Hedge Funds Lecture 2: Risk Management & Portfolio Construction Zhiwu Chen Yale School of Management

Today’s topics Risk management: easier said than done? Ways to control risk: pairs trading? Managing risk with CAPM and APT Managing hedge fund risk with VaR Case: Zebra Capital

Risk and Return: how much can you take? $7860 Small stocks $2279 Large cap $71 Corporate Bonds $17 T-bills Copyright: Ibbotson Associates

Copyright: Ibbotson Associates

The Meaning of Dynamic Risk: Performance of a 1-Stock Portfolio

Performance of a 10-Stock Portfolio

Performance of a Diversified 30-Stock Portfolio

Diversification and Risk Reduction

How do you achieve market-neutrality? Example 1: risk arbitrage “Pfizer Makes Rival Bid For Warner-Lambert” Pfizer said it would offer 2.5 Pfizer shares for each share of Warner-Lambert outstanding. Today’s prices: PFE = $35, WLA = $80 Risk arbitrage: For each 2.5 shares of WLA long, short one share of PFE.

Example 2: pairs trading Long IBM Short eBay? Dollar for dollar?

Pairs Trading is one type of Statistical “Arbitrage” Identify a pair of stocks that move in tandem When they diverge: short the higher one buy the lower one Unwind upon convergence Give me an example?

Who does it? Proprietary trading desks Hedge funds (Long-short) Morgan Stanley Nunzio Tartaglia - 1980’s Other investment banks Hedge funds (Long-short) D.E. Shaw?

Methodology in the Gatev, Goetzmann and Rouwenhorst (1999) Study Two stages: Pairs Formation Pairs Trading Committed Capital full period when-needed no extra leverage

Pairs Formation Period Daily CRSP files Eliminate stocks that missed a day trading in a year Cumulative total return index for each stock Also restrict to same broad industry category: Utilities, Transports, Financials, Industrials

Trading Period Six-month periods: 1962-1997 starting a new “trader” each month closing all positions at end of each six month How many pairs to use? 5, 20 and 20 after first 100, then all pairs under distance metric

Trading Method Open at 2 F (historical F over leading year) Close upon convergence, or end of six- month period Same-day vs. wait one day to control bid- ask effect

Results for Same Day Trading Portfolio of 5 and 20 best pairs earn an average of 6% per six month period. Average size of stocks in pairs: 3rd to 4th decile Utilities predominate

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Example 3: Long-Short Fund Suppose you want To buy 100 stocks with low P/E ratios To short 100 stocks with high P/E ratios, $ for $ What about exposure to systematic and unsystematic risks? How do you match the long and short sides?

One solution: apply the CAPM The risk premium for a stock is a function of its contribution to the risk of the market portfolio A stock’s risk premium is a function of its covariance with the market portfolio. E(rn) - rf = rf + n [E(rm) - rf]

Under the CAPM Each stock’s return follows: rn(t) - rf = rf + n [rm (t) - rf] + en(t) Thus, there is just one source of systematic risk

Implications for Market-Neutral Funds Make the total beta of the Long Portfolio equal the total beta of the Short Portfolio Show me an example? Any problems?

Or, use the APT factor betas Multiple factors constitute “systematic risks”, not just the market portfolio! Show me the equation, please!

What factors? Factors include, in addition to market portfolio, Industrial production growth Interest rates Term premium or yield curve slope Default premium = BBB corporate bond yield - Treasury bond yield Size factor Book/market factor Estimate a beta for each factor using multiple regression

Now, how do I make it market-neutral? You want to make total factor beta of the Long Portfolio == total factor beta of the Short Portfolio, for every known factor!

Problems and Concerns? How many factors are too few? Let the R-square speak! But, ultimately, it is difficult to make the Long & the Short sides exactly match.

Risk Management Keep a profile of your portfolio’s exposure to every known risk factor: Macroeconomic factors: interest rate, inflation, … Industry factors: oil, retail, semiconductor, …. Barra, Ibbotson Associates, Northfield . ...

Alternative method: use Value-at-Risk (VaR) VAR is the maximum loss over a target horizon within a confidence interval (or, under normal market conditions) In other words, if none of the “extreme events” (i.e., low-probability events) occurs, what is my maximum loss over a given time period?

VAR: Example Consider a $100 million portfolio of medium-term bonds. Suppose my confidence interval is 95% (i.e., 95% of possible market events is defined as “normal”.) Then, what is the maximum monthly loss under normal markets over any month? To answer this question, let’s look at the monthly medium-term bond returns from 1953 to 1995: Lowest: -6.5% vs. Highest: 12%

History of Medium Bond Returns

Distribution of Medium Bond Returns

Calculating VAR at 95% Confidence At the 95% confidence interval, the lowest monthly return is -1.7%. (I.e., there is a 5% chance that the monthly medium bond return is lower than -1.7%) That is, there are 26 months out of the 516 for which the monthly returns were lower than -1.7%. VAR = 100 million X 1.7% = $1.7 million (95% of the time, the portfolio’s loss will be no more than $1.7 million!)

Monthly Return Distribution with 5 Stocks

The Case with 20 Stocks

Issues to Ponder on What horizon is appropriate? A day, a month, or a year? What confidence level to consider? * Are you risk averse? The more risk averse => (1) the higher confidence level necessary & (2) the lower VAR desired.