024 $0
Time weighted rate of returnInternal rate of return
r =1
n
nX
i=1
ri (1)
� =
vuut 1
n� 1
nX
i=1
(ri � r)2
For two funds, with monthly returns given by random variables X and Y , thecovariance is obtained as
Cov(X,Y ) =1
n� 1
nX
i=1
�Xi �X
� �Yi � Y
�
and the correlation
⇢(X,Y ) =Cov(X,Y )
�X · �Y
By the Cauchy-Schwartz inequality,
�1 ⇢ 1.
1 +R = (1 + r1)(1 + r2) · · · (1 + rn)
If we make
• investments worth pk
• at time tk ago
• and the final value of the fund is V ,
• the Internal Rate of Return is definedimplicitly by the expression
V =X
k
pk(1 +R)tk
500, 001.024 = 1 · (1 +R)1 + 1, 000, 000, 000 · (1 +R)1/12
R ⇡ �100%
1
26
r =1
n
nX
i=1
ri (1)
� =
vuut 1
n� 1
nX
i=1
(ri � r)2
For two funds, with monthly returns given by random variables X and Y , thecovariance is obtained as
Cov(X,Y ) = E(X � E(X))(Y � E(Y ))
=1
n� 1
nX
i=1
�Xi �X
� �Yi � Y
�
and the correlation
⇢(X,Y ) =Cov(X,Y )
�X · �Y
By the Cauchy-Schwartz inequality,
�1 ⇢ 1.
1 +R = (1 + r1)(1 + r2) · · · (1 + rn)
If we make
• investments worth pk
• at time tk ago
• and the final value of the fund is V ,
• the Internal Rate of Return is definedimplicitly by the expression
V =X
k
pk(1 +R)tk
500, 001.024 = 1 · (1 +R)11 + 1, 000, 000, 000 · (1 +R)1/12
R ⇡ �100%
1
Correlation
27
Heaven
Earth
Portfolio volatility
28
Consider a portfolio ⇧ that allocates wi to assets with returns given by randomvariables Xi, i = 1, . . . , n.If the covariance matrix of Xi is given by V = (�i,j), the portfolio volatility is
�2⇧ =
*nX
i=1
wi ·Xi ,nX
j=1
wi ·Xj
+
=X
i,j
wi wj hXi , Xji
=X
i,j
wi wj �i,j
= w · V · wT .
Therefore,�⇧ =
pw · V · wT .
We seek a portfolio ⇧ that maximizes the probability of exceeding a fixed bench-mark r.If returns are normally distributed:
Prob {⇧ � r} = Prob
⇢⇧� µ
�� r � µ
�
�
= 1� �
✓r � µ
�
◆
µ� r
�
�2 = 0.75 ⇤ 02 + 0.125 ⇤ 1.102 + 0.125 ⇤ 0.92
⇡ 0.25
St =Net Asset Value at time t
Total number of shares at time t
⇢(x) =dF
dx
F
⇢
F (x) = Prob {return x}
2
Implies portfolio diversification:
When correlations are
less then 1
or even negative
portfolio volatility decreases
Marginal value of diversification
29
µ =
Z 1
�1x⇢(x) dx
= E(X)
�2 =
Z 1
�1(x� µ)2⇢(x) dx
= E(X � µ)2
Consider N assets with returns given by random variables Xi, i = 1, . . . , N , anda portfolio ⇧ with allocations wi.For simplicity assume constant pairwise correlations C, equal asset allocations,and also equal means and variances µ and �. Then
�2⇧ =
X
i
w2i �
2i +
X
i 6=j
wiwj�i�j
= C +�2 � C
N
3
Rate of diversification
Adding diversification lowers risk… … but how many funds is “enough”?
Portfolio risk and return• Markowitz proposed, in 1952, that portfolios are
characterized by two parameters:
• Their expected return
• Their standard deviation
• In this framework, portfolio selection is reduced to picking points in a risk-return plane
30
Portfolio Diversification
31
Risk can be lowered combining assets in a portfolio
Efficient Frontier
32
Possible investments
Impossible investmentsBoundary between true and false possible and impossible
Sharpe Ratio
• Markowitz argues that we should invest in the efficient frontier, but does not specify where
• Sharpe tells us where in the frontier we should invest:
33
An employment contract
Independent of the portfolio: Normal (0,1)
Cumulative distribution of the Gaussian
A portfolio manager will be paid a bonus if it makes a benchmark return of r.
Consider a portfolio ⇧ that allocates wi to assets with returns given by randomvariables Xi, i = 1, . . . , n.If the covariance matrix of Xi is given by V = (�i,j , the portfolio volatility is
�2⇧ =
*nX
i=1
wi ·Xi ,nX
j=1
wi ·Xj
+
=X
i,j
wi wj hXi , Xji
=X
i,j
wi wj �i,j
= w · V · wT .
Therefore,�⇧ =
pw · V · wT .
We seek a portfolio ⇧ that maximizes the probability of exceeding a fixed bench-mark r.If returns are normally distributed:
Prob {⇧ � r} = Prob
⇢⇧� µ
�� r � µ
�
�
= 1� �
✓r � µ
�
◆
2
The rational decision is to maximize the Sharpe Ratio with benchmark r
34
Consider a portfolio ⇧ that allocates wi to assets with returns given by randomvariables Xi, i = 1, . . . , n.If the covariance matrix of Xi is given by V = (�i,j , the portfolio volatility is
�2⇧ =
*nX
i=1
wi ·Xi ,nX
j=1
wi ·Xj
+
=X
i,j
wi wj hXi , Xji
=X
i,j
wi wj �i,j
= w · V · wT .
Therefore,�⇧ =
pw · V · wT .
We seek a portfolio ⇧ that maximizes the probability of exceeding a fixed bench-mark r.If returns are normally distributed:
Prob {⇧ � r} = Prob
⇢⇧� µ
�� r � µ
�
�
= 1� �
✓r � µ
�
◆
µ� r
�
2
Another employment contract
Independent of the portfolio: Normal (0,1)
Cumulative distribution of the Gaussian
A portfolio manager will be paid a bonus if it makes a benchmark return of r ABOVE an index Y
The rational decision is to maximize the Information Ratio with
benchmark r
35
µ =
Z 1
�1x⇢(x) dx
= E(X)
�2 =
Z 1
�1(x� µ)2⇢(x) dx
= E(X � µ)2
Consider N assets with returns given by random variables Xi, i = 1, . . . , N , anda portfolio ⇧ with allocations wi.For simplicity assume constant pairwise correlations C, equal asset allocations,and also equal means and variances µ and �. Then
�2⇧ =
X
i
w2i �
2i +
X
i 6=j
wiwj�i�j
= C +�2 � C
N
We seek a portfolio ⇧ that maximizes the probability of exceeding a known, butrandom, benchmark Y + r.If returns are normally distributed:
Prob {⇧ � Y + r} = Prob {(⇧� Y ) � r}
= Prob
⇢⇧� Y � (µ⇧ � µY )
�⇧�Y� r � µ⇧�Y
�⇧�Y
�
= 1� �
✓r � µ⇧�Y
�⇧�Y
◆
µ⇧ � µY � r
�⇧�Y
3
µ =
Z 1
�1x⇢(x) dx
= E(X)
�2 =
Z 1
�1(x� µ)2⇢(x) dx
= E(X � µ)2
Consider N assets with returns given by random variables Xi, i = 1, . . . , N , anda portfolio ⇧ with allocations wi.For simplicity assume constant pairwise correlations C, equal asset allocations,and also equal means and variances µ and �. Then
�2⇧ =
X
i
w2i �
2i +
X
i 6=j
wiwj�i�j
= C +�2 � C
N
We seek a portfolio ⇧ that maximizes the probability of exceeding a known, butrandom, benchmark Y + r.If returns are normally distributed:
Prob {⇧ � Y + r} = Prob {(⇧� Y ) � r}
= Prob
⇢⇧� Y � (µ⇧ � µY )
�⇧�Y� r � µ⇧�Y
�⇧�Y
�
= 1� �
✓r � µ⇧�Y
�⇧�Y
◆
µ⇧ � µY � r
�⇧�Y
3
Tracking error
Portable alpha• Portfolios with return independent of market returns deliver alpha
• Portfolio with returns dependent on markets, indices, etc. deliver beta
• Optimal performance of portfolios benchmarked to indices with futures or forwards can be obtained via portable alpha strategies:
• Construct an optimal portfolio with cash benchmark P
• Add an index futures contract to portfolio P
• This constitutes a direct application of absolute return strategies in the institutional portfolio
36
Alpha - beta
• Hedge funds
• Absolute return strategies
• Active portfolio management
• Prop-desks
37
• Stocks
• Bonds
• Infrastructure
• Private Equity
• Credit
Efficient frontier
38
Alpha
The rationale for including Hedge Funds in a portfolio
Traditional - Alternative
39
© Luis Seco. Not to be distributed without permission.
Hedge fund diversification
Hedge funds are uncorrelated to traditional markets, and internally uncorrelated also.
Correlation histogram for Dow stocks
Correlation histogram for hedge funds
Hedge Funds
Stocks
Less diversification
More diversification
Correlation matrix Stocks
Correlation matrix HF
beta
alpha
Regression analysis• Some investments are directly dependent on the move of
• market variables
• volatilities
• events
• Other investments are truly hedged and not directly dependent of external events
• A way to detect and measure dependencies is through regression analysis
40
Alphas, betas
41
Simultaneous monthly returns
statistical relationship
return differential
low =
meaningful
Mathematical foundation
42
Returns under consideration Benchmark/market/external returns
µ =
Z 1
�1x⇢(x) dx
= E(X)
�2 =
Z 1
�1(x� µ)2⇢(x) dx
= E(X � µ)2
Consider N assets with returns given by random variables Xi, i = 1, . . . , N , anda portfolio ⇧ with allocations wi.For simplicity assume constant pairwise correlations C, equal asset allocations,and also equal means and variances µ and �. Then
�2⇧ =
X
i
w2i �
2i +
X
i 6=j
wiwj�i�j
= C +�2 � C
N
We seek a portfolio ⇧ that maximizes the probability of exceeding a known, butrandom, benchmark Y + r.If returns are normally distributed:
Prob {⇧ � Y + r} = Prob {(⇧� Y ) � r}
= Prob
⇢⇧� Y � (µ⇧ � µY )
�⇧�Y� r � µ⇧�Y
�⇧�Y
�
= 1� �
✓r � µ⇧�Y
�⇧�Y
◆
µ⇧ � µY � r
�⇧�Y
s =E(X � µ)3
�3
Y ⇡ ↵ + �1 ·X1 + �2 ·X2 + · · · + �n ·Xn + Error
3
µ =
Z 1
�1x⇢(x) d
x
= E(X)
�2 =
Z 1
�1(x� µ)
2⇢(x) dx
= E(X � µ)2
Consider N assets w
ith returnsgiven by random
variables Xi, i =
1, . . . , N, and
a portfolio ⇧ with allocat
ions wi.
For simplicity
assume constant pairw
ise correlations C
, equalasset a
llocations,
and also equal means an
d variances µ and �. The
n
�2⇧ =X
i
w2i�
2i+X
i 6=j
wiwj�i�j
= C +�2 � C
N
We seeka portfol
io ⇧ that maximizes the
probability of exce
eding a known, but
random, bench
mark Y + r.
If returns are
normally distributed:
Prob {⇧� Y + r} = Prob {(⇧
� Y ) � r}
= Prob
⇢⇧� Y � (µ⇧ � µY )
�⇧�Y
�r � µ⇧�Y
�⇧�Y
�
= 1� �
✓r � µ⇧�Y
�⇧�Y
◆
µ⇧ � µY � r
�⇧�Y
s =E(X � µ)
3
�3
Y ⇡ ↵ + �1 ·X1 + �2 ·X2 + · · · + �n ·Xn + Error
�i =Cov(Y,
Xi)
�2Y
3
If factors are independent
© Luis Seco. Not to be distributed without permission.
Linear regression
© Luis Seco. Not to be distributed without permission.
Linear regression
Multi-dimensional regression
43
Non-normal returns
44
Gaussian Fit
Events impossible
according to Gaussian Fit
Statistics Time structure
• Moving windows
• EWMA
• ARMA
• GARCH
45
Semi-standard deviation
46
wi@�
@wi= wj
@�
@wj
@�
@wi=
@�
@wj
Using Lagrange multipliers, the minimum variance portfolio is obtained whenr�2 is orthogonal to the plane giving us the total wealth restriction
Pni=1 wi = 1
Therefore the minimum variance portfolio occurs when
@�
@wi=
@�
@wj
wi = wj
Short Interest Ratio =Short interest
Average Daily Volume
Semi-standard (loss) deviation
s� =
vuut 1
n� 1
nX
i=1
(R⇤ � ri)2+
4
Earth
Heaven?
Number of all months? Or
number of months with losses?
Some sort of target return Benchmark
or 0
Only penalize if return is bad (below benchmark..
..or negative)
Volatility of losses
47
wi@�
@wi= wj
@�
@wj
@�
@wi=
@�
@wj
Using Lagrange multipliers, the minimum variance portfolio is obtained whenr�2 is orthogonal to the plane giving us the total wealth restriction
Pni=1 wi = 1
Therefore the minimum variance portfolio occurs when
@�
@wi=
@�
@wj
wi = wj
Short Interest Ratio =Short interest
Average Daily Volume
Semi-standard (loss) deviation
s� =
vuut 1
n� 1
nX
i=1
(R⇤ � ri)2+
Sortino Ratio =µ� r
s�
The standard deviation of negative assets returns is
�loss =
vuutnX
i=1
(ri)2� �
nX
i=1
(ri)�
!2
4
Monthly loss Average loss
Con
sider
aportfolio
⇧that
allocatesw
ito
assetswith
returnsgiven
byran
dom
variables
Xi ,i=
1, . . . , n.
Ifthecovarian
cematrix
ofX
iis
givenby
V=
(�i,j ,
theportfolio
volatilityis
�2⇧=
*nXi=
1
wi ·X
i,
nXj=
1
wi ·X
j +
=Xi,j
wi w
jhX
i,X
j i
=Xi,j
wi w
j�i,j
=w· V
· wT.
Therefore,
�⇧=
pw· V
· wT.
Weseek
aportfolio
⇧that
maxim
izestheprob
ability
ofexceed
ingafixed
bench-
mark
r.Ifretu
rnsare
norm
allydistrib
uted
:
Prob
{⇧�
r}=
Prob
⇢⇧�µ
��
r�µ
�
�
=1��
✓r�µ
�
◆
µ�r
�
�2=
0.75⇤02+0.125
⇤1.10
2+0.125
⇤0.9
2
⇡0.25
St=
Net
Asset
Valu
eat
timet
Total
number
ofshares
attim
et
⇢(x)=
dFdx
F⇢
F(x)=
Prob
{return
x}
2
wi@�
@wi= wj
@�
@wj
@�
@wi=
@�
@wj
Using Lagrange multipliers, the minimum variance portfolio is obtained whenr�2 is orthogonal to the plane giving us the total wealth restriction
Pni=1 wi = 1
Therefore the minimum variance portfolio occurs when
@�
@wi=
@�
@wj
wi = wj
Short Interest Ratio =Short interest
Average Daily Volume
Semi-standard (loss) deviation
s� =
vuut 1
n� 1
nX
i=1
(R⇤ � ri)2+
Sortino Ratio =µ� r
s�
The standard deviation of negative assets returns is
�loss =
vuutnX
i=1
(ri)2� �
nX
i=1
(ri)�
!2
r� =
(r if r 0
0 otherwise
4
Sortino Ratio
48
wi@�
@wi= wj
@�
@wj
@�
@wi=
@�
@wj
Using Lagrange multipliers, the minimum variance portfolio is obtained whenr�2 is orthogonal to the plane giving us the total wealth restriction
Pni=1 wi = 1
Therefore the minimum variance portfolio occurs when
@�
@wi=
@�
@wj
wi = wj
Short Interest Ratio =Short interest
Average Daily Volume
Semi-standard (loss) deviation
s� =
vuut 1
n� 1
nX
i=1
(R⇤ � ri)2+
Sortino Ratio =µ� r
s�
4
Semi-standard deviation
The Sortino ratio is a variation of the Sharpe ratio that differentiates harmful volatility from total overall volatility by using the asset's standard deviation of negative asset returns, called downside deviation. The Sortino ratio takes the asset's return minus the risk-free rate, and divides it by the asset's downside deviation. The ratio was named after Frank A. Sortino.
NO
Moments• Normal distributions are characterized by their first (mean) and second moments (variance, covariances).
• If distributions are not normal, higher moments give additional information on the distribution
• People often use the third and fourth central moments as additional descriptive elements of return distribution: skewness and kurtosis
49
Skewness Kurtosis
50
wi@�
@wi= wj
@�
@wj
@�
@wi=
@�
@wj
Using Lagrange multipliers, the minimum variance portfolio is obtained whenr�2 is orthogonal to the plane giving us the total wealth restriction
Pni=1 wi = 1
Therefore the minimum variance portfolio occurs when
@�
@wi=
@�
@wj
wi = wj
Short Interest Ratio =Short interest
Average Daily Volume
Semi-standard (loss) deviation
s� =
vuut 1
n� 1
nX
i=1
(R⇤ � ri)2+
Sortino Ratio =µ� r
s�The standard deviation of negative assets returns is
�loss =
vuutnX
i=1
(ri)2� �
nX
i=1
(ri)�
!2
r� =
(r if r 0
0 otherwise
s =1
�3
n
(n� 1)(n� 2)
nX
i=1
(ri � r)3
=1
�3E�X �X
�3
=1
�4
n(n+ 1)
(n� 1)(n� 2)(n� 3)
nX
i=1
(ri � r)4 � 3(n� 1)2
(n� 2)(n� 3)
=1
�4E�X �X
�4
1
n
4
Heaven
Earth
Platykurtotic: k<0
Leptokurtotic: k>0
Mesokurtotic: k=0Positive skew s>0 Negative skew s<0
wi@�
@wi= wj
@�
@wj
@�
@wi=
@�
@wj
Using Lagrange multipliers, the minimum variance portfolio is obtained whenr�2 is orthogonal to the plane giving us the total wealth restriction
Pni=1 wi = 1
Therefore the minimum variance portfolio occurs when
@�
@wi=
@�
@wj
wi = wj
Short Interest Ratio =Short interest
Average Daily Volume
Semi-standard (loss) deviation
s� =
vuut 1
n� 1
nX
i=1
(R⇤ � ri)2+
Sortino Ratio =µ� r
s�
The standard deviation of negative assets returns is
�loss =
vuutnX
i=1
(ri)2� �
nX
i=1
(ri)�
!2
r� =
(r if r 0
0 otherwise
s =1
�3
n
(n� 1)(n� 2)
nX
i=1
(ri � r)3
=1
�3E�X �X
�3
=1
�4
n(n+ 1)
(n� 1)(n� 2)(n� 3)
nX
i=1
(ri � r)4 � 3(n� 1)2
(n� 2)(n� 3)
=1
�4E�X �X
�4 � 3
4
Skewness Kurtosis
51
wi@�
@wi= wj
@�
@wj
@�
@wi=
@�
@wj
Using Lagrange multipliers, the minimum variance portfolio is obtained whenr�2 is orthogonal to the plane giving us the total wealth restriction
Pni=1 wi = 1
Therefore the minimum variance portfolio occurs when
@�
@wi=
@�
@wj
wi = wj
Short Interest Ratio =Short interest
Average Daily Volume
Semi-standard (loss) deviation
s� =
vuut 1
n� 1
nX
i=1
(R⇤ � ri)2+
Sortino Ratio =µ� r
s�
The standard deviation of negative assets returns is
�loss =
vuutnX
i=1
(ri)2� �
nX
i=1
(ri)�
!2
r� =
(r if r 0
0 otherwise
s =1
�3
n
(n� 1)(n� 2)
nX
i=1
(ri � r)3
=1
�4
n(n+ 1)
(n� 1)(n� 2)(n� 3)
nX
i=1
(ri � r)4 � 3(n� 1)2
(n� 2)(n� 3)
1
n
4
Morally Morally 3 (the fourth central moment of the gaussian)
wi@�
@wi= wj
@�
@wj
@�
@wi=
@�
@wj
Using Lagrange multipliers, the minimum variance portfolio is obtained whenr�2 is orthogonal to the plane giving us the total wealth restriction
Pni=1 wi = 1
Therefore the minimum variance portfolio occurs when
@�
@wi=
@�
@wj
wi = wj
Short Interest Ratio =Short interest
Average Daily Volume
Semi-standard (loss) deviation
s� =
vuut 1
n� 1
nX
i=1
(R⇤ � ri)2+
Sortino Ratio =µ� r
s�The standard deviation of negative assets returns is
�loss =
vuutnX
i=1
(ri)2� �
nX
i=1
(ri)�
!2
r� =
(r if r 0
0 otherwise
s =1
�3
n
(n� 1)(n� 2)
nX
i=1
(ri � r)3
=1
�3E�X �X
�3
=1
�4
n(n+ 1)
(n� 1)(n� 2)(n� 3)
nX
i=1
(ri � r)4 � 3(n� 1)2
(n� 2)(n� 3)
=1
�4E�X �X
�4
1
n
4
wi@�
@wi= wj
@�
@wj
@�
@wi=
@�
@wj
Using Lagrange multipliers, the minimum variance portfolio is obtained whenr�2 is orthogonal to the plane giving us the total wealth restriction
Pni=1 wi = 1
Therefore the minimum variance portfolio occurs when
@�
@wi=
@�
@wj
wi = wj
Short Interest Ratio =Short interest
Average Daily Volume
Semi-standard (loss) deviation
s� =
vuut 1
n� 1
nX
i=1
(R⇤ � ri)2+
Sortino Ratio =µ� r
s�The standard deviation of negative assets returns is
�loss =
vuutnX
i=1
(ri)2� �
nX
i=1
(ri)�
!2
r� =
(r if r 0
0 otherwise
s =1
�3
n
(n� 1)(n� 2)
nX
i=1
(ri � r)3
=1
�3E�X �X
�3
=1
�4
n(n+ 1)
(n� 1)(n� 2)(n� 3)
nX
i=1
(ri � r)4 � 3(n� 1)2
(n� 2)(n� 3)
=1
�4E�X �X
�4
1
n
4
Estimation problems
52
Outliers have a very dramatic effect in the sample calculation of the historical moments
wi@�
@wi= wj
@�
@wj
@�
@wi=
@�
@wj
Using Lagrange multipliers, the minimum variance portfolio is obtained when
r�2 is orthogonal to the plane giving us the total wealth restriction
Pni=1wi = 1
Therefore the minimum variance portfolio occurs when
@�
@wi=
@�
@wj
wi = wj
Short Interest Ratio =Short interest
Average Daily Volume
Semi-standard (loss) deviation
s� =
vuut 1
n� 1
nX
i=1
(R⇤ � ri)2+
Sortino Ratio =µ� r
s�
The standard deviation of negative assets returns is
�loss =
vuutnX
i=1
(ri)2� �
nX
i=1
(ri)�
!2
r� =
(r if r 0
0 otherwise
s =1
�3
n
(n� 1)(n� 2)
nX
i=1
(ri � r)3
=1
�3E�X �X
�3
=1
�4
n(n+ 1)
(n� 1)(n� 2)(n� 3)
nX
i=1
(ri � r)4 �3(n� 1)2
(n� 2)(n� 3)
=1
�4E�X �X
�4
1
n
4
wi@�@wi
= wj@�@wj
@�@wi
=@�@wjUsing Lagrange multipliers, the minimum variance portfolio is obtained whenr�2 is orthogonal to the plane giving us the total wealth restriction
Pni=1 wi = 1Therefore the minimum variance portfolio occurs when
@�@wi
=@�@wj
wi = wj
Short Interest Ratio = Short interestAverage Daily VolumeSemi-standard (loss) deviation
s� =
vuut 1n� 1
nX
i=1
(R⇤ � ri)2+
Sortino Ratio =µ� rs�The standard deviation of negative assets returns is
�loss =
vuutnX
i=1
(ri)2� �
nX
i=1
(ri)�
!2
r� =
(r if r 00 otherwise
s =1
�3n
(n� 1)(n� 2)
nX
i=1
(ri � r)3
=1
�3E�X �X
�3
=1
�4n(n+ 1)
(n� 1)(n� 2)(n� 3)
nX
i=1
(ri � r)4 � 3(n� 1)2
(n� 2)(n� 3)=
1
�4E�X �X
�4
1
n
4
An outlier 10 times bigger than the next produces a term
1000 times bigger in the estimation of skew and 10,000 times bigger
in the estimation of the kurtosis
HFRX skewness
53
Figure 5: Two year-rolling window standard deviation (yearly) for hedge fund indices and the S&P500
Figure 6: Two year-rolling window skewness (daily) for hedge fund indices and the S&P500
11
HFRX kurtosis
54Figure 7: Two year-rolling window kurtosis (daily) for hedge fund indices and the S&P500
We see that results for hedge fund indices di↵er substantially from the S&P500. While
the two year-rolling window standard deviation of the stock market is much higher than the
hedge funds’ one, the hedge fund indices own a significant higher skewness and kurtosis. This
is the reason why the return of hedge funds are said to be non-gaussian in contrast to the
stock market. The dramatic jumps in skewness and kurtosis are caused by single data points
(outliers) included or excluded from the two year window: The third and fourth moment of
a random variable are very sensitive to outliers. Furthermore the skewness and the kurtosis
di↵er substantially among the di↵erent hedge fund strategies. The Merger Arbitrage Index
has especially during the financial crisis, bigger moments than the other strategy indices,
e.g.. In general it is quite hard to interpret the skewness, because it is a trade-o↵ between
the size and the frequency of ”large events”.
In addition to to the previous analysis we want to compare the Sharpe ratio of the hedge
fund market with the stock market and the fixed income market. To purify the plot hedge
funds are represented by a single index, the HFRX Global Hedge Fund Index. Stock market
is again represented by the S&P500. For the bond market the iShares Core U.S. Aggregate
Bond ETF (AGG) is introduced. The used time series of the AGG Index starts at September
29, 2003. Figure 8 reveals that the Sharpe ratio is much more volatile for hedge funds than
it is for stocks and bonds, although µ/�-pairs spread much more for the S&P500 than for
hedge funds (Figure 9).
12
Uselessness of skewness
55© Luis Seco. Not to be reproduced without permission
Slide 46
Uselessness of skewness
Good average return
High positive skew
Very high volatility
Terrible Performance
300% monthly return
Omega• The Omega Ratio is a risk-return performance measure of an investment asset,
portfolio, or strategy.
• It was devised by Keating & Shadwick in 2002
• Given a benchmark return, it is defined as the probability weighted ratio of gains versus losses for that benchmark return.
• The ratio is an alternative for the widely used Sharpe ratio.
• It is widely believed that it is based on information the Sharpe ratio discards, most notably information on the tails of the distribution, hence it is superior, but this comparison is not accurate.
• The Omega is a function of the benchmark, not a single number. As a function, it is clearly superior to the Sharpe ratio, which is a simple function of two variables: average return and standard deviation.
56
Omega definition
57
1
n
F (x) = Prob {P&L x}
=
Z x
�1⇢(r) dr
Z �VaR↵
�1⇢(r) dr = Prob{Losses � VaR↵}
= 1� ↵
CVaR↵ =1
↵
Z ↵
0VaR1�� d�
= E{Loss | Loss � VaR↵}
⌦(r) =
Z 1
r(1� F (x)) dx
Z r
�1F (x) dx
=
Z 1
r(x� r)⇢(x) dx
Z r
�1(r � x)⇢(x) dx
=Expected out-performance
Expected under-performance
=Dembo’s reward
Dembo’s regret
5
Expected out-performance
Expected under-performance
Dembo’s reward
Dembo’s regret
Price of the call
Price of the putTruncated First Moments
Less tail information
Numerator
Denominator
Hedge Fund Omegas
58
InvestmentRiskManagment 16
The following plot shows the omega curves for daily returns for the S&P 500, the AGG bond
index and the Global Hedge Fund Index. The S&P 500 curve is less steep than the AGG and
the GHFI, which intuitively makes sense considering the higher volatility of stocks in
comparison to bonds and hedge funds. The differences in the steepness mean that stock
investors can be less sure to outperform a low benchmark, but more often are able to
outperform high benchmarks. The bond and the hedge fund curve have a very similar
structure, where the hedge fund has a even higher steepness than the AGG.
A
B
Cumulative distribution function
Histogram
Comparing Omegas
59InvestmentRiskManagment 16
The following plot shows the omega curves for daily returns for the S&P 500, the AGG bond
index and the Global Hedge Fund Index. The S&P 500 curve is less steep than the AGG and
the GHFI, which intuitively makes sense considering the higher volatility of stocks in
comparison to bonds and hedge funds. The differences in the steepness mean that stock
investors can be less sure to outperform a low benchmark, but more often are able to
outperform high benchmarks. The bond and the hedge fund curve have a very similar
structure, where the hedge fund has a even higher steepness than the AGG.
A
B
Equities
Bonds
HF
Correlation risk“Hedge funds are uncorrelated to traditional markets, so they constitute excellent diversification strategies”.
“When things go wrong they all go the same way”
Correlation matrices
61
© Luis Seco. Not to be reproduced without permission
Slide 80
Correlation switching
© Luis Seco. Not to be reproduced without permission
Slide 80
Correlation switching
Correlation matrices
62
© Luis Seco. Not to be reproduced without permission
Slide 78
Normal correlations
© Luis Seco. Not to be reproduced without permission
Slide 79
Distressed correlations
Correlation matrices
63
© Luis Seco. Not to be reproduced without permission
Slide 83
Correlation switching
Trading strategies
Strategy breakdown
65
2016 Asset Breakdown (BarclayHedge)
Convertible Arbitrage $24 1%
Distressed Securities $103 4%
Emerging Markets $253 9%
Equity Long Bias $227 8%
Equity Long/Short $240 9%
Equity Long-Only $136 5%
Equity Market Neutral $84 3%
Event Driven $142 5%
Fixed Income $556 20%
Macro $226 8%
Merger Arbitrage $66 2%
Multi-Strategy $360 13%
Other ** $157 6%
Sector Specific *** $149 5%
Total $2722 100%
Sector Specific ***5%
Other **6%
Multi-Strategy13%
Merger Arbitrage2%
Macro8%
Fixed Income20%
Event Driven5%
Equity Market Neutral3%
Equity Long-Only5%
Equity Long/Short9%
Equity Long Bias8%
Emerging Markets9%
Distressed Securities4%
Convertible Arbitrage1%
$0
$150
$300
$450
$600
Convertible Arbitrage
Distressed Securities
Emerging M
arkets
Equity Long Bias
Equity Long/Short
Equity Long-Only
Equity Market N
eutral
Event Driven
Fixed Income
Macro
Merger Arbitrage
Multi-Strategy
Other **
Sector Specific ***
Equity based trading
66
2016 Asset Breakdown (BarclayHedge)
Convertible Arbitrage $24 1%
Distressed Securities $103 4%
Emerging Markets $253 9%
Equity Long Bias $227 8%
Equity Long/Short $240 9%
Equity Long-Only $136 5%
Equity Market Neutral $84 3%
Event Driven $142 5%
Fixed Income $556 20%
Macro $226 8%
Merger Arbitrage $66 2%
Multi-Strategy $360 13%
Other ** $157 6%
Sector Specific *** $149 5%
Total $2722 100%
The most populous investment style More than 25% of assets Many different
trading styles and characteristics
Long only Long biased Short biased
Short only Long short
Equity Market neutral Quant Equity
etc.
Equity (long)• Fundamental
• Growth
• GARP
• Momentum • look for stocks moving significantly in one
direction on high volume and jump on board to ride the momentum train to a desired profit.
• Technical • Developing trading signals from charts and
graph patterns
67
Netflix surged over 260% to $330 from January to October in 2013, which was way above its valuation.
Its P/E ratio was above 400, while its competitors' were below 20. Momentum traders were trying to profit from the uptrend,
which drove the price even higher. Even Reed Hasting, CEO of Netflix,
admitted that Netflix is a momentum stock during a conference call in October 2013.
Equity (short)• Stock lending: before you can short a stock, someone needs to be willing to lend it
• Financing rate: Fed+spread
• Locate process
• Uptick rule
• Short squeeze: • Return if recalled
68
Short selling stocksMathematically, short selling corresponds to the notion of purchasing negative amount of stock:
• If we short n unites of a stock valued at $S, we receive $nS, and pick up a liability to return the n stocks:
• Whenever we want, or
• If the lender calls the loan, we need to purchase the stocks immediately and return to the lender
• The stock loan is usually done through a broker, who has ample inventory of stocks to lend, and even if one stockholder wants their stock on loan back, the broker can replace the stock with another one
• In some situations, stocks on loan run out, and short sellers are forced out of their shorts, usually causing large losses: this is called a short squeeze
• The most famous short squeeze happened with VW stock in Frankfurt in 2008
• Dividends belong to the lender, and are payable by the short seller.
69
The VW short squeezeMany hedge funds were short the auto industry in 2008.
Many of them were short VW stock
Friday October 24, 2008
Owner Ownership
Porsche 42.6%
Lower Saxony
(Niedersachsen)20.1%
Loans for shorts 13%
Floatavailable for trade
<25%
Sunday October 26, 2008
Owner Ownership
Porsche 42.6%
Lower Saxony
(Niedersachsen)20.1%
Loans for shorts 13%
PorscheCash-settled
options31.5%
> 100%Monday October 27, 2008:
VW roseStock owners wanted to sell
Short sellers were called on the stock loan
Stock rose even more VW stock rose 286%
Short selling strategy• Trades based on negative company information
• SEC filings, litigations, etc.
• Very risky • Takeovers
• Short squeeze
• Unlimited liability
• Short selling gurus
71
wi@�
@wi= wj
@�
@wj
@�
@wi=
@�
@wj
Using Lagrange multipliers, the minimum variance portfolio is obtained when
r�2 is orthogonal to the planegiving us the total wealth restriction
Pni=1wi = 1
Therefore the minimum variance portfolio occurs when
@�
@wi=
@�
@wj
wi = wj
Short Interest Ratio =Short interest
Average Daily Volume
4
Equity long short
Long positions
Single name Short
Index Short: Sector hedge
Index Short: Portfolio hedge
A pairs trade
73
A manager will use $1000 to enter into a pairs trade between two stocks
Long 9 A - Short 9 B
Stock A: high quality, low price: $100 Stock B: low quality, high price: $100
General market directions will have no impact on portfolio valuation
Fundamental Growth GARP
Value driven etc…
Trade financing
74
Conservative trade
Item Cost Future value
Cash 1000 1000
9 Long A 900 990
9 short B -900 -945
Short selling fees 0 9
Total 1000 1036
Performance 3.6%
After one year, the manager is right A rose 10% B rose 5%
Aggressive trade
Item Cost Future value
Cash 500 500
9 Long A 900 990
9 short B -900 -945
Short selling fees 0 9
Total 500 536
Performance 7.2%
Impossible trade
Item Cost Future value
Cash 0 0
9 Long A 900 990
9 short B -900 -945
Short selling fees 0 9
Total 0 36
Performance infinity%
Trade margining
75
Initial margin Long positions: 50% collateral (lower risk profile) Short positions: 80% collateral (higher risk profile)
Optimal trade
Item Cost Future value
Cash 270 270
9 Long A 900 990
9 short B -900 -945
Short selling fees 0 9
Total 270 306
Performance 16.67%
Initial Margin calculation
Item Notional Required margin
9 Long A 900 450
9 short B -900 720
Short selling income income -900
Total 270
Variation margin adjusted as a function of the P&L
Margin call
76
Dangerous trade
Item Cost Internadiate value Future value
Cash 270 270 270
9 Long A 900 850 990
9 short B -900 -900 -945
Short selling fees 0 0 9
Variation margin 0 -50Total 270 -50 306
Performance 16.67%XMargin call
Convertible Arbitrage
Convertible BondsWhen a company needs financing, it can resort to three avenues for fund raising:
• Debt: bonds, loans, lines of credit. Companies first choice is to use some one else's money.
• Creditors have are first in line to collect their dues (after governments and lawyers) in the case of default.
• They achieve this through a fixed payment obligation: no upside.
• If that fails, they issue stock. In doing this, they release control and upside potential.
• Share holders are last in the line up of creditors
• But they are entitled to all that is left: maximum upside.
• When both bonds and stocks fail to raise funding, they can resort to a blend: convertible bonds
• They are structured as bonds, offering bond holders the protection arising from an obligation to pay the coupons and the principal.
• But they also offer upside, allowing the convertible bond holders to exchange their bonds into stocks inside a period of time.
Convertible arbitrage• As a consequence convertible bonds are usually issued by
companies with volatile stock prices and lower credit quality
• Investors want to profit from the upside presented by convertible bonds, but if possible they want to hedge the default and market risks
• Hedging is achieved by shorting the underlying stock.
• Hedging is to be done carefully: stock volatility implies upside potential for stock valuation, but the convertibility properties of the bond mitigate this
• Depending on the view of the manager, more or less of the underlying stock will be done as a hedge, which could totally eliminate the upside potential of the bond
A convertible arbitrage example:
80
Convertible bond sold at $80
Can be converted into 10 shares anytime
Stock price at $7
Annual coupon payment of $4
Interest rates at 4%
Scenario 1: Calm
81
Convertible bond sold at $80
Annual coupon payment of $4
Interest rates if 4%
Now A year later
Convertible bond $80 $80
Stock -$70.00 -$70.00
Coupon $4.00
T-Bill $70.00 $72.80
Fees -$3.50
Total $80.00 $83.30
Performance (4+0.125)%
Scenario 2: Bankruptcy
82
Convertible bond sold at $80
Annual coupon payment of $4
Interest rates if 4%
Now A year later
Convertible bond $80 $50
Stock -$70.00 $0.00
Coupon
T-Bill $70.00 $72.80
Fees -$3.50
Total $80.00 $119.30
Performance (4+46)%
Recovery rate
Scenario 3: Explosive success
83
Convertible bond sold at $80
Annual coupon payment of $4
Interest rates if 4%
Now A year later
Convertible bond $80 $140
Stock -$70.00 -$140.00
Coupon
T-Bill $70.00 $72.80
Fees -$3.50
Total $80.00 $79.30
Performance (4-4)%
After conversion
Scenario 4-5: worsening
84
Convertible bond sold at $80
Annual coupon payment of $4
Interest rates if 4%
Now A year later (1) A year later (2)
Convertible bond
$80 $73 $70
Stock -$70.00 -$60.00 -$60.00
Coupon $4.00 $4.00
T-Bill $70.00 $72.80 $72.80
Fees -$3.50 -$3.50
Total $80.00 $86.30 $83.30
Performance (4+4)% (4+0)%
Normally bonds decrease less than stocks, due to the guarantee
Scenario 6-8: improving
85
Convertible bond sold at $80
Annual coupon payment of $4
Interest rates if 4%
Now A year later (1) A year later (2) A year later (2)
Convertible bond $80 $91 $88 $85
Stock -$70.00 -$80.00 -$80.00 -$80.00
Coupon $4.00 $4.00 $4.00
T-Bill $70.00 $72.80 $72.80 $72.80
Fees -$3.50 -$3.50 -$3.50
Total $80.00 $84.30 $81.30 $78.30
Performance (4+1)% (4-2)% (4-6)%
Exercise: Optimizing the trade
86
20
5.1.5.2 Scenario Analysis
In the following we will examine an example for convertible arbitrage that is constructed as
follows:
Chart 1.20
Scenario assumptions:
o Today: Long in bond (# 1) and short in stocks (# 10)
o Tomorrow: Three different scenarios with probability of 1/3
o Fee: 5%
o Risk-free rate rf: 4%
o Scenario 1: Increase of company value Æ converting the bond into 10 shares
o Scenario 2: Decrease of company value Æ not converting the bond
o Scenario 3: Default of company Æ getting recovery rate
Today S1 S2 S3 Bond 80 140 80 30 Stock -70x -140x -70x 0 T-Bill 70x 72.8x 72.8x 72.8x Coupon 0 4 0 Fee -3.5x -3.5x -3.5x Total 80 140 – 70.7x 84 – 0.7x 30 + 69.3x
Chart 1.21
20
5.1.5.2 Scenario Analysis
In the following we will examine an example for convertible arbitrage that is constructed as
follows:
Chart 1.20
Scenario assumptions:
o Today: Long in bond (# 1) and short in stocks (# 10)
o Tomorrow: Three different scenarios with probability of 1/3
o Fee: 5%
o Risk-free rate rf: 4%
o Scenario 1: Increase of company value Æ converting the bond into 10 shares
o Scenario 2: Decrease of company value Æ not converting the bond
o Scenario 3: Default of company Æ getting recovery rate
Today S1 S2 S3 Bond 80 140 80 30 Stock -70x -140x -70x 0 T-Bill 70x 72.8x 72.8x 72.8x Coupon 0 4 0 Fee -3.5x -3.5x -3.5x Total 80 140 – 70.7x 84 – 0.7x 30 + 69.3x
Chart 1.21
Consider the trade:
1 convertible bond short 10x stocks (0<x<1)
We optimize the hedge as follows
21
Target Function: max (P/V)
In a next step, we want to find the optimal portfolio construction for maximizing the Sharpe
ratio. Therefore we have to calculate the mean return and standard deviation in the future
based on the three scenarios presented before to solve the optimization problem: Max ((P -
rf) / V).
We are looking for the optimal number of stocks x (see Chart 1.x). As result for x we get
7.857 (= 78.57%) as one can see in Chart 1.9 and a maximal Sharpe ratio of 1.72.
Sharpe Ratio dependent on number of stocks
Chart 1.22
Leverage
Once we have assessed the optimal number of stocks, we want to know how much money is
needed to execute the hedge. We are able to examine the trade with leverage under
following conditions / constraints for the collateral:
o Hold back 50% of long position
o Hold back 80% of short position
As we are long in the bond, we have to hold back 50% of 80 (= 40). Additional to that, we are
7.86 short in stocks (assuming we hold the optimal portfolio) with a price of 7 which gives in
total 54.996. 80% of that position gives 43.997. Moreover we have to pay a fee of 5%. So the
overall money we need for executing the hedge is (40 – 55) + 44 + 0.05 * 55 = 31.75 (margin
call). The expected payoff of the collateralized hedge is 84.11 which equals an expected
return of 5.15%. So in all scenarios we have a positive return.
00.20.40.60.8
11.21.41.61.8
2
0 2 4 6 8 10
Sha
rpe
Rat
io
# stocks
Portfolio manager views:
Lead to optimal hedge
Citibank' convertible trade 2009
Friday April 17, 2009
• Citi's convertible preferred shares series T C_pi.N traded at around $34.25
• Terms of Citi's exchange offering, • $50 face value
• is discounted by 15 percent,
• and can then buy shares at $3.25 apiece,
• which translates to about 13.08 shares.
• Citi shares trade at $3.65 a piece, • 13.08 shares are worth about $47.73.
17/04/2009
C
Price $3.65
Borrow rate 25%
C_p Series T
Traded price $34.25
Face value $50.00
Discount 15%
Conversion price $3.25
Conversion ratio $13.08
Implied value $47.73
Market values
Prospectus valuesC
alculated values
$50 x 85%$3.25
Convertibles in History• MCI, the telecommunication contender to ATT, started with a valuation of $161M in 1978 raised
$150M in convertibles in three stages, with stock surging forcing conversion, making bonds disappear from its balance sheet, allowing for more bond issuance: it reached the valuation of $2Bn in 1983,
• In the 1987 stock market crash, many convertible bonds declined more than the stocks into which they were convertible, for liquidity reasons: the market for the stocks being much more liquid than the relatively small market for the bonds. Arbitrageurs gained less from their short stock positions than they lost on their long bond positions.
• Many convertible arbitrageurs suffered losses in early 2005 when the credit of General Motors was downgraded at the same time Kirk Kerkorian was making an offer for GM's stock. Since most arbitrageurs were long GM debt and short the equity, they were hurt on both sides
• Banks used convertible bonds in 2008 to raise capital
• The SEC-FSA surprise short selling ban (on September 18 2008) on financial stocks caused convertible arbitrageurs to flee the market, causing un-intended consequences for the investor appetite for bank convertible bonds
• Convertible bonds in Asia, where short selling is restricted, offer no hedging possibilities and are purchased by investors for their investment upside potential only
Merger/event arbitrageAlcatel Lucent Merger
(April 30, 2001)
Merger Arbitrage tradeItem Future value
Long Lucent Bond +7.57%
Short Alcatel Bonds 0%
Alcatel - Lucent (April 29)Company Rating
Alcatel A-
Lucent BBB-
Convergence Trade
Event Risk: Deal was off a month later
Bonds came back
Fixed income• Exploit pricing differences in fixed income
securities, with the expectation that prices will revert to their true value over time.
• swap-spread arbitrage,
• yield curve arbitrage
• capital structure arbitrage
• Relative value
TED-spreads
91
TED spread = 3-month LIBOR - 3-month T-billTED spread indicator of
credit risk
in the general economy
TED-spread trade
92
T-Bill Eurodollar TradeItem Position Current value Future value
T-bill long 94.2 93.95
Eurodollar futures short -93.1 -92.7
Pairs trade(Relative value) spread 1.1 1.25
Margin = $10 Invested capital = $11.10
P&L = 0.15 Return = 1.35%
Annualized return = 5.4%
OIS Spread
93
OIS spread = difference between London Interbank Offered Rate (LIBOR)
and Overnight Indexed Swap (OIS) rate
Credit risk
No credit risk
Relative value arbitrage
94
1 € =
.6242 Deutsche Marks, 1.332 French francs,
.08784 British pounds, 151.8 Italian Lira,
.2198 Dutch Guilder, 3.431 Belgian Franc,
6.885 Spanish Peseta, .1976 Danish Krone,
.00855 Irish Punt, 1.44 Greek Drachma and
1.393 Portuguese Escudo.
On December 31, 1998, the Euro would officially become a currency. Demand for Euros was so high that one could buy all of the constituent currencies of the Euro
for 98.25% of the value of the Euro
Late in 1998…
Reg-D & PIPEs
95
Regulation D is part of the US Securities Act of 1933 that simplifies filing requirements for companies that sell securities exclusively through
private placements. Private Placements on Public Companies (PIPEs)
allows companies with public shares to issue new restricted shares through private deals,
usually at a heavy discount PIPES cannot be sold for two years,
but companies often file with the exchange before that.
Sarbanes-Oxley has made filing for new issues difficult and slow and has increased the popularity of PIPEs.
Distressed securities
96
Now Corporate actions A year later
Distressed bond $50M Restructure the debt $70M
Coupon $11MHire new management -$10M -$10M
Total $50M $71MPerformance 42%
Key skill-set of the management company
Global MacroAims to profit from variations in securities prices according to forecasts of world economies, political developments and macroeconomic variables
Managers see to profit taking positions on
• Futures
• Currencies
• Indices
• Commodities
• Interest rates
97
Soros and the GBP
98
All European central banks were intervening in the early 90’s to keep
the GBP within the 3% boundary defined by the European monetary system
Macro managers were convinced the British pound had to be devalued.
They took short positions on the GBP against
the continental European currencies in 1992.
When the GBP was finally allowed to move freely, it dropped 20%.
Managed Futures - CTA• Trade in the futures markets only
• Investors provide initial margin or good faith deposit • Variation margin needs to be topped up at the end of every day
• Provides inexpensive access to leverage
• Futures contracts have no value: they represent only a future obligation. Not a security.
• Regulated in the U.S. by the Commodity Futures Trading Commission (CFTC)
99
Futures markets in history• The earliest known futures exchange was established in Japan in 1710 for trading rice futures, although informal futures trading in metals took place in England as far back as 1571
• In the middle ages, buyers and sellers of commodities met annually at trading fairs to lock in future needs and prices. Middlemen provided banking and storage to facilitate trade.
• The Chicago Board of Trade (CBOT) was formed in 1848 and remains one of the largest futures exchanges in the world.
• Permits economic certainty and increased economic activity, acting as insurance
• Liquid, public markets preclude special “inside” information. Extremely regulated. Cannot trade off-exchange.
• Futures exchanges • CBOT (Agro, Interest rates) • the Chicago Mercantile Exchange (CME) (Currencies, Financials) • the New York Mercantile Exchange (NYMEX) (Food and Fiber, Metals) • OneChicago, which trades futures on single stocks and exchange-traded funds (ETFs). • London Metal Exchange (LME) • ICE Futures Europe.
100
Types of CTA• Trend followers
• Short term traders
• Fundamental
• Mechanical
• Discretionary
101
No Credit risk No liquidity risk
No capacity constraints
Model risk: sharp reversals
Leverage Event risk
CTA performance
102
Stock market Crises
Macro-CTA
103
62 © 2016 Preqin Ltd. / www.preqin.com
Macro StrategiesFunds
79%
5%
16%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Pre
-200
020
0020
0120
0220
0320
0420
0520
0620
0720
0820
0920
1020
1120
1220
1320
1420
15
Foreign Exchange
Commodities
Macro
Fig. 5.9: Breakdown of Macro Strategies Fund Launches by Strategy and Year of Inception
Source: Preqin Hedge Fund Analyst
Pro
po
rtio
n o
f Fun
d L
aun
che
s
Year of Inception
77%
14%
9%
Macro
Commodities
Foreign Exchange
Fig. 5.10: Breakdown of Macro Strategies Funds by Strategy
Source: Preqin Hedge Fund Analyst
Key Facts
642642Number of hedge fund managers offering
a macro strategies fund.
Number of active macro strategies funds
in market.
Number of institutions investing in macro
strategies funds.
1,0711,071
+1.32%+1.32%The highest monthly return generated
by macro strategies hedge funds in 2015 (posted in March).
-7.24%-7.24%Commodities hedge funds returned -7.24% in 2015; in comparison, macro
funds made gains of 4.24%.
$64.94bn$64.94bnSize of the largest macro strategies fund,
Bridgewater Pure Alpha Strategy 12%.
2,1002,100
Data Source:
Preqin’s Hedge Fund Analyst provides detailed information on over 1,000 macro strategies hedge funds.
Comprehensive profi les include assets under management, monthly returns, strategy and regional preferences, and much more.
For more information, please visit:
www.preqin.com/hfa
Ð
24%
17%
14%
9%
9%
6%
6%
5%4%
6%
Fund of Hedge FundsManagerFoundation
Private Sector PensionFundEndowment Plan
Public Pension Fund
Family Office
Wealth Manager
Asset Manager
Insurance Company
Other
Fig. 5.11: Breakdown of Investors in Macro Strategies Funds by Type
Source: Preqin Hedge Fund Investor Profi les
5. Overview of the Hedge Fund Industry by Strategy
The 2016 Preqin Global Hedge Fund Report - Sample Pages
62 © 2016 Preqin Ltd. / www.preqin.com
Macro StrategiesFunds
79%
5%
16%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Pre
-200
020
0020
0120
0220
0320
0420
0520
0620
0720
0820
0920
1020
1120
1220
1320
1420
15
Foreign Exchange
Commodities
Macro
Fig. 5.9: Breakdown of Macro Strategies Fund Launches by Strategy and Year of Inception
Source: Preqin Hedge Fund Analyst
Pro
po
rtio
n o
f Fun
d L
aun
che
s
Year of Inception
77%
14%
9%
Macro
Commodities
Foreign Exchange
Fig. 5.10: Breakdown of Macro Strategies Funds by Strategy
Source: Preqin Hedge Fund Analyst
Key Facts
642642Number of hedge fund managers offering
a macro strategies fund.
Number of active macro strategies funds
in market.
Number of institutions investing in macro
strategies funds.
1,0711,071
+1.32%+1.32%The highest monthly return generated
by macro strategies hedge funds in 2015 (posted in March).
-7.24%-7.24%Commodities hedge funds returned -7.24% in 2015; in comparison, macro
funds made gains of 4.24%.
$64.94bn$64.94bnSize of the largest macro strategies fund,
Bridgewater Pure Alpha Strategy 12%.
2,1002,100
Data Source:
Preqin’s Hedge Fund Analyst provides detailed information on over 1,000 macro strategies hedge funds.
Comprehensive profi les include assets under management, monthly returns, strategy and regional preferences, and much more.
For more information, please visit:
www.preqin.com/hfa
Ð
24%
17%
14%
9%
9%
6%
6%
5%4%
6%
Fund of Hedge FundsManagerFoundation
Private Sector PensionFundEndowment Plan
Public Pension Fund
Family Office
Wealth Manager
Asset Manager
Insurance Company
Other
Fig. 5.11: Breakdown of Investors in Macro Strategies Funds by Type
Source: Preqin Hedge Fund Investor Profi les
5. Overview of the Hedge Fund Industry by Strategy
The 2016 Preqin Global Hedge Fund Report - Sample Pages
Blended styles• Friday March 14, 2008, Hedge Funds were selling CDS on Bear Stearns at 10% of notional up front
• Monday March 17, 2008: JPMorgan announces it will purchase Bear Stearns for $2/share
• CDS spreads on Bear Stearns drop to par with JPMorgan’s: 1% up-front
• Hedge funds that sold CDS on Friday start to purchase BS stock to vote in favor of the take over
• Price on BS stock reaches $10/share before the end of the week
104
Investment Products
Investment products• Indices
• Fund of funds
• Leveraged products • Loans
• Options
• CFO’s
• Guaranteed products
• Managed accounts
106
Hedge Fund IndicesThere are two types of hedge fund indices:
• Investable • They are structured as investment vehicles so accredited investors can invest
• The are constructed according to published methodologies but with constraints, as not all hedge funds will accept investments
• Un-sophisticated investors chose them as they provide a way to fulfil an investment mandate without the burden of constructing a HF portfolio
• Different vendors produce different indices with very different return streams
• Around since about 2003
• Non-investable • Published as an average of returns reported by hedge funds
• Used for benchmarking purposes
• Around since about 1993
107
Fund of FundsA fund whose assets are invested in individual hedge funds
• The fund usually seeks manager diversification • Sometimes they seek style diversification
• Other times they seek style themes
• ELS, CTA, etc.
• Risk premia (liquidity, credit, etc.)
• They can invest in a handful to hundred individual funds
• They are pooled funds, and usually require lower investment minimums than individual hedge funds
• They have access to leverage
• Sometimes they are marketing platforms for hedge funds (they market the portfolio)
• Others are fiduciaries and act in the best interest of investors (they manage the portfolio)
108
The fund-of-funds structure
109
Fund of Funds
Investor 1 Investor 1 Investor 1 Investor 1
Management Company
Administrator
Auditor
Bank
Hedge Fund 1
Owners
Service Providers
Conflict free
Hedge Fund 2
Hedge Fund 3
……
Portfolio Construction
1/N• Markowitz theory is very dependent
on the estimations for expected return and standard deviation
• Leads to concentrated portfolios
• Some authors have proposed simpler allocation techniques
• One of them is the one that allocates equal weights to all assets in the portfolio, independent of expected return or risk
• The authors show it outperforms mean variance optimization
111
How Ine�cient is the 1/N Asset-Allocation Strategy?⇤
Victor DeMiguel† Lorenzo Garlappi‡ Raman Uppal§
December 1, 2005
⇤We wish to thank John Campbell and Luis Viceira for their suggestions and for making available theirdata and computer code and Roberto Wessels for making available data on the ten industry sectors ofthe S&P500 index. We also gratefully acknowledge comments from Suleyman Basak, Michael Brennan,Ian Cooper, Bernard Dumas, Bruno Gerard, Francisco Gomes, Eric Jacquier, Chris Malloy, Narayan Naik,Lubos Pastor, Anna Pavlova, Sheridan Titman, Rossen Valkanov, Tan Wang, Yihong Xia, Pradeep Yadav,Zhenyu Wang and seminar participants at BI Norwegian School of Management, HEC Lausanne, HECMontreal, London Business School, University of Mannheim, University of Texas, University of Vienna, theInternational Symposium on Asset Allocation and Pension Management at Copenhagen Business School,the conference on Developments in Quantitative Finance at the Isaac Newton Institute for MathematicalSciences at Cambridge University, Second McGill Conference on Global Asset Management, and the 2005meetings of the Western Finance Association.
†London Business School, 6 Sussex Place, Regent’s Park, London, United Kingdom NW1 4SA; Email:[email protected].
‡McCombs School of Business, The University of Texas at Austin, Austin TX, 78712; Email:[email protected]. Corresponding author.
§London Business School and CEPR; IFA, 6 Sussex Place, Regent’s Park, London, United Kingdom NW14SA; Email: [email protected].
MotivationObjective
MethodologyResults Conclusion
Motivation
I Ancient wisdom
• Rabbi Issac bar Aha (Talmud, 4th Century): Equal allocation
A third in land, a third in merchandise, a third in cash.
I More “recent” wisdom
• de Finetti (2006, 1940), Markowitz (1952);
• Tobin (1958), Sharpe (1964) and Lintner (1965);
• Samuelson (1969), Merton (1969, 1971)
DeMiguel, Garlappi & Uppal1/N
1
Motivation Objective Methodology Results Conclusion
Conclusions
I Empirical analysis shows that
• None of the optimizing models consistently dominate 1/N ;
? 1/N often has a higher Sharpe ratio and CEQ than optimal portfolio strategies;
? 1/N almost always has lower turnover.
I Analytical results indicate that
• Critical length of estimation window needed is unreasonably large;
• Critical length of estimation window needed increases with N .
I Simulation results show that
• Constraints may not help if expected returns need to be estimated (if N is large);
• Critical estimation window for mv similar to other optimal portfolio models.
• High idiosyncratic volatility improves relative performance of optimal models.
DeMiguel, Garlappi & Uppal 1/N 29
Equal Risk Contribution
µ =
Z 1
�1x⇢(x) dx
= E(X)
�2 =
Z 1
�1(x� µ)2⇢(x) dx
= E(X � µ)2
Consider N assets with returns given by random variables Xi, i = 1, . . . , N , anda portfolio ⇧ with allocations wi.For simplicity assume constant pairwise correlations C, equal asset allocations,and also equal means and variances µ and �. Then
�2⇧ =
X
i
w2i �
2i +
X
i 6=j
wiwj�i�j
= C +�2 � C
NWe seek a portfolio ⇧ that maximizes the probability of exceeding a known, butrandom, benchmark Y + r.If returns are normally distributed:
Prob {⇧ � Y + r} = Prob {(⇧� Y ) � r}
= Prob
⇢⇧� Y � (µ⇧ � µY )
�⇧�Y� r � µ⇧�Y
�⇧�Y
�
= 1� �
✓r � µ⇧�Y
�⇧�Y
◆
µ⇧ � µY � r
�⇧�Y
s =E(X � µ)3
�3
Y ⇡ ↵ + �1 ·X1 + �2 ·X2 + · · · + �n ·Xn + Error
�i =Cov(Y,Xi)
�2Y
Consider a portfolio ⇧ that allocates wi to assets with a variance/covariancematrix given by V = {�i,j}
� =pw · V · wT
=nX
i=1
wi@�
@wi
3
Marginal risk contribution
risk contribution of the i’th asset
{wi
@�
@wi= wj
@�
@wj
@�
@wi=
@�
@wj
4
The Equal Risk Contribution portfolio is such that
112
Minimum variance portfolio
113
wi@�
@wi= wj
@�
@wj
@�
@wi=
@�
@wj
Using Lagrange multipliers, the minimum variance portfolio is obtained whenr�2 is orthogonal to the plane giving us the total wealth restriction
Pni=1 wi = 1
Therefore the minimum variance portfolio occurs when
@�
@wi=
@�
@wj
4
Three allocation methodologies
wi@�
@wi= wj
@�
@wj
@�
@wi=
@�
@wj
Using Lagrange multipliers, the minimum variance portfolio is obtained whenr�2 is orthogonal to the plane giving us the total wealth restriction
Pni=1 wi = 1
Therefore the minimum variance portfolio occurs when
@�
@wi=
@�
@wj
wi = wj
4
wi@�
@wi= wj
@�
@wj
@�
@wi=
@�
@wj
Using Lagrange multipliers, the minimum variance portfolio is obtained whenr�2 is orthogonal to the plane giving us the total wealth restriction
Pni=1 wi = 1
Therefore the minimum variance portfolio occurs when
@�
@wi=
@�
@wj
4
wi@�
@wi= wj
@�
@wj
@�
@wi=
@�
@wj
Using Lagrange multipliers, the minimum variance portfolio is obtained whenr�2 is orthogonal to the plane giving us the total wealth restriction
Pni=1 wi = 1
Therefore the minimum variance portfolio occurs when
@�
@wi=
@�
@wj
4
Equal weights
Equal Risk Contributions
Minimum Variance
114
Applications
115
Assume uncorrelated assets
1
n
F (x) = Prob {P&L x}
=
Z x
�1⇢(r) dr
Z �VaR↵
�1⇢(r) dr = Prob{Losses � VaR↵}
= 1� ↵
CVaR↵ =1
↵
Z ↵
0VaR1�� d�
= E{Loss | Loss � VaR↵}
⌦(r) =
Z 1
r(1� F (x)) dx
Z r
�1F (x) dx
=
Z 1
r(x� r)⇢(x) dx
Z r
�1(r � x)⇢(x) dx
=Expected out-performance
Expected under-performance
=Dembo’s reward
Dembo’s regret
Assume uncorrelated assetsXi and a portfolio with individual allocation weightswi. Then
� =
vuutnX
i=1
w2i �
2i
Therefore@�
@wi=
wi�2i
�
5
Uncorrelated assets
Equal weights
Equal Risk Contributions
Minimum Variance
116
1
n
F (x) = Prob {P&L x}
=
Z x
�1⇢(r) dr
Z �VaR↵
�1⇢(r) dr = Prob{Losses � VaR↵}
= 1� ↵
CVaR↵ =1
↵
Z ↵
0VaR1�� d�
= E{Loss | Loss � VaR↵}
⌦(r) =
Z 1
r(1� F (x)) dx
Z r
�1F (x) dx
=
Z 1
r(x� r)⇢(x) dx
Z r
�1(r � x)⇢(x) dx
=Expected out-performance
Expected under-performance
=Dembo’s reward
Dembo’s regret
Assume uncorrelated assetsXi and a portfolio with individual allocation weightswi. Then
� =
vuutnX
i=1
w2i �
2i
Therefore@�
@wi=
wi�2i
�
5
Figure 8: Two year-rolling window Sharpe ratio (yearly) for the Global Hedge Fund Index, the S&P500 and
the AGG Index
Figure 9: µ/� pairs for the Global Hedge Fund Index, the S&P500 and the AGG Index
13
Allocation to stocks, bonds and hedge funds
Equal weights
Equal Risk Contributions
Minimum Variance
117
Standard deviation Allocation
Bonds 15% 33%
Stocks 30% 33%
Hedge Funds 10% 33%
Standard deviation Allocation
Bonds 15% 33%
Stocks 30% 16%
Hedge Funds 10% 50%
Standard deviation Allocation
Bonds 15% 28%
Stocks 30% 7%
Hedge Funds 10% 64%
If they were uncorrelated … which they are not
Leveraged Products
118
Basic HF
Product
Equity
Debt
Structure
funding
funding
return
yield
Leveraged InvestmentsAn investor has $25M to invest in HF; a bank lends her
$75M to invest in a $100M HF portfolio
An investor gives $25M to a bank that will invest $100M in a HF portfolio for the investor
An investor gives $25M to a bank that gives the investor the return of a $100M investment in a HF portfolio
An investor buys a call option on a $100M portfolio from a bank with a $75M strike price, and pays $25M as
premium
119
Leveraged investments
120
Investor: $25M
Bank: $75M Loan
$100M HF Portfolio
Return
Interest
Investor: $25M
Bank: $75M LoanReturn
Return minus
Interest
Investor: $25M
Bank: Warrant
Investment
Investment
Reference portfolio
Return minus
Interest, fees
Investor: $25M
Bank: CPPI Option Constant Delta
Interest rate hedged
Reference portfolio
PremiumOption Pay-off
Liquidity -> Volatility premium
Collateralized Fund Obligations (CFO)
27%
10%
13%
50%
AAA Tranche A Tranche BBB Tranche Equity tranche
Yield
0 2.5 5 7.5 10Case 1: Gains
Case 2: Loses
Probablity of loss
Recovery rate
0 12.5 25 37.5 50
Bond Investors
HF Investor
121
Bonds default as performance degrades
CFO History
122
Amount Rating (S&P) Yield
$125M AAA Libor + 60bps
$32.5M A Libor+160 bps
$26.2 BBB Libor+250bps
$66.3 -
In June 2002, Man-Glenwood Alternative
Strategies issued the first ever CFO, with
$374M in rated notes and
$176M in un-rated notes and shares
Diversified Strategies CFO also launched later in 2002 with Risk Conditions
20% of the funds in separately managed accounts without lockups or reception restrictions At least 25 different funds At least 4 different strategies
Allocation by volatility and market exposure Immediate liquidation when equity tranche loses its value
Equity investors: Access to leverage Bond investors: uncorrelated asset class Expensive
Returnwith
Downside Protection
Guaranteed Products
123
Basic HF
Product
Equity
Guarantee
Structure
funding
return
Premium
Guaranteed notes
124
2%18%
80%
Zero coupon Free capital Fees
100%
Investor Now
100%
Investor later
?
zero coupon Guarantee
Term guaranteed note
Dependent on Portfolio Performance
Invested with leverage
Fund Pool
GraciasThank you