Return Analysis Techniques

Return analysis is the process of assessing the profitability and efficiency of an investment. It is an essential component in financial analysis and investment strategy, especially in algorithmic trading (algotrading). This section will dive into various techniques used for analyzing return on investments in the context of algorithmic trading.

Time-Weighted Rate of Return (TWRR)

The Time-Weighted Rate of Return (TWRR) measures the compound rate of growth of an investment portfolio. TWRR is particularly useful in situations where the investor has made multiple deposits and withdrawals. It neutralizes the impact of cash flows, providing a clearer picture of an investment’s performance.

How to Calculate TWRR:

  1. Break Down the Periods: Divide the overall investment period into sub-periods between each cash flow.
  2. Calculate Individual Period Returns: For each sub-period, compute the return while ignoring the impact of cash flow.
  3. Compound the Returns: Multiply the individual period returns to get the overall TWRR.

[ TWRR = \left( \prod_{i=1}^{n} (1 + R_i) \right)^{\frac{1}{n}} - 1 ]

Where ( R_i ) is the return in sub-period i, and n is the total number of sub-periods.

Money-Weighted Rate of Return (MWRR)

The Money-Weighted Rate of Return (MWRR), also known as the Internal Rate of Return (IRR), takes into account the timing and size of cash flows. This metric is useful for understanding the profitability of an investment from the investor’s perspective.

How to Calculate MWRR:

  1. Identify Cash Flows: List all cash inflows and outflows, including the initial investment.
  2. Solve for IRR: Use a financial calculator or spreadsheet software to find the discount rate that sets the net present value (NPV) of these cash flows to zero.

[ \text{NPV} = \sum_{t=0}^{T} \frac{C_t}{(1 + IRR)^t} = 0 ]

Where ( C_t ) is the cash flow at time t, and T is the total number of periods.

Sharpe Ratio

The Sharpe Ratio is a measure of risk-adjusted return. It indicates how much excess return you receive for the extra volatility you endure for holding a riskier asset.

How to Calculate Sharpe Ratio:

  1. Calculate Excess Return: Subtract the risk-free rate (Rf) from the asset’s return (R).
  2. Calculate Standard Deviation: Compute the standard deviation (σ) of the asset’s return.
  3. Compute Sharpe Ratio: Divide the excess return by the standard deviation.

[ \text{Sharpe Ratio} = \frac{R - R_f}{\sigma} ]

Where R is the average return of the asset, Rf is the risk-free rate, and σ is the standard deviation of the asset’s returns.

Sortino Ratio

The Sortino Ratio is similar to the Sharpe Ratio but focuses only on the downside deviation rather than the standard deviation. This ratio provides a more accurate measure of risk-adjusted return for assets that are asymmetrical in their return distribution, penalizing only the undesirable volatility.

[ \text{Sortino Ratio} = \frac{R - R_f}{\sigma_D} ]

Where σ_D is the standard deviation of the negative asset returns.

Jensen’s Alpha

Jensen’s Alpha measures the abnormal return of an asset, comparing its performance against the expected return based on the Capital Asset Pricing Model (CAPM).

How to Calculate Jensen’s Alpha:

  1. Calculate Expected Return: Using the CAPM, calculate the expected return on the asset:

[ E(R_i) = R_f + \beta_i (R_m - R_f) ]

Where ( E(R_i) ) is the expected return of the asset, ( \beta_i ) is the beta of the asset, ( R_m ) is the market return, and ( R_f ) is the risk-free rate.

  1. Compute Actual Return: Determine the actual return of the asset (R_a).
  2. Find Alpha: Subtract the expected return from the actual return.

[ \text{Alpha} = R_a - E(R_i) ]

Where Alpha represents the asset’s abnormal performance.

Treynor Ratio

The Treynor Ratio provides a measure of risk-adjusted return based on systematic risk, represented by the asset’s beta.

How to Calculate Treynor Ratio:

  1. Calculate Excess Return: Subtract the risk-free rate (Rf) from the asset’s return (R).
  2. Find the Beta: Determine the beta (β) of the asset.
  3. Compute Treynor Ratio: Divide the excess return by the beta.

[ \text{Treynor Ratio} = \frac{R - R_f}{[beta](../b/beta.html)} ]

Where R is the average return of the asset, Rf is the risk-free rate, and β is the beta of the asset.

Information Ratio

The Information Ratio measures the returns of an asset relative to a benchmark, adjusted for tracking error. It helps in assessing the skill of a portfolio manager in generating excess returns compared to a benchmark.

How to Calculate Information Ratio:

  1. Calculate Excess Return: Subtract the benchmark return (R_b) from the asset’s return (R).
  2. Calculate Tracking Error: Find the standard deviation (σ_e) of the excess return.
  3. Compute Information Ratio: Divide the excess return by the tracking error.

[ \text{Information Ratio} = \frac{R - R_b}{\sigma_e} ]

Where R is the average return of the asset, R_b is the average return of the benchmark, and σ_e is the tracking error.

Maximum Drawdown (MDD)

Maximum Drawdown (MDD) measures the largest peak-to-trough decline in the value of a portfolio. It indicates the potential risk and loss investors might face.

How to Calculate MDD:

  1. Identify Peak Values: Find the highest values of the portfolio over time.
  2. Identify Trough Values: For each peak, find the lowest values following the peak until a new peak is reached.
  3. Determine MDD: Calculate the difference between each peak and trough and find the maximum drawdown.

[ \text{MDD} = \frac{\text{Trough Value} - \text{Peak Value}}{\text{Peak Value}} ]

Where the trough value is the lowest value following the peak.

Calmar Ratio

The Calmar Ratio is used to measure risk-adjusted return, considering the Maximum Drawdown (MDD) instead of volatility.

How to Calculate Calmar Ratio:

  1. Calculate Annual Return: Determine the annual return of the asset.
  2. Determine MDD: Find the Maximum Drawdown as described above.
  3. Compute Calmar Ratio: Divide the annual return by the MDD.

[ \text{Calmar Ratio} = \frac{\text{Annual Return}}{\text{MDD}} ]

Where the annual return is the return of the asset over a year and MDD is the Maximum Drawdown.

Omega Ratio

The Omega Ratio compares the probability-weighted gains relative to the probability-weighted losses for a given threshold return. It provides a comprehensive measure of an asset’s return distribution.

How to Calculate Omega Ratio:

  1. Set Threshold Return: Define a minimum acceptable return (MAR).
  2. Calculate Gains and Losses: Integrate the returns above and below the MAR.
  3. Compute Omega Ratio: Find the ratio of weighted gains to weighted losses.

[ \text{Omega Ratio} = \frac{\int_{MAR}^{\infty} (1 - F(R)) dR}{\int_{-\infty}^{MAR} F(R) dR} ]

Where F(R) is the cumulative distribution function of the returns R.

References and Further Reading

By understanding and applying these return analysis techniques, algorithmic traders can enhance their strategies and achieve improved risk-adjusted returns.