Published on October 26, 2018
1. Marco Nicolás Dibo "Risk Management: Maximizing Long Term Growth"
2. Profile 2 • Head of the Quantitative Trading Desk at Argentina Valores S.A. • CEO and Co-founder of Quanticko Trading S.A. • Email • LinkedIn Profile
3. Agenda 3 • Risk Management, real objectives. • Optimal leverage: • Understanding market risk: standard deviation, Sharpe ratio, value-at-risk, maximum drawdown. • Volatility targeting. • Kelly formula. • Stop loss. • Hedging.
4. Risk Management, what does it means? 4 • Loss aversion, we don’t want to lose money. • Real goal: Maximization of long term equity growth. • We should avoid risk only as it interferes with this goal. • Leverage is the key concept in risk management.
5. Optimal Leverage 5 • The objective: Maximize long term wealth that is equivalent to maximizing long term growth of the portfolio. • But be careful, 100% drawdown can never be optimal! • Assumption: probability distribution of returns is Gaussian. • No matter how optimal leverage is determined, it should be kept constant in order to maximize growth.
6. Volatility Targeting 6 The importance of risk targeting • ¿Which is my overall trading risk? 1. Understand our trading system: performance, skew, real sharpe, avoid overconfidence. 2. Understand yourself: Can you lose 20% of you capital in a day?
7. Understanding Market Risk 7 Standard deviation of returns • It’s a measure of risk. • How dispersed some data is around its average. • On unit of standard deviation is called sigma, • It doesn’t increase linearly, but with the square root of time, so the annualized standard deviation is Where is the period return of the asset or strategy and is the period return of some benchmark.
8. Understanding Market Risk 8 Standard deviation of returns
9. Understanding Market Risk 9 Rolling standard deviation
10. Understanding Market Risk 10 Sharpe ratio • The ratio compares the mean average of the excess returns of the asset or strategy with the standard deviation of those returns. Thus a lower volatility of returns will lead to a greater Sharpe ratio, assuming identical returns. Where is the mean average of the excess returns.
11. Understanding Market Risk 11 Sharpe ratio The Sharpe ratio mostly quoted is the Annualized Sharpe, which depends on the trading period on which returns are calculated. Assuming there are N trading periods in a year, For a strategy based on a trading period of days,
12. Understanding Market Risk 12 Rolling Sharpe ratio
13. Understanding Market Risk 13 Value at risk (VAR) • VAR is a risk measure that helps us quantify the risk of our strategy or portfolio. • It provides an estimate, under a given degree of confidence, of the size of a loss from a portfolio over a given period of time. • The given degree of confidence will be a value like 95% or 99%.
14. Understanding Market Risk 14 Value at risk (VAR) Example: A VAR equal to 100,000 USD at a 95% confidence level, for a time period of a day, simply states that there is a 95% probability of losing no more than 100,000 USD in the next day. More generally: where c is the confidence level.
15. Understanding Market Risk 15 Value at risk (VAR) assumptions: • Standard market conditions. • Volatilities and correlations. • Normality of returns. Advantages and Disadvantages: • Easy to calculate, broken down by asset, used as constraints, easy to interpret. • It does not discuss magnitudes, does not take into account extreme events, it takes historical data.
16. Understanding Market Risk 16 Maximum drawdown and drawdown duration The maximum drawdown (md) and drawdown duration (dd) are two measures that traders use to assess the risk in a portfolio. • md quantifies the highest peak-to-trough decline in an equity curve • dd is defined as the number of trading periods over which the md occurs.
17. Volatility Targeting 17 Setting a volatility target • ¿How much risk do I want to take? expected standard deviation = volatility target. It can be measured in % or in cash and over different periods. Example: daily cash volatility target = average expected standard deviation of the daily portfolio returns.
18. Volatility Targeting 18 • Volatility target is the long term average of expected, predictable risk. • This risk depends on the strength of your forecasts and the current correlation of assets prices. • The best proxy for risk, annualized cash volatility target (annualized expected daily standard deviation of returns).
19. Volatility Targeting 19
20. Volatility Targeting 20 Setting your trading capital and volatility target 1) How much can you lose? 2) How much risk can you tolerate? 3) Can you realise that risk? 4) Is this level of risk right for your system?
21. Volatility Targeting 21 1) How much can you lose? • Invest only what you can afford to lose. • Never trade with borrowed money!!!
22. Volatility Targeting 22 2) How much risk can you tolerate? How much risk can you tolerate? Example: • Trading capital = $200.000 • Volatility target = 200% • Annualized cash volatility target = $400.000 • Can you tolerate a $20.000 daily loss? And a cumulative loss or drawdown of $60.000 around 10% of the time?
23. Volatility Targeting 23 3) Can you realize that risk? Buying a short term bond with an expected volatility of 5% a year, then without leverage its impossible to create a portfolio with 50% volatility target. • With no leverage you are restricted to the amount of natural risk that your instruments have*. *Systematic Trading, Robert Carver.
24. Volatility Targeting 24 3) Can you realize that risk? Even if you have access to that kind of leverage to accomplish that volatility target would be an unwise decision. • Ensure that your volatility target won´t wipe out your account.
25. Volatility Targeting 25 4) Is this the right level of risk? • Do not ignore the compounding of returns! • Suppose you have a very profitable strategy, but you lose 90% of you account in the first trade. Then a 190% return in the second trade wont take you even closer to you initial equity.
26. Volatility Targeting 26 4) Is this the right level of risk? Kelly criterion: used by nearly all professional gamblers and professional traders to maximize profits. • If you know your sharpe ratio you can use this formula to determine how you should set your volatility target and leverage.
27. Kelly Formula: Optimal leverage 27 • Assuming that the probability distribution of returns is Gaussian: Where is the optimal leverage, is the mean excess return and is the variance of the excess return. • If returns are normally distributed, the leverage will generate the highest compounded growth rate of equity (assuming reinvestment of all the returns).
28. Kelly Formula: Optimal leverage 28 • Example 1: Ticker: JPM 1. Daily percentage change. 2. Mean return of daily percentage change. 3. Standard deviation of daily percentage change. 4. Mean excess return, rf = risk free rate. 5. Sharpe ratio. 6. Kelly fraction. 7. Levered and unlevered compounded return.
29. Kelly Formula: Optimal leverage 29 • Kelly criterion implies that we should set our volatility target equal to the sharpe ratio. • So, if we think that our sharpe ratio is 0.5, the best performance will be achieved with a 50% volatility target. • Estimation errors can lead to ruin. Dangerous if used by an over confident trader. Difficult to know you exact sharpe ratio. • Even if you know your expected sharpe ratio you could end with a great loss.
30. Kelly Formula: Optimal leverage 30 Recommended percentage volatility targets • Even with a Sharpe ratio of 1 we should not use a 100% volatility target. • The reasons: 1) Backtested Sharpe ratios are hardly achievable in the future. Use a ratio of the observed Sharpe ratio. 2) Kelly criteria is far too aggressive. You could suffer some large drawdowns even if your expected Sharpe ratio is correct. Better use Half-Kelly. 3) Use it as an upper bound.
31. Kelly Formula: Optimal leverage 31 Recommended percentage volatility targets Positive Skew Negative Skew 12% 6% 20% 10% 25% 13% 37% 19% 50% 25%1 or more Realistic Backtested Sharpe Ratio Recommended Volatility Targets 0.25 0.40 0.50 0.75
32. Kelly Formula: BP Allocation 32 • Note: following the Kelly formula requires you to continuously adjust your capital allocation as your equity changes so that it remains optimal. • This continuous updating should occur at least once a day. • We should also periodically update (20 or 60 days lookback?).
33. Kelly Formula: Contagion 33 • Risk management dictates that you should reduce your position size whenever there is a loss, even when it means taking losses. • This is the cause of financial contagion. • Example: summer 2007 meltdown. The frequency of trading to rebalance the portfolio to follow the formula may explain why traders usually use half kelly fraction.
34. Kelly Formula: Contagion 34 • Even half Kelly may not be so conservative. • Another check (additional constraints): A = What is the maximum drawdown you would tolerate? B = What was the maximum loss in one period? (week, day, hour, minute) A/B will tell you the maximum leverage you would tolerate. This can even be smaller than half-kelly.
35. Kelly Formula: BP Allocation 35 • There is another usage of the Kelly formula: buying power allocation between portfolios or strategies. • Let be a column vector consisting of the optimal fractions of our equity that we should allocate to each of the strategies. • The optimal allocation is given by:
36. Kelly Formula: BP Allocation 36 • is the covariance matrix such that is the covariance of the returns of the and strategies. • is the column vector of the mean returns of the strategies. • Example 6.3: Portfolio of 3 stocks
37. Kelly Formula: BP Allocation 37 • Example 2: Portfolio of 3 assets: JPM, C, GS. 1. Daily returns. 2. Excess returns. 3. Excess returns mean. 4. Covariance Matrix. 5. Kelly fraction. 6. Compounded levered return.
38. Kelly Formula: BP Allocation 38 • is the covariance matrix such that is the covariance of the returns of the and strategies. • is the column vector of the mean returns of the strategies. • Example: Portfolio of 3 stocks
39. Stop Loss 39 • There are 2 ways to set stops: 1) Exit whenever your loss is greater than a certain threshold (more common usage). 2) When drawdown drops below a certain threshold (less common, hope we don’t use it never). • If we focus in the first kind, setting a stop loss will depend in the kind of strategy we are running.
40. Stop Loss 40 Mean reversion strategies • Should we set stop loss for these type of strategies? • At first it seems to contradict the idea behind mean reversion: - If price drops, and we expect it will mean reverse, the probability of reversion increases, giving a better opportunity than before.
41. Stop Loss 41 But what happens if mean reversion breaks? • What was true of a price series before may not be true in the future (CHAN). • So a mean reverting process may become a trending process, in this case setting a stop loss will help us. • This can prevent us from suffering a 100% loss. • Be careful with survivorship bias in mean reverting strategies!!!
42. Stop Loss 42 Conclusion: • While a price series remain mean reverting a stop loss will lower the performance. • If price series undergo a regime change, then a stop loss will definitely improve your performance. • Stop loss should be grater than the intraday maximum drawdown from the backtest.
43. Stop Loss 43 Trend following strategies • They benefit from stop loss: - If the strategy is losing, it means momentum is no longer there, we should exit the position. - We can use this change in momentum to open a new position on the other direction and as a stop loss of the previous position. • This is the reason why trend following strategies do not suffer as much tail risk as mean reversion strategies do.
44. Hedging 44 Hedging an Equity Portfolio • We can use index futures to hedge our portfolio by using the capital asset pricing model. • The parameter beta is the slope of the best fit line obtained when excess return on the portfolio over the risk free rate is regressed against the excess return of the index over the risk free rate.
45. Hedging 45 • If 1, the returns of the portfolio mirror the returns of the index. • If 2, the excess return of the portfolio tends to be twice as great as the return on the index. • If , the excess return of the portfolio tends to be half as great as the return on the index. If = 2, our portfolio tends to be as twice as sensitive to movements in the index (HULL).
46. Hedging 46 • The next equation shows the number of futures contracts that we should short, • Where is the current value of the portfolio and is the current value of 1 future contract.
47. Hedging 47 • Example: Value of the S&P 500 index = 2,656 ES[Z] S&P 500 dec. futures price = 2,670 Value of our portfolio = $10,000,000 Risk-free rate = 3% per annum Dividend yield on index = 1% per annum Beta of the portfolio = 1.75
48. Hedging 48 • 1 future contract is for delivery of $250 times the index. • • The number of contracts we should short: contracts
49. Hedging 49 • Suppose the index closes at 2,496 in 2 months and the future price is 2,510. • The gain from the short is:
50. Hedging 50 • The loss in the index is 6%. The index pays a dividend of 1% per annum, or 0.17% per 2 months. • The investor in the index would have earned • Expected return on portfolio – Rf =
51. Hedging 51 • The risk-free rate is 0.75% per 2 months. • The expected return on the portfolio: • The expected value of the portfolio: • The expected value of the hedgers position: $10,029,000
52. Thank you! 52