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

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.

The snow swap

If there is no snow, the city pays the ski resort a fee:

If there is no snow, the ski resort pays the city a fee:

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

$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

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⇧ =

wi ·Xi ,nX

wi ·Xj

wi wj hXi , Xji

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

A diversified fund

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

1-20 Fees

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

Investor 1 Investor 2 Investor 3 Investor 4

Management Company

Administrator

Auditor

Broker (custodian)

Owners

Service Providers

Conflict free

Directors

The fund structure

Management Company

Administrator

Auditor

Broker (custodian)

Owners

Service Providers

Conflict free

Directors

The Master/Feeder structure

Master FUND

Investor 1Investor 2 Investor 3 Investor 4

Management Company

Administrator

AuditorBankBroker

(custodian)

Feeder 1 FUND Feeder 2 FUND

Administrator

Auditor

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

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

Portfolio Return

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

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)

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,

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

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)

return log-return

Share value

In either case, we can collect a time series of portfolio returns Daily Monthly

Portfolio Stats

�2⇧ =

wi ·Xi ,nX

wi ·Xj

wi wj hXi , Xji

wi wj �i,j

= w · V · wT .

Therefore,�⇧ =

pw · V · wT .

⇢⇧� 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

Cumulative return distribution function (CDF)

�2⇧ =

wi ·Xi ,nX

wi ·Xj

wi wj hXi , Xji

wi wj �i,j

= w · V · wT .

Therefore,�⇧ =

pw · V · wT .

⇢⇧� µ

�� r � µ

= 1� �

✓r � µ

µ� r

�2 = 0.75 ⇤ 02 + 0.125 ⇤ 1.102 + 0.125 ⇤ 0.92

⇡ 0.25

⇢(x) =dF

Return density

�2⇧ =

wi ·Xi ,nX

wi ·Xj

wi wj hXi , Xji

wi wj �i,j

= w · V · wT .

Therefore,�⇧ =

pw · V · wT .

⇢⇧� µ

�� r � µ

= 1� �

✓r � µ

µ� r

�2 = 0.75 ⇤ 02 + 0.125 ⇤ 1.102 + 0.125 ⇤ 0.92

⇡ 0.25

⇢(x) =dF

�2⇧ =

wi ·Xi ,nX

wi ·Xj

wi wj hXi , Xji

wi wj �i,j

= w · V · wT .

Therefore,�⇧ =

pw · V · wT .

⇢⇧� µ

�� r � µ

= 1� �

✓r � µ

µ� r

�2 = 0.75 ⇤ 02 + 0.125 ⇤ 1.102 + 0.125 ⇤ 0.92

⇡ 0.25

⇢(x) =dF

F (x) = Prob {return x}

Returns

Statistics - ProbabilityEarth - Heaven

�2⇧ =

wi ·Xi ,nX

wi ·Xj

wi wj hXi , Xji

wi wj �i,j

= w · V · wT .

Therefore,�⇧ =

pw · V · wT .

⇢⇧� µ

�� r � µ

= 1� �

✓r � µ

µ� r

�2 = 0.75 ⇤ 02 + 0.125 ⇤ 1.102 + 0.125 ⇤ 0.92

⇡ 0.25

⇢(x) =dF

Excel: histogramMath: formulas

“On earth as it is in heaven”

Earth and Heaven

Heaven

Portfolio statistics

ri (1)

vuut 1

n� 1

(ri � r)2

ri (1)

vuut 1

n� 1

(ri � r)2

Average return, mean return

Volatility, standard deviation

months (or days)

return

�1x⇢(x) dx

= E(X)

�2 =

�1(x� µ)2⇢(x) dx

= E(X � µ)2

Heaven

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%

Running mean

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

Mean of the prior 2 year time window

2 year window runs ahead daily

Conflict between heaven and earth

Running standard deviation

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

Conflict between heaven and earth

Financial Crisis 2 years after Financial Crisis

Averages over time

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

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

ri (1)

vuut 1

n� 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

�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)

Internal-Rate-of-Return

• Takes into account the amounts invested over time

ri (1)

vuut 1

n� 1

(ri � r)2

Cov(X,Y ) =1

n� 1

�Xi �X

� �Yi � Y

and the correlation

⇢(X,Y ) =Cov(X,Y )

�X · �Y

�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

pk(1 +R)tk

ri (1)

vuut 1

n� 1

(ri � r)2

Cov(X,Y ) = E(X �X)(Y � Y )

n� 1

�Xi �X

� �Yi � Y

and the correlation

⇢(X,Y ) =Cov(X,Y )

�X · �Y

�1 ⇢ 1.

1 +R = (1 + r1)(1 + r2) · · · (1 + rn)

If we make

• at time tk ago

pk(1 +R)tk

500, 001.024 = 1 · (1 +R)11 + 1, 000, 000, 000 · (1 +R)1/12

R ⇡ �100%

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

ri (1)

vuut 1

n� 1

(ri � r)2

Cov(X,Y ) =1

n� 1

�Xi �X

� �Yi � Y

and the correlation

⇢(X,Y ) =Cov(X,Y )

�X · �Y

�1 ⇢ 1.

1 +R = (1 + r1)(1 + r2) · · · (1 + rn)

If we make

• at time tk ago

pk(1 +R)tk

500, 001.024 = 1 · (1 +R)1 + 1, 000, 000, 000 · (1 +R)1/12

R ⇡ �100%

ri (1)

vuut 1

n� 1

(ri � r)2

Cov(X,Y ) = E(X � E(X))(Y � E(Y ))

n� 1

�Xi �X

� �Yi � Y

and the correlation

⇢(X,Y ) =Cov(X,Y )

�X · �Y

�1 ⇢ 1.

1 +R = (1 + r1)(1 + r2) · · · (1 + rn)

If we make

• at time tk ago

pk(1 +R)tk

500, 001.024 = 1 · (1 +R)11 + 1, 000, 000, 000 · (1 +R)1/12

R ⇡ �100%

Correlation

Heaven

Portfolio volatility

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⇧ =

wi ·Xi ,nX

wi ·Xj

wi wj hXi , Xji

wi wj �i,j

= w · V · wT .

Therefore,�⇧ =

pw · V · wT .

⇢⇧� µ

�� r � µ

= 1� �

✓r � µ

µ� r

�2 = 0.75 ⇤ 02 + 0.125 ⇤ 1.102 + 0.125 ⇤ 0.92

⇡ 0.25

⇢(x) =dF

F (x) = Prob {return x}

Implies portfolio diversification:

When correlations are

less then 1

or even negative

portfolio volatility decreases

Marginal value of diversification

�1x⇢(x) dx

= E(X)

�2 =

�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⇧ =

w2i �

wiwj�i�j

= C +�2 � C

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

Portfolio Diversification

Risk can be lowered combining assets in a portfolio

Efficient Frontier

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:

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.

�2⇧ =

wi ·Xi ,nX

wi ·Xj

wi wj hXi , Xji

wi wj �i,j

= w · V · wT .

Therefore,�⇧ =

pw · V · wT .

⇢⇧� µ

�� r � µ

= 1� �

✓r � µ

The rational decision is to maximize the Sharpe Ratio with benchmark r

�2⇧ =

wi ·Xi ,nX

wi ·Xj

wi wj hXi , Xji

wi wj �i,j

= w · V · wT .

Therefore,�⇧ =

pw · V · wT .

⇢⇧� µ

�� r � µ

= 1� �

✓r � µ

µ� r

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

�1x⇢(x) dx

= E(X)

�2 =

�1(x� µ)2⇢(x) dx

= E(X � µ)2

�2⇧ =

w2i �

wiwj�i�j

= C +�2 � C

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

�1x⇢(x) dx

= E(X)

�2 =

�1(x� µ)2⇢(x) dx

= E(X � µ)2

�2⇧ =

w2i �

wiwj�i�j

= C +�2 � C

Prob {⇧ � Y + r} = Prob {(⇧� Y ) � r}

= Prob

⇢⇧� Y � (µ⇧ � µY )

�⇧�Y� r � µ⇧�Y

�⇧�Y

= 1� �

✓r � µ⇧�Y

�⇧�Y

µ⇧ � µY � r

�⇧�Y

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

Alpha - beta

• Hedge funds

• Absolute return strategies

• Active portfolio management

• Prop-desks

• Stocks

• Bonds

• Infrastructure

• Private Equity

• Credit

Efficient frontier

The rationale for including Hedge Funds in a portfolio

Traditional - Alternative

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

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

Alphas, betas

Simultaneous monthly returns

statistical relationship

return differential

meaningful

Mathematical foundation

Returns under consideration Benchmark/market/external returns

�1x⇢(x) dx

= E(X)

�2 =

�1(x� µ)2⇢(x) dx

= E(X � µ)2

�2⇧ =

w2i �

wiwj�i�j

= C +�2 � C

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

Y ⇡ ↵ + �1 ·X1 + �2 ·X2 + · · · + �n ·Xn + Error

�1x⇢(x) d

= E(X)

�2 =

�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

�2⇧ =X

w2i�

wiwj�i�j

= C +�2 � C

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)

Y ⇡ ↵ + �1 ·X1 + �2 ·X2 + · · · + �n ·Xn + Error

�i =Cov(Y,

If factors are independent

Linear regression

Multi-dimensional regression

Non-normal returns

Gaussian Fit

Events impossible

according to Gaussian Fit

Statistics Time structure

• Moving windows

• EWMA

• ARMA

• GARCH

Semi-standard deviation

wi@�

@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

Short Interest Ratio =Short interest

Average Daily Volume

Semi-standard (loss) deviation

s� =

vuut 1

n� 1

(R⇤ � ri)2+

Heaven?

Number of all months? Or

number of months with losses?

Some sort of target return Benchmark

Only penalize if return is bad (below benchmark..

..or negative)

Volatility of losses

wi@�

@wi= wj

Pni=1 wi = 1

wi = wj

s� =

vuut 1

n� 1

(R⇤ � ri)2+

Sortino Ratio =µ� r

The standard deviation of negative assets returns is

�loss =

vuutnX

(ri)2� �

(ri)�

Monthly loss Average loss

aportfolio

⇧that

allocatesw

assetswith

returnsgiven

variables

Xi ,i=

1, . . . , n.

Ifthecovarian

cematrix

givenby

(�i,j ,

theportfolio

volatilityis

�2⇧=

wi ·X

j�i,j

=w· V

· wT.

Therefore,

�⇧=

pw· V

· wT.

Weseek

aportfolio

⇧that

izestheprob

ability

ofexceed

ingafixed

bench-

r.Ifretu

rnsare

allydistrib

{⇧�

⇢⇧�µ

��

r�µ

=1��

✓r�µ

µ�r

0.75⇤02+0.125

⇤1.10

2+0.125

⇤0.9

⇡0.25

number

ofshares

⇢(x)=

{return

wi@�

@wi= wj

Pni=1 wi = 1

wi = wj

s� =

vuut 1

n� 1

(R⇤ � ri)2+

�loss =

vuutnX

(ri)2� �

(ri)�

r� =

(r if r 0

0 otherwise

Sortino Ratio

wi@�

@wi= wj

Pni=1 wi = 1

wi = wj

s� =

vuut 1

n� 1

(R⇤ � ri)2+

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.

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

Skewness Kurtosis

wi@�

@wi= wj

Pni=1 wi = 1

wi = wj

s� =

vuut 1

n� 1

(R⇤ � ri)2+

s�The standard deviation of negative assets returns is

�loss =

vuutnX

(ri)2� �

(ri)�

r� =

(r if r 0

0 otherwise

(n� 1)(n� 2)

(ri � r)3

�3E�X �X

n(n+ 1)

(n� 1)(n� 2)(n� 3)

(ri � r)4 � 3(n� 1)2

(n� 2)(n� 3)

�4E�X �X

Heaven

Platykurtotic: k<0

Leptokurtotic: k>0

Mesokurtotic: k=0Positive skew s>0 Negative skew s<0

wi@�

@wi= wj

Pni=1 wi = 1

wi = wj

s� =

vuut 1

n� 1

(R⇤ � ri)2+

�loss =

vuutnX

(ri)2� �

(ri)�

r� =

(r if r 0

0 otherwise

(n� 1)(n� 2)

(ri � r)3

�3E�X �X

n(n+ 1)

(n� 1)(n� 2)(n� 3)

(ri � r)4 � 3(n� 1)2

(n� 2)(n� 3)

�4E�X �X

�4 � 3

Skewness Kurtosis

wi@�

@wi= wj

Pni=1 wi = 1

wi = wj

s� =

vuut 1

n� 1

(R⇤ � ri)2+

�loss =

vuutnX

(ri)2� �

(ri)�

r� =

(r if r 0

0 otherwise

(n� 1)(n� 2)

(ri � r)3

n(n+ 1)

(n� 1)(n� 2)(n� 3)

(ri � r)4 � 3(n� 1)2

(n� 2)(n� 3)

Morally Morally 3 (the fourth central moment of the gaussian)

wi@�

@wi= wj

Pni=1 wi = 1

wi = wj

s� =

vuut 1

n� 1

(R⇤ � ri)2+

�loss =

vuutnX

(ri)2� �

(ri)�

r� =

(r if r 0

0 otherwise

(n� 1)(n� 2)

(ri � r)3

�3E�X �X

n(n+ 1)

(n� 1)(n� 2)(n� 3)

(ri � r)4 � 3(n� 1)2

(n� 2)(n� 3)

�4E�X �X

wi@�

@wi= wj

Pni=1 wi = 1

wi = wj

s� =

vuut 1

n� 1

(R⇤ � ri)2+

�loss =

vuutnX

(ri)2� �

(ri)�

r� =

(r if r 0

0 otherwise

(n� 1)(n� 2)

(ri � r)3

�3E�X �X

n(n+ 1)

(n� 1)(n� 2)(n� 3)

(ri � r)4 � 3(n� 1)2

(n� 2)(n� 3)

�4E�X �X

Estimation problems

Outliers have a very dramatic effect in the sample calculation of the historical moments

wi@�

@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

wi = wj

s� =

vuut 1

n� 1

(R⇤ � ri)2+

�loss =

vuutnX

(ri)2� �

(ri)�

r� =

(r if r 0

0 otherwise

(n� 1)(n� 2)

(ri � r)3

�3E�X �X

n(n+ 1)

(n� 1)(n� 2)(n� 3)

(ri � r)4 �3(n� 1)2

(n� 2)(n� 3)

�4E�X �X

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

(R⇤ � ri)2+

Sortino Ratio =µ� rs�The standard deviation of negative assets returns is

�loss =

vuutnX

(ri)2� �

(ri)�

r� =

(r if r 00 otherwise

(n� 1)(n� 2)

(ri � r)3

�3E�X �X

�4n(n+ 1)

(n� 1)(n� 2)(n� 3)

(ri � r)4 � 3(n� 1)2

(n� 2)(n� 3)=

�4E�X �X

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

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

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).

Uselessness of skewness

55© Luis Seco. Not to be reproduced without permission

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.

Omega definition

F (x) = Prob {P&L x}

�1⇢(r) dr

Z �VaR↵

�1⇢(r) dr = Prob{Losses � VaR↵}

= 1� ↵

CVaR↵ =1

0VaR1�� d�

= E{Loss | Loss � VaR↵}

⌦(r) =

r(1� F (x)) dx

�1F (x) dx

r(x� r)⇢(x) dx

�1(r � x)⇢(x) dx

=Expected out-performance

Expected under-performance

=Dembo’s reward

Dembo’s regret

Expected out-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

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.

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.

Equities

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

Correlation switching

Normal correlations

Distressed correlations

Trading strategies

Strategy breakdown

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%

Convertible Arbitrage

Distressed Securities

Emerging M

arkets

Equity Long Bias

Equity Long/Short

Equity Long-Only

Equity Market N

eutral

Event Driven

Fixed Income

Merger Arbitrage

Multi-Strategy

Other **

Sector Specific ***

Equity based trading

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

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

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

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.

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

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

wi@�

@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

wi = wj

Equity long short

Long positions

Single name Short

Index Short: Sector hedge

Index Short: Portfolio hedge

A pairs trade

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

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

Cash 500 500

9 Long A 900 990

9 short B -900 -945

Total 500 536

Performance 7.2%

Impossible trade

Cash 0 0

9 Long A 900 990

9 short B -900 -945

Total 0 36

Performance infinity%

Trade margining

Initial margin Long positions: 50% collateral (lower risk profile) Short positions: 80% collateral (higher risk profile)

Optimal trade

Cash 270 270

9 Long A 900 990

9 short B -900 -945

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

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:

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

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

Now A year later

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

Now A year later

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

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

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

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

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

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

0 2 4 6 8 10

# 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

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

TED spread = 3-month LIBOR - 3-month T-billTED spread indicator of

credit risk

in the general economy

TED-spread trade

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

OIS spread = difference between London Interbank Offered Rate (LIBOR)

and Overnight Indexed Swap (OIS) rate

Credit risk

No credit risk

Relative value arbitrage

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

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

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

Soros and the GBP

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)

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.

Types of CTA• Trend followers

• Short term traders

• Fundamental

• Mechanical

• Discretionary

No Credit risk No liquidity risk

No capacity constraints

Model risk: sharp reversals

Leverage Event risk

CTA performance

Stock market Crises

Macro-CTA

Macro StrategiesFunds

Foreign Exchange

Commodities

Fig. 5.9: Breakdown of Macro Strategies Fund Launches by Strategy and Year of Inception

Source: Preqin Hedge Fund Analyst

Year of Inception

Commodities

Foreign Exchange

Fig. 5.10: Breakdown of Macro Strategies Funds by Strategy

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

Fund of Hedge FundsManagerFoundation

Private Sector PensionFundEndowment Plan

Public Pension Fund

Family Office

Wealth Manager

Asset Manager

Insurance Company

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

Macro StrategiesFunds

Foreign Exchange

Commodities

Fig. 5.9: Breakdown of Macro Strategies Fund Launches by Strategy and Year of Inception

Year of Inception

Commodities

Foreign Exchange

Fig. 5.10: Breakdown of Macro Strategies Funds by Strategy

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

Fund of Hedge FundsManagerFoundation

Private Sector PensionFundEndowment Plan

Public Pension Fund

Family Office

Wealth Manager

Asset Manager

Insurance Company

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

Investment Products

Investment products• Indices

• Fund of funds

• Leveraged products • Loans

• Options

• CFO’s

• Guaranteed products

• Managed accounts

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

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)

The fund-of-funds structure

Fund of Funds

Management Company

Administrator

Auditor

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

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:avmiguel@london.edu.

‡McCombs School of Business, The University of Texas at Austin, Austin TX, 78712; Email:lorenzo.garlappi@mccombs.utexas.edu. Corresponding author.

§London Business School and CEPR; IFA, 6 Sussex Place, Regent’s Park, London, United Kingdom NW14SA; Email: ruppal@london.edu.

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

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

�1x⇢(x) dx

= E(X)

�2 =

�1(x� µ)2⇢(x) dx

= E(X � µ)2

�2⇧ =

w2i �

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

Y ⇡ ↵ + �1 ·X1 + �2 ·X2 + · · · + �n ·Xn + Error

�i =Cov(Y,Xi)

Consider a portfolio ⇧ that allocates wi to assets with a variance/covariancematrix given by V = {�i,j}

� =pw · V · wT

wi@�

Marginal risk contribution

risk contribution of the i’th asset

@wi= wj

The Equal Risk Contribution portfolio is such that

Minimum variance portfolio

wi@�

@wi= wj

Pni=1 wi = 1

Three allocation methodologies

wi@�

@wi= wj

Pni=1 wi = 1

wi = wj

wi@�

@wi= wj

Pni=1 wi = 1

wi@�

@wi= wj

Pni=1 wi = 1

Equal weights

Equal Risk Contributions

Minimum Variance

Applications

Assume uncorrelated assets

�1⇢(r) dr

Z �VaR↵

= 1� ↵

CVaR↵ =1

0VaR1�� d�

⌦(r) =

r(1� F (x)) dx

�1F (x) dx

r(x� r)⇢(x) dx

�1(r � x)⇢(x) dx

=Dembo’s reward

Dembo’s regret

Assume uncorrelated assetsXi and a portfolio with individual allocation weightswi. Then

vuutnX

w2i �

Therefore@�

wi�2i

Uncorrelated assets

Equal weights

Minimum Variance

�1⇢(r) dr

Z �VaR↵

= 1� ↵

CVaR↵ =1

0VaR1�� d�

⌦(r) =

r(1� F (x)) dx

�1F (x) dx

r(x� r)⇢(x) dx

�1(r � x)⇢(x) dx

=Dembo’s reward

Dembo’s regret

Assume uncorrelated assetsXi and a portfolio with individual allocation weightswi. Then

vuutnX

w2i �

Therefore@�

wi�2i

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

Allocation to stocks, bonds and hedge funds

Equal weights

Minimum Variance

Standard deviation Allocation

Bonds 15% 33%

Stocks 30% 33%

Hedge Funds 10% 33%

Bonds 15% 33%

Stocks 30% 16%

Hedge Funds 10% 50%

Bonds 15% 28%

Stocks 30% 7%

Hedge Funds 10% 64%

If they were uncorrelated … which they are not

Leveraged Products

Basic HF

Product

Equity

Structure

funding

return

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

Leveraged investments

Investor: $25M

Bank: $75M Loan

$100M HF Portfolio

Return

Interest

Investor: $25M

Bank: $75M LoanReturn

Return minus

Interest

Investor: $25M

Bank: Warrant

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)

AAA Tranche A Tranche BBB Tranche Equity tranche

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

Bonds default as performance degrades

CFO History

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

Basic HF

Product

Equity

Guarantee

Structure

funding

return

Premium

Guaranteed notes

Zero coupon Free capital Fees

Investor Now

Investor later

zero coupon Guarantee

Term guaranteed note

Dependent on Portfolio Performance

Invested with leverage

Fund Pool

GraciasThank you

luis a. seco sigma analysis & management university of toronto risklab

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