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Investment risk management Traditional and alternative products

Luis A. Seco Sigma Analysis & Management

University of Toronto RiskLab

A hedge fund example

The snow swap

!   Track the snow precipitation in late fall and early spring;

!   If the precipitation is high, the ski resort pays to the City of Montreal a prescribed amount.

!   If the precipitation is low, the City pays the resort another pre-determined amount.

!   The dealer keeps a percentage of the cash flows.

A hedge fund example

The snow swap

© Luis Seco. Not to be distributed without permission.

City Resort

Snow

No snow

$10M

The snow fund

!   Modify the snow swap so the City pays when precipitation is low in the city, and the resort pays when precipitation is high in the resort.

!   The fund takes the “spread risk”, and earns a fee for the risk. !   Say the “insurance claim” is $1M. The fund would charge 20%

commission, but assume to take the spread risk. !   Setting aside $2M, and charging $200K, the fund could

–  Lose nothing: 75% –  Make $2M: 12.5% –  Lose $2M: 12.5%

!   Expected return=10%. Std=50%

A diversified fund: a hedge fund.

!   If we do the swap across 100 Canadian cities:

!   Expected return:10% !   Std: 5%. !   Better than investing in

the stock market.

So we create do the snow investment…

… with some of the best known ski resorts:

!   Blue Mountain (Toronto) !   Mountain Creek (New Jersey) !   Panorama Mountain Village (Calgary) !   Snowshoe Mountain (West Virginia) !   Steamboat Ski Resort (Hayden, Denver) !   Stratton Mountain Resort (Vermont) !   Tremblant (Montreal) !   Whistler Blackcomb (Vancouver)

… and then:


Intrawest goes Bankrupt


Hedge Fund: definition

!   An investment partnership; seeks return niches by taking risks, which they may hedge or diversify away (or not).

!   Unregulated !   Bound to an Offering Memorandum !   Seeks returns independent of market

movements !   Reports NAV monthly !   Charges Fees: 1-20

The investment structure

The Management company “the hedge fund”

The Fund legal structure

The Bank Prime Broker The Administrator

Investor 1 Investor 2 Investor 3 Investor 4 Investor n

Risks per strategy

© Luis Seco. Not for dissemination without permission.

Slide 17

Convertible arbitrage

Fig. 1: A graphical analysis of a convertible bond. The different colors indicate different exercise strategies of call and put options.

Risk management for financial institutions (S. Jaschke, O. Reiß, J. Schoenmakers, V. Spokoiny, J.-H. Zacharias-Langhans).

The Galmer Arbitrage GT

Convertible arbitrage

!   The convertible arbitrage strategy uses convertible bonds.

!   Hedge: shorting the underlying common stock. !   Quantitative valuations are overlaid with credit and

fundamental analysis to further reduce risk and increase potential returns.

!   Growth companies with volatile stocks, paying little or no dividend, with stable to improving credits and below investment grade bond ratings.

An convertible arbitrage strategy example

!   Consider a bond selling below par, at $80.00. It has a coupon of $4.00, a maturity date in ten years, and a conversion feature of 10 common shares prior to maturity. The current market price per share is $7.00.

!   The client supplies the $80.00 to the investment manager, who purchases the bond, and immediately borrows ten common shares from a financial institution (at a yearly cost of 1% of the current market value of the shares), sells these shares for $70.00, and invests the $70.00 in T-bills, which yield 4% per year. The cost of selling these common shares and buying them back again after one year is also 1% of the current market value.

Scenario 1

Values of shares and bonds are unchanged:

Today 1 yr later Bonds 80 80 Stock -70 -70 T-Bill +70 +72.8 Coupon 4 Fee -3.5 Total $80 $83.3

Scenario set 2

In the next two examples, the share price has dropped to $6.00, and the bond price has dropped to either $73.00 or $70.00, depending on the reason for the drop in share market values. The net gain to the client is 7.87% and 4.12% respectively, again after deducting costs and fees.

Today 1 yr later (a) 1 yr later (b)

Bonds 80 73 70 Stock -70 -60 -60 T-Bill +70 +72.8 72.8 Coupon 4 4 Fee -3.5 -3.5 Total $80 $86.3 $83.3

Scenario set 3

In the following three examples, the share price increased to $8.00, and the bond price increased either to $91.00, $88.00 or $85.00, depending on the expectations of investors, keeping in mind that we have one less year to maturity. The net gain to the client is 5.37% and 1% in the first two examples, with an unlikely net loss of 2.12% in the last example.

Today 1 yr later(a) 1 yr later(b) 1 yr later(c)

Bonds 80 91 88 85 Stock -70 -80 -80 -80 T-Bill +70 +72.8 +72.8 +72.8 Coupon 4 4 4 Fee -3.5 -3.5 -3.5 Total $80 $84.3 $81.3 $78.3

A Risk Calculation: normal returns

If returns are normal, assume the following:

Bond mean return: 10% Equity mean return: 5% Libor: 4% Bond/equity covariance matrix

(50% correlation):

!   Mean return (gross): 10-5+4=9%

!   Standard deviation:

Long-short equity

William Holbrook Beard (1824-1900)

A long-short pair trade

!   The fund has $1000. The manager is going to purchase stock 9 units of stock A, and sell-short 9 units of stock B. Both are valued at $100 each. After a year, A is worth $110, B is $105.

Assets at Prime Broker

(Before trade)

• $1000


(After trade)

•  $1000

•  -$900 + 9 A

•  +$900 – 9 B


(After one year)

•  $1000

•  990

•  -945

•  -9

$ 1036

A long-short pair trade (v2)

!   The fund has $500. The manager is going to purchase stock 9 units of stock A, and sell-short 9 units of stock B. Both are valued at $100 each. After a year, A is worth $110, B is $105.


(Before trade)

• $500


(After trade)

•  $500

•  -$900 + 9 A

•  +$900 – 9 B


(After one year)

•  $500

•  990

•  -945

•  -9

$ 536

A long-short pair trade (v3)

!   Assumptions: 50% collateral for long trades, 80% collateral for short trades.

Securities at Prime Broker

•  9 A ($900):

•  – 9 B (-$900):

Collateral required:

$450+$720=$1170

Cash from short sale: $900

Cash required: $270

Securities at Prime Broker

•  9 A ($990):

•  – 9 B (-$945):

Profit: $36

Risk and Performance Measurement

Measurement

!   Return: –  from track records

!   Risks: –  Volatility –  Operational risk: due diligence –  Business risk –  Exposures to market factors

© Luis Seco. Not to be reproduced without permission

Slide 31

Return

!   Starting from share value observations Si on a monthly basis, we define the return as

!   Simple Returns: Ri = (Si - Si-1)/Si-1

!   Log Returns Ri = ln(Si/Si-1)


Slide 32

TWR and IRR

!   Over a period of time, the time-weighted-rate of return is defined by

1+TWR = (1+R1)(1+R2)… (1+Rk) !   Over the same period of time, the Internal Rate of

Return is defined as IRR=(1+R)n

where the number R is defined as

and Ni denote the cashflows at month i.

Return statistics


Slide 33

Heaven (Probability)

!   Assumes a probability distribution

!   Assumes total knowledge

!   Expressed with mathematical formulas

Earth (Statistics)

!   Derives distributions from history

!   Only knows the past

!   Implementable on a computer

The portfolio distribution function (CDF)

90% probability that annual returns are less than 3%

7% probability that annual losses exceed 5%

Probability density: histogram

Mean Return

!   Return is usually measured on a monthly basis, and quoted on an annualized basis.

!   If the series of monthly returns (in percentages) is given by numbers ri, where the subindex i denotes every consecutive month, the average monthly return is given by

!   Because returns are expressed in percentages, one has to be careful, as the following example shows.

Mean return estimators


Slide 37

!   Heaven (Probability) !   Earth (Statistics)

Usually measured monthly, and reported annually

Returns: careful.

Imagine a hedge fund with a monthly NAV given by

$1, $2, $1, $2, $1, $2, etc. The monthly return series is given by 100%, -50%, 100%, -50%, 100%, -50%, etc. Its average return (say, after one year) is 25%

monthly, or an annualized return in excess of 300%.

Returns: from monthly to annual

There is no standard method of quoting annualized returns:

One possibility is multiplying returns by 12 (annual return with monthly compounding)

Another, is to annualize using the formula

Portfolio returns

The big advantage of “return”, is that the return of a portfolio is the average of the returns of its constituents.

More precisely, if a portfolio has investments with returns given by

with percentage allocations given by

then, the return of the portfolio is given by

Volatility

!   Like returns, volatility is usually measured on a monthly basis, and quoted on an annual basis.

!   If the series of monthly returns (in percentages) is given by numbers ri, where the subindex i denotes every consecutive month, the monthly volatility is given by

Volatility


Slide 42

!   Heaven (Probability) !   Earth (Statistics)

€

ˆ µ = r = 1n

rii=1

n

∑

σ =1n

(ri − µ)2

i=1

n

∑

s =1

n −1(ri − r )2

i=1

n

∑ Sample s.d.

Population s.d.

Covariances and correlations

!   They measure the joint dependence of uncertain returns. They are applied to pairs of investments.

!   If two investments have monthly return series given by numbers ri and si respectively, where the subindex i denotes every consecutive month, and their average returns are given by r and s, their covariance is given by

!   If they have volatilities given respectively by

!   Then, their correlation is given by

Covariance and correlation matrices

Because correlations and covariances are expressed in terms of pairs of investments, they are usually arranged in matrix form.

If we are given a collection of investments, indexed by i, then the matrix will have the form


Slide 45

Fund-of-Fund Risk: volatility

Volatility of a portfolio with weights w

Portfolio Optimization: Markowitz

Markowitz optimization allows investors to construct portfolios with optimal risk/return characteristics.

  Risk is represented by the portfolio expected return

  Risk is represented by the standard deviation of returns.

The optimization problem thus created is LQ, it is solved using standard techniques.

Risk/return space

A portfolio is represented by a vector θ which represents the number of units it holds in a vector of securities given by S.

Each security Si is assumed a gaussian return profile, with mean µi, and standard deviation given by σi. Correlations are given by a variance/covariance matrix V.

The portfolio return is represented by its return mean

and its risk is given by its standard deviation

The efficient frontier

Risk

Return

Feasible

Region

Efficient Portfolios

Sharpe’s ratio

A way to bring return and risk into one number is by the information ratio, and by the Sharpe’s ratio.

If a certain investment has a return given by r, and a volatility given by σ, then the information ratio is given by r/ σ.

If interest rates are given by i, then Sharpe’s ratio is given by (r-i)/ σ.

It measures the average excess return per unit of risk. Portfolios with higher Sharpe’s ratios are usually better.

Sharpe’s ratio: basic fact

!   Imagine one is looking for the portfolio that has the best chance of optimizing its performance against a benchmark given by LIBOR. That portfolio is the one with the highest Sharpe ratio, as defined in the previous paragraph.

Sharpe Ratio

The objective function to maximize is

Since φ is increasing, our optimization problem becomes that of maximizing

Probability of meeting the benchmark

Cummulative distribution function

of the gaussian

Sharpe vs. Markowitz

Benchmarks

They are reference portfolios against which performance of other portfolios are measured:

!   Bonuses are paid on benchmark-based performance.

!   They can be constant or random

Tracking error

!   It is the standard deviation of the difference between the portfolio returns and the benchmark returns.

!   A performance indicator often times used in traditional investments is

Alpha and beta

Consider a portfolio with returns given by

and a benchmark with returns given by.

Find the linear regression coefficients α, β, such that,

with ε with mean 0 and lowest standard deviation.

VaR and risk budgeting

Assume a portfolio represented by a vector θ which represents the percentage allocated to specific managers or investment instruments.

Each manager or security Si is assumed a gaussian return profile, with mean µi, and standard deviation given by σi. Correlations are given by a variance/covariance matrix V.

VaR and portfolio standard deviation are related to the fundamental expression

Risk budgeting

The previous expression allows us to do a risk allocation to each manager

in such a way that the overall risk of the portfolio is given by

This expression is useful when allocating risk or risk limits to each of the investments in a certain universe.

The normality assumption

Under the normal assumption, a portfolio with a 1% standard deviation will have annual returns which will vary no more than 1%, up or down, from its expected return, with a 65% probability.

If a higher degree of certainty about portfolio performance is desired, then one can say that the portfolio return will vary more than 2% from its expected return only 1% of the times.

These probabilities are linear in the standard deviation; in other words, if the portfolio volatility is 3% (instead of 1% as in the example above), one will expect the returns to oscillate within a 6% band of its average return 99% of the time.


Slide 59

Non-normal returns

Gain/loss deviation

It measures the deviation of portfolio returns from its expected return, taking into account only gains. In other words, portfolio losses are not taken into account with calculating the deviation.

Loss deviation is the corresponding thing when losses only are taken into account in calculating portfolio deviations.

Both of these are used when one is trying to get a feeling as to the asymmetry of the gain/loss distribution. They are not statistically conclusive amounts per se, like standard deviation is.

Semi-standard deviation formula

Target return / benchmark

Gains give a value ot 0

Sortino ratio

It is the substitute of the Sharpe ratio when one looks only at the loss deviation, instead of looking at the combined standard deviation.

Many people believe that by not punishing unusual gains, like the Sharpe ratio does indirectly, one maximizes the upside while maintaining the downside.

There is however no evidence that the Sortino ratio, as such, actually achieves this but it still remains to be a curious quantity to look at.

Moments

One of the criticisms of the use of volatilities and correlations as risk measures is the presence of extreme events in portfolio returns, which will go un-noticed in those calculations.

From a certain viewpoint, volatilities and correlations are magnitudes inherited from normal distributions, according to which events such as the ones in 1987, 1995, 1998, etc. should have never occurred.

One attempt to capture “tail events” is by introducing higher moments to measure large deviations: higher moments are defined as follows:

Skew and kurtosis

!   Skew is a measure of asymmetry. It is the normalized third moment.

!   Kurtosis is a measure of spread. It is the fourth moment, minus 3. Platykurtotic: k<0 Leptokurtotic: k>0 Mesokurtotic: k=0.

Biased estimators

!   The estimator for the skewness and kurtosis introduced earlier is biased: –  Its expected value can even have the opposite sign from the true

skewness (or kurtosis).

!   Intuitively speaking, the third and fourth powers are so large, that one or two events will dominate the value of the formula, making all other observations irrelevant.

!   Skew and kurtosis should not be used in critical situations

Skewness is useless

Uselessness of skewness

L-moments

The Omega

Omega

!   Shadwick introduced the concept of “Omega” a few years ago, as the replacement of the Sharpe ratio when returns are not normally distributed.

!   His aim was to capture the “fat tail” behavior of fund returns.

!   Once the “fat tail” behavior has been captured, one then needs to optimize investment portfolios to maximize the upside, while controlling the downside.

Omega: Shadwick, Keating (2002)

Slide 73

Wins vs. losses: the Omega

Omega tries to capture tail behavior avoiding moments, using the relative proportion of wins over losses:

Wins vs. losses: the Omega

Omega tries to capture tail behavior avoiding moments, using the relative proportion of wins over losses:

Truncated First Moments

The Omega of a heavy tailed distribution

Correlation risk

Slide 77

Hedge fund diversification

Hedge funds are uncorrelated to traditional markets, and internally uncorrelated also.

Correlation histogram for Dow stocks

Correlation histogram for hedge funds

Fact.

Hedge funds are uncorrelated to traditional markets, so they constitute excellent diversification strategies.

Yes, ... and no! Many hedge funds are indeed

uncorrelated to markets, but others are very correlated to simple portfolios of traditional markets, so they add little diversification.

Even those funds which exhibit low correlation to markets and macroeconomic factors, when combined into portfolios, they can be highly correlated to the market.

Hedge Fund Correlation histogram

Normal correlations

Distressed correlations

Correlation switching

Distress analysis

Correlation switching

Correlation risk

We will deal with correlation sensitivity from a mixtures of multivariate gaussian approach

Its density is given by:

GM in pictures

Non-gaussian portfolio theory

Each portfolio is described by four performance numbers: mean and standard deviation, each under normal and distressed market assumptions. They are given by

and

Benchmark satisfaction

The objective function to maximize was

It is possible to have portfolios which are efficient from this point of view, which however are not efficient under either normal or distressed conditions.

Increasing functions

Other risks

!   Backfill bias !   Survivorship bias !   Liquidity risk !   Style risk !   Legal risk !   Non-linear effects: option writing.

Hedge Fund Products

!   Fund-of-funds: Indices !   Options on fund-of-funds !   Warrants !   Non-recourse loans with fund collaterals !   CPPI (Constant proportion portfolio

insurance) !   CFO’s

Hedge Fund indices

!   They offer fund-of-fund investments that try to track the performance of the hedge fund sector (global and style specific) investing in liquid funds with high capacity.

!   The result is a fund that tracks nothing and lags performance.

!   In contrast with equity indices, investors in a fund don’t like it when their fund is included in an index.

Hedge Fund Indices

!   Investable !   Non-investable

Historical comparative analysis

Pro-Forma

Correlation analysis

Guaranteed notes

!   There are two main reasons for a guarantee: –  Regulatory environments –  Risk perceptions (not to confuse with risk appetite)

!   Some guarantees are provided by well-rated banks. Others are not (Portus).

!   Guarantees are obtainable by setting aside an interest-earning portion of the assets, and investing the remainder at higher levels of leverage, through a variety of different instruments.

Anatomy of a guarantee Guarantees principal in

the future: How much is needed is determined by

• Interest rates

• Maturity date of the note

Obtains exposure to the Hedge Funds

The cost of the guarantee

About 2% per year cost

An underlying hedge fund portfolio that produces 6bps/month

Interest rates at 25bps per month A 5 year note that guarantees principal No management or performance fees

Leveraged structures

Loans Options

CPPI

Slide 99

Non-recourse loans

!   The bank lends to the investor and takes the investment in the hedge fund portfolio as collateral.

!   In a low interest rate environment, it allows investors to amplify good hedge fund performance. In high interest rate environments, if hedge fund performance is poor, they can lead to sustained losses.

!   It allows small investors to increase the asset base and diversify the portfolio better; it makes it easier to satisfy the minimum investment requirements of individual hedge funds.

!   The structurer may demand liquidation if performance drops below a certain floor.

Options

!   Options are delta-hedged; the liquidity of the underlying hedge fund portfolio contributes to a volatility spread.

!   They are hard to delta-hedge due to the low liquidity of the underlying portfolio. Implied volatilities will be much higher than historical volatilities.

!   They are path-independent. They are also insensitive to changes in interest rates.

CPPI

!   Investor provide equity to a fund; !   the structurer provides leverage !   Proceeds are invested in a reference portfolio !   If the performance of the reference portfolio is

below a reference curve, the strike price is increased.

!   If performance of the reference portfolio is above another reference curve, the strike price is decreased

CPPI options

Bank

Bond Investor (1)

Bond Investor (2)

Bond Investor (3)

Equity Investor

Fund Pool

Collateralized Fund Obligation (CFO)

A $500M CFO

CFO’s

Advantages !   Equity investors find a way

to obtain leverage. !   Debt holders find an

uncorrelated asset class to invest in.

!   Tranches can be packaged by volume and credit rating.

Disadvantages !   Hard to value !   Very dependent on

correlations amongst the funds constituents

!   Expensive structuring fees makes it difficult to find the equity investor sometimes.

S&P CTA CFO. A case study.


Slide 108

Blow-up risk


Slide 109

The Merton model of default

A double-layer rating system

A B C

A Infrequent, small losses

Frequent, small losses

B

C Infrequent, large losses

Large, probably losses

Rating and Due Diligence

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