performance evaluation & portfolio management

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Shaurya Chandra

Performance Evaluation & Money Management

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Building a Diversified Portfolio?


• Absolute Risk-adjusted Return Measures– Sharpe Ratio– Sortino Ratio– Calmar Ratio– Sterling Ratio

• Relative Return Measures– Up-Capture Ratio / Down-Capture Ratio– Up-Number Ratio / Down-Number Ratio– Proficiency Ratio (Up/Down Percentage Ratio)

• Relative Risk Adjusted Return Measures– Treynor Ratio

• Tail Risk Measures– RAROC-Var

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Strategy Analysis


• Sharpe ratio = (excess return over risk free rate) / (standard deviation of excess returns)

It is a measure of the excess return per unit of standard deviation in an investment asset. Sharpe ratio provides very useful information regarding the return of an asset for a given risk.

Limitations

– Based on historical data , assumes “Future would be same as the Past”

– Poor at estimating tail risk – Normal Distribution Assumption , hence no differentiation between +ve and –ve trades

– Doesn’t account for Draw-downs (leading to low volatility) and transaction costs

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p

fp

σ

RRratio Sharpe

Absolute Risk Adjusted Return Measures

Relative Return Measures

Relative Risk Adjusted Return Measures

Tail Risk Measures

Strategy Analysis


• Sortino ratio = (return over target return) / (downside risk, i.e. semi deviation or variance on the downside only).

<R> = Expected Return Rf = Risk Free Rate

σf = Standard Deviation of Negative Returns

Unlike Sharpe ratio, Sortino ratio does not punish high variance if the returns are on the upside.

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fratio Sortino

RR




Tail Risk Measures

Strategy Analysis


The Flaw of Averages


• Calmar ratio = (return) / (maximum drawdown). Generally quoted for the last 3 years

• Sterling ratio = (return) / (absolute(average drawdown – 10%)). Generally quoted for the last 3 years

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Tail Risk Measures

Strategy Analysis


• Up Capture Ratio = (fund’s return when benchmark return increased) / (benchmark’s return when benchmark increased)

• Up Number Ratio = (number of period when fund’s return increased when benchmark return increased) / (number of periods when benchmark’s return increased)

• Up Percentage Ratio (Proficiency Ratio) = (number of period when fund’s return outperformed the benchmark when benchmark’s return increased) / (number of periods when benchmark’s return increased)

• Likewise Down Capture Ratio, Down Number Ratio, Down Percentage Ratio

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Tail Risk Measures

Strategy Analysis


• Treynor ratio = (excess return over risk free rate)/ beta of portfolio

ri = Average Return of the PortfolioRf = Average Return of the Risk-Free Rateβ = Beta of the Portfolio

It is a measure of returns earned in excess of that which could have been earned on a riskless investment per unit of market risk where β is a measure of market risk. The only difference between Sharpe Ratio and Treynor Ratio is that the later uses β as the measurement of ‘Volatility’.

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Tail Risk Measures

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fi RrratioTreynor

Strategy Analysis


• RAROC (Risk Adjusted Return on Capital) = (expected return) / (Value at Risk)

RE = Expected Return VaR = Value at Risk

In any investment, risk is traded off against benefit. RAROC gives us a picture of the returns on several different investments with varying risk levels.

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Tail Risk Measures

VaR

RAROC ER

Strategy Analysis


Trade Sizing• Ad hoc: trade no larger than lets you sleep at night

• Margin plus drawdown• Fixed Fractional• Fixed Ratio• Hybrid fixed fractional/fixed ratio


Methods that Don’t Work• Martingale methods: increase position size after a loss; decrease it after a win.

• Anti-Martingale methods.• Equity curve methods: increase size when your equity curve falls below its moving average (“reversion to mean”), or increase size when you cross above the moving average (“trade the trend in equity curve”).


Why They Don’t Work• Martingale and equity curve methods assume dependency between trades.

• In most cases, trades are independent of each other. The odds of the next trade being a win are not related to whether the last trade was a win or a loss.

• If trades are independent, you can’t determine the likelihood of the next trade being a win or a loss based on the previous trade.


Kelly’s Criterion for Portfolio ManagementIt is a formula used to determine the optimal size of a series of bets.

Focus : Long Term Capital Growth

Kelly % (f) = ((B + 1)*P -1)/BWhere,f = optimal fixed fractionP = Winning ProbabilityB = Win/Loss Rate ( Average Gain on +ve Trades/ Average Loss on –ve Trades)

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Portfolio Management


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