# Investment Risk Management Slides

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Prof.Luis SecoTRANSCRIPT

Investment risk managementTraditional and alternative products

Luis A. Seco Sigma Analysis & Management University of Toronto RiskLabSlide 1

A hedge fund example

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A hedge fund example

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A hedge fund example

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A hedge fund example

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A hedge fund example

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The snow swap

Track the snow precipitation in late fall and early spring; If the precipitation is high, the ski resort pays to the City of Montreal a prescribed amount. If the precipitation is low, the City pays the resort another pre-determined amount. The dealer keeps a percentage of the cash flows.Slide 7

A hedge fund example

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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.

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: 1-20Slide 11

The investment structureInvestor 1 Investor 2 Investor 3 Investor 4 Investor n

The Fund legal structure

The Administrator The Management company the hedge fund

The Bank Prime Broker

Slide 12

Risks per strategy

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Slide 14 Luis Seco. Not for dissemination without permission.

Convertible arbitrage

Fig. 1: A graphical analysis of a convertible bond. The different colors indicate different exercise strategies of call and put options.Risk management for financial institutions (S. Jaschke, O. Rei, J. Schoenmakers, V. Spokoiny, J.-H. ZachariasLanghans).

The Galmer Arbitrage GT

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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.

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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.

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Scenario 1Values of shares and bonds are unchanged:

Today Bonds Stock T-Bill Coupon Fee Total $80 80 -70 +70

1 yr later 80 -70 +72.8 4 -3.5 $83.3Slide 18

Scenario set 2In 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 Stock T-Bill Coupon Fee Total

80 -70 +70

73 -60 +72.8 4 -3.5

70 -60 72.8 4 -3.5 $83.3Slide 19

$80

$86.3

Scenario set 3In 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 Stock T-Bill Coupon Fee Total

80 -70 +70

91 -80 +72.8 4 -3.5

88 -80 +72.8 4 -3.5 $81.3

85 -80 +72.8 4 -3.5 $78.3Slide 20

$80

$84.3

A Risk Calculation: normal returnsIf 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:

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Long-short equity

William Holbrook Beard (1824-1900)

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A long-short pair trade

The fund has $1000. The manager is going to purchase stock 9 units of stock A, and sell-short 9 units of stock B. Both are valued at $100 each. After a year, A is worth $110, B is $105.Assets at Prime Broker (Before trade) 1000 $ Assets at Prime Broker (After trade) $1000 -$900 + 9 A +$900 9 B Assets at Prime Broker (After one year) $1000 990 -945 -9Slide 23

$ 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.Assets at Prime Broker (Before trade) 500 $ Assets at Prime Broker (After trade) $500 -$900 + 9 A +$900 9 B Assets at Prime Broker (After one year) $500 990 -945 -9Slide 24

$ 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): Securities at Prime Broker 9 A ($990): 9 B (-$945):

Collateral required: $450+$720=$1170 Cash from short sale: $900 Cash required: $270

Profit: $36

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Hedge Fund Correlation histogram

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Risk and Performance Measurement

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Measurement

Return: from track records

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

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Sample Hedge Fund report

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Data Issues (discussion)

www.hedgefundresearch.com www.hedgefund.net www.hedgefund-index.com www.barclaygrp.com www.eurekahedge.com sigma2.fields.utoronto.ca

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The portfolio distribution function (CDF)

90% probability that annual returns are less than 3%

7% probability that annual losses exceed 5%

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Probability density: histogram

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Return

Return is usually measured on a monthly basis, and quoted on an annualized basis. If the series of monthly returns (in percentages) is given by numbers ri, where the subindex i denotes every consecutive month, the average monthly return is given by

Because returns are expressed in percentages, one has to be careful, as the Slide 33 following example shows.

Returns: careful.Imagine a hedge fund with a monthly NAV given by $1, $2, $1, $2, $1, $2, etc. The monthly return series is given by 100%, -50%, 100%, -50%, 100%, -50%, etc. Its average return (say, after one year) is 25% monthly, or an annualized return in excess of 300%.Slide 34

Returns: from monthly to annualThere 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

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Portfolio returnsThe 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 bySlide 37

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

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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

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Covariance and correlation matricesBecause 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

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Portfolio Optimization: MarkowitzMarkowitz optimization allows investors to construct portfolios with optimal risk/return characteristics. Risk is represented by the portfolio expected return Risk is represented by the standard deviation of returns. The optimization problem thus created is LQ, it is solved using standard techniques.Slide 42

Risk/return spaceA portfolio is represented by a vector 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 i, and standard deviation given by i. 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 deviationSlide 43

The efficient frontier

Efficient Portfolios

Risk

Feasible Region

Return

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Sharpes ratioA way to bring return and risk into one number is by the information ratio, and by the Sharpes ratio. If a certain investment has a return given by r, and a volatility given by , then the information ratio is given by r/ . If interest rates are given by i, then Sharpes ratio is given by (r-i)/ . It measures the average excess return per unit of risk. Portfolios with higher Sharpes ratios are usually better.Slide 45

Sharpes 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.

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Sharpe RatioThe objective function to maximize isProbabilityof meetingthe benchmark Cummulative distributionfunction ofthegaussian

Since is increasing, our optimization problem becomes that of maximizingSlide 47

Sharpe vs. Markowitz

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BenchmarksThey are reference portfolios against which performance of other portfolios are measured: Bonuses are paid on benchmark-based performance. They can be constant or random

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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

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Alpha and betaConsider a portfolio with returns given by and a benchmark with returns given by. Find the linear regression coefficients , , such that, with with mean 0 and lowest standard deviation.

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VaR and risk budgetingAssume a portfolio represented by a vector which represents the percentage allocated to specific managers or investment instruments. Each manager or security Si is assumed a gaussian return profile, with mean i, and standard deviation given by i. Correlations are given by a variance/ covariance matrix V. VaR and portfolio standard deviation are related to the fundamental expression

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Risk budgetingThe previous expression allows us to do a risk allocation to each manager in such a way that the overall risk of the portfolio is given by

This expression is useful when allocating risk or risk limits to each of the investments in a certain universe.Slide 53

The normality assumptionUnder the normal assumption, a portfolio with a 1% standard deviation will have annual returns which will vary no more than 1%, up or down, from its expected return, with a 65% probability. If a higher degree of certainty about portfolio performance is desired, then one can say that the portfolio return will vary more than 2% from its expected return only 1% of the times. These probabilities are linear in the standard deviation; in other words, if the portfolio volatility is 3% (instead of 1% as in the example above), one will expect the returns to oscillate within a 6% band of its average return 99% of the time. Slide 54

Non-normal returns

Slide 55 Luis Seco. Not to be reproduced without permission

Gain/loss deviationIt measures the deviation of portfolio returns from its expected return, taking into account only gains. In other words, portfolio losses are not taken into account with calculating the deviation. Loss deviation is the corresponding thing when losses only are taken into account in calculating portfolio deviations. Both of these are used when one is trying to get a feeling as to the asymmetry of the gain/loss distribution. They are not statistically conclusive amounts per se, like standard deviation is.Slide 56

Semi-standard deviation formulaTarget return / benchmark

Gains give a value ot 0

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Sortino ratioIt is the substitute of the Sharpe ratio when one looks only at the loss deviation, instead of looking at the combined standard deviation. Many people believe that by not punishing unusual gains, like the Sharpe ratio does indirectly, one maximizes the upside while maintaining the downside. There is however no evidence that the Sortino ratio, as such, actually achieves this but it still remains to be a curious quantity to look at.Slide 58

MomentsOne 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:

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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: k0 Mesokurtotic: k=0.

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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, ma...

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