Cumulative Returns

Time Series

Easy

Problem

Theory

Cumulative returns represent the total return on an investment over a period, accounting for compounding effects. They show how much an initial investment has grown or declined over time.

R_{\text{cum}}(t) = \prod_{i=1}^{t} (1 + r_i) - 1

where r_i is the return at period i.

The Formula

For a series of periodic returns r_1, r_2, ..., r_T:

R_{\text{cum}} = (1 + r_1)(1 + r_2)...(1 + r_T) - 1

Alternative form (log returns):

If using log returns \ell_i = \ln(1 + r_i):

R_{\text{cum}} = e^{\sum_{i=1}^{T} \ell_i} - 1

Log returns sum to give cumulative return.

Worked Example

Daily returns: [0.02, -0.01, 0.03, 0.01, -0.02]

Step-by-step calculation:

Period 1: R_1 = 1.02 - 1 = 0.02 (2%)

Period 2: R_2 = 1.02 \times 0.99 - 1 = 1.0098 - 1 = 0.0098 (0.98%)

Period 3: R_3 = 1.0098 \times 1.03 - 1 = 1.040094 - 1 = 0.040094 (4.01%)

Period 4: R_4 = 1.040094 \times 1.01 - 1 = 1.0505 - 1 = 0.0505 (5.05%)

Period 5: R_5 = 1.0505 \times 0.98 - 1 = 1.0295 - 1 = 0.0295 (2.95%)

Final cumulative return: 2.95%

Verification:

R_{\text{cum}} = 1.02 \times 0.99 \times 1.03 \times 1.01 \times 0.98 - 1 = 1.0295 - 1 = 0.0295

Simple vs Compounded Returns

Simple sum (incorrect):

0.02 + (-0.01) + 0.03 + 0.01 + (-0.02) = 0.03 = 3\%

Compounded (correct):

2.95\%

Difference: Compounding accounts for returns earning returns.

Small returns: Difference is minimal.

Large returns: Difference is substantial.

Initial Value Normalization

If starting with principal P_0:

V_t = P_0 \prod_{i=1}^{t} (1 + r_i)

Normalized to 1:

V_t = \prod_{i=1}^{t} (1 + r_i)

This is the growth factor. Subtract 1 to get cumulative return.

Interpretation: V_t = 1.0295 means $1 grew to $1.0295.

Cumulative Return Series

Compute cumulative return at each time point:

R_{\text{cum}}(t) = \prod_{i=1}^{t} (1 + r_i) - 1

Example series:

• t=1: R_1 = 0.02
• t=2: R_2 = 0.0098
• t=3: R_3 = 0.040094
• t=4: R_4 = 0.0505
• t=5: R_5 = 0.0295

Visual: Plot shows investment growth trajectory over time.

Maximum Drawdown Connection

Maximum drawdown measures peak-to-trough decline:

\text{MDD} = \max_{t} \left[\max_{s \leq t} R_{\text{cum}}(s) - R_{\text{cum}}(t)\right]

Requires cumulative returns to identify peak and subsequent trough.

Use case: Risk assessment. Shows worst loss from peak.

Annualized Returns

Convert cumulative return to annualized rate:

r_{\text{annual}} = \left(1 + R_{\text{cum}}\right)^{\frac{1}{T}} - 1

where T is the number of years.

Example: 10% cumulative return over 2 years:

r_{\text{annual}} = (1.10)^{0.5} - 1 = 1.0488 - 1 = 0.0488 = 4.88\%

Interpretation: Average annual rate that produces the cumulative return.

Logarithmic vs Arithmetic Returns

Arithmetic returns:

r_t = \frac{P_t - P_{t-1}}{P_{t-1}}

Logarithmic returns:

\ell_t = \ln\left(\frac{P_t}{P_{t-1}}\right) = \ln(1 + r_t)

Cumulative conversion:

Arithmetic: (1 + r_1)(1 + r_2)...(1 + r_T) - 1

Logarithmic: e^{\ell_1 + \ell_2 + ... + \ell_T} - 1

Advantage of log returns: Additive property simplifies calculations.

Portfolio Cumulative Returns

For portfolio with weights w_i and asset returns r_{i,t}:

r_{p,t} = \sum_{i=1}^{N} w_i r_{i,t}

Portfolio cumulative return:

R_{p,\text{cum}} = \prod_{t=1}^{T} (1 + r_{p,t}) - 1

Note: Portfolio cumulative return is NOT the weighted average of individual cumulative returns.

Must compound portfolio period returns.

Time-Weighted vs Money-Weighted Returns

Time-weighted (geometric):

R_{\text{TW}} = \prod_{t=1}^{T} (1 + r_t) - 1

Measures investment performance independent of cash flows.

Money-weighted (IRR):

Solves:

0 = \sum_{t=0}^{T} \frac{CF_t}{(1 + R_{\text{MW}})^t}

where CF_t includes contributions and withdrawals.

Use case: Time-weighted for comparing fund managers. Money-weighted for investor's actual return.

Volatility and Cumulative Returns

Given periodic return volatility \sigma:

Expected cumulative return (approximate):

E[R_{\text{cum}}] \approx T \mu - \frac{T \sigma^2}{2}

where \mu is mean periodic return.

Volatility drag: Higher volatility reduces cumulative returns due to compounding of losses.

Example: 10% average return with 20% volatility produces less than 10% annualized over long periods.

Benchmark Comparison

Relative cumulative return:

R_{\text{rel}} = \frac{1 + R_{\text{asset}}}{1 + R_{\text{benchmark}}} - 1

Interpretation:

• R_{\text{rel}} > 0: Outperformed benchmark
• R_{\text{rel}} < 0: Underperformed benchmark

Example: Asset returned 15%, benchmark returned 10%:

R_{\text{rel}} = \frac{1.15}{1.10} - 1 = 0.0455 = 4.55\%

Reinvestment Assumption

Cumulative returns assume all gains are reinvested:

Dividends: Automatically reinvested at prevailing price.

Interest: Compounded rather than withdrawn.

No withdrawals: Full capital remains invested.

Reality: Actual investor returns may differ due to consumption, taxes, fees.

Multi-Period Decomposition

Break cumulative return into components:

1 + R_{\text{cum}} = (1 + r_1)(1 + r_2)...(1 + r_T)

Attribution analysis: Which periods contributed most to total return?

Example: Identify that 80% of gains occurred in 3 specific months.

Application: Performance attribution, understanding return drivers.

Sharpe Ratio with Cumulative Returns

Sharpe ratio uses periodic returns:

S = \frac{\mu - r_f}{\sigma}

Not directly computable from cumulative return alone.

Need full return series to calculate mean and standard deviation.

Common mistake: Using only starting and ending values loses information about volatility path.

Cumulative Returns in Backtesting

Strategy evaluation:

1. Generate trading signals
2. Compute period returns based on positions
3. Calculate cumulative returns
4. Compare to buy-and-hold

Equity curve: Plot of cumulative returns over time.

Metrics derived:

• Total return
• Maximum drawdown
• Sharpe ratio
• Calmar ratio (return / max drawdown)

Transaction Costs Impact

Each trade incurs cost c:

r_{t,\text{net}} = r_{t,\text{gross}} - c

Cumulative impact:

R_{\text{cum,net}} = \prod_{t=1}^{T} (1 + r_t - c_t) - 1

High-frequency trading: Small per-trade costs compound to significant drag.

Example: 0.1% cost per trade, 100 trades:

(1 - 0.001)^{100} = 0.9048

9.5% loss from costs alone.

Survivorship Bias

Historical cumulative returns often suffer from survivorship bias:

Bias: Only successful assets remain in dataset.

Result: Overstated historical returns.

Example: Mutual fund database includes only funds that survived. Failed funds excluded.

Correction: Include delisted and failed investments.

Distributional Properties

For log returns \ell_t \sim N(\mu, \sigma^2):

\sum_{t=1}^{T} \ell_t \sim N(T\mu, T\sigma^2)

Cumulative return distribution:

R_{\text{cum}} = e^{\sum \ell_t} - 1

follows log-normal distribution (shifted and scaled).

Implications: Positive skew, fat right tail, bounded below at -1.

Real vs Nominal Returns

Nominal return: Raw return without inflation adjustment.

Real return: Inflation-adjusted return.

r_{\text{real}} = \frac{1 + r_{\text{nominal}}}{1 + i} - 1

where i is inflation rate.

Cumulative real return:

R_{\text{real,cum}} = \frac{1 + R_{\text{nominal,cum}}}{\prod_{t=1}^{T} (1 + i_t)} - 1

Interpretation: Actual purchasing power change.

Geometric Mean Return

The per-period geometric mean return:

\bar{r}_g = \left(\prod_{t=1}^{T} (1 + r_t)\right)^{\frac{1}{T}} - 1

Relationship to cumulative return:

1 + R_{\text{cum}} = (1 + \bar{r}_g)^T

Interpretation: Constant per-period return that produces the same cumulative return.

Property: \bar{r}_g \leq \bar{r}_a (geometric mean \leq arithmetic mean).

Excess Returns

Return above risk-free rate:

r_{t,\text{excess}} = r_t - r_{f,t}

Cumulative excess return:

R_{\text{excess,cum}} = \frac{1 + R_{\text{cum}}}{\prod_{t=1}^{T} (1 + r_{f,t})} - 1

Use case: Evaluating active management. Did manager beat risk-free alternative?

Rolling Cumulative Returns

Compute cumulative return over rolling windows:

R_{\text{cum}}(t, w) = \prod_{i=t-w+1}^{t} (1 + r_i) - 1

Example: 12-month rolling cumulative return.

Application: Visualize performance over various time horizons. Identify consistent vs sporadic outperformance.

Examples

Hints

1. Maintain one running wealth value rather than adding returns.
2. Append wealth - 1 immediately after processing each return.

Requirements

• Begin with a wealth factor of 1.0.
• Multiply the running wealth by 1 + r for each period.
• Append the running wealth minus 1 after every update.
• Return a list with the same length as returns.

Constraints

• returns is nonempty.
• Every return is greater than -1.
• Time limit: 300 ms.

Starter Code

def cumulative_returns(returns: list) -> list:
    """
    Returns the compounded cumulative return after every period.
    """
    # Write code here
    pass

Test Cases