Risk-adjusted performance metrics

In financial books Sharpe ratio (and a few others) is defined as excess return (portfolio return rate - risk free rate) divided by volatility.

\[ \textcolor{blue} { Sharpe = \frac{R_p - r_f}{\sigma} } \] but we will keep things simple and only consider portfolio return: \[ \textcolor{blue} { Sharpe = \frac{R_p}{\sigma} } \] where:

• R_p - average portfolio return
• r_f - risk-free rate
• σ - standard deviation (both upside and downside volatility)

Note: There is also Sortino ratio that only takes into account the downside volatility and this is mostly used by long term investors who want upside volatility.

Portfolio return over Maximum Drawdown.

\[ \textcolor{blue} { Calmar = \frac{R_p}{MDD} } \] where:

• R_p - portfolio return
• MDD - maximum drawdown (see below)

Note: this is also called Return over Maximum Drawdown (RoMad) or Risk Return Ratio (RRR) while Sterling ratio is different version of the same MDD-adjusted metric.

Same as Sharpe ratio but it uses Ulcer Index (UI) as risk measure instead of volatility (σ).

\[ \textcolor{blue} { UPI = \frac{R_p}{UI} } \] where:

• R_p - portfolio return
• UI - Ulcer Index (see below)

CAGR is an accurate way to measure how investments have performed over different periods of time but it does not account for risk, only for time.

\[ \textcolor{blue} { CAGR = [ (\frac{C}{P})^{1/n} - 1 ] * 100 } \] where:

• C - compounded end value
• P - investment begin value
• n - compounding periods (eg years)

Take a guess! This is CAGR adjusted for risk.

\[ \textcolor{blue} { MAR = \frac{CAGR}{MDD} } \] where:

• CAGR - compounded annual return
• MDD - maximum drawdown (see below)

Standard deviation of log returns.

\[ \textcolor{blue} { \sigma = \frac{1}{N} * \sqrt{\sum_{i=1}^{N} (r_i - \mu)^2} } \] where:

• r_i - return for period i
• μ - mean of returns
• N - number of periods

\[ \textcolor{blue} { UI = \frac{1}{N} * \sqrt{\sum_{i=1}^{N} {(\frac{C_i - MAX_N}{MAX_N})}^2 * 100} } \] where:

• N - N-period (eg 14-day)
• C_i - current close price
• MAX_N - peak value for given N-period

Maximum Drawdown measures the largest drop, the difference between a peak and trough for a given period.

\[ \textcolor{blue} { MDD = \frac{T_t - P_t}{P_t} } \] where:

• T_t - through value for given period t
• P_t - peak value

It calculates the bang for the buck ratio, the amount of loss (pain) that is "needed" to play the game and make some profit (gain).

\[ \textcolor{blue} { GtP = \frac{\sum_{i=1}^n{r_i}}{abs(\sum_{i=1}^m{r_{i,n}})} } \] where:

• r_i - return for period i (eg monthly)
• r_i,n - negative return only
• n - number of trades
• m - number of negative trades
• abs - absolute value

Profit factor is a bit different because it uses profit/loss for each individual trade/strategy instead of return for given period.

\[ \textcolor{blue} { PF = \frac{\sum_{t=1}^n{p_{t}}}{abs(\sum_{t=1}^m{l_{t}})} } \] where:

• p_t - positive return (profit) for each trade (strategy)
• l_t - negative return (loss)
• n - number of positive trades
• m - number of negative trades

Number of trades in profit vs. total number of trades.

\[ \textcolor{blue} { WR = \frac{\#\ of\ t_w}{\#\ of\ t_t} } \] where:

• t_w - win trades
• t_t - total trades

It compares the average size of win vs. loss trades.

\[ \textcolor{blue} { AWL = \frac{avg(\sum_{i=1}^n{r_{i,p}})}{avg(\sum_{i=1}^n{r_{i,n}})} } \] where:

• r_i,p - positive return for i period
• r_i,n - negative return
• avg - the average